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Open channel hydraulics for engineers. Chapter 6 transitions and energy dissipators

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The term "transition" is introduced whenever a channel's cross-sectional configuration (shape and dimension) changes along its length. Beside it, in the water control design, engineers need to provide for the dissipation of excess kinetic energy possessed by the downstream flow. Formulas for design calculation of transition works and energy dissipators are presented in this chapter

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Nội dung Text: Open channel hydraulics for engineers. Chapter 6 transitions and energy dissipators

  1.  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Chapter TRANSITIONS AND ENERGY DISSIPATORS _________________________________________________________________________ 6.1. Introduction 6.2. Expansions and Contractions 6.3. Drop structures 6.4. Stilling basins 6.5. Other types of energy dissipators _________________________________________________________________________ Summary The term "transition" is introduced whenever a channel's cross-sectional configuration (shape and dimension) changes along its length. Beside it, in the water control design, engineers need to provide for the dissipation of excess kinetic energy possessed by the downstream flow. Formulas for design calculation of transition works and energy dissipators are presented in this chapter. Key words Transition; expansion; contraction; energy dissipator; drop structure; stilling basin _________________________________________________________________________ 6.1. INTRODUCTION A transition may be defined as a change either in the direction, the slope, or the cross section of the channel that produces a change in the state of the flow. Most transitions produce a permanent change in the flow, but some (e.g. channel bends) produce only transient changes, the flow eventually returning to its original state. Practically all transitions of engineering interest are comparatively short features, although they may effect the flow for a great distance upstream or downstream. In the treatment of transitions, as of every other topic in open channel flow, the distinction between subcritical and supercritical flow is of prime importance. It will be seen that design and performance of many transitions are critically dependent on which one of these two flow regimes is operative. In the design of a control structure there is often a need to provide for the dissipation of excess kinetic energy possessed by the downstream flow. The result is that devices known as energy dissipators are a common feature of control structures. The need for them may arise from the occasional discharge of flood waters, as in the spillway of a dam, or from some other factor. In general two methods are in common use to dissipate the energy of the flow. First, there are abrupt transitions or other features, which induce severe turbulence: in this class we can include sudden changes in direction (such as result from the impact at the base of a free overfall) and sudden expansions (such as in the hydraulic jump). In the second class methods are based on throwing the water a long distance as a free jet, in which form it will readily break up into small drops, which are very substantially retarded before they reach any vulnerable surface. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 107
  2. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 6.2. EXPANSIONS AND CONTRACTIONS 6.2.1. The transition problem We know that the equation of the total energy-head H in an open channel may be written as: g V 2 Hz  (6-1)  2g where z is defined as the height of the bed above datum, h is the vertical distance from the bed to the water surface (in case of parallel stream lines). Above equation is to be used in practice. The problem is essentially similar to the elementary one of calculating the discharge in a pipe from the upstream and throat pressure in a Venturi meter. However, in those problems where the depth at some section is not specified in advance, but is to be calculated from our knowledge of some change in the channel cross-section, we encounter the feature of open channel flow that lends it its special difficulty and interest. It is the fact that the depth h plays a dual role: it influences the energy equation, and also the continuity equation, since it helps to determine the cross-sectional area of flow. The problems involved are the best appreciated by considering the two situations shown in Fig. 6.1, each of them amounting to a simple constriction in the flow passage, smooth enough to make energy losses negligible. Suppose that in each case the problem is to determine conditions within the constriction, if the upstream conditions are given. V12 2g V22 ? 2g flow h1 flow h2 ? z (a) Pipe flow (b) Open channel flow Fig. 6.1. The transition problem In the pipe-flow case we can, from the known reduction in area, readily calculate the increase in velocity and in velocity head, and hence the reduction in pressure. The open- channel-flow case, however, is not quite so straightforward. We have a smooth upward step in the otherwise horizontal floor of a channel having a rectangular cross-section. 6.2.2. Expansions and Contractions These features are often required in artificial channels for a variety of practical purposes. As we shall see, supercritical flow in particular brings about certain complex flow phenomena, which make the simplified viewpoints of Chapter 1 and Chapter 2 quite inadequate. As implied by this last remark, the behaviour of expansions and contractions depends on whether the flow is subcritical or supercritical. The following treatment is subdivided accordingly. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 108
