Incompressible Finite Volume Navier Stokes

Lid Driven Cavity Flow

This example solves the INS equations for mass, momentum, and energy in a closed cavity. Because there are no inlet or outlet boundaries, one pressure degree of freedom must be constrained in order to eliminate the nullspace. This is done using the FVScalarLagrangeMultiplier object which implements the mean-zero pressure approach. The finite element theory of the mean-zero approach is described here. The finite volume implementation is completed by simply substituting unity for the test functions.

mu = 1
rho = 1
k = .01
cp = 1
velocity_interp_method = 'rc'
advected_interp_method = 'average'

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 1
    nx = 32
    ny = 32
  []
[]

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
  []
  [vel_y]
    type = INSFVVelocityVariable
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [T_fluid]
    type = INSFVEnergyVariable
  []
  [lambda]
    family = SCALAR
    order = FIRST
  []
[]

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

[AuxKernels]
  [mag]
    type = VectorMagnitudeAux
    variable = U
    x = vel_x
    y = vel_y
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    rho = ${rho}
  []
  [mean_zero_pressure]
    type = FVIntegralValueConstraint
    variable = pressure
    lambda = lambda
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_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_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_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
  []

  [temp_conduction]
    type = FVDiffusion
    coeff = 'k'
    variable = T_fluid
  []

  [temp_advection]
    type = INSFVEnergyAdvection
    variable = T_fluid
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = 'lid_function'
  []

  [no_slip_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'left right bottom'
    function = 0
  []

  [no_slip_y]
    type = INSFVNoSlipWallBC
    variable = vel_y
    boundary = 'left right top bottom'
    function = 0
  []

  [T_hot]
    type = FVDirichletBC
    variable = T_fluid
    boundary = 'bottom'
    value = 1
  []

  [T_cold]
    type = FVDirichletBC
    variable = T_fluid
    boundary = 'top'
    value = 0
  []
[]

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

[Functions]
  [lid_function]
    type = ParsedFunction
    expression = '4*x*(1-x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

Channel Flow

There are examples of both no-slip and free-slip channel flow. The only difference between the two inputs is in the boundary condition block. For the no-slip case, a Dirichlet 0 condition is applied to the tangential velocity (in this case u); this condition is not applied for free-slip. A Dirichlet 0 condition is applied on the pressure at the outlet boundary for both inputs. This is necessary, in particular for the no-slip case, because if a Dirichlet condition is not applied for a finite volume variable at a boundary, then a zero-gradient condition is implicitly applied. This is an invalid boundary condition for no-slip; in fully developed flow we know that pressure, while constant across the channel cross-section, decreases in the direction of flow. (Otherwise without that pressure driving-force, how are you going to drive flow with the walls slowing you down?)

One may reasonably ask why we implicitly apply a zero normal-gradient condition when Dirichlet conditions are not applied. This is so that FVFluxKernels executed along a boundary have a value for the field in the neighboring ghost cell. FVFluxKernels are always executed along a boundary if FVDirichletBCs are active; their execution ensures that that the Dirichlet condition is weakly enforced. When FVDirichletBCs are not active, FVFluxKernels may still be forced to execute along a boundary by specifying force_boundary_execution = true in the respective block. Forcing execution of a FVFluxKernel on a boundary when no Dirichlet BC is present, e.g. with an implicit application of the zero normal-gradient condition, is typically identical to applying a corresponding FVFluxBC because the latter only has access to the cell center value adjacent to the boundary. By directly using a cell-center value in a FVFluxBC (see for example FVConstantScalarOutflowBC), you are implicitly applying the same zero normal-gradient condition. In the future (see MOOSE issue #16169), we hope to be able to apply more appropriate outflow conditions.

