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A roughness-dependent model

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(→Algebraic model for the mixing length, based on (4))
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==Two-equation eddy viscosity model==
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==Two-equation <math>k</math>-<math>\epsilon</math> eddy viscosity model==
<table width="70%"><tr><td>
<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:  
<math> C_{\mu} = 0.09 </math>
<math> C_{\mu} = 0.09 </math>
 +
 +
[http://www.cfd-online.com/Wiki/Standard_k-epsilon_model <math>k</math>-<math>\epsilon</math> model]
==One-equation eddy viscosity model==
==One-equation eddy viscosity model==
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\nu _t  = k^{{1 \over 2}}  l  
\nu _t  = k^{{1 \over 2}}  l  
</math></td><td width="5%">(2)</td></tr></table>
</math></td><td width="5%">(2)</td></tr></table>
 +
 +
[http://www.cfd-online.com/Wiki/Prandtl%27s_one-equation_model One-equation model]
==Algebraic eddy viscosity model==
==Algebraic eddy viscosity model==
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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===
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===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.  
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<math>u_\tau </math> is the shear velocity and <math>A</math> a model parameter.
 +
 
 +
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled <math>k</math>-equation [[#References|[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) [[#References|[Absi (2006)]]]:
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===Algebraic model for the mixing length, based on (4) [[#References|[Absi (2006)]]]===
 
<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>):
 +
<table width="70%"><tr><td>
 +
<math>
 +
l_m(y) = \kappa  A  \left( 1 - e^{\frac{-y}{A}} \right) 
 +
</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}}   
-
</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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 +
For local equilibrium, we are able to find the mean velocity profile <math>u</math> from the turbulent kinetic energy <math>k</math> (equation 4) and the mixing length <math>l_m</math> (equation 5), by:
 +
<table width="70%"><tr><td>
 +
<math>
 +
{{d u} \over {d y}}  = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}}
 +
</math></td><td width="5%">(8)</td></tr></table>
 +
 
 +
Figure (1) shows that the velocity profile obtained from equations (8), (4) and (5) (solid line) is more accurate than the logarithmic velocity profile (dash-dotted line).
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[[Image:fig7a.jpg]]
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[[Image:fig7b.jpg]]
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'''Figure 1''', 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 ([[#References|[Sukhodolov ''et al.'' (1998)]]]). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; (values of <math>y_0 , h, u_\tau</math>  are from  [[#References|[Sukhodolov ''et al.'' (1998)]]]); Figure from [[#References|[Absi (2006)]]].
== References ==
== References ==
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* {{reference-paper|author=Absi, R. |year=2006|title=A roughness and time dependent mixing length equation|rest=Journal of Hydraulic, Coastal and Environmental Engineering, Japan Society of Civil Engineers, (Doboku Gakkai Ronbunshuu B), Vol. 62, No. 4, pp.437-446}}
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* {{reference-paper|author=Absi, R. |year=2006|title=[http://www.jstage.jst.go.jp/article/jscejb/62/4/62_437/_article A roughness and time dependent mixing length equation]|rest=''Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B),'' Japan Society of Civil Engineers, Vol. '''62''', No. 4, pp.437-446}}
 +
 
 +
* {{reference-paper|author=Nezu, I. and Nakagawa, H. |year=1993|title=Turbulence in open-channel flows|rest=A.A. Balkema, Ed. Rotterdam, The Netherlands}}
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 +
* {{reference-paper|author=Sukhodolov, A., Thiele, M. and Bungartz, H. |year=1998|title=Turbulence structure in a river reach with sand bed|rest=''Water Resour. Res.'', Vol. '''34''', pp. 1317-1334}}
 +
 
[[Category:Turbulence models]]
[[Category:Turbulence models]]
{{stub}}
{{stub}}

Latest revision as of 12:47, 22 June 2007

Contents

Two-equation k-\epsilon eddy viscosity model

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

where: C_{\mu} = 0.09

k-\epsilon model

One-equation eddy viscosity model

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

One-equation model

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

For local equilibrium, we are able to find the mean velocity profile u from the turbulent kinetic energy k (equation 4) and the mixing length l_m (equation 5), by:

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

Figure (1) shows that the velocity profile obtained from equations (8), (4) and (5) (solid line) is more accurate than the logarithmic velocity profile (dash-dotted line).

Fig7a.jpg Fig7b.jpg

Figure 1, 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. (1998)]). a) profile 2: y_0 = 0.062 cm; h = 145 cm; u_\tau = 3.82 cm/s. b) profile 4: y_0 = 0.113 cm; h = 164.5 cm; u_\tau = 3.97 cm/s ; (values of y_0 , h, u_\tau are from [Sukhodolov et al. (1998)]); Figure from [Absi (2006)].

References

  • Nezu, I. and Nakagawa, H. (1993), "Turbulence in open-channel flows", A.A. Balkema, Ed. Rotterdam, The Netherlands.
  • Sukhodolov, A., Thiele, M. and Bungartz, H. (1998), "Turbulence structure in a river reach with sand bed", Water Resour. Res., Vol. 34, pp. 1317-1334.


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