  3. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Subcritical flow If for the time being we postpone consideration of wave formation at changes of channel section, this type of flow raises no problems, which are not already implicit in the theory of Chapters 2, 3 and 4. The problem, which still requires explicit consideration, is that of energy loss when the expansion or contraction is abrupt, and we should expect this problem to be tractable by methods similar to those used in the study of pipe flow. For example, consider the abrupt expansion in width of a rectangular channel shown in plan view in Fig. 6.2. By analogy with the pipe-flow case we would treat this case by setting E1 = E2 and F2 = F3, assuming (a) that the depth across section 2 is constant and equal to the depth at section 1; (b) that the width of the jet of moving water at section 2 is equal to b 1. b1 flow b2 1 2 3 Fig. 6.2. Plan view of abrupt channel expansion Manipulation of the resulting equations is much more awkward than in the pipe-flow case, but if it is assumed that Fr1 is small enough for Fr12 and higher powers to be neglected, according to Henderson (1966), the energy loss between sections 1 and 3 is equal to: V12  b1  2Fr1 b1  b 2  b1   2 2 3 E1  E 3  E   1     (6-2) 2g  b 2     b42 The last term inside the brackets is the open-channel-flow term, which vanishes, as Fr1 tends to zero. In this case h1 = h2 = h3, and the situation is equivalent to closed-conduit flow, i.e.  V1  V3  2 E  (6-3) 2g The term containing Fr12 in Eq. (6-2) does not contribute a great deal to the total energy- head loss unless Fr1 > 0.5, or b2/b 1 < 1.5. The former condition is not often fulfilled, and the latter would, if true, make the total head loss very small, in which case little interest would attach to the relative size of its components. Eq. (6-3) can therefore be recommended as safe for most normal circumstances; in fact the experiments of Formica (1955) have indicated an energy-head loss of sudden expansions some 10 percent less than the value given by this equation. Just as in the pipe-flow case, the energy-head loss is reduced by tapering the side walls; when the taper of the line joining tangent points is 1:4, as in the broken lines in Fig. 6.3.a, the head loss is only about one-third of the value given in Eq. (6-2); it is given by some authorities as: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 109
  4. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  V1  V3  2 E  0.3 (6-4) 2g and by other as  V12 V32  E  0.1   (6-5)  2g 2g  The former of these is to be preferred, but over the range 1.5 < V1/V3 < 2.5 the two equations do not give greatly different results. In any event, a more gradual taper does not usually make savings in energy head commensurate with the extra expense, so the amount of 1:4 is the one normally recommended for channel contractions in subcritical flow. Given that this angle of divergence is to be used, the exact form of the sidewalls is not a matter of great importance, provided that they follow reasonably smooth curves without sharp corners, as in the two cases shown in Fig. 6.3. In the first of these, both upstream and downstream sections are rectangular and the sidewalls are generated by vertical lines; in the second case a warped transition is required to transfer from a trapezoidal to a rectangular channel. 4 1 flow 1 (a) Plan view of rectangular channel 4 channel central line flow (b) Warped transition from trapezoidal to rectangular section Fig. 6.3. Channel expansions for subcritical flow Head losses through contractions are smaller than through expansions, just as in the case of pipe flow. An equation could be derived analogous to Eq. (6-2) with section 2 taken at the vena contracta just downstream of the entrance to the narrower channel, and section 3 where the flow has become uniform again downstream. However, direct experimental measurements provide a better approach, for experiment would be needed in any case to determine the contraction coefficient. The results of Formica (1955) indicate energy-head losses up to 0.23 V32/2g for square-edged contractions in rectangular channels and up to 0.11 V32/2g when the edge is rounded – e.g. in the cylinder-quadrant type shown in Fig. 6.4. The results of Yarnell (1934) obtained in connection with an investigation into bridge piers indicated larger coefficients – up to 0.35 and 0.18 for square and rounded edges, respectively. Formica’s results showed that the coefficients increased with the ratio h3/b 2, reaching the above maximum values when this ratio reached a value of about 1.3. When h3/b2  1, these coefficients reduced to about 0.1 and 0.04. Yarnell did not report values of depth : width ratio. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 110
