Chapter 20
Open-channel flow
When one has a flow of water to convey, either to provide
some at a place where there is none, or to drain where there is
too much. One is, almost everywhere, obliged to make the
most water flow with the least possible slope.
The flow in rivers, canals, and pipes that are not flowing
full, that is, where one surface of the liquid is free of solid
boundaries, is called open-channel flow.
This type of flow is often measured by placing an
obstruction across the flow path and metering some
characteristic variable resulting from the flow over or under
the obstruction.
Such dams erected for the purpose of metering the flow of
liquids are called weirs or sluice gates, and the measurable
quantity of liquid is called the head (figure 20.1). these topics
are discussed in the following.
20.1 general relations.
The energy of a liquid flowing between two stations x and y
can be accounted for by the generalized Bernoulli relation[1][3]
 px V

 Z x   Wnet  hloss
 
 w 2g

2
x
 px Vy2

 
 Zy 
 w 2g



(20.1)
where
p = effective static pressure
W = uniform specific weight of liquid
V = average liquid velocity of continuity
Z = effective vertical distance from a consistent horizontal
datum
Wnet = net mechanical energy addition between stations
per pound of flowing liquid
hloss= total energy dissipated between stations per pound of
flowing liquid
If the liquid flows in a horizontal bottomed open channel
such that its upper surface is freely exposed to a uniform
ambient pressure, and in addition the flow is in absence of any
mechanical energy addition between stations, then equation
(20.1)can be expressed more simply as
Ex  Ey  hloss
(20.2)
where E denotes the specific energy of the liquid, and is
defined as
V2
E  D
2g
(20.2)
D is the depth of the liquid at a station measured from the
channel bottom to the free surface, where px=py=pambient
For a given flow rate per unit channel width q where, by
continuity.
q=DV
(20.4)
It follows that the specific energy will reach a minimum
value at special depth
2
q
3
(20.5)
Dc 
g
As indicated by differentiating E with respect to D at
constant q. The critical depth DC will be seen later in this
development to divide the flow into directly from
equations(20.3) and (20.5) are
Vc  ( gD c )1/2
3
Ec  Dc
2
(20.6)
(20.7)
Since a gravity wave is known to propagate in shallow
water at a velocity
G  ( gD)1/2
(20.8)
Another useful quantity, the Froude number Fr, can be
introduced as
V
V
q
Fr  

1/2
G ( gD)
( gD3 )1/2
(20.9)
By combining equations(20.6) and (20.9) we see that
Froude number equals 1at the point of minimum specific
energy, further delineating the flow.
These quantities can be pictured as in figure 20.2, where it
is evident that flow in two regimes is possible for a given
specific energy, with the dividing criterion being the critical
depth.
If the actual depth exceeds DC, the flow is said to be
tranquil. Conversely, when the actual depth is less than DC, the
flow is said to be rapid.
In general it is not possible to go from one regime to the
other without outside influence.
However, the flow may change abruptly with attendant
large losses from the rapid to tranquil regime through the
mechanism of a hydraulic jump.
A hydrostatic force balance (per unit channel width) across
an abrupt jump in depth indicates that there is a maximum
depth attainable in the tranquil regime from a given depth in
the rapid regime for a given flow rate.
In terms of an initial state(2) and a conjugate state(4), this
force balance can be given as
wq
F2  F4  Ma  ( )(V4  V2 )
g
which reduces to
2q 2
D2 D4 ( D2  D4 ) 
g
(20.10)
(20.11)
This quadratic in D4 has the real solution
D2
D4 
[1  (1  8Fr 2 )1/2 ]
2
(20.12)
which can be given in the more explicit form as
D2
16E 2
D4 
[1  (
 15)1/2 ]
2
D2
(20.13)
the minimum head loss across the jump is given as
hloss
jump
( D4  D2 )3
 E2  E4 
4D2 D4
(20.14)
Some of these quantities are pictured in figure 20.3.
Figure20.3 Specific energy-death relations in terms of a hydraulic
jump.Dc-critical depth，D2-initial depth;D3-two possible tailwater
depths;D4-maximum jump depth.
If the actual downstream liquid level (tailwater depth d3)
is maintained at less than the conjugate depth D4, rapid flow
at D2 is initially possible.
If the tailwater depth is also greater than dc, there will be
an abrupt jump to D3. Conversely, if D3 exceeds d4,initially
rapid flow at D2 is not possible.
In terms of a sluice gate, this tailwater-depth-conjugatedepth relationship determines two distinct modes of
operation of the gate.
If D3 is less than D4, the gate will operate with a free
efflux(figure 20.4), whereas if D3 exceeds D4, the gate will
operate with a submerged efflux (figure 20.5).
In the following sections flow under a sluice gate operating
in these situations is discussed, as is the flow over a weir.
20.2 Sluice gate with free efflux
An energy relation between the various terms
pertaining to a vertical sluice gate with free efflux can be
given as
2
2
2
E1  D1 
V
V1
V
 D2  2  hloss1,2  D3  3  hloss1,3
2g
2g
2g
(20.15)
Continuity, for this same flow condition, is expressed as
q  DV
1 1  D2V2  D3V3
(20.16)
These quantities are pictured in figure 20.4.
An ideal flow rate q’ must be defined so that a flow
discharge coefficient C can be particularized by the
relation
q
C
q'
(20.17)
For example, on an analytical basis one could define
the ideal flow rate that is implied by the measurable head
difference D1-H, that is, by continuity,
' '
'
qF'  DV

