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Spalart-Allmaras model

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Spallart-Allmaras model is a one equation model for the turbulent viscosity.
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{{Turbulence modeling}}
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Spalart-Allmaras model is a one equation model which solves a transport equation for a viscosity-like variable <math>\tilde{\nu}</math>. This may be referred to as the Spalart-Allmaras variable.
== Original model ==
== Original model ==
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:<math>
:<math>
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\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
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\begin{matrix}
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\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 \} - \\
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\ & \ & \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 \\
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\end{matrix}
</math>
</math>
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\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}}
</math>
</math>
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where
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:<math>
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S = \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}
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</math>
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:<math>\Omega_{ij} \equiv \frac{1}{2} ( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} )</math>
:<math>
:<math>
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f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)
</math>
</math>
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:d is the distance to the closest surface
The constants are
The constants are
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</math>
</math>
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== 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:
:<math>
:<math>
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</math>
</math>
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== Modifications to original model ==
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[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:
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DES (1999)\\
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:<math>
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S \equiv |\Omega_{ij}| + C_{\rm prod} \; \min \left(0, |S_{ij}| - |\Omega_{ij}| \right)
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</math>
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where
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:<math>C_{\rm prod} = 2.0</math>
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:<math>|\Omega_{ij}| \equiv \sqrt{2 \Omega_{ij} \Omega_{ij}}</math>
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:<math>|S_{ij}| \equiv \sqrt{2 S_{ij} S_{ij}}</math>
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:<math>\Omega_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} - \frac{\partial u_i}{\partial x_j} \right)</math>
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:<math>S_{ij} \equiv \frac{1}{2}\left(\frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right)</math>
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Other models related to the S-A model:
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DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]
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DDES (2006)
DDES (2006)
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== Boundary conditions ==
== Boundary conditions ==
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Boundary conditions are set by defining values of <math>\tilde{\nu}</math>.
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Freestream boundary conditions are discussed in [[Turbulence free-stream boundary conditions|turbulence free-stream boundary conditions]].
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Walls: <math>\tilde{\nu}=0</math>
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Outlet: convective outlet.
== References ==
== References ==
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* {{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}}
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* {{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}}
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[[Category:Turbulence models]]

Latest revision as of 16:35, 19 May 2008

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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.

Contents

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.
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