  5. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- channel flow central line Fig. 6.4. Cylinder-quadrant contraction for subctitical flow Supercritical flow In the preceding discussion on subcritical flow it was assumed that the velocity and the depth remained the same across every section. This assumption is approximately correct, notwithstanding the fact, that within a contraction the velocity may be higher near the sidewalls than it is in midstream. However, when the flow is supercritical there is a further complication in the form of wave motion. This is not confined to supercritical flow, but assumes particular importance when the flow is in that condition. What happens is that any obstacle in the path of the flow generates a surface wave, which moves across the flow and is at the same time carried downstream; the end result is an oblique standing wave, precisely analogous to the Mach waves characteristic of supersonic flow. direction of movement of disturbance A1 A1 A2 A2 shock front successive wave fronts (a) (b) P1 P2 shock front  A2 A1 An (c) Fig. 6.5. Movement of a small disturbance at a speed (a) less than (b) equal to (c) greater than the natural wave velocity ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 111
  6. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- The formation of such waves is illustrated in Fig. 6.5. Consider a mass of stationary fluid, with a solid particle moving through it at a speed V comparable with the natural wave speed c, i.e. the speed with which a disturbance propagates itself through the fluid. When the particle is at A1, it initiates a disturbance which travels outwards at the same velocity in all directions – i.e. at any subsequent instant there is a circular wave front centered at A1. Similar wave fronts are initiated when the particle passes through points A2, A3, etc. When V < c, as in Fig. 6.5a, the particle lags behind the wave fronts; when V = c, as in Fig. 6.5b, the particle moves at the same speed as, and in the same position as, a shock front formed from the accumulated wave fronts generated during the previous motion of the partcle. But when V > c, as in Fig. 6.5c, the particle outstrips the wave fronts. When it reaches An the wave fronts have reached positions such that they can all be enveloped by a common tangent AnP1, which will itself form a distinct wave front. Since a disturbance travels from A1 to P1 in the same time as the particle travels from A1 to An, it follows that:  sin    A1P1 c 1 (6-6) A1A n V Fr  1 V1 V2 shock front Fig. 6.6. Plan view of inclined shock front in supercritical flow A convenient example of a large disturbance is the deflection of a vertical channel wall through a finite angle , as in Fig. 6.6. The oblique wave front then formed will bring about a finite change in depth h, and it is unlikely that the total deflection angle 1 will be given by Eq. (6-6). However we can readily analyze the situation by treating the wave front as a hydraulic jump on which a certain velocity component has been superimposed parallel to the front of the jump; clearly this component must be the same on both sides of the front, for the change in depth h does not bring about any force directed along the font of the jump. We can therefore write, using the terms defined in Fig. 6.6, V1 cos 1  V2 cos  1    (6-7) Considering now the velocity components normal to the wave front, the continuity equation becomes: V1h1 sin 1  V2 h 2 sin  1    (6-8) and the momentum equation must clearly lead to the result: V12 sin 2 1 1 h 2  h 2     1 (6-9) gh1 2 h1  h 1  ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 112
  7. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- which differs from the ordinary hydraulic jump equation (see Eq. (3-15)), only in that V1 is replaced by V1sin1. It follows that; 1 h 2  h2  sin 1    1 1 (6-10) Fr1 2 h1  h1  which reduces to Eq. (6-6) when the disturbance is small and h2/h1 tends to unity. The special case of the small disturbance can be investigated further by eliminating V1/V2 between Eqs. (6-7) and (6-8), leading to the result: tan 1  h2 h1 tan  1    (6-11) Setting h2 = h1 + h, and letting  tend to zero, we obtain tan  V2   dh h d sin  cos  (6-12) g dropping all subscripts. This equation indicates how the depth would increase continuously along a curved wall (Fig. 6.7); each value of  determines a value of h not only at the wall but also a line radiating from the wall as in the figure. A  B surface contours  C positive waves, negative waves, i.e. i.e. increasing depth decreasing depth and converging contours and diverging contours flow Fig. 6.7. Wave patterns due to flow along a curved boundary We may think of this line as representing one of a series of small shocks or wavelets, each originated by a small change in , although in fact there is a continuous change in depth rather than a series of shocks. To be truly consistent with the angle 1 defined in Fig. 6.6,  must be defined as the angle between the boundary tangent and the wave front, as in Fig. 6.7, since the fluid which is about to cross any wave front at any instant is moving parallel to the boundary tangent where that wave front originates; this conclusion is a logical generalization of the picture of events shown in Fig. 6.6. Granted the above definition of , Eq. (6-6) is true, and the second step in Eq. (6-12) is justified. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 113
