$var element = document.getElementById("moose-equation-1f2f36b1-8926-4450-87a9-c7442e19184e");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-1fd7f806-b3a6-48fe-8753-a682bc4193e4");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ Turbulence Modeling in OpenPronghorn

This page documents the Reynolds-averaged Navier–Stokes (RANS) turbulence-modeling capabilities implemented in OpenPronghorn. It focuses on the $var element = document.getElementById("moose-equation-68518235-4b0d-4eb4-9381-2e72a9e653e0");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-09780f78-a82e-495a-a0cc-44730e1518c8");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ family of eddy-viscosity models and practical guidance for choosing and using each model in applications.

For general background on turbulence and RANS closures, see Pope Pope (2000) and Wilcox Wilcox (2006). The baseline $var element = document.getElementById("moose-equation-834c4700-4537-4f98-bcf2-7fbf455f1d29");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-b7ce89ef-5ae8-4a1e-b0ae-ccb2efbec4a6");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ model follows Launder and Spalding Launder and Spalding (1974), with additional options inspired by realizable, low-Reynolds-number, two-layer, curvature/rotation, buoyancy, and compressibility corrections Shih et al. (1995), Spalart and Shur (1997), Yap (1987), Wolfstein (1969), and NORRIS III (1975).

Introduction

Most OpenPronghorn applications use RANS turbulence modeling to represent the effect of unresolved turbulent fluctuations on the mean flow. The main motivations are:

Cost: Direct numerical simulation is generally infeasible at reactor-relevant Reynolds numbers, and even LES can be prohibitively expensive for large, geometrically complex configurations.
Robustness: Two-equation eddy-viscosity models are well established, robust, and widely used in engineering practice.
Integration with thermal–hydraulics: The use of RANS models fits naturally into finite-volume formulations for coupled momentum, energy, and species transport equations.

OpenPronghorn’s turbulence infrastructure is designed to be:

Modular: The same utilities are reused by several $var element = document.getElementById("moose-equation-1d4e8e51-d4a7-49d3-b86c-153c80b3e08b");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-f75f7e69-7bda-4508-8b3e-5914b1c0924e");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ kernels and auxiliary objects.
Extensible: New turbulence models can be added by leveraging the existing functor and finite-volume infrastructure.
Configurable: Users can select a model variant and enable or disable individual corrections (low-Re, two-layer, Yap, nonlinear, curvature, buoyancy, compressibility) through input-file parameters.

Theory of RANS modeling

Reynolds decomposition and RANS equations

Let $var element = document.getElementById("moose-equation-eca3a448-1e0d-4e1b-8338-fd013d23dcf0");katex.render("\\mathbf{u}(\\mathbf{x},t)", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-1e85d1e9-8b48-47e2-a099-7c1374397fc1");katex.render("p(\\mathbf{x},t)", element, {displayMode:false,throwOnError:false});$ denote the instantaneous velocity and pressure of a constant-property Newtonian fluid with density $var element = document.getElementById("moose-equation-e1655dbf-1a71-4748-9b08-503a93ba940a");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ and dynamic viscosity $var element = document.getElementById("moose-equation-99cb530f-0307-4b10-aeb7-21f39c56dfd3");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ . In RANS modeling we write

$var element = document.getElementById("moose-equation-4ce1a5b6-65b0-4f22-9376-5ee1dba3b5e0");katex.render("\\mathbf{u} = \\overline{\\mathbf{u}} + \\mathbf{u}', \\qquad p = \\overline{p} + p',", element, {displayMode:true,throwOnError:false});$

where the overbar denotes a suitable average (e.g., time or ensemble) and primes denote fluctuations Pope (2000).

Substituting into the Navier–Stokes equations and averaging yields the mean-momentum equations

$var element = document.getElementById("moose-equation-a3364f0b-78fc-44b5-b261-b0b8a322a846");katex.render("\\rho \\left( \\frac{\\partial \\overline{\\mathbf{u}}}{\\partial t} + \\overline{\\mathbf{u}} \\cdot \\nabla \\overline{\\mathbf{u}} \\right) = - \\nabla \\overline{p} + \\nabla \\cdot \\left[ \\mu \\left( \\nabla \\overline{\\mathbf{u}} + (\\nabla \\overline{\\mathbf{u}})^T \\right) \\right] - \\nabla \\cdot \\boldsymbol{\\tau}^R + \\rho \\mathbf{g},", element, {displayMode:true,throwOnError:false});$

with the Reynolds-stress tensor

$var element = document.getElementById("moose-equation-e28c617e-fcad-4a08-ab9f-d6333e930769");katex.render("\\boldsymbol{\\tau}^R = - \\rho \\, \\overline{\\mathbf{u}' \\mathbf{u}'}.", element, {displayMode:true,throwOnError:false});$

The appearance of $var element = document.getElementById("moose-equation-3b421ad9-c1f4-45b3-80ea-4db2182e94c3");katex.render("\\boldsymbol{\\tau}^R", element, {displayMode:false,throwOnError:false});$ introduces the closure problem of turbulence: additional equations or models are required to express the Reynolds stresses in terms of mean quantities.

