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

--- 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 implici

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

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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

VgzH

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

2

V

? 2g 

(b) Open channel flow

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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 b1

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 Fr1 and higher powers to be neglected, according to Henderson (1966), the energy loss between sections 1 and 3 is equal to:

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

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

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 V32/2g for square-edged contractions in rectangular channels and up to 0.11 V3/2g 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/b2, 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

(a) Plan view of rectangular channel

(b) Warped transition from trapezoidal to rectangular section

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

- 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:

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,

- which differs from the ordinary hydraulic jump equation (see Eq (3-15)), only in that V1 is replaced by V1sin1 It follows that;

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:

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

flow

positive waves,

i.e increasing depth

and converging contours

negative waves, i.e decreasing depth and diverging contours

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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

ventilation

system

hydraulic jump

tail water level

regions of large bottom pressure fluctuations

air cavity

recirculating

pool of water air entrainment

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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

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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:

h2 < < 1

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 3Vc/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

  (at B) and that on the

upper streamline to Vc (at A) The horizontal forces required to bring each of these to the ultimate value of 3Vc/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

- 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:

except near the brink, where they found hb = 0.676 hc

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

h = 0.715 h can safely be used for flow measurement, with a likely error of only 1 or 2%

Southwell and Vaisey (1946) Hay and Markland (electrolytic tank) (1958) Fraser (Wood’s theorem) (1961)

Rouse (experiment) (1936)