• no slip

mu = 1.1
rho = 1.1
l = 2
U = 1
advected_interp_method = 'average'
velocity_interp_method = 'rc'

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 10
    ymin = ${fparse -l / 2}
    ymax = ${fparse l / 2}
    nx = 100
    ny = 20
  []
  uniform_refine = 0
[]

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 1
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 1
  []
  [pressure]
    type = INSFVPressureVariable
  []
[]

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

  [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_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
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_x
    function = '${U}'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_y
    function = '0'
  []
  [walls-u]
    type = INSFVNoSlipWallBC
    boundary = 'top bottom'
    variable = vel_x
    function = 0
  []
  [walls-v]
    type = INSFVNoSlipWallBC
    boundary = 'top bottom'
    variable = vel_y
    function = 0
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = '0'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  nl_rel_tol = 1e-12
[]

[Preconditioning]
  active = FSP
  [FSP]
    type = FSP
    # It is the starting point of splitting
    topsplit = 'up' # 'up' should match the following block name
    [up]
      splitting = 'u p' # 'u' and 'p' are the names of subsolvers
      splitting_type  = schur
      # Splitting type is set as schur, because the pressure part of Stokes-like systems
      # is not diagonally dominant. CAN NOT use additive, multiplicative and etc.
      #
      # Original system:
      #
      # | Auu Aup | | u | = | f_u |
      # | Apu 0   | | p |   | f_p |
      #
      # is factorized into
      #
      # |I             0 | | Auu  0|  | I  Auu^{-1}*Aup | | u | = | f_u |
      # |Apu*Auu^{-1}  I | | 0   -S|  | 0  I            | | p |   | f_p |
      #
      # where
      #
      # S = Apu*Auu^{-1}*Aup
      #
      # The preconditioning is accomplished via the following steps
      #
      # (1) p* = f_p - Apu*Auu^{-1}f_u,
      # (2) p = (-S)^{-1} p*
      # (3) u = Auu^{-1}(f_u-Aup*p)

      petsc_options_iname = '-pc_fieldsplit_schur_fact_type  -pc_fieldsplit_schur_precondition -ksp_gmres_restart -ksp_rtol -ksp_type'
      petsc_options_value = 'full                            selfp                             300                1e-4      fgmres'
    []
    [u]
      vars = 'vel_x vel_y'
      petsc_options_iname = '-pc_type -pc_hypre_type -ksp_type -ksp_rtol -ksp_gmres_restart -ksp_pc_side'
      petsc_options_value = 'hypre    boomeramg      gmres    5e-1      300                 right'
    []
    [p]
      vars = 'pressure'
      petsc_options_iname = '-ksp_type -ksp_gmres_restart -ksp_rtol -pc_type -ksp_pc_side'
      petsc_options_value = 'gmres    300                5e-1      jacobi    right'
    []
  []
  [SMP]
    type = SMP
    full = true
    petsc_options_iname = '-pc_type -pc_factor_shift_type'
    petsc_options_value = 'lu       NONZERO'
  []
[]

[Outputs]
  print_linear_residuals = true
  print_nonlinear_residuals = true
  [out]
    type = Exodus
    hide = 'Re lin cum_lin'
  []
  [perf]
    type = PerfGraphOutput
  []
[]

[Postprocessors]
  [Re]
    type = ParsedPostprocessor
    function = '${rho} * ${l} * ${U}'
    pp_names = ''
  []
  [lin]
    type = NumLinearIterations
  []
  [cum_lin]
    type = CumulativeValuePostprocessor
    postprocessor = lin
  []
[]

• free slip

mu = 1.1
rho = 1.1
advected_interp_method = 'average'
velocity_interp_method = 'rc'

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

[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 = 1
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 1
  []
  [pressure]
    type = INSFVPressureVariable
  []
[]

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

  [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_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
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_x
    function = '1'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_y
    function = 0
  []
  [walls-u]
    type = INSFVNaturalFreeSlipBC
    boundary = 'top bottom'
    variable = vel_x
    momentum_component = 'x'
  []
  [walls-v]
    type = INSFVNaturalFreeSlipBC
    boundary = 'top bottom'
    variable = vel_y
    momentum_component = 'y'
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = 0
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

Axisymmetric Channel Flow

Channel flow in axisymmetric coordinates is also implemented. Below is an example of solving the no-slip problem using an average interpolation for the velocity:

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

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 1
    ymin = 0
    ymax = 4
    nx = 10
    ny = 40
  []
[]

[Problem]
  coord_type = 'RZ'
[]

