LinearFVAdvectionDiffusionOutflowBC

Description

LinearFVAdvectionDiffusionOutflowBC will contribute to the system matrix and right hand side of a linear finite volume system. The contributions can be derived using the integral of the advective flux over a boundary face ( $var element = document.getElementById("moose-equation-02f45053-0f89-4d06-87c1-d2e79f374348");katex.render("S_b", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) of a boundary element:

var element = document.getElementById("moose-equation-c6343bc2-8f5b-4461-969c-147fb3b1e31c");katex.render("\\int\\limits_{S_b} \\vec{v} \\cdot \\vec{n} u dS \\approx \\vec{v}_f \\cdot \\vec{n} u_f |S_b|~,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-3770a2e4-d068-4520-8dd6-b3b1a02231ef");katex.render("\\vec{v}_f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-b809e3e0-84c4-4256-9eb1-d7661ccd754c");katex.render("\\vec{n}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-8c414d91-076c-4c82-8e77-005272121f40");katex.render("|S_b|", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are the outlet face velocity, outward pointing surface vector and the surface area,

The value of $var element = document.getElementById("moose-equation-19978935-97ee-47c9-aca2-ef84c550c9b2");katex.render("u_f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ can be computed two different ways depending on the setting of the "use_two_term_expansion" parameter. When the two-term expansion is enabled the face value is approximated as:

var element = document.getElementById("moose-equation-12ecb2f9-ae01-4be0-a25e-1bf80c93d9d9");katex.render("u_f = u_C + \\nabla u_C \\cdot \\vec{d}_{Cf}~,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-e71a15f9-44f8-4cd5-b842-486f3710201b");katex.render("u_C", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-e348bacc-64cc-45ca-b76f-14fc4206937b");katex.render("\\nabla u_C", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are the solution value and gradient in the boundary cell, while $var element = document.getElementById("moose-equation-5c5720f4-7942-43c1-9365-514876c99618");katex.render("\\vec{d}_{Cf}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the vector pointing to the face center from the boundary cell centroid. When "use_two_term_expansion" is disabled the following first-order approximation is used:

var element = document.getElementById("moose-equation-880982e6-7230-4c1c-a6a7-0d15b8c94937");katex.render("u_f = u_C.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

This boundary condition assumes zero normal gradient contribution to the diffusion terms.

note

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

Example Syntax

[LinearFVBCs<<<{"href": "../../syntax/LinearFVBCs/index.html"}>>>]
  [inflow]
    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 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
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC<<<{"description": "Adds a boundary condition which represents a surface with outflowing material with a constant velocity. This kernel is only compatible with advection-diffusion problems.", "href": "LinearFVAdvectionDiffusionOutflowBC.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"}>>> = "right"
    use_two_term_expansion<<<{"description": "If an approximate linear expansion should be used to compute the face value."}>>> = false
  []
[]

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
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

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)
use_two_term_expansionFalseIf an approximate linear expansion should be used to compute the face value.
Default:False
C++ Type:bool
Controllable:No
Description:If an approximate linear expansion should be used to compute the face value.

Optional 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_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/advection/advection-1d.i)
(test/tests/linearfvbcs/robin/advection-2d-robin.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/segregated-comparison/segregated-linear.i)
(test/tests/linearfvkernels/advection/advection-2d.i)
(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-2d.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-scalar/channel.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/enthalpy_equation.i)
(test/tests/linearfvkernels/block-restriction/block-restricted-adr.i)
(modules/navier_stokes/test/tests/finite_volume/two_phase/mixture_model/segregated/channel-drift-flux.i)
(test/tests/linearfvkernels/advection/advection-2d-rz.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/block-restricted/block-ke.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated-transient/BFS_ERCOFTAC.i)
(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_LinearFV.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d-heated/fluid.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/turbulence/channel/linear-segregated-transient/channel_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-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/channel/linear-segregated/channel_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/steady-transient-compare/common-blocks.i)
(test/tests/linearfvkernels/diffusion-reaction-advection/advection-diffusion-reaction-1d.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/3d/3d-velocity-pressure.i)
(test/tests/linearfvbcs/robin/advection-1d-robin.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_rob-rob.i)
(modules/navier_stokes/test/tests/finite_volume/ins/cht/conjugate_heat_transfer/cht_neu-dir.i)
(modules/navier_stokes/test/tests/finite_volume/wcns/enthalpy_equation/1d_test_h_fp.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/channel-flow/SIMPLEC/2d/2d-velocity-pressure.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/linear-segregated/BFS_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/linear-segregated/2d/2d-boussinesq-transient.i)

(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
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method = 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'
  petsc_options_value = 'lu       NONZERO               1e-10'
[]

[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
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method = 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
[]

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

(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
  []
[]

[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 = 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
  []
[]

(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
[]

(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/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
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method = 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'
  petsc_options_value = 'lu       NONZERO               1e-10'
[]

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

(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
  []
[]

[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 = 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
  []
[]

(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/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'
[]

(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
  []
[]

[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 = 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
  []
[]

(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
  []
[]

(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
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.0 0.5 0"
    advected_interp_method = 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
  []
[]

(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/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'
  []
[]

(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/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/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/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/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
[]

(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/channel/linear-segregated/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 = ${fparse 0.16*Re^(-1./8.)}
k_init = '${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 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 = 4
    bias_y = 0.7
  []
  [block_2]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = ${fparse -H}
    ymax = 0
    nx = 4
    ny = 4
    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_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 = '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
  []
[]

[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
[]

[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_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure_over_density TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [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 = ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure_over_density TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_radius_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.5 * H} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.5 * H} 0'
    num_points =  ${Mesh/block_1/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/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/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
  []
[]

[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 = 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/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/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/linearfvbcs/robin/advection-1d-robin.i)

##################################################################
c     = 0.1 # advection velocity (+x direction)
amp   = 7.0 # sinusoid amplitude, for u_exact

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

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

u0 = 3 # some positive constant for the solution, > 1

gamma = ${fparse (-alpha*amp*sin(x_l)) + beta*amp*(u0 - cos(x_l))} # RHS of Robin BC, applied at left 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 = '${amp}*(${u0} - cos(x))'
  []
  [source_fn]
    type = ParsedFunction
    expression = '${fparse c*amp}*sin(x)'
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "${c} 0 0"
    advected_interp_method = 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}
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = true
  []
[]

[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/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
[]

(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/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
  []
[]

(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/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/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/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
[]

Description
Example Syntax
Input Parameters
Input Files