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BED EVOLUTION IN CHANNEL BENDS
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| BED EVOLUTION IN CHANNEL BENDS By Chin-lien Yen,1 Member, ASCE, an£ Shin-ya Ho2 ABSTRACT: A numerical model for simulation of temporal bed evolution in chan-nel lends with fixed walls is constructed and verified with laboratory experiments. The model is based on governing equations of flow and sediment motion, with some simplifying assumptions. Numerical experiments are conducted, using the model to investigate influences of various factors on bed evolution. It is found that the process of bed evolution in channel bends with steady discharge and uniform sediment can be approximated by a complementary exponential decay function of a time parameter. A regression equation is established for the coefficient as a func¬tion of flow condition, bend geometry, and sediment size. These relations, the numerical model, or both, may be applied to evaluate evolution of bed topography in river bends. Numerical results arc also verified with laboratory experiments. INTRODUCTION Flow characteristics and sediment movement arc much more complex in channel bends than in straight channels. Rozovskii (1961) and Yen (1965) studied flow characteristics in bends with flat beds, theoretically and ex¬perimentally. In the last two decades, many mathematical models have been developed to study flow characteristics in bends, such as the two-dimen¬sional models of Huang ct al. (1967), De Vriend (1976), Smith and McLean (1984), and Ali (1985), and the k - є models of Leschziner and Rodi (1979) and Tamai and Ikeya (1985). Some of these models have incorporated the effects of bed topography in the flow analyses, but they are basically dealing with the case of fixed topography. Interactions between flow and bed topography in movable-beds have been studied by a large number of investigators, with emphases on problems of sedimentation and formation of bed topography. Yen (1967, J970) investi¬gated equilibrium bed topography and its effects on flow in a channel bend with fixed walls. Engelund (1974) analyzed movement of sediment in bends, and employed sediment continuity and transport formulas to predict equilib¬rium bed topography. Kikkawa et al. (1976) showed that bed evolution in the fully developed region of channel bends can be simulated by an uncou¬pled scheme (with changes in flow and bed elevation computed separately in each time step). Onishi et al. (1976) suggested that bed topography en¬hances nonuniformity of unit water discharge, resulting in greater sediment transport. Zimmermann and Kennedy (1978), Falcon (1979), and Odgaard (1981) analyzed transverse bed slope in the fully developed region of bend, and concluded that the weight of sediment particles and shear force are the dominant factors influencing transverse movement of sediment. Struiksma et al. (1985) investigated the wavelength and amplitude of bend deformation, and also derived a relation for transverse bed profile in the fully developed region of bend. Blondeaux and Scminara (1985) studied the mechanisms of meander initiation and its growth, and found that alternate-bar formation and bend amplification arc due to different mechanisms. Ikeda and Nishimura (1986) showed that inclusion of suspended load can increase the maximum scour depth by as much as 8%, and that secondary flow in a sinuous bend has a phase lag relative to the bend's plan form. Odgaard (1986a, 1936b) considered transverse mass shift to be due to secondary flow and bed to-pography, and employed mass-flux balance to link the equation for equilib¬rium bed profile in bend to the momentum equations, lkeda et al. (1987) found that sorting of bed material can reduce the maximum equilibrium scour depth as much as 30-40% in the fully developed region of uniformly curved bends. From the preceding, it is seen that flow characteristics, sediment move¬ment, and equilibrium bed topography in channel bends have been investi¬gated in great detail. However, until now, temporal evolution of bed topog¬raphy has not been studied to the same extent. Particularly, there appears to be little in the literature concerning bed evolution in the developing region of bends. In addition, the major controlling parameters need to be more fully explored to obtain a better understanding of the bed-evolution process under various flow conditions. The present study looks into the process of bed evolution in channel bend, with particular emphasis on the developing re¬gion. A numerical model, consisting of a depth-averaged flow simulation model and a bed-simulation model, is developed for the analysis. Laboratory experiments arc also conducted to verify the numerical results. From a series of numerical experiments using the model, a time-scale parameter for bed evolution and an equation relating the evolution process to major parameters arc developed. With the bed-evolution equation and the time-scale parameter obtained, it is possible to estimate scour depth and deposition height in the bend under different flow conditions at various times and locations. The numerical model developed herein can also be employed to simulate evo¬lution of bed topography in river bends where detailed information about scouring and deposition may be required in engineering design. THEORETICAL CONSIDERATIONS Flow Conditions In the present study, velocity distribution and depth variation in a channel bend are described in a orthogonal, curvilinear coordinate system, as defined in Fig. 1. The s axis is along the mean flow direction; the n axis is normal to the s axis and positive toward the concave bank; and the z axis is normal to the mean longitudinal bed slope, positive upward. The velocity components in the s and n directions are denoted by us and , respectively. Under the assumptions that: (1) Pressure distribution is hydrostatic; (2) Boussinesq's similarity relation is valid; (3) small-order terms are negligible for the case with h/B < < 1 and h/r << 1, where h , B, and r are the water depth, channel width and bend radius, respectively (Yen 1965); and (4) flow is steady, the depth-averaged equations of motion for s and n directions arc expressed as follows FIG. 1. Definition Sketch (1) (2) in which ; r= local radius of curvature; rc = radius of curvature along the bend centerline; h = water depth; g = gravitational acceleration; , and are water-surface slopes in s and n directions, respectively; and are the s and n components of bed shear, respectively; = mass density of water; and the bar denotes depth-averaged quantity. For flow with a given discharge, the continuity equation can be written as . (3) in which Q = discharge; = s component of depth-averaged velocity; and n0 are n coordinates of inner and outer banks, respectively. In the three preceding equations, Ss and Sn can be expressed in terms of water-surface elevations at various locations. The bed-shear components, and , are related to , h, and the friction factor . Therefore, the unknown variables are , , and h, which, in principle, can be solved by using numerical techniques. However, difficulties often arise from numerical instability and an extremely large volume of computations. In order to simplify the com¬putations, some approximate distributions of us and un are employed in the present study. The approximations, adopted from various published results, are briefly described in the following. 1. The profile of along a vertical is approximated by the power law | |
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