kEpsilonViscosityAux

This is the auxiliary kernel used to compute the dynamic turbulent viscosity

$\mu_t = \rho C_{\mu} k T_e \,,$

where:

$\rho$ is the density,
$C_{\mu} = 0.09$ is a closure parameter,
$k$ is the turbulent kinetic energy,
$\epsilon$ is the turbulent kinetic energy dissipation rate.
$T_e = max( \frac{k}{\epsilon} , \sqrt(\frac{\nu}{\epsilon}) )$ .

By setting parameter "bulk_wall_treatment" to true, the kernel allows us to set the value of the cells on the boundaries specified in "walls" to the dynamic turbulent viscosity predicted from the law of the wall or non-equilibrium wall functions. See INSFVTurbulentViscosityWallFunction for more details about the near-wall implementation.

note

If the boundary conditions for the dynamic turbulent viscosity are already set via INSFVTurbulentViscosityWallFunction, there is no need to add bulk wall treatment, i.e., we should set "bulk_wall_treatment" to false. This type of bulk wall treatment is mainly designed for porous media formulations with large computational cells.

Input Parameters

epsilonCoupled turbulent kinetic energy dissipation rate. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Coupled turbulent kinetic energy dissipation rate. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
muDynamic viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Dynamic viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
rhoDensity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
tkeCoupled turbulent kinetic energy. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Coupled turbulent kinetic energy. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
uThe velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
variableThe name of the variable that this object applies to
C++ Type:AuxVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this object applies to

Required Parameters

C_muCoupled turbulent kinetic energy closure.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Coupled turbulent kinetic energy closure.
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
bulk_wall_treatmentFalseActivate bulk wall treatment.
Default:False
C++ Type:bool
Controllable:No
Description:Activate bulk wall treatment.
check_boundary_restrictedTrueWhether to check for multiple element sides on the boundary in the case of a boundary restricted, element aux variable. Setting this to false will allow contribution to a single element's elemental value(s) from multiple boundary sides on the same element (example: when the restricted boundary exists on two or more sides of an element, such as at a corner of a mesh
Default:True
C++ Type:bool
Controllable:No
Description:Whether to check for multiple element sides on the boundary in the case of a boundary restricted, element aux variable. Setting this to false will allow contribution to a single element's elemental value(s) from multiple boundary sides on the same element (example: when the restricted boundary exists on two or more sides of an element, such as at a corner of a mesh
execute_onLINEAR TIMESTEP_ENDThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:LINEAR TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, FORWARD, ADJOINT, HOMOGENEOUS_FORWARD, ADJOINT_TIMESTEP_BEGIN, ADJOINT_TIMESTEP_END, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM, PRE_DISPLACE
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
mu_t_ratio_max100000Maximum allowable mu_t_ratio : mu/mu_t.
Default:100000
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Maximum allowable mu_t_ratio : mu/mu_t.
scale_limiterstandardThe method used to limit the k-epsilon time scale.'none', 'standard'
Default:standard
C++ Type:MooseEnum
Options:none, standard
Controllable:No
Description:The method used to limit the k-epsilon time scale.'none', 'standard'
vThe velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
wThe velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
wall_treatmenteq_newtonThe method used for computing the wall functions.'eq_newton', 'eq_incremental', 'eq_linearized', 'neq'
Default:eq_newton
C++ Type:MooseEnum
Options:eq_newton, eq_incremental, eq_linearized, neq
Controllable:No
Description:The method used for computing the wall functions.'eq_newton', 'eq_incremental', 'eq_linearized', 'neq'
wallsBoundaries that correspond to solid walls.
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:Boundaries that correspond to solid walls.

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.
newton_solveFalseWhether a Newton nonlinear solve is being used
Default:False
C++ Type:bool
Controllable:No
Description:Whether a Newton nonlinear solve is being used
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
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

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
Unit:(no unit assumed)
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.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-linear-wall.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/BFS_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-inc-wall.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-std-wall.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-non-eq-bulk.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-energy.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-no-wall.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-capped.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-non-eq-wall.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/channel_ERCOFTAC.i)
(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-energy-wall.i)

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Linear wall function formulation (faster runs)
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true

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

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

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

##########################################################
# ERCOFTAC test case foe BFS
# Case Number: 031
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Equilibrium + Newton wall treatement
# SIMPLE solve
##########################################################

Re = 5100

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

advected_interp_method = 'upwind'

pressure_tag = "pressure_grad"

