Cebeci–Smith model

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 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,
• Wilcox, D.C., 1998. Turbulence Modeling for CFD. , 2nd Ed., DCW Industries, Inc.