Spalart-Allmaras Turbulence Model

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

Contents

[edit] 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}

[edit] 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) [1]

DDES (2006)

[edit] 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 ρ 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.

[edit] 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.

[edit] References

  • Spalart, P. R. and Allmaras, S. R., 1992, "A One-Equation Turbulence Model for Aerodynamic Flows" AIAA Paper 92-0439

[edit] External links