A roughness-dependent model
Two-equation - eddy viscosity model
One-equation eddy viscosity model
Algebraic eddy viscosity model
is the mixing length.
Algebraic model for the turbulent kinetic energy
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)]:
, is the hydrodynamic roughness. For a smooth wall ():
the algebraic eddy viscosity model is therefore
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:
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)].
- 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.