From CFD-Wiki
Introduction
The Cebeci-Smith [Cebeci and Smith (1967)] is a two-layer algebraic 0-equation model which gives the eddy viscosity,
, 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}](/W/images/math/2/c/0/2c00d225d371de0cbb8faa306328b2cc.png)
| (1) |
where
is the smallest distance from the surface where
is equal to
:
![y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}](/W/images/math/0/a/e/0aea083a70c0228df853cc09fa4d6aa1.png)
| (2) |
The inner region is given by the Prandtl - Van Driest formula:
![{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|](/W/images/math/5/3/b/53b46ed6398ab5bc23e63f84f566b189.png)
| (3) |
where
![l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)](/W/images/math/8/5/a/85a53dc624249d451e8cdbc23697464a.png)
| (4) |
![\kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}](/W/images/math/d/c/1/dc17c22b20b697ea91c81fe2827b5ef7.png)
| (5) |
![\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}](/W/images/math/9/9/a/99ae36ac6bcb506dd9de94f3ad68fa52.png)
| (5) |
![\Omega_{ij} = \frac{1}{2}
\left(
\frac{\partial u_i}{\partial x_j} -
\frac{\partial u_j}{\partial x_i}
\right)](/W/images/math/f/2/a/f2a00d9b3ab8d744eee92f2309a91247.png)
| (6) |
The outer region is given by:
![{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),](/W/images/math/c/7/d/c7d43fde147c13b34d296ac62885946c.png)
| (7) |
where
,
is the velocity thickness given by
![\delta_v^* = \int_0^\delta (1-U/U_e)dy,](/W/images/math/a/e/c/aec9fb27f61867dd2830217c221ea46d.png)
| (8) |
and
is the Klebanoff intermittency function given by
![F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6
\right]^{-1}](/W/images/math/4/3/3/4331b74159c7e4947a91a3c15e2c8282.png)
| (10) |
Model variants
Performance, applicability and limitations
Implementation issues
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..