FVNeumannBC

Neumann boundary condition for finite volume method.

Overview

A FVNeumannBC may be used to specify a diffusive or an advective flux. For example, to specify a flux boundary condition in the following diffusion problem, a FVNeumannBC with a constant value of $var element = document.getElementById("moose-equation-44a54b74-ab65-4a9d-af78-f9ccc7ec1e12");katex.render("g", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ may be used.

$var element = document.getElementById("moose-equation-ba8db121-928a-40f2-a4d5-5d3c50bf757b");katex.render("\\begin{aligned} -\\nabla^2 u(\\mathbf{M}) &= f(\\mathbf{M}) && \\forall \\mathbf{M} \\in \\Omega & (1)\\\\ \\frac{\\partial u}{\\partial n}(\\mathbf{M}) &= g && \\forall \\mathbf{M} \\in \\partial \\Omega_N & (2) \\\\ u(\\mathbf{M}) &= 1 && \\forall \\mathbf{M} \\in \\partial \\Omega_D & (3) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where $var element = document.getElementById("moose-equation-bb5f370f-a440-4e4b-9d29-6a5d8daefb3a");katex.render("\\Omega \\subset \\mathbb{R}^n", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the domain, $var element = document.getElementById("moose-equation-34d5cb54-bb74-4cf3-acbc-2b888e70ac5e");katex.render("\\partial \\Omega = \\partial \\Omega_D \\cup \\partial \\Omega_N", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is its boundary, and $var element = document.getElementById("moose-equation-f54f2501-8d5a-4933-a993-cf3f475adc37");katex.render("\\mathbf{M}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a point on the domain or its boundary. In this case, a FVNeumannBC object is used to impose the condition (2) on the subset of the boundary denoted by $var element = document.getElementById("moose-equation-aa1ab42a-71f0-4460-a8b4-b41bbffc56e5");katex.render("\\partial \\Omega_D", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . In this case, the value field corresponds to the constant $var element = document.getElementById("moose-equation-cd3341bd-c830-4385-87ec-5edb278be7e1");katex.render("g", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and the user must define one or more sidesets corresponding to the boundary $var element = document.getElementById("moose-equation-f952f0a9-6e2a-4591-b9b6-9343adaa67a5");katex.render("\\partial \\Omega_D", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ to pass to the boundary argument. For this particular problem, an additional boundary condition, for example a FVDirichletBC as in (3) would also be necessary to remove the nullspace.

Likewise, to specify an advective flux of constant value $var element = document.getElementById("moose-equation-cab91e51-bacd-455a-a4be-9580577e9afd");katex.render("g", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ in a 1D advection problem with an advective velocity $var element = document.getElementById("moose-equation-c66ef3a7-a71c-4211-b3c7-4448bca053ec");katex.render("v", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ :

$var element = document.getElementById("moose-equation-be47248b-00df-45d8-8235-068d6841741b");katex.render("\\begin{aligned} \\frac{\\partial u}{\\partial t}(\\mathbf{M}) + v \\frac{\\partial u}{\\partial x}(\\mathbf{M}) &= 0 && \\forall \\mathbf{M} \\in \\Omega & (1)\\\\ v u(\\mathbf{M}) &= g v && \\forall \\mathbf{M} \\in \\partial \\Omega & (2) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

The advective flux, the value to specify to the boundary condition (2), is $var element = document.getElementById("moose-equation-52229a54-2e38-4573-8541-932de8ae761d");katex.render("g v", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

Modeling a multi-dimensional problem will require a FVNeumannBC per component.

note

When using the Navier Stokes module, FVNeumannBC may not be available for use with velocity and pressure, as additional information is required on either the gradient or direction of these variables to model fully developed flow for example. Specific boundary conditions are provided, see for example INSFVOutletPressureBC.

Input Parameters

boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where this boundary condition applies
variableThe name of the variable that this boundary condition applies to
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this boundary condition applies to

Required Parameters

displacementsThe displacements
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
value0The value of the flux crossing the boundary.
Default:0
C++ Type:double
Controllable:No
Description:The value of the flux crossing the boundary.

Optional 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
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

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_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

(test/tests/fvbcs/fv_neumannbc/fv_neumannbc.i)
(test/tests/variables/caching_fv_variables/fv_caching.i)
(test/tests/auxkernels/time_derivative_aux/test_fv.i)
(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_INSFV.i)
(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated-boussinesq.i)
(test/tests/fvkernels/fv_simple_diffusion/neumann.i)
(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-transient.i)
(test/tests/fvkernels/mms/skewness-correction/two_term_extrapol/advection-outflow.i)
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/2d-rc-transient.i)
(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_PINSFV.i)
(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated-effective.i)
(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated.i)
(test/tests/fvkernels/two-var-flux-and-kernel/input.i)

(test/tests/fvbcs/fv_neumannbc/fv_neumannbc.i)

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1 1'
    dy = '1'
    ix = '5 5'
    iy = '5'
    subdomain_id = '1 1'
  []
  [internal_sideset]
    type = ParsedGenerateSideset
    combinatorial_geometry = 'x<1.01 & x>0.99'
    included_subdomain_ids = 1
    new_sideset_name = 'center'
    input = 'mesh'
  []
[]

