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Turbulent velocity profile in fully-developed open channel flows

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The Original Article was published on 28 August 2008

Abstract

The determination of velocity profile in turbulent narrow open channels is a difficult task due to the significant effects of the anisotropic turbulence that involve the Prandtl’s second type of secondary flow occurring in the cross section. With these currents the maximum velocity appears below the free surface that is called dip phenomenon. The well-known logarithmic law describes the velocity distribution in the inner region of the turbulent boundary layer but it is not adapted to define the velocity profile in the outer region of narrow channels. This paper relies on an analysis of the Navier–Stokes equations and yields a new formulation of the vertical velocity profile in the center region of steady, fully developed turbulent flows in open channels. This formulation is able to predict time averaged primary velocity in the outer region of the turbulent boundary layer for both narrow and wide open channels. The proposed law is based on the knowledge of the aspect ratio and involves a parameter CAr depending on the position of the maximum velocity (ξdip). ξdip may be derived, either from measurements or from an empirical equation given in this paper. A wide range of longitudinal velocity profile data for narrow open channels has been used for validating the model. The agreement between the measured and the computed velocities is rather good, despite the simplification used.

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Abbreviations

Ar:

Aspect ratio Ar = b/h

b:

Free surface width (m)

\({\mathcal{F}}= {\rm F +f}\) :

Statistical approach

F:

Mean component of \({\mathcal{F}}\)

f:

Turbulent fluctuations

g:

Gravitational acceleration (ms−2)

h:

Water depth (m)

I:

Turbulence intensity

k:

Turbulent kinetic energy

ks :

Roughness height (m)

κ :

Von-Karman constant

P:

Pressure (Pa)

Rh :

Hydraulic radius (m)

S0 :

Energy slope (mm−1)

U:

Velocity component in the x direction (ms−1)

Ufs :

Velocity component in the x direction at the free surface (ms−1)

V:

Velocity component in the spanwise y direction (ms−1)

W:

Velocity component in the vertical z direction (ms−1)

Us :

Secondary current velocity (ms−1)

u* :

Shear velocity (ms−1)

z0 :

Roughness length of the surface (m)

δij :

Kronecker’s symbol; δij = 1 if i = j; δij = 0 if i ≠  j;

ξdip :

Relative position of the maximum velocity on a vertical profile

ρ :

density (kgm−3)

μ :

Kinematic viscosity (kgm−3 s−1)

ν :

Dynamic viscosity (m2 s−1)

ν t :

Eddy viscosity

τ p :

Wall shear (Pa)

\(\xi =\frac{\rm{z}}{\rm{h}}\) :

Relative distance from the bottom

\({\rm Re}_{\ast} =\frac{{\rm u}_{\ast} {\rm k}_{\rm s}}{\nu}\) :

Roughness Reynolds number

\({\rm Re}=\frac{{\rm UR}_{\rm h}}{\nu}\) :

Reynolds number

\(\overline{{\rm u}_{\rm i} {\rm u}_{\rm j}}\) :

Reynolds stress tensor

\({\rm z}^{+}=\frac{{\rm u}_{\ast} {\rm z}}{\nu}\) :

Relative distance

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Authors and Affiliations

  1. Department of Civil Engineering, University of Razi, Kermanshah, Iran

    Hossein Bonakdari

  2. Water and Environment Division, LCPC, Route de Bouaye, BP 4129, 44341, Bouguenais Cedex, France

    Hossein Bonakdari, Frédérique Larrarte, Laurent Lassabatere & Claude Joannis

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  1. Hossein Bonakdari
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  2. Frédérique Larrarte
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  3. Laurent Lassabatere
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  4. Claude Joannis
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Correspondence to Frédérique Larrarte.

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Bonakdari, H., Larrarte, F., Lassabatere, L. et al. Turbulent velocity profile in fully-developed open channel flows. Environ Fluid Mech 8, 1–17 (2008). https://doi.org/10.1007/s10652-007-9051-6

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  • DOI: https://doi.org/10.1007/s10652-007-9051-6

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