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CHANSON, H. (1996). "Energy Loss at Drops - Discussion." Jl of Hyd. Res., IAHR, Vol. 34, No. 2, pp. 273-278 (ISSN
0022-1686).
Energy Loss at Drops
by N. RAJARATNAM and M.R. CHAMANI
Jl of Hyd. Res., IAHR, Vol. 33, No. 3, pp. 373-384
Discussion by
Hubert CHANSON
Senior Lecturer in Fluid Mechanics, Hydraulics and Environmental Engineering
Department of Civil Engineering, The University of Queensland
Brisbane QLD 4072, AUSTRALIA.
The writer would like to congratulate the authors for their excellent papers. As clearly indicated by them, few
researchers investigated energy losses at drop structures. Most engineering calculations still rely on the work of
MOORE (1943), WHITE (1943) and RAND (1955). The authors' paper provides new experimental results particularly
useful to the profession. For completeness, the writer wishes to indicate that the calculation method developed by the
authors is not exactly new. The writer developed a similar approach (CHANSON 1994 and 1995, pp. 230-235). A
summary is presented below.
Considering an aerated nappe, the flow direction at the edge of the drop is nearly horizontal. Once the fluid leaves the
step, the horizontal acceleration is zero and the vertical acceleration equals minus the gravity acceleration. The
trajectory equations of the nappe centreline are :
x = Vb * t
db 1
y = H + 2 - 2 * g * t2
(1a)
(1b)
where x is the horizontal direction, y is the vertical direction positive upwards, db and Vb is the flow depth and velocity
respectively at the brink of the crest, H is the drop height and t is the time (fig. 1).
From the trajectory equation, the length of the drop Lp equals :
3/2
Lp
H ⎛
H⎞
⎛Yc⎞
*
db * ⎝1 + 2 * db⎠
H = ⎝H⎠
(2)
When the fluid leaves the crest, the time t" taken to reach the free-surface of the pool of recirculating water is given by :
1
⎛ db⎞
Yp = - 2 * g * t"2 + H + 2
(3)
⎝
⎠
where Yp is the height of water in the pool behind the overfalling jet (fig. 1). The nappe velocity V at the intersection
of the falling nappe with the receiving pool is deduced from the trajectory equation :
db
2
H + 2 - Yp
Y
V
⎛ c⎞
(4)
⎜d ⎟ + 2 *
Yc
Vc =
⎝ b⎠
where Vc is the critical velocity (i.e. Vc =
by :
g*Yc). The angle of the falling nappe with the horizontal (fig. 1) is given
CHANSON, H. (1996). "Energy Loss at Drops - Discussion." Jl of Hyd. Res., IAHR, Vol. 34, No. 2, pp. 273-278 (ISSN
0022-1686).
tanϕ =
2*
db
Yc *
db
H + 2 - Yp
Yc
(5)
The pool of water behind the overfalling jet (fig. 1) is important as its weight provides the force parallel to the bottom
surface which is required to change the jet from an angle ϕ to the bottom to parallel to this surface. For a ventilated
nappe, the momentum equation resolved along the step surface is :
1
2 1
2
2 * ρ * g * Yp - 2 * ρ * g * Y1 = ρ * qw * (V1 - V * cosϕ)
(6)
assuming that the edges of the jet fluid have not disintegrated into spray, neglecting the shear forces on the surfaces and
assuming hydrostatic pressure distributions in the pool and downstream of the nappe impact. Y1 and V1 are
respectively the flow depth and velocity downstream of the nappe impact. Assuming that the velocity entering the
control volume is approximately the same as leaving it, it yields:
3
V⎞
⎛
1 + 2 * (1 - cosϕ) * V
Yp
⎝ c⎠
=
Yc
V
Vc
(7)
WHITE (1943) developed the momentum equation at the base of the overfall to estimate the energy loss. His results
indicated that the flow depth downstream of the jet impact equals :
Y1
2
Yc = 1.5
H
3
+
Yc + 2
2
(8)
Discussion
Equations (2), (4), (5), (7) and (8) form a system of five non-linear equations in terms of the drop length, the impact
flow conditions, the pool height and the flow depth Y1.
