PHYSICS OF FLUIDS
VOLUME 15, NUMBER 9
SEPTEMBER 2003
Hysteretic behavior of the flow under a vertical sluice gate
Andrea Definaa) and Francesca Maria Susinb)
Dipartimento di Ingegneria Idraulica, Marittima, Ambientale e Geotecnica (IMAGE), Università di Padova,
via Loredan 20, I-35131 Padova, Italy
共Received 16 July 2002; accepted 5 June 2003; published 31 July 2003兲
In the paper, the phenomenon of hysteresis which can develop in a supercritical channel flow
approaching an obstacle is analyzed, and a simple theory to predict the occurrence of hysteresis is
described. The results of an in-depth theoretical and experimental study of the case of flow under a
vertical sluice gate in a rectangular channel are then presented. Possible flow regimes in the vicinity
of a gate are classified on the basis of the nondimensional gate opening and the Froude number of
the undisturbed approaching flow. It is shown that within a wide range of flow parameters both
undisturbed and free outflow conditions may exist for the same gate opening. Within this range, the
actual regime depends on the previous history of the flow, thus implying the hysteretic character of
the flow. It is worth noting that a subcritical approaching flow may also exhibit such a hysteretic
behavior provided the Froude number is greater than approximately 0.8. This occurrence, which has
not previously been reported in the literature, is probably a result of the contraction affecting the
flow issuing from under the gate. The results of an extensive series of experiments, performed over
a wide range of flow parameters, are detailed in the paper and confirm theoretical predictions.
© 2003 American Institute of Physics. 关DOI: 10.1063/1.1596193兴
I. INTRODUCTION
supercritical flow implies hydraulic hysteresis. Nevertheless,
a unified theory of hysteresis has not been developed yet. In
fact, in Sec. II of the paper a theoretical approach is presented which uses simple relationships expressing energy
balance and momentum conservation to infer the theoretical
conditions in which hysteresis occurs. Section III focuses on
the theoretical description of steady flow regimes under a
vertical sluice gate, for both supercritical and subcritical undisturbed flow conditions. Section IV describes the experimental apparatus used and compares experimental results
with theoretical predictions. Finally, Sec. V is devoted to
some conclusions.
Elementary fluid mechanics problems sometimes have
unsuspected solutions due to the nonlinearity of governing
equations. A well documented but not well known example
of this is the hysteretic behavior of a supercritical channel
flow approaching an obstacle.
The occurrence of hysteresis in a supercritical flow approaching a sill has been recognized and widely investigated
both experimentally and theoretically over the past few
decades.1– 8 All these studies have shown that, in the vicinity
of an obstacle, the flow takes a path which may depend not
only on the height of the obstacle, but on the history of the
flow as well, i.e., on the way in which flow conditions have
evolved up to the current ones. Moreover, simple onedimensional channel flow equations for energy and momentum conservation make it possible to accurately describe hydraulic hysteresis at least within the limits of negligibly
small dissipations.3,6
On the other hand, much less effort has been devoted to
the study of hydraulic hysteresis for other types of obstacles,
such as channel constriction. This might be due to the fact
that the complicated pattern of the flow in the vicinity of a
sudden reduction in the channel width does not allow for the
development of an accurate theoretical formulation based on
the simple one-dimensional approach. Some experimental
studies reported in the literature, however, confirmed the
hysteretic behavior of a supercritical flow through channel
constriction.9,10
The theoretical and experimental results reported in the
studies quoted above suggest that any type of obstacle in a
II. THEORETICAL APPROACH
We consider a one-dimensional supercritical channel
flow of velocity v 0 and depth y 0 impacting a generic obstacle. The frictional effects of the boundaries are considered
and the pressure is assumed to be hydrostatic except in proximity of the obstacle. The bottom slope is assumed to be
negligibly small so that cos(␽)⬇1, sin(␽)⬇tan(␽), ␽ being
the angle of the channel bed to the horizontal.
