PATTERNS OF FLOW IN MEANDERING CHANNELS OF VARYING SINUOSITY

A.M. Ferreira Da Silva¹,  M.S. Yalin¹,  T. El-Tahawy¹,  W.D. Tape²

¹ Department of Civil Engineering, Queen’s University

Kingston, Ontario, Canada K7L 3N6

² Civil and Environmental Engineering, University of Windsor

Windsor, Ontario, Canada N9B 3P4

Abstract: The flow patterns of rough turbulent, steady, meandering flows of varying sinuosity are investigated experimentally. In agreement with the behaviour of large-scale natural alluvial streams, it is assumed that the width-to-depth ratio is large; the channel centreline is a sine-generated curve, the banks are rigid, the channel cross-section is rectangular. In contrast to previous theoretical works, where the computed flow patterns were of the same character , irrespective of the sinuosity of the channel, it is found that every different plan geometry of a meandering flow has its own flow pattern. The deviation angle between the vertically-averaged streamlines and the coordinate lines is used to formulate the variation of flow pattern with sinuosity.

Keywords: meandering flows, sine-generated channels, initial flow patterns, sinuosity

1  INTRODUCTION

This paper concerns the plan patterns of meandering flows, and their relation to erosion-deposition patterns. In agreement with the behaviour of large-scale natural alluvial streams, it is assumed that the meandering flow is rough turbulent, and that the width-to-depth ratio is large. The flow rate Q is constant, the channel centreline is a sine-generated curve, the (parallel) banks are rigid. In accordance with the current approach to the determination of bed topography, it is further assumed that the transport is by bed-load, and that the bed deformation can be attributed mainly to the convective behaviour of the vertically-averaged flow (given by the three field functions ,  and ).

Experiment shows that the location of erosion-deposition zones in meandering streams is mainly a function of the deflection angle  (see Fig. 1). If  is “small”, then the zones of most pronounced erosion-deposition are (in the flow plan) in the neighbourhood of the crossovers  and , as shown in the schematic Fig. 1a. If, however,  is “large”, then the zones of most pronounced erosion-deposition are around the apex  (Fig. 1b). For intermediate values of , the zones of most pronounced erosion-deposition are in an intermediate location with respect to those of small and large values of .

It is also known from experiment that the development of bed topography is mainly due to the growth of deposition “hills” and erosion “deeps” in the vertical direction, their location in the flow plan remaining practically unchanged during the bed development time. But this  means that the location in flow plan of erosion-deposition zones is (practically) determined by the flow at the very beginning of a  run (at the time , when the initial movable bed is still flat). i.e. one can say that the “information” on the (subsequent) bed topography is “locked” in the mechanical structure of the flow at  (which henceforward will be referred to as the initial flow). According to the sediment transport continuity equation, the erosion-deposition zones must be in coincidence with the convergence-divergence zones of the flow. But if this is so, then meandering flows with a flat (initial) bed cannot be of a standard nature applicable to any plan geometry of a sine-generated channel. Rather, and in contrast to previous theoretical works (see e.g. [2], [5], [6]) where the computed flow patterns were of the same character, irrespective of the channel sinuosity, one would expect each different plan geometry of a meandering flow to have its own (initial) flow picture.

Fig. 1

The above stated expectation is supported by the recent measurements in sine-generated channels carried out by Whiting and Dietrich 1993 [9], Silva 1995 [4] (see also Silva and Yalin 1997 [3]), Termini 1996 [8]. In these works, it has been found that the flow pattern in channels having “small”  (shown in the schematic Fig. 1c) is indeed rather different from that in channels having “large”  (Fig. 1d). Observe from Fig. 1c that in the case of “small”  the flow is convectively accelerating at the inner bank (approximately) between the crossover-section  and the apex-section ; while in the case of “large”  (Fig. 1d), the flow is convectively decelerating in the same region (i.e. at the inner bank (approximately) between  and ). In Fig. 1c the maximum velocity occurs near the inner bank (approximately) at the apex-section ; in Fig. 1d, it occurs near the inner bank (approximately) at the crossover-section . However, the aforementioned experiments, being restricted to the (two) limiting cases of “small” and “large” , do not give any information on how the flow pattern varies with . The objective of the present paper is to extend the -range of previous experiments as to enable the formulation of the variation of the flow pattern with .

The measurements of Silva 1995 [4], which were carried out in sine-generated channels having  (representing “small” sinuosity) and  (representing “large” sinuosity), are extended with measurements in sine-generated channels having intermediate values of . An effort is made to find suitable functions expressing the variation of the flow pattern with .