Boussinesq eddy-viscosity hypothesis

In classical eddy-viscosity models we assume that the anisotropic part of the Reynolds stresses is proportional to the mean strain-rate tensor

$var element = document.getElementById("moose-equation-b9b0bf2e-6fa1-4943-8b15-ab6ffbc42ceb");katex.render("\\boldsymbol{\\tau}^R = 2 \\mu_t \\mathbf{S} - \\frac{2}{3} \\rho k \\mathbf{I},", element, {displayMode:true,throwOnError:false});$

where

$var element = document.getElementById("moose-equation-5d07c504-1783-4832-8768-771a73c5852f");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ is the turbulent (eddy) viscosity,
$var element = document.getElementById("moose-equation-d3aedddf-b271-42c5-9478-f0e63d069afa");katex.render("k = \\tfrac{1}{2} \\overline{u'_i u'_i}", element, {displayMode:false,throwOnError:false});$ is the turbulent kinetic energy,
$var element = document.getElementById("moose-equation-23f8e9b8-7497-4fa8-8390-62f6427d1781");katex.render("\\mathbf{S}", element, {displayMode:false,throwOnError:false});$ is the strain-rate tensor with components $var element = document.getElementById("moose-equation-1d2ed86d-e820-42b9-96ab-052283c2075a");katex.render("S_{ij} = \\frac{1}{2} \\left( \\frac{\\partial \\overline{u_i}}{\\partial x_j} + \\frac{\\partial \\overline{u_j}}{\\partial x_i} \\right).", element, {displayMode:true,throwOnError:false});$

In OpenPronghorn, $var element = document.getElementById("moose-equation-6d331017-4a9d-4626-9770-31f720efd6ca");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ is provided by the $var element = document.getElementById("moose-equation-95f20dd6-25f0-4ca8-a1aa-cd00034558b6");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-5249a8d6-66bc-44ef-aebf-9aaa2ca159d1");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ models via kEpsilonViscosity. The Boussinesq hypothesis is reasonably accurate for many wall-bounded shear flows, but is known to underperform in strongly anisotropic, rotating, or curvature-dominated flows Pope (2000) and Wilcox (2006). To mitigate these limitations OpenPronghorn includes:

optional nonlinear stress models (quadratic and cubic) that extend the constitutive relation beyond linear dependence on $var element = document.getElementById("moose-equation-5a944e33-b5c9-4ea9-b96b-ca6bdf1ecf50");katex.render("\\mathbf{S}", element, {displayMode:false,throwOnError:false});$ ;
a curvature/rotation correction that modifies production based on invariants of $var element = document.getElementById("moose-equation-03f2d9ff-084f-407e-93b7-df8040a6c561");katex.render("\\mathbf{S}", element, {displayMode:false,throwOnError:false});$ and the rotation tensor $var element = document.getElementById("moose-equation-f11f244c-37cf-4eed-a4a8-f5056b60185a");katex.render("\\mathbf{W}", element, {displayMode:false,throwOnError:false});$ ;
realizable formulations in which $var element = document.getElementById("moose-equation-e28ea414-07c2-4cbd-a199-9c558dc43306");katex.render("C_\\mu", element, {displayMode:false,throwOnError:false});$ depends on local invariants of $var element = document.getElementById("moose-equation-6278e1b5-823d-4102-8f79-cd8dd7e6b647");katex.render("\\mathbf{S}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-eb08b5e1-fd51-45dc-b9c6-7011ca6e0d2a");katex.render("\\mathbf{W}", element, {displayMode:false,throwOnError:false});$ Shih et al. (1995).

The $var element = document.getElementById("moose-equation-f59ab304-6ea0-4432-96bb-fc2bad7e3d64");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-bcb93ca4-fd6e-49d9-a3a7-d92775bb5793");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ models implemented

General transport equations

The baseline OpenPronghorn implementation follows the two-equation model of Launder and Spalding Launder and Spalding (1974). In conservative form the transport equations for $var element = document.getElementById("moose-equation-57c8e7c8-c473-48b0-bc4a-2449020c32bd");katex.render("k", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-3bc724d8-39c4-4379-a06c-c2b860aefe4f");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ are

$var element = document.getElementById("moose-equation-f9cdf79f-af45-4d04-8e9e-fe363eeb83a7");katex.render("\\frac{\\partial \\rho k}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{u} k) = \\nabla \\cdot \\left[ \\rho \\left( \\nu + \\frac{\\nu_t}{\\sigma_k} \\right) \\nabla k \\right] + \\rho P_k + \\rho G_b - \\rho \\epsilon + \\rho S_k,", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-d9da3350-aff5-465d-9e39-b143db29412a");katex.render("\\frac{\\partial \\rho \\epsilon}{\\partial t} + \\nabla \\cdot (\\rho \\mathbf{u} \\epsilon) = \\nabla \\cdot \\left[ \\rho \\left( \\nu + \\frac{\\nu_t}{\\sigma_\\epsilon} \\right) \\nabla \\epsilon \\right] + \\rho C_{1\\epsilon} f_1 \\frac{\\epsilon}{k} (P_k + C_{3\\epsilon} G_b) - \\rho C_{2\\epsilon} f_2 \\frac{\\epsilon^2}{k} + \\rho S_\\epsilon,", element, {displayMode:true,throwOnError:false});$

where

$var element = document.getElementById("moose-equation-1d6403da-2b62-44d1-90ca-64091a638357");katex.render("\\nu = \\mu/\\rho", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-2df7b032-d823-4eb7-b7d4-55601d12eb20");katex.render("\\nu_t = \\mu_t/\\rho", element, {displayMode:false,throwOnError:false});$ are the molecular and turbulent kinematic viscosities;
$var element = document.getElementById("moose-equation-eca55c03-3d87-44a1-adab-9bac7b4e30a5");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the density, which may have a functional dependency with temperature and pressure;
$var element = document.getElementById("moose-equation-de7de893-e485-42c4-a6ac-685771577fa1");katex.render("P_k", element, {displayMode:false,throwOnError:false});$ is shear production of turbulent kinetic energy;
$var element = document.getElementById("moose-equation-81e06188-15f1-46d1-9dcf-9f4e9c0c2fb2");katex.render("G_b", element, {displayMode:false,throwOnError:false});$ is production (or destruction) of turbulence by buoyancy;
$var element = document.getElementById("moose-equation-35f3b287-1d7c-40f2-828d-fcd21b17b69a");katex.render("S_k", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-19a8bddd-99d3-4e2b-b70c-834aee6e0d9f");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ collect additional user- or model-specific source terms, including compressibility ( $var element = document.getElementById("moose-equation-1ff617d5-c119-4cdb-95f6-06089f9502fb");katex.render("\\gamma_M", element, {displayMode:false,throwOnError:false});$ ), Yap ( $var element = document.getElementById("moose-equation-b1ac7835-35a1-4f1c-9623-3b116e92d47e");katex.render("\\gamma_Y", element, {displayMode:false,throwOnError:false});$ ), and low-Re corrections ( $var element = document.getElementById("moose-equation-24a8e5f6-8d66-4d56-80a9-e58bc1f39863");katex.render("G'", element, {displayMode:false,throwOnError:false});$ );
$var element = document.getElementById("moose-equation-1e189b4a-c2f9-4832-9266-c4120e3fad7f");katex.render("f_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-111dfae3-436b-47e5-96d2-076d662525c9");katex.render("f_2", element, {displayMode:false,throwOnError:false});$ are damping functions (typically $var element = document.getElementById("moose-equation-b311206d-a5a4-40f7-9614-eaa0c9e37a44");katex.render("f_1 = 1", element, {displayMode:false,throwOnError:false});$ in high-Re models, with $var element = document.getElementById("moose-equation-eab09703-5e43-4974-a965-70f36667ae11");katex.render("f_2", element, {displayMode:false,throwOnError:false});$ used in low-Re variants).

