Experiments on Flow in Open-Channel Bends

 

K. BLANCKAERT and W.H. GRAF

 

Laboratoire de Recherches Hydrauliques, Ecole Polytechnique Fédérale

Lausanne, Switzerland

Tel: +41-(0)21-6932375; fax: +41-(0)21-6936767; E-mail: lrh@epfl.ch

 

 

ABSTRACT

Experimental research is reported on flow in a curved open channel, with emphasis on the outer bank region. Simultaneous measurements of the three instantaneous velocity components were made in one section, from which the time-averaged velocity components, , were derived. This paper is concerned mainly with the time-averaged secondary motion, (v_n,v_z), whose measurements on a fine grid and with high precision resulted in a detailed presentation of the flow pattern and permitted to calculate accurately the downstream vorticity component, . The patterns of secondary motion, (v_n,v_z), downstream vorticity, , and downstream velocity, v_s, are presented.

 

Keywords: curved open-channel flow, secondary motion, vorticity

 

INTRODUCTION

A characteristic feature of curved flow is the helical flow pattern. Due to its convective capacity, it plays an essential role in mixing and transport processes. It modifies the velocity distribution, the shear exerted on the flow boundaries and thus the morphology. Most engineering problems in channel bends are concerned with the outer bank region (bank erosion, maximum pool depth). Herewith we report on an experimental study of the dynamics of the helical flow pattern, with emphasis on the outer bank region.

In a section of the bank region, simultaneous measurements of the three instantaneous velocity components, v_i(t) (i=s,n,z), have been made with an Acoustic Doppler Velocity Profiler (ADVP), which allowed to calculate the time-averaged velocity, , and would allow to obtain the Reynolds stresses, . The layout and topographic measurements are shown in fig.1. Attention is focused on the secondary motion, (v_n,v_z). The measurements of both components, (v_n,v_z), resulted in a detailed presentation of the flow pattern and permitted to calculate accurately the downstream vorticity component, . Patterns of the secondary motion and of the downstream vorticity are shown, containing a clockwise turning circulation cell in the center of the channel, accompanied by a weaker counterclockwise cell in the outer bank region. Furthermore, profiles of downstream velocity, v_s, are reported that deviate from the straight-flow profiles.

 

EXPERIMENTAL SET-UP

Experiments were performed in a 40[cm] wide laboratory flume, made of plexiglas, consisting of a 2[m] long straight approach section, followed by a 120[°] bend with a constant radius of curvature of R=2[m] on the centerline. With a nearly uniform sand, d₅₀=2.1[mm] and clear-water scour conditions a typical bar-pool topography was obtained (see fig.1). The transversal bottom and free surface slope show an out-of-phase oscillating behaviour, as predicted by Odgaard (1986). The discharge was Q=17[l/s], the reach-averaged water depth and velocity were h=11.2[cm] and U=Q/Bh=38[cm/s], and the average downstream water surface slope was S_s=1.89[?] on the centerline.

The measuring section was located at 60[°] from the bend entry, being downstream of the end of the point bar (see fig.1). Measurements were made with an Acoustic Doppler Velocity Profiler (ADVP), developed in our laboratory (Lemmin and Rolland, 1997). A central transducer insonifies a water column; the signal backscattered from the targets moving with the water is recorded by four receivers. An analysis of the Doppler information permits to compute instantaneous profiles of the three velocity components. Usually, the instrument is mounted on the water surface. However, near the outer bank, fluctuations of the water surface exist. In order to avoid perturbing the free surface, an installation was developed to measure directly across the plexiglas sidewall. The measuring volume is cylindrical with a diameter of ~1[cm] and height of h~3[mm]. The sampling frequency was 22.3[Hz] and the sampling time was 180[s]. The reproducibility of the data was tested by measuring in two overlapping zones (see fig.2). The results shown in this paper are from zone 1. The installation as well as the measuring grid (n=3[mm] x z=5[mm]) are shown in fig.2. The accuracy on the mean velocity components is estimated to be 2[mm/s] or 1[%].

 

TIME-AVERAGED FLOW FIELD

 

Observations

Profiles of the downstream velocity component, v_s, are presented in fig.3. The measured velocities of v_s~55[cm/s] are higher than the reach-averaged velocity of U=Q/(Bh)=38[cm/s]; this shows that the discharge is concentrated over the deepest part of the cross-section. Contrary to straight uniform flow, the maximum velocity lies in the lower part of the depth, which is due to convective momentum exchanges by the secondary motion. The vertical distribution of v_s is of major importance for the dynamics of the secondary motion (see later).

