Spalart–Allmaras Turbulence Model

In physics, the Spalart–Allmaras model is a one-equation model that solves a modelled transport equation for the kinematic eddy turbulent viscosity. The Spalart–Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in turbomachinery applications.

In its original form, the model is effectively a low-Reynolds number model, requiring the viscosity-affected region of the boundary layer to be properly resolved ( y+ ~1 meshes).
The Spalart–Allmaras model was developed for aerodynamic flows. It is not calibrated for general industrial flows, and does produce relatively larger errors for some free shear flows, especially plane and round jet flows. In addition, it cannot be relied on to predict the decay of homogeneous, isotropic turbulence.

It solves a transport equation for a viscosity-like variable $\tilde$. This may be referred to as the ''Spalart–Allmaras variable''.

Original model

The turbulent eddy viscosity is given by

:$\nu_t = \tilde f_, \quad f_ = \frac, \quad \chi := \frac$

:
\frac + u_j \frac = C_ - f_\tilde \tilde + \frac \ - \left _ f_w - \frac f_\right\left( \frac \right)^2 + f_ \Delta U^2

:$\tilde \equiv S + \frac f_, \quad f_ = 1 - \frac$

:
f_w = g \left \frac \right, \quad g = r + C_(r^6 - r), \quad r \equiv \frac

:
f_ = C_ g_t \exp\left( -C_ \frac d^2 + g^2_t d^2_t\right)

:$f_ = C_ \exp\left(-C_ \chi^2 \right)$

:$S = \sqrt$

The rotation tensor is given by
:$\Omega_ = \frac ( \partial u_i / \partial x_j - \partial u_j / \partial x_i )$
where d is the distance from the closest surface and $\Delta U^2$ is the norm of the difference between the velocity at the trip (usually zero) and that at the field point we are considering.

The constants are

:$\begin \sigma &=& 2/3\\ C_ &=& 0.1355\\ C_ &=& 0.622\\ \kappa &=& 0.41\\ C_ &=& C_/\kappa^2 + (1 + C_)/\sigma \\ C_ &=& 0.3 \\ C_ &=& 2 \\ C_ &=& 7.1 \\ C_ &=& 1 \\ C_ &=& 2 \\ C_ &=& 1.1 \\ C_ &=& 2 \end$

Modifications to original model

According to Spalart it is safer to use the following values for the last two constants:
:$\begin C_ &=& 1.2 \\ C_ &=& 0.5 \end$

Other models related to the S-A model:

DES (1999


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

:$\mu_t = \rho \tilde f_$

where $\rho$ is the local density. The convective terms in the equation for $\tilde$ are modified to

:$\frac + \frac (\tilde u_j)= \mbox$

where the right hand side

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