# 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, ${\displaystyle \mu _{t}}$, 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:

${\displaystyle \mu _{t}={\begin{cases}{\mu _{t}}_{\text{inner}}&{\mbox{if }}y\leq y_{\text{crossover}}\\{\mu _{t}}_{\text{outer}}&{\mbox{if }}y>y_{\text{crossover}}\end{cases}}}$

where ${\displaystyle y_{\text{crossover}}}$ is the smallest distance from the surface where ${\displaystyle {\mu _{t}}_{\text{inner}}}$ is equal to ${\displaystyle {\mu _{t}}_{\text{outer}}}$.

The inner-region eddy viscosity is given by:

${\displaystyle {\mu _{t}}_{\text{inner}}=\rho \ell ^{2}\left[\left({\frac {\partial U}{\partial y}}\right)^{2}+\left({\frac {\partial V}{\partial x}}\right)^{2}\right]^{1/2}}$

where

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

with the von Karman constant ${\displaystyle \kappa }$ usually being taken as 0.4, and with

${\displaystyle A^{+}=26\left[1+y{\frac {dP/dx}{\rho u_{\tau }^{2}}}\right]^{-1/2}}$

The eddy viscosity in the outer region is given by:

${\displaystyle {\mu _{t}}_{\text{outer}}=\alpha \rho U_{e}\delta _{v}^{*}F_{K}}$

where ${\displaystyle \alpha =0.0168}$, ${\displaystyle \delta _{v}^{*}}$ is the displacement thickness, given by

${\displaystyle \delta _{v}^{*}=\int _{0}^{\delta }\left(1-{\frac {U}{U_{e}}}\right)\,dy}$

and FK is the Klebanoff intermittency function given by

${\displaystyle F_{K}=\left[1+5.5\left({\frac {y}{\delta }}\right)^{6}\right]^{-1}}$

## 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.