INSFVTurbulentViscosityWallFunction

Implements wall function boundary condition for the turbulent dynamic viscosity $var element = document.getElementById("moose-equation-b223f164-ff71-4286-9363-8400bfe32c5f");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ .

The boundary conditions are different depending on where the centroid of the cell near the identified boundary lies in the wall function profile. Taking the non-dimensional wall distance as $var element = document.getElementById("moose-equation-8132deff-0df9-40e3-bcc9-a4c767e9cfa3");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ , the three regions of the boundary layer are identified as follows:

Sub-laminar region: $var element = document.getElementById("moose-equation-0d6bf869-cc42-4ae4-966e-15fd6ea68a4f");katex.render("y^+ \\le 5", element, {displayMode:false,throwOnError:false});$
Buffer region: $var element = document.getElementById("moose-equation-034813e6-00b4-410b-b24c-14356340b64c");katex.render("y^+ \\in (5, 30)", element, {displayMode:false,throwOnError:false});$
Logarithmic region: $var element = document.getElementById("moose-equation-3bd7b599-62fd-43b9-9eea-da6f9c4048c7");katex.render("y^+ \\ge 30", element, {displayMode:false,throwOnError:false});$

The wall function goal is to set the total viscosity at the wall $var element = document.getElementById("moose-equation-866f9aac-e397-4581-b34d-a20e7aa7b241");katex.render("\\mu_w", element, {displayMode:false,throwOnError:false});$ , decomposed as $var element = document.getElementById("moose-equation-1f4d132b-a190-4f16-9aa3-26253881a6f3");katex.render("\\mu_w = \\mu + \\mu_t", element, {displayMode:false,throwOnError:false});$ , such that the wall shear stress $var element = document.getElementById("moose-equation-f607d134-7e61-434d-86ac-dcb1c108bfb6");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ is accurately captured without the need of fully resolving the gradients at the near wall region.

$var element = document.getElementById("moose-equation-b8a73cc9-7284-4a49-a0df-d24a9c8267f1");katex.render(" \\tau_w = \\frac{ \\mu_w u_p}{y_p} \\,,", element, {displayMode:true,throwOnError:false});$

where:

$var element = document.getElementById("moose-equation-17a64a1d-e87e-416f-a1e4-e9f6076b97aa");katex.render("\\mu_w = \\mu + \\mu_t", element, {displayMode:false,throwOnError:false});$ is the total viscosity evaluated at the wall face
$var element = document.getElementById("moose-equation-e1ab2bce-2c71-4e08-a898-31de9eb260ff");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ is the turbulent viscosity, evaluated at the wall for the purpose of this boundary condition
$var element = document.getElementById("moose-equation-3cdc15fa-7d9c-484b-848c-3f9333a7d350");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ is the dynamic viscosity, evaluated at the wall for the purpose of this boundary condition
$var element = document.getElementById("moose-equation-d91ece8c-d757-42ee-a715-c9fe8a73a9d8");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ is the wall-shear stress
$var element = document.getElementById("moose-equation-3ff89238-3599-4249-a27a-020b741ef528");katex.render("u_p", element, {displayMode:false,throwOnError:false});$ is the wall-parallel velocity at the centroid
$var element = document.getElementById("moose-equation-c026bd79-7e26-4b48-bbf8-b044e2da2929");katex.render("y_p", element, {displayMode:false,throwOnError:false});$ is the wall normal distance to the centroid

To impose a correct boundary condition for $var element = document.getElementById("moose-equation-2c37cea7-fbb7-471c-b2e1-1ac5c7b24401");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ , as seen in the Equation above, we need to compute $var element = document.getElementById("moose-equation-7ec922c7-5ac4-4457-bdd1-225238671396");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ using analytical relationships between the wall shear stress and the dimensionless wall distance $var element = document.getElementById("moose-equation-eed4b328-3cd1-4e92-81b8-692224bb0240");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ . For this purpose, four different formulations are supported as defined by the "wall_treatment" parameter.

To set the grid spacing for the first cell near the wall in your mesh, we recommend using the RANSYPlusAux auxiliary kernel. to estimate the dimensionless wall distance $var element = document.getElementById("moose-equation-2d3db948-fd93-4d93-8ded-a1a896e01e9f");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ .

Equilibrium wall functions using a Newton solve

This treatment can be enabled by setting the parameter "wall_treatment" to eq_newton. The treatment uses equilibrium wall functions where the following formulation is used for the turbulent viscosity.