The eddy viscosity is given by

$var element = document.getElementById("moose-equation-5bca868a-c427-414b-aa46-ab547fc8c5bd");katex.render("\\mu_t = \\rho \\, C_\\mu^{\\mathrm{eff}} \\frac{k^2}{\\epsilon},", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-1930fbfa-2980-486e-8f66-784da71c2ef8");katex.render("C_\\mu^{\\mathrm{eff}}", element, {displayMode:false,throwOnError:false});$ can be:

the classical constant $var element = document.getElementById("moose-equation-83aec1fc-bbe9-4e33-9272-c513ff3531db");katex.render("C_\\mu = 0.09", element, {displayMode:false,throwOnError:false});$ (standard model),
a realizable value $var element = document.getElementById("moose-equation-a238ae79-8741-417b-ac0f-e7ef1a3ef327");katex.render("C_\\mu^\\text{real}", element, {displayMode:false,throwOnError:false});$ depending on $var element = document.getElementById("moose-equation-e343bfaa-048e-42d1-97c2-d8d197aa3d44");katex.render("S^2", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-ed50e520-90ad-4ddb-8390-75aeca9c5cd4");katex.render("W^2", element, {displayMode:false,throwOnError:false});$ ,
a two-layer or low-Re modified value, which depends on wall distance and Reynolds numbers specific to the chosen near-wall treatment.

Shear production is computed as

$var element = document.getElementById("moose-equation-5f8747ad-c39b-4b97-9f6c-b47b5c8cc09a");katex.render("P_k = G_k = \\mu_t S^2 - \\frac{2}{3} \\mu_t (\\nabla \\cdot \\mathbf{u})^2 - \\frac{2}{3} \\rho k \\, \\nabla \\cdot \\mathbf{u},", element, {displayMode:true,throwOnError:false});$

with $var element = document.getElementById("moose-equation-e8e41edc-86c0-4be3-bce2-b44581d92550");katex.render("S^2 = 2 S_{ij} S_{ij}", element, {displayMode:false,throwOnError:false});$ . The final two terms are optional compressibility corrections that vanish when the flow is strictly incompressible. Buoyancy production is modeled as

$var element = document.getElementById("moose-equation-c941a82d-acc2-4319-9f81-34cf8baec1a7");katex.render("G_b = - \\frac{\\mu_t}{\\Pr_t} \\, \\mathbf{g} \\cdot \\nabla T \\, \\beta(T),", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-f5af42f6-9708-4a18-9f06-32f28606a870");katex.render("\\Pr_t", element, {displayMode:false,throwOnError:false});$ is the turbulent Prandtl number, $var element = document.getElementById("moose-equation-d122e682-8f45-426f-8dba-88f0ccac28bb");katex.render("\\mathbf{g}", element, {displayMode:false,throwOnError:false});$ is gravity, $var element = document.getElementById("moose-equation-cf54dce9-e1bb-4edd-8edc-4dd028bbdcfb");katex.render("T", element, {displayMode:false,throwOnError:false});$ is temperature, and $var element = document.getElementById("moose-equation-1dabefb4-0e43-400e-a0d6-7d5802938f20");katex.render("\\beta(T)", element, {displayMode:false,throwOnError:false});$ is the thermal expansion coefficient.

Model constants (standard high-Re model)

The standard high-Re $var element = document.getElementById("moose-equation-b3fac6d2-ac06-4dc1-af6e-c723fd5af469");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-17f935bb-1c30-44f9-abc2-5725e263800d");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ model uses the classical constants Launder and Spalding (1974):

$var element = document.getElementById("moose-equation-d93b3ca3-9422-4229-96c9-41d81d0bfd35");katex.render("C_\\mu = 0.09", element, {displayMode:false,throwOnError:false});$ ,
$var element = document.getElementById("moose-equation-4826cdf2-811d-47de-baf8-70959c865407");katex.render("C_{1\\epsilon} = 1.44", element, {displayMode:false,throwOnError:false});$ ,
$var element = document.getElementById("moose-equation-e94c992f-caef-4b3c-bffa-8905f666315d");katex.render("C_{2\\epsilon} = 1.92", element, {displayMode:false,throwOnError:false});$ ,
$var element = document.getElementById("moose-equation-142a969f-837a-4fbf-b7c5-4b09fc97c1dc");katex.render("\\sigma_k = 1.0", element, {displayMode:false,throwOnError:false});$ ,
$var element = document.getElementById("moose-equation-56a8827d-d690-4231-bd91-8ed569d46b63");katex.render("\\sigma_\\epsilon = 1.3", element, {displayMode:false,throwOnError:false});$ .

