INSFVTurbulentTemperatureWallFunction

The function sets up the equivalent heat flux of the near-wall boundary layer when the wall temperature is set by the T_w parameter.

The boundary conditions are different depending on whether the centroid of the cell near the identified boundary lies in the wall function profile. Taking the non-dimensional wall distance as $var element = document.getElementById("moose-equation-d77134bd-ff87-4a6a-840b-8eccefb0af64");katex.render("y^+", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , the three regions of the boundary layer are identified as follows:

Sub-laminar region: $var element = document.getElementById("moose-equation-13eb205a-3def-4c90-81b7-014fdbae05be");katex.render("y^+ \\le 5", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$
Buffer region: $var element = document.getElementById("moose-equation-2f8fa8ad-90ff-43d6-b10c-767fb732cf48");katex.render("y^+ \\in (5, 30)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$
Logarithmic region: $var element = document.getElementById("moose-equation-a928d647-8686-472b-b723-383ddd263bc9");katex.render("y^+ \\ge 30", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

For the procedure of determining the non-dimensional wall distance as $var element = document.getElementById("moose-equation-571f8f0d-faf7-4887-91d7-e46272d904f9");katex.render("y^+", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and the friction velocity $var element = document.getElementById("moose-equation-afde342b-d8db-4046-ac15-f7f2a4306000");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ please see INSFVTurbulentViscosityWallFunction.

For the sub-laminar and logarithmic layer, the thermal diffusivity is defined as follows:

$var element = document.getElementById("moose-equation-3a7a4256-bd09-4905-9b63-97ce6475820c");katex.render(" \\alpha = \\begin{cases} \\alpha_l = \\frac{k}{\\rho c_p} & \\text{if } y^+ \\le 5 \\\\ \\alpha_t = \\frac{u_{\\tau} y_p}{Pr_t w_s} & \\text{if } y^+ \\ge 30 \\end{cases}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-45270fb5-259e-4edc-8cd9-4c0df25ae518");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the thermal conductivity
$var element = document.getElementById("moose-equation-7ca7eb7d-359d-45c1-a0cb-ff05f8e6fb65");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density
$var element = document.getElementById("moose-equation-f47b3563-c35a-4806-bd37-a7e497ef00f7");katex.render("c_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the specific heat at constant pressure
$var element = document.getElementById("moose-equation-3570eac6-37d4-460d-b082-033b1f0a4641");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the friction velocity defined by law of the wall
$var element = document.getElementById("moose-equation-aa2ad863-2e85-4363-9c36-0fe045519552");katex.render("y_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the distance from the boundary to the centroid of the near-wall cell
$var element = document.getElementById("moose-equation-b893f366-655f-499f-97ca-5ef5cd0b84b3");katex.render("Pr_t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the turbulent Prandtl number, which typically ranges between 0.3 and 0.9
$var element = document.getElementById("moose-equation-e841315c-ad7d-4e9b-b270-e7ada5324501");katex.render("w_s", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a near-wall scaling factor that is defined as follows:

$var element = document.getElementById("moose-equation-4aca29bf-4771-417d-b6ab-60627d94e028");katex.render(" w_s = \\frac{1}{\\kappa} \\operatorname{ln}(E y^+) + J_k \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-59a741c8-1a17-4df7-9319-64c95c83418a");katex.render("\\kappa = 0.4187", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the von Kármán constant
$var element = document.getElementById("moose-equation-d7b38c18-d25f-4a61-8b05-7ad18e8935c6");katex.render("E = 9.793", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a closure parameter
$var element = document.getElementById("moose-equation-aabb5cbe-3237-48a9-afda-7a6a07eb8371");katex.render("J_k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Jayatilleke wall functions defined as follows:

$var element = document.getElementById("moose-equation-d6e5c61f-5650-4d3c-a7ce-de27a0b62335");katex.render(" J_k = 9.24 \\left[ \\left( \\frac{Pr}{Pr_t} \\right)^{0.75} - 1 \\right] \\left( 1 + 0.28 e^{-0.007 \\frac{Pr}{Pr_t}} \\right) \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-c85a1e15-3cec-4223-b67c-0e65912f64f2");katex.render("Pr", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Prandtl number