The theoretical boundaries of the region of hysteresis for
this supercritical channel flow can be inferred by approaching the problem from a phenomenological point of view. To
describe the occurrence of hysteresis, it is convenient to use
the concept of specific energy H, i.e., mechanical energy per
unit weight of flow relative to the bottom of the channel.
Thus, for the approaching flow
H 0 ⫽y 0 ⫹
a兲
Electronic mail: defina@idra.unipd.it
Electronic mail: susy@idra.unipd.it
b兲
1070-6631/2003/15(9)/2541/8/$20.00
v 20
2g
,
共1兲
where g is gravity.
2541
© 2003 American Institute of Physics
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2542
Phys. Fluids, Vol. 15, No. 9, September 2003
A. Defina and F. M. Susin
Here, F 0 is the Froude number of the undisturbed supercritical flow, given by F 20 ⫽ v 20 /gy 0 .
If we now slightly increase the ratio H 0 /H min so that it
returns to values greater than one, the conditions shown in
Fig. 1共a兲 are not recovered. The flow configuration smoothly
changes towards the one shown in Fig. 1共c兲, where the hydraulic jump has moved downstream and the backwater profile has shortened in order to reduce the gained specific energy ⌬H S . In the limit condition, i.e., when the backwater
profile vanishes, the specific energy of the flow just upstream
of the obstacle is equal to H min⫽H0⫺⌬HJ . It is worth pointing out that in this case the water depth y U , just upstream of
the obstruction, is the conjugate of the undisturbed flow
depth, i.e.,
yU 1
⫽ 共 ⫺1⫹ 冑1⫹8F 20 兲 F ⫺2/3
,
0
yc 2
FIG. 1. Steady flow regimes in the vicinity of an obstacle for different
values of the ratio between the specific energy of the supercritical incoming
flow, H 0 , and the minimum specific energy H min required to pass the obstacle. Free surface profiles are plotted with respect to the bottom of the
channel.
where y c is the critical depth.
Any further increase in the ratio H 0 /H min suddenly restores undisturbed flow conditions.
From the above description of possible flow regimes in
the vicinity of the obstacle, it follows that sufficient and necessary conditions for a supercritical approaching flow to undergo a supercritical to subcritical transition are not the
same, thus implying the hysteretic character of the flow.
In fact, the sufficient condition for transition to occur is
H 0 ⭐H min ,
共5兲
while the necessary condition is given by
H 0 ⫺⌬H J ⭐H min .
We first consider initial flow conditions shown in Fig.
1共a兲, which will from here on be referred to as smooth flow
conditions. The specific energy of the incoming supercritical
flow, H 0 , is greater than the minimum specific energy H min
required to pass the obstacle, and the flow is supercritical
along the channel.
If H 0 /H min is reduced, by either increasing H min or decreasing the flow energy, until H 0 becomes slightly smaller
than H min , then the incoming flow does not have enough
energy to pass the obstacle and, as a consequence, transition
from supercritical to subcritical flow occurs, leading to the
flow configuration shown in Fig. 1共b兲. A stationary hydraulic
jump occurs far from the obstacle and a subcritical backwater profile is established after the jump. The backwater profile, along which H increases, is long enough for the current
to achieve the minimum specific energy required to pass the
obstacle. Hence, the specific energy of the flow just before
the obstacle is
H 0 ⫺⌬H J ⫹⌬H S ⫽H min ,
共2兲
where ⌬H S is the specific energy gained by the flow along
the backwater profile and ⌬H J is the energy loss at the hydraulic jump expressed as
⌬H J ⫽
y 0 共 ⫺3⫹ 冑1⫹8F 20 兲 3
.
16 共 ⫺1⫹ 冑1⫹8F 20 兲
共3兲
共4兲
共6兲
Moreover, for a given undisturbed flow condition and
given geometric characteristics of the obstacle, H min may
depend on the flow regime being established just upstream of
the obstacle. Hence, if the minimum specific energy for supercritical and subcritical regime are denoted with H l and
H L , respectively, as long as the following constraint is met:
H l ⭐H 0 ⭐H L ⫹⌬H J ,
共7兲
both steady flow configurations, i.e., in the presence or absence of an upstream jump, are possible and stable.
