View source for Cebeci–Smith model

The '''Cebeci–Smith model''', developed by [[Tuncer Cebeci]] and [[Apollo M. O. Smith]] in 1967, is a 0-equation [[eddy viscosity]] model used in [[computational fluid dynamics]] analysis of [[turbulence]] in [[boundary layer]] flows. The model gives eddy viscosity, <math>\mu_t</math>, as a function of the local boundary layer velocity profile. The model is suitable for high-speed flows with thin attached boundary layers, typically present in aerospace applications.  Like the [[Baldwin-Lomax model]], it is not suitable for large regions of [[flow separation]] and significant curvature or  rotation.  Unlike the Baldwin-Lomax model, this model requires the determination of a boundary layer edge.

== Equations ==

In a two-layer model, the boundary layer is considered to comprise two layers: inner (close to the surface) and outer. The eddy viscosity is calculated separately for each layer and combined using:

:<math>
\mu_t =
\begin{cases}
{\mu_t}_\text{inner} & \mbox{if } y \le y_\text{crossover} \\ 
{\mu_t}_\text{outer} & \mbox{if } y > y_\text{crossover}
\end{cases}
</math>

where <math>y_\text{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_\text{inner}</math> is equal to <math>{\mu_t}_\text{outer}</math>.

The inner-region eddy viscosity is given by:

:<math>
{\mu_t}_\text{inner} = \rho \ell^2 \left[\left(
 \frac{\partial U}{\partial y}\right)^2 +
 \left(\frac{\partial V}{\partial x}\right)^2
\right]^{1/2}
</math>

where

:<math>
\ell = \kappa y \left( 1 - e^{-y^+/A^+} \right)
</math>

with the von Karman constant <math>\kappa</math> usually being taken as 0.4, and with

:<math>
A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}
</math>

The eddy viscosity in the outer region is given by:

:<math>
{\mu_t}_\text{outer} = \alpha \rho U_e \delta_v^* F_K
</math>

where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the [[displacement thickness]], given by

:<math>
\delta_v^* = \int_0^\delta \left(1 - \frac{U}{U_e}\right)\,dy
</math>

and ''F''<sub>''K''</sub> is the Klebanoff intermittency function given by

:<math>
F_K = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6
  \right]^{-1}
</math>

== References ==

* Smith, A.M.O. and Cebeci, T., 1967. ''Numerical solution of the turbulent boundary layer equations''. Douglas aircraft division report DAC 33735
* Cebeci, T. and Smith, A.M.O., 1974. ''Analysis of turbulent boundary layers''. Academic Press, {{ISBN|0-12-164650-5}}
* Wilcox, D.C., 1998. ''Turbulence Modeling for CFD''. {{ISBN|1-928729-10-X}}, 2nd Ed., DCW Industries, Inc.

== External links ==

* This article was based on the [http://www.cfd-online.com/Wiki/Cebeci-Smith_model Cebeci Smith model] article in [http://www.cfd-online.com/Wiki CFD-Wiki]

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[[Category:Turbulence models]]
[[Category:Fluid dynamics]]
[[Category:Mathematical modeling]]