LinearFVAdvectionDiffusionFunctorDirichletBC

Description

LinearFVAdvectionDiffusionFunctorDirichletBC will specify the value of a field at the boundary. The value will be determined by a Functor (through the "functor" parameter).

note

This boundary condition should only be used for problems which involve advection and/or diffusion problems.

Example Syntax

In this example the functor is defined using a ParsedFunction.

[LinearFVBCs<<<{"href": "../../syntax/LinearFVBCs/index.html"}>>>]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC<<<{"description": "Adds a dirichlet BC which can be used for the assembly of linear finite volume system and whose face values are determined using a functor. This kernel is only designed to work with advection-diffusion problems.", "href": "LinearFVAdvectionDiffusionFunctorDirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this boundary condition applies to"}>>> = u
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = "left right top bottom"
    functor<<<{"description": "The functor for this boundary condition. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number."}>>> = analytic_solution
  []
[]

Input Parameters

boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
functorThe functor for this boundary condition. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The functor for this boundary condition. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
variableThe name of the variable that this boundary condition applies to
C++ Type:LinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this boundary condition applies to

Required Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsrhsThe tag for the vectors this Kernel should fill
Default:rhs
C++ Type:MultiMooseEnum
Options:rhs, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
search_methodnearest_node_connected_sidesChoice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).
Default:nearest_node_connected_sides
C++ Type:MooseEnum
Options:nearest_node_connected_sides, all_proximate_sides
Controllable:No
Description:Choice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).

Advanced Parameters

Input Files

(test/tests/linearfvkernels/diffusion/diffusion-1d_neumann.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC.i)
(test/tests/linearfvbcs/scalar_symmetry/advection-1d-symmetry.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/bulk_heat_transfer/flow-around-square-linear.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-velocity-pressure.i)
(test/tests/time_integrators/implicit-euler/ie-linearfv.i)
(test/tests/vectorpostprocessors/side_value_sampler/side_value_sampler_fv.i)
(modules/heat_transfer/test/tests/radiation_participating_media/rad_T_coupled_medium_1d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_heated-physics.i)
(test/tests/multisystem/picard/linearfv_nonlinearfv/same_input.i)
(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-2d.i)
(test/tests/postprocessors/side_diffusive_flux_integral/side_diffusive_flux_integral_linear_fv.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h_fp.i)
(test/tests/postprocessors/linearfv_flux_integral/linearfv_flux_integral.i)
(test/tests/linearfvkernels/diffusion/diffusion-1d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC_mu_eff.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/linear-segregated/lid-driven-turb-std-wall.i)
(test/tests/linearfvbcs/scalar_symmetry/advection-2d-symmetry.i)
(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-1d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/linear-segregated/lid-driven-turb-non-eq-wall.i)
(test/tests/variables/linearfv/diffusion-1d-aux.i)
(test/tests/linearfvkernels/advection/diagonal-step-2d.i)
(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/channel-drift-flux.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_neu-dir.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_rob-rob.i)
(test/tests/linearfvbcs/robin/diffusion-2d-robin.i)
(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-2d.i)
(test/tests/linearfvbcs/robin/diffusion-1d-robin.i)
(modules/heat_transfer/test/tests/radiation_participating_media/rad_T_coupled_medium_1d_T.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/SIMPLEC/3d/3d-velocity-pressure.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/bulk_heat_transfer/flow-around-square-linear-fluidonly.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated/BFS_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-vortex/2d-vortex.i)
(test/tests/multiapps/linearfv_nonlinearfv/linearfv.i)
(test/tests/linearfvkernels/diffusion/diffusion-2d_neumann.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/rotated-channel.i)
(test/tests/time_steppers/iteration_adaptive/adapt_linear_systems.i)
(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex-rz/2d-symmetric-vortex-rz-spacedependent.i)
(modules/navier_stokes/test/tests/finite_volume/ins/natural_convection/linear_segregated/2d/diff_heated_cavity_linear_segregated.i)
(test/tests/linearfvbcs/scalar_symmetry/diffusion-2d-symmetry.i)
(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC-physics.i)
(test/tests/multisystem/restore_multiapp/parent.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC_symmetric.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/enthalpy_equation.i)
(modules/navier_stokes/test/tests/finite_volume/ins/natural_convection/linear_segregated/2d/diff_heated_cavity_linear_buoyancy.i)
(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion-react.i)
(test/tests/linearfvkernels/advection/diagonal-step-2d-delaunay.i)
(modules/navier_stokes/test/tests/finite_volume/ins/radiation_participating_media/p1_radiation_cht_SIMPLE.i)
(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/rayleigh-bernard-two-phase-physics.i)
(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-vortex/spacedependent_mu/sl.i)
(test/tests/linearfvkernels/advection/advection-2d-rz.i)
(test/tests/time_integrators/bdf2/linearfv.i)
(test/tests/tag/tag-linearfv.i)
(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc_mms_2d.i)
(test/tests/variables/linearfv/diffusion-1d-pp.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/segregated-comparison/segregated-linear.i)
(test/tests/linearfvbcs/robin/advection-2d-robin.i)
(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion-variable-functor-bc.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated-transient/channel_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/3d/3d-velocity-pressure.i)
(test/tests/multisystem/restore_multiapp/sub.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-symmetric/channel.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_scalars-physics.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated-transient/BFS_ERCOFTAC.i)
(test/tests/linearfvkernels/diffusion/diffusion-2d-rz.i)
(test/tests/linearfvkernels/advection/advection-2d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/SIMPLEC/2d/2d-velocity-pressure.i)
(modules/navier_stokes/test/tests/finite_volume/ins/lid-driven/linear-segregated/lid-driven-segregated.i)
(test/tests/transfers/multiapp_copy_transfer/linear_sys_to_aux/linear_sub.i)
(test/tests/linearfvkernels/block-restriction/block-restricted-adr.i)
(test/tests/controls/conditional_functional_enable/linear-fv-bc-control-switch.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-heated/fluid.i)
(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_LinearFV.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/block-restricted/block-ke.i)
(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex/2d-symmetric-vortex.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-scalar/channel.i)
(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/lid-driven-two-phase-physics.i)
(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc_mms.i)
(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex-rz/2d-symmetric-vortex-rz.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq-transient.i)
(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq.i)
(test/tests/linearfvkernels/diffusion/diffusion-2d.i)
(test/tests/linearfvbcs/scalar_symmetry/diffusion-1d-symmetry.i)
(test/tests/linearfvkernels/advection/advection-1d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/steady-transient-compare/common-blocks.i)
(test/tests/outputs/debug/show_execution_linear_fv_flux.i)

Child Objects

(modules/navier_stokes/include/linearfvbcs/LinearFVNormalVelocityFunctorDirichletBC.h)
(modules/navier_stokes/include/linearfvbcs/LinearFVDirichletCHTBC.h)

(test/tests/linearfvkernels/diffusion/diffusion-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right top bottom"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*(1.5-y*y)+2*x*y*(1.5-y*y)+2*(1.5-x*x)+2*x*y*(1.5-x*x)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/diffusion/diffusion-1d_neumann.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right"
    functor = analytic_solution
  []
  [neu]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = u
    boundary = "left"
    functor = analytic_solution_neumann
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '1-x*x'
  []
  [analytic_solution_neumann]
    type = ParsedFunction
    expression = '-(0.5*x)*(-2*x)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC.i)


!include header_and_mesh.i

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    # we use mu_eff directly here to be able to have deviatoric term for both mu and mu_t
    mu = 'mu_eff'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_eff'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []

  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = 'bottom top'
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
[]

[FunctorMaterials]
  [compute_mu_eff]
    type = FunctorEffectiveDynamicViscosity
    property_name = mu_eff
    mu = ${mu}
    mu_t = mu_t
    mu_t_inverse_factor = 1
    execute_on = 'ALWAYS'
  []
[]

!include executioner_regular_channel.i

variables_to_sample = 'vel_x vel_y pressure_over_density TKE TKED'
!include postprocessing.i

(test/tests/linearfvbcs/scalar_symmetry/advection-1d-symmetry.i)

vel = 0.1

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 1
    nx   = 2
    xmin = 0
    xmax = ${fparse pi}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = 'cos(x)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '-${vel}*sin(x)'
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "${vel} 0 0"
    advected_interp_method_name = average
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [rob_l]
    type = LinearFVAdvectionDiffusionScalarSymmetryBC
    variable = u
    boundary = "left"
    use_two_term_expansion = true
  []
  [dirichlet]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right"
    functor = u_exact
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 4
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       NONZERO               1e-10'
  linear_convergence = linear
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/cht/bulk_heat_transfer/flow-around-square-linear.i)

mu = 0.01
rho = 1.1
k = 0.0005
cp = 10
k_s = 3.0
h_conv = 5

power_density = 10000

advected_interp_method = 'upwind'

[Mesh]
  [generated_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = 0
    ymin = 0
    ymax = 0.1
    xmax = 0.1
  []
  [subdomain1]
    type = SubdomainBoundingBoxGenerator
    input = generated_mesh
    block_name = subdomain1
    bottom_left = '0.04 0.04 0'
    block_id = 1
    top_right = '0.06 0.06 0'
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = subdomain1
    primary_block = 0
    paired_block = 1
    new_boundary = interface
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system solid_energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = 0
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.1
    solver_sys = u_system
    block = 0
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.01
    block = 0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
    block = 0
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
    block = 0
  []
  [T_solid]
    type = MooseLinearVariableFVReal
    solver_sys = solid_energy_system
    initial_condition = 500
    block = 1
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = true
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = true
  []

  [solid-conduction]
    type = LinearFVDiffusion
    variable = T_solid
    diffusion_coeff = ${k_s}
    use_nonorthogonal_correction = true
  []
  [solid-source]
    type = LinearFVSource
    variable = T_solid
    source_density = ${power_density}
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '0.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom interface'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom interface'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []

  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    functor = 300
    boundary = 'left'
  []
  [walls_T]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_fluid
    functor = 0.0
    boundary = 'top bottom'
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T_fluid
    use_two_term_expansion = false
    boundary = right
  []

  [fluid_solid]
    type = LinearFVConvectiveHeatTransferBC
    variable = T_fluid
    T_solid = T_solid
    T_fluid = T_fluid
    boundary = interface
    h = ${h_conv}
  []
  [solid_fluid]
    type = LinearFVConvectiveHeatTransferBC
    variable = T_solid
    T_solid = T_solid
    T_fluid = T_fluid
    boundary = interface
    h = ${h_conv}
  []
[]

[FunctorMaterials]
  [rhocpT]
    property_name = 'rhocpT'
    type = ParsedFunctorMaterial
    functor_names = 'T_fluid'
    expression = '${rho}*${cp}*T_fluid'
  []
[]

[Postprocessors]
  [h_in]
    type = VolumetricFlowRate
    boundary = left
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    subtract_mesh_velocity = false
  []
  [h_out]
    type = VolumetricFlowRate
    boundary = right
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    advected_interp_method = upwind
    subtract_mesh_velocity = false
  []
  [power]
    type = ElementIntegralFunctorPostprocessor
    functor = ${power_density}
    block = 1
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  energy_l_abs_tol = 1e-13
  solid_energy_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  solid_energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  solid_energy_system = 'solid_energy_system'
  momentum_equation_relaxation = 0.8
  energy_equation_relaxation = 1.0
  pressure_variable_relaxation = 0.3
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  energy_absolute_tolerance = 1e-10
  solid_energy_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  solid_energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  solid_energy_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-velocity-pressure.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.3'
    dy = '0.3'
    ix = '3'
    iy = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(test/tests/time_integrators/implicit-euler/ie-linearfv.i)

###########################################################
# This is a simple test with a time-dependent problem
# demonstrating the use of the TimeIntegrator system.
#
# Testing a solution that is second order in space
# and first order in time
#
# @Requirement F1.30
###########################################################

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 10
  ny = 10
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
  []
[]

[Functions]
  [forcing_fn]
    type = ParsedFunction
    expression = ((x*x)+(y*y))-(4*t)
  []
  [exact_fn]
    type = ParsedFunction
    expression = t*((x*x)+(y*y))
  []
[]

[LinearFVKernels]
  [ie]
    type = LinearFVTimeDerivative
    variable = u
  []
  [diff]
    type = LinearFVDiffusion
    variable = u
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = forcing_fn
  []
[]

[LinearFVBCs]
  [all]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = '0 1 2 3'
    functor = exact_fn
  []
[]

[Postprocessors]
  [l2_err]
    type = ElementL2Error
    variable = u
    function = exact_fn
  []
[]

[Executioner]
  type = Transient

  system_names = u_sys
  l_tol = 1e-10

  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'

  # Test of the TimeIntegrator System
  scheme = 'implicit-euler'

  start_time = 0.0
  num_steps = 5
  dt = 0.25
[]

[Outputs]
  exodus = true
[]

(test/tests/vectorpostprocessors/side_value_sampler/side_value_sampler_fv.i)

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.5 0.5'
    dy = '1'
    ix = '5 5'
    iy = '10'
    subdomain_id = '1 1'
  []
  # Limited to 1 side to avoid inconsistencies in parallel
  [internal_sideset]
    type = ParsedGenerateSideset
    combinatorial_geometry = 'y<0.51 & y>0.49 & x<0.11'
    new_sideset_name = 'center'
    input = 'mesh'
  []
  # this keeps numbering continuous so tests dont fail on different ids in CSV
  allow_renumbering = false
[]

[Problem]
  nl_sys_names = 'u_sys'
  linear_sys_names = 'v_sys'
[]

[Variables]
  [u]
    type = MooseVariableFVReal
    solver_sys = u_sys
  []
  [v]
    type = MooseLinearVariableFVReal
    solver_sys = v_sys
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = u
    coeff = 1.0
  []
[]

[LinearFVKernels]
  [diff]
    type = LinearFVDiffusion
    variable = v
  []
[]

[FVBCs]
  [all]
    type = FVFunctorDirichletBC
    variable = u
    boundary = 'top bottom left right'
    functor = linear_x
  []
[]

[LinearFVBCs]
  [all]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = 'top bottom left right'
    functor = linear_x
  []
[]

[Functions]
  [linear_x]
    type = ParsedFunction
    expression = '5*x'
  []
[]

[VectorPostprocessors]
  [side_sample]
    type = SideValueSampler
    variable = 'u v'
    boundary = top
    sort_by = x
  []
  [internal_sample]
    type = SideValueSampler
    variable = 'u v'
    boundary = center
    sort_by = 'id'
  []
[]

[Executioner]
  type = Steady
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  system_names = 'u_sys v_sys'
  l_tol = 1e-8
[]

[Outputs]
  execute_on = 'timestep_end'
  [vpp_csv]
    type = CSV
  []
[]

(modules/heat_transfer/test/tests/radiation_participating_media/rad_T_coupled_medium_1d.i)

# The test guarantees that the P1 participating media radiation model in the LinearFV
# system can couple with the energy equation of a semi-transparent fluid with fixed
# temperatures at the left and right walls.
# A simple multi-linear system solver approach is used. Due to the strong coupling between T and G
# multi-system fixed point iterations need to be set.

k = 1e-2
sigma_a = 1.
G_diffusion_coef = ${fparse 1/(3*sigma_a)}
sigma = 5.670374419e-8

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 20
    xmin = 0
    xmax = 1
  []
[]

[Problem]
  linear_sys_names = 'energy_system radiation_system'
  previous_nl_solution_required = true
[]

[Variables]
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = 'energy_system'
    initial_condition = 100
  []
  [G]
    type = MooseLinearVariableFVReal
    solver_sys = 'radiation_system'
    initial_condition = ${fparse sigma*100^4}
  []
[]

[LinearFVKernels]

  [temp_conduction]
    type = LinearFVDiffusion
    diffusion_coeff = ${k}
    variable = T
  []
  [temp_radiation]
    type = LinearFVP1TemperatureSourceSink
    variable = T
    G = 'G'
    absorption_coeff = ${sigma_a}
  []

  [G_diffusion]
    type = LinearFVDiffusion
    variable = G
    diffusion_coeff = ${G_diffusion_coef}
  []
  [source_and_sink]
    type = LinearFVP1RadiationSourceSink
    variable = G
    temperature_radiation = 'T'
    absorption_coeff = ${sigma_a}
  []
[]

[LinearFVBCs]
  [right_bc_G]
    type = LinearFVP1RadiationMarshakBC
    boundary = 'right'
    variable = G
    temperature_radiation = 'T'
    coeff_diffusion = ${G_diffusion_coef}
    boundary_emissivity = 1.0
  []
  [right_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'right'
    functor = 100.
  []
  [left_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = 50.
  []
  [left_bc_G]
    type = LinearFVP1RadiationMarshakBC
    boundary = 'left'
    variable = G
    temperature_radiation = 'T'
    coeff_diffusion = ${G_diffusion_coef}
    boundary_emissivity = 1.0
  []
[]

[Postprocessors]
  [mean_value_G]
    type = ElementIntegralFunctorPostprocessor
    functor = G
  []
  [max_value_G]
    type = ElementExtremeFunctorValue
    functor = G
  []
  [mean_value_T]
    type = ElementIntegralFunctorPostprocessor
    functor = T
  []
  [max_value_T]
    type = ElementExtremeFunctorValue
    functor = T
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 70
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  solve_type = 'Linear' #'NEWTON'
  petsc_options_iname = '-energy_system_pc_type -energy_system_pc_factor_shift_type -radiation_system_pc_type -radiation_system_pc_factor_shift_type'
  petsc_options_value = 'hypre boomeramg hypre boomeramg'
  l_abs_tol = 1e-14
  l_tol = 1e-14
  nl_abs_tol = 1e-14
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
[]

[Outputs]
  #file_base = rad_isothermal_medium_1d_adiabatic
  csv = true
  exodus = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_heated-physics.i)


!include header_and_mesh.i

q = 5
cp = 300

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

# These are only needed when solving for enthalpy
# Alternatively, a FluidProperties user object may be used
[FunctorMaterials]
  active = 'rho_h'
  # These two are for solving for enthalpy without using a FluidProperties object
  [h_from_T]
    type = ParsedFunctorMaterial
    expression = '${cp} * T_fluid'
    functor_names = 'T_fluid'
    property_name = 'h_from_p_T_functor'
  []
  [T_from_h]
    type = ParsedFunctorMaterial
    expression = 'h / ${cp}'
    functor_names = 'h'
    property_name = 'T_from_p_h_functor'
  []
  # This one is for computing the enthalpy density without using a FluidProperties object
  [rho_h]
    type = ParsedFunctorMaterial
    expression = '${rho} * ${cp}  * T_fluid'
    functor_names = 'T_fluid'
    property_name = 'rho_h'
  []
[]

[Physics]
  [NavierStokes]
    [FlowSegregated/flow]
      velocity_variable = 'vel_x vel_y'
      pressure_variable = 'pressure'

      # Initial conditions
      initial_velocity = '${bulk_u} 0 0'
      initial_pressure = '1e-8'

      # Material properties
      density = ${rho}
      dynamic_viscosity = ${mu}

      # Boundary conditions
      inlet_boundaries = 'left'
      momentum_inlet_types = 'fixed-velocity'
      momentum_inlet_functors = '${bulk_u} 0'

      wall_boundaries = 'top bottom'
      momentum_wall_types = 'noslip noslip'

      outlet_boundaries = 'right'
      momentum_outlet_types = 'fixed-pressure'
      pressure_functors = '0'

      # Numerical parameters
      include_deviatoric_stress = true
      orthogonality_correction = false
      pressure_two_term_bc_expansion = false
      momentum_two_term_bc_expansion = false
      momentum_advection_interpolation = ${advected_interp_method}
      system_names = 'u_system v_system pressure_system'
    []
    [FluidHeatTransferSegregated/energy]
      # turbulence Physics coupling picked up automatically
      initial_temperature = 0

      thermal_conductivity = 1.1
      specific_heat = ${cp}

      energy_inlet_types = 'fixed-temperature'
      energy_inlet_functors = '0'
      energy_wall_types = 'heatflux heatflux'
      energy_wall_functors = '0 0'

      external_heat_source = '${q}'
    []
    [TurbulenceSegregated/k-epsilon]
      # Model
      turbulence_handling = 'k-epsilon'
      tke_name = TKE
      tked_name = TKED
      system_names = 'TKE_system TKED_system'
      fluid_heat_transfer_physics = energy

      initial_tke = ${k_init}
      initial_tked = ${eps_init}

      # Model parameters
      mu_t_ratio_max = 1e20
      sigma_k = ${sigma_k}
      sigma_eps = ${sigma_eps}
      C_pl = 1e10
      C1_eps = ${C1_eps}
      C2_eps = ${C2_eps}
      Pr_t = 0.9

      k_t_as_aux_variable = true
      turbulence_walls = ${walls}
      wall_treatment_eps = ${wall_treatment}
      bulk_wall_treatment = ${bulk_wall_treatment}
      use_nonorthogonal_correction = false
    []
  []
[]

# Boundary conditions are not implemented for turbulent quantities at this time
[LinearFVBCs]
  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  energy_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  energy_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.8
  turbulence_equation_relaxation = '0.2 0.2'
  turbulence_field_relaxation = '0.2 0.2'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  energy_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
[]

variables_to_sample = 'vel_x vel_y pressure T_fluid TKE TKED'
!include postprocessing.i

# More postprocessor for temperature
[Postprocessors]
  [outlet_T]
    type = SideAverageValue
    boundary = right
    variable = T_fluid
  []
  [Q_out]
    type = VolumetricFlowRate
    boundary = 'right'
    vel_x = 'vel_x'
    vel_y = 'vel_y'
    advected_quantity = 'rho_h'
    rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  []
  [balance_percent]
    type = ParsedPostprocessor
    expression = '100 * (0 - Q_out + 2 * ${L} * ${H} * ${q}) / Q_out'
    pp_names = 'Q_out'
  []
[]

(test/tests/multisystem/picard/linearfv_nonlinearfv/same_input.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 6
  []
[]

[Problem]
  nl_sys_names = 'v_sys'
  linear_sys_names = 'u_sys'
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    initial_condition = 2.0
    solver_sys = v_sys
  []
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVKernels]
  [diffusion]
    type = FVDiffusion
    variable = v
    coeff = u
  []
  [source]
    type = FVBodyForce
    variable = v
    function = 3
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = v
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 1
  []
[]

[FVBCs]
  [dir]
    type = FVFunctorDirichletBC
    variable = v
    boundary = "left right"
    functor = 2
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = 1
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 6
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = 'v_sys u_sys'
  l_abs_tol = 1e-12
  l_tol = 1e-10
  nl_abs_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(test/tests/linearfvkernels/anisotropic-diffusion/anisotropic-diffusion-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = u
    diffusion_tensor = diffusivity_tensor
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right top bottom"
    functor = analytic_solution
  []
[]

[FunctorMaterials]
  [diff_tensor]
    type = GenericVectorFunctorMaterial
    prop_names = diffusivity_tensor
    prop_values = 'coeff_func_x coeff_func_y 0.0'
  []
[]

[Functions]
  [coeff_func_x]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [coeff_func_y]
    type = ParsedFunction
    expression = '1+x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '(1.5-y*y)*(2+2*x*y)+(1.5-x*x)*(2+4*x*y)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/postprocessors/side_diffusive_flux_integral/side_diffusive_flux_integral_linear_fv.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    solver_sys = 'u_sys'
    type = MooseLinearVariableFVReal
  []
[]

[LinearFVKernels]
  [diff]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diffusivity
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0
  []
  [right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 1
  []
[]

[FunctorMaterials]
  [mat_props]
    type = GenericFunctorMaterial
    prop_names = diffusivity
    prop_values = 1
  []
[]

[AuxVariables]
  [grad]
    family = MONOMIAL_VEC
    order = CONSTANT
  []
[]

[AuxKernels]
  [grad]
    variable = grad
    functor = u
    type = FunctorElementalGradientAux
  []
[]

[Postprocessors]
  [avg_left]
    type = SideAverageValue
    variable = u
    boundary = left
  []
  [avg_right]
    type = SideAverageValue
    variable = u
    boundary = right
  []
  [avg_flux_left]
    type = SideDiffusiveFluxAverage
    variable = u
    boundary = left
    functor_diffusivity = diffusivity
  []
  [avg_flux_right]
    type = SideDiffusiveFluxAverage
    variable = u
    boundary = right
    functor_diffusivity = diffusivity
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h_fp.i)

L = 30
nx = 600
bulk_u = 0.01
p_ref = 101325.0
T_in = 860.
q_source = 20000.
advected_interp_method = 'upwind'

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = ${L}
    nx = ${nx}
  []
  allow_renumbering = false
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
  u = vel_x
[]

[Problem]
  linear_sys_names = 'u_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    pressure = pressure
    rho = 'rho'
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
    initial_condition = ${bulk_u}
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = ${p_ref}
  []
  [h]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = ${fparse 860.*240.}
  []
[]

[AuxVariables]
  [rho_var]
    type = MooseLinearVariableFVReal
  []
  [cp_var]
    type = MooseLinearVariableFVReal
  []
  [mu_var]
    type = MooseLinearVariableFVReal
  []
  [k_var]
    type = MooseLinearVariableFVReal
  []
  [T]
    type = MooseLinearVariableFVReal
    initial_condition = ${T_in}
  []
  [h_aux]
    type = MooseLinearVariableFVReal
  []
[]

[LinearFVKernels]

  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = 'mu'
    momentum_component = 'x'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [temp_advection]
    type = LinearFVEnergyAdvection
    variable = h
  []
  [source]
    type = LinearFVSource
    variable = h
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inlet_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = ${bulk_u}
  []
  [inlet_h]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = h
    boundary = 'left'
    functor = 'h_from_p_T'
  []
  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${T_in}
  []