Assuming that the flow upstream of the drop edge is subcritical, the brink depth is estimated as :
db
Yc = 0.715
(9)
Equation (9) was initially obtained by ROUSE (1936) and verified by other researchers (table 1). With this result, the
writer solved numerically the system of equations (CHANSON 1995). The results can be best fitted by the following
equations :
0.525
Lp
⎛Yc⎞
h = 2.171 * ⎝ H ⎠
V
⎛Yc⎞
Vc = 1.455 * ⎝ H ⎠
(10)
-0.483
(11)
-0.582
⎛Yc⎞
tanϕ = 0.855 * H
⎝ ⎠
(12)
0.675
Yp
⎛Yc⎞
H = 0.998 * ⎝ H ⎠
(13)
CHANSON, H. (1996). "Energy Loss at Drops - Discussion." Jl of Hyd. Res., IAHR, Vol. 34, No. 2, pp. 273-278 (ISSN
0022-1686).
1.326
Y1
⎛Yc⎞
=
0.625
*
H
⎝H⎠
(14)
Equations (12) and (14) are compared with the authors' data and results on figure 2.
In practice, the drop length calculations (eq. (10)) are pessimistic. Experimental investigations (RAND 1955, authors)
indicate that Lp is best estimated as :
0.81
Lp
⎛Yc⎞
(15)
H = 4.30 * ⎝ H ⎠
It must be emphasised that the present calculations were developed for aerated nappes. Complete calculations of unaerated nappe trajectory were developed elsewhere (CHANSON 1995, pp. 235-236).
Further the above calculations (i.e. equations (1) to (7)) apply to both subcritical and supercritical nappe flows although
equations (8) to (15) imply a subcritical flow upstream of the drop brink.
Notations
db
qw
flow depth at the brink of a drop (m);
water discharge per unit width (m2/s);
t, t"
time (s);
Vb
Vc
flow velocity (m/s) at the brink of a drop;
critical velocity (m/s); for a rectangular channel : Vc =
W
channel width (m);
x
horizontal distance (m);
y
vertical distance (m);
g * Yc;
References/Bibliographie
BAUER, S.W., and GRAF, W.H. (1971). "Free Overfall as Flow Measuring Device." Jl of Irrig. and Drain.,
ASCE, Vol. 97, No. IR1, pp. 73-83.
CHANSON, H. (1994). "Hydraulics of Nappe Flow Regime above Stepped Chutes and Spillways." Aust. Civil
Engrg Trans., I.E.Aust., Vol. CE36, No. 1, Jan., pp. 69-76.
CHANSON, H. (1995). "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways." Pergamon,
Oxford, UK, Jan., 292 pages.
CRAYA, A. (1948). "Hauteur d'eau à l'Extrémité d'un Long Déversoir." ('Flow Depth at the Downstream End of a
Long Spillway.') Jl La Houille Blanche, Mar.-Apr., pp. 185-186 (in French).
FERRO, V. (1992). "Flow Measurement with Rectangular Free Overfall." Jl of Irrig. and Drain., ASCE, Vol. 118,
No. 6, pp. 956-964.
KRAIJENHOFF, D.A., and DOMMERHOLT, A. (1977). "Brink Depth Method in Rectangular Channel." Jl of
Irrig. and Drain., ASCE, Vol. 103, No. IR2, pp. 171-177.
CHANSON, H. (1996). "Energy Loss at Drops - Discussion." Jl of Hyd. Res., IAHR, Vol. 34, No. 2, pp. 273-278 (ISSN
0022-1686).
RAJARATNAM, N., and CHAMANI, M.R. (1995). "Energy Loss at Drops." Jl of Hyd. Res., IAHR, Vol. 33, No. 3,
pp. 373-384.
