Langley Research Center

Turbulence Modeling Resource

The Wray-Agarwal Turbulence Model

This web page gives detailed information on the equations for the Wray-Agarwal one-equation turbulence model. It is a linear eddy viscosity models. Linear models use the Boussinesq assumption for the constitutive relation:

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

where the last term is generally ignored for one-equation models like this one because k is not readily available (the term is sometimes ignored for non-supersonic speed flows for other models as well).

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 One-Equation Model (WA-2018)

The WA one-equation model was derived from a k-omega closure, and has gone through several iterations to date. The original Journal reference for it is: Wray, T. J., Agarwal, R. K., "Low-Reynolds-Number One-Equation Turbulence Model Based on a k-omega Closure," AIAA Journal, Vol. 53, No. 8, 2015, pp. 2216-2227, https://doi.org/10.2514/1.J053632 (which had an inconsistency in dimensional units in its arg₁ equation). However, the latest version (as recommended by the authors) is wall-distance free and varies from its description in the Journal. It is described in:

Han, X., Rahman, M. M., Agarwal, R. K., "Development and Application of a Wall Distance Free Wray-Agarwal Turbulence Model (WA2018)," AIAA Paper 2018-0593, January 2018, https://doi.org/10.2514/6.2018-0593.

The model solves for the variable R, using the following equation:

$\frac{\partial R}{\partial t} + \frac{\partial u_j R}{\partial x_j} = \frac{\partial}{\partial x_j} \left[(\sigma_R R + \nu) \frac{\partial R}{\partial x_j} \right] +C_1RS + f_1 C_{2k\omega} \frac{R}{S} \frac{\partial R}{\partial x_j} \frac{\partial S}{\partial x_j} -(1-f_1)\rm{min} \left[ C_{2k\epsilon}R^2 \left(\frac{\frac{\partial S}{\partial x_j} \frac{\partial S}{\partial x_j}}{S^2} \right), C_m\frac{\partial R}{\partial x_j}\frac{\partial R}{\partial x_j} \right]$

The turbulent eddy viscosity is:

$\mu_t = \rho f_{\mu} R$

with $\rho$ the density. S takes on the usual definition for mean strain:

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

The authors used $S = {\rm max}(S,10^{-16} s^{-1})$ to avoid division by zero (private communication). Wall blocking is accounted for by the damping function:

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

where $\chi = R/ \nu$ and $\nu = \mu/ \rho$ . The wall-distance-free switching function is:

$f_1 = \rm{tanh} \left(arg_1^4 \right)$

$arg_1 = \frac{\nu + R}{2} \frac{\eta^2}{C_{\mu} k \omega}$

where

$k = \frac{\nu_t S}{\sqrt{C_{\mu}}}$

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

$\eta = S \rm{max} \left( 1, \left| \frac{W}{S}\right| \right)$

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

The constants are:

$C_{1k\omega}=0.0829$ $C_{1k\epsilon}=0.1284$

$C_1 = f_1(C_{1k\omega} - C_{1k\epsilon}) + C_{1k\epsilon}$

$\sigma_{k \omega} = 0.72$ $\sigma_{k \epsilon} = 1.0$

$\sigma_R = f_1(\sigma_{k \omega} - \sigma_{k \epsilon}) + \sigma_{k \epsilon}$

$C_{2k\omega} = \frac{C_{1k\omega}}{\kappa^2} + \sigma_{k \omega}$ $C_{2k\epsilon} = \frac{C_{1k\epsilon}}{\kappa^2} + \sigma_{k \epsilon}$

$\kappa = 0.41$ $C_w = 8.54$

$C_{\mu} = 0.09$ $C_m = 8.0$

Boundary conditions at solid smooth walls are:

$R_{wall} = 0$

and for the freestream, the authors recommend:

$R_{farfield} = 3 \nu_{\infty} : to : 5 \nu_{\infty}$

Wray-Agarwal One-Equation Model (WA-2017)

The 2017 version of the WA model is described in:

Han, X., Wray, T. J., Agarwal, R. K., "Application of a New DES Model Based on Wray-Agarwal Turbulence Model for Simulation of Wall-Bounded Flows with Separation," AIAA Paper 2017-3966, June 2017, https://doi.org/10.2514/6.2017-3966.

Note that there was a typo in eq (6) of the original version of the above paper, but the online version has been corrected as of 8/10/2017. Note also that although the above reference includes a detached eddy simulation (DES) form of the model, here we only describe the RANS form of the model. This version of the model is different from the (WA-2018) model in the following ways: (1) one of the terms in the R-equation is different, (2) the f₁ is different, and (3) there are a few new/changed constants. For completeness, the WA-2017 model is given here in its entirety.

The model solves for the variable R, using the following equation:

$\frac{\partial R}{\partial t} + \frac{\partial u_j R}{\partial x_j} = \frac{\partial}{\partial x_j} \left[(\sigma_R R + \nu) \frac{\partial R}{\partial x_j} \right] +C_1RS + f_1 C_{2k\omega} \frac{R}{S} \frac{\partial R}{\partial x_j} \frac{\partial S}{\partial x_j} -(1-f_1)C_{2k\epsilon}R^2 \left(\frac{\frac{\partial S}{\partial x_j} \frac{\partial S}{\partial x_j}}{S^2} \right)$

This equation is the same as the (WA-2018) model, except for the last term. The turbulent eddy viscosity is:

$\mu_t = \rho f_{\mu} R$

with $\rho$ the density. S takes on the usual definition for mean strain:

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

The authors used $S = {\rm max}(S,10^{-16} s^{-1})$ to avoid division by zero (private communication). Wall blocking is accounted for by the damping function: