A roughness-dependent model
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==The mean velocity profile== | ==The mean velocity profile== | ||
| - | + | For local equilibrium, we are able to find the mean velocity profile <math>u</math> from the turbulent kinetic energy <math>k</math> (equation 4) and the mixing length <math>l_m</math> (equation 5), by: | |
| - | + | ||
<table width="70%"><tr><td> | <table width="70%"><tr><td> | ||
<math> | <math> | ||
{{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} | {{d u} \over {d y}} = C_{\mu}^{1 \over 4} {{k^{1 \over 2}} \over {l_m}} | ||
</math></td><td width="5%">(8)</td></tr></table> | </math></td><td width="5%">(8)</td></tr></table> | ||
| - | + | ||
| + | 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). | ||
[[Image:fig7a.jpg]] | [[Image:fig7a.jpg]] | ||
[[Image:fig7b.jpg]] | [[Image:fig7b.jpg]] | ||
| - | + | '''Figure 1''', Vertical distribution of mean flow velocity. | |
<math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; | <math>A = {{h} \over {c_1}}</math>; <math>c_1 = 1</math>; | ||
| - | Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data (Sukhodolov et al). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov et al. (1998)]]]. | + | Dash-dotted line: logarithmic profile; solid line: obtained from equation (8); symbols: experimental data ([[#References|[Sukhodolov ''et al.'' (1998)]]]). a) profile 2: <math>y_0 = 0.062 cm</math>; <math>h = 145 cm</math>; <math>u_\tau = 3.82 cm/s</math>. b) profile 4: <math>y_0 = 0.113 cm</math>; <math>h = 164.5 cm</math>; <math>u_\tau = 3.97 cm/s</math> ; (values of <math>y_0 , h, u_\tau</math> are from [[#References|[Sukhodolov ''et al.'' (1998)]]]); Figure from [[#References|[Absi (2006)]]]. |
== References == | == References == | ||
Latest revision as of 12:47, 22 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
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.