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

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

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

[Mesh]
  [gen]
    type = CartesianMeshGenerator
    dim = 2
    dx = '${fparse 10.0*H} ${fparse 20.0*H}'
    dy = '${H} ${fparse 5*H}'
    ix = '8 16'
    iy = '2 8'
    subdomain_id = '
                    2 1
                    1 1
                  '
  []
  [corner_walls]
    type = SideSetsBetweenSubdomainsGenerator
    input = gen
    primary_block = '1'
    paired_block = '2'
    new_boundary = 'wall-side'
  []
  [delete_bottom]
    type = BlockDeletionGenerator
    input = corner_walls
    block = '2'
  []
[]

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

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

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

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

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_x
    functor = '${bulk_u}'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_y
    functor = 0
  []
  [inlet_TKE]
    type = INSFVInletIntensityTKEBC
    boundary = 'left'
    variable = TKE
    u = vel_x
    v = vel_y
    intensity = ${intensity}
  []
  [inlet_TKED]
    type = INSFVMixingLengthTKEDBC
    boundary = 'left'
    variable = TKED
    k = TKE
    characteristic_length = '${fparse 2*H}'
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    functor = 0
  []
  [walls-u]
    type = FVDirichletBC
    boundary = ${walls}
    variable = vel_x
    value = 0
  []
  [walls-v]
    type = FVDirichletBC
    boundary = ${walls}
    variable = vel_y
    value = 0
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = ${walls}
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.3 0.3'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true
[]

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

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Incremental wall function formulation (similar to OpenFOAM)
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false

  continue_on_max_its = true

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

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

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Standard wall functions
# SIMPLE Solve
##########################################################

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

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
    mu_interp_method = 'average'
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
    mu_interp_method = 'average'
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
    coeff_interp_method = 'average'
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
    coeff_interp_method = 'average'
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  continue_on_max_its = true

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false

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

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

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# Standard wall functions with non-equilibrium bulk formaultion
# No wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true

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

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

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-energy.i)

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model with energy transport
# Standard wall functions without temperature wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0
k = 0.01
cp = 10.0
Pr_t = 0.9

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [T_fluid]
    type = INSFVEnergyVariable
    solver_sys = energy_system
    initial_condition = 1.0
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [temp_advection]
    type = INSFVEnergyAdvection
    variable = T_fluid
  []
  [temp_conduction]
    type = FVDiffusion
    coeff = ${k}
    variable = T_fluid
  []
  [temp_turb_conduction]
    type = FVDiffusion
    coeff = 'k_t'
    variable = T_fluid
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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 = 'top'
    value = 1
  []
  [T_cold]
    type = FVDirichletBC
    variable = T_fluid
    boundary = 'bottom'
    value = 0
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[FunctorMaterials]
  [ins_fv]
    type = INSFVEnthalpyFunctorMaterial
    temperature = 'T_fluid'
    rho = ${rho}
    cp = ${cp}
  []
  [k_t]
    type = ADParsedFunctorMaterial
    expression = 'mu_t * cp / Pr_t'
    functor_names = 'mu_t ${cp} ${Pr_t}'
    functor_symbols = 'mu_t cp Pr_t'
    property_name = 'k_t'
  []
[]

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  turbulence_systems = 'TKED_system TKE_system'
  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  energy_equation_relaxation = 0.9
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  energy_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  momentum_l_abs_tol = 1e-14
  energy_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  energy_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.01 0.099 0.0'
  continue_on_max_its = true
[]

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

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-no-wall.i)

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# No wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
  [walls_TKED]
    type = INSFVTKEDWallFunctionBC
    boundary = 'left right top bottom'
    variable = TKED
    u = vel_x
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
  []
  [walls_TKE]
    type = FVDirichletBC
    boundary = 'left right top bottom'
    variable = TKE
    value = ${k_init}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.9 0.9'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true

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

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

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/lid-driven-turb-capped.i)

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model with capped mixing length
# Standard wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

### k-epsilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09
C_pl = 0.1

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C_pl = ${C_pl}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    C_pl = ${C_pl}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false

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

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

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# Standard wall functions with non-equilibrium wall formulation
# No wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []
  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true

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

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

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

##########################################################
# ERCOFTAC test case foe turbulent channel flow
# Case Number: 032
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Equilibrium + Newton wall treatement
# SIMPLE solve
##########################################################

H = 1 #halfwidth of the channel
L = 30

Re = 13700

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

advected_interp_method = 'upwind'

pressure_tag = "pressure_grad"