These can be overridden through the parameters of the relevant kernels (kEpsilonTKESourceSink, kEpsilonTKEDSourceSink, and kEpsilonViscosity) if a different calibration is required.

Variant-specific forms

The implementation supports several model variants. For clarity we summarize the main differences in the $var element = document.getElementById("moose-equation-6150a44e-3847-4b70-afc6-b2f404442ccc");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation and $var element = document.getElementById("moose-equation-cfc4cee3-c4ad-4cc4-b49c-f1a96018dc68");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ definition.

Standard (high-Re)

Eddy viscosity: $var element = document.getElementById("moose-equation-a275e6f4-5ee9-4f5f-8c1f-967d63df9c72");katex.render("\\mu_t = \\rho C_\\mu k^2 / \\epsilon", element, {displayMode:false,throwOnError:false});$ .
$var element = document.getElementById("moose-equation-ada958e7-2f4b-4be2-9f8c-82c93a6bf2dd");katex.render("f_1 = 1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-261531bf-9de1-4638-9544-e3f255b333c5");katex.render("f_2 = 1", element, {displayMode:false,throwOnError:false});$ .
No explicit low-Re damping; near-wall behavior is typically handled by wall functions.
Optional terms:

compressibility correction $var element = document.getElementById("moose-equation-f857cadd-2483-408f-a4e5-cb93d7ef82cc");katex.render("\\gamma_M", element, {displayMode:false,throwOnError:false});$ in $var element = document.getElementById("moose-equation-d410531e-ffaa-4e90-b4c1-faae07f17bca");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ ,
Yap correction $var element = document.getElementById("moose-equation-374c43d3-5493-4beb-8873-085a4cefe9f7");katex.render("\\gamma_Y", element, {displayMode:false,throwOnError:false});$ in $var element = document.getElementById("moose-equation-871f7151-8dc3-4212-87e0-d2ff5346ca51");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ ,
curvature factor $var element = document.getElementById("moose-equation-54dc219e-2558-496a-a21b-babd8815a147");katex.render("f_c", element, {displayMode:false,throwOnError:false});$ multiplying $var element = document.getElementById("moose-equation-0292a132-45c2-4533-8fb8-cbdfdca63f8f");katex.render("P_k", element, {displayMode:false,throwOnError:false});$ ,
nonlinear production $var element = document.getElementById("moose-equation-348b9cfa-153b-4475-a297-c0a859d2d71e");katex.render("G_\\text{nl}", element, {displayMode:false,throwOnError:false});$ added to $var element = document.getElementById("moose-equation-45b36b6d-2886-49cb-956f-c765f4f5724b");katex.render("P_k", element, {displayMode:false,throwOnError:false});$ .

StandardLowRe

Same basic structure as the standard model, but:
$var element = document.getElementById("moose-equation-5b82d502-0b55-4697-9c87-d821f682ab37");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ is multiplied by a low-Re damping function $var element = document.getElementById("moose-equation-0cab740b-a722-4a5c-922f-4c545b5bda86");katex.render("f_\\mu", element, {displayMode:false,throwOnError:false});$ ,
$var element = document.getElementById("moose-equation-40abe383-d3bf-4b6c-bac6-fdbad2ec5004");katex.render("f_2", element, {displayMode:false,throwOnError:false});$ is given by f2_SKE_LRe(Re_t),
an additional $var element = document.getElementById("moose-equation-a7512a37-c077-45c6-9f1a-2a3d17ef37cb");katex.render("G'", element, {displayMode:false,throwOnError:false});$ term may appear in the $var element = document.getElementById("moose-equation-5af849e1-2a91-4704-baef-068cbc30e077");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation, scaled by a low-Re damping factor.
These modifications allow integration to the wall ( $var element = document.getElementById("moose-equation-f059ce7f-a379-4993-9b17-55b7038bd944");katex.render("y^+ \\approx 1", element, {displayMode:false,throwOnError:false});$ ) without wall functions, but require much finer near-wall meshes Wilcox (2006).

StandardTwoLayer

Near the wall, $var element = document.getElementById("moose-equation-352142d7-cdad-433d-bedb-c0c734ad0936");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ and the dissipation length scale $var element = document.getElementById("moose-equation-61f8d7ab-b720-4316-aa0c-6b7b4fd93eba");katex.render("\\ell_\\epsilon", element, {displayMode:false,throwOnError:false});$ are given by one of the two-layer prescriptions:
twoLayerWolfstein Wolfstein (1969),
twoLayerNorrisReynolds NORRIS III (1975),
twoLayerXu Xu et al. (1998).
Away from the wall, the model reverts to the standard high-Re form.
A blending function based on wall-distance Reynolds numbers provides a smooth transition.

Realizable

Uses the realizable form of $var element = document.getElementById("moose-equation-a190ae48-2bbd-4375-bd6d-1a5f0b32e2bf");katex.render("C_\\mu", element, {displayMode:false,throwOnError:false});$ Shih et al. (1995): $var element = document.getElementById("moose-equation-a1e0fb59-c4de-4198-9409-36c43fa4e96a");katex.render("C_\\mu^\\text{real} = \\frac{C_{a0}}{C_{a1} + C_{a2} \\bar{S} + C_{a3} \\bar{W}},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-06ec2624-53a6-476d-968b-031ffc72dc1f");katex.render("\\bar{S} = (k/\\epsilon) \\sqrt{S^2}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-f4ead515-52c3-4730-bb80-0f219f08e9a4");katex.render("\\bar{W} = (k/\\epsilon) \\sqrt{W^2}", element, {displayMode:false,throwOnError:false});$ .
Optionally uses a realizable low-Re damping function f2_RKE.
Improves predictions for rotating, swirling, and separating flows while maintaining realizability of the normal stresses Shih et al. (1995).