HV
1 1
H
(20.18)
Whereas, by energy,
(V1' )2
(VH' )2
D1 
H
2g
2g
(20.19)
Where the primes signify ideal quantities, and the
subscript F stands for free efflux. Equations (20.18) and
(20.19) lead at once to the ideal flow rate
1/2
 2 gD1 
'
qF  H 

1

H
/
D

1
 2 g ( D1  H ) 
H
2
1

(
H
/
D
)

1

1/2
(20.20)
Now according to equations (20.17) and (20.20) and an
experimentally determined flow rate q, a particular discharge
coefficient Cf is defined for the free efflux sluice gate.
Henry [4] provides experimental work on discharge
coefficients for sluice gates, and hence provides a basis by
which the characteristics of the discharge coefficient defined
by equations (20.17) and (20.20) can be examined. Since
Henry uses for an ideal flow rate the arbitrary
'
FH
q
 H[2gD1 ]
1/2
(20.21)
Where the subscript FH stands for free Henry, the relation
between Cf and CFH is simply
'
qF  CF qF'  CFH qFh
Thus
(20.22)
1/2

H
CF  CFH 1  
 D1 
(20.23)
In figure 20.6 both discharge coefficients are shown. Both
are seen to have acceptable characteristics, being easily
formed from measurable depths and being only weak
functions of the flow. The recommended discharge coefficient
for the free efflux sluice gate can be represented quite closely
by the parabola
H
H
CF  0.604  0.02    0.045  
 D1 
 D1 
2
(20.24)
Example 1.
Free efflux under a sluice gate. Find the flow rate of water
for a free efflux under a sluice gate as in figure 20.4, where
H=2 in, D1=20 in, and L is the width of the sluice = 2 ft.
Solution. By equations (20.17), (20.20), and (20.24):
2
 2 
 2 
CF  0.604  0.02    0.045    0.6064
 20 
 20 
2  2  32.2  20 /12 
'
'
QF  LqF  2  

12  1  2 / 20
1/2
 3.292ft 3 /s
.
w  wwnet (CF QF' )  62.4  0.6064  3.292  124.2 Ib / s
Note that this agrees precisely with the flow rate predicted
by Henry’s discharge coefficient.
20.3 sluice gate with submerged efflux
An energy relation between the various terms pertaining to
a vertical sluice gate with submerged efflux is
V32
V12
V22
E1  D1 
 D2 
 hloss1,2  D3 
 hloss1,3
2g
2g
2g
(20.25)
Continuity for this situation is
q  DV
1 1  D2V2  D3V3
(20.26)
These quantities pictured in figure 20.5.
The depth D2 is used to determined flow rate in the
submerged efflux case (rather than D2s) because it represents
the only area (per unit channel width) available for through
flow.
The depth difference D2s-D2 can be looked upon as an
additional pressure head at the vena contract plane, to be
included in the energy accounting of equation (20.25) but not
to be considered in the continuity of equation (20.26).
Since D3>D2, it follows from equation (20.26) that V2>V3,
and hence from equation (20.25) that D2s<D3, as indicated in
figure 20.5. However, in fact D2s may be very close in
magnitude to D3.
Once again the definition of an ideal flow rate, this time for
the submerged efflux case, must be agreed on before a flow
discharge coefficient can be specified by equation (20.17).
One ideal flow rate that appears in the literature has been
defined as
qs'  H[2g (D1  D3 )]1/2
(20.27)
Bakhmeteff [3] indicates that equation (20.27) is to be used
with the simplified discharge coefficient Csb= 0.6 (where the
subscript B stands for Bakhmeteff).
Rouse [1] notes that it is common practice to use the ideal
flow rate of equation (20.27) with the analytical discharge
coefficient
(CvCc ) F [1  CcF ( H / D1 )]1/2
CsR 
[1  (CvCc )2F ( H / D1 )2 ]1/2
(20.28)
Where the subscript R stands for Rouse, although Rouse
expresses misgivings about the lack of information on the
velocity coefficient Cv and contraction coefficient Cc in the
submerged efflux case.
In any case, Henry [4]-[6] again provides the only recent
experimental work in discharge coefficients for sluice gates,
and hence provides a basis by which the characteristics of the
discharge coefficients for submerged efflux sluice gates can
be examined. Henry again employs equation (20.21) to
define his ideal flow rate. Thus ,
q q
'
sH
'
FH
(20.29)
And on the basis
'
qs  Cs qs'  CsH qsH
So that
CsH
Cs 
(1  D3 / D1 )1/2
(20.30)
(20.31)
All of these discharge coefficients are compared with
Henry’s CSH in figure 20.7, where it can be noted that
1 . Some of the submerged efflux coefficients are more
complex than those of the free efflux cases, since they are
functions of H/D3 as well as of H/D1.
2. Csb (of Bakhmeteff) and Csr (of Rouse) do not concide
with Cs of equation (20.31), although they should since all
of the coefficients are to be used with qs’ of equation
(20.27).
This indicates either that the empirical Csb is incorrect,
that (CvCc)≠(CvCc)F, as assumed in Csr or that Henry’s work
is suspect. (in the absence of newer experimental work,
Henry’s data are taken here as correct.)
3. Cs of equation (20.31) varies much less than Henry’s Csh
for a given H/D3 between the same Froude number limits.
The locus of constant Froude number, based on the sluice
gate opening, can be defined for Henry’s data as
1/2
CsH
 1  H 
 FrN p    
 2  D1  
(20.32)
Item 3 above suggests that C, might serve as the basis of an
empirical coefficient of discharge Cse, which will be
independent of H/D3 and yet will yield acceptable flow rates.
A simple parabola that closely approximates the Cs plot in
figure 20.7 is
H
H
CsE  0.49  1.376    1.43  
 D1 
 D1 
2
(20.33)
This is recommended to define the submerged discharge
coefficient when used with equation (20.27).
Example 2.
Submerged efflux under a sluice gate. Find the flow of
water for a submerged efflux under a sluice gate as in figure
20.5, where H= 2 in, D1= 20 in, D3= 16 in, and the width of
the sluice L= 2 ft.
Solution. By equations (20.17), (20.27), and (20.33),
2
 2 
 2 
CsE  0.49  1.376    1.43    0.6133
 20 
 20 
2  2  32.2(20  16) 
Qs'  Lqs'  2  

