CFD Online Logo CFD Online URL
Home > Wiki > A roughness-dependent model

A roughness-dependent model

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
(Algebraic model for the turbulent kinetic energy)
(One-equation eddy viscosity model)
Line 12: Line 12:
\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>
[ One-equation model]
==Algebraic eddy viscosity model==
==Algebraic eddy viscosity model==

Revision as of 15:12, 21 June 2007


Two-equation k-\epsilon eddy viscosity model

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

where:  C_{\mu} = 0.09

One-equation eddy viscosity model

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

One-equation model

Algebraic eddy viscosity model

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

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}} 

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)

\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)  

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}}  

The mean velocity profile

In local equilibrium region, we are able to find the mean velocity u 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}} 

With equations (4) and (5), we obtain [Absi (2006)]:

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_\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)].


  • 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., 34, pp. 1317-1334.

My wiki