A roughness-dependent model
From CFD-Wiki
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 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)]:
 | (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.
