Template:Multiple issues
The Spalart–Allmaras model is a one equation model for turbulent viscosity. It solves a transport equation for a viscosity-like variable
. This may be referred to as the Spalart–Allmaras variable.
Original model
The turbulent eddy viscosity is given by
![{\displaystyle \nu _{t}={\tilde {\nu }}f_{v1},\quad f_{v1}={\frac {\chi ^{3}}{\chi ^{3}+C_{v1}^{3}}},\quad \chi :={\frac {\tilde {\nu }}{\nu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4a179a3304e41363cd43f0f6f1d153b31b0e47)
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5838726d02e09d5b2c46acc4da9fb57082b52f91)
![{\displaystyle {\tilde {S}}\equiv S+{\frac {\tilde {\nu }}{\kappa ^{2}d^{2}}}f_{v2},\quad f_{v2}=1-{\frac {\chi }{1+\chi f_{v1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48d548eae4b21aa29b7eadeb390c75d8c2305247)
![{\displaystyle 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73effbeb918597b6789dd5e35b086174d0f00f32)
![{\displaystyle f_{t1}=C_{t1}g_{t}\exp \left(-C_{t2}{\frac {\omega _{t}^{2}}{\Delta U^{2}}}[d^{2}+g_{t}^{2}d_{t}^{2}]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae6acc5936ec64648ee94e075270e30a6cb6d7e5)
![{\displaystyle f_{t2}=C_{t3}\exp \left(-C_{t4}\chi ^{2}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c47b6f9dd5c61b7e64718ae9202ae26c403ff90)
![{\displaystyle S={\sqrt {2\Omega _{ij}\Omega _{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac773aaa54a46ca4c3b67214496ace2da92d583d)
The rotation tensor is given by
![{\displaystyle \Omega _{ij}={\frac {1}{2}}(\partial u_{i}/\partial x_{j}-\partial u_{j}/\partial x_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e82eefa0aca671c8ec66360527ec47948a6b41c1)
and d is the distance from the closest surface.
The constants are
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326d7e76e4bcfc29f3c54d2f88897c2ec3281f5e)
Modifications to original model
According to Spalart it is safer to use the following values for the last two constants:
![{\displaystyle {\begin{matrix}C_{t3}&=&1.2\\C_{t4}&=&0.5\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a43f8151abc1fb6d87f2987b2d148f8decd666)
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
![{\displaystyle \mu _{t}=\rho {\tilde {\nu }}f_{v1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07a3fda450348a362b05f128122c5be27f444e62)
where
is the local density. The convective terms in the equation for
are modified to
![{\displaystyle {\frac {\partial {\tilde {\nu }}}{\partial t}}+{\frac {\partial }{\partial x_{j}}}({\tilde {\nu }}u_{j})={\mbox{RHS}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9642d37465b9e00f4721de90917107aff6f3e08)
where the right hand side (RHS) is the same as in the original model.
Boundary conditions
Walls:
Freestream:
Ideally
, but some solvers can have problems with a zero value, in which case
can be used.
This is if the trip term is used to "start up" the model. A convenient option is to set
in the freestream. The model then provides "Fully Turbulent" behavior, i.e., it becomes turbulent in any region that contains shear.
Outlet: convective outlet.
References
- Spalart, P. R. and Allmaras, S. R., 1992, "A One-Equation Turbulence Model for Aerodynamic Flows" AIAA Paper 92-0439
External links