Ali Oveisy
M.Sc. E-mail: a_oveisy@yahoo.com
M. A. Banihashemi
Ph.D. E-mail: banihash@shafagh.ut.ac.ir
Dept. of Civil Eng., University of Tehran
P.O. Box 11365/4563, Tehran, Iran
Tel.
Fax. (98) 21 6498981
Abstract: The experimental results on energy loss and shock wave front location at a channel abrupt expansion, assuming one-dimensional concept, are presented. Different streamwise non-dimensional coordinates are introduced for the generalization of experimental observations and the most proper one is determined. A simple method is presented for estimating energy loss and shock wave front location for a supercritical flow at an abrupt expansion.
Further, the observed energy loss is compared with that of the
theoretical gradually varied flow with the same initial energy for each run. The
energy loss, which was calculated using Hager and Mazumder (1992) experimental
observations, is expressed using the new non-dimensional coordinate, and it
shows a fair correlation.
Keywords: abrupt expansion, energy
loss, shock wave, supercritical flow
Abrupt expansion with supercritical flow as a transition in open channels is one of the important elements in hydraulic structures such as outlets, spillways, chutes and flood relief canals.
Supercritical flow investigations are faced with many problems such as large amount of air entrainment which complicates computational methods, and need of large models with considerable discharge for avoiding scale effects in experimental observations. Because of the above-mentioned problems and complicated characteristics of flow passing an abrupt expansion, particular attention should be paid to any simplified and practical methods for explaining these flows. In addition, identifying the rate of the energy loss is essential for computing the surface profile in open channel, and these computations could be made easy using the energy loss in transitions where the flow pattern is too complicated to compute. Therefore, in this paper the rate of the energy loss is investigated experimentally, using a non-dimensional coordinate due to flow pattern which is important in energy loss.
Further, the results are compared with gradually varied flow theory for energy loss differences. In order to develop the new introduced non-dimensional coordinate, it was used for results of Ref. 10, and a fair correlation is showed.
Literature review:
The first study of supercritical flow in
channel expansion was conducted by Rouse et al. (1951). According to their
experimental observations and using dimensional analysis, the free surface for
pure gravity flow was determined as
, where
,
,
streamwise coordinate,
transverse coordinate,
,
and
are approach flow depth, width and
Froude number respectively. In the second part of their paper, they offered a
continuous wall geometry to reduce the shock waves formation in a finite width
expansion.
More studies were conducted by Guenzel (1962), Sherenkov (1965), Herbich and Walsh (1972), Koch (1979), Engelund (1979), Bellos et al. (1991).
Hager and Mazumder (1992) conducted a wide
range experimental investigation on supercritical flow at abrupt expansion with
and expansion ratios of
2, 3, 5 to determine flow pattern downstream of an abrupt expansion. The
pattern was extensively described, and particularities such as the shock front,
the expanding flow portion and the reflection mechanism were discussed. The
non-dimensional Rouse streamwise coordinate
, was used for expressing the results. It was shown that both the axial and
wall flow depth profiles follow distinct curves of X, provided that effects of scale are
insignificant. All relations involving velocity were shown to be strongly
governed by scale effects, such that a simple generalization of results was
impossible. Further, the transverse surface profiles for
were generalized and extended to
include expansion ratios
.
Hager and yasuda (1997) conducted experimental
investigations on unconfined expansion of supercritical water flow with
2, 4, 6 and
. In addition, a simple relation was given between local streamline direction
and local flow depth. The two-dimensional expansion of supercritical water flow
on a horizontal smooth plane was demonstrated to correspond asymptotically to
one-dimensional unsteady, “simple wave problem”, both in terms of free
surface and transverse velocity distributions. In addition, for Froude numbers
in excess of 3, an even simpler approach based on the streamwise coordinate
applied, without any scale effects
observation near zone of jet efflux.
Literature review shows that more experimental investigations are needed for determining flow characteristics, and simple relations are useful for expressing them.
The experiments were conducted in a rectangular
channel b2=1(m) wide
(expanded part) and 12(m) long with a slope of 0.0165. The right wall was made
of Plexiglas and the bed and the left wall were made of very smooth cement
mortar. The jet was produced by a sluice gate, the outflow of which was led to
an approach channel b0 wide and 3(m) long. The walls of the approach
channel were made of Plexiglas and its bed was made of very smooth cement
mortar. The expansion ratios (
b2/bo) were 1/0.30, 1/0.40,
1/0.55, 1/0.70 and the discharge (Q) was varied approximately between
for each expansion ratio. The
discharge was measured by a rectangular sharp-crested weir 330 (mm) wide, at the
end of the channel after a stilling basin. A volume-measuring method was used
for calibrating the weir with
2.5% precision.
