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Cebeci–Smith model

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The Cebeci–Smith model, developed by Tuncer Cebeci and Apollo M. O. Smith in 1967, is a 0-equation eddy viscosity model used in computational fluid dynamics analysis of turbulence in boundary layer flows. The model gives eddy viscosity, μ t {\displaystyle \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, it is not suitable for large regions of flow separation and significant curvature or rotation. Unlike the Baldwin-Lomax model, this model requires the determination of a boundary layer edge.

Equations

In a two-layer model, the boundary layer is considered to comprise two layers: inner (close to the surface) and outer. The eddy viscosity is calculated separately for each layer and combined using:

μ t = { μ t inner if  y ≤ y crossover μ t outer if  y > y crossover {\displaystyle \mu _{t}={\begin{cases}{\mu _{t}}_{\text{inner}}&{\mbox{if }}y\leq y_{\text{crossover}}\\{\mu _{t}}_{\text{outer}}&{\mbox{if }}y>y_{\text{crossover}}\end{cases}}}

where y crossover {\displaystyle y_{\text{crossover}}} is the smallest distance from the surface where μ t inner {\displaystyle {\mu _{t}}_{\text{inner}}} is equal to μ t outer {\displaystyle {\mu _{t}}_{\text{outer}}} .

The inner-region eddy viscosity is given by:

μ t inner = ρ ℓ 2 [ ( ∂ U ∂ y ) 2 + ( ∂ V ∂ x ) 2 ] 1 / 2 {\displaystyle {\mu _{t}}_{\text{inner}}=\rho \ell ^{2}\left^{1/2}}

where

ℓ = κ y ( 1 − e − y + / A + ) {\displaystyle \ell =\kappa y\left(1-e^{-y^{+}/A^{+}}\right)}

with the von Karman constant κ {\displaystyle \kappa } usually being taken as 0.4, and with

A + = 26 [ 1 + y d P / d x ρ u τ 2 ] − 1 / 2 {\displaystyle A^{+}=26\left^{-1/2}}

The eddy viscosity in the outer region is given by:

μ t outer = α ρ U e δ v ∗ F K {\displaystyle {\mu _{t}}_{\text{outer}}=\alpha \rho U_{e}\delta _{v}^{*}F_{K}}

where α = 0.0168 {\displaystyle \alpha =0.0168} , δ v ∗ {\displaystyle \delta _{v}^{*}} is the displacement thickness, given by

δ v ∗ = ∫ 0 δ ( 1 − U U e ) d y {\displaystyle \delta _{v}^{*}=\int _{0}^{\delta }\left(1-{\frac {U}{U_{e}}}\right)\,dy}

and FK is the Klebanoff intermittency function given by

F K = [ 1 + 5.5 ( y δ ) 6 ] − 1 {\displaystyle F_{K}=\left^{-1}}

References

  • Smith, A.M.O. and Cebeci, T., 1967. Numerical solution of the turbulent boundary layer equations. Douglas aircraft division report DAC 33735
  • Cebeci, T. and Smith, A.M.O., 1974. Analysis of turbulent boundary layers. Academic Press, ISBN 0-12-164650-5
  • Wilcox, D.C., 1998. Turbulence Modeling for CFD. ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.

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