The deviation, at a point P, of a vertically-averaged streamline s from the direction of the coordinate lines will be characterized by the deviation angle  (see insert in Fig. 1): if P is on the centreline , then . Clearly, . In a sine-generated channel, the largest cross-sectional  are in the neighbourhood of the centreline. Yet, usually , say, and therefore  is adopted. At the flow boundaries (banks),

2  ALLUVIAL CHANNEL GEOMETRY

As mentioned in the Introduction, the channel centreline is a sine-generated curve [1]. Hence, the variation of  along the channel centreline  is given by

                                             (1)

The meander wavelength  is related to the flow width B by the proportionality

                                             (2)

which is typical of natural alluvial streams (see e.g. [10]). As is well known, in the case of a sine-generated channel the sinuosity  is completely determined by the deflection angle  as

                                            (3)

where  is the Bessel function of the first kind and zero-th order (of the variable ). It should be noticed that when , then  and . However, this can never occur, for when  reaches the value , the meander loops come in contact with each other and the meandering flow pattern is destroyed. Thus  gives the largest practically possible sinuosity of sine-generated channels, viz .

3  EXPERIMENTAL WORK

The present experiments were carried out in two distinct sine-generated laboratory channels having  and , and a rectangular cross-section. With the exception of their -values, the present channels were in every respect (i.e. geometry and hydraulics) similar to those of Silva 1995 [4]. Accordingly, the flow width was ; the meander wavelength was  (Eq. (2)). The channel walls were made of plexiglass; the channel bed was formed by a cohesionless granular material having . The bed surface was scraped so as to have along the channel centreline the desired slope  (the bed slope in the radial direction being zero). The grains of the uppermost layer were then immobilized by spraying the bed surface with a diluted varnish. The geometric characteristics of the present channels and of the channels used by Silva 1995, as well as the experimental conditions for each channel, are summarized in Table 1.

Table   1

The measurement region (effective length) of each channel consisted of three consecutive meander loops, which will be referred to as , , . In each loop, the values of ,  and h were measured in several r-points of eight equally spaced -sections (see Fig. 2 where the measurement sections of the -channel are shown), while  was measured only at the centreline point (i.e. only  was measured). Extensive description of these measurements can be found in Refs. [4], [7].

Fig. 2

4  RESULTS

For each channel, the measured values of  are plotted versus  in Fig. 3a. In this Figure each data-point corresponds to the average of the values of  measured at the “equivalent” -sections of , , . The solid lines in this graph are the sine-curves which best fit the data pattern. Keeping in mind that , and thus  signify the ends of the accelerated/decelerated zones of the flow, Fig. 3a shows that:

(1) For  (“small” ),  maintains its sign throughout the flow region of the length , whose upstream- and downstream-ends are very near (but slightly upstream of) two consecutive apex-sections  and  ((–) sign between  and , and (+) sign between  and ). This type of convective flow is convergent at the inner bank (approximately) between , and divergent at the inner bank (approximately) between .

(2) For  (“large” ),  maintains its positive sign throughout the flow region of the length , whose upstream- and downstream-ends are very near (but slightly downstream of) two consecutive crossover-sections  and  ((+) sign between  and ). This type of convective flow is divergent at the inner bank (approximately) between .

(3) For  and  (“intermediate” ),  maintains its sign in a -long flow region situated somewhere between those mentioned in (a) and (b).

Based on these results, one can assert that the distribution of  along the regions of the length  mentioned can be represented by the curve-family shown in Fig. 3b. In this Figure, the curves  and  are the limiting curves corresponding to  and ; the curves corresponding to a  within the interval  are situated between them. One can say that the -curves “shift” to the left (from  to ) as  increases. The curve-family in Fig. 3b can be represented by

Fig. 3

                                      (4)

both,  and  in Eq. (4) must be expected to be dependent on , and to some extent, on  and  (see [10]). These functions are plotted versus  in Fig. 4 using the data which follows from the experimental runs in Table 1 (which correspond to  and ). Further research should reveal the dependency of  and  on  and .

Fig. 4

5  CONCLUSIONS

It is found that each different plan geometry of a sine-generated meandering flow has its own initial flow picture. The variation of (the vertically-averaged) flow patterns with  (or sinuosity) is formulated in terms of the deviation angle between the vertically-averaged streamlines and the coordinate lines. Further research should be carried out to reveal the influence of  and  on the flow pattern.

Acknowledgements

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is acknowledged.

List of relevant symbols

References

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[3]  Silva, A.M.F., Yalin, M.S. (1997): Laboratory measurements in sine-generated meandering channels. Int. J. Sediment Research, IRTCES, Vol. 12, No. 2, Aug.

[4]  Silva, A.M.F.  (1995): Turbulent flow in sine-generated meandering channels. Ph.D. Thesis, Department of Civil Engineering, Queen’s University, Kingston, Canada.

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[8]  Termini, D. (1996): Evoluzione di un canale meandriforme a fondo inizialmento piano: studio teorico-sperimentale del fondo e le caratteristiche cinematiche iniziali della corrente. Ph.D. Thesis, Dept. Hydraulic and Environmental Applications, University of Palermo, Italy.

[9]  Whiting, P.J., Dietrich, W.E. (1993): Experimental studies of bed topography and flow patterns in large-amplitude meanders. 2. Mechanisms. Water Resources Research, Vol. 29, No. 11, Nov.

[10] Yalin, M.S., Silva, A.M.F.: Fluvial processes. IAHR Monograph, The Netherlands (In Press).