Since the uniformity of outflow near the sluice gate and the expansion was questionable, some transverse sections upstream of the expansion were selected for tracing and observing of the initial flow condition (Fig. 1). Table 1 summarizes the test program with detailed runs, where ho and Fo are the approach flow depth and Froude number respectively.
Different cross-sections were chosen to measure flow characteristics, as shown in Fig. 1. The locations of these cross-sections differed for each run, in order to capture the flow characteristics and shock wave front pattern properly. Each section divided into several parts
Fig.
1 Transverse sections for observation and computing energy
Table 1 Test program with detailed runs
|
Run num. |
Q(m^3) |
b0(m) |
h0(m) |
F0 |
b2(m) |
|
1 |
0.069 |
0.300 |
0.076 |
3.51 |
1.000 |
|
2 |
0.060 |
0.300 |
0.073 |
3.28 |
1.000 |
|
3 |
0.049 |
0.300 |
0.070 |
2.80 |
1.000 |
|
4 |
0.037 |
0.300 |
0.066 |
2.33 |
1.000 |
|
5 |
0.022 |
0.300 |
0.049 |
2.12 |
1.000 |
|
6 |
0.069 |
0.400 |
0.065 |
3.33 |
1.000 |
|
7 |
0.055 |
0.400 |
0.062 |
2.88 |
1.000 |
|
8 |
0.048 |
0.400 |
0.060 |
2.63 |
1.000 |
|
9 |
0.037 |
0.400 |
0.044 |
2.24 |
1.000 |
|
10 |
0.025 |
0.400 |
0.044 |
2.13 |
1.000 |
|
11 |
0.070 |
0.550 |
0.042 |
4.65 |
1.000 |
|
12 |
0.048 |
0.550 |
0.039 |
3.10 |
1.000 |
|
13 |
0.041 |
0.550 |
0.039 |
3.10 |
1.000 |
|
14 |
0.035 |
0.550 |
0.040 |
2.68 |
1.000 |
|
15 |
0.023 |
0.550 |
0.040 |
1.67 |
1.000 |
|
16 |
0.066 |
0.700 |
0.044 |
3.30 |
1.000 |
|
17 |
0.048 |
0.700 |
0.040 |
2.68 |
1.000 |
|
18 |
0.033 |
0.700 |
0.033 |
2.54 |
1.000 |
|
19 |
0.019 |
0.700 |
0.024 |
2.32 |
1.000 |
such
as
, where n was depends on the uniformity of flow in each section. In each part,
velocity in x direction (
) and depth (
) were measured. The flow depth was measured by a point gauge to the nearest mm
and the velocity was measured by a Pitot tube. Repetitive observations showed
that the errors are confined to
1 (mm) for depth, and
2% for velocity.
The flow jet efflux contacts with the channel walls at a location called First Contact (FC). The shock waves are created and distributed from FC and cross each other and form a rooster tail shape, this location is called Rooster Tail (RT). The shock waves contact with the channel walls again at the location, which was called Second Contact (SC).
Since vertical acceleration was not considerable after FC except
for the front wave, which was a small part of the flow and could be neglected,
the flow pressure distribution was assumed hydrostatic. Therefore, the total
energy head was computed by equation 1 according to the Fig. 2, using the
average velocity
; the average depth
and the bed height from zero level, Z at each section.
Fig. 2 The algorithm of energy computation
According to
Ref. 10, it is assumed that the energy of the flow at FC is equal to the
energy of the approaching flow, and
(Coriolis coefficient) was ignored
for computing the energy of other sections downstream of FC. So at FC,
is greater, and for downstream sections it is smaller than the actual energy.
These differences are not considerable, because the value of
after the FC is relatively small.
(Refer to
values, which were presented in Ref. 9).
(1)
A non-dimensional method is used for
generalizing the results. For energy loss, “
” is used (
is the energy loss, and
is the specific energy for approach flow.), and for “ X “ (the non-dimensional coordinate
parameter) several trials are made some of which are shown in equations 2 to 7.
