Spalart-Allmaras model
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Spallart-Allmaras model is a one equation model for the turbulent viscosity.
Contents |
Original model
The turbulent eddy viscosity is given by
![\frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} = C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} + \frac{1}{\sigma} \{ \nabla \cdot [(\nu + \tilde{\nu}) \nabla \tilde{\nu}] + C_{b2} | \nabla \nu |^2 \} - \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 + f_{t1} \Delta U^2](/W/images/math/4/7/6/476216889d1ea0604979b921eae3428c.png)
![f_w = g \left[ \frac{ 1 + C_{w3}^6 }{ g^6 + C_{w3}^6 } \right]^{1/6}, \quad g = r + C_{w2}(r^6 - r), \quad r \equiv \frac{\tilde{\nu} }{ \tilde{S} \kappa^2 d^2 }](/W/images/math/0/9/b/09b19885ed6dffe3dec850e2516f1696.png)
![f_{t1} = C_{t1} g_t \exp\left( -C_{t2} \frac{\omega_t^2}{\Delta U^2} [ d^2 + g^2_t d^2_t] \right)](/W/images/math/9/9/3/993b41d0ba795e0b3877d041c4cff1cb.png)
The constants are
According to Spalart it is safer to use the following values for the last two constants:
- Failed to parse (unknown function\begin): \begin{math} C_{t3} &=& 1.2 \\ C_{t4} &=& 0.5 \end{matrix}
Modifications to original model
Model for compressible flows
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from
where 
is the local density. The convective terms in the equation for 
are modified to
where the right hand side (RHS) is the same as in the original model.
Boundary conditions
References
- Spalart, P. R. and Allmaras, S. R. (1992), "A One-Equation Turbulence Model for Aerodynamic Flows", AIAA Paper 92-0439.
