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

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

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