A roughness-dependent 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. | ||
- | 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. | + | 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=== | ===Algebraic model for the mixing length=== |
Revision as of 15:09, 21 June 2007
Contents |
Two-equation - eddy viscosity model
(1) |
where:
One-equation eddy viscosity model
(2) |
Algebraic eddy viscosity model
(3) |
is the mixing length.
Algebraic model for the turbulent kinetic energy
(4) |
is the shear velocity and a model parameter.
For steady open channel flows in local equilibrium, where the energy production is balanced by the dissipation, from the modeled -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)]:
(5) |
, is the hydrodynamic roughness. For a smooth wall ():
(6) |
the algebraic eddy viscosity model is therefore
(7) |
The mean velocity profile
In local equilibrium region, we are able to find the mean velocity profile from the mixing length and the turbulent kinetic energy by:
(8) |
With equations (4) and (5), we obtain [Absi (2006)]:
Fig. Vertical distribution of mean flow velocity. ; ; Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: ; ; . b) profile 4: ; ; ; values of are from [Sukhodolov et al. (1998)].
References
- Absi, R. (2006), "A roughness and time dependent mixing length equation", Journal of Hydraulic, Coastal and Environmental Engineering, (Doboku Gakkai Ronbunshuu B), Japan Society of Civil Engineers, Vol. 62, No. 4, pp.437-446.
- 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.