  8. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 6.3. DROP STURURES 6.3.1. Introduction The simplest case of drop structures is a vertical drop in a wide horizontal channel as presented in Fig. 6.8. In the following sections, we shall assume that the air cavity below the free-falling nappe is adequately ventilated. A drop structure is also called a vertical weir. overfall free falling jet hc upper nappe hydraulic jump lower z nappe tail water h1 level hp h2 Ld roller length air cavity nappe impact hydraulic jump ventilation system tail water level recirculating pool of water air entrainment regions of large bottom pressure fluctuations Fig. 6.8. Sketch of a drop structure 6.3.2. Free overfall In this situation, shown in Fig. 6.9, flow takes place over a drop, which is sharp enough for the lowermost streamline to part company with the channel bed. It has been previously mentioned as a special case (P = 0), see Chapter 5, of the sharp-crested weir, but it is of enough importance to warrant individual treatment. Clearly, an important feature of the flow is the strong departure from hydrostatic pressure distribution, which must exist near the brink, induced by strong vertical components of acceleration in the neighbourhood. The form of this pressure distribution at the brink B will evidently be somewhat as shown in Fig. 6.9, with a mean pressure considerably less than hydrostatic. It should also be clear that at some section A, quite a short distance back from the brink, the vertical accelerations will be small and the pressure will be hydrostatic. Experiment confirms the conclusion suggested by intuition, that from A to B there is pronounced acceleration and reduction in depth, as in Fig. 6.9. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 114
  9. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 0.215 hb A pressure – head distribution hc hb 45 D B C A 3 - 4 hc Fig. 6.9. The free overfall If the upstream channel is steep, the flow at A will be supercritical and determined by upstream conditions. If on the other hand the channel slope is mild, horizontal, or adverse, the flow at A will be critical. This can readily be seen to be true by returning to Chapter 4 where it is stated that flow is critical at the transition from a mild (or horizontal, or adverse) slope to a steep slope. Imagine now that in this case the steep slope is gradually made even steeper, until the lower streamline separates and the overfall condition is reached. The critical section cannot disappear; it simply retreats upstream into the region of hydrostatic pressure, i.e. to A in Fig. 6.9. The local effects of the brink are therefore confined to the region AB; experiment shows this section to be quite short, of the order of 3 – 4 times the depth. Upstream of A the profile will be one of the normal types determined by channel slope and roughness (see Chapter 4); if our interest is confined to longitudinal profiles, the local effect of the brink may be neglected because AB is so short compared with the channel lengths normally considered in profile computations. However, our interest may center on the overfall itself, because of its use either as a form of spillway or as a means of flow measurement, the latter arising from the unique relationship between brink depth and the discharge. Apart from these matters of practical interest, the problem, like that of the sharp-crested weir, continues to attract the exasperated interest of theoreticians who find it difficult to believe that a complete theoretical solution can really be as elusive as it has so far proved to be. In the following discussion, it is convenient to subdivide the flow into two regions of interest; first, the brink itself, and the falling jet, which we may call the “head” of the overfall; and second, the base of the overfall where the jet strikes some lower bed level and proceeds downstream after the dissipation of some energy. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 115
  10. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 6.3.3. The head of the overfall The simplest case is that of a rectangular channel with sidewalls continuing downstream on either side of the free jet, so that the atmosphere has access only to the upper and lower streamlines, not to the sides. This is a two-dimensional case and it is only in this form of the problem that serious attempts have been made at a complete theoretical solution. Consider section C (in Fig. 6.9), a vertical section through the jet far enough downstream for the pressure throughout the jet to be atmospheric, and the horizontal velocity to be constant. If we simplify the problem further by assuming a horizontal channel bed with no resistance, and apply the momentum equation to sections A and C, it can easily be shown (Henderson, 1966) that: 2Fr12  h2 (6-13) h1 1  2Fr12 where the subscripts 1 and 2 characterize sections A and C, respectively; if the flow is critical at section A the above equation becomes:  h2 2 (6-14) hc 3 which sets a lower limit on the brink depth hb; since there is some residual pressure at the brink, hb must be greater than h2. It follows that: 2 h < b