[Variables]
  [u]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    block = 1
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = u
    coeff = 1
  []
[]

[FVBCs]
  inactive = 'center'
  [left]
    type = FVDirichletBC
    variable = u
    boundary = left
    value = 1
  []
  [right]
    type = FVNeumannBC
    variable = u
    boundary = right
    value = 4
  []
  # Internal center sideset, should cause erroring out
  [center]
    type = FVNeumannBC
    variable = u
    boundary = center
    value = 0
  []
[]

[Executioner]
  type = Steady
  solve_type = 'Newton'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  exodus = true
[]

(test/tests/variables/caching_fv_variables/fv_caching.i)

[Mesh]
  [cmg]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1.5 2.4 0.1'
    dy = '1.3 0.9'
    ix = '2 1 1'
    iy = '2 3'
    subdomain_id = '0 1 1 2 2 2'
  []
[]

[Variables]
  [u]
    type = MooseVariableFVReal
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = u
    coeff = 1
  []

  [adv]
    type = FVMatAdvection
    variable = u
    vel = v_mat
  []

  [body_force]
    type = FVBodyForce
    variable = u
    value = 10
  []
[]

[FVBCs]
  [left]
    type = FVDirichletBC
    variable = u
    boundary = 'left'
    value = 1
  []

  [right]
    type = FVDirichletBC
    variable = u
    boundary = 'right'
    value = 1
  []

  [top]
    type = FVNeumannBC
    variable = u
    value = 1
    boundary = 'top'
  []
[]

[Materials]
  [v_mat]
    type = ADGenericVectorFunctorMaterial
    prop_names = 'v_mat'
    prop_values = '4 0 0'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'PJFNK'
[]

[Outputs]
  exodus = true
[]

(test/tests/auxkernels/time_derivative_aux/test_fv.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 6
  ny = 6
[]

[Variables]
  [u]
    type = MooseVariableFVReal
    initial_condition = 2
  []
[]

[FVKernels]
  [time]
    type = FVTimeKernel
    variable = u
  []
  [reaction]
    type = FVReaction
    variable = u
    rate = 2.0
  []
  [diffusion]
    type = FVDiffusion
    variable = u
    coeff = 0.1
  []
[]

[FVBCs]
  [left]
    type = FVNeumannBC
    variable = u
    value = 5
    boundary = 'left'
  []
[]

[AuxVariables]
  inactive = 'variable_derivative'
  [variable_derivative]
    family = MONOMIAL
    order = CONSTANT
  []
  [variable_derivative_fv]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  []
[]

[AuxKernels]
  # Time derivative of a FV variable using the functor system
  [function_derivative_element]
    type = ADTimeDerivativeAux
    variable = variable_derivative_fv
    functor = 'u'
    factor = 2
  []
  # this places the derivative of a FV variable in a FE one
  # let's output a warning
  inactive = 'function_derivative_element_fv_fe'
  [function_derivative_element_fv_fe]
    type = ADTimeDerivativeAux
    variable = variable_derivative
    functor = 'u'
    factor = 2
  []
[]

[Executioner]
  type = Transient
  dt = 0.1
  num_steps = 2
  nl_abs_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_INSFV.i)

mu=1
rho=1
advected_interp_method='average'
velocity_interp_method='rc'

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

[UserObjects]
  [rc]
    type = INSFVRhieChowInterpolator
    u = u
    v = v
    pressure = pressure
  []
[]

[Mesh]
  inactive = 'mesh internal_boundary_bot internal_boundary_top'
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1'
    dy = '1 1 1'
    ix = '5'
    iy = '5 5 5'
    subdomain_id = '1
                    2
                    3'
  []
  [internal_boundary_bot]
    type = SideSetsBetweenSubdomainsGenerator
    input = mesh
    new_boundary = 'internal_bot'
    primary_block = 1
    paired_block = 2
  []
  [internal_boundary_top]
    type = SideSetsBetweenSubdomainsGenerator
    input = internal_boundary_bot
    new_boundary = 'internal_top'
    primary_block = 2
    paired_block = 3
  []
  [diverging_mesh]
    type = FileMeshGenerator
    file = 'expansion_quad.e'
  []
[]

[Variables]
  [u]
    type = INSFVVelocityVariable
    initial_condition = 0
  []
  [v]
    type = INSFVVelocityVariable
    initial_condition = 1
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [temperature]
    type = INSFVEnergyVariable
  []
[]

[AuxVariables]
  [advected_density]
    type = MooseVariableFVReal
    initial_condition = ${rho}
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    rho = ${rho}
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = u
    mu = ${mu}
    force_boundary_execution = true
    momentum_component = 'x'
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = v
    mu = ${mu}
    force_boundary_execution = true
    momentum_component = 'y'
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
  []

  [temp_advection]
    type = INSFVEnergyAdvection
    variable = temperature
    advected_interp_method = 'upwind'
  []
  [temp_source]
    type = FVBodyForce
    variable = temperature
    function = 10
    block = 1
  []
[]

