INSFVTKESourceSink

The object computes the turbulent source and sink term for the turbulent kinetic energy equation.

Two terms are computed destruction = $var element = document.getElementById("moose-equation-41212570-684e-4389-8b8a-35f8b1715a06");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and production = $var element = document.getElementById("moose-equation-0d60b6a0-9da0-4c99-a567-e5e4c3a979f4");katex.render("G_k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and the term $var element = document.getElementById("moose-equation-aae90e8f-6afa-4042-8350-b5febf48dc95");katex.render("\\epsilon - G_k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is passed to the residual. A different treatment is used for the bulk and the near wall regions.

Bulk formulation:

The turbulent production $var element = document.getElementById("moose-equation-b77a4041-a93c-45c9-999d-fa41fdd65cae");katex.render("G_k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is modeled as:

$var element = document.getElementById("moose-equation-ccffedcf-e05e-4522-b293-20d581cdba8f");katex.render("G_k = \\mu_t S^2 \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-7e68c720-acb7-4f49-9392-ca0c1fd08bec");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the turbulent dynamic viscosity,
$var element = document.getElementById("moose-equation-e4d2ccc2-520a-47c4-8218-2b15b4c34068");katex.render("S", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the shear strain tensor internal norm, defined as $var element = document.getElementById("moose-equation-c9a1b943-89c5-4cbf-b5cc-619e2037d41f");katex.render("S = \\sqrt{2\\mathbf{S}:\\mathbf{S}}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ with the shear strain tensor defined as $var element = document.getElementById("moose-equation-5bb38552-5401-4887-b54c-169eae99a61f");katex.render("\\mathbf{S} = \\frac{1}{2} [\\nabla \\vec{u} + (\\nabla \\vec{u})^T]", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

The turbulent kinetic energy dissipation rate $var element = document.getElementById("moose-equation-8780df9d-6d5d-4b93-968b-39459ec8f540");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is generally coming from a coupled transport equation for $var element = document.getElementById("moose-equation-4d83bc00-fc8f-4e9b-96aa-37c0cf23e9c3");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . However, for canonical or measured cases, e.g., isotropic decaying turbulence, the user can utilize predefined fields through functors in MOOSE.

To avoid the overproduction of turbulent kinetic energy in stagnation zones \cite{durbin1996k}, a production limiter is imposed in relation to the dissipation using the formulation in \cite{menter1994two}:

$var element = document.getElementById("moose-equation-270b4343-6927-4ef8-abd9-425be5aca0eb");katex.render("G_k = min \\left( G_k , C_{PL} \\rho \\epsilon \\right) \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-6848bebc-4ad7-4fb4-b5bf-79f8a9949b12");katex.render("C_{PL}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ it the limiter constant, and set by default to a recommended value of 10 \cite{durbin1996k}.

Wall formulation:

All cells in contact with a boundary identified in the "walls" list are applied a different treatment for production and destruction. A different formulation is used for the sub-laminar and logarithmic boundary layers. The determination of whether the near-wall cell lies in the laminar or logarithmic region is performed via the non-dimensional wall distance $var element = document.getElementById("moose-equation-d0b60790-8a8e-42ef-bc84-8ac536ca6aa7");katex.render("y^+", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . The non-dimensional wall distance is defined as

$var element = document.getElementById("moose-equation-076b51a7-0e5e-49e4-85f2-3592da1fe339");katex.render("y^+ = \\frac{\\rho y_p u_{\\tau}}{\\mu} \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-4c2be423-6d6e-44a1-be6e-0cddb7a45e67");katex.render("\\rho", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the density,
$var element = document.getElementById("moose-equation-b2d86a46-4d43-497e-86cf-2022da8cb42e");katex.render("y_p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the distance to the wall to the centroid of the next-to-wall cell,
$var element = document.getElementById("moose-equation-e5fa6e04-916e-493e-8db6-5ff21cc639b6");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the friction velocity, defined as $var element = document.getElementById("moose-equation-c9845a8d-2c10-495c-b56e-d4f4141b9ffc");katex.render("u_{\\tau} = \\sqrt{\\frac{\\tau_w}{\\rho}}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ with $var element = document.getElementById("moose-equation-465ffb7c-8083-4793-a462-731a55f7fa13");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ the shear stress at the wall for which the condition is applied,
$var element = document.getElementById("moose-equation-1f52ef18-22ba-409e-9a8c-16ac9d5d5206");katex.render("\\mu", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the dynamic molecular viscosity.

