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Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS

107
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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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

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
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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
 


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



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.

 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
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Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS

111

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

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:


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


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


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,
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

regions of large
bottom pressure fluctuations
air cavity

recirculating
pool of water
air entrainment

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Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS

115

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
