# 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:

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

${\mu _{t}}_{\text{outer}}=\alpha \rho U_{e}\delta _{v}^{*}F_{K}$ $\delta _{v}^{*}=\int _{0}^{\delta }\left(1-{\frac {U}{U_{e}}}\right)\,dy$ and FK is the Klebanoff intermittency function given by

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