  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = ${p_ref}
  []
  [outlet_h]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = h
    use_two_term_expansion = false
    boundary = 'right'
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = 'right'
  []
[]

[FluidProperties]
  [lead]
    type = LeadFluidProperties
  []
[]

[FunctorMaterials]
  [enthalpy_material]
    type = LinearFVEnthalpyFunctorMaterial
    pressure = ${p_ref}
    T_fluid = T
    h = h
    fp = lead
  []
  [fluid_props_to_mat_props]
    type = GeneralFunctorFluidProps
    fp = lead
    pressure = ${p_ref}
    T_fluid = 'T'
    speed = 1
    porosity = 1
    characteristic_length = 1
  []
  [source_func]
    type = ADParsedFunctorMaterial
    property_name = source_func
    functor_names = 'rho'
    expression = ${q_source}
  []
[]

[AuxKernels]
  [rho_out]
    type = FunctorAux
    functor = 'rho'
    variable = 'rho_var'
    execute_on = 'NONLINEAR'
  []
  [cp_out]
    type = FunctorAux
    functor = 'cp'
    variable = 'cp_var'
    execute_on = 'NONLINEAR'
  []
  [mu_out]
    type = FunctorAux
    functor = 'mu'
    variable = 'mu_var'
    execute_on = 'NONLINEAR'
  []
  [k_out]
    type = FunctorAux
    functor = 'k'
    variable = 'k_var'
    execute_on = 'NONLINEAR'
  []
  [T_from_h_functor_aux]
    type = FunctorAux
    functor = 'T_from_p_h'
    variable = 'T'
    execute_on = 'NONLINEAR'
  []
  [h_from_T_functor_aux]
    type = FunctorAux
    functor = 'h_from_p_T'
    variable = 'h_aux'
    execute_on = 'NONLINEAR'
  []
[]

[Postprocessors]
  [T_out_sim]
    type = ElementalVariableValue
    variable = T
    elementid = ${fparse nx-1}
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-12
  pressure_l_abs_tol = 1e-12
  energy_l_abs_tol = 1e-12
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.95
  num_iterations = 100
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  energy_absolute_tolerance = 1e-6
  print_fields = false
  momentum_l_max_its = 200

  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'

  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  continue_on_max_its = true
[]

[Outputs]
  [out]
    type = CSV
  []
[]

(test/tests/postprocessors/linearfv_flux_integral/linearfv_flux_integral.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 100
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.25
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 0.5
    use_nonorthogonal_correction = false
  []
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = '0.5 0 0'
    advected_interp_method_name = upwind
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 1
  []
  [right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 0
  []
[]

[Postprocessors]
  [flux_left_total]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'diffusion advection'
  []
  [flux_left_diff]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'diffusion'
  []
  [flux_left_adv]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'advection'
  []
  [flux_right_total]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'diffusion advection'
  []
  [flux_right_diff]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'diffusion'
  []
  [flux_right_adv]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'advection'
  []
  [flux_left_sum_error]
    type = ParsedPostprocessor
    expression = 'f_total - f_diff - f_adv'
    pp_names = 'flux_left_total flux_left_diff flux_left_adv'
    pp_symbols = 'f_total f_diff f_adv'
  []
  [flux_right_sum_error]
    type = ParsedPostprocessor
    expression = 'f_total - f_diff - f_adv'
    pp_names = 'flux_right_total flux_right_diff flux_right_adv'
    pp_symbols = 'f_total f_diff f_adv'
  []
  [flux_balance]
    type = ParsedPostprocessor
    expression = 'f_left + f_right'
    pp_names = 'flux_left_total flux_right_total'
    pp_symbols = 'f_left f_right'
  []
  # These are the analytic solutions, when we refine we get closer and closer
  [analytic_flux_left_adv]
    type = ParsedPostprocessor
    expression = '-v'
    constant_names = 'v'
    constant_expressions = '0.5'
  []
  [analytic_flux_left_diff]
    type = ParsedPostprocessor
    expression = '-v / (exp(v / k * L) - 1)'
    constant_names = 'v k L'
    constant_expressions = '0.5 0.5 1'
  []
  [analytic_flux_right_adv]
    type = ParsedPostprocessor
    expression = '0'
  []
  [analytic_flux_right_diff]
    type = ParsedPostprocessor
    expression = 'v * exp(v / k * L) / (exp(v / k * L) - 1)'
    constant_names = 'v k L'
    constant_expressions = '0.5 0.5 1'
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  l_abs_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(test/tests/linearfvkernels/diffusion/diffusion-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [geom]
    type = FVGeometricAverage
  []
  [harm]
    type = FVHarmonicAverage
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [coeff_pos_func]
    type = ParsedFunction
    expression = '1+0.5*x'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '1-x*x'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC_mu_eff.i)

!include header_and_mesh.i

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_eff'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_eff'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_eff_k'
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_eff_eps'
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []

  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = 'bottom top'
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
[]

[FunctorMaterials]
  [compute_mu_eff]
    type = FunctorEffectiveDynamicViscosity
    property_name = mu_eff
    mu = ${mu}
    mu_t = mu_t
    mu_t_inverse_factor = 1
    execute_on = 'ALWAYS'
  []
  [compute_mu_eff_k]
    type = FunctorEffectiveDynamicViscosity
    property_name = mu_eff_k
    mu = ${mu}
    mu_t = mu_t
    mu_t_inverse_factor = ${sigma_k}
    execute_on = 'ALWAYS'
  []
  [compute_mu_eff_eps]
    type = FunctorEffectiveDynamicViscosity
    property_name = mu_eff_eps
    mu = ${mu}
    mu_t = mu_t
    mu_t_inverse_factor = ${sigma_eps}
    execute_on = 'ALWAYS'
  []
[]

!include executioner_regular_channel.i

variables_to_sample = 'vel_x vel_y pressure_over_density TKE TKED'
!include postprocessing.i

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/linear-segregated/lid-driven-turb-std-wall.i)

### Thermophysical Properties ###
mu = 1e-3
rho = 1.0

### Operation Conditions ###
lid_velocity = 1.0
side_length = 0.1

### Initial Conditions ###
intensity = 0.01
k_init = '${fparse 1.5*(intensity * lid_velocity)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / side_length}'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'left top right bottom'
wall_treatment = 'eq_newton' # Options: eq_newton, eq_incremental, eq_linearized, neq

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = 'upwind'
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${lid_velocity}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = true
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = true
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = true
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = true
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = true
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = true
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = true
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [top_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'top'
    functor = 1
  []
  [no_slip_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'left right bottom'
    functor = 0
  []
  [no_slip_y]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = 'left right top bottom'
    functor = 0
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'left right top bottom'
    variable = pressure
    use_two_term_expansion = true
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = ${walls}
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.5 0.5'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.01 0.099 0.0'
[]

[Outputs]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(test/tests/linearfvbcs/scalar_symmetry/advection-2d-symmetry.i)

vel_x = -0.1
two_term_bc=true

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
    xmin = 0
    xmax = ${fparse pi/3}
    ymin = 0
    ymax = ${fparse pi/3}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.02
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = 'cos(x)*cos(y)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '-${vel_x}*sin(x)*cos(y)'
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "${vel_x} 0 0"
    advected_interp_method_name = average
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [symm]
    type = LinearFVAdvectionDiffusionScalarSymmetryBC
    variable = u
    boundary = "bottom"
    use_two_term_expansion = ${two_term_bc}
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "left"
    use_two_term_expansion = ${two_term_bc}
  []
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right top"
    functor = u_exact
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = 'TIMESTEP_END'
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = 'TIMESTEP_END'
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-9
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       NONZERO               1e-10'
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
  exodus = true
[]

(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diff_coeff_func
    use_nonorthogonal_correction = false
  []
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = average
  []
  [reaction]
    type = LinearFVReaction
    variable = u
    coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  inactive = "outflow neumann"
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = true
  []
  [neumann]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = u
    boundary = "left"
    functor = analytic_solution_neumann_left
    diffusion_coeff = diff_coeff_func
  []
[]

[Functions]
  [diff_coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x'
  []
  [coeff_func]
    type = ParsedFunction
    expression = '1+1/(1+x)'
  []
  [source_func]
    type = ParsedFunction
    expression = '(1+1/(x+1))*(sin(pi/2*x)+1.5)+0.25*pi*pi*(0.5*x+1)*sin(pi/2*x)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(pi/2*x)+1.5'
  []
  [analytic_solution_neumann_left]
    type = ParsedFunction
    expression = '-(1+0.5*x)*cos(pi/2*x)*pi/2'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_rtol'
  petsc_options_value = 'hypre boomeramg 1e-10'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/linear-segregated/lid-driven-turb-non-eq-wall.i)

### Thermophysical Properties ###
mu = 1e-3
rho = 1.0

### Operation Conditions ###
lid_velocity = 1.0
side_length = 0.1

### Initial Conditions ###
intensity = 0.01
k_init = '${fparse 1.5*(intensity * lid_velocity)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / side_length}'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'left top right bottom'
wall_treatment = 'neq' # Options: eq_newton, eq_incremental, eq_linearized, neq

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = 'upwind'
[]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${lid_velocity}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [top_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'top'
    functor = 1
  []
  [no_slip_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'left right bottom'
    functor = 0
  []
  [no_slip_y]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = 'left right top bottom'
    functor = 0
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'left right top bottom'
    variable = pressure
    use_two_term_expansion = true
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = 'bottom top'
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.5 0.5'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.01 0.099 0.0'
[]

[Outputs]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(test/tests/variables/linearfv/diffusion-1d-aux.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[AuxVariables]
  [v_volume]
    type = MooseLinearVariableFVReal
    initial_condition = 50
  []
  [v_functor]
    type = MooseLinearVariableFVReal
    initial_condition = 25
  []
  [v_parsed]
    type = MooseLinearVariableFVReal
    initial_condition = 12.5
  []
[]

[AuxKernels]
  [volume]
    type = VolumeAux
    variable = v_volume
  []
  [functor]
    type = FunctorAux
    variable = v_functor
    functor = u
  []
  [parsed]
    type = ParsedAux
    variable = v_parsed
    coupled_variables = 'v_volume v_functor'
    expression = '0.5*v_volume+0.5*v_functor'
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '1-x*x'
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
  execute_on = TIMESTEP_END
[]

(test/tests/linearfvkernels/advection/diagonal-step-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 51
    ny = 51
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.5
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 0.75
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 0.75
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "1 1 0"
    advected_interp_method_name = nvd_minmod
  []
[]

[LinearFVBCs]
  [left_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0.5
  []
  [bottom_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = bottom
    functor = 1
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = 'right top'
    use_two_term_expansion = true
  []
[]

[VectorPostprocessors]
  [diag_sample]
    type = LineValueSampler
    variable = u
    start_point = '1.0 0 0'
    end_point = '0 1.0 0'
    num_points = 101
    sort_by = id
    warn_discontinuous_face_values = false
    execute_on = TIMESTEP_END
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 50
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  multi_system_fixed_point_relaxation_factor = 0.3
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  csv = true
  exodus = true
  execute_on = TIMESTEP_END
[]

(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/channel-drift-flux.i)

mu = 1.0
rho = 10.0
mu_d = 0.1
rho_d = 1.0
l = 2
# 'average' leads to slight oscillations, upwind may be preferred
# This method is selected for consistency with the original nonlinear input
advected_interp_method = 'average'

# TODO remove need for those
cp = 1
k = 1
cp_d = 1
k_d = 1

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = '${fparse l * 5}'
    ymin = '${fparse -l / 2}'
    ymax = '${fparse l / 2}'
    nx = 10
    ny = 4
  []
  uniform_refine = 0
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system phi_system'
  previous_nl_solution_required = true
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
    initial_condition = 1
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
  []
  [phase_2]
    type = MooseLinearVariableFVReal
    solver_sys = phi_system
  []
[]

[LinearFVKernels]
  [flow_p_diffusion]
    type = LinearFVAnisotropicDiffusion
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
    variable = pressure
  []
  [flow_HbyA_divergence]
    type = LinearFVDivergence
    face_flux = HbyA
    force_boundary_execution = true
    variable = pressure
  []
  [flow_ins_momentum_flux_x]
    type = LinearWCNSFVMomentumFlux
    advected_interp_method = ${advected_interp_method}
    momentum_component = x
    mu = mu_mixture
    rhie_chow_user_object = ins_rhie_chow_interpolator
    u = vel_x
    use_deviatoric_terms = false
    use_nonorthogonal_correction = false
    v = vel_y
    variable = vel_x
  []
  [flow_ins_momentum_flux_y]
    type = LinearWCNSFVMomentumFlux
    advected_interp_method = ${advected_interp_method}
    momentum_component = y
    mu = mu_mixture
    rhie_chow_user_object = ins_rhie_chow_interpolator
    u = vel_x
    use_deviatoric_terms = false
    use_nonorthogonal_correction = false
    v = vel_y
    variable = vel_y
  []
  [mixture_drift_flux_x]
    type = LinearWCNSFV2PMomentumDriftFlux
    density_interp_method = average
    fraction_dispersed = phase_2
    momentum_component = x
    rhie_chow_user_object = ins_rhie_chow_interpolator
    rho_d = ${rho_d}
    u_slip = vel_slip_x
    v_slip = vel_slip_y
    variable = vel_x
  []
  [mixture_drift_flux_y]
    type = LinearWCNSFV2PMomentumDriftFlux
    density_interp_method = average
    fraction_dispersed = phase_2
    momentum_component = y
    rhie_chow_user_object = ins_rhie_chow_interpolator
    rho_d = ${rho_d}
    u_slip = vel_slip_x
    v_slip = vel_slip_y
    variable = vel_y
  []
  [flow_ins_momentum_pressure_x]
    type = LinearFVMomentumPressure
    momentum_component = x
    pressure = pressure
    variable = vel_x
  []
  [flow_ins_momentum_pressure_y]
    type = LinearFVMomentumPressure
    momentum_component = y
    pressure = pressure
    variable = vel_y
  []
  [flow_momentum_friction_0_x]
    type = LinearFVMomentumFriction
    Darcy_name = Darcy_coefficient_vec
    momentum_component = x
    mu = mu_mixture
    variable = vel_x
  []
  [flow_momentum_friction_0_y]
    type = LinearFVMomentumFriction
    Darcy_name = Darcy_coefficient_vec
    momentum_component = y
    mu = mu_mixture
    variable = vel_y
  []

  # Mixture phase equation
  [mixture_ins_phase_2_advection]
    type = LinearFVScalarAdvection
    advected_interp_method = upwind
    rhie_chow_user_object = ins_rhie_chow_interpolator
    u_slip = vel_slip_x
    v_slip = vel_slip_y
    variable = phase_2
  []
  [mixture_phase_interface_reaction]
    type = LinearFVReaction
    coeff = 0.1
    variable = phase_2
  []
  [mixture_phase_interface_source]
    type = LinearFVSource
    scaling_factor = 0.1
    source_density = phase_1
    variable = phase_2
  []
[]

[LinearFVBCs]
  [vel_x_left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = left
    functor = 1
    variable = vel_x
  []
  [vel_y_left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = left
    functor = 0
    variable = vel_y
  []
  [pressure_extrapolation_inlet_left]
    type = LinearFVExtrapolatedPressureBC
    boundary = left
    use_two_term_expansion = true
    variable = pressure
  []
  [vel_x_right]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = right
    use_two_term_expansion = true
    variable = vel_x
  []
  [vel_y_right]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = right
    use_two_term_expansion = true
    variable = vel_y
  []
  [pressure_right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = right
    functor = 0
    variable = pressure
  []
  [vel_x_bottom]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = bottom
    functor = 0
    variable = vel_x
  []
  [vel_y_bottom]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = bottom
    functor = 0
    variable = vel_y
  []
  [vel_x_top]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = top
    functor = 0
    variable = vel_x
  []
  [vel_y_top]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = top
    functor = 0
    variable = vel_y
  []
  [pressure_extrapolation_top_bottom]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'top bottom'
    use_two_term_expansion = true
    variable = pressure
  []

  [phase_2_left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = left
    functor = 0.1
    variable = phase_2
  []
  [phase_2_right]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = right
    use_two_term_expansion = true
    variable = phase_2
  []
[]

[FunctorMaterials]
  [flow_ins_speed_material]
    type = ADVectorMagnitudeFunctorMaterial
    execute_on = ALWAYS
    outputs = none
    vector_magnitude_name = speed
    x_functor = vel_x
    y_functor = vel_y
  []

  [mixture_phase_1_fraction]
    type = ParsedFunctorMaterial
    execute_on = ALWAYS
    expression = '1 - phase_2'
    functor_names = phase_2
    output_properties = phase_1
    outputs = all
    property_name = phase_1
  []
  [mixture_mixture_material]
    type = WCNSLinearFVMixtureFunctorMaterial
    execute_on = ALWAYS
    limit_phase_fraction = true
    outputs = all
    phase_1_fraction = phase_2
    phase_1_names = '${rho_d} ${mu_d} ${cp_d} ${k_d}'
    phase_2_names = '${rho}   ${mu}   ${cp}   ${k}'
    prop_names = 'rho_mixture mu_mixture cp_mixture k_mixture'
  []
  [mixture_slip_x]
    type = WCNSFV2PSlipVelocityFunctorMaterial
    execute_on = ALWAYS
    gravity = '0 0 0'
    linear_coef_name = Darcy_coefficient
    momentum_component = x
    mu = mu_mixture
    outputs = all
    particle_diameter = 0.01
    rho = ${rho}
    rho_d = ${rho_d}
    slip_velocity_name = vel_slip_x
    u = vel_x
    v = vel_y
  []
  [mixture_slip_y]
    type = WCNSFV2PSlipVelocityFunctorMaterial
    execute_on = ALWAYS
    gravity = '0 0 0'
    linear_coef_name = Darcy_coefficient
    momentum_component = y
    mu = mu_mixture
    outputs = all
    particle_diameter = 0.01
    rho = ${rho}
    rho_d = ${rho_d}
    slip_velocity_name = vel_slip_y
    u = vel_x
    v = vel_y
  []
  [mixture_dispersed_drag]
    type = NSFVDispersePhaseDragFunctorMaterial
    drag_coef_name = Darcy_coefficient
    execute_on = ALWAYS
    mu = mu_mixture
    outputs = all
    particle_diameter = 0.01
    rho = rho_mixture
    u = vel_x
    v = vel_y
  []
[]

[UserObjects]
  [ins_rhie_chow_interpolator]
    type = RhieChowMassFlux
    p_diffusion_kernel = flow_p_diffusion
    pressure = pressure
    rho = rho_mixture
    u = vel_x
    v = vel_y
  []
[]

[Executioner]
  type = SIMPLE
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'

  # Systems
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  active_scalar_systems = 'phi_system'
  momentum_equation_relaxation = 0.8
  active_scalar_equation_relaxation = '0.7'
  pressure_variable_relaxation = 0.3

  # We need to converge the problem to show conservation
  num_iterations = 200
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  active_scalar_absolute_tolerance = '1e-10'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  active_scalar_petsc_options_iname = '-pc_type -pc_hypre_type'
  active_scalar_petsc_options_value = 'hypre boomeramg'
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  active_scalar_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  active_scalar_l_tol = 0
  # print_fields = true
  continue_on_max_its = true
[]

[Outputs]
  csv = true
[]

[Postprocessors]
  [Re]
    type = ParsedPostprocessor
    expression = '10.0 * 2 * 1'
  []
  [average_phase2]
    type = ElementAverageValue
    variable = phase_2
  []
  [dp]
    type = PressureDrop
    boundary = 'left right'
    downstream_boundary = right
    pressure = pressure
    upstream_boundary = left
  []
  [max_phase2]
    type = ElementExtremeValue
    variable = phase_2
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_neu-dir.i)

### benchmark sources:
### https://doi.org/10.1016/j.compfluid.2018.06.016
### https://doi.org/10.1016/0017-9310(74)90087-8

b = 0.01 # plate thickness
l = 0.2 # plate length

nxi = 24 # nx in the inlet/entrance region
nyf = 18 # ny in fluid
nxf = 24 # nx in the main fluid region
nys = 8 # ny in the solid domain

fx1_bias = 1.00 # bdry layer bias - fluid
fx2_bias = '${fparse 1.0/1.00}' # bdry layer bias - solid
fy_bias = 1.20 # bdry layer bias - fluid
sy_bias = '${fparse 1.0/1.05}' # bdry layer bias - solid

k_s = 0.2876
rho = 0.3525
mu = 3.95e-5
k = 0.06808
cp = 1142.6

vin = 12.0
Tin = 1000.0
T_s_bottom = 600.0
P_out = 1.03e5

h_s = 0.0

advected_interp_method = 'upwind'

[Mesh]
  [fluid_channel]
    type = GeneratedMeshGenerator
    dim = 2
    nx = ${nxf}
    ny = ${nyf}
    xmin = 0
    xmax = ${l}
    ymin = 0
    ymax = '${fparse 10.0*b}'
    subdomain_ids = '1'
    subdomain_name = 'fluid'
    bias_x = '${fx1_bias}'
    bias_y = '${fparse fy_bias}'
    boundary_name_prefix = 'fluid'
  []
  [solid_base]
    type = GeneratedMeshGenerator
    dim = 2
    nx = ${nxf}
    ny = ${nys}
    xmin = 0
    xmax = ${l}
    ymin = '${fparse -b}'
    ymax = 0
    subdomain_ids = '2'
    subdomain_name = 'solid'
    bias_x = ${fx1_bias}
    bias_y = '${fparse sy_bias}'
    boundary_id_offset = 10
    boundary_name_prefix = 'solid'
  []
  [entrance]
    type = GeneratedMeshGenerator
    dim = 2
    nx = '${fparse 2.0*nxi}'
    ny = ${nyf}
    xmin = '${fparse -2.0*l}'
    xmax = 0
    ymin = 0
    ymax = '${fparse 10.0*b}'
    subdomain_ids = '0'
    subdomain_name = 'entrance'
    bias_x = ${fx2_bias}
    bias_y = '${fparse fy_bias}'
    boundary_id_offset = 20
    boundary_name_prefix = 'ent'
  []
  [smg]
    type = StitchedMeshGenerator
    inputs = 'entrance fluid_channel solid_base'
    stitch_boundaries_pairs = 'ent_right fluid_left;
                              fluid_bottom solid_top'
    prevent_boundary_ids_overlap = false
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'smg'
    primary_block = 'fluid'
    paired_block = 'solid'
    new_boundary = interface
  []
  [symmetry_transform]
    type = SymmetryTransformGenerator
    input = interface
    mirror_point = '0 0 0'
    mirror_normal_vector = '0 1 0'
  []
  inactive = 'symmetry_transform'
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system solid_energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = '0 1'
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${vin}
    solver_sys = u_system
    block = '0 1'
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
    block = '0 1'
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = ${P_out}
    block = '0 1'
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = ${Tin}
    block = '0 1'
  []
  [T_solid]
    type = MooseLinearVariableFVReal
    solver_sys = solid_energy_system
    initial_condition = ${T_s_bottom}
    block = 2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []
  [solid-conduction]
    type = LinearFVDiffusion
    variable = T_solid
    diffusion_coeff = ${k_s}
    use_nonorthogonal_correction = false
  []
[]

[LinearFVBCs]
  # velocity BCs
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_left'
    variable = vel_x
    functor = ${vin}
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_left'
    variable = vel_y
    functor = '0.000'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_bottom interface'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_bottom interface'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'fluid_right'
    variable = pressure
    functor = ${P_out}
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_right'
    variable = vel_y
    use_two_term_expansion = false
  []

  # freestream BCs for top of fluid domain
  [freestream_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_top ent_top'
    variable = vel_x
    use_two_term_expansion = false
  []
  [freestream_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_top ent_top'
    variable = vel_y
    use_two_term_expansion = false
  []
  [freestream_p]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    boundary = 'fluid_top ent_top'
    variable = pressure
    functor = 0
  []

  # temperature BCs
  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    functor = ${Tin}
    boundary = 'ent_left'
  []
  [heated_wall_solid]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_solid
    functor = ${T_s_bottom}
    boundary = 'solid_bottom'
  []
  [insulated_fluid]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_fluid
    functor = 0
    boundary = 'ent_bottom ent_top fluid_top'
  []
  [insulated_solid]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_solid
    functor = 0
    boundary = 'solid_left solid_right'
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T_fluid
    use_two_term_expansion = false
    boundary = 'fluid_right'
  []

  [fluid_solid]
    type = LinearFVDirichletCHTBC
    variable = T_fluid
    boundary = interface
    functor = interface_temperature_solid_interface
  []
  [solid_fluid]
    type = LinearFVRobinCHTBC
    variable = T_solid
    boundary = interface
    h = ${h_s}
    thermal_conductivity = ${k_s}
    incoming_flux = heat_flux_to_solid_interface
    surface_temperature = interface_temperature_fluid_interface
  []
[]

[FunctorMaterials]
  [rhocpT]
    property_name = 'rhocpT'
    type = ParsedFunctorMaterial
    functor_names = 'T_fluid'
    expression = '${rho}*${cp}*T_fluid'
  []
[]

[Postprocessors]
  [h_in]
    type = VolumetricFlowRate
    boundary = 'ent_left'
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    subtract_mesh_velocity = false
  []
  [h_out]
    type = VolumetricFlowRate
    boundary = 'fluid_right fluid_top ent_top interface'
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    advected_interp_method = upwind
    subtract_mesh_velocity = false
  []
[]

