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The '''Spalart–Allmaras model''' is a one equation model for [[turbulent]] [[viscosity]]. It solves a [[transport equation]] for a viscosity-like variable <math>\tilde{\nu}</math>. This may be referred to as the Spalart–Allmaras variable.
 
== Original model ==
The turbulent [[Viscosity|eddy viscosity]] is given by
 
:<math>
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}
</math>
 
:<math>
\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
</math>
 
:<math>
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}
</math>
 
:<math>
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 }
</math>
 
:<math>
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)
</math>
 
:<math>
f_{t2} = C_{t3} \exp\left(-C_{t4} \chi^2 \right)
</math>
 
:<math>
S = \sqrt{2 \Omega_{ij} \Omega_{ij}}
</math>
 
The [[rotation]] [[tensor]] is given by
:<math>
\Omega_{ij} = \frac{1}{2} ( \partial u_i / \partial x_j - \partial u_j / \partial x_i )
</math>
and d is the distance from the closest surface.
 
The [[Constant (mathematics)|constants]] are
 
:<math>
\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}
</math>
 
== Modifications to original model ==
According to Spalart it is safer to use the following values for the last two constants:
:<math>
\begin{matrix}
C_{t3} &=& 1.2 \\
C_{t4} &=& 0.5
\end{matrix}
</math>
 
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 flow]]s. In the first approach, the turbulent dynamic viscosity is computed from
 
:<math>
\mu_t = \rho \tilde{\nu} f_{v1}
</math>
 
where <math>\rho</math> is the local density. The [[convective]] terms in the equation for <math>\tilde{\nu}</math> are modified to
 
:<math>
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}
</math>
 
where the [[Sides of an equation|right hand side]] (RHS) is the same as in the original model.
 
== Boundary conditions ==
Walls: <math>\tilde{\nu}=0</math>
 
Freestream:
 
Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problems with a zero value, in which case <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.
 
This is if the trip term is used to "start up" the model. A convenient option is to set <math>\tilde{\nu}=5{\nu}</math> in the [[freestream]].  The model then provides "Fully Turbulent" behavior, i.e., it becomes turbulent in any region that contains [[shear stress|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]
* [http://www.kxcad.net/STAR-CCM/online/138-spalartAllmarasTurbulence-02.html What Are the Spalart-Allmaras Turbulence Models?] from kxcad.net
* [http://turbmodels.larc.nasa.gov/spalart.html The Spalart-Allmaras Turbulence Model] at NASA's Langley Research Center Turbulence Modelling Resource site
 
{{DEFAULTSORT:Spalart-Allmaras turbulence model}}
[[Category:Turbulence models]]

Revision as of 07:26, 23 May 2013

Template:Multiple issues

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

Original model

The turbulent eddy viscosity is given by

The rotation tensor is given by

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) [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

where is the local density. The convective terms in the equation for are modified to

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

Boundary conditions

Walls:

Freestream:

Ideally , 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 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