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A roughness-dependent model

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==Two-equation eddy viscosity model==
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==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==
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<table width="70%"><tr><td>
<math>  
<math>  
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\nu _t  = C_{\mu} {{k^2 } \over \varepsilon }
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\nu _t  = C_{\mu} {{k^2 } \over \epsilon }
</math></td><td width="5%">(1)</td></tr></table>
</math></td><td width="5%">(1)</td></tr></table>
where:  
where:  

Revision as of 20:31, 19 June 2007

Contents

Two-equation k-\epsilon eddy viscosity model

\nu _t = C_{\mu} {{k^2 } \over \epsilon } (1)

where: C_{\mu} = 0.09

One-equation eddy viscosity model

\nu _t = k^{{1 \over 2}} l (2)

Algebraic eddy viscosity model

\nu _t(y) = {C_{\mu}}^{{1 \over 4}} l_m(y) k^{{1 \over 2}}(y) (3)

l_m is the mixing length.

Algebraic model for the turbulent kinetic energy

k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}} u_\tau e^{\frac{-y}{A}} (4)

u_\tau is the shear velocity and A a model parameter.

Algebraic model for the mixing length, based on (4) [Absi (2006)]

l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) (5)

\kappa = 0.4, y_0 is the hydrodynamic roughness

the algebraic eddy viscosity model is therefore

\nu _t(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right) u_\tau e^{\frac{-y}{A}} (6)

for a smooth wall (y_0 = 0):

\nu _t(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) u_\tau e^{\frac{-y}{A}} (7)

References


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