Langley Research Center

Turbulence Modeling Resource

K-gamma 2-equation Transitional Model

Note: this model page was contributed by Jatinder Pal Singh Sandhu of IIT Madras, India.

This web page gives detailed information on the equations for the k-gamma two-equation turbulence+transition model. The model given on this page is a linear eddy viscosity model, which uses the Boussinesq assumption for the constitutive relation:

$\tau_{ij} = 2 \mu_t \left(S_{ij} - \frac{1}{3} \frac{\partial u_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

Unless otherwise stated, for compressible flow with heat transfer this model is implemented as described on the page Implementing Turbulence Models into the Compressible RANS Equations, with perfect gas assumed and Pr = 0.72, Pr_t = 0.90, and Sutherland's law for dynamic viscosity.

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K-gamma 2-equation Transition Model (K-gamma-2021)

The reference for the k-gamma two-equation turbulence/transition model is:

Sandhu, J. P. S., and Ghosh, S., "A local correlation-based zero-equation transition model," Computers & Fluids, Vol. 214, Jan 2021, p. 104758, https://doi.org/10.1016/j.compfluid.2020.104758.

The transport equations of the k-gamma model are similar to the SST-2003 model with a few additional source terms for the k transport equation. The model (written in conservation form) is given as:

$\frac{\partial(\rho k)}{\partial t}+\frac{\partial\left(\rho u_{j} k\right)}{\partial x_{j}}=P-\text{max}(\widetilde{\gamma},0.1)\beta^{*} \rho \omega k+\frac{\partial}{\partial x_{j}}\left[\left(\mu+\sigma_{k} \mu_{t}\right) \frac{\partial k}{\partial x_{j}}\right] + \Psi_{k_\gamma}$

$\frac{\partial(\rho \omega)}{\partial t}+\frac{\partial\left(\rho u_{j} \omega\right)}{\partial x_{j}}=\frac{\gamma}{\nu_{t}} P-\beta \rho \omega^{2}+\frac{\partial}{\partial x_{j}}\left[\left(\mu+\sigma_{\omega} \mu_{t}\right) \frac{\partial \omega}{\partial x_{j}}\right]+2\left(1-F_{1}\right) \frac{\rho \sigma_{\omega 2}}{\omega} \frac{\partial k}{\partial x_{j}} \frac{\partial \omega}{\partial x_{j}}$

where

$\Psi_{k_\gamma} = P_{k_\gamma}+P_{k_\gamma}^{lim}- E_{k_\gamma}+ D_{k_\gamma}$

$P_{k_\gamma} = F_{length} \phantom{I} \rho S (1-\widetilde{\gamma})F_{onset} k$

$E_{k_\gamma} = \widetilde{\gamma} C_{e1}\ \rho\Omega F_{turb} k$

$D_{k_\gamma} = -C_{d1}(\mu+\mu_t)\frac{\partial k}{\partial x_j}\frac{\partial \widetilde{\gamma}}{\partial x_j}$

$P_k^{lim} = 5\left[\text{max}(\widetilde{\gamma} - 0.2, 0)(1-\widetilde{\gamma})F_{on}^{lim}\text{max}(3\mu - \mu_t,0)S\Omega\right]$

Here,

$F_{on}^{lim} = \text{min}\left(\text{max}\left(\frac{Re_v}{2.2Re_{\theta c}^{lim}}-1, 0\right),3\right)$

and,

$Re_{\theta c}^{lim} = 1100$

The approximated intermittency $\widetilde{\gamma}$ is modeled as:

$\widetilde{\gamma} = \left(1-e^{-R_T}\right)^3$

and its gradient is computed as:

$\frac{\partial \widetilde{\gamma}}{\partial x_i} = \frac{n\rho\widetilde{\gamma}^{2/3}}{\mu \omega} \left(\frac{\partial k}{\partial x_i} - \frac{k}{\omega}\frac{\partial \omega}{\partial x_i}\right)e^{-R_T}$

where

$R_T = \frac{\rho k}{\omega \mu}$

is the turbulent Reynolds number. In order to obtain the gradient of approximated intermittency, the gradient of density and molecular viscosity have been ignored. The functions which control transition are given as:

$F_{turb} = e^{-\left(\frac{R_T}{2}\right)^4}$

$F_{onset} = \text{max}(F_{onset2} - F_{onset3}, 0.0)$

where

$F_{onset2} = \text{min}(F_{onset1}, 2.0)$

and

$F_{onset3} = \text{max}\left( 1 - \left(\frac{R_T}{3.5}\right)^3, 0\right)$

The term $F_{onset1}$ is defined as,

$F_{onset1} = \frac{Re_v}{2.2Re_{\theta c}}$

where

$Re_\text{v} = \frac{\rho d_w^2 S}{\mu}$

is the local strain-rate Reynolds number. Also, $Re_{\theta c}$ is calculated using the following correlation:

$Re_{\theta c} = 100.0 + 1000.0 ~exp(-Tu_L F_{PG})$

The effects of pressure gradient and local turbulence intensity are accounted for using the following functions:

$F_{PG} = \left\{ \begin{array}{ll} min(1 + 14.68\lambda_{\theta L}, 1.5) , &\quad\lambda_{\theta L} \geq 0\\ min(1 - 7.34\lambda_{\theta L}, 3.0) , &\quad\lambda_{\theta L} < 0 \end{array}\right.$

$Tu_L = min\left(100\frac{\sqrt{2k/3}}{\omega d_w}, 100\right)$

Here, $d_w$ is the distance to the nearest wall. The pressure gradient parameter, $\lambda_{\theta L}$ , is defined as

$\lambda_{\theta L} = -7.57. 10^{-3}\frac{d V}{dy}\frac{d_w^2}{\nu} + 0.0128$

The term $\frac{dV}{dy}$ can be computed as:

$\frac{dV}{dy} = \nabla (\vec{n}.\vec{V}).\vec{n}$

where

$\vec{n} = \frac{\nabla (d_w)}{|\nabla (d_w)|}$

The production term of the original transport equation is modified as

$P = \widetilde{\gamma} \mu_t S\Omega$

where the turbulent viscosity is defined as:

$\mu_{t}=\frac{\rho a_{1} k}{\max \left(a_{1} \omega, \Omega F_{2}\right)}$

and

$F_{2}=\tanh \left(\arg _{2}^{2}\right)$

where

$\arg _{2}=\max \left(2 \frac{\sqrt{k}}{\beta^{*} \omega d}, \frac{500 \nu}{d^{2} \omega}\right)$

The recommended production limiter is

$\min \left(P, 20 \beta^{*} \rho \omega k\right)$

In the SST model, the constants are a blend of an inner (1) and an outer (2) constant, blended via

$\phi=F_{1} \phi_{1}+\left(1-F_{1}\right) \phi_{2}$

The blending function F₁ is given as

$F_1 = max \left( F_{1,SST},F_3 \right)$

$F_{1,SST}=\tanh \left(a r g_{1}^{4}\right)$

where

$\arg _{1}=\min \left[\max \left(\frac{\sqrt{k}}{\beta^{*} \omega d}, \frac{500 \nu}{d^{2} \omega}\right), \frac{4 \rho \sigma_{\omega 2} k}{C D_{k \omega} d^{2}}\right]$

and

$F_3 = exp \left[ - \left( \frac{R_y}{120} \right)^8 \right]$

where

$R_y = \frac{\rho d \sqrt{k}}{\mu}$

The cross-diffusion term is defined as:

$C D_{k \omega}=\max \left(2 \rho \sigma_{\omega 2} \frac{1}{\omega} \frac{\partial k}{\partial x_{j}} \frac{\partial \omega}{\partial x_{j}}, 10^{-10}\right)$

The strain rate magnitude $S$ and vorticity magnitude $\Omega$ are defined as:

$S = \sqrt{2S_{ij}S_{ij}}$

and

$\Omega = \sqrt{2W_{ij}W_{ij}}$

where

$S_{ij}=\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$

and

$W_{ij}=\frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$

The model constants are:

$\begin{array}{l} F_{length} = 1.0\\[0.5cm] C_{e1} = 0.03\\[0.5cm] C_{d1} = 0.02 \end{array}$

The unchanged original SST-2003 model constants are:

$\gamma_{1}=\frac{\beta_{1}}{\beta^{*}}-\frac{\sigma_{\omega 1} \kappa^{2}}{\sqrt{\beta^{*}}}, \; \qquad \gamma_{2}=\frac{\beta_{2}}{\beta^{*}}-\frac{\sigma_{\omega 2} \kappa^{2}}{\sqrt{\beta^{*}}}$

$\[\arraycolsep=6pt\def\arraystretch{3.2} \begin{array}{lll} \sigma_{k 1}=0.85 &\quad \sigma_{\omega 1}=0.5 &\quad \beta_{1}=0.075 \\[0.5cm] \sigma_{k 2}=1.0 &\quad \sigma_{\omega 2}=0.856 &\quad \quad \beta_{2}=0.0828 \\[0.5cm] \beta^{*}=0.09 &\quad \kappa=0.41 &\quad a_{1}=0.31 \end{array}$

For boundary conditions, similar to the SST-2003 turbulence model, the following values are specified at a wall:

$k_{wall} = 0$

$\omega_{wall} = 10 \frac{6 \nu}{\beta_1 (\Delta d_1)^2}$

The freestream values of k and omega should be set according to the desired (or specified) freestream levels of Tu and $\mu_t$ , using

$Tu_{freestream}\% = 100 \sqrt{\frac{2}{3} \frac{k_{freestream}}{U_{ref}^2}}$

$\mu_{t, freestream} = \rho_{freestream}\frac{k_{freestream}}{\omega_{freestream}}$

A point to note is that although the use of turbulent Reynolds number (R_T) in approximated intermittency is an obvious choice, it makes the model sensitive to initial conditions for low freestream turbulence intensity cases. The model is calibrated for freestream initial conditions.

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Last Updated: 11/08/2021