# Spalart-Allmaras model

(Difference between revisions)
 Revision as of 19:32, 1 September 2006 (view source)Odlopez (Talk | contribs) (→References)← Older edit Revision as of 16:35, 19 May 2008 (view source)Toh24 (Talk | contribs) Newer edit → (12 intermediate revisions not shown) Line 1: Line 1: - Spallart-Allmaras model is a one equation model for the turbulent viscosity. + {{Turbulence modeling}} + Spalart-Allmaras model is a one equation model which solves a transport equation for a viscosity-like variable $\tilde{\nu}$. This may be referred to as the Spalart-Allmaras variable. == Original model == == Original model == Line 9: Line 10: :$:[itex] - \frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} = C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} + \frac{1}{\sigma} \{ \nabla \cdot [(\nu + \tilde{\nu}) \nabla \tilde{\nu}] + C_{b2} | \nabla \nu |^2 \} - \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 + f_{t1} \Delta U^2 + \begin{matrix} + \frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} & = & C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} + \frac{1}{\sigma} \{ \nabla \cdot [(\nu + \tilde{\nu}) \nabla \tilde{\nu}] + C_{b2} | \nabla \nu |^2 \} - \\ + \ & \ & \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 + f_{t1} \Delta U^2 \\ + \end{matrix}$ [/itex] Line 15: Line 19: \tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}} \tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}} [/itex] [/itex] + + where + + :$+ S = \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}} +$ + + :$\Omega_{ij} \equiv \frac{1}{2} ( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} )$ :$:[itex] Line 27: Line 39: f_{t2} = C_{t3} \exp(-C_{t4} \chi^2) f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)$ [/itex] + + :d is the distance to the closest surface The constants are The constants are Line 47: Line 61: [/itex] [/itex] + == Modifications to original model == According to Spalart it is safer to use the following values for the last two constants: According to Spalart it is safer to use the following values for the last two constants: :$:[itex] Line 55: Line 70:$ [/itex] - == Modifications to original model == + [Dacles-Mariani et. al. 1995] proposed a modification of the model which also accounts for the effect of mean strain rate on turbulence production. This modification instead prescribes: - DES (1999) + + :$+ S \equiv |\Omega_{ij}| + C_{\rm prod} \; \min \left(0, |S_{ij}| - |\Omega_{ij}| \right) +$ + + where + + :$C_{\rm prod} = 2.0$ + :$|\Omega_{ij}| \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}$ + :$|S_{ij}| \equiv \sqrt{2 S_{ij} S_{ij}}$ + :$\Omega_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} \right)$ + :$S_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right)$ + + Other models related to the S-A model: + + DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29] DDES (2006) DDES (2006) Line 76: Line 106: == Boundary conditions == == Boundary conditions == - [[Walls:]] $\tilde{\nu}=0$ + Boundary conditions are set by defining values of $\tilde{\nu}$. - [[Freestream:]] Ideally $\tilde{\nu}=0$, but some solvers can have problem with that so $\tilde{\nu}<=\frac{\nu}{2}$ can be used. + Freestream boundary conditions are discussed in [[Turbulence free-stream boundary conditions|turbulence free-stream boundary conditions]]. - [[Outlet:]] convective outlet. + Walls: $\tilde{\nu}=0$ + + Outlet: convective outlet. == References == == References == + + * {{reference-paper|author=Dacles-Mariani, J., Zilliac, G. G., Chow, J. S. and Bradshaw, P.|year=1995|title=Numerical/Experimental Study of a Wingtip Vortex in the Near Field|rest=AIAA Journal, 33(9), pp. 1561-1568, 1995}} * {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=AIAA Paper 92-0439}} * {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=AIAA Paper 92-0439}} * {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1994|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=La Recherche Aerospatiale n 1, 5-21}} * {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1994|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=La Recherche Aerospatiale n 1, 5-21}} + + [[Category:Turbulence models]]

## Revision as of 16:35, 19 May 2008

Spalart-Allmaras model is a one equation model which 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

$\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}$
$\begin{matrix} \frac{\partial \tilde{\nu}}{\partial t} + u_j \frac{\partial \tilde{\nu}}{\partial x_j} & = & C_{b1} [1 - f_{t2}] \tilde{S} \tilde{\nu} + \frac{1}{\sigma} \{ \nabla \cdot [(\nu + \tilde{\nu}) \nabla \tilde{\nu}] + C_{b2} | \nabla \nu |^2 \} - \\ \ & \ & \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right] \left( \frac{\tilde{\nu}}{d} \right)^2 + f_{t1} \Delta U^2 \\ \end{matrix}$
$\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}$

where

$S = \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}$
$\Omega_{ij} \equiv \frac{1}{2} ( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} )$
$f_w = g \left[ \frac{ 1 + C_{w3}^6 }{ g^6 + C_{w3}^6 } \right]^{1/6}, \quad g = r + C_{w2}(r^6 - r), \quad r \equiv \frac{\tilde{\nu} }{ \tilde{S} \kappa^2 d^2 }$
$f_{t1} = C_{t1} g_t \exp\left( -C_{t2} \frac{\omega_t^2}{\Delta U^2} [ d^2 + g^2_t d^2_t] \right)$
$f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)$
d is the distance to 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:

$\begin{matrix} C_{t3} &=& 1.2 \\ C_{t4} &=& 0.5 \end{matrix}$

[Dacles-Mariani et. al. 1995] proposed a modification of the model which also accounts for the effect of mean strain rate on turbulence production. This modification instead prescribes:

$S \equiv |\Omega_{ij}| + C_{\rm prod} \; \min \left(0, |S_{ij}| - |\Omega_{ij}| \right)$

where

$C_{\rm prod} = 2.0$
$|\Omega_{ij}| \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}$
$|S_{ij}| \equiv \sqrt{2 S_{ij} S_{ij}}$
$\Omega_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} \right)$
$S_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right)$

Other models related to the S-A 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

Boundary conditions are set by defining values of $\tilde{\nu}$.

Freestream boundary conditions are discussed in turbulence free-stream boundary conditions.

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

Outlet: convective outlet.

## References

• Dacles-Mariani, J., Zilliac, G. G., Chow, J. S. and Bradshaw, P. (1995), "Numerical/Experimental Study of a Wingtip Vortex in the Near Field", AIAA Journal, 33(9), pp. 1561-1568, 1995.
• Spalart, P. R. and Allmaras, S. R. (1992), "A One-Equation Turbulence Model for Aerodynamic Flows", AIAA Paper 92-0439.
• Spalart, P. R. and Allmaras, S. R. (1994), "A One-Equation Turbulence Model for Aerodynamic Flows", La Recherche Aerospatiale n 1, 5-21.