Cebeci–Smith model

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:

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

where ${\displaystyle y_{\text{crossover}}}$  is the smallest distance from the surface where ${\displaystyle {\mu _{t}}_{\text{inner}}}$  is equal to ${\displaystyle {\mu _{t}}_{\text{outer}}}$ .

The inner-region eddy viscosity is given by:

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

where

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

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

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

The eddy viscosity in the outer region is given by:

${\displaystyle {\mu _{t}}_{\text{outer}}=\alpha \rho U_{e}\delta _{v}^{*}F_{K}}$ 

where ${\displaystyle \alpha =0.0168}$ , ${\displaystyle \delta _{v}^{*}}$  is the displacement thickness, given by

${\displaystyle \delta _{v}^{*}=\int _{0}^{\delta }\left(1-{\frac {U}{U_{e}}}\right)\,dy}$ 

and F_K is the Klebanoff intermittency function given by

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

• 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.

• This article was based on the Cebeci Smith model article in CFD-Wiki