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Analytical models of mean secondary velocities and stream functions under different bed-roughness configurations in wide open-channel turbulent flows

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Abstract

Abstract

Turbulence induced secondary currents are commonly present in straight natural as well as artificial open-channels. Different structures of cellular secondary currents can be seen in open-channel flows due to various bed configurations. In our study, mathematical models of turbulence-induced secondary currents in the vertical and transverse directions within a straight open rectangular channel with alternate rough and smooth longitudinal bed strips are proposed. The proposed models are derived using theoretical and mathematical analysis with appropriate boundary condition. Most of the previous models of secondary currents in the literature are proposed empirically and without proper mathematical derivation including effects of fluid viscosity and eddy diffusivity. The inclusion of these effects make the present study more practical. Initially the governing equation for vertical secondary flow velocity is derived from continuity and the Reynolds-Averaged Navier Stokes equations. Then, the proposed problem is divided into two sub-considerations, corresponding to the base flow and perturbed flow. Finally, these sub-problems are analytically solved using method of separation of variables with suitable boundary conditions. Different models to consider two different types of bed-roughness configurations (i.e. equal and unequal lengths of smooth and rough longitudinal bed strips) are obtained. Apart from velocity formulations, models of the stream function are proposed for these two types of bed configurations. All proposed models are validated using existing experimental data for the various bed configurations in open-channel flows and satisfactory results have been obtained. These present models are also compared with empirical models from the literature and they are found to be more effective in representing both types of bed-roughness configurations. The effects of bed configuration on the streamlines of settling velocity are also investigated. Results show that laterally-skewed secondary cells (which occurs due to unequal smooth and rough bed strips), have significant effects on the closed \(\omega\)-streamlines in terms of shape and location of the center of these closed streamlines. More precisely, it is found that the area of the downflow zone proportionally increases with the length of rough-bed strips.

Article highlights

  • Complete analytical models of secondary flow velocities and stream functions along vertical and transverse directions are proposed.

  • Under two different bed configurations, models gives satisfactory results.

  • Area of the downflow zone proportionally increases with the length of rough bed strips which significantly influence particle settlement.

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Availability of data and materials

All data used in this study is electronically available.

Abbreviations

B :

Channel width (m)

\(C_1\) :

Parameter

Er :

Absolute error value

g :

Gravitational acceleration (m/s)

h :

Flow height (m)

J :

Slope of the channel

K :

Some constant

\(K_b\) :

Amplitude of perturbation

M :

Parameter [=\((4\tilde{\overline{\nu }}_t)/\varLambda\)]

MKE :

Mean Kinetic Energy

N :

Total number of data points

P :

Pressure

\(PS(\tilde{y})\) :

Sinusoidal function

\(PC(\tilde{y})\) :

Derivative of \(PS(\tilde{y})\)

p, q, r :

Mean velocity components along x, y and z directions

\(p',q',r'\) :

Fluctuating velocity components

RANS:

Reynolds-averaged Navier-Stokes equation

\(\tilde{r}_1\) :

Idealized secondary flow

\(\tilde{r}_2\) :

Perturbed flow related to bed configuration

S50, S75:

Run numbers of [48] experiment

SC:

Secondary current

\(U_m\) :

Mean value of primary flow

\(u_*\) :

Shear velocity

\(\overline{u_*}\) :

Averaged shear velocity

\(v_{observe}\) :

Observed value

\(v_{computed}\) :

Observed value

\(W_{max}\) :

Maximum upwelling velocity (m/s)

\(\widetilde{W}_{max}\) :

Dimensionless maximum upwelling velocity (m/s)

x, y, z :

Streamwise, longitudianl and vertical co-ordinates respectively

\(\tilde{x}, \tilde{y}, \tilde{z}\) :

Dimensionless co-ordinates

\(z_0\) :

Zero velocity level (main velocity)

\(\tilde{z}_0\) :

Dimensionless zero velocity level

\(\alpha\) :

Dip correction parameter

\(\alpha _0\) :

Empirical constant

\(\varTheta\) :

Coefficient [=\(\tilde{\nu } +\tilde{\nu }_t\)]

\(\kappa\) :

von Karman constant \((\approx 0.4)\)

\(\varLambda\) :

Coefficient [=\(2(\tilde{\nu } -\tilde{\nu }_t)\)]

\(\lambda\) :

Average width of the strips

\(\lambda _{dn}, \lambda _{up}\) :

Widths of down-flow and up-flow zones

\(\lambda _r\) :

Length of rough bed-strip

\(\lambda _s\) :

Length of smooth bed-strip

\(\nu\) :

Kinematic viscosity

\(\nu _t\) :

Turbulent eddy viscosity

\(\tilde{\nu }_t\) :

Dimensionless turbulent eddy viscosity

\(\bar{\nu }_t\) :

Depth-averaged eddy viscosity

\(\varPi\) :

Cole’s wake parameter \((\approx 0.19)\)

\(\rho\) :

Density of fluid

\(\tau _b\) :

Local bed shear stress

\(\bar{\tau _b}\) :

Average bed shear stress [=\(\rho \overline{u}^2_*\) ]

\(\varphi\) :

A function of y and z

\(\psi\) :

Stream function

\(\tilde{\psi }\) :

Dimensionless stream function

\(\varOmega\) :

Vorticity vector

\(\omega _0\) :

Particle settling velocity in still fluid

\(\overrightarrow{\omega _s}\) :

Settling velocity vector

\(\nabla\) :

Laplacian operator

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Acknowledgements

Authors are very much thankful to the Associate Editor Dr. Eric Pardyjak and the Reviewers for their fruitful and constructive comments which improves the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, NIT Jamshedpur, Jamshedpur, Jharkhand, 831014, India

    Snehasis Kundu & Titas Chattopadhyay

  2. Faculty of Engineering and Informatics, University of Bradford, Bradford, UK

    Jaan H. Pu

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  1. Snehasis Kundu
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  2. Titas Chattopadhyay
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  3. Jaan H. Pu
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Contributions

Dr. K originally formulated the idea of the work and derived the model with solution methods. Ms. C contributed to the preliminary concept of the study under the supervision of Dr. K. Dr. P modified and gave comments of the model. Afterward, Ms. C derived the preliminary solutions and Dr. K modified and corrected them. Ms. C completes the data analysis, modeling, programming and fitting parts and prepared the initial draft of the paper. Dr. K and Dr. P edited, revised and completed the final draft. Finally, all authors approved the final version.

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Correspondence to Snehasis Kundu.

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Kundu, S., Chattopadhyay, T. & Pu, J.H. Analytical models of mean secondary velocities and stream functions under different bed-roughness configurations in wide open-channel turbulent flows. Environ Fluid Mech 22, 159–188 (2022). https://doi.org/10.1007/s10652-022-09835-8

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