# Cebeci-Smith model

(Difference between revisions)
 Revision as of 15:25, 6 May 2006 (view source)Jasond (Talk | contribs) (further clean-up, still needs work)← Older edit Latest revision as of 12:13, 18 December 2008 (view source)Peter (Talk | contribs) m (Reverted edits by RicgeTcnac (Talk) to last version by Merrifj) (7 intermediate revisions not shown) Line 1: Line 1: - == Introduction == + {{Turbulence modeling}} - + 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. 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 == -
+
:$:[itex] \mu_t = \mu_t = \begin{cases} \begin{cases} {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ {\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ - {\mu_t}_{outer} & \mbox{if} y > y_{crossover} + {\mu_t}_{outer} & \mbox{if } y > y_{crossover} \end{cases} \end{cases}$(1)
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(1)
Line 16: 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 ${\mu_t}_{outer}$: -
+
:$:[itex] y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer} y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer} Line 23: Line 22: The inner region is given The inner region is given - + :[itex] :[itex] - {\mu_t}_{inner} = \rho l^2 l \left[\left( + {\mu_t}_{inner} = \rho l^2 \left[\left( \frac{\partial U}{\partial y}\right)^2 + \frac{\partial U}{\partial y}\right)^2 + \left(\frac{\partial V}{\partial x}\right)^2 \left(\frac{\partial V}{\partial x}\right)^2 Line 33: Line 32: where where - + :[itex] :[itex] l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right) l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right) Line 40: Line 39: with the constant [itex]\kappa = 0.4$ and with the constant $\kappa = 0.4$ and -
+
:$:[itex] A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}. A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}. Line 47: Line 46: The outer region is given by: The outer region is given by: - + :[itex] :[itex] {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta), {\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),$(6)
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- where $\alpha=0.0168$, $\delta_v^*$ is the velocity thickness given by + where $\alpha=0.0168$ and $\delta_v^*$ is the velocity thickness given by -
+
:$:[itex] - \delta_v^* = \int_0^\delta (1-U/U_e)dy, + \delta_v^* = \int_0^\delta (1-U/U_e)dy.$(7)
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- and $F_{KLEB}$ is the Klebanoff intermittency function given by + $F_{KLEB}$ is the Klebanoff intermittency function given by
Line 66: Line 65: \right]^{-1} \right]^{-1} [/itex](8)
[/itex]
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- == Model variants == == Model variants == Line 81: Line 79: * {{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-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.}} + + [[Category:Turbulence models]] + + {{stub}}

## Latest revision as of 12:13, 18 December 2008

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

 ${\mu_t}_{inner} = \rho l^2 \left[\left( \frac{\partial U}{\partial y}\right)^2 + \left(\frac{\partial V}{\partial x}\right)^2 \right]^{1/2},$ (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 $\alpha=0.0168$ and $\delta_v^*$ is the velocity thickness given by

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

$F_{KLEB}$ 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..