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

OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 107 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. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 108 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: g2 Vg zH 2     (6-1) 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. 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. flow (a) Pipe flow flow h 1 h 2 ? z 2 1 V 2g 2 2 V ? 2g  (b) Open channel flow OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 109 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 E 1 = E 2 and F 2 = F 3 , 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 . 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 Fr 1 is small enough for Fr 1 2 and higher powers to be neglected, according to Henderson (1966), the energy loss between sections 1 and 3 is equal to:   2 2 3 2 1 1 2 1 1 1 1 3 4 2 2 2Fr b b b V b E E E 1 2g b b                     (6-2) The last term inside the brackets is the open-channel-flow term, which vanishes, as Fr 1 tends to zero. In this case h 1 = h 2 = h 3 , and the situation is equivalent to closed-conduit flow, i.e.   2 1 3 V V E 2g    (6-3) The term containing Fr 1 2 in Eq. (6-2) does not contribute a great deal to the total energy- head loss unless Fr 1 > 0.5, or b 2 /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: 3 2 1 b 1 b 2 flow OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 110   2 1 3 V V E 0.3 2g    (6-4) and by other as 22 31 VV E 0.1 2g 2g          (6-5) The former of these is to be preferred, but over the range 1.5 < V 1 /V 3 < 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. 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 V 3 2 /2g for square-edged contractions in rectangular channels and up to 0.11 V 3 2 /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 h 3 /b 2 , reaching the above maximum values when this ratio reached a value of about 1.3. When h 3 /b 2  1, these coefficients reduced to about 0.1 and 0.04. Yarnell did not report values of depth : width ratio. (a) Plan view of rectangular channel 1 4 flow flow 1 4 channel central line (b) Warped transition from trapezoidal to rectangular section OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 111 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. Fig. 6.5. Movement of a small disturbance at a speed (a) less than (b) equal to (c) greater than the natural wave velocity flow channel central line A 2 A 1 A 2 A 1 direction of movement of disturbance shock front successive wave fronts (a) (b) A 2 A 1 P 2 P 1 A n  shock front (c) OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 112 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 A 1 , 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 A 1 . Similar wave fronts are initiated when the particle passes through points A 2 , A 3 , 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 A n the wave fronts have reached positions such that they can all be enveloped by a common tangent A n P 1 , which will itself form a distinct wave front. Since a disturbance travels from A 1 to P 1 in the same time as the particle travels from A 1 to A n , it follows that: 1 1 1 n A P c 1 sin A A V Fr     (6-6) 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,   1 1 2 1 V cos V cos       (6-7) Considering now the velocity components normal to the wave front, the continuity equation becomes:   1 1 1 2 2 1 V h sin V h sin       (6-8) and the momentum equation must clearly lead to the result: 2 2 1 1 2 2 1 1 1 V sin 1 h h 1 gh 2 h h          (6-9)   1 V 1 V 2 shock front OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 113 which differs from the ordinary hydraulic jump equation (see Eq. (3-15)), only in that V 1 is replaced by V 1 sin 1 . It follows that; 2 2 1 1 1 1 1 1 h h sin 1 Fr 2 h h          (6-10) which reduces to Eq. (6-6) when the disturbance is small and h 2 /h 1 tends to unity. The special case of the small disturbance can be investigated further by eliminating V 1 /V 2 between Eqs. (6-7) and (6-8), leading to the result:   2 1 1 1 h tan h tan       (6-11) Setting h 2 = h 1 + h, and letting  tend to zero, we obtain 2 dh h V tan d sin cos g       (6-12) 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. 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. A B C   surface contours flow positive waves, i.e. increasing depth and converging contours negative waves, i.e. decreasing depth and diverging contours OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 114 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. 