[VectorPostprocessors]
  [y_vs_ts]
    type = LineValueSampler
    variable = 'T_solid'
    start_point = '0.05 -1e-9 0' # making sure we are always in the domain
    end_point = '0.05 ${fparse -b+1e-9} 0'
    num_points = 8
    sort_by = id
    warn_discontinuous_face_values = false
  []
  [y_vs_tf]
    type = LineValueSampler
    variable = 'T_fluid'
    start_point = '0.05 1e-9 0' # making sure we are always in the domain
    end_point = '0.05 ${fparse b-1e-9} 0'
    num_points = 12
    sort_by = id
    warn_discontinuous_face_values = false
  []
[]

[Executioner]
  type = SIMPLE

  num_iterations = 1000

  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  rhie_chow_user_object = 'rc'
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  momentum_equation_relaxation = 0.9
  pressure_variable_relaxation = 0.3
  momentum_absolute_tolerance = 1e-7
  pressure_absolute_tolerance = 1e-7
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_system = 'energy_system'
  solid_energy_system = 'solid_energy_system'
  energy_l_abs_tol = 1e-10
  solid_energy_l_abs_tol = 1e-10
  energy_l_tol = 0
  solid_energy_l_tol = 0
  energy_equation_relaxation = 1.0
  energy_absolute_tolerance = 1e-7
  solid_energy_absolute_tolerance = 1e-7
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  solid_energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  solid_energy_petsc_options_value = 'hypre boomeramg'

  cht_interfaces = 'interface'
  cht_solid_flux_relaxation = 0.4
  cht_fluid_flux_relaxation = 0.4
  cht_solid_temperature_relaxation = 0.4
  cht_fluid_temperature_relaxation = 0.4
  max_cht_fpi = 3

  print_fields = false
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_rob-rob.i)

### benchmark sources:
### https://doi.org/10.1016/j.compfluid.2018.06.016
### https://doi.org/10.1016/0017-9310(74)90087-8

b = 0.01 # plate thickness
l = 0.2 # plate length

nxi = 24 # nx in the inlet/entrance region
nyf = 18 # ny in fluid
nxf = 24 # nx in the main fluid region
nys = 8 # ny in the solid domain

fx1_bias = 1.00 # bdry layer bias - fluid
fx2_bias = '${fparse 1.0/1.00}' # bdry layer bias - solid
fy_bias = 1.20 # bdry layer bias - fluid
sy_bias = '${fparse 1.0/1.05}' # bdry layer bias - solid

k_s = 0.2876
rho = 0.3525
mu = 3.95e-5
k = 0.06808
cp = 1142.6

vin = 12.0
Tin = 1000.0
T_s_bottom = 600.0
P_out = 1.03e5

h_s = 1.0
h_f = 1.0

advected_interp_method = 'upwind'

[Mesh]
  [fluid_channel]
    type = GeneratedMeshGenerator
    dim = 2
    nx = ${nxf}
    ny = ${nyf}
    xmin = 0
    xmax = ${l}
    ymin = 0
    ymax = '${fparse 10.0*b}'
    subdomain_ids = '1'
    subdomain_name = 'fluid'
    bias_x = '${fx1_bias}'
    bias_y = '${fparse fy_bias}'
    boundary_name_prefix = 'fluid'
  []
  [solid_base]
    type = GeneratedMeshGenerator
    dim = 2
    nx = ${nxf}
    ny = ${nys}
    xmin = 0
    xmax = ${l}
    ymin = '${fparse -b}'
    ymax = 0
    subdomain_ids = '2'
    subdomain_name = 'solid'
    bias_x = ${fx1_bias}
    bias_y = '${fparse sy_bias}'
    boundary_id_offset = 10
    boundary_name_prefix = 'solid'
  []
  [entrance]
    type = GeneratedMeshGenerator
    dim = 2
    nx = '${fparse 2.0*nxi}'
    ny = ${nyf}
    xmin = '${fparse -2.0*l}'
    xmax = 0
    ymin = 0
    ymax = '${fparse 10.0*b}'
    subdomain_ids = '0'
    subdomain_name = 'entrance'
    bias_x = ${fx2_bias}
    bias_y = '${fparse fy_bias}'
    boundary_id_offset = 20
    boundary_name_prefix = 'ent'
  []
  [smg]
    type = StitchedMeshGenerator
    inputs = 'entrance fluid_channel solid_base'
    stitch_boundaries_pairs = 'ent_right fluid_left;
                              fluid_bottom solid_top'
    prevent_boundary_ids_overlap = false
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = 'smg'
    primary_block = 'fluid'
    paired_block = 'solid'
    new_boundary = interface
  []
  [symmetry_transform]
    type = SymmetryTransformGenerator
    input = interface
    mirror_point = '0 0 0'
    mirror_normal_vector = '0 1 0'
  []
  inactive = 'symmetry_transform'
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system solid_energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = '0 1'
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${vin}
    solver_sys = u_system
    block = '0 1'
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
    block = '0 1'
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = ${P_out}
    block = '0 1'
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = ${Tin}
    block = '0 1'
  []
  [T_solid]
    type = MooseLinearVariableFVReal
    solver_sys = solid_energy_system
    initial_condition = ${T_s_bottom}
    block = 2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []

  [solid-conduction]
    type = LinearFVDiffusion
    variable = T_solid
    diffusion_coeff = ${k_s}
    use_nonorthogonal_correction = false
  []
[]

[LinearFVBCs]
  # velocity BCs
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_left'
    variable = vel_x
    functor = ${vin}
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_left'
    variable = vel_y
    functor = '0.000'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_bottom interface'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'ent_bottom interface'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'fluid_right'
    variable = pressure
    functor = ${P_out}
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_right'
    variable = vel_y
    use_two_term_expansion = false
  []

  # freestream BCs for top of fluid domain
  [freestream_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_top ent_top'
    variable = vel_x
    use_two_term_expansion = false
  []
  [freestream_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'fluid_top ent_top'
    variable = vel_y
    use_two_term_expansion = false
  []
  [freestream_p]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    boundary = 'fluid_top ent_top'
    variable = pressure
    functor = 0
  []

  # temperature BCs
  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    functor = ${Tin}
    boundary = 'ent_left'
  []
  [heated_wall_solid]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_solid
    functor = ${T_s_bottom}
    boundary = 'solid_bottom'
  []
  [insulated_fluid]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_fluid
    functor = 0
    boundary = 'ent_top ent_bottom fluid_top'
  []
  [insulated_solid]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_solid
    functor = 0
    boundary = 'solid_left solid_right'
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T_fluid
    use_two_term_expansion = false
    boundary = 'fluid_right'
  []

  [fluid_solid]
    type = LinearFVRobinCHTBC
    variable = T_fluid
    boundary = interface
    h = ${h_f}
    incoming_flux = heat_flux_to_fluid_interface
    surface_temperature = interface_temperature_solid_interface
    thermal_conductivity = ${k}
  []
  [solid_fluid]
    type = LinearFVRobinCHTBC
    variable = T_solid
    boundary = interface
    h = ${h_s}
    incoming_flux = heat_flux_to_solid_interface
    surface_temperature = interface_temperature_fluid_interface
    thermal_conductivity = ${k_s}
  []
[]

[FunctorMaterials]
  [rhocpT]
    property_name = 'rhocpT'
    type = ParsedFunctorMaterial
    functor_names = 'T_fluid'
    expression = '${rho}*${cp}*T_fluid'
  []
[]

[Postprocessors]
  [h_in]
    type = VolumetricFlowRate
    boundary = 'ent_left'
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    subtract_mesh_velocity = false
  []
  [h_out]
    type = VolumetricFlowRate
    boundary = 'fluid_right fluid_top ent_top interface'
    vel_x = vel_x
    vel_y = vel_y
    rhie_chow_user_object = rc
    advected_quantity = 'rhocpT'
    advected_interp_method = upwind
    subtract_mesh_velocity = false
  []
[]

[VectorPostprocessors]
  [y_vs_ts]
    type = LineValueSampler
    variable = 'T_solid'
    start_point = '0.05 -1e-9 0' # making sure we are always in the domain
    end_point = '0.05 ${fparse -b+1e-9} 0'
    num_points = 8
    sort_by = id
    warn_discontinuous_face_values = false
  []
  [y_vs_tf]
    type = LineValueSampler
    variable = 'T_fluid'
    start_point = '0.05 1e-9 0' # making sure we are always in the domain
    end_point = '0.05 ${fparse b-1e-9} 0'
    num_points = 12
    sort_by = id
    warn_discontinuous_face_values = false
  []
[]

[Executioner]
  type = SIMPLE

  num_iterations = 1000

  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  rhie_chow_user_object = 'rc'
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  momentum_equation_relaxation = 0.9
  pressure_variable_relaxation = 0.3
  momentum_absolute_tolerance = 1e-7
  pressure_absolute_tolerance = 1e-7
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_system = 'energy_system'
  solid_energy_system = 'solid_energy_system'

  energy_l_abs_tol = 1e-10
  solid_energy_l_abs_tol = 1e-10
  energy_l_tol = 0
  solid_energy_l_tol = 0
  energy_equation_relaxation = 1.0
  energy_absolute_tolerance = 1e-7
  solid_energy_absolute_tolerance = 1e-7
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  solid_energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  solid_energy_petsc_options_value = 'hypre boomeramg'

  cht_interfaces = 'interface'
  cht_solid_flux_relaxation = 1.0
  cht_fluid_flux_relaxation = 1.0
  cht_solid_temperature_relaxation = 1.0
  cht_fluid_temperature_relaxation = 1.0
  max_cht_fpi = 2

  print_fields = false
[]

[Outputs]
  exodus = true
  csv = true
  execute_on = timestep_end
[]

(test/tests/linearfvbcs/robin/diffusion-2d-robin.i)

#################################################################
k     = 7.0 # diffusion coeff.
amp   = 3.6 # sinusoid amplitude, for u_exact

x1   = ${fparse 0.1*pi}
x2   = ${fparse 1.0*pi}
y1   = ${fparse 0.0*pi}
y2   = ${fparse 1.0*pi}

alpha = 2.000 # robin BC coeff for gradient term
beta  = 5.000 # robin BC coeff for variable term

nx = 2
ny = 2
##################################################################

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 2
    nx   = ${nx}
    ny   = ${ny}
    xmin = ${x1}
    xmax = ${x2}
    ymin = ${y1}
    ymax = ${y2}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = '(${amp}*sin(x)*sin(y))'
  []
  [source_fn]
    type = ParsedFunction
    expression = '${fparse k*amp}*2.0*sin(x)*sin(y)'
  []
  [gamma_fn]
    type = ParsedFunction
    expression = '${fparse -amp*alpha}*cos(x)*sin(y) + ${beta} * u_e'
    symbol_names = 'u_e'
    symbol_values = 'u_exact'
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = true
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right"
    functor = u_exact
  []
  [top]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "top"
    functor = u_exact
  []
  [bottom]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "bottom"
    functor = u_exact
  []
  [robin_left]
    type = LinearFVAdvectionDiffusionFunctorRobinBC
    variable = u
    boundary = "left"
    alpha = ${alpha}
    beta  = ${beta}
    gamma = gamma_fn
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 4
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  linear_convergence = linear
[]

(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diff_coeff_func
    use_nonorthogonal_correction = false
  []
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = average
  []
  [reaction]
    type = LinearFVReaction
    variable = u
    coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  inactive = "outflow neumann"
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right top bottom"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = true
  []
  [neumann]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = u
    boundary = "top"
    functor = analytic_solution_neumann_top
    diffusion_coeff = diff_coeff_func
  []
[]

[Functions]
  [diff_coeff_func]
    type = ParsedFunction
    expression = '1.0+0.5*x*y'
  []
  [coeff_func]
    type = ParsedFunction
    expression = '1.0+1.0/(1+x*y)'
  []
  [source_func]
    type = ParsedFunction
    expression = '-1.0*x*pi*sin((1/2)*x*pi)*cos(2*y*pi) - 0.25*y*pi*sin(2*y*pi)*cos((1/2)*x*pi) + (1.0 + 1.0/(x*y + 1))*(sin((1/2)*x*pi)*sin(2*y*pi) + 1.5) + (17/4)*pi^2*(0.5*x*y + 1.0)*sin((1/2)*x*pi)*sin(2*y*pi) + 0.25*pi*sin(2*y*pi)*cos((1/2)*x*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin((1/2)*x*pi)*sin(2*y*pi) + 1.5'
  []
  [analytic_solution_neumann_top]
    type = ParsedFunction
    expression = '(1.0+0.5*x*y)*sin((1/2)*x*pi)*cos(2*y*pi)*2*pi'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvbcs/robin/diffusion-1d-robin.i)

##################################################################
k     = 7.0 # diffusion coeff.
amp   = 3.6 # sinusoid amplitude, for u_exact

x_l   = ${fparse 0.0*pi} # domain bound (left)
x_r   = ${fparse 0.9*pi}     # domain bound (right)

alpha = 5.000 # robin BC coeff for gradient term
beta  = 2.000 # robin BC coeff for variable term
gamma = ${fparse (alpha*amp*cos(x_r) ) + (beta*amp*sin(x_r))} # RHS of Robin BC, applied at right boundary

npts = 2
##################################################################

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 1
    nx   = ${npts}
    xmin = ${x_l}
    xmax = ${x_r}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = '${amp}*sin(x)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '${fparse k*amp}*sin(x)'
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = False
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [dir_r]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = 0
  []
  [rob_l]
    type = LinearFVAdvectionDiffusionFunctorRobinBC
    variable = u
    boundary = "right"
    alpha = ${alpha}
    beta  = ${beta}
    gamma = ${gamma}
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 4
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  linear_convergence = linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

(modules/heat_transfer/test/tests/radiation_participating_media/rad_T_coupled_medium_1d_T.i)

# The test guarantees that the P1 radiation model in the Linear FV system using the linear system solver
# can be used with MultiApps. The solution update of the energy system is relaxed to improve convergence
# of the coupled energy and radiative systems.
# This simulation is a 1D test with Dirichlet BCs on the left and right of the domain. Marshak BCs are
# applied at the boundaries for G.


k = 1.0
sigma_a = 0.33333
r_wall_temp = 400.0
l_wall_temp = 300.0

[Mesh]
  [salt_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 100
    xmin = -0.5
    xmax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'energy_system'
  previous_nl_solution_required = true
[]

[Variables]
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = 'energy_system'
    initial_condition = 350
  []
[]

[AuxVariables]
  [G]
    type = MooseLinearVariableFVReal
    initial_condition = 3400
  []
[]

[LinearFVKernels]
  [temp_conduction]
    type = LinearFVDiffusion
    diffusion_coeff = ${k}
    variable = T
  []
  [temp_radiation]
    type = LinearFVP1TemperatureSourceSink
    variable = T
    G = 'G'
    absorption_coeff = ${sigma_a}
  []
[]

[LinearFVBCs]
  [left_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${l_wall_temp}
  []
  [right_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'right'
    functor = ${r_wall_temp}
  []
[]

[Postprocessors]
  [mean_value_G]
    type = ElementIntegralFunctorPostprocessor
    functor = G
  []
  [max_value_G]
    type = ElementExtremeFunctorValue
    functor = G
  []
  [mean_value_T]
    type = ElementIntegralFunctorPostprocessor
    functor = T
  []
  [max_value_T]
    type = ElementExtremeFunctorValue
    functor = T
  []
[]

[Executioner]
  type = Steady
  solve_type = 'Linear'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  l_abs_tol = 1e-14
  l_tol = 1e-14
  nl_abs_tol = 1e-14

  # MultiApp fixed point iteration parameters
  fixed_point_min_its = 2
  fixed_point_max_its = 500
  relaxation_factor = 0.9
  transformed_variables = 'T'
  fixed_point_rel_tol = 1e-12
  fixed_point_abs_tol = 1e-12
[]

[Outputs]
  csv = true
  exodus = true
  execute_on = timestep_end
[]

[MultiApps]
  [sub_app]
    type = FullSolveMultiApp
    input_files = 'rad_T_coupled_medium_1d_G.i'
  []
[]

[Transfers]
  [push]
    type = MultiAppCopyTransfer
    to_multi_app = sub_app
    source_variable = 'T'
    variable = 'T'
  []
  [pull]
    type = MultiAppCopyTransfer
    from_multi_app = sub_app
    source_variable = 'G'
    variable = 'G'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/SIMPLEC/3d/3d-velocity-pressure.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 3
    dx = '1.0'
    dy = '0.3'
    dz = '0.3'
    ix = '10'
    iy = '3'
    iz = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system w_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    w = vel_z
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    pressure_projection_method = consistent
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [vel_z]
    type = MooseLinearVariableFVReal
    solver_sys = w_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [w_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_z
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'z'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [w_pressure]
    type = LinearFVMomentumPressure
    variable = vel_z
    pressure = pressure
    momentum_component = 'z'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [inlet-w]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_z
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_y
    functor = 0.0
  []
  [walls-w]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_z
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_w]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_z
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system w_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 1.0
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type -pc_hypre_boomeramg_agg_nl -pc_hypre_boomeramg_agg_num_paths -pc_hypre_boomeramg_truncfactor -pc_hypre_boomeramg_strong_threshold -pc_hypre_boomeramg_coarsen_type -pc_hypre_boomeramg_interp_type'
  momentum_petsc_options_value = 'hypre boomeramg 4 1 0.1 0.6 HMIS ext+i'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type -pc_hypre_boomeramg_agg_nl -pc_hypre_boomeramg_agg_num_paths -pc_hypre_boomeramg_truncfactor -pc_hypre_boomeramg_strong_threshold -pc_hypre_boomeramg_coarsen_type -pc_hypre_boomeramg_interp_type'
  pressure_petsc_options_value = 'hypre boomeramg 2 1 0.1 0.6 HMIS ext+i'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(modules/navier_stokes/test/tests/finite_volume/ins/cht/bulk_heat_transfer/flow-around-square-linear-fluidonly.i)

mu = 0.01
rho = 1.1
k = 0.0005
cp = 10
h_conv = 5

advected_interp_method = 'upwind'

[Mesh]
  [generated_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
    xmin = 0
    ymin = 0
    ymax = 0.1
    xmax = 0.1
  []
  [subdomain1]
    type = SubdomainBoundingBoxGenerator
    input = generated_mesh
    block_name = subdomain1
    bottom_left = '0.04 0.04 0'
    block_id = 1
    top_right = '0.06 0.06 0'
  []
  [interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = subdomain1
    primary_block = 0
    paired_block = 1
    new_boundary = interface
  []
  [delete]
    type = BlockDeletionGenerator
    input = interface
    block = 1
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = 0
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.1
    solver_sys = u_system
    block = 0
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.01
    block = 0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
    block = 0
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
    block = 0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = true
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = true
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '0.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom interface'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom interface'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []

  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    functor = 300
    boundary = 'left'
  []
  [walls_T]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_fluid
    functor = 0.0
    boundary = 'top bottom'
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T_fluid
    use_two_term_expansion = false
    boundary = right
  []

  [fluid_solid]
    type = LinearFVConvectiveHeatTransferBC
    variable = T_fluid
    T_solid = boundary_value
    T_fluid = T_fluid
    boundary = interface
    h = ${h_conv}
  []
[]

[FunctorMaterials]
  [rhocpT]
    property_name = 'rhocpT'
    type = ParsedFunctorMaterial
    functor_names = 'T_fluid'
    expression = '${rho}*${cp}*T_fluid'
  []
[]

[Functions]
  [boundary_value]
    type = ConstantFunction
    value = 350
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  energy_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.8
  energy_equation_relaxation = 1.0
  pressure_variable_relaxation = 0.3
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  energy_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated/BFS_ERCOFTAC.i)

Re = 5100

rho = 1.0
bulk_u = 1.0
H = 1.0
mu = '${fparse rho * bulk_u * H / Re}'

advected_interp_method = 'upwind'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Initial and Boundary Conditions ###
intensity = 0.01
k_init = '${fparse 1.5*(intensity * bulk_u)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / H}'

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'bottom wall-side top'
wall_treatment = 'eq_incremental' # Options: eq_newton, eq_incremental, eq_linearized, neq

[Mesh]
  [gen]
    type = CartesianMeshGenerator
    dim = 2
    dx = '${fparse 10.0*H} ${fparse 20.0*H}'
    dy = '${H} ${fparse 5*H}'
    ix = '8 16'
    iy = '2 8'
    subdomain_id = '
                    2 1
                    1 1
                  '
  []
  [corner_walls]
    type = SideSetsBetweenSubdomainsGenerator
    input = gen
    primary_block = '1'
    paired_block = '2'
    new_boundary = 'wall-side'
  []
  [delete_bottom]
    type = BlockDeletionGenerator
    input = corner_walls
    block = '2'
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []

  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []

  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_y
    functor = 0.0
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = ${walls}
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init + mu}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.5 0.5'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  csv = true
  [console]
    type = Console
    outlier_variable_norms = false
  []
[]


[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_entry_channel_wall]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 1.00001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 1.00001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_entry_channel]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 2.25001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 2.25001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-vortex/2d-vortex.i)

mu = 1
rho = 1
advected_interp_method = 'average'

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]


[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
  []
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]


[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_forcing]
    type = LinearFVSource
    variable = vel_x
    source_density = forcing_u
  []
  [v_forcing]
    type = LinearFVSource
    variable = vel_y
    source_density = forcing_v
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
    use_nonorthogonal_correction_on_boundary = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [no-slip-wall-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top bottom'
    variable = vel_x
    functor = '0'
  []
  [no-slip-wall-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top bottom'
    variable = vel_y
    functor = '0'
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'left right top bottom'
    variable = pressure
    use_two_term_expansion = true
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'x^2*(1-x)^2*(2*y-6*y^2+4*y^3)'
  []
  [exact_v]
    type = ParsedFunction
    expression = '-y^2*(1-y)^2*(2*x-6*x^2+4*x^3)'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'x*(1-x)'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '-4*mu*(-1+2*y)*(y^2-6*x*y^2+6*x^2*y^2-y+6*x*y-6*x^2*y+3*x^2-6*x^3+3*x^4)+1-2*x+rho*4*x^3'
            '*y^2*(2*y^2-2*y+1)*(y-1)^2*(-1+2*x)*(x-1)^3'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '4*mu*(-1+2*x)*(x^2-6*y*x^2+6*x^2*y^2-x+6*x*y-6*x*y^2+3*y^2-6*y^3+3*y^4)+rho*4*y^3*x^2*(2'
            '*x^2-2*x+1)*(x-1)^2*(-1+2*y)*(y-1)^3'
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-8
  pressure_l_abs_tol = 1e-8
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 2000
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false

  pin_pressure = true
  pressure_pin_value = 0.25
  pressure_pin_point = '0.5 0.5 0.0'
[]

[Outputs]
  exodus = true
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2u]
    type = ElementL2FunctorError
    approximate = vel_x
    exact = exact_u
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2v]
    type = ElementL2FunctorError
    approximate = vel_y
    exact = exact_v
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2p]
    approximate = pressure
    exact = exact_p
    type = ElementL2FunctorError
    outputs = 'csv'
    execute_on = FINAL
  []
[]

(test/tests/multiapps/linearfv_nonlinearfv/linearfv.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 6
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[AuxVariables]
  [diff_var]
    type = MooseVariableFVReal
    initial_condition = 2.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diff_var
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 1
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = 1
  []
[]

[MultiApps]
  inactive = 'nonlinear'
  [nonlinear]
    type = FullSolveMultiApp
    input_files = nonlinearfv.i
    execute_on = timestep_begin
    no_restore = true
  []
[]

[Transfers]
  inactive = 'from_nonlinear to_nonlinear'
  [from_nonlinear]
    type = MultiAppCopyTransfer
    from_multi_app = nonlinear
    source_variable = 'v'
    variable = 'diff_var'
    execute_on = timestep_begin
  []
  [to_nonlinear]
    type = MultiAppCopyTransfer
    to_multi_app = nonlinear
    source_variable = 'u'
    variable = 'diff_var'
    execute_on = timestep_begin
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_type'
  petsc_options_value = 'lu mumps'
  fixed_point_rel_tol = 1e-10
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(test/tests/linearfvkernels/diffusion/diffusion-2d_neumann.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
  [neu_bottom]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = u
    boundary = "bottom"
    functor = analytic_solution_neumann_bottom
    diffusion_coeff = coeff_func
  []
  [neu_top]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = u
    boundary = "top"
    functor = analytic_solution_neumann_top
    diffusion_coeff = coeff_func
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*(1.5-y*y)+2*x*y*(1.5-y*y)+2*(1.5-x*x)+2*x*y*(1.5-x*x)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
  [analytic_solution_neumann_bottom]
    type = ParsedFunction
    expression = '-(1+0.5*x*y)*(1.5-x*x)*(-2*y)'
  []
  [analytic_solution_neumann_top]
    type = ParsedFunction
    expression = '(1+0.5*x*y)*(1.5-x*x)*(-2*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/rotated-channel.i)

mu = 1
rho = 1
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 5
    nx = 20
    ny = 4
  []
  [transform]
    type = TransformGenerator
    transform = rotate
    vector_value = '45 0 0'
    input = mesh
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVNormalVelocityFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    normal_velocity = '-1'
    component = 'x'
  []
  [inlet-v]
    type = LinearFVNormalVelocityFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    normal_velocity = '-1'
    component = 'y'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(test/tests/time_steppers/iteration_adaptive/adapt_linear_systems.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 50
  ny = 2
  xmax = 5
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [time]
    type = LinearFVTimeDerivative
    variable = 'u'
  []
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 5
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 2
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = 12
  []
[]

