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

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

$\frac{\partial \tilde{\nu }}{\partial t}+u_j\frac{\partial \tilde{\nu }}{\partial x_j}=C_{b1}[1-f_{t2}]\tilde{S}\tilde{\nu }+\frac{1}{\sigma }\{\mathrm{\nabla }\cdot [(\nu +\tilde{\nu })\mathrm{\nabla }\tilde{\nu }]+C_{b2}|\mathrm{\nabla }\nu {|}^2\}-[C_{w1}{f}_w-\frac{C_{b1}}{{\kappa }^2}{f}_{t2}]{\left(\frac{\tilde{\nu }}{d}\right)}^2+f_{t1}\mathrm{\Delta }{U}^2$

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

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

$f_{t1}=C_{t1}{g}_t\exp (-C_{t2}\frac{{\omega }_t^2}{\mathrm{\Delta }{U}^2}[d^2+g_t^2{d}_t^2])$

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

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

The rotation tensor is given by

${\mathrm{\Omega }}_{ij}=\frac{1}{2}(\partial u_i/\partial x_j-\partial u_j/\partial x_i)$

and d is the distance from the closest surface.

The constants are

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

Modifications to original model

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

$\begin{array}{ccc}{C}_{t3}& =& 1.2\\ C_{t4}& =& 0.5\end{array}$

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

$\frac{\partial \tilde{\nu }}{\partial t}+\frac{\partial }{\partial x_j}(\tilde{\nu }{u}_j)=\text{RHS}$

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