fist simulation: RAS turbulence model

The challenge for using simplified Navier Stokes equations is the new generated term: the correlation \(\overline{u_i^{\prime} u_j^{\prime}}\). Spalart and Allmaras (Spalart and Allmaras 1993) proposed that the above term should be governed by \(\overline{u_i u_j}=-2 \nu_t S_{i j}\). Following is what it looks:

Kinematic Eddy Viscosity: \[ \nu_T=\tilde{\nu} f_{v 1} \]

Eddy Viscosity Equation:

\begin{aligned} & \frac{\partial \tilde{\nu}}{\partial t}+U_j \frac{\partial \tilde{\nu}}{\partial x_j}=c_{b 1} \tilde{S} \tilde{\nu}-c_{w 1} f_w\left(\frac{\tilde{\nu}}{d}\right)^2 \\ & +\frac{1}{\sigma} \frac{\partial}{\partial x_k}\left[(\nu+\tilde{\nu}) \frac{\partial \tilde{\nu}}{\partial x_k}\right]+\frac{c_{b 2}}{\sigma} \frac{\partial \tilde{\nu}}{\partial x_k} \frac{\partial \tilde{\nu}}{\partial x_k} \end{aligned}

Closure Coefficients:

\begin{array}{r} c_{b 1}=0.1355, \quad c_{b 2}=0.622, \quad c_{v 1}=7.1, \quad \sigma=2 / 3 \\ c_{w 1}=\frac{c_{b 1}}{\kappa^2}+\frac{\left(1+c_{b 2}\right)}{\sigma}, \quad c_{w 2}=0.3, \quad c_{w 3}=2, \quad \kappa=0.41 \end{array}

Auxiliary Relations:

\begin{gathered} f_{v 1}=\frac{\chi^3}{\chi^3+c_{v 1}^3}, \quad f_{v 2}=1-\frac{\chi}{1+\chi f_{v 1}}, \quad f_w=g\left[\frac{1+c_{w 3}^6}{g^6+c_{w 3}^6}\right]^{1 / 6} \\ \chi=\frac{\tilde{\nu}}{\nu}, \quad g=r+c_{w 2}\left(r^6-r\right), \quad r=\frac{\tilde{\nu}}{\tilde{S} \kappa^2 d^2} \\ \tilde{S}=S+\frac{\tilde{\nu}}{\kappa^2 d^2} f_{v 2}, \quad S=\sqrt{2 \Omega_{i j} \Omega_{i j}} \end{gathered}

Reference(s)