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 -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., Vol. 34, pp. 1317-1334.