# Spalart-Allmaras model

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

## Original model

The turbulent eddy viscosity is given by

$\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}$
$\begin{matrix} \frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} & = & C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} + \frac{1}{\sigma} \{ \nabla \cdot [(\nu + \tilde{\nu}) \nabla \tilde{\nu}] + C_{b2} | \nabla \nu |^2 \} - \\ \ & \ & \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 + f_{t1} \Delta U^2 \\ \end{matrix}$
$\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}$

where

$S = \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}$
$\Omega_{ij} \equiv \frac{1}{2} ( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} )$
$f_w = g \left[ \frac{ 1 + C_{w3}^6 }{ g^6 + C_{w3}^6 } \right]^{1/6}, \quad g = r + C_{w2}(r^6 - r), \quad r \equiv \frac{\tilde{\nu} }{ \tilde{S} \kappa^2 d^2 }$
$f_{t1} = C_{t1} g_t \exp\left( -C_{t2} \frac{\omega_t^2}{\Delta U^2} [ d^2 + g^2_t d^2_t] \right)$
$f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)$
d is the distance to 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:

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

[Dacles-Mariani et. al. 1995] proposed a modification of the model which also accounts for the effect of mean strain rate on turbulence production. This modification instead prescribes:

$S \equiv |\Omega_{ij}| + C_{\rm prod} \; \min \left(0, |S_{ij}| - |\Omega_{ij}| \right)$

where

$C_{\rm prod} = 2.0$
$|\Omega_{ij}| \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}$
$|S_{ij}| \equiv \sqrt{2 S_{ij} S_{ij}}$
$\Omega_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} \right)$
$S_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right)$

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

Boundary conditions are set by defining values of $\tilde{\nu}$. This may be refered to as the Spalart-Allmaras variable.

Walls: $\tilde{\nu}=0$

Freestream: Ideally $\tilde{\nu}=0$, but some solvers can have problem with that so $\tilde{\nu}<=\frac{\nu}{2}$ 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 results and any regions like boundary layers that contain shear become fully turbulent.

Outlet: convective outlet.

To calculate boundary conditions for this model from turbulence intensity and length-scale see turbulence free-stream boundary conditions.

## References

• Dacles-Mariani, J., Zilliac, G. G., Chow, J. S. and Bradshaw, P. (1995), "Numerical/Experimental Study of a Wingtip Vortex in the Near Field", AIAA Journal, 33(9), pp. 1561-1568, 1995.
• Spalart, P. R. and Allmaras, S. R. (1992), "A One-Equation Turbulence Model for Aerodynamic Flows", AIAA Paper 92-0439.
• Spalart, P. R. and Allmaras, S. R. (1994), "A One-Equation Turbulence Model for Aerodynamic Flows", La Recherche Aerospatiale n 1, 5-21.