SUBCRITICAL OPEN CHANNEL EXPANSIONS WITH RIGID AND MOVABLE BEDS

Abstract

The flow in a suddenly expanding rectangular channel with rigid and movable beds is studied experimentally in this thesis. The optimum shape of transition for rigid bed expansions is also studied analytically. Flow through sudden expansion separates from the boundary, causing zones of separation and a zone of decelerating flow. Earlier investi gators have found that the flow in two-dimensional rigid bed sudden expansion is unsymmetrical and unstable at expansion ratios \/\ greater than 1.5. Here Bx and B2 are channel widths before and after the expansions. Asymmetric flow in expansion gives unequal reattach ment lengths and hence the line joining the points of maximum velocity is inclined w. r.t. centre line of expansion. The shift in location of maximum velocity is higher for larger expansion ratios. The flow in sudden expansions with movable bed has been earlier studied to a very limited extent. The flow in sudden expansion with movable bed is more symmetrical and stable than the flow in rigid bed expansion £4,5]. Some work has been done on the optimum shape of transi tions in expanding channels £9 ], however there is no theore tical analysis of this problem with a view to relating the transition shape with the flow conditions. The experiments were carried out using three symmetri cal expansions in rigid bed open channels. With B^/B, equal to 1.5, 2.0, 3.0 and four symmetrical expansions in movable bed open channels with EU/B. equal to 1.5, 2.0, 3,0 and 4.5. The range of Reynolds number in unexpanded channel was 2.49x10 to 6.5x10 . The experiments were planned and carried out in such a way as to provide detailed information on velo city distribution, head loss, bed shear stress on the smooth rigid bed in sudden expansion and flow and scour characteris tics in expanding movable channels. The form loss in a short length dx and friction loss in this reach have been computed using equation hQ = |~ (U-^ - uS and the Mannings equation. The momentum equation in the form of back water equation is also satisfied and total energy loss is minimised. In this procedure new values of dB/dx and hence width Bx at any distance x are obtained. The procedure is repeated for newly obtained profile and the iteration is stopped when at every point dB/dx changes in successive iterations is negligibly small. This is achieved by using gradient (steepest descent) minimization technique. The resulting profile followed LebedeV s equation with n^ equal to 0.55. From the experimental results it is found that in the separation zone the power law velocity distribution is suitable for velocity distribution in vertical and the exponent in this law increases from 4.3 to 6.5 in the expansion in the down stream direction. For both the rigid and the movable bed expansions the mean velocity profiles in transverse sections in the zone of established flow, follow a cosine-type distri bution namely, hxz in . z \0.45 ,,* — = (C0S kvF") d) XO V in which kv is the coefficient which is found to have a value of 1.32. Here U is the mean velocity at any value of x and for lateral distance z measured from the axis of jet. The law seems to be valid for expansion ratios less than 3.0, for larger expansion ratios, i.e. B^B, = 4.5, the Gussian distri bution was found to be valid in movable bed expansion. The shape of separating stream lines has been studied using Lebedev* s relation. The value of n, given by Lebedev is 0.5. In present study for rigid bed, the mean value of n, for the short length eddy is found to be 0.62 which is close to value of nfe (0.6) suggested by Garde et al. £l5 ]. For movable bed n^, obtained is 0.68. The bed shear distribution at different horizontal cross-sections of rigid bed expansions is found to be similar, following cosine-type distribution. The variation of skin friction coefficient Cf along the length and across the width has been studied. Scour depth profiles in the case of movable bed expan sion are found to conform to the cosine-law of velocity distribution or 522. =(cos 1.32 f-)0.45 (0. xo v ^d) Here Sv„ is depth of scour at any intermediate horizontal cross section below the original bed and S is maximum value XO of S at the centre line of the channel. S /S is found XZ XO XOO to be related to x/x where S ^ is defined the maximum of ' o xoo the maximum scour depth occuring at distance x from the beginning of expansion which is governed by the following relationship itt. 0.039 (-^=^(^4^)0-44 (3) in which U-, and Y-, are average velocity and depth at the beginning of expansion and d is the median sediment size. Further the deposition bar, its dimensions and loca tion have been studied. The average height of deposition bar and its location were found to be governed by the relations? AS=0.01 (J!^ ( V^O.575 Of and (4) 4 2Z_P_ r-i 1 ^B2~B1\ / x_n0.63 /•-* •-*• = L1 + o (—") (—) (5) °1 \ B2_B1 in which a S and z are the average height of deposition bar and distance measured from centre line of the channel up to the bar.