The energies H l and H L which define the boundaries of
the hysteresis region can often be determined by simply imposing critical conditions at the obstacle. It is worth pointing
out that H l and H L are defined at a section immediately
upstream of the obstacle, where the flow can be assumed to
be one-dimensional regardless of the type of obstacle.
We can consider, for example, the flow over a sill. In this
case, the threshold values for specific energy are
H l ⫽⌬⫹H c ⫹⌬H l ,
H L ⫽⌬⫹H c ⫹⌬H L ,
共8兲
where ⌬ is the height of the sill, H c is the critical specific
energy for a given flow rate and channel width, and ⌬H l and
⌬H L are the energy losses just before the section at which
critical condition attains for supercritical and subcritical regime, respectively.
For a sill with a smooth shape, the energy losses ⌬H l
and ⌬H L can be neglected. Thus, from 共8兲, H l ⫽H L , and the
amplitude of the hysteresis region is given by ⌬H J . 2,6,7
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Phys. Fluids, Vol. 15, No. 9, September 2003
Hysteretic behavior of the flow
2543
Fig. 2共d兲. In Figs. 2共a兲 and 2共c兲, characterized by the same
undisturbed flow conditions and geometry of the obstacle but
by different previous flow regimes, the flow exhibits two
different stable states, thus confirming the hysteretic character of the flow–gate interaction.
For the case of a vertical sluice gate in a rectangular
channel, the theoretical approach outlined in Sec. II can be
particularized as follows.
The upper threshold H L can be expressed in terms of the
gate opening by applying the Bernoulli equation between the
upstream flow and the vena contracta
H L ⫽a L c c ⫹q 2 / 共 2g 共 a L c c 兲 2 兲 ,
FIG. 2. Sketch of the hysteretic behavior of a supercritical flow approaching
a vertical sluice gate. 共a兲 Initial undisturbed conditions with a⬎y 0 ; 共b兲 the
gate is lowered to a⬍y 0 and the free outflow regime is established; 共c兲 the
gate is raised back to a⬎y 0 , but free outflow still persists; 共d兲 the gate is
further raised until undisturbed conditions are restored. Cases 共a兲 and 共c兲
correspond to the same gate opening but different previous states of the
flow.
On the contrary, for a sharp crested sill ⌬H l must be
accounted for and its value can usually be inferred by using
the momentum equation.4 In this case, the amplitude of the
hysteresis region is reduced to ⌬H J ⫺⌬H l .
As stated above, the constraint expressed by Eq. 共7兲 defines the hysteresis region. However, for practical purposes,
this condition is usually expressed in terms of the significant
flow parameters and geometrical characteristics of the
obstruction.3–5,7,8 In the next section, condition 共7兲 will be
developed for the case of flow under a vertical sluice gate in
terms of a nondimensional gate opening and Froude number
of the approaching flow.
III. FLOW UNDER A SLUICE GATE
The hysteretic behavior of the flow under a vertical
sluice gate concerns the possibility that, for a given height of
the gate opening, a supercritical approaching flow may either
continue undisturbed or undergo a supercritical to subcritical
transition upstream of the gate. The state that is actually
obtained is determined by the past history of the flow.
