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


$\displaystyle \tilde S_{ij} = \frac{1}{2} \left[ \left(\frac{\partial U_i}{\par...
..._k} \delta_{ij} \: ,\qquad
\qquad S^*=\sqrt{2 \tilde S_{ij} \tilde S_{ij}} \; ,$     (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)
$\displaystyle \frac{ D\tilde\nu_t}{ Dt}
-\frac{\partial }{\partial x_k} \left[ ...
...}{Pr_{\tilde\nu_t}} \right] \frac{\tilde\nu_t ^2}{l_n ^2}}_{\rm Dissipation}\:.$     (2)

The employed damping-functions and coefficients read as follows:
$\displaystyle f_{\nu 1} = \frac{(\nu_t^+)^3}{C_{\nu 1}^3 + (\nu_t^+)^3} \: ,\qq...
...ight]^{1/6} \:, \qquad
g = r \left[ 1+ C_{w2} \left(r^5 -1 \right) \right] \: ,$      
$\displaystyle r= 1.6 \, \tanh \left[ 0.7 \left( \frac{\Psi}{\tilde S} \right) \...
...\rho_0}{\rho}} \; \left( \frac{ \tilde\nu_t }{ \kappa^2 \: l_n^2 } \right) \: ,$      
$\displaystyle \quad C_{b2}= 0.622, \quad
C_{\nu 1}= 7.1 ,\quad
C_{w2} =0.3, \quad
C_{w3} =2, \quad
\kappa=0.41, \quad
Pr_{\tilde\nu_t}=\frac{2}{3} \: .$     (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
$\displaystyle \tilde S=S^* \left[ \left( \frac{1}{\nu_t^+} \right) + f_{\nu 1} \right] ,
\qquad \nu_t^+$ $\displaystyle =$ $\displaystyle \frac{\tilde\nu_t}{\nu}
\; .$ (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.



Subsections
next up previous
Next: Production-Term Modification Up: J26222 Previous: Introduction
Markus Schatz 2004-02-10