The Spalart–Allmaras model is a one equation model for turbulent viscosity. It solves a transport equation for a viscositylike 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}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \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} \{ \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

\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \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 } \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\left(C_{t4} \chi^2 \right)

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

\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}
Other models related to the SA 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
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 OneEquation Turbulence Model for Aerodynamic Flows" AIAA Paper 920439
External links

This article was based on the SpalartAllmaras model article in CFDWiki

What Are the SpalartAllmaras Turbulence Models? from kxcad.net

The SpalartAllmaras Turbulence Model at NASA's Langley Research Center Turbulence Modelling Resource site
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