Spalart–Allmaras turbulence model

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) http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29

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.

References