Figure 2 shows four different steady flow configurations
in the vicinity of a gate for different gate openings and/or
different previous states of the flow. In Fig. 2共a兲 the height of
the gate opening is larger than the undisturbed water depth
and the smooth flow conditions are assumed to exist along
the channel. As soon as the edge of the gate touches or
slightly goes under the water surface, as shown in Fig. 2共b兲,
the free outflow condition is established 共i.e., the fluid issues
from under the gate as a jet of supercritical flow with a free
surface open to the atmosphere兲 and the transition from supercritical to subcritical regime takes place upstream of the
gate. If, starting from the state shown in Fig. 2共b兲, the gate is
raised back to the same level as Fig. 2共a兲, transition may still
persist 关Fig. 2共c兲兴. The only differences between Figs. 2共b兲
and 2共c兲 are the jump location and the length of the backwater profile. Undisturbed smooth conditions are only restored
for sufficiently high values of the gate opening, as in case
共9兲
where a L is the height of the gate opening at the upper limit,
q the flow rate per unit width, and c c the contraction coefficient 共i.e., the ratio of the flow depth at the vena contracta to
the height of the gate opening兲.
The lower boundary of the hysteresis domain can also be
inferred from Eq. 共9兲, simply by assuming that c c ⫽1 since
the flow is undisturbed. Therefore, the lower threshold specific energy is related to gate opening by
H l ⫽a l ⫹q 2 / 共 2ga 2l 兲 ,
共10兲
where a l is the height of the gate opening at the lower limit.
Substituting Eqs. 共9兲 and 共10兲 in the constraint 共7兲 gives
the boundaries of the hysteresis region in terms of the height
of the gate opening and Froude number of the undisturbed
flow. In fact, the lower limit (H l ⫽H 0 ) corresponds to a l
⫽y 0 or, in terms of nondimensional variables, to
a l /y c ⫽F ⫺2/3
.
0
共11兲
Moreover, by combining Eqs. 共9兲, 共7兲, and 共3兲, and recalling that
H 0 /y c ⫽F ⫺2/3
⫹F 4/3
0
0 /2,
共12兲
the following expression can be easily found for the upper
boundary a L /y c as a function of the undisturbed Froude
number
冉 冊 冉 冊
aL
1 aL
cc ⫹
c
yc
2 yc c
⫺2
1
⫽ 共 ⫺1⫹ 冑1⫹8F 20 兲 F ⫺2/3
0
2
⫹
2F 4/3
0
共 ⫺1⫹ 冑1⫹8F 20 兲 2
.
共13兲
In Eq. 共13兲 the actual value of a L /y c depends on the
contraction coefficient c c , which is computed using the following parametric equation
c c ⫽1⫺r 共 ␽ 兲 sin共 ␽ 兲 ,
a L /y U ⫽1⫺r 共 ␽ 兲关 1⫺cos共 ␽ 兲兴 ,
共14兲
r 共 ␽ 兲 ⫽0.153␽ 2 ⫺0.451␽ ⫹0.727.
Equation 共14兲 is obtained by fitting the present experimental
data in the range 0⬍ ␽ ⬍2.5, ␽ being a dummy parameter
共see Sec. IV A兲.
The plot of Eqs. 共11兲 and 共13兲, shown in Fig. 3, displays
the existence of three different regions in the plane
(a/y c ,F 0 ). The first region, extending above the curve
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2544
Phys. Fluids, Vol. 15, No. 9, September 2003
A. Defina and F. M. Susin
FIG. 3. Steady flow regimes in the vicinity of a vertical sluice gate for a
supercritical approaching flow on the F 0 -a/yc diagram.
a L /y c , corresponds to undisturbed flow conditions where
the flow does not interact with the gate at all. In the second
region, below the curve a l /y c , the flow necessarily interacts
with the obstacle and undergoes a supercritical to subcritical
transition upstream of the gate. Finally, the hysteresis region,
in which both stable states are possible, lies between these
two curves.
It is worth pointing out that the conditions described by
the branch AM of the curve a L /y c cannot be experimentally
recovered and this branch of the curve must be replaced with
the horizontal line A ⬘ M : a/y c ⫽(a/y c ) max⬵1.15 共a discussion of this important point can be found in the Appendix兲.
As a consequence, the two boundaries of the hysteresis region do not merge at F 0 ⫽1, thus suggesting that the bistable
hysteretic behavior may also extend to subcritical undisturbed approaching flows.