[FVBCs]
  inactive = 'noslip-u noslip-v'
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = u
    function = 0
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = v
    function = 1
  []
  [noslip-u]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = u
    function = 0
  []
  [noslip-v]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = v
    function = 0
  []
  [free-slip-u]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = u
    momentum_component = 'x'
  []
  [free-slip-v]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = v
    momentum_component = 'y'
  []
  [axis-u]
    type = INSFVSymmetryVelocityBC
    boundary = 'left'
    variable = u
    u = u
    v = v
    mu = ${mu}
    momentum_component = x
  []
  [axis-v]
    type = INSFVSymmetryVelocityBC
    boundary = 'left'
    variable = v
    u = u
    v = v
    mu = ${mu}
    momentum_component = y
  []
  [axis-p]
    type = INSFVSymmetryPressureBC
    boundary = 'left'
    variable = pressure
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'top'
    variable = pressure
    function = 0
  []
  [inlet_temp]
    type = FVNeumannBC
    boundary = 'bottom'
    variable = temperature
    value = 300
  []
[]

[Materials]
  [ins_fv]
    type = INSFVEnthalpyMaterial
    temperature = 'temperature'
    rho = ${rho}
  []
  [advected_material_property]
    type = ADGenericFunctorMaterial
    prop_names = 'advected_rho cp'
    prop_values ='${rho} 1'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      200                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-12
[]

[Postprocessors]
  [inlet_mass_variable]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = advected_density
  []
  [inlet_mass_constant]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [inlet_mass_matprop]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = 'advected_rho'
  []
  [mid1_mass]
    type = VolumetricFlowRate
    boundary = internal_bot
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [mid2_mass]
    type = VolumetricFlowRate
    boundary = internal_top
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [outlet_mass]
    type = VolumetricFlowRate
    boundary = top
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []

  [inlet_momentum_x]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = u
  []

  [inlet_momentum_y]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = v
  []

  [mid1_advected_energy]
    type = VolumetricFlowRate
    boundary = internal_bot
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
  [mid2_advected_energy]
    type = VolumetricFlowRate
    boundary = internal_top
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
  [outlet_advected_energy]
    type = VolumetricFlowRate
    boundary = top
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
[]

[Outputs]
  exodus = true
  csv = true
[]

(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated-boussinesq.i)

mu=1
rho=1
k=1e-3
cp=1
v_inlet=1
T_inlet=200
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = 0
    ymax = 10
    nx = 20
    ny = 100
  []
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

[UserObjects]
  [rc]
    type = PINSFVRhieChowInterpolator
    u = superficial_vel_x
    v = superficial_vel_y
    pressure = pressure
    porosity = porosity
  []
[]

[Variables]
  [superficial_vel_x]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 1e-6
  []
  [superficial_vel_y]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = ${v_inlet}
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [T_fluid]
    type = INSFVEnergyVariable
  []
[]

[AuxVariables]
  [T_solid]
    family = 'MONOMIAL'
    order = 'CONSTANT'
    fv = true
    initial_condition = 100
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 0.4
  []
[]

[FVKernels]
  [mass]
    type = PINSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []

  [u_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_x
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_x
    mu = ${mu}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_x
    momentum_component = 'x'
    pressure = pressure
    porosity = porosity
  []
  [u_gravity]
    type = PINSFVMomentumGravity
    variable = superficial_vel_x
    rho = ${rho}
    gravity = '0 -9.81 0'
    momentum_component = 'x'
    porosity = porosity
  []
  [u_boussinesq]
    type = PINSFVMomentumBoussinesq
    variable = superficial_vel_x
    T_fluid = 'T_fluid'
    rho = ${rho}
    ref_temperature = 150
    gravity = '0 -9.81 0'
    momentum_component = 'x'
    alpha_name = 'alpha_b'
    porosity = porosity
  []

  [v_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_y
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_y
    mu = ${mu}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_y
    momentum_component = 'y'
    pressure = pressure
    porosity = porosity
  []
  [v_gravity]
    type = PINSFVMomentumGravity
    variable = superficial_vel_y
    rho = ${rho}
    gravity = '-0 -9.81 0'
    momentum_component = 'y'
    porosity = porosity
  []
  [v_boussinesq]
    type = PINSFVMomentumBoussinesq
    variable = superficial_vel_y
    T_fluid = 'T_fluid'
    rho = ${rho}
    ref_temperature = 150
    gravity = '0 -9.81 0'
    momentum_component = 'y'
    alpha_name = 'alpha_b'
    porosity = porosity
  []

  [energy_advection]
    type = PINSFVEnergyAdvection
    variable = T_fluid
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
  [energy_diffusion]
    type = PINSFVEnergyDiffusion
    k = ${k}
    variable = T_fluid
    porosity = porosity
  []
  [energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_fluid
    is_solid = false
    T_fluid = T_fluid
    T_solid = T_solid
    h_solid_fluid = 'h_cv'
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_x
    function = 0
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_y
    function = ${v_inlet}
  []
  [inlet-T]
    type = FVNeumannBC
    variable = T_fluid
    value = ${fparse v_inlet * rho * cp * T_inlet}
    boundary = 'bottom'
  []

  [no-slip-u]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = superficial_vel_x
    function = 0
  []
  [no-slip-v]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = superficial_vel_y
    function = 0
  []