  11. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- The foregoing discussion, although of some general interest, does not lead to specific conclusions. For these we depend on experiment and on approximate analysis. The experiments of Rouse (1936) showed that the brink section has a depth of 0.715 hc. Rouse also pointed out that combination of the weir Eq. (5-6), in Chapter 5, with the critical flow equation: q  Vc h c  h c gh c (6-16) setting H = hc and Vo = Vc, led to the result hb = 0.715 hc, although there might be some doubt about the physical significance of the result. Although no complete theoretical solution has yet been obtained, a number of solutions based on trial or approximate methods have been advanced, most of them offering remarkably close confirmation of Rouse’s result hb = 0.715 hc. Southwell and Vaisey (1946) used relaxation methods to plot the complete flow pattern, finding in the process a value of hb of approximately 0.705 hc. Jaeger (1948) and Roy (1949) used ingenious approximations to obtain near-complete solutions in the neighbourhood of the brink; each found that hb = 0.72 hc. Fraser (1961) used an iterative method due to Woods (1945) to trace the upper and lower streamlines, concluding that hb = 0.71 hc. Hay and Markland (1958) used the electrical analogy to determine an experimental solution in an electrolytic plotting tank. The profile they deduced was very close to Southwell and Vaisey’s (1946) except near the brink, where they found hb = 0.676 hc. 1.0 h hc 0.5 - 1.0 - 0.5 0 0.5 1.0 1.5 x 0.0 hc - 0.5 Southwell and Vaisey (1946) Hay and Markland (electrolytic tank) (1958) Fraser (Wood’s theorem) (1961) Rouse (experiment) (1936) Fig. 6.10. Flow profiles at the free overfall The conclusion suggested by all this work, summarized in Fig. 6.10, is that the brink depth hb = 0.715 hc can safely be used for flow measurement, with a likely error of only 1 or 2%. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 117
  12. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 6.3.4. The base of the overfall The situation is illustrated in Fig. 6.11, which shows a complete “drop structure” such as is installed at intervals in steep channels in order to dissipate energy without scouring the channel. We are concerned here with the events occurring where the jet strikes the floor and turns downstream at section 1. hc vent V zo A Vm Q3 h2 Q3  Q1 B 1 h2 6 Ld Lj 1 2 Fig.6.11. The drop structure 14 E1 Eq. (6-17), (White) yc 12 Experiment (Moore) 10 z o E2 Ej EL 8 hc hc hc hc initial energy 6 4 2 E 0 2 4 6 8 10 12 14 hc Fig. 6.12. Energy dissipation at the base of the free overfall ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 118
  13. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- In this region there will be a great deal of energy loss because of circulation induced by the jet in the pool, which forms beneath the nappe. The function of this pool is to supply the horizontal thrust required to turn the jet into the horizontal direction. The amount of the energy loss has been determined by the experiments of Moore (1943), results of which are plotted in Fig. 6.12. Interesting comment on these results is provided by the analysis of White, who in a discussion on Moore’s paper assumed the following mechanism by which the jet sets up circulation in the pool; near the point A, a thin layer of water, having negligible momentum, is entrained into the jet; it mixes with the jet, the two streams merging into a single one having a uniform velocity Vm. The effect is that the jet becomes thicker and slower moving; then when the jet strikes the floor at B it divides into a main stream moving forward with velocity V1 = Vm, and into a smaller stream which returns to the pool, where it dissipates the momentum which it acquired from the jet. The discharge rate Q3 of this smaller stream is of course equal to the rate of entrainment at A. Application of the momentum equation to this situation leads to the result:  h1 2 (6-17) z o 3 1.06   hc hc 2 by means of which the specific energy at section 1 is readily obtained from the equation: E1 h1 h 2   c2 (6-18) h c h c 2h1 A curve combining the results of Eqs. (6-17) and (6-18) is plotted in Fig. 6.12 and it is seen that the agreement with experiment is remarkably good considering the approximations that must be inherent in this formulation of the problem. 6.3.5. The drop structure Fig. 6.12 shows that the energy loss EL at the base of an overfall may be 50 percent or more of the initial energy, referred to the basin floor as datum. If, as in Fig. 6.11, there is a hydraulic jump downstream of section 1 dissipating further energy, the energy loss in the entire “drop structure” may be very substantial. The loss due to the hydraulic jump is readily calculated in terms of the parameters of Fig. 6.12 (Henderson, 1966), and a curve is plotted on that figure displaced to the left of the E1/hc curve by the amount Ej/hc, where Ej is the loss in the jump. This left-hand curve then indicates the remaining specific energy E2 downstream of the jump. It is seen that the ratio E2/hc does not vary greatly with zo/hc; this suggests that a value of E2/hc may form a satisfactory basis for a preliminary design. Rand (1955) assembled the results of experimental measurements made by himself, by Moore (1943), and others, and from them he obtained the following exponential equations, which fit the data with errors of 5 percent or less:  h  1.275  0.54  c  h1 (6-19) z o  z o   h  0.275  0.54  c  h1 or (6-20) hc  z o  ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 119