RealizableTwoLayer

Combines the realizable $var element = document.getElementById("moose-equation-355def50-b22b-4429-ab7b-cacbf40d9499");katex.render("C_\\mu", element, {displayMode:false,throwOnError:false});$ formulation with a two-layer near-wall treatment.
Often a good default for complex internal flows when mesh resolution allows a near-wall cluster with $var element = document.getElementById("moose-equation-b578b32f-5050-4ffd-8f83-ed979b863b9e");katex.render("1 \\lesssim y^+ \\lesssim 50", element, {displayMode:false,throwOnError:false});$ .

Corrections implemented in the $var element = document.getElementById("moose-equation-b78b2c58-1de0-410c-8f66-1bdc41deec2a");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-2b016673-8748-4db4-aed1-60e4756e473f");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ models

All correction functions are defined in TurbulenceMethods: complete list of utilities. This section provides a summary of the corrections implemented in the $var element = document.getElementById("moose-equation-f77772f8-fb90-491f-a106-a220c15ced67");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-ea0c8984-1b3a-4548-8ba5-db247089c332");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ models.

This section explains the available corrections, their motivation, and how they are used.

Low-Reynolds-number corrections

Motivation: Standard high-Re models with wall functions assume an equilibrium logarithmic layer and cannot accurately represent the viscous sublayer and buffer region. Low-Re formulations include damping functions that allow integration to the wall at $var element = document.getElementById("moose-equation-8f0117e2-94b3-40bb-9dcc-2956941131fe");katex.render("y^+ \\approx 1", element, {displayMode:false,throwOnError:false});$ Wilcox (2006).
Implementation: (a) $var element = document.getElementById("moose-equation-1a8d1295-ae66-4d94-971d-69c75a27e2a4");katex.render("\\mu_t \\gets f_\\mu \\, \\rho k^2/\\epsilon", element, {displayMode:false,throwOnError:false});$ with fmu_SKE_LRe (standard) or combined realizable damping, (b) The $var element = document.getElementById("moose-equation-be061f58-361f-4f30-8215-7e07ed4b1950");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation uses $var element = document.getElementById("moose-equation-42412ea5-1bb0-4b6b-b5c8-fd997e2c8c4d");katex.render("f_2 = f2_SKE_LRe", element, {displayMode:false,throwOnError:false});$ or f2_RKE, (c) An additional production term $var element = document.getElementById("moose-equation-38befe41-b6c6-46df-a9a2-0eb031259528");katex.render("G'", element, {displayMode:false,throwOnError:false});$ (via computeGprime) can be added to $var element = document.getElementById("moose-equation-5b2a1caa-7043-4f6d-b932-d7f99a282755");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ .
Activation: (a) k_epsilon_variant = 'StandardLowRe', (b) use_low_re_Gprime = true/false, (c) near-wall mesh with $var element = document.getElementById("moose-equation-5d0c62c4-f594-477d-ad77-ba05d85e7d0f");katex.render("y^+ \\approx 1", element, {displayMode:false,throwOnError:false});$ .

Two-layer near-wall models

Motivation: Two-layer models avoid the need to fully resolve the viscous sublayer while improving accuracy over simple wall functions. Near the wall, $var element = document.getElementById("moose-equation-e8a186c9-e1c2-4e1a-9da2-1f02e283c2e0");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b8e34ca2-6252-420b-8fbc-4bd98748e2ea");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ are driven by empirical formulas; away from the wall, the standard or realizable $var element = document.getElementById("moose-equation-6b813289-cafd-4d8e-bb01-d2b0068a4329");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-ebf623d1-1bcf-47db-b1c5-f0e55dc3b879");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equations are used Wolfstein (1969) and NORRIS III (1975).
Implementation: (a) The wall distance $var element = document.getElementById("moose-equation-4752dc14-100c-44c4-9367-956e8fa072d4");katex.render("d", element, {displayMode:false,throwOnError:false});$ is obtained from WallDistanceAux, (b) twoLayerWolfstein, twoLayerNorrisReynolds, or twoLayerXu compute $var element = document.getElementById("moose-equation-70f86c27-e65b-4706-8852-9a34e79d0b71");katex.render("\\ell_\\epsilon", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-edf961bf-176d-4925-9680-9e9f588bb96f");katex.render("\\mu_t/\\mu", element, {displayMode:false,throwOnError:false});$ in the near-wall region, (c) A blending function transitions between near-wall and outer-layer behavior.
Activation: (a) k_epsilon_variant = 'StandardTwoLayer' or 'RealizableTwoLayer', (b) two_layer_flavor = 'Wolfstein' | 'NorrisReynolds' | 'Xu', (c) wall_distance auxiliary variable and consistent walls list.

Realizable formulation

Motivation: The classical $var element = document.getElementById("moose-equation-138597de-2567-4520-b6f1-d6b554b9b8c9");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-11b78081-391f-40ea-a6d4-b726d6a28bc3");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ model can violate realizability constraints (e.g., negative normal stresses). The realizable formulation introduces a new expression for $var element = document.getElementById("moose-equation-88cf61d0-ca39-4326-ae01-80a7ad12d7d9");katex.render("C_\\mu", element, {displayMode:false,throwOnError:false});$ and a modified $var element = document.getElementById("moose-equation-9910171a-8d33-400e-b436-60e5d4b575d8");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation to enforce these constraints, improving predictions in rotating and separating flows Shih et al. (1995).
Implementation: (a) Cmu_realizable provides $var element = document.getElementById("moose-equation-20173045-083d-4666-87dd-33a0489e3be0");katex.render("C_\\mu^\\text{real}", element, {displayMode:false,throwOnError:false});$ based on $var element = document.getElementById("moose-equation-2656b213-7168-4444-9438-ba813041f389");katex.render("S^2", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-44b1ae47-e473-40d9-9240-430de2ca1e0e");katex.render("W^2", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-2388aa8f-2e2b-4841-9970-c11364fbbd79");katex.render("k", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-49438f97-c362-446a-8b27-4958c898138e");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ . (b) Optional low-Re damping via f2_RKE may be applied.
Activation: k_epsilon_variant = 'Realizable' or 'RealizableTwoLayer'.