[Executioner]
  type = Transient
  system_names = u_sys
  start_time = 0.0
  end_time = 19
  n_startup_steps = 2
  dtmax = 6.0
  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 8
    dt = 1.0
  []
  verbose = true
  linear_convergence = much_logic
[]

[Convergence]
  [force_grow]
    type = IterationCountConvergence
    min_iterations = 0
    max_iterations = 4
    converge_at_max_iterations = true
  []
  [force_shrink]
    type = IterationCountConvergence
    min_iterations = 12
    max_iterations = 13
    converge_at_max_iterations = true
  []
  [much_logic]
    type = ParsedConvergence
    convergence_expression = 'if(time < 5, force_grow, force_shrink)'
    symbol_names = 'time force_grow force_shrink'
    symbol_values = 'time force_grow force_shrink'
  []
[]

[Postprocessors]
  [_dt]
    type = TimestepSize
  []
  [time]
    type = TimePostprocessor
  []
[]

[Outputs]
  csv = true
[]

(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex-rz/2d-symmetric-vortex-rz-spacedependent.i)

mu_ref = 1.2
mu_r = 0.1
mu_x = 0.2
rho_ref = 1.4
rho_r = 0.15
rho_x = 0.25

use_dev = true

advected_interp_method = 'average'

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
  []
  coord_type = 'RZ'
  rz_coord_axis = x
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = rho_fun
    p_diffusion_kernel = p_diffusion
    pressure_projection_method = consistent
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = mu_fun
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = ${use_dev}
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = mu_fun
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = ${use_dev}
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_forcing]
    type = LinearFVSource
    variable = vel_x
    source_density = forcing_u
  []
  [v_forcing]
    type = LinearFVSource
    variable = vel_y
    source_density = forcing_v
  []
  [v_viscous_forcing]
    type = LinearFVRZViscousSource
    variable = vel_y
    mu = mu_fun
    momentum_component = 'y'
    u = vel_x
    v = vel_y
    use_deviatoric_terms = ${use_dev}
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = 'Ainv'
    use_nonorthogonal_correction = false
    use_nonorthogonal_correction_on_boundary = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = 'HbyA'
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [no-slip-wall-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_x
    functor = exact_u
  []
  [no-slip-wall-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_y
    functor = exact_v
  []
  [symmetry-u]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_x
    u = vel_x
    v = vel_y
    momentum_component = x
  []
  [symmetry-v]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_y
    u = vel_x
    v = vel_y
    momentum_component = y
  []

  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'top left right'
    variable = pressure
    use_two_term_expansion = true
  []
  [pressure-symmetry]
    type = LinearFVPressureSymmetryBC
    boundary = 'bottom'
    variable = pressure
    HbyA_flux = 'HbyA'
  []
[]

[AuxVariables]
  [vel_x_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
  [vel_y_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
  [pressure_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
[]

[AuxKernels]
  [vel_x_function_aux]
    type = FunctionAux
    variable = vel_x_aux
    function = exact_u
    execute_on = TIMESTEP_END
  []
  [vel_y_function_aux]
    type = FunctionAux
    variable = vel_y_aux
    function = exact_v
    execute_on = TIMESTEP_END
  []
  [pressure_function_aux]
    type = FunctionAux
    variable = pressure_aux
    function = exact_p
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  # --------------------------------------------------
  # Material property functions
  # --------------------------------------------------
  [rho_fun]
    type = ParsedFunction
    symbol_names = 'rho_ref rho_r rho_x'
    symbol_values = '${rho_ref} ${rho_r} ${rho_x}'
    expression = 'rho_ref + rho_x*x + rho_r*y'
  []
  [mu_fun]
    type = ParsedFunction
    symbol_names = 'mu_ref mu_r mu_x'
    symbol_values = '${mu_ref} ${mu_r} ${mu_x}'
    expression = 'mu_ref + mu_x*x + mu_r*y'
  []
  # --------------------------------------------------
  # Convenience reciprocals
  # --------------------------------------------------
  [S]
    type=ParsedFunction
    symbol_names = 'rho_fun'
    symbol_values = 'rho_fun'
    expression='1.0/rho_fun'
  []
  [S2]
    type=ParsedFunction
    symbol_names = 'rho_fun'
    symbol_values = 'rho_fun'
    expression='(1.0/rho_fun)^2'
  []
  [S3]
    type=ParsedFunction
    symbol_names = 'rho_fun'
    symbol_values = 'rho_fun'
    expression='(1.0/rho_fun)^3'
  []
  # --------------------------------------------------
  # Building blocks
  # --------------------------------------------------
  [A]
    type=ParsedFunction
    expression='x^2*(1-x)^2'
  []
  [Ap]
    type=ParsedFunction
    expression='2*x - 6*x^2 + 4*x^3'
  []
  [App]
    type=ParsedFunction
    expression='2 - 12*x + 12*x^2'
  []
  [Appp]
    type=ParsedFunction
    expression='-12 + 24*x'
  []
  [Q]
    type=ParsedFunction
    expression='4*y^2 - 10*y^3 + 6*y^4'
  []
  [Qp]
    type=ParsedFunction
    expression='8*y - 30*y^2 + 24*y^3'
  []
  [Qpp]
    type=ParsedFunction
    expression='8 - 60*y + 72*y^2'
  []
  [R]
    type=ParsedFunction
    expression='y^3 - 2*y^4 + y^5'
  []
  [Rp]
    type=ParsedFunction
    expression='3*y^2 - 8*y^3 + 5*y^4'
  []
  [Rpp]
    type=ParsedFunction
    expression='6*y - 24*y^2 + 20*y^3'
  []
  # --------------------------------------------------
  # Exact solutions
  # --------------------------------------------------
  [exact_u]
    type=ParsedFunction
    symbol_names = 'A Q rho_fun'
    symbol_values = 'A Q rho_fun'
    expression='A*Q / rho_fun'
  []
  [exact_v]
    type=ParsedFunction
    symbol_names = 'Ap R rho_fun'
    symbol_values = 'Ap R rho_fun'
    expression='-Ap*R / rho_fun'
  []
  [exact_p]
    type=ParsedFunction
    expression='x*y^2'
  []
  # --------------------------------------------------
  # First derivatives
  # --------------------------------------------------
  [ux]
    type=ParsedFunction
    symbol_names = 'A Ap Q rho_fun rho_x'
    symbol_values = 'A Ap Q rho_fun ${rho_x}'
    expression='(Ap*Q)/rho_fun - (A*Q*rho_x)/rho_fun^2'
  []
  [ur]
    type=ParsedFunction
    symbol_names = 'A Q Qp rho_fun rho_r'
    symbol_values = 'A Q Qp rho_fun ${rho_r}'
    expression='(A*Qp)/rho_fun - (A*Q*rho_r)/rho_fun^2'
  []
  [vx]
    type=ParsedFunction
    symbol_names = 'App Ap R rho_fun rho_x'
    symbol_values = 'App Ap R rho_fun ${rho_x}'
    expression='-(App*R)/rho_fun + (Ap*R*rho_x)/rho_fun^2'
  []
  [vr]
    type=ParsedFunction
    symbol_names = 'Ap Rp R rho_fun rho_r'
    symbol_values = 'Ap Rp R rho_fun ${rho_r}'
    expression='-(Ap*Rp)/rho_fun + (Ap*R*rho_r)/rho_fun^2'
  []
  # --------------------------------------------------
  # Second derivatives
  # --------------------------------------------------
  [uxx]
    type = ParsedFunction
    symbol_names  = 'rho_x App Q S Ap S2 A S3'
    symbol_values = '${rho_x} App Q S Ap S2 A S3'
    expression = 'App*Q*S - 2*Ap*Q*rho_x*S2 + 2*A*Q*(rho_x^2)*S3'
  []
  [uxr]
    type = ParsedFunction
    symbol_names  = 'rho_x rho_r Ap Qp S Q S2 A S3'
    symbol_values = '${rho_x} ${rho_r} Ap Qp S Q S2 A S3'
    expression = 'Ap*Qp*S - Ap*Q*rho_r*S2 - A*Qp*rho_x*S2 + 2*A*Q*(rho_x*rho_r)*S3'
  []
  [urr]
    type = ParsedFunction
    symbol_names  = 'rho_r A Qpp S Qp S2 Q S3'
    symbol_values = '${rho_r} A Qpp S Qp S2 Q S3'
    expression = 'A*Qpp*S - 2*A*Qp*rho_r*S2 + 2*A*Q*(rho_r^2)*S3'
  []
  [vxx]
    type = ParsedFunction
    symbol_names  = 'rho_x Appp R S App S2 Ap S3'
    symbol_values = '${rho_x} Appp R S App S2 Ap S3'
    expression = '-Appp*R*S + 2*App*R*rho_x*S2 - 2*Ap*R*(rho_x^2)*S3'
  []
  [vxr]
    type = ParsedFunction
    symbol_names  = 'rho_x rho_r App Rp S R Ap S2 S3'
    symbol_values = '${rho_x} ${rho_r} App Rp S R Ap S2 S3'
    expression = '-App*Rp*S + (App*R*rho_r + Ap*Rp*rho_x)*S2 - 2*Ap*R*(rho_x*rho_r)*S3'
  []
  [vrr]
    type = ParsedFunction
    symbol_names  = 'rho_r Ap Rpp S Rp S2 R S3'
    symbol_values = '${rho_r} Ap Rpp S Rp S2 R S3'
    expression = '-Ap*Rpp*S + 2*Ap*Rp*rho_r*S2 - 2*Ap*R*(rho_r^2)*S3'
  []
  # --------------------------------------------------
  # Derivatives of theta
  # --------------------------------------------------
  [theta]
    type = ParsedFunction
    symbol_names = 'ux vr exact_v'
    symbol_values = 'ux vr exact_v'
    expression = 'ux + vr + (exact_v/y)'
  []
  [thetax]
    type = ParsedFunction
    symbol_names = 'uxx vxr vx'
    symbol_values = 'uxx vxr vx'
    expression = 'uxx + vxr + (vx/y)'
  []
  [thetar]
    type = ParsedFunction
    symbol_names = 'uxr vrr vr exact_v'
    symbol_values = 'uxr vrr vr exact_v'
    expression = 'uxr + vrr + (vr/y) - (exact_v/(y^2))'
  []
  # --------------------------------------------------
  # Stress components
  # --------------------------------------------------
  [tau_xx]
    type=ParsedFunction
    symbol_names = 'ux theta mu_fun'
    symbol_values = 'ux theta mu_fun'
    expression='2*mu_fun*ux - (2.0/3.0)*mu_fun*theta'
  []
  [tau_rr]
    type=ParsedFunction
    symbol_names = 'vr theta mu_fun'
    symbol_values = 'vr theta mu_fun'
    expression='2*mu_fun*vr - (2.0/3.0)*mu_fun*theta'
  []
  [tau_xr]
    type=ParsedFunction
    symbol_names = 'ur vx mu_fun'
    symbol_values = 'ur vx mu_fun'
    expression='mu_fun*(ur + vx)'
  []
  [tau_tt]
    type=ParsedFunction
    symbol_names = 'exact_v theta mu_fun'
    symbol_values = 'exact_v theta mu_fun'
    expression='2*mu_fun*(exact_v/y) - (2.0/3.0)*mu_fun*theta'
  []
  # --------------------------------------------------
  # Convective fluxes
  # --------------------------------------------------
  [Fxx]
    type=ParsedFunction
    symbol_names = 'exact_u rho_fun'
    symbol_values = 'exact_u rho_fun'
    expression='rho_fun*exact_u*exact_u'
  []
  [Fxr]
    type=ParsedFunction
    symbol_names = 'exact_u exact_v rho_fun'
    symbol_values = 'exact_u exact_v rho_fun'
    expression='rho_fun*exact_u*exact_v'
  []
  [Frx]
    type=ParsedFunction
    symbol_names = 'exact_u exact_v rho_fun'
    symbol_values = 'exact_u exact_v rho_fun'
    expression='rho_fun*exact_v*exact_u'
  []
  [Frr]
    type=ParsedFunction
    symbol_names = 'exact_v rho_fun'
    symbol_values = 'exact_v rho_fun'
    expression='rho_fun*exact_v*exact_v'
  []
  # --------------------------------------------------
  # Stress derivatives
  # --------------------------------------------------
  [tau_xx_x]
    type = ParsedFunction
    symbol_names  = 'mu_x ux mu_fun uxx theta thetax'
    symbol_values = '${mu_x} ux mu_fun uxx theta thetax'
    expression = '2*mu_x*ux + 2*mu_fun*uxx - (2.0/3.0)*(mu_x*theta + mu_fun*thetax)'
  []
  [tau_xr_x]
    type = ParsedFunction
    symbol_names  = 'mu_x ur vx mu_fun uxr vxx'
    symbol_values = '${mu_x} ur vx mu_fun uxr vxx'
    expression = 'mu_x*(ur + vx) + mu_fun*(uxr + vxx)'
  []
  [tau_xr_r]
    type = ParsedFunction
    symbol_names  = 'mu_r ur vx mu_fun urr vxr'
    symbol_values = '${mu_r} ur vx mu_fun urr vxr'
    expression = 'mu_r*(ur + vx) + mu_fun*(urr + vxr)'
  []
  [tau_rr_r]
    type = ParsedFunction
    symbol_names  = 'mu_r vr mu_fun vrr theta thetar'
    symbol_values = '${mu_r} vr mu_fun vrr theta thetar'
    expression = '2*mu_r*vr + 2*mu_fun*vrr - (2.0/3.0)*(mu_r*theta + mu_fun*thetar)'
  []
  # --------------------------------------------------
  # Conservative flux derivatives
  # --------------------------------------------------
  [Fxx_x]
    type = ParsedFunction
    symbol_names  = 'rho_x A Ap Q S S2'
    symbol_values = '${rho_x} A Ap Q S S2'
    expression = '2*A*Ap*Q^2*S - A^2*Q^2*rho_x*S2'
  []
  [Frx_r]
    type = ParsedFunction
    symbol_names  = 'rho_r A Ap Qp R Q Rp S S2'
    symbol_values = '${rho_r} A Ap Qp R Q Rp S S2'
    expression = '-A*Ap*((Qp*R + Q*Rp)*S - Q*R*rho_r*S2)'
  []
  [Fxr_x]
    type = ParsedFunction
    symbol_names  = 'rho_x Ap A App Q R S S2'
    symbol_values = '${rho_x} Ap A App Q R S S2'
    expression = '-((Ap^2 + A*App)*Q*R*S - A*Ap*Q*R*rho_x*S2)'
  []
  [Frr_r]
    type = ParsedFunction
    symbol_names  = 'rho_r Ap R Rp S S2'
    symbol_values = '${rho_r} Ap R Rp S S2'
    expression = 'Ap^2*(2*R*Rp*S - R^2*rho_r*S2)'
  []
  # --------------------------------------------------
  # Pressure gradients
  # --------------------------------------------------
  [px]
    type=ParsedFunction
    expression='y^2'
  []
  [pr]
    type=ParsedFunction
    expression='2*x*y'
  []
  # --------------------------------------------------
  # Forcing terms
  # --------------------------------------------------
  [forcing_u]
    type = ParsedFunction
    symbol_names  = 'Fxx_x Frx Frx_r px tau_xx_x tau_xr tau_xr_r'
    symbol_values = 'Fxx_x Frx Frx_r px tau_xx_x tau_xr tau_xr_r'
    expression = 'Fxx_x + (Frx/y) + Frx_r + px - ( tau_xx_x + (tau_xr/y) + tau_xr_r )'
  []
  [forcing_v]
    type = ParsedFunction
    symbol_names  = 'Fxr_x Frr Frr_r pr tau_xr_x tau_rr tau_rr_r tau_tt'
    symbol_values = 'Fxr_x Frr Frr_r pr tau_xr_x tau_rr tau_rr_r tau_tt'
    expression = 'Fxr_x + (Frr/y) + Frr_r + pr - ( tau_xr_x + (tau_rr/y) + tau_rr_r - (tau_tt)/y )'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  momentum_l_max_its = 500
  pressure_l_max_its = 500
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.85
  pressure_variable_relaxation = 1.0
  num_iterations = 10000
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.125
  pressure_pin_point = '0.5 0.5 0.0'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = TIMESTEP_END
  []
  [L2u]
    type = ElementL2FunctorError
    approximate = vel_x
    exact = exact_u
    execute_on = TIMESTEP_END
  []
  [L2v]
    type = ElementL2FunctorError
    approximate = vel_y
    exact = exact_v
    execute_on = TIMESTEP_END
  []
  [L2p]
    approximate = pressure
    exact = exact_p
    type = ElementL2FunctorError
    execute_on = TIMESTEP_END
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/natural_convection/linear_segregated/2d/diff_heated_cavity_linear_segregated.i)

################################################################################
# MATERIAL PROPERTIES
################################################################################
rho = 3279.
T_0 = 875.0
mu = 1.
k_cond = 38.0
cp = 640.
alpha = 3.26e-5

walls = 'right left top bottom'

[GlobalParams]
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  advected_interp_method = 'upwind'
  u = vel_x
  v = vel_y
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

################################################################################
# GEOMETRY
################################################################################

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
    nx = 15
    ny = 15
  []
[]

################################################################################
# EQUATIONS: VARIABLES, KERNELS & BCS
################################################################################

[UserObjects]
  [ins_rhie_chow_interpolator]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    body_force_kernel_names = "u_buoyancy; v_buoyancy"
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = pressure_system
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 875
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [u_buoyancy]
    type = LinearFVMomentumBoussinesq
    variable = vel_x
    T_fluid = T_fluid
    gravity = '0 -9.8 0'
    rho = ${rho}
    ref_temperature = ${T_0}
    alpha_name = ${alpha}
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [v_buoyancy]
    type = LinearFVMomentumBoussinesq
    variable = vel_y
    T_fluid = T_fluid
    gravity = '0 -9.81 0'
    rho = ${rho}
    ref_temperature = ${T_0}
    alpha_name = ${alpha}
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = false
  []

   ####### FUEL ENERGY EQUATION #######

  [heat_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${fparse k_cond}
  []
[]


[LinearFVBCs]
  [no-slip-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = ${walls}
    functor = 0
  []
  [no-slip-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = ${walls}
    functor = 0
  []
  [T_cold]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    boundary = 'right'
    functor = 870.0
  []
  [T_hot]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    boundary = 'left'
    functor = 880.0
  []
  [T_all]
    type = LinearFVAdvectionDiffusionExtrapolatedBC
    variable = T_fluid
    boundary = 'top bottom'
    use_two_term_expansion = false
  []
  [pressure]
    type = LinearFVPressureFluxBC
    boundary = 'top bottom left right'
    variable = pressure
    HbyA_flux = HbyA
    Ainv = Ainv
  []
[]

[FunctorMaterials]
  [constant_functors]
    type = GenericFunctorMaterial
    prop_names = 'cp alpha_b'
    prop_values = '${cp} ${alpha}'
  []
[]

################################################################################
# EXECUTION / SOLVE
################################################################################

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-11
  pressure_l_abs_tol = 1e-11
  energy_l_abs_tol = 1e-11
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.9
  num_iterations = 1500
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  energy_absolute_tolerance = 1e-8
  print_fields = false
  momentum_l_max_its = 300

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.5 0.0 0.0'

  # momentum_petsc_options = '-ksp_monitor'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'

  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'

  continue_on_max_its = true
[]

################################################################################
# SIMULATION OUTPUTS
################################################################################

[Outputs]
  #exodus = true
  [out]
    type = Exodus
    file_base = 'diff_heated_cavity_linear_segregated_out'
  []
[]

(test/tests/linearfvbcs/scalar_symmetry/diffusion-2d-symmetry.i)

diff = 0.1 # diffusion coeff.

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 2
    nx   = 2
    ny   = 2
    xmin = 0
    xmax = ${fparse pi/3}
    ymin = 0
    ymax = ${fparse pi/3}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = 'cos(x)*cos(y)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '2*${diff}*cos(x)*cos(y)'
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = ${diff}
    use_nonorthogonal_correction = true
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right top"
    functor = u_exact
  []
  [symm]
    type = LinearFVAdvectionDiffusionScalarSymmetryBC
    variable = u
    boundary = "bottom left"
    use_two_term_expansion = true
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 10
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  linear_convergence = linear
[]

(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc.i)

# 1D steady-state heat conduction with a radiative BC on the right boundary.
# The domain is [0,1] with:
#   Left:  Dirichlet temperature T_L (fixed)
#   Right: LinearFVFunctorRadiativeBC, q = sigma * eps * (T^4 - Tinf^4)
# A pseudo-transient approach (large dt steps) iteratively updates the
# lagged linearization coefficients until convergence to the nonlinear solution.

T_L   = 1000.0   # Left boundary temperature [K]
T_inf = 300.0    # Far-field radiation temperature [K]
eps   = 1.0      # Emissivity
k     = 1.0      # Thermal conductivity [W/(m-K)]

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 50
    xmin = 0
    xmax = 1
  []
[]

[Problem]
  linear_sys_names = 'heat_system'
[]

[Variables]
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = 'heat_system'
    initial_condition = ${T_L}
  []
[]

[LinearFVKernels]
  [time]
    type = LinearFVTimeDerivative
    variable = T
  []
  [diffusion]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${T_L}
  []
  [right]
    type = LinearFVFunctorRadiativeBC
    variable = T
    boundary = 'right'
    emissivity = ${eps}
    Tinfinity = ${T_inf}
    diffusion_coeff = ${k}
  []
[]

[Postprocessors]
  [T_avg]
    type = ElementAverageValue
    variable = T
  []
  [T_max]
    type = ElementExtremeValue
    variable = T
    value_type = max
  []
  [T_min]
    type = ElementExtremeValue
    variable = T
    value_type = min
  []
[]

[Executioner]
  type = Transient
  system_names = heat_system
  scheme = 'implicit-euler'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  dt = 1e6
  num_steps = 50
  l_tol = 1e-10
[]

[Outputs]
  file_base = 'linear_fv_functor_radiative_bc_hot_body_out'
  [csv]
    type = CSV
    execute_on = final
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC-physics.i)


!include header_and_mesh.i

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[Physics]
  [NavierStokes]
    [FlowSegregated/flow]
      velocity_variable = 'vel_x vel_y'
      pressure_variable = 'pressure'

      # Initial conditions
      initial_velocity = '${bulk_u} 0 0'
      initial_pressure = '1e-8'

      # Material properties
      density = ${rho}
      dynamic_viscosity = ${mu}

      # Boundary conditions
      inlet_boundaries = 'left'
      momentum_inlet_types = 'fixed-velocity'
      momentum_inlet_functors = '${bulk_u} 0'

      wall_boundaries = 'top bottom'
      momentum_wall_types = 'noslip noslip'

      outlet_boundaries = 'right'
      momentum_outlet_types = 'fixed-pressure'
      pressure_functors = '0'

      # Numerical parameters
      include_deviatoric_stress = true
      orthogonality_correction = false
      pressure_two_term_bc_expansion = false
      momentum_two_term_bc_expansion = false
      momentum_advection_interpolation = ${advected_interp_method}
      system_names = 'u_system v_system pressure_system'
      # consistency with kernel syntax input, allows to use the same executioner block
      rhie_chow_uo_name = 'rc'
    []
    [TurbulenceSegregated/k-epsilon]
      # Model
      turbulence_handling = 'k-epsilon'
      tke_name = TKE
      tked_name = TKED
      system_names = 'TKE_system TKED_system'

      initial_tke = ${k_init}
      initial_tked = ${eps_init}

      # Model parameters
      mu_t_ratio_max = 1e20
      sigma_k = ${sigma_k}
      sigma_eps = ${sigma_eps}
      C_pl = 1e10
      C1_eps = ${C1_eps}
      C2_eps = ${C2_eps}

      turbulence_walls = ${walls}
      wall_treatment_eps = ${wall_treatment}
      bulk_wall_treatment = ${bulk_wall_treatment}
      use_nonorthogonal_correction = false
    []
  []
[]

# Boundary conditions are not implemented for turbulent quantities at this time
[LinearFVBCs]
  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
[]

!include executioner_regular_channel.i

variables_to_sample = 'vel_x vel_y pressure_over_density TKE TKED'
!include postprocessing.i

(test/tests/multisystem/restore_multiapp/parent.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 6
  []
[]

[Problem]
  nl_sys_names = 'v_sys'
  linear_sys_names = 'u_sys'
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    initial_condition = 2.0
    solver_sys = v_sys
  []
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVKernels]
  [diffusion]
    type = FVDiffusion
    variable = v
    coeff = u
  []
  [source]
    type = FVBodyForce
    variable = v
    function = 3
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = v
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 1
  []
[]

[FVBCs]
  [dir]
    type = FVFunctorDirichletBC
    variable = v
    boundary = "left right"
    functor = 2
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = 1
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 6
    converge_at_max_iterations = true
  []
[]

[MultiApps]
  [sub]
    type = FullSolveMultiApp
    input_files = sub.i
    execute_on = TIMESTEP_END
    keep_solution_during_restore = true
    # The choice does not matter here, but conceptually there is no need to override
    # the old solution when the subapp is a Steady solve
    update_old_solution_when_keeping_solution_during_restore = false
  []
[]

[Executioner]
  type = Steady
  system_names = 'v_sys u_sys'
  l_abs_tol = 1e-12
  l_tol = 1e-10
  nl_abs_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  fixed_point_max_its = 2
  fixed_point_min_its = 2
  #disable_fixed_point_residual_norm_check = true
  accept_on_max_fixed_point_iteration = true
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_ERCOFTAC_symmetric.i)