$var element = document.getElementById("moose-equation-2c369b7b-1942-442f-aac9-e1ed3dc61e29");katex.render(" \\mu_t = \\begin{cases} 0 & \\text{if } y^+ \\le 5 \\\\ \\frac{\\rho u_{\\tau}^2 y_p}{u_p} - \\mu & \\text{if } y^+ \\ge 30 \\,, \\end{cases}", element, {displayMode:true,throwOnError:false});$

where:

$var element = document.getElementById("moose-equation-13dc2446-2d46-4d30-9a89-7b90146371be");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density
$var element = document.getElementById("moose-equation-f0fc2259-256a-4bb5-ab4f-20a449c2da2a");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ is the dynamic viscosity
$var element = document.getElementById("moose-equation-56e9e579-89ad-4faa-95ba-e577bbdb2d63");katex.render("u_{\\tau} = \\sqrt{\\frac{\\tau_w}{\\rho}}", element, {displayMode:false,throwOnError:false});$ is the friction velocity and $var element = document.getElementById("moose-equation-d4cb6547-0dff-492c-a6c8-5682d2d96216");katex.render("\\tau_w", element, {displayMode:false,throwOnError:false});$ is the wall friction
$var element = document.getElementById("moose-equation-790eb21c-1bb3-4aee-9e1c-305df36ea1fa");katex.render("y_p", element, {displayMode:false,throwOnError:false});$ is the distance from the boundary to the center of the near-wall cell
$var element = document.getElementById("moose-equation-3d6519cb-9739-4061-a909-802dbd2487b4");katex.render("u_p", element, {displayMode:false,throwOnError:false});$ is the parallel velocity to the boundary computed at the center of the near-wall cell

For the buffer layer, a linear blending method is used that defines the turbulent viscosity as follows:

$var element = document.getElementById("moose-equation-b922273d-55df-4170-931d-dac0f9986c94");katex.render(" \\mu_t = \\mu_t(y^+=30) \\frac{(y^+ - 5)}{25}", element, {displayMode:true,throwOnError:false});$

Note that for $var element = document.getElementById("moose-equation-9a6fc460-23af-421e-9587-a949a1b1cf62");katex.render("y^+ = 5", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-52b2f4e1-a1fe-42c9-95c7-afa11d4348cb");katex.render("y^+ = 30", element, {displayMode:false,throwOnError:false});$ we recover the sub-laminar and logarithmic profiles, respectively.

Here the standard or equilibrium law of the wall defines $var element = document.getElementById("moose-equation-105c6a3f-3803-40dc-a8a0-d54ccaf97dc2");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-c5e54662-1d9b-4292-9e81-41bfaaf50c1a");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ as follows:

$var element = document.getElementById("moose-equation-b9997c34-9e4a-4245-887d-7faec903cb24");katex.render(" \\frac{u_p}{u_{\\tau}} = \\frac{1}{\\kappa} \\operatorname{ln}(E y^+) \\,,", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-92691203-a628-48b1-a38c-24e401884b44");katex.render(" y^+ = \\frac{\\rho u_{\\tau} y_p}{\\mu} \\,,", element, {displayMode:true,throwOnError:false});$

where:

$var element = document.getElementById("moose-equation-cf4c811f-130b-433c-8058-44f713fd24c8");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ is the molecular dynamic viscosity
$var element = document.getElementById("moose-equation-0ca567d1-e00c-41f9-8500-2e4a8013cc63");katex.render("E = 9.793", element, {displayMode:false,throwOnError:false});$ is a closure parameter
$var element = document.getElementById("moose-equation-497788a7-2172-42a9-b078-cd342307c36b");katex.render("\\kappa = 0.4187", element, {displayMode:false,throwOnError:false});$ is the von Kármán constant

In this method, we iterate on the wall function and $var element = document.getElementById("moose-equation-9911f646-8f04-47a5-b973-49cdc49d6018");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ to find $var element = document.getElementById("moose-equation-55fbaf9a-05c6-41ca-94c7-ac14999de0ae");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ via a Newton solve. Once $var element = document.getElementById("moose-equation-dd1d3a7a-6448-4ec2-9fb5-4e89fad8315d");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ is defined, $var element = document.getElementById("moose-equation-6aa35907-2a11-42c3-b009-21213171d62d");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ is computed followed by the determination of the boundary turbulent viscosity.

note

eq_newton solve will converge the fastest for simple flow geometries but it may diverge for more complicated flows. Also, the code will run if the center of the near wall cells are in the buffer layer. However, using a mesh that contains nodes in the buffer layer is not recommended.

