A roughness-dependent model
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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
For local equilibrium, we are able to find the mean velocity profile from the turbulent kinetic energy (equation 4) and the mixing length (equation 5), by:
(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).
Figure 1, Vertical distribution of mean flow velocity. ; ; Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data ([Sukhodolov et al. (1998)]). a) profile 2: ; ; . b) profile 4: ; ; ; (values of are from [Sukhodolov et al. (1998)]); Figure from [Absi (2006)].
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., Vol. 34, pp. 1317-1334.