12 
12
1/2
 1.544ft 3 / s
w  wwsser (CsE Qs' )  62.4  0.6133 1.544  59.1Ib / s
Note that this is within 3% of the flow rate predicted by
Henry’s more complex discharge coefficient.
20.4 weirs
In determining the ideal flow rate over a rectangular
weir, the approach velocities are considered to be
negligible and the sheet of liquid flowing over the weir
(the nappe) is assumed to be surrounded by atmospheric
pressure. Hence the nappe is treated as a free-falling
body. Thus according to figure 20.1,
Vh  (2 gh)1/2
(20.34)
And the ideal volumetric flow rate as given by equations
(19.3) and (19.4) is
dQideal  Vh dA
(20.35)
Substituting equation (20.34) and using the dimensions
shown in figure 20.8a, we have
dQideal  (2gh)1/2 ( Ldh)
(20.36)
After integration between zero and H.
Qideal 
2
L(2 g )1/2 H 3/2
3
(20.37)
Which shows that the volumetric flow rate should be
proportional to the three-half power of the head for rectangular
weirs. Similarly, the ideal flow rate for a triangular weir
(figure 20.8b) can be shown to be
Qideal
4L

(2 g )1/2 H 5/2
15H
(20.38)
Weight flow rates are related to volumetric flow rates
by equation (19.7), so that
w  wQ
(20.39)
Empirical equations are always used for actual flow
rates, but they are based on ideal equations given above.
For example, the fluid velocity of approach has an effect
on flow rates. This could be introduced in the ideal
equations, but it has been omitted here for the sake of
brevity.
The actual installation particulars and shape of the crest
of the weir also are important. One of the most widely
accepted empirical equations for rectangular weirs is the
Francis formula.
2 3/2
2 3/2 

 V0 
V0 
nH  

Q  3.33  L 
 
 
  H 
10  
2g 

 2 g  

(20.40)
Where n is the number of lateral contractions of weir and V0
is the approach velocity.
If the weir does not extend over the full width of the
approach channel, lateral contractions occur. On the other
hand, if all lateral contractions are eliminated, the weir is said
to be suppressed (n=0). In this case, if the velocity of approach
is negligible, equation (20.40) becomes
Q  3.33LH 3/2
(20.41)
Which agrees well with equation (20.37), since the actual
flow rate must be less than the ideal one because of losses.
Example 3. water flow over rectangular weir. Find flow
rate of water in lb/s for a fully contracted weir as in figure
20.8a, where n=2 for L=4 ft, H=3 ft, and the approach area
is 56 ft2.
Solution.
For a first try, assume the velocity of approach to be
negligible. From equation (20.40),
2  3  3/2

3
Q  3.33  4 
3

58.83ft
/s

10 

At this approximate flow rate and with an approach area of
56 ft2, the approach velocity is, by equation (20.35),
V0 
58.83
 1.05ft / s
56
And the approach velocity head is
V02 1.052

 0.024ft
2 g 64.34
Which certainly can be considered to be negligible
compared to the 3-ft head.
Hence Q=58.83 ft3/s. For the flow rate of water by
equation (20.39), w=62.4*58.83=2671 lb/s.