CFD Online URL
[Sponsors]
Home > Wiki > Cebeci-Smith model

Cebeci-Smith model

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
m (Reverted edits by RicgeTcnac (Talk) to last version by Merrifj)
 
(5 intermediate revisions not shown)
Line 4: Line 4:
== Equations ==
== Equations ==
-
<table width="100%"><tr><td>
+
<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>
Line 15: Line 15:
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>:
-
<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}
Line 22: Line 22:
The inner region is given
The inner region is given
-
<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
-
{\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 32: Line 32:
where
where
-
<table width="100%"><tr><td>
+
<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)
Line 39: Line 39:
with the constant <math>\kappa = 0.4</math> and
with the constant <math>\kappa = 0.4</math> and
-
<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
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:
-
<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),
</math></td><td width="5%">(6)</td></tr></table>
</math></td><td width="5%">(6)</td></tr></table>
-
where <math>\alpha=0.0168</math>, <math>\delta_v^*</math> is the velocity thickness given by  
+
where <math>\alpha=0.0168</math> and <math>\delta_v^*</math> is the velocity thickness given by  
-
<table width="100%"><tr><td>
+
<table width="70%"><tr><td>
:<math>
:<math>
-
\delta_v^* = \int_0^\delta (1-U/U_e)dy,
+
\delta_v^* = \int_0^\delta (1-U/U_e)dy.
</math></td><td width="5%">(7)</td></tr></table>
</math></td><td width="5%">(7)</td></tr></table>
-
and <math>F_{KLEB}</math> is the Klebanoff intermittency function given by
+
<math>F_{KLEB}</math> is the Klebanoff intermittency function given by
<table width="100%"><tr><td>
<table width="100%"><tr><td>
Line 65: Line 65:
   \right]^{-1}
   \right]^{-1}
</math></td><td width="5%">(8)</td></tr></table>
</math></td><td width="5%">(8)</td></tr></table>
-
 
== Model variants ==
== Model variants ==
Line 82: Line 81:
[[Category:Turbulence models]]
[[Category:Turbulence models]]
 +
 +
{{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..


My wiki