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, $\mu _{t}$ , 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\leq 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 F_K 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

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