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Cebeci-Smith model
2008-12-18T12:06:59Z
<p>RicgeTcnac: labaserdrond</p>
<hr />
<div>letoaloucotr<br />
{{Turbulence modeling}}<br />
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.<br />
<br />
== Equations ==<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
\mu_t =<br />
\begin{cases}<br />
{\mu_t}_{inner} & \mbox{if } y \le y_{crossover} \\ <br />
{\mu_t}_{outer} & \mbox{if } y > y_{crossover}<br />
\end{cases}<br />
</math></td><td width="5%">(1)</td></tr></table><br />
<br />
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>:<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
y_{crossover} = MIN(y) \ : \ {\mu_t}_{inner} = {\mu_t}_{outer}<br />
</math></td><td width="5%">(2)</td></tr></table><br />
<br />
The inner region is given<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
{\mu_t}_{inner} = \rho l^2 \left[\left(<br />
\frac{\partial U}{\partial y}\right)^2 +<br />
\left(\frac{\partial V}{\partial x}\right)^2<br />
\right]^{1/2},<br />
</math></td><td width="5%">(3)</td></tr></table><br />
<br />
where<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
l = \kappa y \left( 1 - e^{\frac{-y^+}{A^+}} \right)<br />
</math></td><td width="5%">(4)</td></tr></table><br />
<br />
with the constant <math>\kappa = 0.4</math> and<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
A^+ = 26\left[1+y\frac{dP/dx}{\rho u_\tau^2}\right]^{-1/2}.<br />
</math></td><td width="5%">(5)</td></tr></table><br />
<br />
The outer region is given by:<br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
{\mu_t}_{outer} = \alpha \rho U_e \delta_v^* F_{KLEB}(y;\delta),<br />
</math></td><td width="5%">(6)</td></tr></table><br />
<br />
where <math>\alpha=0.0168</math> and <math>\delta_v^*</math> is the velocity thickness given by <br />
<br />
<table width="70%"><tr><td><br />
:<math><br />
\delta_v^* = \int_0^\delta (1-U/U_e)dy.<br />
</math></td><td width="5%">(7)</td></tr></table><br />
<br />
<math>F_{KLEB}</math> is the Klebanoff intermittency function given by<br />
<br />
<table width="100%"><tr><td><br />
:<math><br />
F_{KLEB}(y;\delta) = \left[1 + 5.5 \left( \frac{y}{\delta} \right)^6<br />
\right]^{-1}<br />
</math></td><td width="5%">(8)</td></tr></table><br />
<br />
== Model variants ==<br />
<br />
<br />
== Performance, applicability and limitations ==<br />
<br />
<br />
== Implementation issues ==<br />
<br />
<br />
== References ==<br />
<br />
* {{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}}<br />
* {{reference-book|author=Wilcox, D.C. |year=1998|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}<br />
<br />
[[Category:Turbulence models]]<br />
<br />
{{stub}}</div>
RicgeTcnac