ROUSE, H. (1936). "Discharge Characteristics of the Free Overfall." Civil Engineering, Vol. 6, April, p. 257.
Table 1 - Nappe geometry calculations for aerated nappes
Variable
(1)
db (a)
Lp
Formula
(2)
db = 0.715 * Yc
Ref.
(3)
[RO]
Remarks
(4)
Experimental data.
db = 0.65 * Yc
[CR]
Analytical solution.
db = 0.781 * Yc
[BA]
db = 0.714 * Yc
[KR]
0.72 < db/Yc < 0.93
db = 0.76 * Yc
[GI]
Model data. Yc < 0.124 m.
W = 0.46 m.
Model data. 0.021 < Yc < 0.068 m.
W = 0.5 m.
Model data. 0.075 < Yc/H < 0.45.
[FE]
Model data. 0.15 < Yc/H < 0.93.
Lp
⎛Yc⎞0.525
=
2.17
*
H
⎝H⎠
Lp
⎛Yc⎞0.81
=
4.30
*
H
H
[CH95] Best fit of equation (2).
V
-0.483
V
⎛Yc⎞
Vc = 1.455 * ⎝ H ⎠
[CH94] Best fit of equation (4).
tanϕ
⎛Yc⎞-0.582
tanϕ = 0.855 * H
⎝ ⎠
[CH94] Best fit of equation (5).
[RA]
Model data. 0.045 < Yc/H < 1.
⎝ ⎠
Yp
[MO]
Analytical solution.
Yp
⎛Yc⎞0.66
=
H
⎝H⎠
[RA]
Model data. 0.045 < Yc/H < 1.
Yp
⎛Yc
⎞0.697
=
1.067
*
0.0016
H
⎝H
⎠
Yp
⎛dc⎞0.719
=
1.107
*
h
H
[GI]
Model data. 0.075 < Yc/H < 0.45.
[RC]
Solution of non-linear equations.
Yp
Yc =
Yc
⎛Y1⎞2
⎜Y ⎟ + 2 * Y - 3
1
⎝ c⎠
⎝ ⎠
Y1
Yp
⎛Yc⎞0.675
=
0.998
*
H
⎝H⎠
Y1
2
Yc = 1.5
H
3
+
Yc + 2
2
Y1
⎛Yc⎞1.275
=
0.54
*
H
⎝H⎠
Y1
⎛Yc⎞1.326
=
0.625
*
H
H
⎝ ⎠
Notes :
[CH95] Best fit of equation (7)
[WH]
Analytical solution.
[RA]
Model data. 0.045 < Yc/H < 1.
[CH95] Best fit of equation (8).
CHANSON, H. (1996). "Energy Loss at Drops - Discussion." Jl of Hyd. Res., IAHR, Vol. 34, No. 2, pp. 273-278 (ISSN
0022-1686).
(a) :
subcritical flow in rectangular channel
[BA] BAUER and GRAF (1971); [CH94] CHANSON (1994); [CH95] CHANSON (1995); [CR] CRAYA (1948); [FE]
FERRO (1992); [GI] GILL (1979); [KR] KRAIJENHOFF and DOMMERHOLT (1977); [MO] MOORE (1943); [RC]
RAJARATNAM and CHAMANI (1995); [RA] RAND (1955); [RO] ROUSE (1936); [WH] WHITE (1943).
Fig. 1 - Definition sketch of a drop structure
db
Over fall
Vent
ϕ
V
Yp
Y1
Air
entrainm ent
Lp
Fig. 2 - Comparison between predicted and experimental values
ϕ
degrees
90
Y1/H
0.2
80
70
0.15
60
ϕ
Data
Authors
50
Writer EQ. (12)
0.1
40
Data
Authors
30
Writer EQ. (14)
20
0.05
Y1/H
10
0
0
0
0.1
0.2
Yc/H
0.3
0.4