# Wray-Agarwal(WA) Turbulence Model

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## Introduction

The WA 2017m model is a one-equation model that was derived from two-equation $k-\omega$ closure. It combines the most desirable characteristics of the one-equation $k-\epsilon$ model and one-equation $k-\omega$ model, analogous to the SST $k-\omega$ model which combines best features of two-equation $k-\epsilon$ and $k-\omega$ models.

## WA Model

The turbulent eddy viscosity is given by:

${{\mu }_{t}}=\rho {{f}_{\mu }}R$

The model solves for the variable $R (= k / \omega)$ using the following equation:

$\begin{matrix} \frac{\partial R}{\partial t}+\frac{\partial u_{j} R}{\partial x_{j}}=& \frac{\partial}{\partial x_{j}}\left[\left(\sigma_{R} R+v\right) \frac{\partial R}{\partial x_{j}}\right]+C_{1} R S+f_{1} C_{2 k \omega} \frac{R}{S} \frac{\partial R}{\partial x_{j}} \frac{\partial S}{\partial x_{j}}\\ &-\left(1-f_{1}\right) \min \left[C_{2 k \varepsilon} R^{2}\left(\frac{\frac{\partial S}{\partial x_{j}} \frac{\partial S}{\partial x_{j}}}{S^{2}}\right), C_{m} \frac{\partial R}{\partial x_{j}} \frac{\partial R}{\partial x_{j}}\right] \end{matrix}$

Where:

$\begin{matrix} S=~\sqrt{2{{S}_{ij}}{{S}_{ij}}}~,~~{{S}_{ij}}=\frac{1}{2}\left( \frac{\partial {{u}_{i}}}{\partial {{x}_{j}}}+\frac{\partial {{u}_{j}}}{\partial {{x}_{i}}} \right) \end{matrix}$
$\begin{matrix} {{f}_{\mu }}=\frac{{{\chi }^{3}}}{{{\chi }^{3}}+C_{w}^{3}},~~\chi =\frac{R}{\nu },~~\nu =\frac{\mu }{\rho } \end{matrix}$

$\begin{matrix} {{f}_{1}}=\min \left( \tanh \left( arg_{1}^{4} \right),0.9 \right),~~ar{{g}_{1}}=\frac{1+\frac{d\sqrt{RS}}{\nu }}{1+{{\left[ \frac{max\left( d\sqrt{RS},1.5R \right)}{20\nu } \right]}^{2}}} \end{matrix}$

and d is the minimum distance to the nearest wall. The model constants are:

${{C}_{1k\omega }}=0.0829$
${{C}_{1k\varepsilon }}=0.1127$
${{C}_{1}}={{f}_{1}}\left( {{C}_{1k\omega }}-{{C}_{1k\varepsilon }} \right)+{{C}_{1k\varepsilon }}$
${{\sigma }_{k\omega }}=0.72$
${{\sigma }_{k\varepsilon }}=1.0$
${{C}_{2k\omega }}=\frac{{{C}_{1k\omega }}}{{{\kappa }^{2}}}+{{\sigma }_{kw}}$
${{C}_{2k\varepsilon }}=\frac{{{C}_{1k\varepsilon }}}{{{\kappa }^{2}}}+{{\sigma }_{k\varepsilon }}$
$\kappa =0.41$
${{C}_{w}}=8.54$
$C_{m} = 8.0$

## Boundary Conditions

Solid smooth wall:

${{R}_{wall}}=0$

Freestream:

${{R}_{farfield}}=3{{v}_{\infty }}:to:5{{v}_{\infty }}$

## References

• X. Han, T. J. Wray, and R. K. Agarwal. (2017), "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.