H = 1 #halfwidth of the channel
L = 100

Re = 13700

rho = 1
bulk_u = 1
mu = '${fparse rho * bulk_u * 2 * H / Re}'

advected_interp_method = 'upwind'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Initial and Boundary Conditions ###
intensity = '${fparse 0.16*Re^(-1./8.)}'
k_init = '${fparse 1.5*(intensity * bulk_u)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / (2*H)}'

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'top'
wall_treatment = 'eq_newton' # Options: eq_newton, eq_incremental, eq_linearized, neq

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = 0
    ymax = ${H}
    nx = 4
    ny = 4
    bias_y = 0.7
  []
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_y
    functor = 0.0
  []
  [symmetry-u]
    type = LinearFVVelocitySymmetryBC
    variable = vel_x
    boundary = 'bottom'
    momentum_component = 'x'
    u = 'vel_x'
    v = 'vel_y'
  []
  [symmetry-v]
    type = LinearFVVelocitySymmetryBC
    variable = vel_y
    boundary = 'bottom'
    momentum_component = 'y'
    u = 'vel_x'
    v = 'vel_y'
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [symmetry-p]
    type = LinearFVPressureSymmetryBC
    variable = pressure
    boundary = 'bottom'
    HbyA_flux = 'HbyA' # Functor created in the RhieChowMassFlux UO
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []

  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = ${walls}
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.2 0.2'
  turbulence_field_relaxation = '0.2 0.2'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  csv = true
  execute_on = 'TIMESTEP_END'
[]

[AuxVariables]
  [pressure_over_density]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
[]

[AuxKernels]
  [compute_pressure_over_density]
    type = ParsedAux
    variable = pressure_over_density
    coupled_variables = 'pressure'
    expression = 'pressure/${rho}'
  []
[]

[VectorPostprocessors]
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure_over_density TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_center_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.0001} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.0001} 0'
    num_points = ${Mesh/gmg/nx}
    variable = 'vel_x vel_y pressure_over_density TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_radius_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.5 * H} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.5 * H} 0'
    num_points = ${Mesh/gmg/nx}
    variable = 'vel_x vel_y pressure_over_density TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/enthalpy_equation.i)

H = 0.015 #halfwidth of the channel, 10 cm of channel height
L = 1
bulk_u = 0.01
p_ref = 101325.0

advected_interp_method = 'upwind'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = -${H}
    ymax = ${H}
    nx = 30
    ny = 15
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = 'rho'
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
    initial_condition = ${bulk_u}
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = ${p_ref}
  []
  [h]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 44000 # 1900 is an approx of cp(T)
  []
[]

[AuxVariables]
  [rho_var]
    type = MooseLinearVariableFVReal
  []
  [cp_var]
    type = MooseLinearVariableFVReal
  []
  [mu_var]
    type = MooseLinearVariableFVReal
  []
  [k_var]
    type = MooseLinearVariableFVReal
  []
  [T]
    type = MooseLinearVariableFVReal
    initial_condition = 777.
  []
[]

[LinearFVKernels]

  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = 'mu'
    momentum_component = 'x'
    use_nonorthogonal_correction = false
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    mu = 'mu'
    momentum_component = 'y'
    use_nonorthogonal_correction = false
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [temp_conduction]
    type = LinearFVDiffusion
    diffusion_coeff = 'alpha'
    variable = h
  []
  [temp_advection]
    type = LinearFVEnergyAdvection
    variable = h
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
[]

[LinearFVBCs]
  [inlet_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = ${bulk_u} #${bulk_u} #'fully_developed_velocity'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = 0
  []
  [inlet_h]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = h
    boundary = 'left'
    functor = h_from_p_T # ${fparse 1900.*860.}
  []
  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = 860.
  []

  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'top bottom'
    functor = 0.
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = 'top bottom'
    functor = 0.
  []
  [walls_h]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = h
    boundary = 'top bottom'
    functor = h_from_p_T # ${fparse 1900. * 950}
  []
  [walls_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'top bottom'
    functor = 950.
  []
  [walls_p]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'top bottom'
    variable = pressure
    use_two_term_expansion = false
  []

  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = ${p_ref}
  []
  [outlet_h]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = h
    use_two_term_expansion = false
    boundary = 'right'
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[FluidProperties]
  [lead]
    type = LeadFluidProperties
  []
[]

[FunctorMaterials]
  [fluid_props_to_mat_props]
    type = GeneralFunctorFluidProps
    fp = lead
    pressure = ${p_ref}
    T_fluid = 'T'
    speed = 1
    porosity = 1
    characteristic_length = 1
  []
  [alpha]
    type = ADParsedFunctorMaterial
    property_name = 'alpha'
    functor_names = 'k cp'
    expression = 'k/cp'
  []
  [enthalpy_material]
    type = LinearFVEnthalpyFunctorMaterial
    pressure = ${p_ref}
    T_fluid = T
    h = h
    fp = lead
  []
[]

[AuxKernels]
  [rho_out]
    type = FunctorAux
    functor = 'rho'
    variable = 'rho_var'
    execute_on = 'NONLINEAR'
  []
  [cp_out]
    type = FunctorAux
    functor = 'cp'
    variable = 'cp_var'
    execute_on = 'NONLINEAR'
  []
  [mu_out]
    type = FunctorAux
    functor = 'mu'
    variable = 'mu_var'
    execute_on = 'NONLINEAR'
  []
  [k_out]
    type = FunctorAux
    functor = 'k'
    variable = 'k_var'
    execute_on = 'NONLINEAR'
  []
  [T_from_h_functor]
    type = FunctorAux
    functor = 'T_from_p_h'
    variable = 'T'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-6
  pressure_l_abs_tol = 1e-6
  energy_l_abs_tol = 1e-8
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.9
  num_iterations = 200
  pressure_absolute_tolerance = 1e-6
  momentum_absolute_tolerance = 1e-6
  energy_absolute_tolerance = 1e-6
  print_fields = false
  momentum_l_max_its = 1000

  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'

  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = 'TIMESTEP_BEGIN FINAL'
  [out]
    type = Checkpoint
    execute_on = 'TIMESTEP_END'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/natural_convection/linear_segregated/2d/diff_heated_cavity_linear_buoyancy.i)

################################################################################
# MATERIAL PROPERTIES
################################################################################
rho_0 = 3279.
mu = 1.0
k_cond = 38.0
cp = ${fparse 640}
alpha_b = 3.26e-5
T_0 = 875.0

walls = 'right left top bottom'

[GlobalParams]
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  advected_interp_method = 'upwind'
  u = vel_x
  v = vel_y
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

################################################################################
# GEOMETRY
################################################################################

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 1
    xmax = 2.0
    ymin = 1.0
    ymax = 2.0
    nx = 15
    ny = 15
  []
[]

################################################################################
# EQUATIONS: VARIABLES, KERNELS & BCS
################################################################################

[UserObjects]
  [ins_rhie_chow_interpolator]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = 'rho'
    p_diffusion_kernel = p_diffusion
    body_force_kernel_names = 'u_buoyancy; v_buoyancy'
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = pressure_system
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 875
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [u_buoyancy]
    type = LinearFVMomentumBuoyancy
    variable = vel_x
    rho = 'rho'
    reference_rho = ${rho_0}
    gravity = '0 -9.81 0'
    momentum_component = 'x'
  []

  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [v_buoyancy]
    type = LinearFVMomentumBuoyancy
    variable = vel_y
    rho = 'rho'
    reference_rho = ${rho_0}
    gravity = '0 -9.81 0'
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = false
  []

   ####### FUEL ENERGY EQUATION #######

  [heat_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k_cond}
  []
[]

[LinearFVBCs]
  [no-slip-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = ${walls}
    functor = 0
  []
  [no-slip-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = ${walls}
    functor = 0
  []
  [T_cold]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    boundary = 'right'
    functor = 870.0
  []
  [T_hot]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    boundary = 'left'
    functor = 880.0
  []
  [T_all]
    type = LinearFVAdvectionDiffusionFunctorNeumannBC
    variable = T_fluid
    boundary = 'top bottom'
    functor = 0.0
  []
  [pressure]
    type = LinearFVPressureFluxBC
    boundary = 'top bottom left right'
    variable = pressure
    HbyA_flux = HbyA
    Ainv = Ainv
  []
[]

[FunctorMaterials]
  [rho_function]
    type = ParsedFunctorMaterial
    property_name = 'rho'
    functor_names = 'T_fluid'
    expression = '${rho_0}*(1-${alpha_b}*(T_fluid-${T_0})) '
  []
[]

################################################################################
# EXECUTION / SOLVE
################################################################################

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-11
  pressure_l_abs_tol = 1e-11
  energy_l_abs_tol = 1e-11
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.9
  num_iterations = 1500
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  energy_absolute_tolerance = 1e-8
  print_fields = false
  momentum_l_max_its = 300

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '1.5 1.5 0.0'

  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'

  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'

  continue_on_max_its = true
[]

################################################################################
# SIMULATION OUTPUTS
################################################################################

[Outputs]
  exodus = true
[]

(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion-react.i)

source=1
diff_coeff=2
reac_coeff=3

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 1
    dx = '0.5 0.5'
    ix = '20 20'
    subdomain_id = '1 2'
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = ${diff_coeff}
    use_nonorthogonal_correction = false
    block = 1
  []
  [reaction]
    type = LinearFVReaction
    variable = u
    coeff = ${reac_coeff}
    block = 2
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = ${source}
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = 0
  []
[]

[Functions]
  [analytic_solution]
    type = ParsedFunction
    expression = 'if(x<0.5, -x*x*S/2/D+(S/C+0.5*0.5/2/D*S)/0.5*x, S/C)'
    symbol_names = 'S D C'
    symbol_values = '${source} ${diff_coeff} ${reac_coeff}'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = TIMESTEP_END
    block = 2
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = TIMESTEP_END
    block = 2
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type -ksp_rtol'
  petsc_options_value = 'hypre boomeramg 1e-10'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
[]

(test/tests/linearfvkernels/advection/diagonal-step-2d-delaunay.i)

[Mesh]
  [file_mesh]
    type = FileMeshGenerator
    file = diagonal-step-2d-delaunay_in.e
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.5
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 0.75
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 0.75
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "1 1 0"
    advected_interp_method_name = muscl_venkat
  []
[]

[LinearFVBCs]
  [left_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0.5
  []
  [bottom_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = bottom
    functor = 1
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = 'right top'
    use_two_term_expansion = false
  []
[]

[VectorPostprocessors]
  [diag_sample]
    type = LineValueSampler
    variable = u
    start_point = '1.0 0 0'
    end_point = '0 1.0 0'
    num_points = 101
    sort_by = id
    warn_discontinuous_face_values = false
    execute_on = TIMESTEP_END
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 200
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  multi_system_fixed_point_relaxation_factor = 0.3
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  csv = true
  exodus = true
  execute_on = TIMESTEP_END
[]

(modules/navier_stokes/test/tests/finite_volume/ins/radiation_participating_media/p1_radiation_cht_SIMPLE.i)

# Coupled conduction and radiative heat transfer test in the linear FV system using the SIMPLE executioner.
# A 1D P1 radiation model is solved in the fluid region. Energy is solved in the
# fluid and solid region using two different variables. The radiative and conductive flux
# is maintained at the fluid-solid interface using conjugate heat transfer, Robin BCs for
# the solid side and Dirichlet BCs for the fluid side. Fixed temperatures are set at the solid and
# fluid external boundaries.
### Benchmark sources:
### https://hal.science/hal-02070285/document

nx = 10
Tw_left = 1.0
Tw_right = 0.0
sigma = 5.670374419e-8

sigma_a = 1.0

N = 0.1 # Stark number N = k sigma_a / (4*sigma*pow(T_w,3))
k_salt = ${fparse N*4*sigma*pow(Tw_left,3)/sigma_a}
k_solid = ${fparse 4.0*k_salt}
diffusion_coef = ${fparse 1/(3*sigma_a)}
b_eps = 1.0

rho = 1.
mu = 5.
h_s = 0.0

advected_interp_method = 'upwind'

[Mesh]
  [salt_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = ${nx}
    xmin = 0
    xmax = 1
    subdomain_ids = 0
  []
  [solid_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = ${nx}
    xmin = 1
    xmax = 2
    subdomain_ids = 1
  []
  [give_name_solid]
    type = RenameBlockGenerator
    input = solid_mesh
    old_block = 1
    new_block = 'solid'
  []
  [give_name_salt]
    type = RenameBlockGenerator
    input = salt_mesh
    old_block = 0
    new_block = 'salt'
  []
  [stitch]
    type = StitchMeshGenerator
    inputs = 'give_name_salt give_name_solid'
    stitch_boundaries_pairs = 'right left'
  []
  [interface]
    input = stitch
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = 'solid'
    paired_block = 'salt'
    new_boundary = interface
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system pressure_system energy_system solid_energy_system radiation_system '
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = 'salt'
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-6
    solver_sys = 'u_system'
    block = 'salt'
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = 'pressure_system'
    initial_condition = 1e-6
    block = 'salt'
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = 'energy_system'
    initial_condition = ${fparse Tw_left/2}
    block = 'salt'
  []
  [G]
    type = MooseLinearVariableFVReal
    solver_sys = 'radiation_system'
    initial_condition = ${fparse 4*sigma*pow(Tw_left/2,4)}
    block = 'salt'
  []
  [T_solid]
    type = MooseLinearVariableFVReal
    solver_sys = 'solid_energy_system'
    initial_condition = ${fparse Tw_left/2}
    block = 'solid'
  []
[]

[Postprocessors]
  [T_fluid_iface]
    type = ElementalVariableValue
    variable = T_fluid
    elementid = ${fparse nx-1}
  []
  [T_solid_iface]
    type = ElementalVariableValue
    variable = T_solid
    elementid = ${fparse nx}
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [fluid_conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k_salt}
    use_nonorthogonal_correction = false
    block = salt
  []
  [fluid_radiation]
    type = LinearFVP1TemperatureSourceSink
    variable = T_fluid
    G = 'G'
    absorption_coeff = ${sigma_a}
    block = salt
  []

  [G_diffusion]
    type = LinearFVDiffusion
    variable = G
    diffusion_coeff = ${diffusion_coef}
    block = salt
  []
  [G_source_and_sink]
    type = LinearFVP1RadiationSourceSink
    variable = G
    temperature_radiation = 'T_fluid'
    absorption_coeff = ${sigma_a}
    block = salt
  []

  [solid_conduction]
    type = LinearFVDiffusion
    variable = T_solid
    diffusion_coeff = ${k_solid}
    use_nonorthogonal_correction = false
    block = solid
  []
[]

[LinearFVBCs]
  # velocity BCs
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left interface'
    variable = vel_x
    functor = 0.0
  []

  # temperature BCs
  [left_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    boundary = 'left'
    functor = ${Tw_left}
  []
  [right_bc_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_solid
    boundary = 'right'
    functor = ${Tw_right}
  []

  # G
  [right_bc_G]
    type = LinearFVP1RadiationMarshakBC
    boundary = 'interface'
    variable = G
    temperature_radiation = 'T_fluid'
    coeff_diffusion = ${diffusion_coef}
    boundary_emissivity = ${b_eps}
  []
  [left_bc_G]
    type = LinearFVP1RadiationMarshakBC
    boundary = 'left'
    variable = G
    temperature_radiation = ${Tw_left}
    coeff_diffusion = ${diffusion_coef}
    boundary_emissivity = ${b_eps}
  []

  # cht
  [fluid_solid]
    type = LinearFVDirichletCHTBC
    variable = T_fluid
    boundary = interface
    functor = interface_temperature_solid_interface
  []
  [solid_fluid]
    type = LinearFVRobinCHTBC
    variable = T_solid
    boundary = interface
    h = ${h_s}
    thermal_conductivity = ${k_solid}
    incoming_flux = heat_flux_to_solid_interface
    surface_temperature = interface_temperature_fluid_interface
  []
[]

[Executioner]
  type = SIMPLE

  num_iterations = 2000

  momentum_systems = 'u_system '
  pressure_system = 'pressure_system'
  rhie_chow_user_object = 'rc'
  momentum_l_abs_tol = 1e-8
  pressure_l_abs_tol = 1e-7
  momentum_l_tol = 0
  pressure_l_tol = 0
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  momentum_absolute_tolerance = 1e-5
  pressure_absolute_tolerance = 1e-5
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  #active_scalar_systems = 'radiation_system'
  pm_radiation_systems = 'radiation_system'
  energy_system = 'energy_system'
  solid_energy_system = 'solid_energy_system'
  #active_scalar_l_abs_tol = 1e-14
  pm_radiation_l_abs_tol = 1e-11
  energy_l_abs_tol = 1e-11
  solid_energy_l_abs_tol = 1e-11
  #active_scalar_l_tol = 0
  pm_radiation_l_tol = 0
  energy_l_tol = 0
  solid_energy_l_tol = 0
  #active_scalar_equation_relaxation = 0.95
  pm_radiation_equation_relaxation = 1.0
  energy_equation_relaxation = 0.9
  energy_absolute_tolerance = 1e-10
  solid_energy_absolute_tolerance = 1e-10
  #active_scalar_absolute_tolerance = 1e-14
  #active_scalar_petsc_options_iname = '-pc_type -pc_hypre_type'
  #active_scalar_petsc_options_value = 'hypre boomeramg'
  pm_radiation_absolute_tolerance = 1e-10
  pm_radiation_petsc_options_iname = '-pc_type -pc_hypre_type'
  pm_radiation_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  solid_energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  solid_energy_petsc_options_value = 'hypre boomeramg'

  cht_interfaces = 'interface'
  cht_solid_flux_relaxation = 0.3
  cht_fluid_flux_relaxation = 0.3
  cht_solid_temperature_relaxation = 0.3
  cht_fluid_temperature_relaxation = 0.3
  cht_heat_flux_tolerance = 1e-4
  max_cht_fpi = 10

  print_fields = false
  continue_on_max_its = true

[]

[Outputs]
  csv = true
  execute_on = timestep_end
  exodus = true
[]

(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/rayleigh-bernard-two-phase-physics.i)

mu = 1.0
rho = 1e3
mu_d = 0.3
rho_d = 1.0
dp = 0.01
U_lid = 0.0
g = -9.81
advected_interp_method = 'upwind'

# Currently required
k = 1
k_d = 1
cp = 1
cp_d = 1

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = .1
    ymin = 0
    ymax = .1
    nx = 11
    ny = 11
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system phi_system'
[]

[Physics]
  [NavierStokes]
    [FlowSegregated]
      [flow]
        compressibility = 'incompressible'

        density = 'rho_mixture'
        dynamic_viscosity = 'mu_mixture'
        gravity = '0 ${g} 0'

        # Initial conditions
        initial_velocity = '1e-12 1e-12 0'
        initial_pressure = 0.2

        wall_boundaries = 'top left right bottom'
        momentum_wall_types = 'noslip noslip noslip noslip'
        momentum_wall_functors = '${U_lid} 0; 0 0; 0 0; 0 0'

        orthogonality_correction = false
        pressure_two_term_bc_expansion = true
        momentum_advection_interpolation = ${advected_interp_method}
      []
    []
    [TwoPhaseMixtureSegregated]
      [mixture]
        system_names = 'phi_system'
        phase_1_fraction_name = 'phase_1'
        phase_2_fraction_name = 'phase_2'

        add_phase_transport_equation = true
        phase_advection_interpolation = '${advected_interp_method}'
        phase_fraction_diffusivity = 1e-3

        # We could consider adding fixed-value-yet-not-an-inlet
        # boundary conditions to the TwoPhaseMixture physics

        # Base phase material properties
        phase_1_density_name = ${rho}
        phase_1_viscosity_name = ${mu}
        phase_1_specific_heat_name = ${cp}
        phase_1_thermal_conductivity_name = ${k}

        # Other phase material properties
        phase_2_density_name = ${rho_d}
        phase_2_viscosity_name = ${mu_d}
        phase_2_specific_heat_name = ${cp_d}
        phase_2_thermal_conductivity_name = ${k_d}
        output_all_properties = true

        # Friction model, not actually used!
        use_dispersed_phase_drag_model = true
        particle_diameter = ${dp}
        add_advection_slip_term = false
        # To match Rayleigh Bernard nonlinear test setup
        add_gravity_term_in_slip_velocity = false
      []
    []
  []
[]

[LinearFVBCs]
  [botttom-phase-2]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = phase_2
    functor = '1'
  []
  [top-phase-2]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top'
    variable = phase_2
    functor = '0'
  []
[]

[Executioner]
  type = PIMPLE
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'

  end_time = 1e8
  [TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 10
    iteration_window = 2
    growth_factor = 2
    cutback_factor = 0.5
    dt = 1e-3
  []

  # Systems
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  active_scalar_systems = 'phi_system'
  momentum_equation_relaxation = 0.8
  active_scalar_equation_relaxation = '0.7'
  pressure_variable_relaxation = 0.3

  # We need to converge the problem to show conservation
  num_iterations = 200
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  active_scalar_absolute_tolerance = '1e-10'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  active_scalar_petsc_options_iname = '-pc_type -pc_factor_shift_type' # -pc_hypre_type'
  active_scalar_petsc_options_value = 'lu NONZERO'
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  active_scalar_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  active_scalar_l_tol = 0
  # print_fields = true
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.0 0.0 0.0'
[]

[Outputs]
  exodus = false
  [out]
    type = CSV
    execute_on = 'FINAL'
  []
[]

[AuxVariables]
  [U]
    order = CONSTANT
    family = MONOMIAL
    fv = true
  []
[]

[Postprocessors]
  [average_void]
    type = ElementAverageValue
    variable = 'phase_2'
  []
  [max_y_velocity]
    type = ElementExtremeValue
    variable = 'vel_y'
    value_type = max
  []
  [min_y_velocity]
    type = ElementExtremeValue
    variable = 'vel_y'
    value_type = min
  []
  [max_x_velocity]
    type = ElementExtremeValue
    variable = 'vel_x'
    value_type = max
  []
  [min_x_velocity]
    type = ElementExtremeValue
    variable = 'vel_x'
    value_type = min
  []
  [max_x_slip_velocity]
    type = ElementExtremeFunctorValue
    functor = 'vel_slip_x'
    value_type = max
  []
  [max_y_slip_velocity]
    type = ElementExtremeFunctorValue
    functor = 'vel_slip_y'
    value_type = max
  []
  [max_drag_coefficient_x]
    type = ElementExtremeFunctorValue
    functor = 'Darcy_coefficient_vec_out_x'
    value_type = max
  []
  [max_drag_coefficient_y]
    type = ElementExtremeFunctorValue
    functor = 'Darcy_coefficient_vec_out_y'
    value_type = max
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-vortex/spacedependent_mu/sl.i)

rho = 1
advected_interp_method = 'average'

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
  []
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_forcing]
    type = LinearFVSource
    variable = vel_x
    source_density = forcing_u
  []
  [v_forcing]
    type = LinearFVSource
    variable = vel_y
    source_density = forcing_v
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
    use_nonorthogonal_correction_on_boundary = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [no-slip-wall-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top bottom'
    variable = vel_x
    functor = '0'
  []
  [no-slip-wall-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top bottom'
    variable = vel_y
    functor = '0'
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'left right top bottom'
    variable = pressure
    use_two_term_expansion = true
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'x^2*(1-x)^2*(2*y-6*y^2+4*y^3)'
  []
  [exact_v]
    type = ParsedFunction
    expression = '-y^2*(1-y)^2*(2*x-6*x^2+4*x^3)'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'x*(1-x)'
  []
  [mu]
    type = ParsedFunction
    expression = '1+(x-1)*x*(y-1)*y'
  []
  [forcing_u]
    type = ParsedFunction
    expression = '-(2*x-1)*y*(y-1)*(2*x-6*x^2+4*x^3)*(2*y-6*y^2+4*y^3)'
                 '-(1+x*(x-1)*y*(y-1))*(2*y-6*y^2+4*y^3)*(2-12*x+12*x^2)'
                 '-(2*y-1)*x*(x-1)*(x^2*(1-x)^2*(2-12*y+12*y^2))'
                 '-(1+x*(x-1)*y*(y-1))*(x^2*(1-x)^2*(-12+24*y))'
                 '+1-2*x+rho*4*x^3*y^2*(2*y^2-2*y+1)*(y-1)^2*(-1+2*x)*(x-1)^3'
    symbol_names = 'rho'
    symbol_values = '${rho}'
  []
  [forcing_v]
    type = ParsedFunction
    expression = '(2*y-1)*x*(x-1)*(2*y-6*y^2+4*y^3)*(2*x-6*x^2+4*x^3)'
                 '+(1+x*(x-1)*y*(y-1))*(2-12*y+12*y^2)*(2*x-6*x^2+4*x^3)'
                 '+(2*x-1)*y*(y-1)*(y^2*(1-y)^2*(2-12*x+12*x^2))'
                 '+(1+x*(x-1)*y*(y-1))*(y^2*(1-y)^2*(-12+24*x))'
                 '+rho*4*y^3*x^2*(2*x^2-2*x+1)*(x-1)^2*(-1+2*y)*(y-1)^3'
    symbol_names = 'rho'
    symbol_values = '${rho}'
  []
  [forcing_u_deviatoric]
    type = ParsedFunction
    expression = '-2*(2*x-1)*y*(y-1)*(2*x-6*x^2+4*x^3)*(2*y-6*y^2+4*y^3)'
                 '-2*(1+x*(x-1)*y*(y-1))*(2*y-6*y^2+4*y^3)*(2-12*x+12*x^2)'
                 '-(2*y-1)*x*(x-1)*(x^2*(1-x)^2*(2-12*y+12*y^2)-y^2*(1-y)^2*(2-12*x+12*x^2))'
                 '-(1+x*(x-1)*y*(y-1))*(x^2*(1-x)^2*(-12+24*y)-(2*y-6*y^2+4*y^3)*(2-12*x+12*x^2))'
                 '+1-2*x+rho*4*x^3*y^2*(2*y^2-2*y+1)*(y-1)^2*(-1+2*x)*(x-1)^3'
    symbol_names = 'rho'
    symbol_values = '${rho}'
  []
  [forcing_v_deviatoric]
    type = ParsedFunction
    expression = '2*(2*y-1)*x*(x-1)*(2*y-6*y^2+4*y^3)*(2*x-6*x^2+4*x^3)'
                 '+2*(1+x*(x-1)*y*(y-1))*(2-12*y+12*y^2)*(2*x-6*x^2+4*x^3)'
                 '-(2*x-1)*y*(y-1)*(x^2*(1-x)^2*(2-12*y+12*y^2)-y^2*(1-y)^2*(2-12*x+12*x^2))'
                 '-(1+x*(x-1)*y*(y-1))*(-y^2*(1-y)^2*(-12+24*x)+(2*x-6*x^2+4*x^3)*(2-12*y+12*y^2))'
                 '+rho*4*y^3*x^2*(2*x^2-2*x+1)*(x-1)^2*(-1+2*y)*(y-1)^3'
    symbol_names = 'rho'
    symbol_values = '${rho}'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 2000
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false

  pin_pressure = true
  pressure_pin_value = 0.25
  pressure_pin_point = '0.5 0.5 0.0'
[]

[Outputs]
  exodus = true
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2u]
    type = ElementL2FunctorError
    approximate = vel_x
    exact = exact_u
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2v]
    type = ElementL2FunctorError
    approximate = vel_y
    exact = exact_v
    outputs = 'csv'
    execute_on = FINAL
  []
  [L2p]
    approximate = pressure
    exact = exact_p
    type = ElementL2FunctorError
    outputs = 'csv'
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/advection-2d-rz.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny= 1
    ymax = 0.5
  []
  coord_type = RZ
  rz_coord_axis = Y
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.0 0.5 0"
    advected_interp_method_name = average
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right bottom"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "top"
    use_two_term_expansion = true
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '1.0*pi*sin(x*pi)*cos(2*y*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 10
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       NONZERO               1e-10'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/time_integrators/bdf2/linearfv.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 4
  ny = 4
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
  []
[]

[LinearFVKernels]
  [timeu]
    type = LinearFVTimeDerivative
    variable = u
  []
  [diffu]
    type = LinearFVDiffusion
    variable = u
  []
  [forceu]
    type = LinearFVSource
    variable = u
    source_density = force
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    expression = 't^3*x*y'
  []
  [force]
    type = ParsedFunction
    expression = '3*x*y*t^2'
  []
[]

[LinearFVBCs]
  [allu]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    functor = exact
    variable = u
    boundary = 'left right top bottom'
  []
[]

[Executioner]
  type = Transient
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'hypre'
  dt = 1
  end_time = 3
  scheme = 'bdf2'
[]

[Postprocessors]
  [L2u]
    type = ElementL2Error
    function = exact
    variable = u
  []
[]

[Outputs]
  csv = true
[]

(test/tests/tag/tag-linearfv.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 3
  []
[]

[Problem]
  linear_sys_names = 'u_sys v_sys'
  extra_tag_matrices = 'mat_tag_u; mat_tag_v'
  extra_tag_vectors = 'vec_tag_u; vec_tag_v'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    initial_condition = 1.0
    solver_sys = u_sys
  []
  [v]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = v_sys
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 2.0
  []
  [reaction]
    type = LinearFVReaction
    variable = u
    coeff = 3.0
    matrix_tags = 'system mat_tag_u'
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 60.0
    vector_tags = 'rhs vec_tag_u'
  []
  [diffusion_v]
    type = LinearFVDiffusion
    variable = v
    diffusion_coeff = 1.0
  []
  [reaction_v]
    type = LinearFVReaction
    variable = v
    coeff = 1.5
    matrix_tags = 'system mat_tag_v'
  []
  [source_v]
    type = LinearFVSource
    variable = v
    source_density = 20.0
    vector_tags = 'rhs vec_tag_v'
  []
[]

[LinearFVBCs]
  [left_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 1.0
  []
  [right_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 3.0
  []
  [left_v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = left
    functor = 1.0
  []
  [right_v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = right
    functor = 3.0
  []
[]

[AuxVariables]
  [soln_u_dof]
    type = MooseLinearVariableFVReal
  []
  [soln_u]
    type = MooseLinearVariableFVReal
  []
  [rhs_u_dof]
    type = MooseLinearVariableFVReal
  []
  [rhs_u]
    type = MooseLinearVariableFVReal
  []
  [vector_tag_u]
    type = MooseLinearVariableFVReal
  []
  [matrix_u_diag]
    type = MooseLinearVariableFVReal
  []
  [soln_v_dof]
    type = MooseLinearVariableFVReal
  []
  [soln_v]
    type = MooseLinearVariableFVReal
  []
  [rhs_v_dof]
    type = MooseLinearVariableFVReal
  []
  [rhs_v]
    type = MooseLinearVariableFVReal
  []
  [vector_tag_v]
    type = MooseLinearVariableFVReal
  []
  [matrix_v_diag]
    type = MooseLinearVariableFVReal
  []
[]

[AuxKernels]
  [soln_u_dof]
    type = TagVectorDofValueAux
    variable = soln_u_dof
    v = u
    vector_tag = 'solution'
  []
  [soln_u]
    type = TagVectorAux
    variable = soln_u
    v = u
    vector_tag = 'solution'
  []
  [rhs_u_dof]
    type = TagVectorDofValueAux
    variable = rhs_u_dof
    v = u
    vector_tag = 'rhs'
  []
  [rhs_u]
    type = TagVectorAux
    variable = rhs_u
    v = u
    vector_tag = 'rhs'
  []
  [extra_vector_u_dof]
    type = TagVectorDofValueAux
    variable = vector_tag_u
    v = u
    vector_tag = 'vec_tag_u'
  []
  [extra_vector_u]
    type = TagVectorAux
    variable = vector_tag_u
    v = u
    vector_tag = 'vec_tag_u'
  []
  [extra_matrix_u]
    type = TagMatrixAux
    variable = matrix_u_diag
    v = u
    matrix_tag = 'mat_tag_u'
  []
  [soln_v_dof]
    type = TagVectorDofValueAux
    variable = soln_v_dof
    v = v
    vector_tag = 'solution'
  []
  [soln_v]
    type = TagVectorAux
    variable = soln_v
    v = v
    vector_tag = 'solution'
  []
  [rhs_v_dof]
    type = TagVectorDofValueAux
    variable = rhs_v_dof
    v = v
    vector_tag = 'rhs'
  []
  [rhs_v]
    type = TagVectorAux
    variable = rhs_v
    v = v
    vector_tag = 'rhs'
  []
  [extra_vector_v_dof]
    type = TagVectorDofValueAux
    variable = vector_tag_v
    v = v
    vector_tag = 'vec_tag_v'
  []
  [extra_vector_v]
    type = TagVectorAux
    variable = vector_tag_v
    v = v
    vector_tag = 'vec_tag_v'
  []
  [extra_matrix_v]
    type = TagMatrixAux
    variable = matrix_v_diag
    v = v
    matrix_tag = 'mat_tag_v'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'LINEAR'
  system_names = 'u_sys v_sys'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc_mms_2d.i)

# 2D MMS convergence test for LinearFVFunctorRadiativeBC.
#
# Exact solution: T(x,y) = T_L + B_x*(exp(x) - 1) + B_y*x*sin(pi*y)
# where B_x = (T_R_mms - T_L) / (e - 1).
#
# The solution combines a 1D exponential profile with a sinusoidal 2D
# perturbation B_y*x*sin(pi*y).  The sinusoidal y-dependence is
# non-polynomial, ensuring the FV scheme has genuine O(h^2) truncation
# error in the y-direction.
#
# At the radiative boundary (x=1):
#   T(1,y) = T_R_mms + B_y*sin(pi*y)
#   dT/dx(1,y) = B_x*e + B_y*sin(pi*y)
#
# A spatially varying T_inf(y) is computed analytically so the radiative
# BC is satisfied exactly at every boundary face:
#   T_inf(y)^4 = T(1,y)^4 + k*dT/dx(1,y)/(sigma*eps)
# At y=0,1 this recovers the 1D far-field temperature T_inf ~ 300 K.
#
# Boundary conditions:
#   Left (x=0):   Dirichlet T = T_L             (constant, since x=0)
#   Right (x=1):  Radiative Robin with T_inf(y)  (spatially varying)
#   Bottom (y=0): Dirichlet T = T_L + B_x*(exp(x) - 1)  (sin(0)=0)
#   Top (y=1):    Dirichlet T = T_L + B_x*(exp(x) - 1)  (sin(pi)=0)
#
# Source: f(x,y) = -k*(B_x*exp(x) - B_y*pi^2*x*sin(pi*y))
#
# T_R_mms must be supplied via CLI args, as the root of:
#   k*(T_L - T_R)*e/(e-1) = sigma*eps*(T_R^4 - T_inf^4)
# (computed in the Python convergence script).

T_L    = 1000.0
k      = 1.0
eps    = 1.0
sigma  = 5.670374419e-8
B_y    = 200.0

T_R_mms = 0  # set via CLI: T_R_mms=<value>
B_x = '${fparse (T_R_mms - T_L) / (exp(1.0) - 1.0)}'
B_x_e = '${fparse B_x * exp(1.0)}'
flux_scale = '${fparse k / (sigma * eps)}'
pi_val = '${fparse pi}'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10  # overridden by CLI
    ny = 10  # overridden by CLI
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
  []
[]

[Problem]
  linear_sys_names = 'heat_system'
[]

[Variables]
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = 'heat_system'
    initial_condition = ${T_L}
  []
[]

[Functions]
  [exact_T]
    type = ParsedFunction
    expression = 'T_L + Bx * (exp(x) - 1.0) + By * x * sin(pv * y)'
    symbol_names = 'T_L Bx By pv'
    symbol_values = '${T_L} ${B_x} ${B_y} ${pi_val}'
  []
  [source_fn]
    type = ParsedFunction
    expression = '-k * Bx * exp(x) + k * By * pv * pv * x * sin(pv * y)'
    symbol_names = 'k Bx By pv'
    symbol_values = '${k} ${B_x} ${B_y} ${pi_val}'
  []
  [T_inf_fn]
    type = ParsedFunction
    expression = 'pow(pow(TR + By * sin(pv * y), 4) + fs * (Bxe + By * sin(pv * y)), 0.25)'
    symbol_names = 'TR By fs Bxe pv'
    symbol_values = '${T_R_mms} ${B_y} ${flux_scale} ${B_x_e} ${pi_val}'
  []
  [bottom_top_T]
    type = ParsedFunction
    expression = 'T_L + Bx * (exp(x) - 1.0)'
    symbol_names = 'T_L Bx'
    symbol_values = '${T_L} ${B_x}'
  []
[]

[LinearFVKernels]
  [time]
    type = LinearFVTimeDerivative
    variable = T
  []
  [diffusion]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
  []
  [source]
    type = LinearFVSource
    variable = T
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${T_L}
  []
  [right]
    type = LinearFVFunctorRadiativeBC
    variable = T
    boundary = 'right'
    emissivity = ${eps}
    Tinfinity = T_inf_fn
    diffusion_coeff = ${k}
  []
  [bottom]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'bottom'
    functor = bottom_top_T
  []
  [top]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = 'T'
    boundary = 'top'
    functor = bottom_top_T
  []
[]

[Postprocessors]
  [l2_error]
    type = ElementL2FunctorError
    approximate = T
    exact = exact_T
    execute_on = TIMESTEP_END
  []
[]

[Executioner]
  type = Transient
  system_names = heat_system
  scheme = 'implicit-euler'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  dt = 1e6
  num_steps = 100
  l_tol = 1e-12
[]

[Outputs]
  [out]
    type = CSV
    execute_on = final
  []
[]

(test/tests/variables/linearfv/diffusion-1d-pp.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 50
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '1-x*x'
  []
[]

[Postprocessors]
  [average]
    type = ElementAverageValue
    variable = u
    execute_on = TIMESTEP_END
    outputs = csv
  []
  [min]
    type = ElementExtremeValue
    variable = u
    value_type = min
    execute_on = TIMESTEP_END
    outputs = csv
  []
  [max]
    type = ElementExtremeValue
    variable = u
    value_type = max
    execute_on = TIMESTEP_END
    outputs = csv
  []
  [num_dofs]
    type = NumDOFs
    execute_on = TIMESTEP_END
    outputs = csv
  []
  [elem_value]
    type = ElementalVariableValue
    variable = u
    elementid = 10
    execute_on = TIMESTEP_END
    outputs = csv
  []
  [point_value]
    type = PointValue
    variable = u
    point = '0.33333 0 0'
    execute_on = TIMESTEP_END
    outputs = csv
  []
[]

[VectorPostprocessors]
  [line-sample]
    type = LineValueSampler
    variable = u
    start_point = '0.13333 0 0'
    end_point = '0.766666 0 0'
    num_points = 9
    sort_by = x
    execute_on = TIMESTEP_END
    outputs = vpp_csv
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
  [vpp_csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/segregated-comparison/segregated-linear.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.3'
    dy = '0.3'
    ix = '3'
    iy = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 2
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Outputs]
  exodus = true
  execute_on = TIMESTEP_END
[]

(test/tests/linearfvbcs/robin/advection-2d-robin.i)

##################################################################
c     = 0.01 # advection velocity (+x direction)
amp   = 1.0 # sinusoid amplitude, for u_exact
u0    = 1.2 # any positive constant > 1.0

x_l   = ${fparse 0.0*pi}
x_r   = ${fparse pi}
y_l   = ${fparse 0.0*pi}
y_r   = ${fparse 1.0*pi}

alpha = 5.000 # robin BC coeff for gradient term
beta  = 2.000 # robin BC coeff for variable term

nx = 2
ny = 2
##################################################################

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 2
    nx   = ${nx}
    ny   = ${ny}
    xmin = ${x_l}
    xmax = ${x_r}
    ymin = ${y_l}
    ymax = ${y_r}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.01
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = '${amp} * (${u0} - cos(x)) * sin(y)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '${fparse c*amp} * sin(x) * sin(y)'
  []
  [gamma_fn]
    type = ParsedFunction
    expression = '(${fparse -amp*alpha}*sin(x)*sin(y)) + (${beta} * u_e)'
    symbol_names = 'u_e'
    symbol_values = 'u_exact'
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "${c} 0 0"
    advected_interp_method_name = average
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [rob_l]
    type = LinearFVAdvectionDiffusionFunctorRobinBC
    variable = u
    boundary = "left"
    alpha = ${alpha}
    beta  = ${beta}
    gamma = gamma_fn
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = true
  []
  [top]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "top"
    functor = 0.0
  []
  [bottom]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "bottom"
    functor = 0.0
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 4
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       NONZERO               1e-10'
  linear_convergence = linear
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion-variable-functor-bc.i)

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 1
    dx = '0.5 0.5'
    ix = '5 5'
    subdomain_id = '1 2'
  []
  [create_interface]
    type = SideSetsBetweenSubdomainsGenerator
    input = cmg
    new_boundary = interface
    primary_block = 2
    paired_block = 1
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
    block = 2
  []
[]

[AuxVariables]
  [v]
    type = MooseLinearVariableFVReal
    initial_condition = 1.0
    block = 1
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 1
    use_nonorthogonal_correction = false
    block = 2
  []
[]

[LinearFVBCs]
  [interface]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = interface
    functor = v
  []
  [right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 0
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[VectorPostprocessors]
  [u_sample]
    type = LineValueSampler
    variable = u
    start_point = '0.55 0 0'
    end_point = '0.95 0 0'
    num_points = 5
    sort_by = x
  []
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated-transient/channel_ERCOFTAC.i)

H = 1 #halfwidth of the channel
L = 100

Re = 13700

rho = 1
bulk_u = 1
mu = '${fparse rho * bulk_u * 2 * H / Re}'

advected_interp_method = 'upwind'

### k-epsilon Closure Parameters ###
sigma_k =1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Initial and Boundary Conditions ###
intensity = 0.01
k_init = '${fparse 1.5*(intensity * bulk_u)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / (2*H)}'

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'top bottom'
wall_treatment = 'eq_newton'  # Options: eq_newton, eq_incremental, eq_linearized, neq

[Mesh]
  [block_1]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = 0
    ymax = ${H}
    nx = 4
    ny = 3
    bias_y = 0.7
  []
  [block_2]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = ${fparse -H}
    ymax = 0
    nx = 4
    ny = 3
    bias_y = ${fparse 1/0.7}
  []
  [smg]
    type = StitchedMeshGenerator
    inputs = 'block_1 block_2'
    clear_stitched_boundary_ids = true
    stitch_boundaries_pairs = 'bottom top'
    merge_boundaries_with_same_name = true
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_time]
    type = LinearFVTimeDerivative
    variable = vel_x
    factor = ${rho}
  []
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_time]
    type = LinearFVTimeDerivative
    variable = vel_y
    factor = ${rho}
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_time]
    type = LinearFVTimeDerivative
    variable = TKE
    factor = ${rho}
  []
  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_time]
    type = LinearFVTimeDerivative
    variable = TKED
    factor = ${rho}
  []
  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []

  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = 'bottom top'
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = PIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.25 0.25'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
  dt = 1.0
  num_steps = 2
[]

[Outputs]
  csv = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_center_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.0001} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.0001} 0'
    num_points = ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_radius_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.5 * H} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.5 * H} 0'
    num_points =  ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/3d/3d-velocity-pressure.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 3
    dx = '0.3'
    dy = '0.3'
    dz = '0.3'
    ix = '3'
    iy = '3'
    iz = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system w_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    w = vel_z
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [vel_z]
    type = MooseLinearVariableFVReal
    solver_sys = w_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [w_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_z
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    w = vel_z
    momentum_component = 'z'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [w_pressure]
    type = LinearFVMomentumPressure
    variable = vel_z
    pressure = pressure
    momentum_component = 'z'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [inlet-w]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_z
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_y
    functor = 0.0
  []
  [walls-w]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom back front'
    variable = vel_z
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_w]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_z
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system w_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type -pc_hypre_boomeramg_agg_nl -pc_hypre_boomeramg_agg_num_paths -pc_hypre_boomeramg_truncfactor -pc_hypre_boomeramg_strong_threshold -pc_hypre_boomeramg_coarsen_type -pc_hypre_boomeramg_interp_type'
  momentum_petsc_options_value = 'hypre boomeramg 4 1 0.1 0.6 HMIS ext+i'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type -pc_hypre_boomeramg_agg_nl -pc_hypre_boomeramg_agg_num_paths -pc_hypre_boomeramg_truncfactor -pc_hypre_boomeramg_strong_threshold -pc_hypre_boomeramg_coarsen_type -pc_hypre_boomeramg_interp_type'
  pressure_petsc_options_value = 'hypre boomeramg 2 1 0.1 0.6 HMIS ext+i'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(test/tests/multisystem/restore_multiapp/sub.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 6
  []
[]

[Problem]
  nl_sys_names = 'v_sys'
  linear_sys_names = 'u_sys'
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    initial_condition = 2.0
    solver_sys = v_sys
  []
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVKernels]
  [diffusion]
    type = FVDiffusion
    variable = v
    coeff = u
  []
  [source]
    type = FVBodyForce
    variable = v
    function = 3
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = v
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = 1
  []
[]

[FVBCs]
  [dir]
    type = FVFunctorDirichletBC
    variable = v
    boundary = "left right"
    functor = 2
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = 1
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 6
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  #type = Transient
  #steady_state_detection = true
  system_names = 'v_sys u_sys'
  l_abs_tol = 1e-12
  l_tol = 1e-10
  nl_abs_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
  #execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-symmetric/channel.i)

mu = 2.6
rho = 1.2
advected_interp_method = 'upwind'
u_inlet = 1.1
half_width = 0.2

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.0'
    dy = '${half_width}'
    ix = '30'
    iy = '15'
    subdomain_id = '0'
  []
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  inactive = v_viscous_forcing
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [v_viscous_forcing]
    type = LinearFVRZViscousSource
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = ${u_inlet}
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top'
    variable = vel_y
    functor = 0.0
  []
  [symmetry-u]
    type = LinearFVVelocitySymmetryBC
    variable = vel_x
    momentum_component = x
    u = vel_x
    v = vel_y
    boundary = 'bottom'
  []
  [symmetry-v]
    type = LinearFVVelocitySymmetryBC
    variable = vel_y
    momentum_component = y
    u = vel_x
    v = vel_y
    boundary = 'bottom'
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = 'right'
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = 'right'
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'top left'
    variable = pressure
    use_two_term_expansion = true
  []
  [symmetry-p]
    type = LinearFVPressureSymmetryBC
    variable = pressure
    boundary = 'bottom'
    HbyA_flux = 'HbyA' # Functor created in the RhieChowMassFlux UO
  []
[]

[Functions]
  [u_parabolic_profile]
    type = ParsedFunction
    expression = '3/2*${u_inlet}*(1 - pow(y/${half_width}, 2))' # Poiseuille profile
  []
  [u_parabolic_profile_rz]
    type = ParsedFunction
    expression = '2*${u_inlet}*(1 - pow(y/${half_width}, 2))' # Cylindrical profile
  []
[]

[AuxVariables]
  [vel_exact]
    type = MooseLinearVariableFVReal
  []
[]

[AuxKernels]
  [assign_vel_exact]
    type = FunctionAux
    variable = vel_exact
    function = u_parabolic_profile
    execute_on = TIMESTEP_END
  []
[]

[VectorPostprocessors]
  [outlet_velocity_profile]
    type = SideValueSampler
    variable = 'vel_x vel_exact'
    boundary = 'right'
    sort_by = 'y'
    execute_on = TIMESTEP_END
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-12
  pressure_l_abs_tol = 1e-12
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.5
  pressure_variable_relaxation = 0.3
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/linear-segregated/channel_scalars-physics.i)


!include header_and_mesh.i

qs = 5

[Problem]
  linear_sys_names = 'u_system v_system pressure_system scalar_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[Physics]
  [NavierStokes]
    [FlowSegregated/flow]
      velocity_variable = 'vel_x vel_y'
      pressure_variable = 'pressure'

      # Initial conditions
      initial_velocity = '${bulk_u} 0 0'
      initial_pressure = '1e-8'

      # Material properties
      density = ${rho}
      dynamic_viscosity = ${mu}

      # Boundary conditions
      inlet_boundaries = 'left'
      momentum_inlet_types = 'fixed-velocity'
      momentum_inlet_functors = '${bulk_u} 0'

      wall_boundaries = 'top bottom'
      momentum_wall_types = 'noslip noslip'

      outlet_boundaries = 'right'
      momentum_outlet_types = 'fixed-pressure'
      pressure_functors = '0'

      # Numerical parameters
      include_deviatoric_stress = true
      orthogonality_correction = false
      pressure_two_term_bc_expansion = false
      momentum_two_term_bc_expansion = false
      momentum_advection_interpolation = ${advected_interp_method}
      system_names = 'u_system v_system pressure_system'
    []
    [ScalarTransportSegregated/scalar]
      # turbulence Physics coupling picked up automatically
      passive_scalar_names = 'advected_scalar'

      passive_scalar_inlet_types = 'fixed-value'
      passive_scalar_inlet_functors = '0'

      passive_scalar_source = '${qs}'
      system_names = 'scalar_system'
    []
    [TurbulenceSegregated/k-epsilon]
      # Model
      turbulence_handling = 'k-epsilon'
      tke_name = TKE
      tked_name = TKED
      system_names = 'TKE_system TKED_system'
      scalar_transport_physics = scalar

      initial_tke = ${k_init}
      initial_tked = ${eps_init}

      # Model parameters
      mu_t_ratio_max = 1e20
      sigma_k = ${sigma_k}
      sigma_eps = ${sigma_eps}
      C_pl = 1e10
      C1_eps = ${C1_eps}
      C2_eps = ${C2_eps}
      Sc_t = 0.9

      turbulence_walls = ${walls}
      wall_treatment_eps = ${wall_treatment}
      bulk_wall_treatment = ${bulk_wall_treatment}
      use_nonorthogonal_correction = false
    []
  []
[]

# Boundary conditions are not implemented for turbulent quantities at this time
[LinearFVBCs]
  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  passive_scalar_systems = 'scalar_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  passive_scalar_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  passive_scalar_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  passive_scalar_equation_relaxation = 0.8
  turbulence_equation_relaxation = '0.2 0.2'
  turbulence_field_relaxation = '0.2 0.2'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  passive_scalar_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  passive_scalar_petsc_options_iname = '-pc_type -pc_hypre_type'
  passive_scalar_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
[]

variables_to_sample = 'vel_x vel_y pressure advected_scalar TKE TKED'
!include postprocessing.i

# More postprocessor for scalar conservation analysis
[Postprocessors]
  [outlet_scalar]
    type = SideAverageValue
    boundary = right
    variable = advected_scalar
  []
  [S_out]
    type = VolumetricFlowRate
    boundary = 'right'
    vel_x = 'vel_x'
    vel_y = 'vel_y'
    advected_quantity = 'advected_scalar'
    rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  []
  [balance_percent]
    type = ParsedPostprocessor
    expression = '100 * (0 - S_out + 2 * ${L} * ${H} * ${qs}) / S_out'
    pp_names = 'S_out'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h.i)