For every next-to-wall cell and every iteration step, $var element = document.getElementById("moose-equation-0a1996ed-a556-42a1-b435-33b6bdf43269");katex.render("y^+", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is found via an incremental fixed-point search algorithm. The cells belonging to the sub-laminar boundary layers are defined as those for which $var element = document.getElementById("moose-equation-514e0d17-8ef5-44f8-a1c0-4e3a1a9f42d0");katex.render("y^+ < 11.25", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . The ones belonging to the logarithmic boundary layer are those for which $var element = document.getElementById("moose-equation-22176320-2d99-45f0-aeb3-510c88de91a3");katex.render("y^+ \\ge 11.25", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ . The imposed threshold of $var element = document.getElementById("moose-equation-38a502de-4de7-4551-87d5-3e12a72e771d");katex.render("y^+ = 11.25", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is given by the value of $var element = document.getElementById("moose-equation-cb527a0a-f42a-4946-b8b9-6ae150c8fdf4");katex.render("y^+", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ at which the sub-laminar and logarithmic boundary profiles intersect.

In the sub-laminar region production of turbulent kinetic energy is negligible, therefore, if $var element = document.getElementById("moose-equation-54b1be17-0067-4620-b451-77fbf8bc0220");katex.render("y^+ \\lt 11.25", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ :

$var element = document.getElementById("moose-equation-c4b9e90b-45da-4933-b28f-afeea6d8cdfe");katex.render("G_k = 0.0 \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

In the logarithmic boundary layers the production term is no longer negligible and is defined as:

$var element = document.getElementById("moose-equation-c2cb3f94-0325-4001-bb3b-d2a6a79df1a6");katex.render("G_k = \\tau_w ||\\nabla \\vec{u}|| = \\mu_w ||\\nabla \\vec{u}|| \\frac{ C_{\\mu}^{0.25} \\sqrt(k)}{\\kappa y_p} \\,,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-ce53906a-9922-44ab-b766-989d4068cd2b");katex.render("C_{\\mu} = 0.09", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a closure parameter,
$var element = document.getElementById("moose-equation-7e771c7a-9a9d-406a-9ff0-9d913cbd10df");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the turbulent kinetic energy,
$var element = document.getElementById("moose-equation-efeb4d0e-6ffc-491f-99f0-611b68ae54d7");katex.render("||\\nabla \\vec{u}||", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the near wall velocity gradient norm, which is defined as $var element = document.getElementById("moose-equation-c47ced28-d714-41ed-9ae7-3258d2ffb8c1");katex.render("||\\nabla \\vec{u}|| = (\\nabla \\vec{u} \\cdot \\hat{n}) \\cdot \\hat{n}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ,
$var element = document.getElementById("moose-equation-92d83e72-c191-4524-af94-b9a3ca3aff13");katex.render("\\kappa = 0.41", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the von Kármán constant.

The formulation assumes that the near wall value is already imposed in the $var element = document.getElementById("moose-equation-cbb0da45-0eac-484d-9f73-c40c36551584");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ functor.

When solving a linear problem, instead of the nonlinear formulation, the production term is formulated as:

$var element = document.getElementById("moose-equation-d14649fb-cd3d-426c-a851-39927c957bd2");katex.render("G_k = \\mu_w ||\\nabla \\vec{u}|| \\frac{ C_{\\mu}^{0.25} k}{\\sqrt{k_{old}} \\kappa y_p} \\,.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where:

$var element = document.getElementById("moose-equation-1e4823a8-b6f2-4231-adde-4c682714b5e6");katex.render("k_{old}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the value of the turbulent kinetic energy in the previous iteration.

