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Incompressible Finite Volume Navier Stokes

MOOSE's Incompressible Navier Stokes Finite Volume (INSFV) implementation uses a colocated grid. To suppress the checkerboard pattern in the pressure field, INSFV objects support a Rhie-Chow interpolation for the velocity. Users can get a feel for INSFV by looking at some tests.

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
vel = 'velocity'
velocity_interp_method = 'rc'
advected_interp_method = 'average'

[GlobalParams]
  two_term_boundary_expansion = true
[]

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

[Variables]
  [u]
    type = INSFVVelocityVariable
  []
  [v]
    type = INSFVVelocityVariable
  []
  [pressure]
    type = INSFVPressureVariable
  []
  [T]
    type = INSFVEnergyVariable
  []
  [lambda]
    family = SCALAR
    order = FIRST
  []
[]

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

[AuxKernels]
  [mag]
    type = VectorMagnitudeAux
    variable = U
    x = u
    y = v
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    vel = ${vel}
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    u = u
    v = v
    pressure = pressure
    mu = ${mu}
    rho = ${rho}
  []
  [mean_zero_pressure]
    type = FVScalarLagrangeMultiplier
    variable = pressure
    lambda = lambda
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    advected_quantity = 'rhou'
    vel = ${vel}
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [u_viscosity]
    type = FVDiffusion
    variable = u
    coeff = ${mu}
  []

  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    advected_quantity = 'rhov'
    vel = ${vel}
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [v_viscosity]
    type = FVDiffusion
    variable = v
    coeff = ${mu}
  []

  [v_pressure]
    type = INSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
  []

  [temp-condution]
    type = FVDiffusion
    coeff = 'k'
    variable = T
  []

  [temp_advection]
    type = INSFVEnergyAdvection
    variable = T
    vel = ${vel}
    velocity_interp_method = ${velocity_interp_method}
    advected_interp_method = ${advected_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
[]

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

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

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

  [T_hot]
    type = FVDirichletBC
    variable = T
    boundary = 'bottom'
    value = 1
  []

  [T_cold]
    type = FVDirichletBC
    variable = T
    boundary = 'top'
    value = 0
  []
[]

[Materials]
  [const]
    type = ADGenericConstantMaterial
    prop_names = 'k cp'
    prop_values = '${k} ${cp}'
  []
  [ins_fv]
    type = INSFVMaterial
    u = 'u'
    v = 'v'
    pressure = 'pressure'
    temperature = 'T'
    rho = ${rho}
  []
[]

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

[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      300                lu           NONZERO'
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]
(modules/navier_stokes/test/tests/finite_volume/ins/lid-driven/lid-driven-with-energy.i)

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
advected_interp_method='average'
velocity_interp_method='rc'

[GlobalParams]
  two_term_boundary_expansion = true
[]

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

[Problem]
  fv_bcs_integrity_check = true
[]

[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}
    vel = 'velocity'
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    advected_quantity = 'rhou'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [u_viscosity]
    type = FVDiffusion
    variable = u
    coeff = ${mu}
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    advected_quantity = 'rhov'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [v_viscosity]
    type = FVDiffusion
    variable = v
    coeff = ${mu}
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = u
    function = '1'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = v
    function = '0'
  []
  [walls-u]
    type = INSFVNoSlipWallBC
    boundary = 'top bottom'
    variable = u
    function = 0
  []
  [walls-v]
    type = INSFVNoSlipWallBC
    boundary = 'top bottom'
    variable = v
    function = 0
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = '0'
  []
[]

[Materials]
  [ins_fv]
    type = INSFVMaterial
    u = 'u'
    v = 'v'
    pressure = 'pressure'
    rho = ${rho}
  []
[]

[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
[]

[Outputs]
  exodus = true
  csv = true
[]
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/2d-rc-no-slip.i)

  • 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
  []
[]

[Problem]
  fv_bcs_integrity_check = true
[]

[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}
    vel = 'velocity'
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    advected_quantity = 'rhou'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [u_viscosity]
    type = FVDiffusion
    variable = u
    coeff = ${mu}
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    advected_quantity = 'rhov'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [v_viscosity]
    type = FVDiffusion
    variable = v
    coeff = ${mu}
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = v
    momentum_component = 'y'
    pressure = pressure
  []
[]

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

[Materials]
  [ins_fv]
    type = INSFVMaterial
    u = 'u'
    v = 'v'
    pressure = 'pressure'
    rho = ${rho}
  []
[]

[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
[]

[Outputs]
  exodus = true
  csv = true
[]
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/2d-rc.i)

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]
  two_term_boundary_expansion = true
[]

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

[Problem]
  fv_bcs_integrity_check = true
  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}
    vel = 'velocity'
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    advected_quantity = 'rhou'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [u_viscosity]
    type = FVDiffusion
    variable = u
    coeff = ${mu}
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    advected_quantity = 'rhov'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [v_viscosity]
    type = FVDiffusion
    variable = v
    coeff = ${mu}
  []
  [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
  []
[]

[Materials]
  [ins_fv]
    type = INSFVMaterial
    u = 'u'
    v = 'v'
    pressure = 'pressure'
    rho = ${rho}
  []
[]

[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 -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
[]

[Outputs]
  exodus = true
  csv = true
[]
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/cylindrical/2d-average-no-slip.i)

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]
  fv_bcs_integrity_check = true
  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}
    vel = 'velocity'
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []

  [u_advection]
    type = INSFVMomentumAdvection
    variable = u
    advected_quantity = 'rhou'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [u_viscosity]
    type = FVDiffusion
    variable = u
    coeff = ${mu}
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = u
    momentum_component = 'x'
    pressure = pressure
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = v
    advected_quantity = 'rhov'
    vel = 'velocity'
    advected_interp_method = ${advected_interp_method}
    velocity_interp_method = ${velocity_interp_method}
    pressure = pressure
    u = u
    v = v
    mu = ${mu}
    rho = ${rho}
  []
  [v_viscosity]
    type = FVDiffusion
    variable = v
    coeff = ${mu}
  []
  [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
  []
  [free-slip-wall-v]
    type = INSFVNaturalFreeSlipBC
    boundary = 'right'
    variable = v
  []
  [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
  []
[]

[Materials]
  [ins_fv]
    type = INSFVMaterial
    u = 'u'
    v = 'v'
    pressure = 'pressure'
    rho = ${rho}
  []
[]

[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'
[]

[Outputs]
  exodus = true
  csv = true
[]
(modules/navier_stokes/test/tests/finite_volume/ins/channel-flow/cylindrical/2d-rc-slip.i)