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]\displaystyle{ \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]\displaystyle{ \mu_t = \begin{cases} {\mu_t}_\text{inner} & \mbox{if } y \le y_\text{crossover} \\ {\mu_t}_\text{outer} & \mbox{if } y \gt y_\text{crossover} \end{cases} }[/math]

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

The inner-region eddy viscosity is given by:

[math]\displaystyle{ {\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]\displaystyle{ \ell = \kappa y \left( 1 - e^{-y^+/A^+} \right) }[/math]

with the von Karman constant [math]\displaystyle{ \kappa }[/math] usually being taken as 0.4, and with

[math]\displaystyle{ 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]\displaystyle{ {\mu_t}_\text{outer} = \alpha \rho U_e \delta_v^* F_K }[/math]

where [math]\displaystyle{ \alpha=0.0168 }[/math], [math]\displaystyle{ \delta_v^* }[/math] is the displacement thickness, given by

[math]\displaystyle{ \delta_v^* = \int_0^\delta \left(1 - \frac{U}{U_e}\right)\,dy }[/math]

and F_K is the Klebanoff intermittency function given by

[math]\displaystyle{ 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 Cebeci Smith model article in CFD-Wiki