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

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

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = 0
    ymax = ${H}
    nx = 20
    ny = 5
    bias_y = 0.7
  []
[]

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

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

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

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

[FVKernels]

  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
  []
[]

[FVBCs]
  [inlet-u]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_x
    function = '${bulk_u}'
  []
  [inlet-v]
    type = INSFVInletVelocityBC
    boundary = 'left'
    variable = vel_y
    function = 0
  []
  [walls-u]
    type = FVDirichletBC
    boundary = 'top'
    variable = vel_x
    value = 0
  []
  [walls-v]
    type = FVDirichletBC
    boundary = 'top'
    variable = vel_y
    value = 0
  []
  [outlet_p]
    type = INSFVOutletPressureBC
    boundary = 'right'
    variable = pressure
    function = 0
  []
  [inlet_TKE]
    type = INSFVInletIntensityTKEBC
    boundary = 'left'
    variable = TKE
    u = vel_x
    v = vel_y
    intensity = ${intensity}
  []
  [inlet_TKED]
    type = INSFVMixingLengthTKEDBC
    boundary = 'left'
    variable = TKED
    k = TKE
    characteristic_length = '${fparse 2*H}'
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'top'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment}
  []
  [sym-u]
    type = INSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = vel_x
    u = vel_x
    v = vel_y
    mu = 'mu_t'
    momentum_component = x
  []
  [sym-v]
    type = INSFVSymmetryVelocityBC
    boundary = 'bottom'
    variable = vel_y
    u = vel_x
    v = vel_y
    mu = 'mu_t'
    momentum_component = y
  []
  [symmetry_pressure]
    type = INSFVSymmetryPressureBC
    boundary = 'bottom'
    variable = pressure
  []
  [symmetry_TKE]
    type = INSFVSymmetryScalarBC
    boundary = 'bottom'
    variable = TKE
  []
  [symmetry_TKED]
    type = INSFVSymmetryScalarBC
    boundary = 'bottom'
    variable = TKED
  []
[]

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

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

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  turbulence_systems = 'TKED_system TKE_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.25 0.25'
  num_iterations = 1000
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'

  momentum_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  continue_on_max_its = true
[]

[Outputs]
  exodus = true
[]

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

##########################################################
# Lid-driven cavity test
# Reynolds: 5,000
# Author: Dr. Mauricio Tano
# Last Update: November, 2023
# Turbulent model using:
# k-epsilon model
# Standard wall functions with temperature wall functions
# SIMPLE Solve
##########################################################

### Thermophysical Properties ###
mu = 2e-5
rho = 1.0
k = 0.01
cp = 10.0
Pr_t = 0.9

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

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

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

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

pressure_tag = "pressure_grad"

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

[Mesh]
  [gen]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${side_length}
    ymin = 0
    ymax = ${side_length}
    nx = 12
    ny = 12
  []
[]

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

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    solver_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    solver_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [T_fluid]
    type = INSFVEnergyVariable
    solver_sys = energy_system
    initial_condition = 1.0
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    solver_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
  []
[]

[FVKernels]
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
  []
  [u_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = 'mu_t'
    momentum_component = 'x'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [u_pressure]
    type = INSFVMomentumPressure
    variable = vel_x
    momentum_component = 'x'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
  []
  [v_viscosity_turbulent]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = 'mu_t'
    momentum_component = 'y'
    complete_expansion = true
    u = vel_x
    v = vel_y
  []
  [v_pressure]
    type = INSFVMomentumPressure
    variable = vel_y
    momentum_component = 'y'
    pressure = pressure
    extra_vector_tags = ${pressure_tag}
  []

  [p_diffusion]
    type = FVAnisotropicDiffusion
    variable = pressure
    coeff = "Ainv"
    coeff_interp_method = 'average'
  []
  [p_source]
    type = FVDivergence
    variable = pressure
    vector_field = "HbyA"
    force_boundary_execution = true
  []

  [temp_advection]
    type = INSFVEnergyAdvection
    variable = T_fluid
  []
  [temp_conduction]
    type = FVDiffusion
    coeff = ${k}
    variable = T_fluid
  []
  [temp_turb_conduction]
    type = FVDiffusion
    coeff = 'k_t'
    variable = T_fluid
  []

  [TKE_advection]
    type = INSFVTurbulentAdvection
    variable = TKE
    rho = ${rho}
  []
  [TKE_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = ${mu}
  []
  [TKE_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKE
    coeff = 'mu_t'
    scaling_coef = ${sigma_k}
  []
  [TKE_source_sink]
    type = INSFVTKESourceSink
    variable = TKE
    u = vel_x
    v = vel_y
    epsilon = TKED
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    walls = ${walls}
    wall_treatment = ${wall_treatment_v}
  []

