Mathematics of cyclic redundancy checks

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:

${\mu }_t=\{\begin{array}{ll}{{\mu }_t}_{\text{inner}}& \text{if }y\le y_{\text{crossover}}\\ {{\mu }_t}_{\text{outer}}& \text{if }y>y_{\text{crossover}}\end{array}$

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:

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

where

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

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

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

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

${\delta }_v^{*}=\underset{0}{\overset{\delta }{\int }}(1-\frac{U}{U_e})dy$

and F_K is the Klebanoff intermittency function given by

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

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