In order to verify this possibility, the theoretical analysis
of the flow under a sluice gate was extended to subcritical
undisturbed flow conditions. In this case, three different regimes may be present in the vicinity of the gate, namely:
undisturbed flow conditions, free outflow, and submerged
outflow 共Fig. 4兲.
Undisturbed flow conditions are recovered as long as
a/y c ⭓a l /y c , where the behavior of a l /y c is expressed in
Eq. 共11兲. The boundary curve 共hereafter denoted with a L /y c )
between free and submerged outflow is given by the equation
冉 冊
F ⫺2/3
aL
0
cc ⫽
共 ⫺1⫹ 冑1⫹8F 20 兲 ,
yc
2
FIG. 4. Sketch of steady flow regimes in the vicinity of a vertical sluice gate
for a subcritical approaching flow. 共a兲 Undisturbed conditions; 共b兲 free outflow; 共c兲 submerged outflow.
The behavior of the boundary curves a l /y c and a L /y c
for F 0 ⭐1 is shown in Fig. 5. Here, the branch AM ⬘ of the
curve a L /y c is replaced with the horizontal line A ⬘ M ⬘ :
a/y c ⬵1.15 as was the case for the supercritical approaching
flow 共see the Appendix兲. The two above curves intercept at
B, highlighting the region AA ⬘ B, which is the extension of
the hysteresis region into the subcritical domain. Therefore,
in approximately the range 0.8⬍F 0 ⭐1, the hysteretic behavior regards the possibility that, for a given height of the gate
opening, either smooth conditions persist along the channel
or free outflow conditions are established, depending on the
previous history of the flow.
The hysteretic character of the flow in the vicinity of a
sluice gate can be adequately described by plotting the behavior of the water depth just upstream of the gate, y u , as a
function of the gate opening a. In the example shown in Fig.
6, corresponding to F 0 ⫽2, the initial steady flow is undisturbed with a⬎y 0 ⫽y u 共point I兲. The gate is then lowered
共15兲
which states that the flow depth at the vena contracta, a L c c ,
is equal to the conjugate depth of the downstream undisturbed subcritical flow. In other words, Eq. 共15兲 expresses the
momentum balance between the vena contracta and the
downstream flow.
The contraction coefficient in Eq. 共15兲 is computed using
Eq. 共14兲, where the water depth just upstream of the gate, y u
is calculated from the Bernoulli equation
冉 冊 冉 冊 冉 冊 冉 冊
aL
1 aL
c ⫹
c
yc c
2 yc c
⫺2
⫽
yu
1 yu
⫹
yc
2 yc
⫺2
.
共16兲
FIG. 5. Steady flow regimes in the vicinity of a vertical sluice gate for a
subcritical approaching flow on the F 0 -a/y c diagram.
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Phys. Fluids, Vol. 15, No. 9, September 2003
Hysteretic behavior of the flow
2545
FIG. 7. Upper boundary of the hysteresis region computed from Eq. 共13兲
adopting different theoretical expressions for the contraction coefficient c c :
Cisotti 共Ref. 11兲 and von Mises 共Ref. 12兲 共dotted line兲 and Marchi 共Ref. 14兲
共solid line兲.
FIG. 6. Upstream depth y u /y c as a function of the gate opening a/y c describing the hysteresis loop for flow with Froude number F 0 ⫽2. Point I:
Initial undisturbed flow conditions. Point VI: Restored undisturbed flow
conditions. Black arrows and white arrows denote the gate lowering and
gate lifting stages, respectively.
共points from I to II兲, and as it touches the water surface
共point II兲 free outflow conditions establish and the upstream
water depth suddenly increases due to the transition from
supercritical to subcritical flow upstream of the gate 共point
III兲. This flow regime persists not only as the gate is lowered
and/or raised along the branch from III to IV (a⬍y 0 and
y u ⬎y 0 ), but also when the gate is raised along the branch
from III to V to levels higher than the undisturbed flow depth
(a⬎y 0 and y u ⬎y 0 ). When the limit point V is reached, i.e.,
a⫽a L , undisturbed conditions are suddenly restored
in the channel 共point VI兲, thus closing the hysteresis
loop II-III-V-VI.