  [symmetry-u]
    type = PINSFVSymmetryVelocityBC
    boundary = 'left'
    variable = superficial_vel_x
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'x'
  []
  [symmetry-v]
    type = PINSFVSymmetryVelocityBC
    boundary = 'left'
    variable = superficial_vel_y
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [symmetry-p]
    type = INSFVSymmetryPressureBC
    boundary = 'left'
    variable = pressure
  []

  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'top'
    variable = pressure
    function = 0
  []
[]

[Materials]
  [constants]
    type = ADGenericFunctorMaterial
    prop_names = 'h_cv alpha_b'
    prop_values = '1e-3 8e-4'
  []
  [functor_constants]
    type = ADGenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []

  [ins_fv]
    type = INSFVEnthalpyMaterial
    rho = ${rho}
    temperature = 'T_fluid'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      100                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-12
[]

# Some basic Postprocessors to examine the solution
[Postprocessors]
  [inlet-p]
    type = SideAverageValue
    variable = pressure
    boundary = 'top'
  []
  [outlet-v]
    type = SideAverageValue
    variable = superficial_vel_y
    boundary = 'top'
  []
  [outlet-temp]
    type = SideAverageValue
    variable = T_fluid
    boundary = 'top'
  []
[]

[Outputs]
  exodus = true
  csv = false
[]

(test/tests/fvkernels/fv_simple_diffusion/neumann.i)

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

[Variables]
  [u]
  []
  [v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = u
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = v
    coeff = coeff
  []
[]

[FVBCs]
  [left]
    type = FVNeumannBC
    variable = v
    boundary = left
    value = 5
  []
  [right]
    type = FVDirichletBC
    variable = v
    boundary = right
    value = 42
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '1'
  []
[]

[BCs]
  [left]
    type = NeumannBC
    variable = u
    boundary = left
    value = 5
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 42
  []
[]

[Executioner]
  type = Steady
  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-transient.i)

# Fluid properties
mu = 1
rho = 1
cp = 1
k = 1e-3

# Solid properties
cp_s = 2
rho_s = 4
k_s = 1e-2
h_fs = 10

# Operating conditions
u_inlet = 1
T_inlet = 200
p_outlet = 10
top_side_temperature = 150

# Numerical scheme
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 10
    ymin = 0
    ymax = 1
    nx = 100
    ny = 20
  []
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

[UserObjects]
  [rc]
    type = PINSFVRhieChowInterpolator
    u = superficial_vel_x
    v = superficial_vel_y
    pressure = pressure
    porosity = porosity
  []
[]

[Variables]
  [superficial_vel_x]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = ${u_inlet}
  []
  [superficial_vel_y]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 1e-6
  []
  [pressure]
    type = INSFVPressureVariable
    initial_condition = ${p_outlet}
  []
  [T_fluid]
    type = INSFVEnergyVariable
  []
  [T_solid]
    type = MooseVariableFVReal
    initial_condition = 100
  []
[]

[AuxVariables]
  [porosity]
    type = MooseVariableFVReal
    initial_condition = 0.5
  []
[]

[FVKernels]
  [mass]
    type = PINSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []

  [u_time]
    type = INSFVMomentumTimeDerivative
    variable = superficial_vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_x
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_x
    mu = ${mu}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_x
    momentum_component = 'x'
    pressure = pressure
    porosity = porosity
  []

  [v_time]
    type = INSFVMomentumTimeDerivative
    variable = superficial_vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_y
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_y
    mu = ${mu}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_y
    momentum_component = 'y'
    pressure = pressure
    porosity = porosity
  []

  [energy_time]
    type = PINSFVEnergyTimeDerivative
    variable = T_fluid
    cp = ${cp}
    rho = ${rho}
    is_solid = false
    porosity = porosity
  []
  [energy_advection]
    type = PINSFVEnergyAdvection
    variable = T_fluid
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
  [energy_diffusion]
    type = PINSFVEnergyDiffusion
    variable = T_fluid
    k = ${k}
    porosity = porosity
  []
  [energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_fluid
    is_solid = false
    T_fluid = 'T_fluid'
    T_solid = 'T_solid'
    h_solid_fluid = 'h_cv'
  []

  [solid_energy_time]
    type = PINSFVEnergyTimeDerivative
    variable = T_solid
    cp = ${cp_s}
    rho = ${rho_s}
    is_solid = true
    porosity = porosity
  []
  [solid_energy_diffusion]
    type = FVDiffusion
    variable = T_solid
    coeff = ${k_s}
  []
  [solid_energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_solid
    is_solid = true
    T_fluid = 'T_fluid'
    T_solid = 'T_solid'
    h_solid_fluid = 'h_cv'
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = superficial_vel_x
    function = ${u_inlet}
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = superficial_vel_y
    function = 0
  []
  [inlet-T]
    type = FVNeumannBC
    variable = T_fluid
    value = ${fparse u_inlet * rho * cp * T_inlet}
    boundary = 'left'
  []

  [no-slip-u]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = superficial_vel_x
    function = 0
  []
  [no-slip-v]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = superficial_vel_y
    function = 0
  []
  [heated-side]
    type = FVDirichletBC
    boundary = 'top'
    variable = 'T_solid'
    value = ${top_side_temperature}
  []

