# Cebeci-Smith model

## Introduction

The Cebeci-Smith [Cebeci and Smith (1967)] is a two-layer algebraic 0-equation model which gives the 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 (see below). Unlike the Baldwin-Lomax model, this model requires the determination of of a boundary layer edge.

## Equations

 $\mu_t = \begin{cases} {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ {\mu_t}_{outer} & \mbox{if} y > y_{crossover} \end{cases}$ (1)

where $y_{crossover}$ is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$:

 $y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}$ (2)

The inner region is given by the Prandtl - Van Driest formula:

 ${\mu_t}_{inner} = \rho l^2 \left| \Omega \right|$ (3)

where

 $l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)$ (4)
 $\kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}$ (5)
 $\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}$ (5)
 $\Omega_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right)$ (6)

The outer region is given by:

 ${\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),$ (7)

where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by

 $\delta_v^* = \int_0^\delta (1-U/U_e)dy,$ (8)

and $F_{KLEB}$ is the Klebanoff intermittency function given by

 $F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 \right]^{-1}$ (10)

## References

• Smith, A.M.O. and Cebeci, T. Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.
• Wilcox, D.C. (1998), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..