Nonlinear eddy-viscosity model

Motivation: Linear eddy-viscosity models cannot reproduce some important features of anisotropic turbulence (secondary flows, normal stress differences) Pope (2000). Nonlinear constitutive relations add higher-order dependencies on $var element = document.getElementById("moose-equation-ae7b7f64-db28-4fb0-a3c7-5eaed6508bb4");katex.render("\\mathbf{S}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-66939500-3e79-42fa-be2e-63eaf8576d79");katex.render("\\mathbf{W}", element, {displayMode:false,throwOnError:false});$ .
Implementation: (a) computeTRANS_NL constructs nonlinear tensor bases, (b) computeGnl computes additional production $var element = document.getElementById("moose-equation-37315f62-ca72-4581-b73b-426be8231f62");katex.render("G_\\text{nl}", element, {displayMode:false,throwOnError:false});$ , (c) Cmu_nonlinear keeps $var element = document.getElementById("moose-equation-2573de93-a70c-4fcb-96cc-0b710b0dd9e5");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ consistent with the nonlinear stress model.
Activation: nonlinear_model = 'quadratic' or 'cubic' (in the $var element = document.getElementById("moose-equation-a024185b-7208-4985-92d1-2af86207918a");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-db416815-2cd3-4ea9-9d9f-4161f299b394");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ kernels).

Yap correction

Motivation: In separated and impinging flows, classical $var element = document.getElementById("moose-equation-40771c71-9222-44ae-a0c8-da886bd20bba");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-14c83f95-7663-42cd-9080-3f79e28e9b33");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ models may overpredict the turbulence length scale, leading to excessive eddy viscosity and overly diffused separation zones. Yap’s correction adds an extra sink term for $var element = document.getElementById("moose-equation-c09b3d63-3e8e-47ed-bc50-dfa8a084feed");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ to limit the growth of the length scale Yap (1987).
Implementation: (a) computeGammaY computes $var element = document.getElementById("moose-equation-f94bbf87-fd02-4c3e-88e1-539b63d9c2a0");katex.render("\\gamma_Y", element, {displayMode:false,throwOnError:false});$ based on $var element = document.getElementById("moose-equation-126f8e6d-2856-4a2b-8796-1352b30c3497");katex.render("\\ell", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-95fc204f-16a5-485c-8b25-454931261c5d");katex.render("\\ell_\\epsilon", element, {displayMode:false,throwOnError:false});$ , and model constants, (b) $var element = document.getElementById("moose-equation-925b60f3-325a-4847-be4c-95175bf1daa6");katex.render("\\gamma_Y", element, {displayMode:false,throwOnError:false});$ is added to $var element = document.getElementById("moose-equation-b418e2a6-fb5a-4056-bdf0-d92ec2966d48");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ in kEpsilonTKEDSourceSink when enabled.
Activation: (a) use_yap = true, (b) consistent wall-distance and two-layer settings if applicable.

Buoyancy production

Motivation: In natural or mixed convection, buoyancy can generate or suppress turbulence, especially in strongly stratified regions Wilcox (2006).
Implementation: (a) computeGb provides $var element = document.getElementById("moose-equation-df739106-6447-482f-bc91-a88a54fcc6b3");katex.render("G_b", element, {displayMode:false,throwOnError:false});$ using gravity, temperature gradient, and expansion coefficient $var element = document.getElementById("moose-equation-21d9872b-26b8-412f-9f2b-ddba71ee53fb");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ , (b) $var element = document.getElementById("moose-equation-33a0fb51-341f-424a-ad94-197ce495b8fd");katex.render("G_b", element, {displayMode:false,throwOnError:false});$ is added to the $var element = document.getElementById("moose-equation-5f28b9af-7406-4628-b558-9e74b07a0418");katex.render("k", element, {displayMode:false,throwOnError:false});$ equation and appears in the $var element = document.getElementById("moose-equation-9006c20d-e10a-414f-8514-edc6af693c0f");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation multiplied by $var element = document.getElementById("moose-equation-387e4352-d87e-4796-8d5f-39dad1f60a41");katex.render("C_{3\\epsilon}", element, {displayMode:false,throwOnError:false});$ .
Activation: (a) use_buoyancy = true, (b) gravity, T_fluid, beta, and Pr_t set appropriately.

Compressibility correction

Motivation: At high Mach numbers, dilatational effects can alter the turbulence structure and time scales. Compressibility corrections add an extra term to the $var element = document.getElementById("moose-equation-ee90d0c0-abef-463c-b0dc-f7f8787dd33d");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equation to account for these effects Wilcox (2006).
Implementation: (a) computeGammaM computes $var element = document.getElementById("moose-equation-5de594fa-8dd4-4ccd-be96-fe54097bbbf6");katex.render("\\gamma_M", element, {displayMode:false,throwOnError:false});$ based on $var element = document.getElementById("moose-equation-ce7a8dec-eed0-47d2-96df-33c4b6ce60e8");katex.render("k", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-fea314d5-9dd0-4814-9cd5-e36fc27608c1");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ , and local speed of sound, (b) $var element = document.getElementById("moose-equation-9699eeb7-3b97-4211-b54b-b26fbc01851c");katex.render("\\gamma_M", element, {displayMode:false,throwOnError:false});$ is added to $var element = document.getElementById("moose-equation-d7f710ba-648a-4f10-837c-a3bb0f8e04fd");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ when use_compressibility = true.
Activation: (a) use_compressibility = true, (b) a speed_of_sound functor is provided and consistent with the equation of state.