L = 30
nx = 600
bulk_u = 0.01
q_source = 50000.
A_cp = 976.78
B_cp = 1.0634
T_in = 860.
p_ref = 101325.0
rho = 2000.
advected_interp_method = 'upwind'

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = ${L}
    nx = ${nx}
  []
  allow_renumbering = false
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
  u = vel_x
[]

[Problem]
  linear_sys_names = 'u_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    pressure = pressure
    rho = 'rho'
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    solver_sys = u_system
    initial_condition = ${bulk_u}
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = ${p_ref}
  []
  [h]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = ${fparse 860.*1900.}
  []
[]

[AuxVariables]
  [rho_var]
    type = MooseLinearVariableFVReal
  []
  [cp_var]
    type = MooseLinearVariableFVReal
  []
  [mu_var]
    type = MooseLinearVariableFVReal
  []
  [k_var]
    type = MooseLinearVariableFVReal
  []
  [T]
    type = MooseLinearVariableFVReal
    initial_condition = 860.
  []
  [h_aux]
    type = MooseLinearVariableFVReal
  []
[]

[LinearFVKernels]

  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    mu = 'mu'
    momentum_component = 'x'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [temp_advection]
    type = LinearFVEnergyAdvection
    variable = h
  []
  [source]
    type = LinearFVSource
    variable = h
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inlet_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = ${bulk_u} #${bulk_u} #'fully_developed_velocity'
  []
  [inlet_h]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = h
    boundary = 'left'
    functor = 'h_from_p_T'
  []
  [inlet_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${T_in}
  []

  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = ${p_ref}
  []
  [outlet_h]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = h
    use_two_term_expansion = false
    boundary = 'right'
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = 'right'
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = ${q_source}
  []
  [T_analytical]
    type = ParsedFunction
    expression = ${fparse (-A_cp+sqrt(A_cp^2-2*B_cp*(-q_source/rho/bulk_u*L-A_cp*T_in-B_cp/2*T_in*T_in)))/B_cp}
  []
[]

[FunctorMaterials]
  [enthalpy_material]
    type = LinearFVEnthalpyFunctorMaterial
    pressure = ${p_ref}
    T_fluid = T
    h = h
    h_from_p_T_functor = h_from_p_T_functor
    T_from_p_h_functor = T_from_p_h_functor
  []
  [h_from_p_T_functor]
    type = ParsedFunctorMaterial
    property_name = 'h_from_p_T_functor'
    functor_names = 'T'
    expression = '${A_cp}*T+${B_cp}/2*(T^2)'
  []
  [T_from_p_h_functor]
    type = ParsedFunctorMaterial
    property_name = 'T_from_p_h_functor'
    functor_names = 'h'
    expression = '(-${A_cp}+sqrt(${A_cp}^2+2*h*${B_cp}))/${B_cp}'
  []
  [rho]
    type = ADParsedFunctorMaterial
    property_name = 'rho'
    functor_names = 'T'
    expression = ${rho}
  []
  [cp]
    type = ADParsedFunctorMaterial
    property_name = 'cp'
    functor_names = 'T'
    expression = '${A_cp}+${B_cp}*T'
  []
  [mu]
    type = ADParsedFunctorMaterial
    property_name = 'mu'
    functor_names = 'T'
    expression = '4.5e-3'
  []
  [k]
    type = ADParsedFunctorMaterial
    property_name = 'k'
    functor_names = 'T'
    expression = 0.7
  []
[]

[AuxKernels]
  [rho_out]
    type = FunctorAux
    functor = 'rho'
    variable = 'rho_var'
    execute_on = 'NONLINEAR'
  []
  [cp_out]
    type = FunctorAux
    functor = 'cp'
    variable = 'cp_var'
    execute_on = 'NONLINEAR'
  []
  [mu_out]
    type = FunctorAux
    functor = 'mu'
    variable = 'mu_var'
    execute_on = 'NONLINEAR'
  []
  [k_out]
    type = FunctorAux
    functor = 'k'
    variable = 'k_var'
    execute_on = 'NONLINEAR'
  []
  [T_from_h_functor_aux]
    type = FunctorAux
    functor = 'T_from_p_h'
    variable = 'T'
    execute_on = 'NONLINEAR'
  []
  [h_from_T_functor_aux]
    type = FunctorAux
    functor = 'h_from_p_T'
    variable = 'h_aux'
    execute_on = 'NONLINEAR'
  []
[]

[Postprocessors]
  [T_out_sim]
    type = ElementalVariableValue
    variable = T
    elementid = ${fparse nx-1}
  []
  [T_out_analytic]
    type = FunctionValuePostprocessor
    function = T_analytical
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-12
  pressure_l_abs_tol = 1e-12
  energy_l_abs_tol = 1e-12
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.95
  num_iterations = 100
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  energy_absolute_tolerance = 1e-6
  print_fields = false
  momentum_l_max_its = 200

  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'

  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  continue_on_max_its = true
[]

[Outputs]
  [out]
    type = CSV
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated-transient/BFS_ERCOFTAC.i)

Re = 5100

rho = 1.0
bulk_u = 1.0
H = 1.0
mu = '${fparse rho * bulk_u * H / Re}'

advected_interp_method = 'upwind'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Initial and Boundary Conditions ###
intensity = 0.01
k_init = '${fparse 1.5*(intensity * bulk_u)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / H}'

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'bottom wall-side top'
wall_treatment = 'neq' # Options: eq_newton, eq_incremental, eq_linearized, neq

[Mesh]
  [gen]
    type = CartesianMeshGenerator
    dim = 2
    dx = '${fparse 10.0*H} ${fparse 20.0*H}'
    dy = '${H} ${fparse 5*H}'
    ix = '8 16'
    iy = '2 8'
    subdomain_id = '
                    2 1
                    1 1
                  '
  []
  [corner_walls]
    type = SideSetsBetweenSubdomainsGenerator
    input = gen
    primary_block ='1'
    paired_block ='2'
    new_boundary = 'wall-side'
  []
  [delete_bottom]
    type = BlockDeletionGenerator
    input = corner_walls
    block ='2'
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[LinearFVKernels]
  [u_time]
    type = LinearFVTimeDerivative
    variable = vel_x
    factor = ${rho}
  []
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []

  [v_time]
    type = LinearFVTimeDerivative
    variable = vel_y
    factor = ${rho}
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_time]
    type = LinearFVTimeDerivative
    variable = TKE
    factor = ${rho}
  []
  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_time]
    type = LinearFVTimeDerivative
    variable = TKED
    factor = ${rho}
  []
  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKE
    functor = '${k_init}'
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = TKED
    functor = '${eps_init}'
  []

  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKE
    use_two_term_expansion = false
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'right'
    variable = TKED
    use_two_term_expansion = false
  []

  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = ${walls}
    variable = vel_y
    functor = 0.0
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = ${walls}
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
  []
  [yplus]
    type = MooseLinearVariableFVReal
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init + mu}'
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = PIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.5 0.5'
  num_iterations = 20
  pressure_absolute_tolerance = 1e-6
  momentum_absolute_tolerance = 1e-6
  turbulence_absolute_tolerance = '1e-6 1e-6'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
  dt = 10.0
  num_steps = 10
  num_piso_iterations = 2
[]

[Outputs]
  csv = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_entry_channel_wall]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 1.00001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 1.00001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_entry_channel]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 2.25001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 2.25001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(test/tests/linearfvkernels/diffusion/diffusion-2d-rz.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
  coord_type = RZ
  rz_coord_axis = Y
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = true
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right top bottom"
    functor = analytic_solution
  []
  [symmetry]
    type = LinearFVAdvectionDiffusionScalarSymmetryBC
    variable = u
    boundary = "left"
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '-(-1.0*x^2*y*(1.5 - x^2) + x*(1.5 - x^2)*(-1.0*x*y - 2))/x - (-1.0*x^2*y*(1.5 - y^2) - 4*x*(1.5 - y^2)*(0.5*x*y + 1))/x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
  [exo]
    type = Exodus
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/advection-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny= 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left top bottom"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*pi*sin(2*y*pi)*cos(x*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/SIMPLEC/2d/2d-velocity-pressure.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.0'
    dy = '0.1'
    ix = '20'
    iy = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    pressure_projection_method = consistent
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 1.0
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(modules/navier_stokes/test/tests/finite_volume/ins/lid-driven/linear-segregated/lid-driven-segregated.i)

mu = .01
rho = 1
advected_interp_method = 'average'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = .1
    ymin = 0
    ymax = .1
    nx = 3
    ny = 3
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = v_system
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [top_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'top'
    functor = 1
  []
  [no_slip_x]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_x
    boundary = 'left right bottom'
    functor = 0
  []
  [no_slip_y]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = vel_y
    boundary = 'left right top bottom'
    functor = 0
  []
  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'left right top bottom'
    variable = pressure
    use_two_term_expansion = true
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 500
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.05 0.05 0.0'
[]

[Outputs]
  exodus = true
  csv = false
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

(test/tests/transfers/multiapp_copy_transfer/linear_sys_to_aux/linear_sub.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '1-x*x'
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
  execute_on = TIMESTEP_END
[]

(test/tests/linearfvkernels/block-restriction/block-restricted-adr.i)

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.1 1 0.1'
    dy = '0.1 0.5 0.1'
    ix = '1 2 1'
    iy = '1 1 1'
    subdomain_id = '1 1 1 1 2 3 1 1 1'
  []
  [transform]
    type = TransformGenerator
    input = cmg
    transform = TRANSLATE
    vector_value = '-0.1 -0.1 0.0'
  []
  [create_sides]
    type = SideSetsBetweenSubdomainsGenerator
    input = transform
    new_boundary = sides
    primary_block = 2
    paired_block = 1
  []
  [create_outlet]
    type = SideSetsBetweenSubdomainsGenerator
    input = create_sides
    new_boundary = outlet
    primary_block = 2
    paired_block = 3
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
    block = 2
  []
[]

[FVInterpolationMethods]
  [average]
    type = FVGeometricAverage
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diff_coeff_func
    use_nonorthogonal_correction = false
  []
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = average
  []
  [reaction]
    type = LinearFVReaction
    variable = u
    coeff = coeff_func
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  inactive = "outflow"
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "sides outlet"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = true
  []
[]

[Functions]
  [diff_coeff_func]
    type = ParsedFunction
    expression = '1.0+0.5*x*y'
  []
  [coeff_func]
    type = ParsedFunction
    expression = '1.0+1.0/(1+x*y)'
  []
  [source_func]
    type = ParsedFunction
    expression = '-1.0*x*pi*sin((1/2)*x*pi)*cos(2*y*pi) - 0.25*y*pi*sin(2*y*pi)*cos((1/2)*x*pi) + (1.0 + 1.0/(x*y + 1))*(sin((1/2)*x*pi)*sin(2*y*pi) + 1.5) + (17/4)*pi^2*(0.5*x*y + 1.0)*sin((1/2)*x*pi)*sin(2*y*pi) + 0.25*pi*sin(2*y*pi)*cos((1/2)*x*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin((1/2)*x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
    block = 2
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
    block = 2
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/controls/conditional_functional_enable/linear-fv-bc-control-switch.i)

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 20
    xmin = 0
    xmax = 1
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0
  []
  # "Off" right BC: u=0, initially enabled; disabled by Control
  [right_off]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 0
    enable = true
  []
  # "On" right BC: u=1, initially disabled; enabled by Control
  [right_on]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 1
    enable = false
  []
[]

[Functions]
  [always_on]
    type = ParsedFunction
    expression = '1'
  []
[]

[Controls]
  [switch_right_bc]
    type = ConditionalFunctionEnableControl
    conditional_function = always_on
    enable_objects = 'LinearFVBCs::right_on'
    disable_objects = 'LinearFVBCs::right_off'
  []
[]

[Postprocessors]
  [side_avg_right]
    type = SideAverageValue
    variable = u
    boundary = right
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  [out]
    type = CSV
    execute_on = 'TIMESTEP_END'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-heated/fluid.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'upwind'
cp = 1000
k = 1

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.25 0.25'
    dy = '0.2'
    ix = '5 5'
    iy = '5'
    subdomain_id = '0 1'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
  [T_fluid]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T_fluid
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T_fluid
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []
  [heat_exchange]
    type = LinearFVVolumetricHeatTransfer
    variable = T_fluid
    h_solid_fluid = 100
    T_fluid = T_fluid
    T_solid = T_solid
    is_solid = false
    block = 1
  []
[]

[FunctorMaterials]
  [constant_functors]
    type = GenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_wall_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T_fluid
    functor = 300
    boundary = 'left top bottom'
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T_fluid
    use_two_term_expansion = false
    boundary = right
  []
[]

[AuxVariables]
  [T_solid]
    type = MooseLinearVariableFVReal
    initial_condition = 300
    block = 1
  []
[]

[MultiApps]
  inactive = 'solid'
  [solid]
    type = FullSolveMultiApp
    input_files = solid.i
    execute_on = timestep_begin
    no_restore = true
  []
[]

[Transfers]
  inactive = 'from_solid to_solid'
  [from_solid]
    type = MultiAppGeneralFieldShapeEvaluationTransfer
    from_multi_app = solid
    source_variable = 'T_solid'
    variable = 'T_solid'
    execute_on = timestep_begin
    from_blocks = 1
  []
  [to_solid]
    type = MultiAppGeneralFieldShapeEvaluationTransfer
    to_multi_app = solid
    source_variable = 'T_fluid'
    variable = 'T_fluid'
    execute_on = timestep_begin
    to_blocks = 1
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  energy_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.8
  energy_equation_relaxation = 0.9
  pressure_variable_relaxation = 0.3
  num_iterations = 20
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  energy_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
[]

(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_LinearFV.i)

mu = 2.6
rho = 1.1
advected_interp_method='average'

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.3'
    dy = '0.3'
    ix = '3'
    iy = '3'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 100
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Postprocessors]
  [inlet_mass]
    type = VolumetricFlowRate
    boundary = left
    rhie_chow_user_object = rc
    advected_quantity = ${rho}
    vel_x = vel_x
    vel_y = vel_y
    subtract_mesh_velocity = false
  []
  [outlet_mass]
    type = VolumetricFlowRate
    boundary = right
    rhie_chow_user_object = rc
    advected_quantity = ${rho}
    vel_x = vel_x
    vel_y = vel_y
    subtract_mesh_velocity = false
  []
[]

[Outputs]
  csv = true
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/block-restricted/block-ke.i)

H = 1 #halfwidth of the channel
L = 100

Re = 13700

rho = 1
bulk_u = 1
mu = '${fparse rho * bulk_u * 2 * H / Re}'

advected_interp_method = 'upwind'

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Initial and Boundary Conditions ###
intensity = '${fparse 0.16*Re^(-1./8.)}'
k_init = '${fparse 1.5*(intensity * bulk_u)^2}'
eps_init = '${fparse C_mu^0.75 * k_init^1.5 / (2*H)}'

### Modeling parameters ###
bulk_wall_treatment = false
walls = 'walls'
wall_treatment = 'eq_newton' # Options: eq_newton, eq_incremental, eq_linearized, neq

[Mesh]
  [block_left]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = '${fparse -2*H}'
    xmax = '${fparse -H}'
    ymin = 0
    ymax = ${L}
    nx = 3
    ny = 4
  []
  [block_1]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = '${fparse -H}'
    xmax = 0
    ymin = 0
    ymax = ${L}
    nx = 4
    ny = 4
    bias_x = '${fparse 1/0.7}'
  []
  [block_2]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${H}
    ymin = 0
    ymax = ${L}
    nx = 4
    ny = 4
    bias_x = 0.7
  []
  [block_right]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = ${H}
    xmax = '${fparse 2*H}'
    ymin = 0
    ymax = ${L}
    nx = 3
    ny = 4
  []
  [smg]
    type = StitchedMeshGenerator
    inputs = 'block_left block_1 block_2 block_right'
    stitch_boundaries_pairs = 'right left; right left; right left'
  []
  [middle]
    input = smg
    type = SubdomainBoundingBoxGenerator
    block_id = 1
    bottom_left = '${fparse -H} 0 0'
    top_right = '${H} ${L} 50'
  []
  [walls]
    type = SideSetsBetweenSubdomainsGenerator
    primary_block = 1
    paired_block = 0
    new_boundary = walls
    input = middle
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system TKE_system TKED_system'
  previous_nl_solution_required = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
  advected_interp_method = ${advected_interp_method}
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    block = 1
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
    block = 1
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    initial_condition = ${bulk_u}
    solver_sys = v_system
    block = 1
  []
  [pressure]
    type = MooseLinearVariableFVReal
    initial_condition = 1e-8
    solver_sys = pressure_system
    block = 1
  []
  [TKE]
    type = MooseLinearVariableFVReal
    solver_sys = TKE_system
    initial_condition = ${k_init}
    block = 1
  []
  [TKED]
    type = MooseLinearVariableFVReal
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    block = 1
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [u_diffusion]
    type = LinearFVDiffusion
    variable = vel_x
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = 'mu_t'
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
    use_deviatoric_terms = yes
  []
  [v_diffusion]
    type = LinearFVDiffusion
    variable = vel_y
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [TKE_advection]
    type = LinearFVTurbulentAdvection
    variable = TKE
  []
  [TKE_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
  []
  [TKE_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKE
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_k}
    use_nonorthogonal_correction = false
  []
  [TKE_source_sink]
    type = LinearFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []

  [TKED_advection]
    type = LinearFVTurbulentAdvection
    variable = TKED
    walls = ${walls}
  []
  [TKED_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = ${mu}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_turb_diffusion]
    type = LinearFVTurbulentDiffusion
    variable = TKED
    diffusion_coeff = 'mu_t'
    scaling_coeff = ${sigma_eps}
    use_nonorthogonal_correction = false
    walls = ${walls}
  []
  [TKED_source_sink]
    type = LinearFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = vel_x
    functor = 0
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = vel_y
    functor = '${bulk_u}'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'walls'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'walls'
    variable = vel_y
    functor = 0.0
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'top'
    variable = vel_x
    use_two_term_expansion = false
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'top'
    variable = vel_y
    use_two_term_expansion = false
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top'
    variable = pressure
    functor = 0.0
  []

  [inlet_TKE]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = TKE
    functor = '${k_init}'
  []
  [outlet_TKE]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'top'
    variable = TKE
    use_two_term_expansion = false
  []
  [inlet_TKED]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = TKED
    functor = '${eps_init}'
  []
  [outlet_TKED]
    type = LinearFVAdvectionDiffusionOutflowBC
    boundary = 'top'
    variable = TKED
    use_two_term_expansion = false
  []
  [walls_mu_t]
    type = LinearFVTurbulentViscosityWallFunctionBC
    boundary = 'walls'
    variable = 'mu_t'
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    tke = TKE
    wall_treatment = ${wall_treatment}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
    block = 1
  []
  [yplus]
    type = MooseLinearVariableFVReal
    block = 1
  []
  [mu_eff]
    type = MooseLinearVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
    block = 1
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = TKE
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_mu_eff]
    type = ParsedAux
    variable = 'mu_eff'
    coupled_variables = 'mu_t'
    expression = 'mu_t + ${mu}'
    execute_on = 'NONLINEAR'
  []
[]

[Executioner]
  type = SIMPLE

  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKE_system TKED_system'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_tol = 1e-14
  pressure_l_tol = 1e-14
  turbulence_l_tol = 1e-14

  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.2 0.2'
  turbulence_field_relaxation = '0.2 0.2'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  turbulence_petsc_options_iname = '-pc_type -pc_hypre_type'
  turbulence_petsc_options_value = 'hypre boomeramg'

  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
  csv = true
[]

[VectorPostprocessors]
  [line_wall]
    type = LineValueSampler
    start_point = '${fparse 0.99 * H} ${fparse 0.125 * L} 0'
    end_point = '${fparse 0.99 * H} ${fparse 0.875 * L} 0'
    num_points = ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_center_channel]
    type = LineValueSampler
    start_point = ' ${fparse 0.0001 * H} ${fparse 0.125 * L} 0'
    end_point = '${fparse 0.0001 * H} ${fparse 0.875 * L} 0'
    num_points = ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_radius_channel]
    type = LineValueSampler
    start_point = '${fparse 0.51 * H} ${fparse 0.125 * L} 0'
    end_point = '${fparse 0.51 * H} ${fparse 0.875 * L} 0'
    num_points =  ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex/2d-symmetric-vortex.i)

mu = 1.2
rho = 1.5
advected_interp_method = 'average'

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
  []
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    body_force_kernel_names = "u_forcing; v_forcing"
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_forcing]
    type = LinearFVSource
    variable = vel_x
    source_density = forcing_u
  []
  [v_forcing]
    type = LinearFVSource
    variable = vel_y
    source_density = forcing_v
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = 'Ainv' # Functor created in the RhieChowMassFlux UO
    use_nonorthogonal_correction = false
    use_nonorthogonal_correction_on_boundary = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = 'HbyA' # Functor created in the RhieChowMassFlux UO
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [no-slip-wall-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_x
    functor = exact_u
  []
  [no-slip-wall-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_y
    functor = exact_v
  []
  [symmetry-u]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_x
    u = vel_x
    v = vel_y
    momentum_component = x
  []
  [symmetry-v]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_y
    u = vel_x
    v = vel_y
    momentum_component = y
  []
  [pressure-extrapolation]
    type = LinearFVPressureFluxBC
    boundary = 'top left right'
    variable = pressure
    HbyA_flux = HbyA
    Ainv = Ainv
  []
  [pressure-symmetry]
    type = LinearFVPressureSymmetryBC
    boundary = 'bottom'
    variable = pressure
    HbyA_flux = 'HbyA' # Functor created in the RhieChowMassFlux UO
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = 'x^2*(1-x)^2*(8*y^3 - 9*y^2 + 1)'
  []
  [exact_v]
    type = ParsedFunction
    expression = '-(2*x - 6*x^2 + 4*x^3)*y*(1-y)^2*(2*y+1)'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'y^2'
  []
  [forcing_u]
    type = ParsedFunction
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
    expression = 'rho*( (x^2*(1-x)^2)*(2*x - 6*x^2 + 4*x^3)*
                        ( (8*y^3 - 9*y^2 + 1)^2
                          - (2*y^4 - 3*y^3 + y)*(24*y^2 - 18*y) ) )
                  - mu*( (2 - 12*x + 12*x^2)*(8*y^3 - 9*y^2 + 1)
                         + (x^2*(1-x)^2)*(48*y - 18) )'
  []
  [forcing_v]
    type = ParsedFunction
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
    expression = 'rho*( (8*y^3 - 9*y^2 + 1)*(2*y^4 - 3*y^3 + y)*
                        ( (2*x - 6*x^2 + 4*x^3)^2
                          - (x^2*(1-x)^2)*(2 - 12*x + 12*x^2) ) )
                  + mu*( (24*x - 12)*(2*y^4 - 3*y^3 + y)
                         + (2*x - 6*x^2 + 4*x^3)*(24*y^2 - 18*y) )
                  + 2*y'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-8
  pressure_l_abs_tol = 1e-8
  momentum_l_tol = 0
  pressure_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  num_iterations = 10000
  pressure_absolute_tolerance = 1e-7
  momentum_absolute_tolerance = 1e-7
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.25
  pressure_pin_point = '0.5 0.5 0.0'
[]

[Outputs]
  exodus = true
  [csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = TIMESTEP_END
  []
  [L2u]
    type = ElementL2FunctorError
    approximate = vel_x
    exact = exact_u
    execute_on = TIMESTEP_END
  []
  [L2v]
    type = ElementL2FunctorError
    approximate = vel_y
    exact = exact_v
    execute_on = TIMESTEP_END
  []
  [L2p]
    approximate = pressure
    exact = exact_p
    type = ElementL2FunctorError
    execute_on = TIMESTEP_END
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-scalar/channel.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'upwind'
k1 = 0.1
k2 = 0.2

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.25 0.25'
    dy = '0.2'
    ix = '5 5'
    iy = '5'
    subdomain_id = '0 1'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system s1_system s2_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
  [scalar1]
    type = MooseLinearVariableFVReal
    solver_sys = s1_system
    initial_condition = 1.1
  []
  [scalar2]
    type = MooseLinearVariableFVReal
    solver_sys = s2_system
    initial_condition = 3
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [s1_advection]
    type = LinearFVScalarAdvection
    variable = scalar1
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [s1_diffusion]
    type = LinearFVDiffusion
    variable = scalar1
    diffusion_coeff = ${k1}
    use_nonorthogonal_correction = false
  []
  [s2_diffusion]
    type = LinearFVScalarAdvection
    variable = scalar2
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [s2_conduction]
    type = LinearFVDiffusion
    variable = scalar2
    diffusion_coeff = ${k2}
    use_nonorthogonal_correction = false
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_wall_scalar1]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = scalar1
    functor = 1
    boundary = 'left'
  []
  [outlet_scalar1]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = scalar1
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_wall_scalar2]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = scalar2
    functor = 2
    boundary = 'left'
  []
  [outlet_scalar2]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = scalar2
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  passive_scalar_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  passive_scalar_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  passive_scalar_systems = 's1_system s2_system'
  momentum_equation_relaxation = 0.8
  passive_scalar_equation_relaxation = '0.9 0.9'
  pressure_variable_relaxation = 0.3
  # We need to converge the problem to show conservation
  num_iterations = 200
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  # The solution being flat, the normalization factor based on fluxes goes to
  # 0. The convergence criteria being multiplied by said factor, we won't do any
  # better than this. For a non-flat solution, use tighter tolerances
  passive_scalar_absolute_tolerance = '1e-2 1e-2'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  passive_scalar_petsc_options_iname = '-pc_type -pc_hypre_type'
  passive_scalar_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
  execute_on = timestep_end
  hide = 'pressure vel_x vel_y'
[]