  14. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  h  0.81  1.66  c  h2 (6-21) z o  z o   h  0.09  4.30  c  Ld (6-22) z o  z o  L j  6.9  h 2  h1  (6-23) where Ld and Lj are horizontal distances covered by the jet and the hydraulic jump, respectively, as shown in Fig. 6.11. With the help of these equations the designer can proportion the simple drop structure completely. The upward step of h2/6 at the end of the structure, shown in Fig. 6.11, is a standard design feature, which helps to localize the jump immediately below the overfall. It has been assumed in the preceding discussion that the flow at the brink of the overfall is critical, i.e. that the upstream slope is mild. This is often true of the drop structure, whose very purpose is usually to allow the main channel to be laid on the mild slope. However, steep upstream slopes sometimes occur, leading to supercritical flow at the brink. The drop structure described above constitutes a two-dimensional problem and is the simplest type in use. Many variants of this design are used in practice; e.g. a basin may be used that is shorter than Eqs. (6-22) and (6-23) would indicate, embodying special means of forming and locating the jump. A common type is the box-inlet drop structure, shown in Fig. 6.13, which has the advantage of dissipating more energy by making three streams meet at the foot of the drop, (Blaisdell & Donelly, 1956). Fig. 6.13. The box inlet drop structure ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 120
  15. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 6.4. STILLING BASINS 6.4.1. Concept of stilling basin There are structures designed to contain the hydraulic jump, which is used as the energy dissipation device. They also allow a secondary, but no less important function of the jump, that is to decrease the flow velocity at the exit of the basin with respect to the entry velocity. In this way the scour below the stilling basin is kept under control. A stilling basin is a short length of paved channel placed at the foot of a spillway or any other source of supercritical flow. The aim of the designer is to make a hydraulic jump form within the basin, so that the flow is converted to subcritical before it reaches the exposed and unpaved riverbed downstream. Desirable features of the stilling basin are those that tend to promote the formation of the jump, to make it stable in one position, and to make it as short as possible. It is very difficult to classify the possible stilling basin designs, as there is so much variation in the purpose, size and constrains of each type. There are literally hundreds of designs, most of which have been developed by careful testing in model form. It may be said that, unless the energy dissipation structure under study is of relatively small size and importance, this structure is almost always tested in model form, a study that will define the sensitivity to tailwater variations and to asymmetry in inflow conditions. It is also quite likely that a study of the scour pattern below the basin will be made. 6.4.2. Simple stilling basin design for canals Considering first the situation of relatively minor energy dissipation, or scour control such as might be desirable in irrigation canals with a small Froude number, there are several simple types of hydraulic jump controls that may be employed in practice. One of the simplest is the straight drop, which may be of constant width or may coincide with a channel width variation at the drop location. In the present situation, such drops are preceded by subcritical flow, but a hydraulic jump is formed at the toe of the drop, in a location that depends primarily on the tailwater level. A very considerable body of experiment and analysis on this type of energy dissipator has been presented by Moore (1943), Rand (1955) and Dominguez (1958, 1974). The depth of the supercritical flow forming below the drop can be obtained from Fig. 6.14, which summarizes the experimental data of Rand and Dominguez. The upper branch of this curve is the conjugate depth of the hydraulic jump, for the case of a rectangular, horizontal channel. The distance d between the vertical drop and the toe of the jump is well represented by the simple relation:  a  0.30  3.0   d (6-24) hc  hc  Rand, (1943) suggested the alternative expression:  a  0.19  4.30   d (6-25) hc  hc  Using the length of the jump from the equation proposed by Dominguez (1974): ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 121
  16. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  18  20 1 L h (6-26) hc hc which agrees closely with the USBR results, and assuming that the length of the jump should be the minimum length given to the stilling basin, one obtains:  a   h  0.30  3.0    18  20 1  LB (6-27) hc  hc   hc  The energy dissipation characteristics of these basins can be computed by noting that the energy at the brink section, measured with respect to the downstream canal level is given by: Eo = a + 1.5 hc (6-28) Rand 2 h2 Dominguez hc 1.66 hc h2 a h1 1 d (a) conjugate depths in the jump 0.54 h1 K hc a 0 1 2 3 4 5 6 7 8 hc 1.208 d Rand 6 hc 5 Dominguez 4 (b) distance from the drop at 3 K which the jump begins a hc 2 0 1 2 3 4 5 6 7 8 Fig. 6.14. Conjugate depths and distance from vertical drop at which the hydraulic jump begins. Experimental data from Rand (1955) and Dominguez (1958) Example 6.1: A rectangular canal with a width of 3 m has a flow of 2.5 m3/s. A downward step of 0.50 m stabilizes the jump. It is required to determine the maximum tailwater depth consistent with a free jump at the bottom of the drop, the distance of the downstream section of the jump below the drop and the total energy dissipation in the basin. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 122
