Grain sorting process and scour in a curved channel with coarse gravel bed 

Anton Schleiss, Daniel Hersberger

Laboratory of Hydraulic Constructions, Swiss Federal Institute of Technology Lausanne
EPFL-LCH, CH-1015 Lausanne, Switzerland

Phone: +41 21 693 23 85; Fax: +41 21 693 22 64;

E-mail: secretariat.lch@epfl.ch;http://lchwww.epfl.ch

 Abstract: In a curved channel with a movable bed, systematic laboratory tests were carried out by using wide grain size distribution and rather high longitudinal slopes as found in pre-alpine rivers in Europe. Scour formations at the outer side of the bend and the transversal grain sorting process were studied in detail. The scour hole is armored by grain sorting. At equilibrium condition of the scour, this armoring layer has a mean grain size which is typically almost the double of the initial mixture. At the inner side of the bend, a bank of fine material is formed with a mean diameter of about 60% of the initial grain size distribution. Grain sorting process seems to highly influence the maximum scour depth. A comparison with some empirical scour formulas revealed that in general these underestimate transversal bed slopes and maximum scour for such conditions.

 Keywords: sediment transport, curved channel, scouring at bends, grain sorting process, coarse gravel bed, grain size distribution.

1    Introduction

In the framework of a research project, which deals with the roughness effect of outside protection walls on flow and scouring in river bends, systematic laboratory tests in a flume were carried out using a coarse, wide grain size distribution. In the following, some key observations made in a channel with vertical smooth walls are presented, focussing on the lateral grain sorting process and the scour formation in the bend.

2    Experimental set-up and basic parameters

The experimental installation consists of an about 30 m long, 1 m wide flume with a 90° curve of 6 m radius. The mobile flume bed was composed of a coarse sand-gravel mixture. The mixture was selected according to observed grain size distributions in typical pre-alpine rivers in Europe. The grain sizes vary between 2 mm and 32 mm with a mean diameter

where d_i is the mean diameter of a fraction i and Δp_i the weight of the fraction i. The standard deviation of the grain size distribution, defined as  for the used mixture was about 1.8.

The range of discharges investigated varies from 150 l/s to 210 l/s. All test series were performed for three bed slopes 0.35%, 0.50% and 0.70%, for which a subcritical regime in the straight approach stretch upstream of the bend could be maintained. Furthermore, the tests have been carried out under equilibrium conditions for the bed load transport in the straight upstream entry reach. The quantity of the sediments supplied into the channel was adjusted according to discharge and to bed slope in order to maintain this dynamic equilibrium condition.

Before the real scouring tests a rather small discharge (about 70 l/s), which was not yet strong enough to produce scour in the bend, was applied to form a certain armouring of the channel bed. Then the discharge was increased to 150 l/s, 180 l/s and 210 l/s respectively and maintained constant until the scour stabilized and the sediment supply in the channel was equal to the outfall at the end of the flume. The main parameters as well as the duration of the test are given in Table 1.

Table 1    Hydraulic characteristics and duration of tests; locations of scour holes in 90° end

3    Number and location of scour holes

As already observed by others (Garbrecht, 1953; Peter, 1986), two scour holes occurred systematically in the bend. The deepest point of the upstream hole is located at the entrance of the bend with an angle varying from 30° to 43° with an average angle of 37°. The downstream hole is located at the exit of the bend at an angle varying from 86° to 97° (= 1 m after the bend) with an average angle of 90° (i.e. at the end of the bend). For higher slopes and increasing discharges, the scour moves slightly downstream.

For the estimation of the location of the scour holes, Peter (1986) proposed the following formulas based on the standard deviation

upstream scour:  ;    downstream scour:   

For the initial mixture with s = 1.82, according to Peter (1986) the first hole should develop at 46°, the second at 114°, which is too far downstream compared to the observed scour holes.

Transversal grain sorting process at scour holes

For all the tests, area samples were taken at the inner and outer sides of the channel at the upstream and downstream scours. All grains located at the surface in an area 15 cm wide and 60 cm long were marked with colour spray, then sampled by standard sieve analysis. The obtained grain size distributions for the upstream scour are presented in Fig.1. Derived characteristic values over all tests are given in Table 2.

Fig. 1    Grain size distributions at outer and inner sides of the channel at upstream scour (area sample) compared to initial mixture (volume sample)

The grain size distributions in the downstream scour hole are very similar to the one given on Fig.1.

Table 2    Characteristic values (mean values and standard-deviations) of grain size distributions at scours

It can be noted that by using a wide grain size distribution, as observed in natural rivers, a significant transversal grain sorting process at the scour occurs. The coarse grains are moved towards the outer wall into the scour hole, the fine sediments towards the inner side forming a bank. An armouring layer is formed by the coarse grains in the scour hole, which limits its maximum depth.

