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| | Hello buddy. Let me introduce myself. I am Luther Aubrey. What she loves doing is taking part in croquet and she is attempting to make it a profession. He currently life in Idaho and his parents reside close by. My occupation is a messenger.<br><br>Feel free to visit my web site; [http://Vollrausch-gaming.de/index.php?mod=users&action=view&id=2209 Vollrausch-gaming.de] |
| {{context|date=May 2011}}
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| {{technical|date=May 2011}}
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| }}
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| The '''Spalart–Allmaras model''' is a one equation model for [[turbulent]] [[viscosity]]. It solves a [[transport equation]] for a viscosity-like variable <math>\tilde{\nu}</math>. This may be referred to as the Spalart–Allmaras variable.
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| == Original model ==
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| The turbulent [[Viscosity|eddy viscosity]] is given by
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| :<math>
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| \nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}
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| </math>
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| :<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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| </math>
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| :<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}}
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| </math>
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| :<math>
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| 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 }
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| </math>
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| :<math>
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| 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)
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| </math>
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| :<math>
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| f_{t2} = C_{t3} \exp\left(-C_{t4} \chi^2 \right)
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| </math>
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| :<math>
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| S = \sqrt{2 \Omega_{ij} \Omega_{ij}}
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| </math>
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| The [[rotation]] [[tensor]] is given by
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| :<math>
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| \Omega_{ij} = \frac{1}{2} ( \partial u_i / \partial x_j - \partial u_j / \partial x_i )
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| </math>
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| and d is the distance from the closest surface. | |
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| The [[Constant (mathematics)|constants]] are
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| :<math>
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| \begin{matrix}
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| \sigma &=& 2/3\\
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| C_{b1} &=& 0.1355\\
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| C_{b2} &=& 0.622\\
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| \kappa &=& 0.41\\
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| C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\
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| C_{w2} &=& 0.3 \\
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| C_{w3} &=& 2 \\
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| C_{v1} &=& 7.1 \\
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| C_{t1} &=& 1 \\
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| C_{t2} &=& 2 \\
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| C_{t3} &=& 1.1 \\
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| C_{t4} &=& 2
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| \end{matrix}
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| </math>
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| == Modifications to original model ==
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| According to Spalart it is safer to use the following values for the last two constants:
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| :<math>
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| \begin{matrix}
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| C_{t3} &=& 1.2 \\
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| C_{t4} &=& 0.5
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| \end{matrix}
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| </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)
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| == Model for compressible flows ==
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| There are two approaches to adapting the model for [[compressible flow]]s. In the first approach, the turbulent dynamic viscosity is computed from
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| :<math>
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| \mu_t = \rho \tilde{\nu} f_{v1}
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| </math>
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| where <math>\rho</math> is the local density. The [[convective]] terms in the equation for <math>\tilde{\nu}</math> are modified to
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| :<math>
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| \frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}
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| </math>
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| where the [[Sides of an equation|right hand side]] (RHS) is the same as in the original model.
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| == Boundary conditions ==
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| Walls: <math>\tilde{\nu}=0</math>
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| Freestream:
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| Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problems with a zero value, in which case <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.
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| This is if the trip term is used to "start up" the model. A convenient option is to set <math>\tilde{\nu}=5{\nu}</math> in the [[freestream]]. The model then provides "Fully Turbulent" behavior, i.e., it becomes turbulent in any region that contains [[shear stress|shear]].
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| Outlet: convective outlet.
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| == References == | |
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| * ''Spalart, P. R. and Allmaras, S. R.'', 1992, '''"A One-Equation Turbulence Model for Aerodynamic Flows"''' ''AIAA Paper 92-0439''
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| == External links ==
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| * This article was based on the [http://www.cfd-online.com/Wiki/Spalart-Allmaras_model Spalart-Allmaras model] article in [http://www.cfd-online.com/Wiki CFD-Wiki]
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| * [http://www.kxcad.net/STAR-CCM/online/138-spalartAllmarasTurbulence-02.html What Are the Spalart-Allmaras Turbulence Models?] from kxcad.net
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| * [http://turbmodels.larc.nasa.gov/spalart.html The Spalart-Allmaras Turbulence Model] at NASA's Langley Research Center Turbulence Modelling Resource site
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| {{DEFAULTSORT:Spalart-Allmaras turbulence model}}
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| [[Category:Turbulence models]]
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Hello buddy. Let me introduce myself. I am Luther Aubrey. What she loves doing is taking part in croquet and she is attempting to make it a profession. He currently life in Idaho and his parents reside close by. My occupation is a messenger.
Feel free to visit my web site; Vollrausch-gaming.de