The first and the second ones have been used by Rouse at el. (1965), Hager and
Mazumder (1992) and Hager and Yashuda (1997). Other parameters are introduced to
consider the effect of both ho and bo simultaneously.
(2)
(3)
(4)
(5)
(6)
(7)
Since shock waves are important in the process of the energy loss, the locations of FC, RT and SC were studied using above-mentioned non-dimensional coordinates.
The calculated energy loss obtained from
measurements of Ref. 10 were made dimensionless, using these new
nondimensional parameters. For these calculation the average depth, the
discharge and
were used and it is assumed that the pressure distribution is hydrostatic.
The nondimensional results were compared and
the following expression (equation 7)
was found to give the best
correlation for both the energy loss and the wave front locations, as shown in
Fig. 3 to 5(“R” is regression coefficient).
In the expansion ratio of 1/0.7, it is difficult to define the location of FC, because of shallow dead water zones next to the jet borderlines. The energy of the flow with average depth upstream of FC is smaller than the energy of the flow with observed transverse profile (see Fig. 6). Thus, some virtual energy loss was shown in Fig. 4 near the expansion.
The length of the approach channel was 3m, so the depth of the flow was closed to the normal depth. Therefore, there is a weak relation between ho and Fo, and it may make some differences between the present investigation and that of Ref. 10.
Fig. 3 The experimental data of the energy loss and its trendline for 19 runs, and trend line for 19 gradually varied flows
Fig. 4 Energy loss from Ref. 9 and it’s trendline (b0=0.5 and b2=1.5)
According to Fig. 3 and experimental observations, the flow can be divided into four parts after the abrupt expansion, with special mechanisms, which are important for energy loss in each part. These parts are described in the following.
(1) From the beginning of the expansion to FC,
there is not any dead water zone, as opposed to the case of subcritical flow.
However, small vortices were observed next to the jet boundaries. Because of the
small size of the vortices and high depth of the flow at the channel axis, the
energy loss due to vortices and friction is small, so energy loss in this part
can be neglected. The large values of
in Ref. 9 shows that the energy of
the approaching flow is used to accelerate the flow in transverse direction.
Fig.
5 location of FC, RTand SC for present experiments
Fig.
6 (a) Transverse profile of jet efflux. (b) Average depth for
section (a)
(2) From FC to RT, the energy that is used to accelerate the flow in transverse direction in previous section is dissipated along the shock waves. The bed friction is also important in this part except for the main flow zone that showed in Fig. 1 (Ref. 10), so the energy loss trend line in Fig. 3 is steeper at this interval.
(3) From RT to SC, less turbulence is observed, so the energy dissipation due to the shock waves is less than the previous section, and the friction loss tends to be more important.
(4) From the Second contact to the end of the channel, friction is the main factor which dissipates the energy. In this part, the flow can be assumed as a gradually varied flow.
Gradually varied flow theory was used to compute the energy loss of the flow in the expanded part, with the same initial energy of the approach flow for each experimental run. In these computations, the flow was assumed one-dimensional. A trend line for these computational results and a comparison of the energy loss differences between the experimental observations and this simple computational method are shown in Fig. 3. As it is shown, after the SC the energy loss and the energy loss gradient obtained from both methods are approximately equal.
(1) The main part of the energy dissipates gradually after FC, and it is mainly due to shock wave formation and bed friction. Between FC and RT the shock waves are more important in energy loss, but after RT the bed friction is going to be main agent of the energy loss.
(2) The best dimensionless coordinate for
interpretation of the energy loss and different locations of the front waves is
found to be
.
(3) It seems that beyond the jet efflux, energy loss for experimental observations is approximately equal to that of computed from gradually varied flows theory and after SC the main factor of the energy loss is bed friction.
(4) Taking in to account the wide range of initial flow conditions, more investigations are needed to achieve more reliable expressions for explaining complicated characteristics of supercritical flow passing an abrupt expansion.
Appendix
Notation
b = channel width; E = the flow specific energy;
F =Froude number; g = gravitational acceleration;
H= total energy head; h = flow depth;
n = number of width divisions; Q = discharge;
V = velocity; X = streamwise non-dimensional coordinate;
x = streamwise coordinate; Z = height from zero level;
= Coriolis coefficient;
= b2/bo
= energy loss
FC = First Contact
RT = Rooster Tail SC = Second Contact
Subscript
o = for approach flow; 2 = for expanded part;
x = characteristic of x direction;
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