The measured pattern of the secondary motion, (v_n,v_z), is shown in fig.4. Near the channel center, a clockwise circulation cell with outward velocities near the surface and inward velocities near the bottom is observed. On the average, the secondary velocities (v_n,v_z) are about 4[cm/s]. This cell does not extent into the zone near the outer bank, where a counterclockwise circulation cell (named outer bank cell) is discernable. This one is weaker than the major cell and contains average secondary velocities (v_n,v_z) of about 1[cm/s]. These observations can be summarised as:

 

and U < v_s

 

The outer bank cell could not be visualised with dye injection. An explanation for this can be found in the behaviour of the instantaneous transversal velocity, v_n(t). Fig.6 shows a record of 27[s] for points A and B (see fig.4) in the eye of the major circulation cell and the outer bank cell. In point A in the eye of the major cell, the time-averaged transversal velocity was v_n=4.98[cm/s] and its fluctuation component was =2.06[cm/s]. In point B in the eye of the outer bank cell, the corresponding values were v_n=-0.52[cm/s] and =2.17[cm/s]. These important fluctuations cause the major cell to be highly unsteady, but always turning in the clockwise sense. In the weaker outer bank cell, however, frequent sign reversal of the transversal velocity is noticeable. It is only after long averaging periods (>10[s]) that the outer bank cell becomes visible.

A major contribution of this paper is the presentation of the downstream vorticity field,

.

Although the secondary motion, (v_n,v_z), was measured with high precision on a fine grid, the evaluation of the derivatives by finite-differences was not sufficiently accurate. The following procedure was adopted. The random fluctuations in the measured profiles, of the order of the measuring precision, were filtered by fitting an analytical function to the data. Smooth profiles of the downstream vorticity, , were consequently obtained by evaluating the derivatives of the analytical functions. The resulting cross-sectional distribution of the downstream vorticity component, , shows clearly the two circulation cells (see fig.5). The maximum vorticity, appearing in the eye of the cells, is =4[s^-1] for the clockwise cell and =-1[s^-1] for the outer bank cell; this agrees with the relative strength of both cells as derived from the average secondary motion (v_n,v_z) (see fig.4). The definition of downstream vorticity, , yields an objective criterion to delimit both cells. The outer bank cell approximately occupies a square of size 2/3h_max in the corner formed by the free surface and the outer bank.

 

COMMENTARY

The center region cell - resulting in a helical flow pattern - has been studied extensively in curved channel flow for a wide range of parameters such as curvature, Froude-number, roughness, fixed or movable bed, etc. (Rozovskii, 1957, and others). Under simplifying assumptions, analytical expressions for the distribution of the velocity vector, (v_n,v_z) have been derived. These investigations explain the center region cell as follows. While the outward centrifugal force, v_s²/R, has a vertical distribution, the transversal pressure gradient - induced by the superelevation of the free surface - is quasi-uniformly distributed over the depth. It is the local non-equilibrium between both that gives rise to the center region cell. Since this mechanism depends on the centrifugal force, it is not encountered in straight flow. It is sensitive to the vertical distribution of v_s (see fig.3), which is amplified in the vertical distribution of the centrifugal force.

Contrary to the center region cell, the outer bank cell has not always been observed in previous experiments on flow in channel bends. Possibly it is not well documented because in most investigations the measuring grid was too coarse and the accuracy too low to measure the small velocities (<1[cm/s]). The most detailed observations of the outer bank cell in bends were made in field measurements by Bathurst et al. (1979), who remarked its importance for the bank stability. It is not clear under what conditions this circulation cell can persist beside the major circulation cell. The outer bank cell resembles the secondary circulation cells encountered in straight flow. These cells, which have been extensively studied, have typically a strength of about 2[%] of the maximum downstream velocity and cover a width of about twice the local water depth (Nezu and Nakagawa, 1993, ch.5.3). The observed outer bank cell has a comparable strength but occupies a narrower width. This could be explained by the outward velocities near the surface contained in the center region cell, which deplace the outer bank cell. More research is needed to study the mechanics and the conditions of existance of this cell.

