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Spalart-Allmaras model

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* {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=AIAA Paper 92-0439}}
* {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=AIAA Paper 92-0439}}
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* {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=La Recherche Aerospatiale 1994 n 1, 5-21}}
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* {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1994|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=La Recherche Aerospatiale n 1, 5-21}}

Revision as of 19:32, 1 September 2006

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)

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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