The above theoretical behavior was verified for both supercritical and subcritical undisturbed initial flows by performing laboratory experiments designed for this purpose, as
described in the following section.
A. Evaluation of the contraction coefficient
As stated in Sec. III, the behavior of the curve a L /y c is
very sensitive to the contraction coefficient. This is clearly
shown in Fig. 7, where the theoretical upper boundary of the
hysteresis region in the supercritical domain is computed using Eq. 共13兲 adopting two different formulations for c c ,
namely the one given by Cisotti11 and von Mises,12 as reported by Gentilini,13 and the one proposed by Marchi.14
These two expressions for c c , which were inferred assuming gravity-free and gravity-affected flow, respectively,
are plotted in Fig. 8 as a function of a/y u . Further theoretical
formulations for c c have been proposed in the literature,16 –18
and some of them are plotted in Fig. 8 for the sake of comparison.
On the contrary, only a few experimental measurements
of the contraction coefficient are reported in the
literature.15,16 Moreover, the experimental data available for
gravity-affected flows mainly pertain to the range 0⬍a/y u
⬍0.5 whereas, in the present study, the typical range was
approximately 0.4⬍a/y u ⬍0.85 共see Fig. 7兲. For this reason,
IV. EXPERIMENTS
The experiments were performed in a 0.38 m wide, 0.5
m high, and 20 m long tilting flume with Plexiglas walls,
whose bottom slope could be adjusted to a maximum of 5%.
Water was recirculated through the channel via a constant
head tank which maintained very steady flow conditions.
A vertical sharp crested gate was placed at the test section, which was located approximately 5 m from the downstream end of the channel in supercritical undisturbed flow
experiments, and approximately 7 m downstream of the
channel inlet in subcritical undisturbed flow experiments. In
the latter case, the flume used a downstream weir to achieve
quasi-uniform flow conditions at the test section.
A gear system was used to raise and lower the gate. It
was possible to set the height of the gate opening with an
accuracy of ⫾0.2 mm.
Two ultra-sonic transducers, movable along the channel
axis upstream and downstream of the gate, measured flow
depths with an accuracy of ⫾0.5 mm.
FIG. 8. Contraction coefficient c c as a function of a/y u as theoretically
predicted by different authors.
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2546
Phys. Fluids, Vol. 15, No. 9, September 2003
FIG. 9. Experimental behavior of the contraction coefficient c c plotted as a
function of a/y u .
and in order to perform a reliable comparison between theory
and experiments of the phenomena of hysteresis, a preliminary set of experiments was carried out to measure the contraction coefficient.
These experiments were performed using the abovementioned apparatus. The downstream ultra-sonic transducer
was slowly moved along the channel axis from the gate to a
distance of about 10–15 times the gate opening, in order to
detect the position of the vena contracta and to measure the
corresponding water depth y d . At the same time, the upstream ultra-sonic transducer measured the flow depth y u .
The contraction coefficient c c ⫽y d /a was evaluated as a
function of a/y u in different conditions of the flow rate,
flume slope, and gate opening. The experimental points
(a/y u ,c c ) are plotted in Fig. 9, which includes the fitting
curve given by Eq. 共14兲 and the theoretical predictions given
by Marchi14 and Vanden-Broeck.17 Major discrepancies can
be seen in the range 0⬍a/y u ⬍0.3, possibly due to the developing boundary layer which, in this range, is thick if compared with the gate opening, thus significantly affecting the
free outflow profile. Note that experimental findings by
Benjamin16 and those reported by Anwar15 show even greater
values for c c .
