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 \le 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 FK is the Klebanoff intermittency function given by



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