Spalart-Allmaras Turbulence Model

Spalart-Allmaras model is a one equation model for the turbulent viscosity.

Original model

The turbulent eddy viscosity is given by

:$u\_t\; =\; ilde\{\; u\}\; f\_\{v1\},\; quad\; f\_\{v1\}\; =\; frac\{chi^3\}\{chi^3\; +\; C^3\_\{v1,\; quad\; chi\; :=\; frac\{\; ilde\{\; u\{\; u\}$

:$frac\{partial\; ilde\{\; u\{partial\; t\}\; +\; u\_j\; frac\{partial\; ilde\{\; u\{partial\; x\_j\}\; =\; C\_\{b1\}\; [1\; -\; f\_\{t2\}]\; ilde\{S\}\; ilde\{\; u\}\; +\; frac\{1\}\{sigma\}\; \{\; abla\; cdot\; [(\; u\; +\; ilde\{\; u\})\; abla\; ilde\{\; u\}]\; +\; C\_\{b2\}\; |\; abla\; u\; |^2\; \}\; -\; left\; [C\_\{w1\}\; f\_w\; -\; frac\{C\_\{b1\{kappa^2\}\; f\_\{t2\}\; ight]\; left(\; frac\{\; ilde\{\; u\{d\}\; ight)^2\; +\; f\_\{t1\}\; Delta\; U^2$

:$ilde\{S\}\; equiv\; S\; +\; frac\{\; ilde\{\; u\}\; \}\{\; kappa^2\; d^2\; \}\; f\_\{v2\},\; quad\; f\_\{v2\}\; =\; 1\; -\; frac\{chi\}\{1\; +\; chi\; f\_\{v1$

:$f\_w\; =\; g\; left\; [\; frac\{\; 1\; +\; C\_\{w3\}^6\; \}\{\; g^6\; +\; C\_\{w3\}^6\; \}\; ight]\; ^\{1/6\},\; quad\; g\; =\; r\; +\; C\_\{w2\}(r^6\; -\; r),\; quad\; r\; equiv\; frac\{\; ilde\{\; u\}\; \}\{\; ilde\{S\}\; kappa^2\; d^2\; \}$

:$f\_\{t1\}\; =\; C\_\{t1\}\; g\_t\; expleft(\; -C\_\{t2\}\; frac\{omega\_t^2\}\{Delta\; U^2\}\; [\; d^2\; +\; g^2\_t\; d^2\_t]\; ight)$

:$f\_\{t2\}\; =\; C\_\{t3\}\; expleft(-C\_\{t4\}\; chi^2\; ight)$

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

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

:

Modifications to original model

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

Other models related to the S-A model:

DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]

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

:$mu\_t\; =\; ho\; ilde\{\; u\}\; f\_\{v1\}$

where $ho$ is the local density. The convective terms in the equation for $ilde\{\; u\}$ are modified to

:$frac\{partial\; ilde\{\; u\{partial\; t\}\; +\; frac\{partial\}\{partial\; x\_j\}\; (\; ilde\{\; u\}\; u\_j)=\; mbox\{RHS\}$

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

Boundary conditions

Walls: $ilde\{\; u\}=0$

Freestream:

Ideally $ilde\{\; u\}=0$, but some solvers can have problems with a zero value, in which case can be used.

This is if the trip term is used to "start up" the model. A convenient option is to set $ilde\{\; u\}=5\{\; u\}$ 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 [http://www.cfd-online.com/Wiki/Spalart-Allmaras_model Spalart-Allmaras model] article in [http://www.cfd-online.com/Wiki CFD-Wiki]