  [symmetry-u]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_x
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'x'
  []
  [symmetry-v]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_y
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [symmetry-p]
    type = INSFVSymmetryPressureBC
    boundary = 'bottom'
    variable = pressure
  []

  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = ${p_outlet}
  []
[]

[Materials]
  [constants]
    type = ADGenericFunctorMaterial
    prop_names = 'h_cv'
    prop_values = '${h_fs}'
  []
  [functor_constants]
    type = ADGenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []
  [ins_fv]
    type = INSFVEnthalpyMaterial
    rho = ${rho}
    temperature = 'T_fluid'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      100                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-12

  end_time = 1.5
[]

# Some basic Postprocessors to examine the solution
[Postprocessors]
  [inlet-p]
    type = SideAverageValue
    variable = pressure
    boundary = 'left'
  []
  [outlet-u]
    type = SideAverageValue
    variable = superficial_vel_x
    boundary = 'right'
  []
  [outlet-temp]
    type = SideAverageValue
    variable = T_fluid
    boundary = 'right'
  []
  [solid-temp]
    type = ElementAverageValue
    variable = T_solid
  []
[]

[Outputs]
  exodus = true
  csv = false
[]

(test/tests/fvkernels/mms/skewness-correction/two_term_extrapol/advection-outflow.i)

diff=1
a=1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = FileMeshGenerator
    file = skewed.msh
  [../]
[]

[Variables]
  [./v]
    type = MooseVariableFVReal
    face_interp_method = 'skewness-corrected'
    cache_face_gradients = false
    cache_face_values = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [./diffusion]
    type = FVDiffusion
    variable = v
    coeff = coeff
  [../]
  [./body]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  [../]
[]

[FVBCs]
  [left]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = v
  []
  [top]
    type = FVNeumannBC
    boundary = 'top'
    value = 0
    variable = v
  []
  [bottom]
    type = FVNeumannBC
    boundary = 'bottom'
    value = 0
    variable = v
  []
  [right]
    type = FVConstantScalarOutflowBC
    variable = v
    velocity = '${a} 0 0'
    boundary = 'right'
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '${diff}'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'cos(x)'
  []
  [forcing]
    type = ParsedFunction
    value = 'cos(x) - sin(x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type -snes_linesearch_minlambda'
  petsc_options_value = 'hypre boomeramg 1e-9'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/2d-rc-transient.i)

# Fluid properties
mu = 1.1
rho = 1.1
cp = 1.1
k = 1e-3

# Operating conditions
u_inlet = 1
T_inlet = 200
T_solid = 190
p_outlet = 10
h_fs = 0.01

# Numerical scheme
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 5
    ymin = -1
    ymax = 1
    nx = 50
    ny = 20
  []
[]

[Problem]
  fv_bcs_integrity_check = true
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

[UserObjects]
  [rc]
    type = INSFVRhieChowInterpolator
    u = vel_x
    v = vel_y
    pressure = pressure
  []
[]

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = ${u_inlet}
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 1e-12
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [T_fluid]
    type = INSFVEnergyVariable
    initial_condition = ${T_inlet}
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []
  [u_time]
    type = INSFVMomentumTimeDerivative
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
  []

  [v_time]
    type = INSFVMomentumTimeDerivative
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
  []

  [energy_time]
    type = INSFVEnergyTimeDerivative
    variable = T_fluid
    cp = ${cp}
    dcp_dt = 0.0
    rho = ${rho}
  []
  [energy_advection]
    type = INSFVEnergyAdvection
    variable = T_fluid
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
  [energy_diffusion]
    type = FVDiffusion
    variable = T_fluid
    coeff = ${k}
  []
  [energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_fluid
    is_solid = false
    T_fluid = 'T_fluid'
    T_solid = 'T_solid'
    h_solid_fluid = 'h_cv'
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_x
    function = '1'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_y
    function = 0
  []
  [inlet-T]
    type = FVNeumannBC
    variable = T_fluid
    value = ${fparse u_inlet * rho * cp * T_inlet}
    boundary = 'left'
  []

  [no-slip-u]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = vel_x
    function = 0
  []
  [no-slip-v]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = vel_y
    function = 0
  []

  [symmetry-u]
    type = INSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = vel_x
    u = vel_x
    v = vel_y
    mu = ${mu}
    momentum_component = 'x'
  []
  [symmetry-v]
    type = INSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = vel_y
    u = vel_x
    v = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [symmetry-p]
    type = INSFVSymmetryPressureBC
    boundary = 'bottom'
    variable = pressure
  []

  [outlet_u]
    type = INSFVMomentumAdvectionOutflowBC
    variable = vel_x
    u = vel_x
    v = vel_y
    boundary = 'right'
    momentum_component = 'x'
    rho = ${rho}
  []
  [outlet_v]
    type = INSFVMomentumAdvectionOutflowBC
    variable = vel_y
    u = vel_x
    v = vel_y
    boundary = 'right'
    momentum_component = 'y'
    rho = ${rho}
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = '${p_outlet}'
  []
[]

