Eddy-Viscosity Transport Model

The proposed Strain-Adaptive Linear Spalart-Allmaras (SALSA) model complies in most parts with the original SA model. The present model is based on the eddy-viscosity principle for weakly compressible media with negligible density fluctuations, viz.

$$\overline {u_iu_j} = \frac{2}{3} k \delta_{ij} - \nu_t \tilde S_{... ...d \qquad k = \frac{S^* \nu_t }{\sqrt{c_\mu}} \; , \qquad \qquad c_\mu=0.09 \; ,$$

$$\tilde S_{ij} = \frac{1}{2} \left[ \left(\frac{\partial U_i}{\par... ..._k} \delta_{ij} \: ,\qquad \qquad S^*=\sqrt{2 \tilde S_{ij} \tilde S_{ij}} \; ,$$ (1)

where $U_i$, $\overline {u_iu_j}$ and $k=\frac{1}{2} \overline {u_iu_i}$ denote to the mean velocity, kinematic Reynolds stresses and turbulence energy, respectively. The eddy-viscosity principle (1) is supplemented by a transport equation for the undamped turbulent (eddy) viscosity $\tilde \nu_t$ defined in eq. (2)

$$\frac{ D\tilde\nu_t}{ Dt} -\frac{\partial }{\partial x_k} \left[ ... ...}{Pr_{\tilde\nu_t}} \right] \frac{\tilde\nu_t ^2}{l_n ^2}}_{\rm Dissipation}\:.$$ (2)

The employed damping-functions and coefficients read as follows:

$$f_{\nu 1} = \frac{(\nu_t^+)^3}{C_{\nu 1}^3 + (\nu_t^+)^3} \: ,\qq... ...ight]^{1/6} \:, \qquad g = r \left[ 1+ C_{w2} \left(r^5 -1 \right) \right] \: ,$$

$$r= 1.6 \, \tanh \left[ 0.7 \left( \frac{\Psi}{\tilde S} \right) \... ...\rho_0}{\rho}} \; \left( \frac{ \tilde\nu_t }{ \kappa^2 \: l_n^2 } \right) \: ,$$

$$\quad C_{b2}= 0.622, \quad C_{\nu 1}= 7.1 ,\quad C_{w2} =0.3, \quad C_{w3} =2, \quad \kappa=0.41, \quad Pr_{\tilde\nu_t}=\frac{2}{3} \: .$$ (3)

Here, $l_n$ is the wall-normal distance determined in a customary manner. The definition of the effective velocity gradient $\tilde S$ is given by

$$\tilde S=S^* \left[ \left( \frac{1}{\nu_t^+} \right) + f_{\nu 1} \right] , \qquad \nu_t^+ = \frac{\tilde\nu_t}{\nu} \; .$$ (4)

The choice of the near-wall parameter $r$ follows a route outlined by Edwards [3] and is motivated by the more robust behaviour experienced for complex industrial applications. It should be noted, that the adopted near-wall model slightly alters the predicted skin friction in equilibrium flows. For a flat-plate boundary layer at $Re_\theta=10^4$ the predicted shape factor $H_{12}$ increases by 1.8 % and the skin friction decreases by the same amount when compared to the SA model.

The specific closure of the production-term $P_{\tilde \nu_t}$ is outlined in the next section.