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Cebeci-Smith model

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(Copied from B-L model, still pretty rough)
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== References ==
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== Introduction ==
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*<b>Smith, A.M.O. and Cebeci, T.</b> Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.
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The Cebeci-Smith [[#References|[Cebeci and Smith (1967)]]] is a two-layer algebraic 0-equation model which gives the eddy viscosity, <math>\mu_t</math>, 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.
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== Equations ==
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----
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<table width="100%"><tr><td>
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<i> Return to [[Turbulence modeling]] </i>
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:<math>
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\mu_t =
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\begin{cases}
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{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\
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{\mu_t}_{outer} & \mbox{if} y > y_{crossover}
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\end{cases}
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</math></td><td width="5%">(1)</td></tr></table>
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where <math>y_{crossover}</math> is the smallest distance from the surface where <math>{\mu_t}_{inner}</math> is equal to <math>{\mu_t}_{outer}</math>:
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<table width="100%"><tr><td>
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:<math>
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y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
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</math></td><td width="5%">(2)</td></tr></table>
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The inner region is given by the Prandtl - Van Driest formula:
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<table width="100%"><tr><td>
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:<math>
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{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|
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</math></td><td width="5%">(3)</td></tr></table>
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where
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<table width="100%"><tr><td>
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:<math>
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l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
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</math></td><td width="5%">(4)</td></tr></table>
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<table width="100%"><tr><td>
 +
:<math>
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\kappa = 0.4, A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}
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</math></td><td width="5%">(5)</td></tr></table>
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<table width="100%"><tr><td>
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:<math>
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\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
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</math></td><td width="5%">(5)</td></tr></table>
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<table width="100%"><tr><td>
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:<math>
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\Omega_{ij} = \frac{1}{2}
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\left(
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\frac{\partial u_i}{\partial x_j} -
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\frac{\partial u_j}{\partial x_i}
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\right)
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</math></td><td width="5%">(6)</td></tr></table>
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The outer region is given by:
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<table width="100%"><tr><td>
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:<math>
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{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),
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</math></td><td width="5%">(7)</td></tr></table>
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where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by
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<table width="100%"><tr><td>
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:<math>
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\delta_v^* = \int_0^\delta (1-U/U_e)dy,
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</math></td><td width="5%">(8)</td></tr></table>
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and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by
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<table width="100%"><tr><td>
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:<math>
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F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6
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  \right]^{-1}
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</math></td><td width="5%">(10)</td></tr></table>
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== Model variants ==
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== Performance, applicability and limitations ==
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== Implementation issues ==
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== References ==
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*<b>Smith, A.M.O. and Cebeci, T.</b> Numerical solution of the turbulent boundary layer equations, Douglas aircraft division report DAC 33735.
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* {{reference-book|author=Wilcox, D.C. |year=1998|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}

Revision as of 18:22, 5 May 2006

Contents

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)


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..
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