Equilibrium wall functions using a fixed-point solve

This treatment is enabled by setting parameter "wall_treatment" to eq_incremental. The method uses the same equilibrium wall treatment than the Newton solve. However, the main difference is that, instead of computing $var element = document.getElementById("moose-equation-4ba7776d-c545-4117-8f2c-8195feaa236f");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ for the near wall cells, a fixed point iteration is performed in the wall functions to find $var element = document.getElementById("moose-equation-715402a2-ae80-47f7-b444-4bcfa43917c4");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ .

note

eq_incremental has a larger convergence radius than the Newton solve and internal controls are added to avoid issues converging the wall function at the buffer layer. However, it will take more iterations than the Newton solve to converge. Using a mesh that contains nodes in the buffer layer is not recommended.

Equilibrium wall functions using linearized wall function

This treatment is enabled by setting parameter "wall_treatment" to eq_linearized. The treatment uses a linearized version of the wall function, in which a linear Taylor approximation is used for the natural logarithm. This approximation results in a quadratic equation that is solved directly for $var element = document.getElementById("moose-equation-f9d5c125-88a7-4a7f-875e-c32333be1358");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ . Then, $var element = document.getElementById("moose-equation-23d5cb10-6815-41b7-83d3-19dc7a2b8bcf");katex.render("y^+", element, {displayMode:false,throwOnError:false});$ is computed from $var element = document.getElementById("moose-equation-9d8a3d45-3168-407e-ab9a-2398f447a753");katex.render("u_{\\tau}", element, {displayMode:false,throwOnError:false});$ .

note

eq_linearized will work fast as there is no nonlinear solve at the near-wall region. However, the method may introduce significant near-wall errors. The method is designed to be used in conjunction with porous media treatment and not necessarily for free flow.

Non-equilibrium wall functions

This treatment is enabled by setting parameter "wall_treatment" to neq. In this case, the non-dimensional wall distance is computed from the turbulent kinetic energy near the wall as follows:

$var element = document.getElementById("moose-equation-104ed21f-2c56-40b1-b6f3-4c7aff9fd365");katex.render(" y^+ = \\frac{\\rho y_p C_{\\mu}^{0.25} \\sqrt{k_p}}{\\mu} \\,,", element, {displayMode:true,throwOnError:false});$

where:

$var element = document.getElementById("moose-equation-b1ee0d7b-f02d-4816-bc35-0261ec3fb3ad");katex.render("C_{\\mu} = 0.09", element, {displayMode:false,throwOnError:false});$ is a fitting parameter
$var element = document.getElementById("moose-equation-69c27f85-5924-4f41-bef1-14c0cd08ca2b");katex.render("k_p", element, {displayMode:false,throwOnError:false});$ is the turbulent kinetic energy at the centroid of the near-wall cell

Then, the turbulent viscosity is defined as follows:

$var element = document.getElementById("moose-equation-c7c1b081-7d58-4514-88f6-4a93b30d8e1b");katex.render(" \\mu_t = \\begin{cases} 0 & \\text{if } y^+ \\le 5 \\\\ \\mu \\left[ \\frac{\\kappa y^+}{\\operatorname{ln}(E y^+)} - 1.0 \\right] & \\text{if } y^+ \\ge 30 \\end{cases}", element, {displayMode:true,throwOnError:false});$

For the buffer layer, a linear blending method is used that defines the turbulent viscosity as follows:

$var element = document.getElementById("moose-equation-41d3ab35-1ecc-4dae-ae03-2b12b52ea096");katex.render(" \\mu_t = \\mu_t(y^+=30) \\frac{(y^+ - 5)}{25}", element, {displayMode:true,throwOnError:false});$

note

neq should mainly be used for detached flow or other cases for which equilibrium wall functions are not valid. One should try to use equilibrium wall functions when possible.

Input Parameters

boundaryThe list of boundary IDs from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where this object applies
muDynamic viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Dynamic viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
mu_tThe turbulent viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The turbulent viscosity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
rhoDensity. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:Density. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
uThe velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the x direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
variableThe name of the variable that this boundary condition applies to
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this boundary condition applies to

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.
displacementsThe displacements
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacements
tkeThe turbulent kinetic energy. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The turbulent kinetic energy. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
vThe velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the y direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
wThe velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
C++ Type:MooseFunctorName
Unit:(no unit assumed)
Controllable:No
Description:The velocity in the z direction. A functor is any of the following: a variable, a functor material property, a function, a post-processor, or a number.
wall_treatmentneqThe method used for computing the wall functions
Default:neq
C++ Type:MooseEnum
Options:eq_newton, eq_incremental, eq_linearized, neq
Controllable:No
Description:The method used for computing the wall functions

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
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

Equilibrium wall functions using a Newton solve
Equilibrium wall functions using a fixed - point solve
Equilibrium wall functions using linearized wall function
Non - equilibrium wall functions
Input Parameters