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Cebeci-Smith model

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== Introduction ==
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{{Turbulence modeling}}
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The Cebeci-Smith [[#References|[Smith and Cebeci (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.  Unlike the [[Baldwin-Lomax model]], this model requires the determination of of a boundary layer edge.
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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 ==
== Equations ==
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
\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}
</math></td><td width="5%">(1)</td></tr></table>
</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>:
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>
+
<table width="70%"><tr><td>
:<math>
:<math>
y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}
</math></td><td width="5%">(2)</td></tr></table>
</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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The inner region is given
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
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{\mu_t}_{inner} = \rho l^2 \left| \Omega \right|
+
{\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},
</math></td><td width="5%">(3)</td></tr></table>
</math></td><td width="5%">(3)</td></tr></table>
where
where
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<table width="100%"><tr><td>
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<table width="70%"><tr><td>
:<math>
:<math>
l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)
</math></td><td width="5%">(4)</td></tr></table>
</math></td><td width="5%">(4)</td></tr></table>
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<table width="100%"><tr><td>
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with the constant <math>\kappa = 0.4</math> and
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:<math>
+
-
\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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<table width="70%"><tr><td>
:<math>
:<math>
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\left| \Omega \right| = \sqrt{2 \Omega_{ij} \Omega_{ij}}
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A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.
</math></td><td width="5%">(5)</td></tr></table>
</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}
 
-
\left(
 
-
\frac{\partial u_i}{\partial x_j} -
 
-
\frac{\partial u_j}{\partial x_i}
 
-
\right)
 
-
</math></td><td width="5%">(6)</td></tr></table>
 
The outer region is given by:
The outer region is given by:
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<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
{\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),
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</math></td><td width="5%">(7)</td></tr></table>
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</math></td><td width="5%">(6)</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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where <math>\alpha=0.0168</math> and <math>\delta_v^*</math> is the velocity thickness given by  
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<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
-
\delta_v^* = \int_0^\delta (1-U/U_e)dy,
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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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</math></td><td width="5%">(7)</td></tr></table>
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and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by
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<math>F_{KLEB}</math> is the Klebanoff intermittency function given by
<table width="100%"><tr><td>
<table width="100%"><tr><td>
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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}
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</math></td><td width="5%">(10)</td></tr></table>
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</math></td><td width="5%">(8)</td></tr></table>
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== Model variants ==
== Model variants ==
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== References ==
== 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-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.}}
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[[Category:Turbulence models]]
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{{stub}}

Latest revision as of 12:13, 18 December 2008

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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.

Contents

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)

Model variants

Performance, applicability and limitations

Implementation issues

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..


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