Langley Research Center

Turbulence Modeling Resource

One-Equation Wray-Agarwal Algebraic Transitional Model

This web page gives detailed information on the equations for the WA-AT transitional turbulence model. This model is a linear eddy viscosity model. Linear models use the Boussinesq assumption for the constitutive relation:

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

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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One-Equation Wray-Agarwal Algebraic Transitional Model (WA-AT)

This transition model is based on the WA-2018 one-equation model. Without any modification, WA-2018 alone cannot predict transition from laminar flow to turbulent flow. Following the work of the SA-BCM model by Cakmakcioglu et al, the WA-2018 turbulence model is coupled with an algebraic turbulence intermittency $\gamma$ equation to provide the capability to predict transition. (However, note that there are differences in some of the calibrated terms between SA-BCM and WA-AT.)

The reference is:

Xue, Y., Wen, T., and Agarwal, R. K., "Development of a New Transitional Flow Model Integrating the One-Equation Wray-Agarwal Turbulence Model with an Algebraic Intermittency Transport Term," AIAA Paper 2021-2712, August 2021, https://doi.org/10.2514/6.2021-2712.

The baseline WA-2018 model is modified to include the $\gamma$ term through multiplication with the kinetic energy production term $C_1RS$ :

$\frac{\partial R}{\partial t} + \frac{\partial u_j R}{\partial x_j} = \frac{\partial}{\partial x_j} \left[(\sigma_{{}_R} R+\nu)\frac{\partial R}{\partial x_j}\right]+C_1\gamma RS + f_1 C_{2kw}\frac{R}{S}\frac{\partial R}{\partial x_j}\frac{\partial S}{\partial x_j} -(1-f_1)\min\left[C_{2kw}R^2\left(\frac{\frac{\partial S}{\partial x_j}\frac{\partial S}{\partial x_j}}{S^2}\right), C_m\frac{\partial R}{\partial x_j}\frac{\partial R}{\partial x_j}\right]$

The value of $\gamma$ is 0 in laminar flow, and 1 in fully turbulent flow. In WA-2018, the eddy viscosity is given by:

$\nu_t=f_\mu R$

The intermittency term $\gamma$ is formulated as:

$\gamma=1-\exp\left(-\sqrt{Term_1}-\sqrt{Term_2}\right)$

Here $Term_1$ is designated to trigger the transition location, and $Term_2$ helps the intermittency to penetrate into the boundary layer. $Term_1$ is given by:

$Term_1=\frac{\max(1.2 R_{e_\theta}-R_{e_{\theta c}},0.0)}{\chi_1 R_{e_{\theta c}}}$

where

$R_{e_\theta}=\frac{R_{e_v}}{2.193}$

$R_{e_v}=\frac{\rho d^2}{\mu}\Omega$

and d is the wall distance. The local turbulence intensity is set to a constant value, based solely on freestream turbulence intensity:

$R_{e_{\theta c}}=803.73(Tu_\infty + 0.6067)^{-1.027}$

$Term_2$ is given by:

$Term_2=\max\left(\frac{\nu_t}{\nu}\chi_2, 0.0\right)$

The $chi_1$ and $chi_2$ are calibrated constants given by: $\chi_1 = 0.02$ and $\chi_2 = 50$ .

The damping function is designed to account for wall blocking effect. It is given by:

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

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

where $\nu$ is the kinematic viscosity and $R = k/\omega$ . S and W are the mean strain rate and mean vorticity, given by:

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

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

$S_{ij}=\frac 12\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$

$W_{ij}=\frac 12\left(\frac{\partial u_i}{\partial x_j}-\frac{\partial u_j}{\partial x_i}\right)$

This model combines the features of standard $k-\omega$ and $k-\epsilon$ models. The switching function $f_1$ triggers the switch:

$f_1=\tanh(arg^4_1)$

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

where

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

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

$\eta=S \max\left(1,\left|\frac WS\right|\right)$

The model constants are the same as in WA-2018:

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

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

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

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

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

$\kappa = 0.41$ $C_w = 8.54$

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

Boundary conditions at solid smooth walls are:

$R_{wall} = 0$

and for the freestream, the authors recommend:

$R_{farfield} = 0.002 \nu_{\infty}$

(The influence of $Tu_\infty$ comes in through $R_{e_{\theta c}}$ , and not through $R_{farfield}$ .)

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Ramesh Agarwal and his students Y. Xue and T. Wen are acknowledged for helping with this webpage.

Recent significant updates:
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Last Updated: 01/17/2023