Open channel hydraulics for engineers. Chapter 6 transitions and energy dissipators
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
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter TRANSITIONS AND
ENERGY DISSIPATORS
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6.1. Introduction
6.2. Expansions and Contractions
6.3. Drop structures
6.4. Stilling basins
6.5. Other types of energy dissipators
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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
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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.
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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 V 2
Hz (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.
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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 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:
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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.
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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
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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
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which differs from the ordinary hydraulic jump equation (see Eq. (3-15)), only in that V1 is
replaced by V1sin1. 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.
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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.
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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.
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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:
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
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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%.
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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
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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
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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
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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):
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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.
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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
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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.
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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
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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
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Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS 126