Looking at the standard variation d₈₄/d₁₆ of the grain size distributions, this value is strongly decreased both at the inner and outer bank of the bend in comparison to the initial mixture of the bed (see Table 2). The grain sorting process is somewhat more significant in the region of the first, upstream scour hole. Once scouring has started, the transversal sorting of grains in the bend is almost independent of discharge initial longitudinal bed slope as it can be clearly seen on Fig.1. Analysing all the tests, the following approximate relationship of the sorted mean grain sizes at different locations in the bed and the mean grain size of the initial mixture (bed material) can be given:

Upstream scour:     outer side:   d_{m uo}= 1.9 d_m，  inner side:  d_{m ui} = 0.7 d_m

Downstream scour   outer side:   d_{m do}= 1.8 d_m，  inner side:  d_{m di} = 0.6 d_m

With a somewhat narrower grain size distribution (s = 3.21), Peter (1986) observed for the deepest scour similar relationships (d_{m uo} » 2 d_m and d_{m di} » 0.8…1 d_m).

4    Comparison of measured scour depth with some formulas

At steady state conditions most formulas are based on equilibrium considerations at the grains of the bed. Hence the transversal slope b of the bed is given as a function of flow depth h, local radius r and a factor of proportionality K:

                                (1)

The different formulas for the estimation of the transversal bed slope are characterized mainly by the definition of the proportionality factor K. Assuming that the water surface is almost horizontal (sin b » dh/dr) and K is independent from the radius r, the transversal bed geometry can be obtained by integration of (1)

                              (2)

where h is the local water depth at a radius r for a mean water depth h_m and a bend radius R at the centre line. In Tables 3 and 4 the experimental data are compared with 5 prediction formulas by indicating mean transversal bed slope and medium depth for deepest scour.

Table 3    Calculated mean transversal bed slope at the location of the deepest scour compared to experimental data

Table 4    Calculated maximum scour depth compared to experimental data

Applied to the conditions of the laboratory tests,

Table 3 and Fig.2 (left) indicate that all formulas except Peter's (1986) predict transversal mean slope significantly smaller.

Fig. 2    Comparison of computed results and experimental dataset (left: lateral bed slope [°]; right: scour depth [mm])

It can be seen from Table 4 and Fig.2 right that compared to the test series carried out, all formulas underestimate the maximum scour depth except the formula of Peter (1986). This formula overestimates the maximum scour depth but follows rather close the measured bed level within 60% of the channel width measured from the centreline (see Fig.3).

Fig. 3    Calculated transversal bed elevation compared to experimental data for Q=210 l/s and J_f =0.7%

The difference compared to Engelund's formula (1976) is probably due to the fact that he used slightly meandering channels with a small longitudinal slope to establish his formula. Furthermore, grain size is not considered. Kikkawa et al.'s (1976) formula was developed for rivers in plains, which means mainly for sand bed rivers. Bridge's (1997) approach is similar to Engelund's (1976) but with a somewhat increased transversal slope due to a higher radial force. Odgaard's formula is in principle applicable to a coarse grain bed but adjusted with field data of Sacramento River with a large bend radius / water depth ratio and rather small longitudinal bed slope. Despite armouring, the scour depths are underestimated by most formulas. This is probably due to the fact Bridge and Odgaard based their formulas on measurements in a natural river where scour often doesn't reach its final depth. The formula of Peter (1986) was developed for rather wide grain size distributions, but not very coarse material (d_{90 max} = 7.7mm).

5    Conclusions

In the bend of a curved channel, a wide grain size distribution has an important influence on the scour process. Due to important grain sorting, the scour holes at the outer side of the bend, are armoured by coarse sediments and at the inner side deposits of fine grains can be observed. Most analysed scour formulas (Engelund (1976), Kikkawa et al. (1976), Bridge (1976), Odgaard (1981)) underestimate the lateral bed slope as well as the scour depth, when compared to the systematic laboratory tests carried out by using a wide grain size distribution at rather high longitudinal slopes. This is probably due to the fact that they either don't consider all governing parameters or that they were based on measurements in natural rivers where it is very difficult to measure ultimate scour depth occurred during floods. Furthermore the coarser material in the scour hole allows steeper transversal bed slopes and the growth of the bank at the inner side is facilitated by the fine grains. The formula of Peter (1986) overestimates the scour depth and the lateral bed slope but approaches the bed topography quite well in the central 60% of the channel.

References

Bridge, J.S. (1976): Bed topography and grain size in open channel bends; Sedimentology 23, 407-414.

Engelund, F. (1976): Flow and bed topography in channel bends; Journal of the Hydraulics Division; ASCE 102, 1631-1648.

Garbrecht, G. (1953): Wasserabfluss in gekrümmten Gerinnen; Wasserwirtschaft 2 and 3,44

Kikkawa, H., Ikeda, S. and Kitagawa, A. (1976): Flow and bed topography in curved open channels; Journal of the Hydraulics Division, ASCE 102, 1327-1342.

Odgaard, A.J. (1981): Transverse bed slope in alluvial channel bends; Journal of the Hydraulics Division; ASCE 107, 1677-1694.

Peter, W. (1986): Kurvenkolk –Untersuchungen über die Sohlenausbildung in Flusskrümmungen; Mitteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, Nr.85 – VAW, ETHZ, Zürich, Switzerland.