Whether the linear or nonlinear formulation is used can be controlled by the "linearized_model" parameter.

For the destruction, formulation is different for the sub-laminar and logarithmic layers. For the sub-laminar layer, the destruction is defined as follows:

$var element = document.getElementById("moose-equation-282f3ab6-772f-4863-85a2-31457513807b");katex.render("\\epsilon = \\frac{2 \\mu k}{y_p ^2} \\,.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

For the logarithmic layer, the destruction is defined as follows:

$var element = document.getElementById("moose-equation-2e939fd7-9794-4f2e-9b66-d3ce69f98907");katex.render("\\epsilon = C_{\\mu}^{0.75} \\frac{\\rho k^{\\frac{3}{2}}}{\\kappa y_p} \\,.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

note

When the wall treatment is specified in this kernel, any boundary condition for $var element = document.getElementById("moose-equation-dfc9a0a9-2ee1-40a1-b710-125bd28f7fc5");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ will be ignored. In other words, there is no need to impose boundary conditions for $var element = document.getElementById("moose-equation-c550cc7d-f3f1-4999-bdb2-e7deb2765703");katex.render("k", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ when the wall treatment is specified in his kernel.

note

When using near-wall treatment, we assume that the $var element = document.getElementById("moose-equation-ee7a22e2-5b00-4d5e-9823-4552361b7383");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ functor is computed by an object that performs near-wall treatment. Otherwise, the results obtained won't not physically correct

Input Parameters

epsilonCoupled turbulent kinetic energy dissipation rate. A functor is any of the following: a variable, a functor material property, a function, a postprocessor 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 postprocessor or a number.
muDynamic viscosity. A functor is any of the following: a variable, a functor material property, a function, a postprocessor 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 postprocessor or a number.
mu_tTurbulent viscosity. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Turbulent viscosity. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
rhofluid density. A functor is any of the following: a variable, a functor material property, a function, a postprocessor or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:fluid density. A functor is any of the following: a variable, a functor material property, a function, a postprocessor 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 postprocessor 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 postprocessor or a number.
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

C_mu0.09Coupled turbulent kinetic energy closure.
Default:0.09
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Coupled turbulent kinetic energy closure.
C_pl10Production Limiter Constant Multiplier.
Default:10
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Production Limiter Constant Multiplier.
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
linearized_modelTrueBoolean to determine if the problem should be use in a linear or nonlinear solve.
Default:True
C++ Type:bool
Controllable:No
Description:Boolean to determine if the problem should be use in a linear or nonlinear solve.
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
vThe velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a postprocessor 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 postprocessor 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 postprocessor 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 postprocessor or a number.
wall_treatmentneqThe method used for computing the wall functions 'eq_newton', 'eq_incremental', 'eq_linearized', 'neq'
Default:neq
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

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

Contribution To Tagged Field Data 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
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

ghost_layers2The number of layers of elements to ghost.
Default:2
C++ Type:unsigned short
Controllable:No
Description:The number of layers of elements to ghost.
use_point_neighborsFalseWhether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

