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A roughness-dependent model

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(the algebraic eddy viscosity model is therefore)
(Algebraic model for the turbulent kinetic Energy)
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<math>l_m</math> is the mixing length.  
<math>l_m</math> is the mixing length.  
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===Algebraic model for the turbulent kinetic Energy===
+
===Algebraic model for the turbulent kinetic energy===
<table width="70%"><tr><td>
<table width="70%"><tr><td>
<math>
<math>
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}}  u_\tau  e^{\frac{-y}{A}}  
k^{{1 \over 2}}(y) = {1 \over {C_{\mu}}^{{1 \over 4}}}  u_\tau  e^{\frac{-y}{A}}  
</math></td><td width="5%">(4)</td></tr></table>
</math></td><td width="5%">(4)</td></tr></table>
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<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.  
+
<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]===  
===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]===  

Revision as of 15:30, 19 June 2007

Contents

Two-equation eddy viscosity model

 
\nu _t  = C_{\mu} {{k^2 } \over \varepsilon }
(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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