(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/lid-driven-two-phase-physics.i)

mu = 1.0
rho = 1.0e3
mu_d = 0.3
rho_d = 5e2
dp = 0.01
U_lid = 0.1
advected_interp_method = 'upwind'

k = 1
k_d = 1
cp = 1
cp_d = 1

[Mesh]
  inactive = 'make_skew'
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = .1
    ymin = 0
    ymax = .1
    nx = 10
    ny = 10
  []
  [make_skew]
    type = MoveNodeGenerator
    input = 'gen'
    node_id = 0
    shift_position = '0.007 0.0021 0'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system phi_system'
[]

[Physics]
  [NavierStokes]
    [FlowSegregated]
      [flow]
        compressibility = 'incompressible'

        density = 'rho_mixture'
        dynamic_viscosity = 'mu_mixture'

        # Initial conditions
        initial_velocity = '1e-12 1e-12 0'
        initial_pressure = 0.2

        wall_boundaries = 'top left right bottom'
        momentum_wall_types = 'noslip noslip noslip noslip'
        momentum_wall_functors = '${U_lid} 0; 0 0; 0 0; 0 0'

        orthogonality_correction = false
        pressure_two_term_bc_expansion = true
        momentum_advection_interpolation = ${advected_interp_method}
      []
    []
    [TwoPhaseMixtureSegregated]
      [mixture]
        system_names = 'phi_system'
        phase_1_fraction_name = 'phase_1'
        phase_2_fraction_name = 'phase_2'

        add_phase_transport_equation = true
        phase_advection_interpolation = '${advected_interp_method}'
        phase_fraction_diffusivity = 1e-3

        # We could consider adding fixed-value-yet-not-an-inlet
        # boundary conditions to the TwoPhaseMixture physics

        # Base phase material properties
        phase_1_density_name = ${rho}
        phase_1_viscosity_name = ${mu}
        phase_1_specific_heat_name = ${cp}
        phase_1_thermal_conductivity_name = ${k}

        # Other phase material properties
        phase_2_density_name = ${rho_d}
        phase_2_viscosity_name = ${mu_d}
        phase_2_specific_heat_name = ${cp_d}
        phase_2_thermal_conductivity_name = ${k_d}
        output_all_properties = true

        # Friction model, not actually used!
        use_dispersed_phase_drag_model = true
        particle_diameter = ${dp}
        add_advection_slip_term = false
      []
    []
  []
[]

[LinearFVBCs]
  [botttom-phase-2]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'bottom'
    variable = phase_2
    functor = '0'
  []
  [top-phase-2]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top'
    variable = phase_2
    functor = '1'
  []
[]

[Executioner]
  type = PIMPLE
  rhie_chow_user_object = 'ins_rhie_chow_interpolator'
  dt = 1
  end_time = 10

  # Systems
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  active_scalar_systems = 'phi_system'
  momentum_equation_relaxation = 0.8
  active_scalar_equation_relaxation = '0.7'
  pressure_variable_relaxation = 0.3

  # We need to converge the problem to show conservation
  num_iterations = 200
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  active_scalar_absolute_tolerance = '1e-10'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  active_scalar_petsc_options_iname = '-pc_type -pc_factor_shift_type' # -pc_hypre_type'
  active_scalar_petsc_options_value = 'lu NONZERO'
  momentum_l_abs_tol = 1e-13
  pressure_l_abs_tol = 1e-13
  active_scalar_l_abs_tol = 1e-13
  momentum_l_tol = 0
  pressure_l_tol = 0
  active_scalar_l_tol = 0
  # print_fields = true
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.05 0.05 0.0'
[]

[Outputs]
  exodus = true
  [out]
    type = CSV
    execute_on = 'FINAL'
  []
[]

[AuxVariables]
  [drag_coefficient]
    type = MooseVariableFVReal
  []
[]

[AuxKernels]
  [populate_cd]
    type = FunctorAux
    variable = drag_coefficient
    functor = 'Darcy_coefficient'
    execute_on = 'TIMESTEP_END'
  []
[]

[Postprocessors]
  [average_void]
    type = ElementAverageValue
    variable = 'phase_2'
  []
  [max_y_velocity]
    type = ElementExtremeValue
    variable = 'vel_y'
    value_type = max
  []
  [min_y_velocity]
    type = ElementExtremeValue
    variable = 'vel_y'
    value_type = min
  []
  [max_x_velocity]
    type = ElementExtremeValue
    variable = 'vel_x'
    value_type = max
  []
  [min_x_velocity]
    type = ElementExtremeValue
    variable = 'vel_x'
    value_type = min
  []
  [max_x_slip_velocity]
    type = ElementExtremeFunctorValue
    functor = 'vel_slip_x'
    value_type = max
  []
  [max_y_slip_velocity]
    type = ElementExtremeFunctorValue
    functor = 'vel_slip_y'
    value_type = max
  []
  [max_drag_coefficient]
    type = ElementExtremeFunctorValue
    functor = 'drag_coefficient'
    value_type = max
  []
[]

(modules/heat_transfer/test/tests/linearfvbcs/linear_fv_functor_radiative_bc/linear_fv_functor_radiative_bc_mms.i)

# MMS convergence test for LinearFVFunctorRadiativeBC.
#
# Exact solution: T_exact(x) = T_L + B * (exp(x) - 1)
# where B = (T_R_mms - T_L) / (e - 1).
#
# This satisfies:
#   Left BC:  T(0) = T_L                          (Dirichlet)
#   Right BC: -k * T'(1) = sigma*eps*(T_R_mms^4 - T_inf^4)  (radiative)
#             with T'(1) = B*e = (T_R_mms - T_L)*e/(e-1)
#             i.e. k*(T_L - T_R_mms)*e/(e-1) = sigma*eps*(T_R_mms^4 - T_inf^4)
#   Source:   f(x) = -k * T_exact'' = -k * B * exp(x)  (spatially varying)
#
# The exponential profile produces non-trivial interior discretization error,
# ensuring the convergence test exercises both the FV interior scheme and the
# Robin boundary condition simultaneously.
#
# T_R_mms must be supplied via CLI args, as the root of:
#   k*(T_L - T_R)*e/(e-1) = sigma*eps*(T_R^4 - T_inf^4)
# (computed in the Python convergence script).

T_L    = 1000.0
T_inf  = 300.0
k      = 1.0
eps    = 1.0
T_R_mms = 0  # set via CLI: T_R_mms=<value>
B_mms = '${fparse (T_R_mms - T_L) / (exp(1.0) - 1.0)}'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 10  # overridden by CLI: Mesh/gen/nx=<N>
    xmin = 0
    xmax = 1
  []
[]

[Problem]
  linear_sys_names = 'heat_system'
[]

[Variables]
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = 'heat_system'
    initial_condition = ${T_L}
  []
[]

[Functions]
  [exact_T]
    type = ParsedFunction
    expression = 'T_L + B_mms * (exp(x) - 1.0)'
    symbol_names = 'T_L B_mms'
    symbol_values = '${T_L} ${B_mms}'
  []
  [source_fn]
    type = ParsedFunction
    expression = '-k * B_mms * exp(x)'
    symbol_names = 'k B_mms'
    symbol_values = '${k} ${B_mms}'
  []
[]

[LinearFVKernels]
  [time]
    type = LinearFVTimeDerivative
    variable = T
  []
  [diffusion]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
  []
  [source]
    type = LinearFVSource
    variable = T
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    boundary = 'left'
    functor = ${T_L}
  []
  [right]
    type = LinearFVFunctorRadiativeBC
    variable = T
    boundary = 'right'
    emissivity = ${eps}
    Tinfinity = ${T_inf}
    diffusion_coeff = ${k}
  []
[]

[Postprocessors]
  [l2_error]
    type = ElementL2FunctorError
    approximate = T
    exact = exact_T
    execute_on = TIMESTEP_END
  []
[]

[Executioner]
  type = Transient
  system_names = heat_system
  scheme = 'implicit-euler'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  dt = 1e6
  num_steps = 100
  l_tol = 1e-12
[]

[Outputs]
  [out]
    type = CSV
    execute_on = final
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/mms/linear-segregated/2d-symmetric-vortex-rz/2d-symmetric-vortex-rz.i)

mu = 1.2
rho = 1.5
advected_interp_method = 'average'

[Problem]
  linear_sys_names = 'u_system v_system pressure_system'
  previous_nl_solution_required = true
[]

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 2
  []
  coord_type = 'RZ'
  rz_coord_axis = x
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
    pressure_projection_method = consistent
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.0
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_forcing]
    type = LinearFVSource
    variable = vel_x
    source_density = forcing_u
  []
  [v_forcing]
    type = LinearFVSource
    variable = vel_y
    source_density = forcing_v
  []
  [v_viscous_forcing]
    type = LinearFVRZViscousSource
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = 'Ainv'
    use_nonorthogonal_correction = false
    use_nonorthogonal_correction_on_boundary = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = 'HbyA'
    force_boundary_execution = true
  []
[]

[LinearFVBCs]
  [no-slip-wall-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_x
    functor = exact_u
  []
  [no-slip-wall-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left right top'
    variable = vel_y
    functor = exact_v
  []
  [symmetry-u]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_x
    u = vel_x
    v = vel_y
    momentum_component = x
  []
  [symmetry-v]
    type = LinearFVVelocitySymmetryBC
    boundary = 'bottom'
    variable = vel_y
    u = vel_x
    v = vel_y
    momentum_component = y
  []

  [pressure-extrapolation]
    type = LinearFVExtrapolatedPressureBC
    boundary = 'top left right'
    variable = pressure
    use_two_term_expansion = true
  []
  [pressure-symmetry]
    type = LinearFVPressureSymmetryBC
    boundary = 'bottom'
    variable = pressure
    HbyA_flux = 'HbyA'
  []
[]

[AuxVariables]
  [vel_x_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
  [vel_y_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
  [pressure_aux]
    type = MooseLinearVariableFVReal
    initial_condition = 0.0
  []
[]

[AuxKernels]
  [vel_x_function_aux]
    type = FunctionAux
    variable = vel_x_aux
    function = exact_u
    execute_on = TIMESTEP_END
  []
  [vel_y_function_aux]
    type = FunctionAux
    variable = vel_y_aux
    function = exact_v
    execute_on = TIMESTEP_END
  []
  [pressure_function_aux]
    type = FunctionAux
    variable = pressure_aux
    function = exact_p
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_u]
    type = ParsedFunction
    expression = '4*x^2*y^2 - 20*x^2*y^3 - 8*x^3*y^2 + 36*x^2*y^4 + 40*x^3*y^3 + 4*x^4*y^2 - 28*x^2*y^5 - 72*x^3*y^4 - 20*x^4*y^3 + 8*x^2*y^6 + 56*x^3*y^5 + 36*x^4*y^4 - 16*x^3*y^6 - 28*x^4*y^5 + 8*x^4*y^6'
  []
  [exact_v]
    type = ParsedFunction
    expression = '-2*x*y^3*(x-1)*(2*x-1)*(y-1)^4'
  []
  [exact_p]
    type = ParsedFunction
    expression = 'x*y^2'
  []
  [forcing_u]
    type = ParsedFunction
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
    expression = 'rho*(8*x^3*y^4*(x-1)^3*(2*x-1)*(y-1)^6*(4*y^2 - 5*y + 2)) - mu*(4*(y-1)*(72*x^4*y^3 - 103*x^4*y^2 + 41*x^4*y - 4*x^4 - 144*x^3*y^3 + 206*x^3*y^2 - 82*x^3*y + 8*x^3 + 24*x^2*y^5 - 60*x^2*y^4 + 120*x^2*y^3 - 115*x^2*y^2 + 41*x^2*y - 4*x^2 - 24*x*y^5 + 60*x*y^4 - 48*x*y^3 + 12*x*y^2 + 4*y^5 - 10*y^4 + 8*y^3 - 2*y^2)) + y^2'
  []
  [forcing_v]
    type = ParsedFunction
    symbol_names = 'mu rho'
    symbol_values = '${mu} ${rho}'
    expression = 'rho*4*x^2*y^5*(x-1)^2*(y-1)^7*(2*(2*x^2 - 2*x + y) + (2*x - 1)^2*(y - 1)) + mu * (4*y*(2*x-1)*(y-1)^2*(24*x^2*y^2 - 22*x^2*y + 4*x^2 - 24*x*y^2 + 22*x*y - 4*x + 3*y^4 - 6*y^3 + 3*y^2)) + 2*x*y'
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  momentum_l_max_its = 500
  pressure_l_max_its = 500
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  momentum_equation_relaxation = 0.85
  pressure_variable_relaxation = 1.0
  num_iterations = 10000
  pressure_absolute_tolerance = 1e-8
  momentum_absolute_tolerance = 1e-8
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true

  pin_pressure = true
  pressure_pin_value = 0.125
  pressure_pin_point = '0.5 0.5 0.0'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = TIMESTEP_END
  []
  [L2u]
    type = ElementL2FunctorError
    approximate = vel_x
    exact = exact_u
    execute_on = TIMESTEP_END
  []
  [L2v]
    type = ElementL2FunctorError
    approximate = vel_y
    exact = exact_v
    execute_on = TIMESTEP_END
  []
  [L2p]
    approximate = pressure
    exact = exact_p
    type = ElementL2FunctorError
    execute_on = TIMESTEP_END
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq-transient.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'
cp = 300
k = 10
alpha_b = 1e-4

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.'
    dy = '0.2'
    ix = '10'
    iy = '5'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
  []
[]

[LinearFVKernels]
  [u_time]
    type = LinearFVTimeDerivative
    variable = vel_x
    factor = ${rho}
  []
  [v_time]
    type = LinearFVTimeDerivative
    variable = vel_y
    factor = ${rho}
  []
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_x
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'x'
  []
  [v_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_y
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_time]
    type = LinearFVTimeDerivative
    variable = T
    factor = ${fparse rho*cp}
  []
  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []
[]

[FunctorMaterials]
  [constant_functors]
    type = GenericFunctorMaterial
    prop_names = 'cp alpha_b'
    prop_values = '${cp} ${alpha_b}'
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_top_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = 300.0
    boundary = 'left top'
  []
  [bottom_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = wall-temperature
    boundary = bottom
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T
    use_two_term_expansion = false
    boundary = right
  []
[]

[Functions]
  [wall-temperature]
    type = ParsedFunction
    expression = '350 + 50 * sin(6.28*t)'
  []
[]

[Executioner]
  type = PIMPLE
  momentum_l_abs_tol = 1e-12
  pressure_l_abs_tol = 1e-12
  energy_l_abs_tol = 1e-12
  momentum_l_tol = 1e-12
  pressure_l_tol = 1e-12
  energy_l_tol = 1e-12
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.3
  energy_equation_relaxation = 0.9
  num_iterations = 100
  pressure_absolute_tolerance = 1e-11
  momentum_absolute_tolerance = 1e-11
  energy_absolute_tolerance = 1e-11
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  print_fields = false
  continue_on_max_its = true
  dt = 0.01
  num_steps = 6
  num_piso_iterations = 0
[]


[Outputs]
  exodus = true
[]

(test/tests/linearfvkernels/block-restriction/block-restricted-diffusion.i)

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 2
    dx = '0.1 1 0.1'
    dy = '0.1 0.5 0.1'
    ix = '1 2 1'
    iy = '1 1 1'
    subdomain_id = '1 1 1 1 2 3 1 1 1'
  []
  [transform]
    type = TransformGenerator
    input = cmg
    transform = TRANSLATE
    vector_value = '-0.1 -0.1 0.0'
  []
  [create_sides]
    type = SideSetsBetweenSubdomainsGenerator
    input = transform
    new_boundary = sides
    primary_block = 2
    paired_block = 1
  []
  [create_outlet]
    type = SideSetsBetweenSubdomainsGenerator
    input = create_sides
    new_boundary = outlet
    primary_block = 2
    paired_block = 3
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
    block = 2
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = diff_coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "sides outlet"
    functor = analytic_solution
  []
[]

[Functions]
  [diff_coeff_func]
    type = ParsedFunction
    expression = '1.0+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '-1.0*x*pi*sin(x*pi)*cos(2*y*pi) - 0.5*y*pi*sin(2*y*pi)*cos(x*pi) + 5*pi^2*(0.5*x*y + 1.0)*sin(x*pi)*sin(2*y*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
    block = 2
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
    block = 2
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'
cp = 300
k = 20
alpha_b = 1e-4

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.'
    dy = '0.2'
    ix = '10'
    iy = '5'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_x
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'x'
  []
  [v_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_y
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []
[]

[FunctorMaterials]
  [constant_functors]
    type = GenericFunctorMaterial
    prop_names = 'cp alpha_b'
    prop_values = '${cp} ${alpha_b}'
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_top_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = ${fparse 300.0}
    boundary = 'left top'
  []
  [bottom_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = ${fparse 350.0}
    boundary = bottom
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T
    use_two_term_expansion = false
    boundary = right
  []
[]

[Executioner]
  type = SIMPLE
  momentum_l_abs_tol = 1e-10
  pressure_l_abs_tol = 1e-10
  energy_l_abs_tol = 1e-10
  momentum_l_tol = 0
  pressure_l_tol = 0
  energy_l_tol = 0
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  momentum_equation_relaxation = 0.8
  energy_equation_relaxation = 0.9
  pressure_variable_relaxation = 0.3
  num_iterations = 200
  pressure_absolute_tolerance = 1e-10
  momentum_absolute_tolerance = 1e-10
  energy_absolute_tolerance = 1e-10
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  print_fields = false
[]

[Outputs]
  exodus = true
[]

(test/tests/linearfvkernels/diffusion/diffusion-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny = 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = coeff_func
    use_nonorthogonal_correction = false
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [dir]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right top bottom"
    functor = analytic_solution
  []
[]

[Functions]
  [coeff_func]
    type = ParsedFunction
    expression = '1+0.5*x*y'
  []
  [source_func]
    type = ParsedFunction
    expression = '2*(1.5-y*y)+2*x*y*(1.5-y*y)+2*(1.5-x*x)+2*x*y*(1.5-x*x)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '(1.5-x*x)*(1.5-y*y)'
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvbcs/scalar_symmetry/diffusion-1d-symmetry.i)

diff = 0.1

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim  = 1
    nx   = 2
    xmin = 0
    xmax = ${fparse pi}
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[Functions]
  [u_exact]
    type = ParsedFunction
    expression = 'cos(x)'
  []
  [source_fn]
    type = ParsedFunction
    expression = '${diff}*cos(x)'
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = ${diff}
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_fn
  []
[]

[LinearFVBCs]
  [rob_l]
    type = LinearFVAdvectionDiffusionScalarSymmetryBC
    variable = u
    boundary = "left"
    use_two_term_expansion = true
  []
  [dirichlet]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "right"
    functor = u_exact
  []
[]

[Postprocessors]
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = u_exact
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 4
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-7
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
  linear_convergence = linear
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/advection-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '0.5+0.5*x*x'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  multi_system_fixed_point_relaxation_factor = 1.0
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/steady-transient-compare/common-blocks.i)

mu = 2.6
rho = 1.0
advected_interp_method = 'average'
cp = 300
k = 20
alpha_b = 1e-4

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.'
    dy = '0.2'
    ix = '10'
    iy = '5'
  []
[]

[Problem]
  linear_sys_names = 'u_system v_system pressure_system energy_system'
  previous_nl_solution_required = true
[]

[UserObjects]
  [rc]
    type = RhieChowMassFlux
    u = vel_x
    v = vel_y
    pressure = pressure
    rho = ${rho}
    p_diffusion_kernel = p_diffusion
  []
[]

[Variables]
  [vel_x]
    type = MooseLinearVariableFVReal
    initial_condition = 0.5
    solver_sys = u_system
  []
  [vel_y]
    type = MooseLinearVariableFVReal
    solver_sys = v_system
    initial_condition = 0.0
  []
  [pressure]
    type = MooseLinearVariableFVReal
    solver_sys = pressure_system
    initial_condition = 0.2
  []
  [T]
    type = MooseLinearVariableFVReal
    solver_sys = energy_system
    initial_condition = 300
  []
[]

[LinearFVKernels]
  [u_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'x'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [v_advection_stress]
    type = LinearWCNSFVMomentumFlux
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    mu = ${mu}
    u = vel_x
    v = vel_y
    momentum_component = 'y'
    rhie_chow_user_object = 'rc'
    use_nonorthogonal_correction = false
  []
  [u_pressure]
    type = LinearFVMomentumPressure
    variable = vel_x
    pressure = pressure
    momentum_component = 'x'
  []
  [v_pressure]
    type = LinearFVMomentumPressure
    variable = vel_y
    pressure = pressure
    momentum_component = 'y'
  []
  [u_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_x
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'x'
  []
  [v_boussinesq]
    type = LinearFVMomentumBoussinesq
    variable = vel_y
    rho = ${rho}
    gravity = '0 -9.81 0'
    alpha_name = ${alpha_b}
    ref_temperature = 300.0
    T_fluid = T
    momentum_component = 'y'
  []

  [p_diffusion]
    type = LinearFVAnisotropicDiffusion
    variable = pressure
    diffusion_tensor = Ainv
    use_nonorthogonal_correction = false
  []
  [HbyA_divergence]
    type = LinearFVDivergence
    variable = pressure
    face_flux = HbyA
    force_boundary_execution = true
  []

  [h_advection]
    type = LinearFVEnergyAdvection
    variable = T
    advected_quantity = temperature
    cp = ${cp}
    advected_interp_method = ${advected_interp_method}
    rhie_chow_user_object = 'rc'
  []
  [conduction]
    type = LinearFVDiffusion
    variable = T
    diffusion_coeff = ${k}
    use_nonorthogonal_correction = false
  []
[]

[FunctorMaterials]
  [constant_functors]
    type = GenericFunctorMaterial
    prop_names = 'cp alpha_b'
    prop_values = '${cp} ${alpha_b}'
  []
[]

[LinearFVBCs]
  [inlet-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_x
    functor = '1.1'
  []
  [inlet-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'left'
    variable = vel_y
    functor = '0.0'
  []
  [walls-u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_x
    functor = 0.0
  []
  [walls-v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'top bottom'
    variable = vel_y
    functor = 0.0
  []
  [outlet_p]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    boundary = 'right'
    variable = pressure
    functor = 1.4
  []
  [outlet_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_x
    use_two_term_expansion = false
    boundary = right
  []
  [outlet_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = vel_y
    use_two_term_expansion = false
    boundary = right
  []
  [inlet_top_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = ${fparse 300.0}
    boundary = 'left top'
  []
  [bottom_T]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = T
    functor = ${fparse 350.0}
    boundary = bottom
  []
  [outlet_T]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = T
    use_two_term_expansion = false
    boundary = right
  []
[]

[Outputs]
  exodus = true
  execute_on = FINAL
[]

(test/tests/outputs/debug/show_execution_linear_fv_flux.i)

[Mesh]
  [gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = 10
    nx = 50
  []
  [left]
    type = ParsedSubdomainMeshGenerator
    input = 'gen_mesh'
    combinatorial_geometry = 'x < 0.5'
    block_id = '1'
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
  []
[]

[LinearFVKernels]
  [diffusion_1]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 1.5
    block = 0
  []
  [diffusion_2]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 2.5
    block = 1
  []
  [source_1]
    type = LinearFVSource
    variable = u
    source_density = 3.5
    block = 0
  []
  [source_2]
    type = LinearFVSource
    variable = u
    source_density = 4.5
    block = 1
  []
[]

[LinearFVBCs]
  [left_bc]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = 'left right'
    functor = 0
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
[]

[Debug]
  show_execution_order = 'NONLINEAR'
[]

(modules/navier_stokes/include/linearfvbcs/LinearFVNormalVelocityFunctorDirichletBC.h)

// This file is part of the MOOSE framework
// https://mooseframework.inl.gov
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "LinearFVAdvectionDiffusionFunctorDirichletBC.h"

/**
 * Adds a dirichlet BC for prescribing a velocity parallel to the boundary face normal direction;
 * this boundary condition is meant for the velocity component variables for the Navier-Stokes
 * momentum equation. A positive dirichlet value would denote outflow, while negative denotes inflow
 */
class LinearFVNormalVelocityFunctorDirichletBC : public LinearFVAdvectionDiffusionFunctorDirichletBC
{
public:
  /**
   * Class constructor.
   * @param parameters The InputParameters for the object
   */
  LinearFVNormalVelocityFunctorDirichletBC(const InputParameters & parameters);

  static InputParameters validParams();

  virtual Real computeBoundaryValue() const override;

protected:
  /// The velocity component this Dirichlet condition object is for
  const enum Component : int { X = 0, Y, Z } _component;
};

(modules/navier_stokes/include/linearfvbcs/LinearFVDirichletCHTBC.h)

// This file is part of the MOOSE framework
// https://mooseframework.inl.gov
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "LinearFVCHTBCInterface.h"
#include "LinearFVAdvectionDiffusionFunctorDirichletBC.h"

/**
 * Conjugate heat transfer BC for Dirichlet boundary condition-based
 * coupling. Differs from regular BCs due to the need of error checking.
 */
class LinearFVDirichletCHTBC : public LinearFVAdvectionDiffusionFunctorDirichletBC,
                               public LinearFVCHTBCInterface
{
public:
  /**
   * Class constructor.
   * @param parameters The InputParameters for the object
   */
  LinearFVDirichletCHTBC(const InputParameters & parameters);

  static InputParameters validParams();
};

Description
Example Syntax
Input Parameters
Input Files
Child Objects