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

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/bfs/segregated/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'
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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 = ${intensity}
  []
  [inlet_TKED]
    type = INSFVMixingLengthTKEDBC
    boundary = 'left'
    variable = TKED
    tke = TKE
    characteristic_length = '${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'
    tke = 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}
    tke = 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]
  csv = true
  [console]
    type = Console
    outlier_variable_norms = false
  []
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_entry_channel_wall]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 1.00001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 1.00001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_entry_channel]
    type = LineValueSampler
    start_point = '${fparse 0.5 * H} ${fparse 2.25001 * H} 0'
    end_point = '${fparse 29.5 * H} ${fparse 2.25001 * H} 0'
    num_points = 24
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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 = ${side_length}
    nx = 12
    ny = 12
  []
  [break_symmetries]
    type = ParsedNodeTransformGenerator
    input = gen
    constant_names = 'side_length'
    constant_expressions = '${side_length}'
    x_function = 'if(x<side_length*1.001 / 2 & x > side_length * 0.999 / 2, x * 1.05, x)'
    y_function = 'if(y<side_length*1.001 / 2 & y > side_length * 0.999 / 2, y * 1.05, y)'
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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
    solver_sys = u_system
    two_term_boundary_expansion = false
  []
  [vel_y]
    type = INSFVVelocityVariable
    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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[ICs]
  [vx]
    type = FunctionIC
    variable = vel_x
    function = 'if(y>0.09, 0.1, -0.001)'
  []
  [vy]
    type = FunctionIC
    variable = vel_y
    function = 'if(x>0.05, -0.001, 0.001)'
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/channel/segregated/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 = 100

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 / (2*H)}'

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

[Mesh]
  [block_1]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = 0
    ymax = ${H}
    nx = 4
    ny = 3
    bias_y = 0.7
  []
  [block_2]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = ${L}
    ymin = ${fparse -H}
    ymax = 0
    nx = 4
    ny = 3
    bias_y = ${fparse 1/0.7}
  []
  [smg]
    type = StitchedMeshGenerator
    inputs = 'block_1 block_2'
    clear_stitched_boundary_ids = true
    stitch_boundaries_pairs = 'bottom top'
    merge_boundaries_with_same_name = true
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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 = no
    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 = no
    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}
    C_pl = 1e10
  []

  [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
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    wall_treatment = ${wall_treatment}
    C_pl = 1e10
  []
[]

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

  [inlet_TKE]
    type = FVDirichletBC
    boundary = 'left'
    variable = TKE
    value = '${k_init}'
  []
  [inlet_TKED]
    type = FVDirichletBC
    boundary = 'left'
    variable = TKED
    value = '${eps_init}'
  []
  [walls_mu_t]
    type = INSFVTurbulentViscosityWallFunction
    boundary = 'bottom top'
    variable = mu_t
    u = vel_x
    v = vel_y
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    tke = 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
  []
  [yplus]
    type = MooseVariableFVReal
    two_term_boundary_expansion = false
  []
[]

[AuxKernels]
  [compute_mu_t]
    type = kEpsilonViscosityAux
    variable = mu_t
    C_mu = ${C_mu}
    tke = 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'
    mu_t_ratio_max = 1e20
  []
  [compute_y_plus]
    type = RANSYPlusAux
    variable = yplus
    tke = 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 = 'TKE_system TKED_system'

  pressure_gradient_tag = ${pressure_tag}
  momentum_equation_relaxation = 0.7
  pressure_variable_relaxation = 0.3
  turbulence_equation_relaxation = '0.2 0.2'
  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]
  csv = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_center_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.0001} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.0001} 0'
    num_points = ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [line_quarter_radius_channel]
    type = LineValueSampler
    start_point = '${fparse 0.125 * L} ${fparse 0.5 * H} 0'
    end_point = '${fparse 0.875 * L} ${fparse 0.5 * H} 0'
    num_points =  ${Mesh/block_1/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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}
    tke = 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}
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/newton/lid-driven-turb-std-wall-nonlinear.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
# Newton 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 ###
walls = ''
linearized_model = false

[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 = 10
    ny = 10
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  previous_nl_solution_required = true
[]

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

[Variables]
  [vel_x]
    type = INSFVVelocityVariable
    initial_condition = 1e-10
  []
  [vel_y]
    type = INSFVVelocityVariable
    initial_condition = 1e-10
  []
  [pressure]
    type = INSFVPressureVariable
    initial_condition = 0.2
  []
  [TKE]
    type = INSFVEnergyVariable
    initial_condition = ${k_init}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
  [lambda]
    family = SCALAR
    order = FIRST
  []
[]