It should be noted that there exists an important difference between the secondary motion, (v_n,v_z), and the secondary circulation, contained in the downstream vorticity component, . This can be clarified by the following decomposition of the secondary motion:

 

and

 

<v_n>_h and <v_z>_b are the depth-averaged transversal velocity and the width-averaged vertical velocity. They represent the irrotational redistribution of mass over the cross-section, mainly due to non-uniformities, such as a varying bottom topography or a change in curvature. For flow in equilibrium with the curvature - which means that it is uniform in downstream direction - these components are identically zero and the secondary motion reduces to (v_n*,v_z*). This indicates that these components, (<v_n>_h, <v_z>_b) are not directly related to curvature, but that the interaction between the curvature and the flow field is contained in the "equilibrium" secondary motion, (v_n*,v_z*). Moreover, (v_n*,v_z*) contains the same vorticity as the total secondary motion, (v_n,v_z):

 

 

In a non-uniform curved flow, such as is always the case over a natural bed topography, the velocities <v_n>_h and <v_z>_b are often important in magnitude, and render it difficult to recognise the different circulation cells in the (v_n,v_z)-pattern. Therefore, in a future study, we will concentrate on the "equilibrium" part of the secondary motion by considering the scalar downstream vorticity, , instead of the vectorial velocities (v_n*,v_z*). The downstream vorticity equation will be used to investigate the dynamics of the secondary circulation. This equation, which has been derived in a curvilinear reference system, shows the different mechanisms generating secondary circulation in curved flow (Blanckaert and Graf, 1997).

 

CONCLUSION

The flow field, , in one section at 60[°] of a channel bend has been measured in detail, focusing on the time-averaged secondary motion, (v_n,v_z). A pattern consisting of a clockwise circulation cell in the center of the channel accompanied by a weaker counterclockwise circulation cell near the outer bank has been observed (see figs.4, 5). The distribution of the downstream vorticity component, , which is of major importance for the study of the dynamics of the secondary motion, has been presented (see fig.5). Furthermore, the vertical distribution of the downstream velocity component, v_s, is shown (see fig.3). It is seen that it is modified by the presence of the helical motion.

The results reported in this paper come from a preliminary experiment, which was designed to test the use of the ADVP instrument in flow around bends. In the future, this research will be enlarged, notably by optimising the experimental infrastructure and comparing different experimental conditions such as varying radius, a fixed horizontal bed against an equilibrium bed, etc.

 

ACKNOWLEDGEMENTS

This research is being sponsored by the Swiss National Science Foundation under grant Nr.2100-052257.97/1. D. Hurther and U. Lemmin are acknowledged for help with the ADVP instrument.

 

REFERENCES

Bathurst, J.C., Thorne, C.R. and Hey R.D. (1979). "Secondary flow and shear stress at river bends." ASCE, J. Hydr. Div. , 105(HY10), 1277-1295.

Blanckaert, K. and Graf, W.H. (1997). "Flow in alluvial channel bends: mathematical framework." Rapport annuel, Lab. Rech. Hydr., Lausanne.

Lemmin, U. and Rolland, T. (1997). "Acoustic velocity profiler for laboratory and field studies." ASCE, J. Hydr. Eng. , 123(12), 1089-1098.

Nezu, I. and Nakagawa, H. (1993). Turbulence in Open-Channel Flows . A. Balkema, Rotterdam.

Odgaard, J.A. (1986). "Meander flow model. I: Development." ASCE, J. Hydr. Eng. , 112(12), 1117-1136.

Rozovskii, I.L. (1957). Flow of Water in Bends of Open Channels . Ac. Sc. Ukr. SSR; Isr. Progr. Sc. Transl., Jerusalem, 1961.

 

 

Fig.1: Channel layout, bottom topography and reference system.

 

Fig.2: Configuration of the ADVP and its measuring zones and grid.

 

Fig.3: Transv. and vert. profiles of the downstream velocity component, v_s [m/s], at 60[°].

 

Fig.4: Velocity vector of the secondary motion, (v_n,v_z) [m/s], at 60[°].

 

Fig.5: Isolines of the downstream vorticity, [1/s], at 60[°].

 

Fig.6: Time series of instantaneous transversal velocity, v_n(t), for points A and B in the eye of the center region and the outer bank cell.