Cebeci–Smith model
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The Cebeci–Smith model is a 0-equation eddy viscosity model used in computational fluid dynamics analysis of turbulent boundary layer flows. The model gives eddy viscosity, \(\mu_t\), 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, this model is not suitable for cases with large separated regions and significant curvature/rotation effects. Unlike the Baldwin-Lomax model, this model requires the determination of a boundary layer edge.
The model was developed by Tuncer Cebeci and Apollo M. O. Smith, in 1967.
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:
\[ \mu_t = \begin{cases} {\mu_t}_\text{inner} & \mbox{if } y \le y_\text{crossover} \\ {\mu_t}_\text{outer} & \mbox{if } y > y_\text{crossover} \end{cases} \]
where \(y_\text{crossover}\) is the smallest distance from the surface where \({\mu_t}_\text{inner}\) is equal to \({\mu_t}_\text{outer}\).
The inner-region eddy viscosity is given by:
\[ {\mu_t}_\text{inner} = \rho \ell^2 \left[\left( \frac{\partial U}{\partial y}\right)^2 + \left(\frac{\partial V}{\partial x}\right)^2 \right]^{1/2} \]
where
\[ \ell = \kappa y \left( 1 - e^{-y^+/A^+} \right) \]
with the von Karman constant \(\kappa\) usually being taken as 0.4, and with
\[ A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2} \]
The eddy viscosity in the outer region is given by:
\[ {\mu_t}_\text{outer} = \alpha \rho U_e \delta_v^* F_K \]
where \(\alpha=0.0168\), \(\delta_v^*\) is the displacement thickness, given by
\[ \delta_v^* = \int_0^\delta \left(1 - \frac{U}{U_e}\right)\,dy \]
and FK is the Klebanoff intermittency function given by
\[ F_K = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 \right]^{-1} \]
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