[FVKernels]
  [mass]
    type = INSFVMassAdvection
    variable = pressure
    rho = ${rho}
  []
  [mean_zero_pressure]
    type = FVIntegralValueConstraint
    variable = pressure
    lambda = lambda
  []
  [u_time]
    type = INSFVMomentumTimeDerivative
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_advection]
    type = INSFVMomentumAdvection
    variable = vel_x
    rho = ${rho}
    momentum_component = 'x'
  []
  [u_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_x
    mu = ${mu}
    momentum_component = 'x'
    mu_interp_method = average
  []
  [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
  []
  [v_advection]
    type = INSFVMomentumAdvection
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []

  [v_time]
    type = INSFVMomentumTimeDerivative
    variable = vel_y
    rho = ${rho}
    momentum_component = 'y'
  []
  [v_viscosity]
    type = INSFVMomentumDiffusion
    variable = vel_y
    mu = ${mu}
    momentum_component = 'y'
    mu_interp_method = average
  []
  [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
  []

  [TKE_time]
    type = FVFunctorTimeKernel
    variable = TKE
  []
  [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}
    linearized_model = ${linearized_model}
  []

  [TKED_time]
    type = FVFunctorTimeKernel
    variable = TKED
  []
  [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
    tke = TKE
    rho = ${rho}
    mu = ${mu}
    mu_t = 'mu_t'
    C1_eps = ${C1_eps}
    C2_eps = ${C2_eps}
    walls = ${walls}
    linearized_model = ${linearized_model}
  []
[]

[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_TKE]
    type = FVDirichletBC
    boundary = 'left right top bottom'
    variable = TKE
    value = ${k_init}
  []
  [walls_TKED]
    type = FVDirichletBC
    boundary = 'left right top bottom'
    variable = TKED
    value = ${eps_init}
  []
[]

[FunctorMaterials]
  [mu_t_material]
    type = INSFVkEpsilonViscosityFunctorMaterial
    tke = TKE
    epsilon = TKED
    rho = ${rho}
  []
[]

[Executioner]
  type = Transient
  end_time = 200
  dt = 0.01
  steady_state_detection = true
  steady_state_tolerance = 1e-3
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_factor_shift_type -snes_linesearch_damping'
  petsc_options_value = 'lu        NONZERO               0.5'
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-8
  nl_max_its = 50
  line_search = none
[]


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

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

(modules/navier_stokes/test/tests/finite_volume/ins/turbulence/lid-driven/segregated/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
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[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}
    two_term_boundary_expansion = false
  []
  [TKED]
    type = INSFVEnergyVariable
    solver_sys = TKED_system
    initial_condition = ${eps_init}
    two_term_boundary_expansion = false
  []
[]

[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
    tke = 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'
    tke = 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}
    tke = 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]
  csv = true
  perf_graph = false
  print_nonlinear_residuals = false
  print_linear_residuals = true
[]

[VectorPostprocessors]
  [side_bottom]
    type = SideValueSampler
    boundary = 'bottom'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_top]
    type = SideValueSampler
    boundary = 'top'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [side_left]
    type = SideValueSampler
    boundary = 'left'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [side_right]
    type = SideValueSampler
    boundary = 'right'
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
  [horizontal_center]
    type = LineValueSampler
    start_point = '${fparse 0.01 * side_length} ${fparse 0.499 * side_length} 0'
    end_point = '${fparse 0.99 * side_length} ${fparse 0.499 * side_length} 0'
    num_points = ${Mesh/gen/nx}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'x'
    execute_on = 'timestep_end'
  []
  [vertical_center]
    type = LineValueSampler
    start_point = '${fparse 0.499 * side_length} ${fparse 0.01 * side_length} 0'
    end_point = '${fparse 0.499 * side_length} ${fparse 0.99 * side_length} 0'
    num_points =  ${Mesh/gen/ny}
    variable = 'vel_x vel_y pressure TKE TKED'
    sort_by = 'y'
    execute_on = 'timestep_end'
  []
[]

Bulk formulation :
Wall formulation :
Input Parameters
Input Files