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# Cebeci-Smith model

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, μt, 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 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:

$\mu_t = \begin{cases} {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ {\mu_t}_{outer} & \mbox{if } y > y_{crossover} \end{cases}$

where ycrossover is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$.

The inner-region eddy viscosity is given by:

${\mu_t}_{inner} = \rho l^2 l \left[\left( \frac{\partial U}{\partial y}\right)^2 + \left(\frac{\partial V}{\partial x}\right)^2 \right]^{1/2}$

where

$l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)$

with the von Karman constant κ 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:

${\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{K}$

where α = 0.0168, $\delta_v^*$ is the displacement thickness, given by

$\delta_v^* = \int_0^\delta (1-U/U_e)dy$

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
• Wilcox, D.C., 1998. Turbulence Modeling for CFD. ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.