[Materials]
  [constants]
    type = ADGenericFunctorMaterial
    prop_names = 'h_cv T_solid'
    prop_values = '${h_fs} ${T_solid}'
  []
  [functor_constants]
    type = ADGenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []
  [ins_fv]
    type = INSFVEnthalpyMaterial
    rho = ${rho}
    temperature = 'T_fluid'
  []
[]

[Executioner]
  type = Transient
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      100                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 7e-13
  dt = 0.4
  end_time = 0.8
[]

[Outputs]
  exodus = true
  csv = true
[]

(modules/navier_stokes/test/tests/postprocessors/flow_rates/conservation_PINSFV.i)

mu=1
rho=1
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  inactive = 'mesh internal_boundary_bot internal_boundary_top'
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '1'
    dy = '1 1 1'
    ix = '5'
    iy = '5 5 5'
    subdomain_id = '1
                    2
                    3'
  []
  [internal_boundary_bot]
    type = SideSetsBetweenSubdomainsGenerator
    input = mesh
    new_boundary = 'internal_bot'
    primary_block = 1
    paired_block = 2
  []
  [internal_boundary_top]
    type = SideSetsBetweenSubdomainsGenerator
    input = internal_boundary_bot
    new_boundary = 'internal_top'
    primary_block = 2
    paired_block = 3
  []
  [diverging_mesh]
    type = FileMeshGenerator
    file = 'expansion_quad.e'
  []
[]

[Problem]
  fv_bcs_integrity_check = true
[]

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

[UserObjects]
  [rc]
    type = PINSFVRhieChowInterpolator
    u = u
    v = v
    pressure = pressure
    porosity = porosity
  []
[]

[Variables]
  [u]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 0
  []
  [v]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 1
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [temperature]
    type = INSFVEnergyVariable
  []
[]

[AuxVariables]
  [advected_density]
    order = CONSTANT
    family = MONOMIAL
    fv = true
    initial_condition = ${rho}
  []
  [porosity]
    order = CONSTANT
    family = MONOMIAL
    fv = true
    initial_condition = 0.5
  []
[]

[FVKernels]
  [mass]
    type = PINSFVMassAdvection
    variable = pressure
    rho = ${rho}
  []

  [u_advection]
    type = PINSFVMomentumAdvection
    variable = u
    rho = ${rho}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_viscosity]
    type = PINSFVMomentumDiffusion
    variable = u
    force_boundary_execution = true
    porosity = porosity
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_pressure]
    type = PINSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
    porosity = porosity
  []

  [v_advection]
    type = PINSFVMomentumAdvection
    variable = v
    rho = ${rho}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_viscosity]
    type = PINSFVMomentumDiffusion
    variable = v
    force_boundary_execution = true
    porosity = porosity
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_pressure]
    type = PINSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
    porosity = porosity
  []

  [temp_advection]
    type = PINSFVEnergyAdvection
    variable = temperature
    advected_interp_method = 'upwind'
  []
  [temp_source]
    type = FVBodyForce
    variable = temperature
    function = 10
    block = 1
  []
[]

[FVBCs]
  inactive = 'noslip-u noslip-v'
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = u
    function = 0
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = v
    function = 1
  []
  [noslip-u]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = u
    function = 0
  []
  [noslip-v]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = v
    function = 0
  []
  [free-slip-u]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = u
    momentum_component = 'x'
  []
  [free-slip-v]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = v
    momentum_component = 'y'
  []
  [axis-u]
    type = PINSFVSymmetryVelocityBC
    boundary = 'left'
    variable = u
    u = u
    v = v
    mu = ${mu}
    momentum_component = x
  []
  [axis-v]
    type = PINSFVSymmetryVelocityBC
    boundary = 'left'
    variable = v
    u = u
    v = v
    mu = ${mu}
    momentum_component = y
  []
  [axis-p]
    type = INSFVSymmetryPressureBC
    boundary = 'left'
    variable = pressure
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'top'
    variable = pressure
    function = 0
  []
  [inlet_temp]
    type = FVNeumannBC
    boundary = 'bottom'
    variable = temperature
    value = 300
  []
[]

[Materials]
  [ins_fv]
    type = INSFVEnthalpyMaterial
    temperature = 'temperature'
    rho = ${rho}
  []
  [advected_material_property]
    type = ADGenericFunctorMaterial
    prop_names = 'advected_rho cp'
    prop_values ='${rho} 1'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      200                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-12
[]

[Postprocessors]
  [inlet_mass_variable]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = advected_density
  []
  [inlet_mass_constant]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [inlet_mass_matprop]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = 'advected_rho'
  []
  [mid1_mass]
    type = VolumetricFlowRate
    boundary = internal_bot
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [mid2_mass]
    type = VolumetricFlowRate
    boundary = internal_top
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []
  [outlet_mass]
    type = VolumetricFlowRate
    boundary = top
    vel_x = u
    vel_y = v
    advected_quantity = ${rho}
  []

  [inlet_momentum_x]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = u
  []

  [inlet_momentum_y]
    type = VolumetricFlowRate
    boundary = bottom
    vel_x = u
    vel_y = v
    advected_quantity = v
  []

