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 $$\tilde{\nu}$$. 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 \tilde{\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 ) $$ where d is the distance from the closest surface and $$\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{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 S-A model:

DES (1999)

DDES (2006)

Model for compressible flows
There are several approaches to adapting the model for compressible flows.

In all cases, the turbulent dynamic viscosity is computed from



\mu_t = \rho \tilde{\nu} f_{v1} $$

where $$\rho$$ is the local density.

The first approach applies the original equation for $$\tilde{\nu}$$.

In the second approach, 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.

The third approach involves inserting the density inside some of the derivatives on the LHS and RHS.

The second and third approaches are not recommended by the original authors, but they are found in several solvers.

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

Freestream:

Ideally $$\tilde{\nu}=0$$, but some solvers can have problems with a zero value, in which case $$\tilde{\nu} \leq \frac{\nu}{2}$$ 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.