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. overfall free falling jet upper nappe lower nappe hydraulic jump tail water level roller length L d h c h p h 2 h 1 z nappe impact ventilation system hydraulic jump tail water level regions of large bottom pressure fluctuations air cavity recirculating pool of water air entrainment OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 115 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. A A C B D 3 - 4 h c h c h b 0.215 h b pressure – head distribution 45  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 116 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: 2 2 1 2 1 1 h 2Fr h 1 2Fr   (6-13) where the subscripts 1 and 2 characterize sections A and C, respectively; if the flow is critical at section A the above equation becomes: 2 c h 2 h 3  (6-14) which sets a lower limit on the brink depth h b ; since there is some residual pressure at the brink, h b must be greater than h 2 . It follows that: b c h2 < < 1 3 h (6-15) Actually there is no part of the jet, however far downstream, where the pressure is completely atmospheric; if there were, the streamlines would all become parabolas, and these curves cannot exhibit the property of asymptotic convergence, which the streamlines actually possess. However, the point is a somewhat academic one, for it can be shown (see Henderson, 1966) that the internal pressure in the jet tends to zero much faster than does the width of the jet, as the jet moves further downstream. In the limit, when the jet has fallen infinitely far below the brink, Eq. (6-14) will be true and the horizontal velocity will be equal to 3V c /2. From this last fact it can be seen that the internal pressure of the jet plays a decisive role in developing the ultimate form of the jet, for the horizontal velocity on the lower streamline is originally equal to c c 3h 2g 3V 2        (at B) and that on the upper streamline to V c (at A). The horizontal forces required to bring each of these to the ultimate value of 3V c /2 are supplied by the pressure gradients at either end of pressure profiles such as that shown on the horizontal section BD. The form of the pressure distribution at B has already been referred to (see Section 6.3.2); the pressure profiles just upstream must be of the form indicated in Fig. 6.9, with inflexions as shown. These are necessary in order to return the pressure distribution to hydrostatic at the bed, where the vertical acceleration must be zero. One consequence of this property of the profiles is that the pressure on the bed must remain finite very close to the brink; the longitudinal pressure gradient there must therefore be infinite, and the same is true of the lateral pressure gradient, as indicated by the pressure profile at B. It follows that the radius of the curvature of the lower streamline must momentarily be zero just downstream of B. This is a well-recognized property of all free jets. [...]... Lj 1 2 Fig .6. 11 The drop structure 14 Eq (6- 17), (White) E1 yc 12 Experiment (Moore) 10 z o hc 8 EL hc Ej E2 hc hc initial energy 6 4 2 0 2 4 6 8 10 12 14 E hc Fig 6. 12 Energy dissipation at the base of the free overfall Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 118 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ... 6: TRANSITIONS AND ENERGY DISSIPATORS 119 OPEN CHANNEL HYDRAULICS FOR ENGINEERS -  h  h2  1 .66  c  z o  z o  0.81  h  Ld  4.30  c  z o  z o  (6- 21) 0.09 L j  6. 9  h 2  h1  (6- 22) (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... 19 56) Fig 6. 13 The box inlet drop structure Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 120 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. .. STILLING BASIN DOWNSTREAM ELEVATION Fig 6. 17 The Saint Anthony Falls (SAF) Stilling basin, after Blaisdell (1948) Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 127 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 6. 5 OTHER TYPES OF ENERGY DISSIPATORS 6. 5.1 Stepped spillways Many concrete... 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 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - The falling jet is divided into a number of thin sheets of water which show excellent energy. .. (experiment) (19 36) 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 OPEN CHANNEL HYDRAULICS FOR ENGINEERS ... the step height Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 128 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 6. 5.2 Bucket-type and Ski-Jump The bucket-type energy dissipator illustrated in Fig 6. 19 is normally cheaper than the stilling basin but is remarkably effective Essentially... the downstream section of the jump below the drop and the total energy dissipation in the basin Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 122 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - Solution: Given: discharge Q = 2.5 m3/s, channel width: b = 3 m downward step: a = 0.50... by Dominguez (1974): Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 121 OPEN CHANNEL HYDRAULICS FOR ENGINEERS - L h  18  20 1 hc hc (6- 26) which agrees closely with the USBR results, and assuming that the length of the jump should be the minimum length given to the stilling... Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 124 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 . ----------------------------------------------------------------------------------------------------------------------------------- Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 107 Chapter  TRANSITIONS AND ENERGY DISSIPATORS _________________________________________________________________________. Southwell and Vaisey (19 46) Hay and Markland (electrolytic tank) (1958) Fraser (Wood’s theorem) (1 961 ) Rouse (experiment) (19 36) OPEN CHANNEL HYDRAULICS FOR ENGINEERS