A Modified Theory for Rectangular Vertical Drop
Structures
MEHDI AZHDARY MOGHADDAM
Civil Engineering. Department
University of Sistan & Baluchestan
Nikbakht Faculty of Engineering, Zahedan
IRAN
Abstract: Vertical drop structures, or free-overfalls, are a simple form of energy dissipator
commonly employed in urban stormwater drainage channels and irrigation schemes where the local
topography is steeply-sloping. Free-overfalls can also develop naturally in degrading alluvial
channels at rock outcrops in the channel bed. Several theories described the hydraulics of
rectangular vertical drop structures. These theories may be applied to estimate spillway energy loss
when the flow on a stepped spillway conforms to a nappe-type flow regime. A modified theory is
proposed which addresses certain limitations associated with some earlier theories. The modified
relationships are verified based on experimental data.
Key-words: Vertical Drop Structures, Energy loss, stepped spillways, momentum
1 Introduction
Vertical drop structures, or free-overfalls, are
a simple form of energy dissipator commonly
employed in urban stormwater drainage
channels and irrigation schemes where the
local topography is steeply-sloping. Freeoverfalls can also develop naturally in
degrading alluvial channels at rock outcrops
in the channel bed. The hydraulics of vertical
drop structures has been studied by, among
others: Moore (1943), White(1943), Rand
(1955), Gill (1979), Chanson (1994), and
Rajaratnam and Chamani (1995). The various
theories developed by these researchers reflect
the different assumptions made regarding the
hydraulics of the system. Two types of flow
regime can occur on a stepped spillway,
namely: nappe-flow and skimming-flow
regimes (Fig. 1).
For a nappe-type regime, the flow across
a single spillway step can be considered
as being equivalent to that for a freelydischarging vertical drop structure, (Fig.
2). Accordingly, in these circumstances
stepped-spillway energy losses may be
estimated using expressions developed
for vertical drops. However, certain
assumptions made in the development of
earlier design relationships for vertical
drop structures are considered suspect.
Previous studies on the hydraulics of
vertical drop structures include, among
others, those of Moore (1943), White
(1943), Rand (1955), Gill (1979),
Chanson (1994), and Rajaratnam and
Chamani (1995).
Moore's (1943)
laboratory study investigated the energy
loss produced at the base of vertical
drops, using two model structures of
0.15m- and 0.45m-height respectively.
He established that energy loss was a
function of the relative fall h/yc , where h
= height of drop, and yc = critical depth.
Moreover he observed that, under freedischarge conditions, the depth of water
standing behind the falling jet, (i.e. the
pool depth), yp was greater than the
depth of water shooting from the base of
the fall into the downstream channel, y1
(Fig. 2).
Based on Moore's and his own
experimental
data,
Rand
(1955)
developed the following relationships for
vertical drops:
Moore's expression for pool depth takes
the form:
1.275
2
2
 yp 
y 
y 
  =  1 +2 c - 3
y 
y 
y 
 1
 c
 c
where
yp = pool depth; and
y1 = depth at section 1-1 (see Fig. 2).
(1)
E1 =
yc
2
h
1.06 +
+ 1.5
yc
(4)
0.66
(5)
0.81
Ld = 4.30  yc 
(6)
 
h
h
where
y2 = conjugate depth (of y1); and
Ld = distance from the wall to position y1 ,
(Fig. 2).
Gill (1979) modified White's (1943)
assumptions. He assumed that:
(i)
uniform velocity occurs in the
thinner layer below the pool;
(ii)
velocity is not constant in the
mixing zone;
(iii)
the effect of the pool in the
calculation of velocity is expressed as:
V = 2 g ( h + 1.5 y c - y p ) ; and
(iv)
the effect of the pool in regards to
energy loss is neglected.
Gill's relationships are:
0.283
y  y
y1

= 0.524 c  c - 0.0053 
h
h  h

(7)
0.697
 y

= 1.067  c - 0.0016 
h
 h

yp
2
(8)
0.305
(2)
where
E1 = energy at section 1-1.
0.81
y 
=  c 
h h
These assumptions were questioned by
Moore and others, particularly in regards
to items (ii) and (v). White's (1943)
expression for energy at section 1-1 is:

h
+ 1.5 

yc

4
(3)
y2
y 
= 1.66  c 
h
h
yp
In commenting on Moore's study, White
(1943) made the following assumptions
regarding the hydraulics of the system:
(i)
the recirculating flow in the pool
at the bottom of the drop was the
same as the backwater flow;
(ii)
the velocity of the supercritical
stream immediately downstream
of the drop was the same as the
uniform velocity in the thicker
stream at the side of the pool;
(iii)
the angle of the falling jet was not
affected by the presence of the
pool;
(iv)
the energy loss at the drop was
due to mixing between the jet
and the pool; and
(v)
the effect of the pool in
calculating the velocity of jet,
V, was neglected.