  [mid1_advected_energy]
    type = VolumetricFlowRate
    boundary = internal_bot
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
  [mid2_advected_energy]
    type = VolumetricFlowRate
    boundary = internal_top
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
  [outlet_advected_energy]
    type = VolumetricFlowRate
    boundary = top
    vel_x = u
    vel_y = v
    advected_quantity = 'rho_cp_temp'
    advected_interp_method = 'upwind'
  []
[]

[Outputs]
  exodus = true
  csv = true
[]

(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated-effective.i)

mu=1
rho=1
k=1e-3
cp=1
u_inlet=1
T_inlet=200
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 10
    ymin = 0
    ymax = 1
    nx = 100
    ny = 20
  []
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

[UserObjects]
  [rc]
    type = PINSFVRhieChowInterpolator
    u = u
    v = v
    pressure = pressure
    porosity = porosity
  []
[]

[Variables]
  inactive = 'temp_solid'
  [u]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = ${u_inlet}
  []
  [v]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 1e-6
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [temperature]
    type = INSFVEnergyVariable
  []
  [temp_solid]
    family = 'MONOMIAL'
    order = 'CONSTANT'
    fv = true
  []
[]

[AuxVariables]
  [temp_solid]
    family = 'MONOMIAL'
    order = 'CONSTANT'
    fv = true
    initial_condition = 100
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 0.5
  []
[]

[FVKernels]
  inactive = 'solid_energy_diffusion solid_energy_convection'
  [mass]
    type = PINSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []

  [u_advection]
    type = PINSFVMomentumAdvection
    variable = u
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_viscosity]
    type = PINSFVMomentumDiffusion
    variable = u
    mu = ${mu}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_pressure]
    type = PINSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
    porosity = porosity
  []

  [v_advection]
    type = PINSFVMomentumAdvection
    variable = v
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_viscosity]
    type = PINSFVMomentumDiffusion
    variable = v
    mu = ${mu}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_pressure]
    type = PINSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
    porosity = porosity
  []

  [energy_advection]
    type = PINSFVEnergyAdvection
    variable = temperature
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
  [energy_diffusion]
    type = PINSFVEnergyAnisotropicDiffusion
    kappa = 'kappa'
    variable = temperature
    porosity = porosity
  []
  [energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = temperature
    is_solid = false
    T_fluid = temperature
    T_solid = temp_solid
    h_solid_fluid = 'h_cv'
  []

  [solid_energy_diffusion]
    type = FVDiffusion
    coeff = ${k}
    variable = temp_solid
  []
  [solid_energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = temp_solid
    is_solid = true
    T_fluid = temperature
    T_solid = temp_solid
    h_solid_fluid = 'h_cv'
  []
[]

[FVBCs]
  inactive = 'heated-side'
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = u
    function = ${u_inlet}
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = v
    function = 0
  []
  [inlet-T]
    type = FVNeumannBC
    variable = temperature
    value = ${fparse u_inlet * rho * cp * T_inlet}
    boundary = 'left'
  []

  [no-slip-u]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = u
    function = 0
  []
  [no-slip-v]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = v
    function = 0
  []
  [heated-side]
    type = FVDirichletBC
    boundary = 'top'
    variable = 'temp_solid'
    value = 150
  []

  [symmetry-u]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = u
    u = u
    v = v
    mu = ${mu}
    momentum_component = 'x'
  []
  [symmetry-v]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = v
    u = u
    v = v
    mu = ${mu}
    momentum_component = 'y'
  []
  [symmetry-p]
    type = INSFVSymmetryPressureBC
    boundary = 'bottom'
    variable = pressure
  []

  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = 0.1
  []
[]

[Materials]
  [constants]
    type = ADGenericFunctorMaterial
    prop_names = 'h_cv'
    prop_values = '1'
  []
  [functor_constants]
    type = ADGenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []
  [kappa]
    type = ADGenericVectorFunctorMaterial
    prop_names = 'kappa'
    prop_values = '1e-3 1e-2 1e-1'
  []
  [ins_fv]
    type = INSFVEnthalpyMaterial
    rho = ${rho}
    temperature = 'temperature'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      100                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-12
[]

# Some basic Postprocessors to examine the solution
[Postprocessors]
  [inlet-p]
    type = SideAverageValue
    variable = pressure
    boundary = 'left'
  []
  [outlet-u]
    type = SideAverageValue
    variable = u
    boundary = 'right'
  []
  [outlet-temp]
    type = SideAverageValue
    variable = temperature
    boundary = 'right'
  []
  [solid-temp]
    type = ElementAverageValue
    variable = temp_solid
  []
[]

[Outputs]
  exodus = true
  csv = false
[]

(modules/navier_stokes/test/tests/finite_volume/pins/channel-flow/heated/2d-rc-heated.i)

mu=1
rho=1
k=1e-3
cp=1
u_inlet=1
T_inlet=200
advected_interp_method='average'
velocity_interp_method='rc'

[Mesh]
  [mesh]
    type = CartesianMeshGenerator
    dim = 2
    dx = '5 5'
    dy = '1.0'
    ix = '50 50'
    iy = '20'
    subdomain_id = '1 2'
  []
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

[UserObjects]
  [rc]
    type = PINSFVRhieChowInterpolator
    u = superficial_vel_x
    v = superficial_vel_y
    pressure = pressure
    porosity = porosity
  []
[]

