water

Article
Characteristics of Positive Surges in a
Rectangular Channel
Feidong Zheng 1,2 , Yun Li 1,3, *, Guoxiang Xuan 1,3 , Zhonghua Li 1,3 and Long Zhu 1,3
1 Nanjing Hydraulic Research Institute, Nanjing 210029, Jiangsu, China; feidongzheng@126.com (F.Z.);
xuan@nhri.cn (G.X.); zhli@nhri.cn (Z.L.); zhulong@nhri.cn (L.Z.)
2 College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098,
Jiangsu, China
3 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing 210029,
Jiangsu, China
* Correspondence: yli_nhri@126.com; Tel.: +86-025-8582-8022




Received: 17 September 2018; Accepted: 16 October 2018; Published: 19 October 2018


Abstract: A positive surge is an unsteady open channel flow motion characterized by an increase
of flow depth. In previous experimental studies, a positive surge was typically induced by either
a sudden increase of discharge in a channel or by the rapid closure of a downstream sluice gate,
thus leading to a steep initial profile. However, in many instances, the evolution of a positive surge is
of a progressive manner (e.g., in the downstream navigation canal during the emptying operation
of lock chambers). In the present work, the inception and development of a positive surge induced
by a progressive increase of discharge was investigated in a rectangular channel with a smooth bed.
Both undular and breaking surges were studied. The results demonstrate that the maximum wave
height at the first wave crest of an undular surge is in very close agreement with the McCowan theory.
Additionally, the wave amplitude essentially shows a linearly increasing trend with an increasing
surge Froude number up to Fr0 = 1.26 to 1.28, whereas it tends to suggest a power law reduction for
larger surge Froude numbers. Moreover, the dispersion of undular surges is consistent with the linear
wave theory only for surge Froude numbers close to unity. Overall, the present study demonstrates
the unique features of positive surges induced by a progressive increase of discharge.

Keywords: undular surge; breaking surge; progressive; dispersion characteristics; surge

Froude number

1. Introduction
A positive surge is an unsteady open channel flow motion characterized by an increase of flow
depth [1]. Positive surges are widely observed in man-made and natural channels. They may be
induced by the emptying of lock chambers during lock operations [2] or by sluice gates installed along
water supply canals for irrigation and power purposes [3]. For some flow conditions, the front of
a positive surge is followed by a train of undulations (i.e., an undular surge) and even behaves as
a nearly vertical water wall (i.e., a breaking surge). The passage of undular and breaking surges is
associated with significant variations in hydrodynamics (i.e., flow velocity, water level and Reynolds
stresses), and therefore they can be expected to affect the mixing and advection processes of sediments
in channels [4,5]. In the case of navigation channels, these surges also potentially influence tow’s
maneuverability, thereby affecting the navigation safety and functional efficiency of locks [6].
The first systematic study of positive waves was performed by Favre [7], who investigated the
characteristics of undular surges in a rectangular flume. In the experimental studies of Benet and
Cunge [8], and Treske [6], the focus was on the properties of both undular and breaking surges in

Water 2018, 10, 1473; doi:10.3390/w10101473 www.mdpi.com/journal/water

Water 2018, 10, 1473 2 of 12

Water 2018, 10, x FOR PEER REVIEW 2 of 12

trapezoidal channels. New experiments were carried out by Soares Frazão and Zech [9] to investigate
the characteristics
the characteristics of of undular
undular surges
surges and and to
to validate
validate thethe proposed
proposed numerical
numerical scheme.
scheme. MoreMore recent
recent
experimental work of the same kind has been performed by Chanson and co-workers [10–15]. KochKoch
experimental work of the same kind has been performed by Chanson and co-workers [10–15]. and
and Chanson
Chanson [10] focused
[10] focused on theon the free-surface
free-surface and turbulent
and turbulent flow properties
flow properties of both and
of both undular undular and
breaking
breakingHydraulic
surges. surges. Hydraulic
studies bystudies
Chanson by [11]
Chanson [11] andand
and Gualtieri Gualtieri
Chanson and[12]
Chanson [12] were
were mostly mostly
concerned
concerned with the effect of bed roughness on the wave height attenuation
with the effect of bed roughness on the wave height attenuation and dispersion of an undular surge. and dispersion of an
undular
An attemptsurge. An attempt
was made by Chansonwas made[13] toby Chanson
compare the [13] to compare
free-surface theoffree-surface
profile the undulations profile
withofboth
the
undulations with both the sinusoidal and cnoidal wave functions. In the
the sinusoidal and cnoidal wave functions. In the studies by Gualtieri and Chanson [14] and Leng and studies by Gualtieri and
Chanson [15],
Chanson [14] and Leng and Chanson
the free-surface properties [15], thecompared
were free-surface to properties
the analyticalwere compared
solutions to the analytical
of Lemoine [16] and
solutions of
Anderson Lemoine
[17], based on [16]the
and Anderson
linear [17], based
wave theory and theon Boussinesq
the linear wave theoryrespectively.
equations, and the Boussinesq
In some
equations,
other studies,respectively.
the upstream In propagation
some other studies,
of positivethesurges
upstream propagation
in a sloping of positive
rectangular channelsurges in a
was also
sloping rectangular
investigated [18]. channel was also investigated [18].
Undular and
Undular and breaking
breaking surges
surges in in previous
previous experimental
experimental studies
studies have
have typically
typically been
been generated
generated in in
two different ways. One way is to induce a sudden increase in discharge
two different ways. One way is to induce a sudden increase in discharge in a channel, while anotherin a channel, while another
way is
way is to
to generate
generate undular
undular and and breaking
breaking surges
surges by by the
the rapid
rapid closure
closure of
of a downstream sluice
a downstream sluice gate,
gate,
thus leading to steep initial water profiles. However, in many instances,
thus leading to steep initial water profiles. However, in many instances, these surges occurred these surges occurred in a
progressive manner. For example, in the case of the emptying operation
in a progressive manner. For example, in the case of the emptying operation of lock chambers, of lock chambers, a positive
asurge withsurge
positive a rather
withsmooth water
a rather profilewater
smooth is firstprofile
generated in the
is first downstream
generated in thechannel;
downstreamsubsequently,
channel;
the front of the positive surge steepens and undulations slowly grow
subsequently, the front of the positive surge steepens and undulations slowly grow at the front at the front head during
head
during propagation; and finally, an undular surge with some breaking or a breaking surge may form,
propagation; and finally, an undular surge with some breaking or a breaking surge may form,
depending on
depending on the
the flow
flow conditions.
conditions.
This work is focused
This work is focused on on thethe inception
inception andand development
development of of aa positive
positive surge
surge induced
induced by by the
the
progressive opening
progressive opening of of an
an upstream
upstream plate plate gate.
gate. The
The free-surface
free-surface characteristics
characteristics were
were systematically
systematically
investigated during the downstream propagation
investigated during the downstream propagation of positive surges. of positive surges.

2. Experimental Setup and Experiment

The experimental arrangement is schematically shown in Figure Figure 1.
1. The experimental facility
reservoir, aa rectangular
consisted of a rectangular upstream reservoir, rectangular flume
flume and a pipeline connecting the two
parts together.
parts together. Both
Both the
the length
length and
and width
width of the upstream
upstream reservoir
reservoir were
were 11 m. The horizontal channel
was made
was made of a smooth poly (vinyl chloride)
chloride) (PVC) bed and Plexiglass
Plexiglass sidewalls,
sidewalls, 38 m long and 0.3 m
wide, with
wide, with aa height
height of
of 0.35 m. The flume was connected to a pipeline of the same width with a curved
entrance at the upstream
upstream end
end ofof the
the channel
channelbed.
bed.TheThedownstream
downstreamend endofofthe
theflume
flumewas
wasclosed
closedbybya
vertical rigid wall. The pipeline system was equipped with a plate gate and its shaft.
a vertical rigid wall. The pipeline system was equipped with a plate gate and its shaft. The width The width and
height
and of the
height gategate
of the were 0.3 0.3
were m and
m and 0.10.1
m,m,
respectively.
respectively.The
Thegate
gatewas
wasdriven
drivenvertically
vertically by
by a stepper
motor, which
motor, which could
could be
be used
used to exactly
exactly reproduce
reproduce anyany desirable
desirable opening
opening function.
function. The total traveling
equal to
height of the gate was equal to its
its height.
height.

Figure 1. Sketch of the experimental facility.

facility.

The
The initial
initialflow
flowdepths were
depths recorded
were withwith
recorded pointpoint
gauges in thein
gauges upstream reservoir
the upstream and the channel.
reservoir and the
The unsteady water depth in the channel was measured using a set of displacement
channel. The unsteady water depth in the channel was measured using a set of displacement meters spaced
meters
along the channel centerline, and a sampling frequency of 200 Hz was used for all the
spaced along the channel centerline, and a sampling frequency of 200 Hz was used for all the measurements.
measurements. For all experiments, flow measurements were started 60 s before the gate opening,
Water 2018, 10, 1473 3 of 12

For all experiments, flow measurements were started 60 s before the gate opening, and were stopped
when the surge reached the downstream rigid wall to avoid wave reflection interference.
It was assumed that the plate gate was raised at a constant speed until complete gate opening.
The drop height H was defined as the initial difference in elevation between the water surfaces of
the upstream reservoir and the channel. Hydraulic considerations indicated that the unsteady water
depth h in the channel should essentially depend on the initial water depth in channel h0 , the drop
height H, and the amount of time needed to fully open the gate tv . In the present study, the initial
water depth in the channel was set to 8 cm by referring to the studies of Treske [6] and Koch and
Chanson [10]. A summary of the experimental measurements is shown in Table 1. Overall, a large
number of experimental runs were carried out in four series, denoted as A, B, C, and D, corresponding
to four different times taken to fully open the gate of 20, 25, 30, and 40 s with four different drop
heights. The experimental conditions were selected to generate both undular and breaking surges,
taking into account the influence of undulation shape deviations on the characteristics of surges with
close Froude numbers, which is defined in the system of coordinates in translation with the surge and
is given by
S
Fr0 = p (1)
gh0
where S = surge front celerity positive downstream and g = gravity acceleration.

Table 1. Experimental series.

Series h0 (m) tv (s) H (m)

A 0.08 20 0.1, 0.2, 0.3, 0.4
B 0.08 25 0.1, 0.2, 0.3, 0.4
C 0.08 30 0.1, 0.2, 0.3, 0.4
D 0.08 40 0.1, 0.2, 0.3, 0.4

3. Results and Discussion

3.1. Basic Flow Patterns

A positive surge was generated in the flume by raising the plate gate at a constant speed. During
the downstream propagation of the positive surge, different wave types may occur at the front head.
Here, the overall developing processes observed at the front of positive surges have been consolidated
into four major flow stages, and are described below. It should be noted that the number of flow stages
for each experimental run was variable, depending on the drop height and the time taken to fully open
the gate.

• In Stage 1, the front of the positive wave is rather smooth.

• Stage 2 corresponds to the generating process of the undular surges, and can be divided into
two sub-stages: Stage2a and Stage2b. In Stage 2a, the front head of the positive wave gradually
becomes “rough” and develops into a series of cascading steps at the end of this stage; in Stage 2b,
the cascading steps develop into a train of well-formed undulations (i.e., non-breaking undular
surges) (Figure 2). Additionally, some sidewall shock waves can be observed in this stage that
develop upstream of the first wave crest and intersect next to the first crest. This stage ends when
some small waves break at the first wave crest (i.e., breaking undular surges) (Figure 3).
• In Stage 3, the wave amplitude decreases and the free-surface undulations become flatter.
This stage ends with the disappearance of fluctuating characteristics of water depth at the
surge front.
• During Stage 4, the surge front behaves as a nearly vertical water wall (i.e., breaking surges)
(Figure 4).
 The tests indicate that transitions between different flow stages can be characterized solely with
the surge Froude number (Table 2). Moreover, the surge Froude numbers corresponding to these
transitions are close to those of Chanson [11,13] and Leng and Chanson [15] for both smooth and
rough beds,
Water 2018, although these studies focused on the positive surges induced by the rapid closure4 of
10, 1473 a
of 12
downstream sluice gate.

(a) (b)
Figure
Figure 2. Non-breaking
Non-breaking undular
undular surges
surges in
in Stage
Stage 2b:
2b: (a)
(a) Lateral
Lateral view;
view; (b)
(b) looking
looking upstream
upstream at
at the
the
incoming
incoming wave.
wave.

(a) (b)
Figure
Figure 3.3.Breaking undular
Breaking surges
undular in Stage
surges 3: (a) 3:
in Stage Lateral view; (b)
(a) Lateral looking
view; (b) upstream at the incoming
looking upstream at the
wave.
incoming wave.

(a) (b)
Figure
Figure 4.
4. Breaking
Breaking surges
surges in
in Stage
Stage 4:
4: (a)
(a)Lateral
Lateral view;
view; (b)
(b) looking
looking upstream
upstream at
at the
the incoming
incoming wave.
wave.

The tests indicate that 2.

Table transitions between
Surge Froude different
number flow
values for flowstages
stage can be characterized solely with
transitions.
the surge Froude number (Table 2). Moreover, the surge Froude numbers corresponding to these
Flow Stage Transition Surge Froude Number Fr00
transitions are close to those of Chanson [11,13] and Leng and Chanson [15] for both smooth and
rough beds, although these studies 1–2a ≈1.03 induced by the rapid closure of a
focused on the positive surges
downstream sluice gate. 2a–2b 1.07–1.10
2b–3 1.26–1.28
Table 2. Surge Froude number values for1.45–1.50
3–4 flow stage transitions.

Flow Stage Transition Surge Froude Number Fr0

1–2a ≈1.03
2a–2b 1.07–1.10
2b–3 1.26–1.28
3–4 1.45–1.50
Water 2018, 10, x FOR PEER REVIEW 5 of 12
Water 2018, 10, 1473 5 of 12

3.2. Free-Surface Properties

3.2. Free-Surface Properties
3.2.1. Ratio of Conjugate Depths
3.2.1.In
Ratio of Conjugate
a rectangular Depths channel, when neglecting friction loss, the combined application of
horizontal
the continuity and momentum
In a rectangular horizontalequations acrossneglecting
channel, when the surge front yields
friction loss, the combined application of
the continuity and momentum equations across the surge front yields
hconj 1
(
= q1 + 8Fr02 − 1
h0 = 12 1 + 8Fr2 − 1
hconj ) (2)
(2)
h0 2 0
where hconj and h0 are the conjugate and initial depths, respectively. Equation (2) is the famous
where
Bélanger hconj and h0 In
equation. arethethepresent
conjugate study, andtheinitial depths,
conjugate respectively.
depth Equation
hconj was defined (2) average
as the is the famous
of the
Bélanger
first wave equation.
crest and In the
the present
trough study,depthsthefor conjugate
the undular hconj was
depthsurges, defined
while as the
it was theaverage
water of the
depth
first wave crestbehind
immediately and thethetrough
surge depths
front forforthe undularsurges.
breaking surges,Thewhile it was
ratio of the water depth
conjugate immediately
depths, hconj/h0, is
behind
plotted the surgethe
against front for breaking
surge Froude number, surges. The Fr0, ratio of conjugate
in Figure depths,
5 for both hconj /h
undular and0 , is plotted against
breaking surges.
the surge Froude
Although number, Fr
the experimental trend
0 , inisFigure
comparable5 for both
to undular
that produced and bybreaking
Equation surges.
(2), the Although
entire the
dataset
experimental
shows generally trend is comparable
higher values ofto that
hconj /h0produced
for 1.07 < by Fr0Equation (2), may
< 1.57. This the entire dataset attributed
be partially shows generally
to the
higher values
uncertainty of of hconj /h0 for
estimating the1.07 < Fr0 <depth;
conjugate 1.57. This
that is,may be partially
in the attributed
undular surges, the to the uncertainty
conjugate depth was of
estimating the conjugate depth; that is, in the undular surges, the conjugate
calculated based on a symmetrical undulation profile which is inconsistent with the experimental depth was calculated
based on a symmetrical
data, while the measurementundulation
of theprofile
conjugate which is inconsistent
depth was affected with the experimental
adversely by the large data, while the
free-surface
measurement
fluctuations behindof the the
conjugate
surge frontdepthinwas the affected
breakingadversely
surges. It by the large
is worth free-surface
noting fluctuations
that the data in flow
behind
Stage 2bthe aresurge
morefront in thethan
scattered breakingin flow surges.
Stages It is worth
3 and noting
4 for that surge
a given the data in flow
Froude Stage 2b
number, are
which
more
mightscattered
imply largerthan deviations
in flow Stagesof wave3 andshapes
4 for afor
given surge Froude
non-breaking number,
surges than which might undular
for breaking imply larger
and
deviations of wave shapes for non-breaking surges than for breaking undular and breaking surges.
breaking surges.

