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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


\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}

\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

\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}

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 }

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)

f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)

The constants are


\begin{matrix}
\sigma &=& 2/3\\
C_{b1} &=& 0.1355\\
C_{b2} &=& 0.622\\
\kappa &=& 0.41\\
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\
C_{w2} &=& 0.3 \\
C_{w3} &=& 2 \\
C_{v1} &=& 7.1 \\
C_{t1} &=& 1 \\
C_{t2} &=& 2 \\
C_{t3} &=& 1.1 \\
C_{t4} &=& 2
\end{matrix}

According to Spalart it is safer to use the following values for the last two constants:


\begin{matrix}
C_{t3} &=& 1.2 \\
C_{t4} &=& 0.5
\end{matrix}

Modifications to original model

DES (1999) [1]

DDES (2006)

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


\mu_t = \rho \tilde{\nu} f_{v1}

where \rho is the local density. The convective terms in the equation for \tilde{\nu} are modified to


\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}

where the right hand side (RHS) is the same as in the original model.

Boundary conditions

Walls: \tilde{\nu}=0

Freestream: Ideally \tilde{\nu}=0, but some solvers can have problem with that so \tilde{\nu}<=\frac{\nu}{2} can be used.

Outlet: convective outlet.

References

  • Spalart, P. R. and Allmaras, S. R. (1992), "A One-Equation Turbulence Model for Aerodynamic Flows", AIAA Paper 92-0439.
  • Spalart, P. R. and Allmaras, S. R. (1994), "A One-Equation Turbulence Model for Aerodynamic Flows", La Recherche Aerospatiale n 1, 5-21.
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