  17. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Solution: Given: discharge Q = 2.5 m3/s, channel width: b = 3 m downward step: a = 0.50 m Discharge per unit width: Q q= = 0.83 m2/s b Depth of water for minimum specific energy or critical depth:  q2  1 hc =   3  g  = 0.414 m   Hence, applying Fig. 6.13. with: K    1.208  2  1.66 approximately. a 0.5 h h c 0.414 hc This is the maximum tailwater depth consistent with a free jump. Ans. If this value is exceeded, then the jump is pushed against the drop, without necessarily destroying the critical flow above the step (this depends, of course, on the height of the drop; further details on this particular case may be found in Dominguez (1974)). The toe of the jump will be located between 3.2 to 4.5 critical depths below the drop, depending whether one follows the trend of Dominguez’s or Rand’s experiment. The length of the jump is computed from the information given by Fig. 6.14, where the supercritical flow depth is found to be 1  0.54 that is h1 = 0.224 m. Hence the distance below the drop for h hc parallel subcritical flow from Eq. (6-27) is 4.3 m. Ans. The energy dissipation is given by the difference between the upstream energy and that of the parallel flow downstream of the jump and amounts to a difference in energy head of : Eo = a + 1.5 hc = 0.22 m. This represents 20% of the energy above the drop. Ans. Other types of energy dissipators for slow flowing canals have been described by Peterka (1964). Among them there is one that uses a principle similar to the straight drop, but lets the flow through a longitudinal rack, as in the case of diversion weirs. This type of energy dissipator is illustrated in Fig. 6.15. h b L Fig. 6.15: Drop type of energy dissipator developed by the US Bureau of Reclamation ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 123
  18. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- The falling jet is divided into a number of thin sheets of water which show excellent energy dissipation and provide a wave-free downstream flow. Peterka recommends that the coefficient of discharge Cd = 0.245 be used in the determination of the length of the racks. The length of the rack is given by: L Q (6-29) C d bN 2gh where N is the number of openings, b their width, and h the upstream water depth of the longitudinal rack weir as shown in Fig. 6.15. 6.4.3. Specially designed stilling basins A series of highly successful stilling basin designs have originated with the US Bureau of Reclamation (USBR), as described by Peterka (1964), and from the Saint Anthony Falls Hydraulics Laboratory (SAF) (see Blaisdell, 1948). The principle of most of these special designs is to enhance the control of the hydraulic jump by placing baffles and sills on the bottom of the basin (Montes, 1998). A noteworthy feature of these basins is that they are, in general, shorter than the length of the uncontrolled jump for the design discharge, which is a specially attractive economic consideration in a rather complex hydraulic structure. The design and placement of the baffles and sills has received very careful attention in the USBR- and SAF studies. Due to the very high velocity of the flow at the entrance of the stilling basins (which may exceed 30 m/s in the case of high dams), it is necessary that the blocks be structurally strong. In some occasions, the upstream face is armour-plated with steel. A stilling basin conceived for use in high dam and earth dam spillway is shown in Fig. 6.16a (Basin II of the USBR). In the design the high speed flow over the sloping apron of the dam is channeled by a series of chute blocks located at the entrance to the basin. These blocks lift the high velocity from the bottom and in doing so induce the formation of strong eddies that aid the energy dissipation and the stability of the jump when the tailwater level descends below the conjugate depth of the uncontrolled jump. The optimum proportions of the chute blocks were determined by analysis of existing structures. The height, width and spacing of the blocks is recommended to be equal to the supercritical depth ho of the design discharge. At the end of the stilling basin, a sill is placed whose purpose is to lift further the jet above the floor. The USBR-studies found that a suitable shape is a dentated profile with slope of 1V:2H, and with the proportions as indicated in Fig. 6.16. The length of the jump L in this type of basins is practically independent of the initial Froude number. The experimental data of Peterka (1964), indicate that:  3.4 to 4.2 L (for Fro = 4 to 16) (6-30) h1 The stilling basin design has undergone considerable testing in prototype structures and in laboratory models. It appears to be a safe and reliable design within the limits of initial velocity and flow intensity of the structures tested: velocities up to 33 m/s and unit discharges up to 46 m3/s per m width. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 124