Curvature / rotation correction

Motivation: Simple eddy-viscosity models tend to mispredict turbulence levels in rotating channels, swirling jets, and flows over curved surfaces. Spalart and Shur proposed a correction factor $var element = document.getElementById("moose-equation-a7d38003-f86a-4d4e-96fa-9147e6698a93");katex.render("f_c", element, {displayMode:false,throwOnError:false});$ that modifies production based on curvature/rotation Spalart and Shur (1997).
Implementation: (a) computeCurvatureFactor uses $var element = document.getElementById("moose-equation-5f76397d-73e5-4e5f-a369-3d1cf8da3716");katex.render("S^2", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-4be48c51-4808-419f-acd7-8902956bec95");katex.render("W^2", element, {displayMode:false,throwOnError:false});$ to build $var element = document.getElementById("moose-equation-6ca11d37-bac2-4806-8325-6a6b202d7b83");katex.render("f_c", element, {displayMode:false,throwOnError:false});$ , (b) $var element = document.getElementById("moose-equation-b69233ce-7f54-4beb-a1f1-1a2093a59eda");katex.render("P_k", element, {displayMode:false,throwOnError:false});$ is replaced by $var element = document.getElementById("moose-equation-364a4f7c-ea87-4ec3-a77d-7e1dc8ad5366");katex.render("f_c P_k", element, {displayMode:false,throwOnError:false});$ in both the $var element = document.getElementById("moose-equation-4b91f52c-1146-480b-ab6e-78c14dd36a1b");katex.render("k", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-e93caab7-5207-4698-a41c-2ebdefb90452");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ equations when curvature corrections are enabled.
Activation: (a) use_curvature_correction = true, (b) curvature_model = 'standard' (or similar option as implemented).

Practical guidance for using each model

Summary table

The table below provides a practical summary for choosing a $var element = document.getElementById("moose-equation-6fadd336-a9f2-4464-be89-efa1e8f37136");katex.render("k", element, {displayMode:false,throwOnError:false});$ – $var element = document.getElementById("moose-equation-1f7f9da7-faad-40b8-b0d8-60c08fc515ed");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ variant in OpenPronghorn.

Variant	Near-wall treatment	Typical first-cell $var element = document.getElementById("moose-equation-7d511eb8-9e7e-464c-8694-cbb8bb72a852");katex.render("y^+", element, {displayMode:false,throwOnError:false});$	Mesh requirements	Typical domain of application	Example applications (OpenPronghorn context)	Comments
`Standard`	High-Re, wall functions	$var element = document.getElementById("moose-equation-9e0e4e84-0b91-42a6-9473-7f73c9ea280c");katex.render("30 \\lesssim y^+ \\lesssim 300", element, {displayMode:false,throwOnError:false});$	Coarse first cell in log layer; few cells across BL	Fully turbulent attached flows, mild pressure gradients	Straight channels, simple pipe flows, external bluff body with mild separation	Fast and robust; good initial choice for simple internal flows.
`StandardLowRe`	Low-Re damping ( $var element = document.getElementById("moose-equation-a6ee9237-51d1-45f7-9388-5b8e3c7f5ab9");katex.render("f_\\mu", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-a6230eeb-4c05-4502-8992-5bbc8edf22d9");katex.render("f_2", element, {displayMode:false,throwOnError:false});$ )	$var element = document.getElementById("moose-equation-41e30e61-50b9-4bb7-9159-829d13479a6a");katex.render("y^+ \\approx 1", element, {displayMode:false,throwOnError:false});$	Strong wall clustering; many cells in viscous sublayer and buffer region	Heat-transfer-dominated internal flows, strong adverse pressure gradients	High-fidelity channel-flow benchmarks, heated rod bundles (resolved near wall)	More expensive; sensitive to near-wall mesh quality and wall-distance accuracy.
`StandardTwoLayer`	Two-layer (Wolfstein/NorrisReynolds/Xu)	$var element = document.getElementById("moose-equation-e3754b25-a581-47d1-8a7d-8df783df679e");katex.render("1 \\lesssim y^+ \\lesssim 100", element, {displayMode:false,throwOnError:false});$	Moderate wall clustering; smooth near-wall blending	Industrial internal flows, moderate separation, HVAC and ductwork	Reactor auxiliary systems, plenum mixing with moderate separation	Good compromise between cost and accuracy; pick `two_layer_flavor` based on target data.
`Realizable`	High-Re, wall functions	$var element = document.getElementById("moose-equation-578821a7-f869-48c9-b706-ef57d6d90f94");katex.render("30 \\lesssim y^+ \\lesssim 300", element, {displayMode:false,throwOnError:false});$	Similar to `Standard`	Rotating, swirling, and separating flows	Mixing tees, swirling jets, complex manifolds with recirculation	Often preferred to `Standard` when rotation or separation is important.
`RealizableTwoLayer`	Realizable + two-layer near-wall model	$var element = document.getElementById("moose-equation-3177237d-f530-4f4c-86e2-544073a8066a");katex.render("1 \\lesssim y^+ \\lesssim 100", element, {displayMode:false,throwOnError:false});$	Moderate wall clustering plus two-layer blending	General-purpose model for complex internal flows with heat transfer	Reactor primary loops with moderate mesh resolution near walls	Recommended default in many industrial-like configurations when mesh allows $var element = document.getElementById("moose-equation-a074ccc0-a60f-4eee-b63a-39911c926207");katex.render("y^+ \\lesssim 50", element, {displayMode:false,throwOnError:false});$ .

Additional configuration guidelines

Wall-distance field

Use WallDistanceAux to compute the distance to solid walls.
Ensure that the walls list is consistent across WallDistanceAux, kEpsilonTKESourceSink, kEpsilonTKEDSourceSink, and kEpsilonViscosity.

