# Cebeci-Smith model

(Difference between revisions)
 Revision as of 09:39, 12 June 2007 (view source)Jola (Talk | contribs)← Older edit Revision as of 09:41, 12 June 2007 (view source)Jola (Talk | contribs) Newer edit → Line 15: Line 15: where $y_{crossover}$ is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$: where $y_{crossover}$ is the smallest distance from the surface where ${\mu_t}_{inner}$ is equal to ${\mu_t}_{outer}$: -
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:$:[itex] y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer} y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer} Line 22: Line 22: The inner region is given The inner region is given - + :[itex] :[itex] {\mu_t}_{inner} = \rho l^2 l \left[\left( {\mu_t}_{inner} = \rho l^2 l \left[\left( Line 32: Line 32: where where - + :[itex] :[itex] l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right) l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right) Line 39: Line 39: with the constant [itex]\kappa = 0.4$ and with the constant $\kappa = 0.4$ and -
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:$:[itex] A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}. A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}. Line 46: Line 46: The outer region is given by: The outer region is given by: - + :[itex] :[itex] {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), Line 53: Line 53: where [itex]\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by -
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:[itex] :[itex] \delta_v^* = \int_0^\delta (1-U/U_e)dy, \delta_v^* = \int_0^\delta (1-U/U_e)dy, Line 82: Line 82: [[Category:Turbulence models]] [[Category:Turbulence models]] + + {{stub}}

## Revision as of 09:41, 12 June 2007

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..