Langley Research Center

Turbulence Modeling Resource

WA-gamma 2-equation Transitional Model

Note: this model page was contributed by Ramesh Agarwal and Jonathan Richter of Washington University in St. Louis.

This web page gives detailed information on the equations for the WA-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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Wray-Agarwal-gamma 2-equation Transition Model (WA-gamma)

The Wray-Agarwal (WA) model is a one-equation eddy-viscosity model derived from the $k-\omega$ closure. An important distinction between the WA model and previous one-equation models based on $k-\omega$ closure is the inclusion of the cross diffusion term in the $R = k/\omega$ equation and a blending function, which allows smooth switching between the two destruction terms. The model determines $R = k/\omega$ by a transport equation. However, this model alone cannot model the transition and is therefore modified to include a correlation based intermittency equation for $\gamma$ , employing the local correlation-based transition-modeling concept. In this respect, the modeling philosophy behind the two equation WA-gamma model is similar to that of the four equation Shear-Stress Transport (SST) transition model SST-2003-LM2009. The source code for the WA-gamma model is available at https://github.com/Ada810.

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

Nagapetyan, H. and Agarwal, R. K., "Development of a New Transitional Flow Model Integrating the Wray-Agarwal Turbulence Model with an Intermittency Transport Equation," AIAA Paper 2018-3384, June 2018, https://doi.org/10.2514/6.2018-3384.

The transport equations of the WA-gamma model (written in conservation form) are:

$\frac{\partial \rho R}{\partial t}+\frac{\partial \rho u_{j}R}{\partial x_{j}}=\frac{\partial }{\partial x_{j}}\lbrack (\mu +\sigma _{R}\mu _{T})\frac{\partial R}{\partial x_{j}}\rbrack +\gamma \rho C_{1}RS+\rho f_{1}C_{2k\omega }\frac{\partial R}{\partial x_{j}}\frac{\partial S}{\partial x_{j}}\frac{R}{S}+ P^{\lim }_{R} \\ -(1-f_{1})\rho C_{2k\varepsilon }min(\frac{R^{2}}{S^{2}}\frac{\partial S}{\partial x_{j}}\frac{\partial S}{\partial x_{j}}, C_m\frac{\partial R}{\partial x_{j}}\frac{\partial R}{\partial x_{j}})$

$\frac{\partial \rho \gamma }{\partial t}+\frac{\partial \rho u_{j}\gamma }{\partial x_{j}}=\frac{\partial }{\partial x_{j}}\lbrack (\mu +\frac{\mu _{T}}{\sigma _{\gamma }})\frac{\partial \gamma }{\partial x_{j}}\rbrack +F_{length}\rho S\gamma (1-\gamma )F_{onset} \\ -\rho c_{a2}\Omega \gamma F_{turb}(c_{e2}\gamma -1)$

In the wall-distance-free Wray-Agarwal One-Equation Model, the eddy viscosity is given by:

$\nu_{T} = f_{\mu}R$

The term $P_{R}^{\lim}$ is used to ensure proper generation of R for very low values of turbulent intensity Tu:

$P_{R}^{\lim }=1.5W\max (\gamma -0.2, 0)(1.0-\gamma )\min (\max (\frac{Re_{v}}{2420}-1, 0), 3)max(3\nu- \nu_{T}, 0)$

F_onset is used to trigger the intermittency production and is a function of $R_{T}$ , $Re_{v}$ , and $Re_{\theta c}$ as given in the following equations:

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

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

$F_{onset3}=\max (1-(\frac{R_{T}}{3.5})^{3}, 0)$

$F_{onset}=\max (F_{onset2}-F_{onset3},0)$

$F_{turb}=e^{-(\frac{R_{T}}{2})}$

$R_{T}=\frac{\mu_{t}}{\mu}$

$Re_{v}=\frac{\rho d_{w}^{2}S}{\mu}$

The model constants for the intermittency equation are as follows:

$F_{length}=100$

$c_{e2}=50$

$c_{a2}=0.06$

$\sigma_{\gamma}=1.0$

The local turbulence intensity $Tu_{L}$ is given by:

$Tu_{L}=min(100 \frac{\sqrt{\frac{2R}{3}}}{\sqrt{\frac{S}{0.3}}d_{w}}, 100)$

where $d_w$ is the wall distance. In the original formulation of $Tu_{L}$ (from Menter, F., Smirnov, P., Liu, T., and Avancha, R., "A One-Equation Local Correlation-Based Transition Model," Flow, Turbulence and Combustion, Vol. 5, No. 4, 2015, pp. 583-619 (https://doi.org/10.1007/s10494-015-9622-4), R replaces turbulent kinetic energy k (note that $R = k/\omega$ ) and $\omega$ in the original formulation is replaced by $\omega$ \approx S/0.3$ .