  [TKED_advection]
    type = INSFVTurbulentAdvection
    variable = TKED
    rho = ${rho}
    walls = ${walls}
  []
  [TKED_diffusion]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = ${mu}
    walls = ${walls}
  []
  [TKED_diffusion_turbulent]
    type = INSFVTurbulentDiffusion
    variable = TKED
    coeff = 'mu_t'
    scaling_coef = ${sigma_eps}
    walls = ${walls}
  []
  [TKED_source_sink]
    type = INSFVTKEDSourceSink
    variable = TKED
    u = vel_x
    v = vel_y
    k = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment_v}
  []
[]

[FVBCs]
  [top_x]
    type = INSFVNoSlipWallBC
    variable = vel_x
    boundary = 'top'
    function = ${lid_velocity}
  []
  [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 = INSFVTurbulentTemperatureWallFunction
    variable = T_fluid
    boundary = 'top'
    T_w = 1
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    cp = ${cp}
    kappa = ${k}
    k = TKE
    wall_treatment = ${wall_treatment_T}
  []
  [T_cold]
    type = INSFVTurbulentTemperatureWallFunction
    variable = T_fluid
    boundary = 'bottom'
    T_w = 0
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    cp = ${cp}
    kappa = ${k}
    k = TKE
    wall_treatment = ${wall_treatment_T}
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'left right top bottom'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    k = TKE
    wall_treatment = ${wall_treatment_v}
  []
[]

[AuxVariables]
  [mu_t]
    type = MooseVariableFVReal
    initial_condition = '${fparse rho * C_mu * ${k_init}^2 / eps_init}'
    two_term_boundary_expansion = false
  []
  [k_t]
    type = MooseVariableFVReal
    initial_condition = 1.0
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    k = TKE
    epsilon = TKED
    mu = ${mu}
    rho = ${rho}
    u = vel_x
    v = vel_y
    bulk_wall_treatment = ${bulk_wall_treatment}
    walls = ${walls}
    wall_treatment = ${wall_treatment_v}
    execute_on = 'NONLINEAR'
  []
  [compute_k_t]
    type = TurbulentConductivityAux
    variable = k_t
    Pr_t = ${Pr_t}
    cp = ${cp}
    mu_t = 'mu_t'
    execute_on = 'NONLINEAR'
  []
[]

[FunctorMaterials]
  [ins_fv]
    type = INSFVEnthalpyFunctorMaterial
    temperature = 'T_fluid'
    rho = ${rho}
    cp = ${cp}
  []
[]

[Executioner]
  type = SIMPLENonlinearAssembly
  rhie_chow_user_object = 'rc'
  momentum_systems = 'u_system v_system'
  pressure_system = 'pressure_system'
  energy_system = 'energy_system'
  turbulence_systems = 'TKED_system TKE_system'
  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.8
  pressure_variable_relaxation = 0.5
  energy_equation_relaxation = 0.9
  turbulence_equation_relaxation = '0.8 0.8'
  num_iterations = 500
  pressure_absolute_tolerance = 1e-12
  momentum_absolute_tolerance = 1e-12
  energy_absolute_tolerance = 1e-12
  turbulence_absolute_tolerance = '1e-12 1e-12'
  momentum_petsc_options_iname = '-pc_type -pc_hypre_type'
  momentum_petsc_options_value = 'hypre boomeramg'
  pressure_petsc_options_iname = '-pc_type -pc_hypre_type'
  pressure_petsc_options_value = 'hypre boomeramg'
  energy_petsc_options_iname = '-pc_type -pc_hypre_type'
  energy_petsc_options_value = 'hypre boomeramg'
  momentum_l_abs_tol = 1e-14
  energy_l_abs_tol = 1e-14
  pressure_l_abs_tol = 1e-14
  turbulence_l_abs_tol = 1e-14
  momentum_l_max_its = 30
  pressure_l_max_its = 30
  momentum_l_tol = 0.0
  energy_l_tol = 0.0
  pressure_l_tol = 0.0
  turbulence_l_tol = 0.0
  print_fields = false
  pin_pressure = true
  pressure_pin_value = 0.0
  pressure_pin_point = '0.01 0.099 0.0'
  continue_on_max_its = true
[]

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

Input Parameters
Input Files