[Variables]
  inactive = 'T_solid'
  [superficial_vel_x]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = ${u_inlet}
  []
  [superficial_vel_y]
    type = PINSFVSuperficialVelocityVariable
    initial_condition = 1e-6
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [T_fluid]
    type = INSFVEnergyVariable
  []
  [T_solid]
    family = 'MONOMIAL'
    order = 'CONSTANT'
    fv = true
  []
[]

[AuxVariables]
  [T_solid]
    family = 'MONOMIAL'
    order = 'CONSTANT'
    fv = true
    initial_condition = 100
  []
  [porosity]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 0.5
  []
[]

[FVKernels]
  inactive = 'solid_energy_diffusion solid_energy_convection'
  [mass]
    type = PINSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []

  [u_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_x
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_x
    mu = ${mu}
    porosity = porosity
    momentum_component = 'x'
  []
  [u_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_x
    momentum_component = 'x'
    pressure = pressure
    porosity = porosity
  []

  [v_advection]
    type = PINSFVMomentumAdvection
    variable = superficial_vel_y
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_viscosity]
    type = PINSFVMomentumDiffusion
    variable = superficial_vel_y
    mu = ${mu}
    porosity = porosity
    momentum_component = 'y'
  []
  [v_pressure]
    type = PINSFVMomentumPressure
    variable = superficial_vel_y
    momentum_component = 'y'
    pressure = pressure
    porosity = porosity
  []

  [energy_advection]
    type = PINSFVEnergyAdvection
    variable = T_fluid
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
  [energy_diffusion]
    type = PINSFVEnergyDiffusion
    k = ${k}
    variable = T_fluid
    porosity = porosity
  []
  [energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_fluid
    is_solid = false
    T_fluid = 'T_fluid'
    T_solid = 'T_solid'
    h_solid_fluid = 'h_cv'
  []

  [solid_energy_diffusion]
    type = FVDiffusion
    coeff = ${k}
    variable = T_solid
  []
  [solid_energy_convection]
    type = PINSFVEnergyAmbientConvection
    variable = T_solid
    is_solid = true
    T_fluid = 'T_fluid'
    T_solid = 'T_solid'
    h_solid_fluid = 'h_cv'
  []
[]

[FVBCs]
  inactive = 'heated-side'
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = superficial_vel_x
    function = ${u_inlet}
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = superficial_vel_y
    function = 0
  []
  [inlet-T]
    type = FVNeumannBC
    variable = T_fluid
    value = ${fparse u_inlet * rho * cp * T_inlet}
    boundary = 'left'
  []

  [no-slip-u]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = superficial_vel_x
    function = 0
  []
  [no-slip-v]
    type = INSFVNoSlipWallBC
    boundary = 'top'
    variable = superficial_vel_y
    function = 0
  []
  [heated-side]
    type = FVDirichletBC
    boundary = 'top'
    variable = 'T_solid'
    value = 150
  []

  [symmetry-u]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_x
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'x'
  []
  [symmetry-v]
    type = PINSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = superficial_vel_y
    u = superficial_vel_x
    v = superficial_vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [symmetry-p]
    type = INSFVSymmetryPressureBC
    boundary = 'bottom'
    variable = pressure
  []

  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = 0.1
  []
[]

[Materials]
  [constants]
    type = ADGenericFunctorMaterial
    prop_names = 'h_cv'
    prop_values = '1'
  []
  [functor_constants]
    type = ADGenericFunctorMaterial
    prop_names = 'cp'
    prop_values = '${cp}'
  []
  [ins_fv]
    type = INSFVEnthalpyMaterial
    rho = ${rho}
    temperature = 'T_fluid'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -ksp_gmres_restart -sub_pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      100                lu           NONZERO'
  line_search = 'none'
  nl_rel_tol = 1e-14
[]

# Some basic Postprocessors to examine the solution
[Postprocessors]
  [inlet-p]
    type = SideAverageValue
    variable = pressure
    boundary = 'left'
  []
  [outlet-u]
    type = SideAverageValue
    variable = superficial_vel_x
    boundary = 'right'
  []
  [outlet-temp]
    type = SideAverageValue
    variable = T_fluid
    boundary = 'right'
  []
  [solid-temp]
    type = ElementAverageValue
    variable = T_solid
  []
[]

[Outputs]
  exodus = true
  csv = false
[]

(test/tests/fvkernels/two-var-flux-and-kernel/input.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 20
[]

[Variables]
  [u]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  []
  [v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  []
[]

[FVKernels]
  [diff_u]
    type = FVDiffusion
    variable = u
    coeff = coeff
  []
  [diff]
    type = FVDiffusion
    variable = v
    coeff = coeff
  []
[]

[FVBCs]
  [left_u]
    type = FVNeumannBC
    variable = u
    boundary = left
    value = 0
  []
  [right_u]
    type = FVDirichletBC
    variable = u
    boundary = right
    value = 42
  []
  [left]
    type = FVDirichletBC
    variable = v
    boundary = left
    value = 7
  []
  [right]
    type = FVDirichletBC
    variable = v
    boundary = right
    value = 42
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '1'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  exodus = true
[]

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