The formula for the pressure gradient parameter can be written as

$\lambda _{\theta L}=-7.57\cdot 10^{-3}\frac{dV}{dy}\frac{d_{w}^{2}}{\nu }+0.0128$

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 $\lambda_{\theta L}$ term is bounded by $-1.0 \leq \lambda _{\theta L} \leq 1.0$ for numerical robustness. The $Re_{\theta c}$ correlation is given by:

$Re_{\theta c}=100.0+1000.0exp[-1.0 \cdot Tu_{L} \cdot F_{PG}]$

where $F_{PG}$ is a correlation function of $\lambda_{\theta L}$ :

$F_{PG}= \left\{ \begin{array}{ll} \min (1+C_{PG1}\lambda _{\theta L}, C_{PG1}^{\lim }), & \lambda _{\theta L}\ge 0; \\ \min (1+C_{PG2}\lambda _{\theta L}+C_{PG3}min[\lambda _{\theta L}+0.0681, 0], C_{PG2}^{\lim }), & \lambda _{\theta L}<0 \end{array} \right.$

$C_{PG1}=14.68$

$C_{PG2}=-7.34$

$C_{PG3}=0.0$

$C_{PG1}^{\lim }=1.5$

$C_{PG2}^{\lim }=3.0$

$F_{PG}$ is limited in order to avoid negative values:

$F_{PG}=max(F_{PG}, 0)$

The wall blocking effect is accounted for by the damping function

$f_{\mu }=\frac{\chi ^{3}}{\chi ^{3}+C_{w}^{3}}$

$\chi =\frac{R}{\nu }$

S is the mean strain given by

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

$S_{ij}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}})$

W is the mean vorticity given by

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

$W_{ij}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}-\frac{\partial u_{j}}{\partial x_{i}})$

While the $C_{2k\omega}$ term is active, the transport equation for R behaves as a one-equation model based on the standard $k-\omega$ equations. The inclusion of the cross diffusion term in the derivation causes the additional $C_{2k\epsilon}$ term to appear. This term corresponds to the destruction term in one equation models derived from the standard $k-\epsilon$ closure. The presence of both terms allows the WA model to behave as either a one-equation $k-\omega$ or one equation $k-\epsilon$ model based on the switching function $f_1$ The blending function was designed so that the $k-\omega$ destruction term is active near solid boundaries and the $k-\epsilon$ destruction term becomes active away from the wall near the end of the log-layer. This function was modified from the original Wray-Agarwal model to remove its dependence on the wall distance. The following equations describe the formulation of $f_1$ for the wall-distance-free WA model WA-2018 (AIAA-2018-0593).

$f_{1}=\text{tanh}(\text{arg}_{1}^{4})$

$\text{arg}_{1}=\frac{R+v}{2}\cdot\frac{\eta ^{2}}{C_{\mu }k\omega }$

$k=\frac{v_{T}S}{\sqrt{C_{\mu }}}$

$\omega =\frac{S}{\sqrt{C_{\mu }}}$

$\eta =S\max (1, \vert \frac{W}{S}\vert )$

Other model constants and equations are:

$C_{1}=f_{1}(C_{1k\omega }-C_{1k\epsilon })+C_{1k\omega }$

$C_{2k_{\epsilon}}=\frac{C_{1k\epsilon }}{\kappa ^{2}}+ \sigma _{k\epsilon }$

$C_{2k\omega }=\frac{C_{1k\omega }}{\kappa ^{2}}+ \sigma _{k\omega }$

$C_{1k\omega }=0.0829$

$C_{1k\epsilon }=0.1284$

$\sigma _{R}= f_{1}(\sigma _{k\omega }- \sigma _{k\epsilon })+\sigma_{k\epsilon }$

$\sigma _{k\omega }=0.72$

$\sigma _{k\epsilon }=1.0$

$\kappa =0.41$

$C_{\omega }=8.54$

$C_{m}=8.0$

Boundary conditions are not explicitly described in the above reference. However, the authors recommend the following BCs at walls:

$R_{wall} = 0$

$\frac{\partial \gamma}{\partial n} \vert _{wall} = 0$

and the following BCs at the freestream:

$R_{farfield}=\nu_{T,farfield}$

$\gamma_{farfield} = 1$

where the farfield $\nu_T$ is either known from experiment, or assumed.

(Alternately, one could specify

$R_{farfield}=k_{farfield}/\omega_{farfield}$

where $k_{farfield}$ comes from the freestream turbulence intensity

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

and $\omega_{farfield}$ can be computed from known experimental quantities, or assumed.)

Wray-Agarwal-gamma 2-equation Transition Model for Rough Walls (WA-gamma-rough)

The reference for the WA-gamma two-equation turbulence/transition model for rough walls is:

Shuai, S. and Agarwal, R. K., "Modification of Wray-Agarwal Transition Model for Computing Rough Wall Flows," AIAA Paper 2019-0333, January 2019, https://doi.org/10.2514/6.2019-0333.

The roughness creates a shift in the log layer of the turbulent boundary layer. To account for this shift, the value of $d_w$ in various correlations used in the intermittency transport equation is replaced with $d_{new}$ :

$d_{new}=d_{w}+0.03k_{s}$

where $k_s$ is the equivalent sand-grain roughness height.

The viscous damping must also be modified to match the viscous sublayer and buffer layer profiles in the presence of surface roughness. Thus the equation for $f_{\mu}$ is replaced by

$f_{\mu }=\frac{\chi ^{3}}{\chi ^{3}+C_{w}^{3}}$

$\chi =\frac{R}{\nu} +Cr_{0}\frac{k_{s}}{d_{new}}$

where C_r0 = 0.4.

Furthermore, it turns out that the modification of the viscous damping term above does not provide a large enough increase in the eddy viscosity near the wall, especially for high roughness values. To further increase the eddy viscosity, the destruction coefficient $C_{2k\omega}$ is replaced by:

$C_{2k\omega }=(\frac{C_{1k\omega }}{\kappa ^{2}}+\sigma _{k\omega })(\frac{1}{1+\frac{Cr_{1}k_{s}}{d_{new}}})$

with C_r1 = 0.006.

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Last Updated: 03/24/2021