Coupling $var element = document.getElementById("moose-equation-30dd9b56-1abe-4d92-be78-03dcf1044b2a");katex.render("k", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-1bb484d9-e22d-4d00-b7de-6b4a1bf7b7b1");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-e11408aa-a38c-414d-9f7b-88709e31aee8");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$

Always include all three components: kEpsilonTKESourceSink (for $var element = document.getElementById("moose-equation-02b8bc9b-08e2-4644-9b73-277d2405c00b");katex.render("k", element, {displayMode:false,throwOnError:false});$ ), kEpsilonTKEDSourceSink (for $var element = document.getElementById("moose-equation-67ca0f28-5a08-4a0e-83a7-b7c72d05229e");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$ ), and kEpsilonViscosity (for $var element = document.getElementById("moose-equation-d5c92874-dab7-4679-9664-c7fd1981d17a");katex.render("\\mu_t", element, {displayMode:false,throwOnError:false});$ ).
The k_epsilon_variant parameter should be the same in all of them.

Incompressible vs compressible

For incompressible or low-Mach flows, set use_compressibility = false and omit speed_of_sound.
For compressible flows, provide a consistent speed_of_sound functor and consider tighter nonlinear and linear solver tolerances.

Buoyancy-driven and stratified flows

Pay attention to Pr_t, beta, and the reference temperature/state.
Post-process $var element = document.getElementById("moose-equation-d15552fa-28fc-4ab1-95d8-4afbceaeb76f");katex.render("P_k", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-43ed1fc8-12b6-4567-84cf-b1300f7e14b2");katex.render("G_b", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-7a767930-9c62-4092-81e6-6b8b99e746fd");katex.render("S_\\epsilon", element, {displayMode:false,throwOnError:false});$ terms where possible to verify the qualitative behavior of the model.

Enabling corrections judiciously

Nonlinear and curvature corrections add complexity and stiffness. Use them when there is clear physical motivation (strong curvature, swirl, or secondary flows) and when validation data are available.
Yap and low-Re corrections are valuable in separated flows and near-wall heat transfer, but may not be necessary in simple attached turbulence.

Validation and calibration

The default constants are consistent with widely used industrial models Launder and Spalding (1974) and Shih et al. (1995).
For critical design calculations, validate against experiments or high-fidelity simulations (DNS/LES) Pope (2000) and Wilcox (2006).
If any model coefficients are retuned, document these changes and the validation cases used.

References

Brian E. Launder and David B. Spalding. The numerical computation of turbulent flows. Computer Methods in Applied Mechanics and Engineering, 3(2):269–289, 1974.

@article{launder1974numerical,
    author = "Launder, Brian E. and Spalding, David B.",
    title = "The numerical computation of turbulent flows",
    journal = "Computer Methods in Applied Mechanics and Engineering",
    volume = "3",
    number = "2",
    pages = "269--289",
    year = "1974"
}

H LEE NORRIS III. Turbulent channel flow with a moving wavy boundary. Stanford University, 1975.

Stephen B. Pope. Turbulent Flows. Cambridge University Press, 2000.

Tsan-Hsing Shih, William W. Liou, Aamir Shabbir, Zhigang Yang, and Jiang Zhu. A new k-ε eddy-viscosity model for high reynolds number turbulent flows: model development and validation. Computers & Fluids, 24(3):227–238, 1995.

@article{shih1995realizable,
    author = "Shih, Tsan-Hsing and Liou, William W. and Shabbir, Aamir and Yang, Zhigang and Zhu, Jiang",
    title = "A new k-$\epsilon$ eddy-viscosity model for high Reynolds number turbulent flows: model development and validation",
    journal = "Computers \\& Fluids",
    volume = "24",
    number = "3",
    pages = "227--238",
    year = "1995"
}

P. R. Spalart and M. Shur. On the sensitization of turbulence models to rotation and curvature. Aerospace Science and Technology, 1(5):297–302, 1997.

@article{spalart1997rotation,
    author = "Spalart, P. R. and Shur, M.",
    title = "On the sensitization of turbulence models to rotation and curvature",
    journal = "Aerospace Science and Technology",
    volume = "1",
    number = "5",
    pages = "297--302",
    year = "1997"
}

David C. Wilcox. Turbulence Modeling for CFD. DCW Industries, 3 edition, 2006.

M. Wolfstein. The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and pressure gradient. International Journal of Heat and Mass Transfer, 12(3):301–318, 1969.

@article{wolfstein1969velocity,
    author = "Wolfstein, M.",
    title = "The velocity and temperature distribution in one-dimensional flow with turbulence augmentation and pressure gradient",
    journal = "International Journal of Heat and Mass Transfer",
    volume = "12",
    number = "3",
    pages = "301--318",
    year = "1969"
}

Wen Xu, Q Chen, and FTM Nieuwstadt. A new turbulence model for near-wall natural convection. International Journal of Heat and Mass Transfer, 41(21):3161–3176, 1998.

@article{xu1998new,
    author = "Xu, Wen and Chen, Q and Nieuwstadt, FTM",
    title = "A new turbulence model for near-wall natural convection",
    journal = "International Journal of Heat and Mass Transfer",
    volume = "41",
    number = "21",
    pages = "3161--3176",
    year = "1998"
}

Christopher R Yap. Turbulent heat and momentum transfer in recirculating and impinging flow. The University of Manchester (United Kingdom), 1987.

@book{yap1987correction,
    author = "Yap, Christopher R",
    title = "Turbulent heat and momentum transfer in recirculating and impinging flow",
    year = "1987",
    publisher = "The University of Manchester (United Kingdom)"
}

Introduction
Theory of RANS modeling
The – models implemented var element = document.getElementById("moose-equation-f59ab304-6ea0-4432-96bb-fc2bad7e3d64");katex.render("k", element, {displayMode:false,throwOnError:false}); var element = document.getElementById("moose-equation-bcb93ca4-fd6e-49d9-a3a7-d92775bb5793");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});
Corrections implemented in the – models var element = document.getElementById("moose-equation-b78b2c58-1de0-410c-8f66-1bdc41deec2a");katex.render("k", element, {displayMode:false,throwOnError:false}); var element = document.getElementById("moose-equation-2b016673-8748-4db4-aed1-60e4756e473f");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});
Practical guidance for using each model
References