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Effect of lateral bed roughness variation on particle suspension in open channels

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Abstract

In this paper the effect of variation of bed roughness along lateral direction on suspension concentration distribution in open channel turbulent flows was investigated. Starting from the mass and momentum conservation equations, this study demonstrates that both the Reynolds shear stress \((-\overline{u'v'})\) and sediment diffusivity depends on bed roughness. From the theoretical analysis, it is found that both the Reynolds shear stress and the sediment diffusivity increase over smooth bed surfaces and decrease over rough bed surfaces. At the junction of smooth and rough bed surface, the effect of bed roughness on the Reynolds shear stress and sediment diffusion is almost negligible. Including this effect, suspension concentration distribution is also studied and from the Hunt’s diffusion equation, an analytical model for predicting suspension concentration is proposed. Apart from this effect, the effects of moveable bed roughness and stratification are also considered in the model. It is observed that the Rouse equation is obtained from the proposed model as a special case when the flow is considered as single phase and there is no effect of secondary current, stratification and bed roughness variation. On the basis of experimental data available in literature, the proposed model is validated and also compared with the Rouse equation. To get a quantitative idea about the goodness of fit, weighted relative error is calculated. The comparison results and calculated errors indicate that the present model is capable of describing the suspension concentration distribution more accurately than Rouse model throughout the flow depth in open channel flow.

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Abbreviations

\(A+1\) :

\((=\rho _{\rm s}/\rho _{\rm f})\) Specific gravity of particles

Ar:

\((=b/h)\) Channel aspect ratio

\({\rm Ar}_{\rm crit}\) :

Critical aspect ratio

\(B_{\rm s}\) :

Log law constant

b :

Channel width

C :

Volumetric sediment concentration

\(C_{\rm a}\) :

Reference concentration

\(C_{\rm d}\) :

Drag coefficient

\(C_{\rm m}\) :

Maximum volumetric concentration

d :

Particle diameter

\(d_*\) :

Dimensionless particle diameter

g :

Gravitational acceleration

h :

Flow depth

\(k_{\rm s}\) :

Bed roughness height

\(l_{\rm r}\) :

\((=k_{\rm s}/h)\) Relative bed roughness height

\(M_*\) :

Density coefficient of bed material

n :

Number of data points

\(P_*\) :

Grain size percentage

\({\rm Re}_*\) :

Roughness Reynolds number

\({\rm R}_i\) :

Richardson number

S :

Channel slope

\(S_*\) :

Fluid-sediment parameter

\(S_1\) :

Sum of residuals

\(S_{\rm c}\) :

(\(1/\gamma \)) Schmidt number

\(S_{\rm co}\) :

Computed concentration

\(S_{\rm o}\) :

Observed concentration

u :

Streamwise velocity

\(u'\) :

Fluctuating part of u

\(u_*\) :

Shear velocity

\(u_{\rm surf}\) :

Fluid surface velocity

v :

Vertical component of velocity

\(v'\) :

Fluctuating part of v

\(V_{\rm wind}\) :

Wind velocity

w :

Lateral component of velocity

\(w'\) :

Fluctuating part of w

\(\overline{u'v'}\), \(\overline{u'w'}\) :

Reynolds shear stresses

x :

Longitudinal co-ordinate

y :

Vertical co-ordinate

\(y_0\) :

Moveable bed roughness height

z :

Lateral co-ordinate

\(\alpha \) :

A parameter

\(\beta \) :

Parameter

\(\beta _1\) :

Stratification parameter

\(\varepsilon _{\rm s}\) :

Sediment diffusion coefficient

\(\varepsilon _{\rm sn}\) :

Sediment diffusivity in neutral flow

\(\varepsilon _{\rm m}\) :

Momentum diffusion coefficient

\(\varepsilon _{\rm mn}\) :

Momentum diffusivity in neutral flow

\(\gamma \) :

Proportionality constant

\(\kappa \) :

von Karman coefficient

\(\mu \) :

Dynamic viscosity

\(\mu _*\) :

Relative viscosity

\(\nu \) :

Kinematic viscosity

\(\psi \) :

Shields parameter

\(\psi _*\) :

Critical shields parameter

\(\phi \), \(\varPhi \), \(\varPsi \) :

functions

\(\rho _{\rm f}\) :

Density of fluid

\(\rho _{\rm air}\) :

Density of air

\(\theta \) :

Angle for channel slope

\(\tau \) :

Total shear stress

\(\tau _{\rm b}\) :

Bed shear stress

\(\tau _{xy}\), \(\tau _{xz}\) :

Components of total shear stresses

\(\eta \) :

\((=z/h)\) Dimensionless lateral coordinate

\(\xi \) :

\((=y/h\)) Dimensionless depth

\(\omega _0\) :

Particle settling velocity

\(\omega _*\) :

Relative settling velocity

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  1. Department of Mathematics, NIST, Berhampur, Orissa, 761008, India

    Snehasis Kundu

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Kundu, S. Effect of lateral bed roughness variation on particle suspension in open channels. Environ Earth Sci 75, 631 (2016). https://doi.org/10.1007/s12665-016-5418-7

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