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https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:43:16Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
== Modifications to original model ==<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
Other models related to the S-A model:<br />
<br />
DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:42:21Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
== Modifications to original model ==<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
Other models related to the S-A model:<br />
DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:41:57Z
<p>Odlopez: /* Original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:41:15Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999) [http://www.cfd-online.com/Wiki/Detached_eddy_simulation_%28DES%29]<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:32:53Z
<p>Odlopez: /* References */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:32:30Z
<p>Odlopez: /* References */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
<br />
* {{reference-paper|author=Spalart, P. R. and Allmaras, S. R.|year=1992|title=A One-Equation Turbulence Model for Aerodynamic Flows|rest=La Recherche Aerospatiale 1994 n 1, 5-21}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:27:50Z
<p>Odlopez: /* Boundary conditions */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
[[Walls:]] <math>\tilde{\nu}=0</math><br />
<br />
[[Freestream:]] Ideally <math>\tilde{\nu}=0</math>, but some solvers can have problem with that so <math>\tilde{\nu}<=\frac{\nu}{2}</math> can be used.<br />
<br />
[[Outlet:]] convective outlet.<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:24:04Z
<p>Odlopez: /* Boundary conditions */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<math><br />
\nu<br />
</math><br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:22:22Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:20:58Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)\\<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:20:46Z
<p>Odlopez: /* Modifications to original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
DES (1999)<br />
DDES (2006)<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:19:30Z
<p>Odlopez: /* Original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{matrix}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez
https://www.cfd-online.com/Wiki/Spalart-Allmaras_model
Spalart-Allmaras model
2006-09-01T19:18:56Z
<p>Odlopez: /* Original model */</p>
<hr />
<div>Spallart-Allmaras model is a one equation model for the turbulent viscosity.<br />
<br />
== Original model ==<br />
The turbulent eddy viscosity is given by<br />
<br />
:<math><br />
\nu_t = \tilde{\nu} f_{v1}, \quad f_{v1} = \frac{\chi^3}{\chi^3 + C^3_{v1}}, \quad \chi := \frac{\tilde{\nu}}{\nu}<br />
</math><br />
<br />
:<math><br />
\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<br />
</math><br />
<br />
:<math><br />
\tilde{S} \equiv S + \frac{ \tilde{\nu} }{ \kappa^2 d^2 } f_{v2}, \quad f_{v2} = 1 - \frac{\chi}{1 + \chi f_{v1}}<br />
</math><br />
<br />
:<math><br />
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 }<br />
</math><br />
<br />
:<math><br />
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)<br />
</math><br />
<br />
:<math><br />
f_{t2} = C_{t3} \exp(-C_{t4} \chi^2)<br />
</math><br />
<br />
The constants are<br />
<br />
:<math><br />
\begin{matrix}<br />
\sigma &=& 2/3\\<br />
C_{b1} &=& 0.1355\\<br />
C_{b2} &=& 0.622\\<br />
\kappa &=& 0.41\\<br />
C_{w1} &=& C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma \\<br />
C_{w2} &=& 0.3 \\<br />
C_{w3} &=& 2 \\<br />
C_{v1} &=& 7.1 \\<br />
C_{t1} &=& 1 \\<br />
C_{t2} &=& 2 \\<br />
C_{t3} &=& 1.1 \\<br />
C_{t4} &=& 2<br />
\end{matrix} <br />
</math><br />
<br />
According to Spalart it is safer to use the following values for the last two constants:<br />
:<math><br />
\begin{math}<br />
C_{t3} &=& 1.2 \\<br />
C_{t4} &=& 0.5<br />
\end{matrix}<br />
</math><br />
<br />
== Modifications to original model ==<br />
<br />
== Model for compressible flows ==<br />
There are two approaches to adapting the model for compressible flows. In the first approach the turbulent dynamic viscosity is computed from<br />
<br />
:<math><br />
\mu_t = \rho \tilde{\nu} f_{v1}<br />
</math><br />
<br />
where <math>\rho</math> is the local density. The convective terms in the equation for <math>\tilde{\nu}</math> are modified to<br />
<br />
:<math><br />
\frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j} (\tilde{\nu} u_j)= \mbox{RHS}<br />
</math><br />
<br />
where the right hand side (RHS) is the same as in the original model.<br />
<br />
== Boundary conditions ==<br />
<br />
== References ==<br />
<br />
* {{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}}</div>
Odlopez