Musean hypernumber

The Spalart–Allmaras model is a one equation model for turbulent viscosity. It solves a transport equation for a viscosity-like variable ${\tilde {\nu }}$. This may be referred to as the Spalart–Allmaras variable.

Original model

The turbulent eddy viscosity is given by

$\nu _{t}={\tilde {\nu }}f_{v1},\quad f_{v1}={\frac {\chi ^{3}}{\chi ^{3}+C_{v1}^{3}}},\quad \chi :={\frac {\tilde {\nu }}{\nu }}$

${\frac {\partial {\tilde {\nu }}}{\partial t}}+u_{j}{\frac {\partial {\tilde {\nu }}}{\partial x_{j}}}=C_{b1}[1-f_{t2}]{\tilde {S}}{\tilde {\nu }}+{\frac {1}{\sigma }}\{\nabla \cdot [(\nu +{\tilde {\nu }})\nabla {\tilde {\nu }}]+C_{b2}|\nabla \nu |^{2}\}-\left[C_{w1}f_{w}-{\frac {C_{b1}}{\kappa ^{2}}}f_{t2}\right]\left({\frac {\tilde {\nu }}{d}}\right)^{2}+f_{t1}\Delta U^{2}$

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

$f_{w}=g\left[{\frac {1+C_{w3}^{6}}{g^{6}+C_{w3}^{6}}}\right]^{1/6},\quad g=r+C_{w2}(r^{6}-r),\quad r\equiv {\frac {\tilde {\nu }}{{\tilde {S}}\kappa ^{2}d^{2}}}$

$f_{t1}=C_{t1}g_{t}\exp \left(-C_{t2}{\frac {\omega _{t}^{2}}{\Delta U^{2}}}[d^{2}+g_{t}^{2}d_{t}^{2}]\right)$

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

$S={\sqrt {2\Omega _{ij}\Omega _{ij}}}$

The rotation tensor is given by

$\Omega _{ij}={\frac {1}{2}}(\partial u_{i}/\partial x_{j}-\partial u_{j}/\partial x_{i})$

and d is the distance from the closest surface.

The constants are

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

Modifications to original model

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

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

Other models related to the S-A 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 problems with a zero value, in which case ${\tilde {\nu }}<={\frac {\nu }{2}}$ can be used.

This is if the trip term is used to "start up" the model. A convenient option is to set ${\tilde {\nu }}=5{\nu }$ in the freestream. The model then provides "Fully Turbulent" behavior, i.e., it becomes turbulent in any region that contains shear.

Outlet: convective outlet.

References

• Spalart, P. R. and Allmaras, S. R., 1992, "A One-Equation Turbulence Model for Aerodynamic Flows" AIAA Paper 92-0439

External links

• This article was based on the Spalart-Allmaras model article in CFD-Wiki
• What Are the Spalart-Allmaras Turbulence Models? from kxcad.net
• The Spalart-Allmaras Turbulence Model at NASA's Langley Research Center Turbulence Modelling Resource site