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

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

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

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

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = u
    function = 0
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = v
    function = 1
  []
  [no-slip-wall-u]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = u
    function = 0
  []
  [no-slip-wall-v]
    type = INSFVNoSlipWallBC
    boundary = 'right'
    variable = v
    function = 0
  []
  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'top'
    variable = pressure
    function = 0
  []
  [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
  []
[]

[Postprocessors]
  [in]
    type = SideIntegralVariablePostprocessor
    variable = v
    boundary = 'bottom'
  []
  [out]
    type = SideIntegralVariablePostprocessor
    variable = v
    boundary = 'top'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

The Rhie-Chow interpolation can be used simply by specifying velocity_interp_method='rc' either in the input file or from the command line. An axisymmetric example with free slip conditions, using the Rhie-Chow interpolation is shown below:

mu = 1.1
rho = 1.1
advected_interp_method = 'average'
velocity_interp_method = 'rc'

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = 0
    ymax = 10
    nx = 10
    ny = 50
  []
[]

[Problem]
  coord_type = 'RZ'
[]

[GlobalParams]
  rhie_chow_user_object = 'rc'
[]

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

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

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

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

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

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = u
    function = 0
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'bottom'
    variable = v
    function = 1
  []
  [free-slip-wall-u]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = u
    momentum_component = 'x'
  []
  [free-slip-wall-v]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = v
    momentum_component = 'y'
  []
  [outlet-p]
    type = INSFVOutletPressureBC
    boundary = 'top'
    variable = pressure
    function = 0
  []
  [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
  []
[]

[Postprocessors]
  [in]
    type = SideIntegralVariablePostprocessor
    variable = v
    boundary = 'bottom'
    outputs = 'csv'
  []
  [out]
    type = SideIntegralVariablePostprocessor
    variable = v
    boundary = 'top'
    outputs = 'csv'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
  csv = true
[]

Skewness-correction

The skewness-correction of different variables can be enabled by defining the face_interp_method=skewness-corrected parameter for the INSFVVariables and selecting it as an option in the advection kernels. It has proven to increase accuracy on unstructured grids. In case of a skewed 2D triangulation, it increases the order of the error from to for velocity, and from to for pressure. For an example see:

mu = 1.0
rho = 1.0

[Problem]
  error_on_jacobian_nonzero_reallocation = true
[]

[Mesh]
  [gen_mesh]
    type = FileMeshGenerator
    file = skewed.msh
  []
  coord_type = 'XYZ'
[]

[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 = 1
    face_interp_method = 'skewness-corrected'
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 1
    face_interp_method = 'skewness-corrected'
  []
  [pressure]
    type = INSFVPressureVariable
    face_interp_method = 'skewness-corrected'
  []
  [lambda]
    family = SCALAR
    order = FIRST
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    advected_interp_method = 'skewness-corrected'
    velocity_interp_method = 'rc'
    rho = ${rho}
  []
  [mean_zero_pressure]
    type = FVIntegralValueConstraint
    variable = pressure
    lambda = lambda
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    advected_interp_method = 'skewness-corrected'
    velocity_interp_method = 'rc'
    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
  []
  [u_forcing]
    type = INSFVBodyForce
    variable = vel_x
    functor = forcing_u
    momentum_component = 'x'
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    advected_interp_method = 'skewness-corrected'
    velocity_interp_method = 'rc'
    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
  []
  [v_forcing]
    type = INSFVBodyForce
    variable = vel_y
    functor = forcing_v
    momentum_component = 'y'
  []
[]

[FVBCs]
  [no-slip-wall-u]
    type = INSFVNoSlipWallBC
    boundary = 'left right top bottom'
    variable = vel_x
    function = '0'
  []
  [no-slip-wall-v]
    type = INSFVNoSlipWallBC
    boundary = 'left right top bottom'
    variable = vel_y
    function = '0'
  []
[]

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

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type'
  petsc_options_value = 'lu NONZERO'
  nl_rel_tol = 1e-8
[]

[Outputs]
  [out]
    type = Exodus
    hide = lambda
  []
  csv = true
[]

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

Cell-centered vector field reconstruction using face fluxes

References