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

:$$egin{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} &=& 2end{matrix}

Modifications to original model

According to Spalart it is safer to use the following values for the last two constants::$$egin{matrix}C_{t3} &=& 1.2 \C_{t4} &=& 0.5end{matrix}

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 $$ilde{ u}=0, but some solvers can have problems with a zero value, in which case $$ilde{ u}<=frac{ u}{2} 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]