Nevertheless, the range of the gate openings which were
of interest in this study extends from a/y u ⫽0.4 to a/y u
⫽0.8– 0.85. In this range, the present experimental data are
in fairly good agreement with recent theoretical predictions
given by Vanden-Broeck.17
B. Evaluation of the boundaries of hysteresis region
The second set of experiments allowed for both qualitative and quantitative comparisons between the experimental
and theoretical upper boundary of the hysteresis region. The
experimental strategy was to investigate possible flow regimes for different values of the gate opening while maintaining a fixed flow rate. Undisturbed conditions with the
Froude number in the range F 0 ⫽0.72– 4.57 were investigated, and each run was performed according to the following procedure. Starting from a quasi-uniform flow, the gate
was lowered until it touched the free surface. For F 0 ⬎0.8, a
free outflow regime was always established as soon as the
gate touched the undisturbed stream. It is worth noting that
A. Defina and F. M. Susin
FIG. 10. Comparison between the theoretical and experimental hysteresis
loop in the a/y c ⫺y u /y c diagram. The Froude number of the undisturbed
flow is F 0 ⫽0.98. The sequence of experimental points is clockwise from I
共initial undisturbed flow兲 to R 共restored undisturbed flow兲. The circles and
diamonds denote free outflow and undisturbed flow, respectively; the full
and open symbols denote the lowering stage and the raising stage, respectively.
the sudden appearance of free outflow for F 0 ⫽0.81– 0.98
substantiates the physical meaning of the branch A ⬘ M ⬘ in
Fig. 5.
When steady conditions for a/y 0 ⬍1 were attained, i.e.,
when the hydraulic jump reached its equilibrium position,
the gate was gradually raised at small steps of approximately
0.5 or 1.0 mm. The time interval between the adjustments
was long enough for the flow to reach steady state conditions. This time interval ranged from approximately 10 to 30
minutes, depending on the undisturbed Froude number. The
gradual lifting of the gate was protracted until the water surface suddenly detached from the upstream face of the gate
and uniform flow conditions were restored. The experimental
value of the gate opening a L was estimated to be the average
of the two openings measured just before and after the flow
detached from the gate, respectively.
As theoretically predicted, for F 0 ⬎0.8 all the experiments showed the hysteretic behavior of the flow, i.e., both
the undisturbed state and the free outflow regime were possible for the same gate opening depending on the previous
flow configuration in the vicinity of the obstacle. The example of hysteresis loop shown in Fig. 10 confirms the good
qualitative and quantitative agreement between theoretical
and experimental results.
A comprehensive plot of all the experimental conditions
at which flow reversion from a free outflow regime to an
undisturbed state occurred is given in Fig. 11, together with
the theoretical curves delineating the hysteresis region. The
experimental results are in good agreement with the theoretical predictions. In particular, they clearly exhibit the horizontal trend in the range 0.81⬍F 0 ⬍1.65. Discrepancies between the theoretical and experimental boundaries of the
hysteresis region 共in the order of 5%兲 are very small when
considering that the theory does not include a number of
physical processes 共e.g., finite length of the jump兲 that take
place during the experiments.
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Phys. Fluids, Vol. 15, No. 9, September 2003
Hysteretic behavior of the flow
2547
FIG. 11. Comparison between the theoretical 共
兲
and experimental 共ⴰ兲 upper boundary of the hysteresis
region. The lower boundary is also plotted for completeness.
V. CONCLUSIONS
A simple theoretical approach to predict conditions for
the occurrence of hydraulic hysteresis and to evaluate the
boundaries of the hysteresis region for a supercritical flow
impacting an obstacle was proposed.
The theoretical approach was described in detail for the
case of a vertical sluice gate in a rectangular channel. It was
shown that the hysteresis region extends up to subcritical
approaching flows provided that the Froude number of the
incoming flow is greater than approximately 0.8. This is the
first time, to the knowledge of the authors, that hysteresis is
found to affect undisturbed subcritical flows. This occurrence
is due to the contraction experienced by the flow issuing
from under the sharp crested gate. Therefore, it can be speculated that subcritical flows may also exhibit hysteresis when
approaching a channel constriction where some lateral contraction might affect the flow path.