For the buffer layer, i.e., in $var element = document.getElementById("moose-equation-a8ef2dac-0bc8-4fcf-a145-38a69a3ac080");katex.render("y^+ \\in (5, 30)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , the thermal diffusivity is defined via a linear blending function as follows:

$var element = document.getElementById("moose-equation-7dc11191-cc7a-4a87-a4f3-93c19012ae9e");katex.render(" \\alpha_b = \\alpha_t \\frac{(y^+ - 5)}{25} + \\alpha_l \\left[ 1 - \\frac{(y^+ - 5)}{25} \\right]", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Finally, using the thermal diffusivity, the heat flux at the wall is defined as follows:

$var element = document.getElementById("moose-equation-73459dd2-f385-40db-a154-19f229f2feee");katex.render(" q''_w = - \\rho c_p \\alpha \\frac{T_w - T_p}{y_p} \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-111d24fe-7ded-4111-b326-b1d4e96ea4a4");katex.render("T_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the temperature at the centroid of the near-wall cell
$var element = document.getElementById("moose-equation-b22d125b-df10-4f9f-9087-ef67274ff130");katex.render("y_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the distance from the wall to the centroid of the near-wall cell

note

The thermal wall functions are only valid for regions in which the equilibrium momentum wall functions are valid, i.e., no flow detachment, recirculation, etc. For resolving non-equilibrium phenomena, we recommend refining the mesh.

Input Parameters

T_wThe wall temperature. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Controllable:No
Description:The wall temperature. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
cpThe spcific heat at constant pressure. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Controllable:No
Description:The spcific heat at constant pressure. 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
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
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.
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
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 boundary condition applies to
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this boundary condition applies to

Required Parameters

Pr_t0.58The turbulent Prandtl number. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
Default:0.58
C++ Type:MooseFunctorName
Controllable:No
Description:The turbulent Prandtl number. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
displacementsThe displacements
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements
kappaThe thermal conductivity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Controllable:No
Description:The thermal conductivity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
linearized_yplusFalseBoolean to indicate if yplus must be estimate locally for the blending functions.
Default:False
C++ Type:bool
Controllable:No
Description:Boolean to indicate if yplus must be estimate locally for the blending functions.
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
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.
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
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
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.

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

Input Files

(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-energy-wall.i)

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

### Thermophsyical 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-epslilon Closure Parameters ###
sigma_k = 1.0
sigma_eps = 1.3
C1_eps = 1.44
C2_eps = 1.92
C_mu = 0.09

### Modeling parameters ###
non_equilibrium_treatment = false
bulk_wall_treatment = false
walls = 'left top right bottom'
max_mixing_length = 1e10
linearized_yplus_mu_t = false
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
    nl_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 0.0
    nl_sys = v_system
    two_term_boundary_expansion = false
  []
  [pressure]
    type = INSFVPressureVariable
    nl_sys = pressure_system
    initial_condition = 0.2
    two_term_boundary_expansion = false
  []
  [T_fluid]
    type = INSFVEnergyVariable
    nl_sys = energy_system
    initial_condition = 1.0
    two_term_boundary_expansion = false
  []
  [TKE]
    type = INSFVEnergyVariable
    nl_sys = TKE_system
    initial_condition = ${k_init}
  []
  [TKED]
    type = INSFVEnergyVariable
    nl_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}
    non_equilibrium_treatment = ${non_equilibrium_treatment}
    max_mixing_length = ${max_mixing_length}
  []

  [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}
    non_equilibrium_treatment = ${non_equilibrium_treatment}
    max_mixing_length = ${max_mixing_length}
  []
[]

[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}
  []
  [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}
  []
  [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
  []
  [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}
    linearized_yplus = ${linearized_yplus_mu_t}
    non_equilibrium_treatment = ${non_equilibrium_treatment}
    execute_on = 'NONLINEAR'
  []
  [compute_k_t]
    type = TurbulentConductivityAux
    variable = k_t
    Pr_t = ${Pr_t}
    cp = ${cp}
    mu_t = 'mu_t'
    execute_on = 'NONLINEAR'
  []
[]

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

[Executioner]
  type = SIMPLE
  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'
[]

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

Input Parameters
Input Files