 1.06 +

+
y1
y 
= 0.54  c 
h
h
 y

(9)
cos  = 0.968  c - 0.0058 
 h

β = angle of jet where it impacts the bed.
While Gill addressed certain weaknesses
in White's assumptions, assuming the
pool immediately downstream of the drop
would not affect energy loss was
incorrect.
Chanson (1994), rather than using a
momentum approach, used the trajectory
equations of the nappe centreline to
develop the following expressions:
1.326
y1
y 
= 0.625  c 
h
h
(10)
0.675
y 
= 0.998  c 
h
h
yp
(11)
0.525
 yc 
Ld = 2.171  
h
(12)
-0.582
y 
(13)
tan  = 0.855  c 
h
Rajaratnam and Chamani (1995)
examined experimentally the energy loss
produced at the base of a vertical drop
structure. Their relationships take the
form:
E
-0.766
y 
(14)
= 0.896  c 
Ec
h
0.719
yp
 yc 
(15)
= 1.107  
h
h
where
ΔE = energy loss across the drop (= EcE1); where
Ec = ( h + 1.5 yc ); and
E1 = ( y1 + V12/2g ).
Applying the momentum principle
between sections 0-0 and 1-1 gives:
1
1
 q V cos  +  y 2p =  q V 1 +  y12 (16)
2
2
where
q = unit-width discharge;
γ = specific weight of water; and
ρ = mass density of water.
Critical depth, yc, occurs a short distance
upstream from the brink. Once again,
applying the momentum principle
between sections c-c and 0-0 gives:
1
 y c2 =
(17)
2
Assuming no energy loss occurs between
sections c-c and 0-0, then
2
2
V
V
h+ c = yp+
(18)
2g
2g
Now, critical velocity V c = g y c 18
therefore eq. (18) becomes:
2
V
h + 1.5 y c = y p +
(19)
2g
Applying the grid-search method to
Moore's (1943), Rand's (1955), and
Rajaratnam and Chamani's (1995) data
produces the following relationship:
1.222
y1
y 
(20)
= 0.505  c 
h
h
Solving eqs. (16), (17), (19), and (20)
gives:
0.598
yp
 yc

= 1.026  - 0.03 
(21)
h
h

2 Modified Theory
The following conditions are assumed
regarding the vertical drop shown in Fig.
2:
(i)
fluid shear on the downstream
channel bed is negligible; and
(ii)
the angles of the jet, where it
impacts the pool and the channel
bed, are the same.
0.466
y 
cos  = 0.99  c 
h
E
-0.813
(22)
y 
(23)
= 0.077  c 
Ec
h
Figs. 3, 4, 5, and 6 compare the above
equations with the experimental data and
the data of others. Fig. 4 shows that, with
increasing discharge the pool depth
formed at the foot of the drop structure
will also increase. When yc = h, then eq.
(21) indicates that the pool depth yp = h,
i.e. the air cavity below the lower nappe is
filled entirely.
Fig. 5 shows the
relationship between yc/h and cos β,
which indicates that β decreases with
increasing discharge. Fig. 6 presents the
energy loss across a vertical drop
structure, as described in eq. (23). It can
be seen that as discharge increases the
energy loss decreases. When yc = h, the
energy loss ratio (ΔE/Ec) would be the
least (0.1). As noted in Figs. 3through 6,
generally good agreement was observed
between eqs. (20), (21), (22), (23) and
Moore's (1943), Rand's (1955), and
Rajaratnam and Chamani's (1995) data.
[4] Gill, M.A., (1979), "Hydraulics of
Rectangular Vertical Drop Structures," J.
Hydr. Res., 17(4), 289-302
[5] Moore, W.L., (1943), "Energy Loss at
the Base of a free Overfall," Trans.
ASCE, 108, 1343-1360.
[6] Rajaratnam, N., Chamani, M.R.,
(1995), "Energy Loss at Drops," J. Hydr.
Res., 33(3), 373-384. Discussions: (1)
Chanson, H., (1996), J. Hydr. Res., 34(2),
273-278.
[7] Rand, W., (1955), "Flow Geometry at
Straight Drop Spillways," Proceedings,
ASCE, 81E, Paper 791, 1-13.
[8] White, M.P., (1943): Discussion
appearing in Moore, W.L., (1943)
3 Conclusion
In this research a modified theory is
proposed which addresses certain
limitations associated with earlier theories
for the hydraulics of rectangular vertical
drop structures. The modified theory
establishes the hydraulic parameters of
vertical rectangular drop structures
through eqs. (20), (21), (22), and (23). As
indicated in Figs. 3 through 6, a generally
good agreement was obtained between
these equations and the experimental data
of others.
References:
[1] Chanson, H., (1996): Discussion
appearing in Rajaratnam, N., Chamani,
M.R., (1995)
[2] Chanson, H., (1994), "Comparison of
Energy Dissipation between Nappe and
Skimming Flow Regimes on Stepped
Chutes," J. Hydr. Res., 32(2), 213-218.
[3] Chow, V.T., (1959), "Open Channel
Hydraulics," McGraw- Hill, New York,
N.Y.
Fig. 1. Characteristic flows over a
stepped spillway:
(a) nappe flow with fully developed
hydraulic jump
(b) nappe flow with partially
developed hydraulic jump
(c) skimming flow
cosB
Rajaratnam & Chmani
(1995)
Eq. 22
0
Fig. 2. Simple vertical drop structure
0.1
0.2
Yc/h
0.3
0.4
Fig.5. Comparison of eq. (22) with
experimental data
0.3
0.8
0.25
EQ. (23)
0.6
EQ. (20)
0.15
Rajaratnam &
Chmani (1995)
Moore (1943)
h=45.72 cm
Moore (1943)
h=15.24 cm
0.1
0.05
0
0
0.1
0.2
0.3
Yc / h
0.4
0.5
0.6
Del E / E0
Y1 / h
0.2
Moore (1943)
Rajaratnam & Chmani (1995)
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Yc / h
Fig.3 Comparison of eq. (20) with
experimental data
1
Yp / h
0.8
0.6
EQ. (21)
0.4
Rand (1955)
0.2
Rajaratnam &
Chamani's (1995)
Moore (1943)
0
0
0.2
0.4 Yc / h 0.6
0.8
1
Fig. 4. Comparison of eq. (21) with
experimental data
Fig.6. Comparison of eq. (23) with
experimental data