It is also worth pointing out that, even at moderate
Froude number 共i.e., F 0 ⬇1.5– 3), the amplitude of the hysteresis region is large, as it ranges from a⫽y 0 to approximately a⫽2•y 0 . Therefore, hysteresis can be expected to
manifest itself in many practical cases.
The above theoretical results were tested through an extensive series of experiments. A first set of experiments
served to measure the contraction coefficient in the case of a
gravity affected flow under a vertical sluice gate. This had to
be done in order to accurately predict the upper boundary of
the hysteresis region.
A second set of experiments was then carried out and
confirmed the hysteretic character of the flow under a sluice
gate for both supercritical and subcritical approaching flows.
Moreover, the measured extent of the hysteresis region was
found to agree quite satisfactorily with the predicted one. In
particular, differences between the computed and measured
upper boundary of the hysteresis region were found to fall
within experimental uncertainty.
gram, PRIN 2002, Influenza di vorticità e turbolenza nelle
interazioni dei corpi idrici con gli elementi al contorno e
ripercussioni sulle progettazioni idrauliche.
APPENDIX: BEHAVIOR OF THE UPPER BOUNDARY
OF HYSTERESIS FOR F 0 CLOSE TO 1
In Sec. III we stated that the conditions represented by
the branch AM of the curve plotted in Fig. 3 cannot be
achieved experimentally. The reason for this is discussed
here in detail. This discussion uses the standard relationship
between gate opening a and the specific energy at the gate,
H for a given flow rate q. This is plotted in Fig. 12, where
point L denotes the upper limit condition for hysteresis to
occur, and point M the maximum of the curve a-H.
An initial undisturbed flow with F 0 ⬎1 and a⬎y 0 is
considered here. In order to produce free outflow conditions,
the gate is lowered to a⭐y 0 ⬍y c , and the specific energy at
the gate for this flow configuration is denoted with H B 共Fig.
12兲. Note that point B necessarily lies on the right branch of
the curve 共i.e., H B ⬎H M ).
Starting from point B, raising the gate moves the flow
conditions towards point L. In the case of H L ⭓H M 关Fig.
ACKNOWLEDGMENTS
The authors wish to thank Guido Sutto and Gabriela
Ronzato for their valuable contribution to the experimental
investigation. This work was partially supported by MURST
and University of Padova under the National Research Pro-
FIG. 12. Flow energy H as a function of the height of the gate opening, a,
for fixed flow rate q. 共a兲 H L ⬎H M ; 共b兲 H L ⬍H M .
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2548
Phys. Fluids, Vol. 15, No. 9, September 2003
12共a兲兴, once point L is reached, any further increase in the
gate opening produces a sudden reversion to undisturbed
flow conditions.
One may ask what happens when H L ⬍H M 关Fig. 12共b兲兴.
In this case, starting from point B, the gate can be raised
until point M . From here, and in order to move along the
curve towards point H⫽H L , the gate opening should be reduced. However, if this is done, flow conditions do not move
along the branch AM of the curve, but rather back towards
point B.
The reasons for this behavior are not yet fully understood and deserve further investigation. However, as far as
the upper boundary of the hysteresis region is concerned, we
can conclude that, when H L ⬍H M , the gate can be raised
until a⫽a M . Any further raising of the gate does not have a
solution on the curve a-H and, therefore, undisturbed flow
conditions are necessarily restored. In other words, when
H L ⬍H M , the upper boundary of the hysteresis region is
a L ⫽a M and the branch AM of the curve must be replaced
with the horizontal line A ⬘ M as shown in Fig. 3.
Similar reasoning suggests the branch AM ⬘ be replaced
with the horizontal line A ⬘ M ⬘ in the plot in Fig. 5 which
regards to subcritical undisturbed flow conditions.
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