# Cebeci-Smith model

(Difference between revisions)
 Revision as of 18:22, 5 May 2006 (view source)Jasond (Talk | contribs) (Copied from B-L model, still pretty rough)← Older edit Revision as of 15:25, 6 May 2006 (view source)Jasond (Talk | contribs) (further clean-up, still needs work)Newer edit → Line 1: Line 1: == Introduction == == Introduction == - The Cebeci-Smith [[#References|[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. + The Cebeci-Smith [[#References|[Smith and Cebeci (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.  Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge. == Equations == == Equations == Line 21: Line 21: [/itex]
(2)
[/itex]
(2)
- The inner region is given by the Prandtl - Van Driest formula: + The inner region is given
:$:[itex] - {\mu_t}_{inner} = \rho l^2 \left| \Omega \right| + {\mu_t}_{inner} = \rho l^2 l \left[\left( + \frac{\partial U}{\partial y}\right)^2 + + \left(\frac{\partial V}{\partial x}\right)^2 + \right]^{1/2},$(3)
[/itex]
(3)
Line 35: Line 38: [/itex]
(4)
[/itex]
(4)
-
+ with the constant $\kappa = 0.4$ and - :$+ - \kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2} + -$(5)
+
:$:[itex] - \left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}} + A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.$(5)
[/itex]
(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: The outer region is given by: Line 59: Line 50: :$:[itex] {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), -$
(7)
+ [/itex]
(6)
where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by Line 66: Line 57: :$:[itex] \delta_v^* = \int_0^\delta (1-U/U_e)dy, \delta_v^* = \int_0^\delta (1-U/U_e)dy, -$
(8)
+ [/itex]
(7)
and $F_{KLEB}$ is the Klebanoff intermittency function given by and $F_{KLEB}$ is the Klebanoff intermittency function given by Line 74: Line 65: F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 \right]^{-1} \right]^{-1} - [/itex]
(10)
+ [/itex]
(8)
Line 88: Line 79: == References == == References == - *Smith, A.M.O. and Cebeci, T. Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735. + * {{reference-paper|author=Smith, A.M.O. and Cebeci, T. |year=1967|title=Numerical solution of the turbulent boundary layer equations|rest=Douglas aircraft division report DAC 33735}} * {{reference-book|author=Wilcox, D.C. |year=1998|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}} * {{reference-book|author=Wilcox, D.C. |year=1998|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}

## Introduction

The Cebeci-Smith [Smith and Cebeci (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. 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

 ${\mu_t}_{inner} = \rho l^2 l \left[\left( \frac{\partial U}{\partial y}\right)^2 + \left(\frac{\partial V}{\partial x}\right)^2 \right]^{1/2},$ (3)

where

 $l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)$ (4)

with the constant $\kappa = 0.4$ and

 $A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.$ (5)

The outer region is given by:

 ${\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),$ (6)

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

 $\delta_v^* = \int_0^\delta (1-U/U_e)dy,$ (7)

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

 $F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6 \right]^{-1}$ (8)

## References

• Smith, A.M.O. and Cebeci, T. (1967), "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..