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A roughness-dependent model

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(→Algebraic eddy viscosity model)
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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.
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===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]===
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For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [[#References|[Nezu and Nakagawa (1993)]]] obtained a similar semi-theoretical equation.
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===Algebraic model for the mixing length===
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For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (4) [[#References|[Absi (2006)]]]:
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<table width="70%"><tr><td>
<table width="70%"><tr><td>
<math>
<math>
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)
l_m(y) = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)
</math></td><td width="5%">(5)</td></tr></table>
</math></td><td width="5%">(5)</td></tr></table>
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<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness
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<math>\kappa = 0.4</math>, <math>y_0</math> is the hydrodynamic roughness.
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For a smooth wall (<math>y_0 = 0</math>):
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<table width="70%"><tr><td>
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<math>
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l_m(y) = \kappa  A  \left( 1 - e^{\frac{-y}{A}} \right) 
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</math></td><td width="5%">(6)</td></tr></table>
===the algebraic eddy viscosity model is therefore===  
===the algebraic eddy viscosity model is therefore===  
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\nu _t(y)  = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)
\nu _t(y)  = \kappa \left( A - \left(A - y_0\right) e^{\frac{-(y-y_0)}{A}} \right)
  u_\tau  e^{\frac{-y}{A}}   
  u_\tau  e^{\frac{-y}{A}}   
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</math></td><td width="5%">(6)</td></tr></table>
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</math></td><td width="5%">(7)</td></tr></table>
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==The mean velocity profile==
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In local equilibrium region, we are able to find the mean velocity profile from the mixing length <math>l_m</math> and the turbulent kinetic energy <math>k</math> by:
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for a smooth wall (<math>y_0 = 0</math>):
 
<table width="70%"><tr><td>
<table width="70%"><tr><td>
<math>  
<math>  
-
\nu _t(y) = \kappa  A  \left( 1 - e^{\frac{-y}{A}} \right)
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{{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}}  
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u_\tau  e^{\frac{-y}{A}}
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</math></td><td width="5%">(8)</td></tr></table>  
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</math></td><td width="5%">(7)</td></tr></table>
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With equations (4) and (5), we obtain:
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[[Image:fig7a.jpg]]
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[[Image:fig7b.jpg]]
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Fig. Vertical distribution of mean flow velocity.
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<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>;
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Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>U_f = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>U_f = 3.97 cm/s</math> ; values of <math>y_0 , h, U_f</math>  are from  [[#References|[Sukhodolov et al. (1998)]]].
== References ==
== References ==

Revision as of 12:12, 21 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.

For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled k-equation [Nezu and Nakagawa (1993)] obtained a similar semi-theoretical equation.

Algebraic model for the mixing length

For local equilibrium, an extension of von Kármán’s similarity hypothesis allows to write, with equation (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. For a smooth wall (y_0 = 0):

l_m(y) = \kappa A \left( 1 - e^{\frac{-y}{A}} \right) (6)

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}} (7)


The mean velocity profile

In local equilibrium region, we are able to find the mean velocity profile from the mixing length l_m and the turbulent kinetic energy k by:

{{d U} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} (8)

With equations (4) and (5), we obtain:

Fig7a.jpg Fig7b.jpg

Fig. Vertical distribution of mean flow velocity. A = {{h} \over {c_1}}; c_1 = 1; Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: y_0 = 0.062 cm; h = 145 cm; U_f = 3.82 cm/s. b) profile 4: y_0 = 0.113 cm; h = 164.5 cm; U_f = 3.97 cm/s ; values of y_0 , h, U_f are from [Sukhodolov et al. (1998)].

References


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