  19. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- In the case of smaller discharges and more modest initial velocities the USBR Basin II design is considered to be too conservative, in the sense of requiring a stilling basin that is too long for maximum economy. Several attempts have been made to shorten the necessary length of the stilling basin by incorporating devices in the bottom that enhance the energy dissipation. The purpose can be accomplished by providing an extra set of baffle blocks at the center of the stilling basin (Fig. 6.16b or Fig 6.17). The USBR Basin III design and the SAF type are of this type. They perform satisfactorily for initial velocities up to 20 m/s and unit discharges up to 18 m2/s. The shape and spacing of the baffle blocks was extensively studied by the USBR with the following conclusions:  The upstream face of the block must be vertical or even slope forward. Any backward slope diminishes severely the efficiency of the baffle block.  It is important that the edges of the block remain sharp. Rounding, either by design or by abrasion reduces the effectiveness of the block. Baffle blocks perform best when suitably spaced. Those shown in Fig. 6.16 were considered the most efficient, but cubical blocks with the same spacing were also deemed acceptable. The sill located at the end of the basin is not as critical in shape and a simple continuous sill with a 1V:2H slope was found adequate. The combination of chute blocks, baffle blocks and sills allows a close control of the jump. For the USBR Basin III design, the length of the jump is also independent of the initial Froude number and is approximately:  2.8 L (for Fro = 4 to 14) (6-31) h2 The central baffle blocks create a water surface profile of singular shape, as shown in Fig. 6.16. The depth before the blocks is about one half of the tailwater depth h2. The USBR recommends that the basin be operated with a tailwater level not less than the conjugate jump height h1. The appurtenances (chute blocks, baffle blocks and end sills) act as a supplementary safety device. This type of structure should not be operated at an initial Froude number less than about 4.5. It appears that the central baffle arrangement is not as effective at low Froude numbers, because of the characteristic of the high velocity jet to fluctuate between the bottom and the surface. Many of these conclusions were independently confirmed by Blaisdell (1948). In his design of the SAF Basin for small outlet works he found that the use of a central baffle block (or floor block) allowed a substantial reduction of the length of the basin within acceptable limits. A curve approximating Blaisdell’s test results indicates that the length of the basin Lb should be no less than:  0.76 Lb 4.5 (6-32) h1 Fro Hence, for the range of Froude numbers of the USBR Basin III design, we have extremely short basin lengths, of the order of half the USBR value. This type of basin will produce a high wave, which extends above the tailwater elevation. However, the careful SAF Laboratory tests indicated that even basins as short as 70% of the downstream depth were still safe of excessive scour. The proportions of the SAF basin are shown in Fig. 6.17. Jumps generated by initial Froude numbers in the range 2.5 to 4.5 are difficult to control properly, as the jet in these jumps tends to oscillate. When the jet strikes the surface it ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 125
  20. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- creates a wave which can travel without much change for considerable distances. Peterka notes further that the waves reflected from piers, channel transitions, etc. may actually enhance the initial wave formation and lead to appreciable scour on the banks of the channel. surface profile h2 ho 50 h2 0.15 h2 ho 0.15 h2 ho  ho 0.2 h2 4 h2 (a) The USBR Basin II for high dams and earth dam spillways, after Peterka (1964) surface profile ho h2 ¼h3 0.5 h2 ¾h3 ¾h3  0 .6  0.17 F ro h3  h3 s3 ho  1.0  0 .05 5 Fro s3 ho 0.8 h2 1.6 to 2.0 h2 (b) The USBR Basin III for small outlet works and canal structures, after Peterka (1964) Fig. 6.16. Stilling basins designed for efficient energy dissipation at high Froude numbers A stilling basin designed to contain efficiently this type of initial flow was designed by the USBR after protracted experimentation. The best way to deal with the phenomenon of wave formation was found to increase the Froude number of the supercritical flow of the jump by providing a vertical drop at the entrance. Furthermore, the roller of the jump was intensified by the jets springing from the chute blocks located over the drop, Fig. 6.17. In this case, the placement of baffle blocks at the center of the basin was not desirable. As a consequence the length of the jump tends to be longer than in the case of the USBR Basin III design or SAF design discussed previously. The recommended length of the basin is:  2.42  1.27Fro - 0.11Fro2 Lb (6-33) h1 ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 126
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