Figure Variationofofhconj
Figure5.5.Variation hconj/h
/h00 with Fr00for
with Fr forboth
bothundular
undularand
andbreaking
breakingsurges.
surges.

3.2.2. Free-Surface Undulation Characteristics

3.2.2. Free-Surface Undulation Characteristics
(1) Wave height of the first wave crest
(1) Wave height of the first wave crest
The wave height of the first wave crest (h1c − h0 ) is plotted in a dimensionless form against Fr0
The wave height of the first wave crest (h1c − h0) is plotted in a dimensionless form against Fr0 in
in Figure 6. For non-breaking undular surges in Stage 2b, the values of (h1c − h0 )/h0 show a trend
Figure 6. For non-breaking undular surges in Stage 2b, the values of (h1c − h0)/h0 show a trend of linear
of linear increase with Fr0 in general. For breaking undular surges in Stage 3, a sharp decrease in
increase with Fr0 in general. For breaking undular surges in Stage 3, a sharp decrease in wave height
wave height is observed shortly after the appearance of some wave breaking at the first wave crest,
is observed shortly after the appearance of some wave breaking at the first wave crest, and
and subsequently, (h1c − h0 )/h0 exhibits the same increasing trend as the data in flow Stage 2b, but at a
subsequently, (h1c − h0)/h0 exhibits the same increasing trend as the data in flow Stage 2b, but at a
much smaller rate. It is worth noting that in flow Stage 2b, (h1c − h0 )/h0 is close to the solitary wave
much smaller rate. It is worth noting that in flow Stage 2b, (h1c − h0)/h0 is close to the solitary wave
theory for surge Froude numbers Fr0 less than 1.1, while the data show consistently higher values of
theory for surge Froude numbers Fr0 less than 1.1, while the data show consistently higher values of
(h1c − h0 )/h0 for larger surge Froude numbers.
(h1c − h0)/h0 for larger surge Froude numbers.
 The wave height data of the first wave crest are further compared with previous experimental
studies in Figure 7. It is found that present data are consistent with the trend observed by Peregrine
[19], Koch and Chanson [10] and Chanson [11]. In Figure 7, the data are also compared with the
McCowan
Water 2018, 10,theory,
1473 which shows the maximum wave heights attained by solitary waves. It can be6seen
of 12
that
Waterthere is an
2018, 10, excellent
x FOR agreement between this paper’s results and McCowan’s theory [20]. 6 of 12
PEER REVIEW

The wave height data of the first wave crest are further compared with previous experimental
studies in Figure 7. It is found that present data are consistent with the trend observed by Peregrine
[19], Koch and Chanson [10] and Chanson [11]. In Figure 7, the data are also compared with the
McCowan theory, which shows the maximum wave heights attained by solitary waves. It can be seen
that there is an excellent agreement between this paper’s results and McCowan’s theory [20].

Figure6.6.Variation
Figure 1c1c−
Variationofof(h(h − h00)/h withFrFr0 0for
)/h0 0with forundular
undularsurges.
surges.

The wave height data of the first wave crest are further compared with previous experimental
studies in Figure 7. It is found that present data are consistent with the trend observed by Peregrine [19],
Koch and Chanson [10] and Chanson [11]. In Figure 7, the data are also compared with the McCowan
theory, which shows the maximum wave heights attained by solitary waves. It can be seen that there
is an excellent agreement between
Figure this paper’s
6. Variation of (h1c −results andFrMcCowan’s
h0)/h0 with theory [20].
0 for undular surges.

Figure 7. Wave height at the first wave crest for undular surges.

(2) Wave amplitude, length and steepness

The wave amplitude, length and steepness are shown in Figure 8 in dimensionless forms (i.e.,
aw/h0, Lw/h0, and aw/Lw). Both wave amplitude and wave length were calculated according to the
definitions by Koch and Chanson [10]; that is, the wave amplitude, aw, was half of the difference
Wave height at the the first
first wave
between the water depth Figure
at 7.
the first wave crest andwaveat thecrest for
firstfor
crest undular surges.
undular
wave surges.
trough, while the wave length,
Lw, was defined between the first and second crests. The data are compared with the linear wave
(2)
(2) Wave
Wave amplitude,
amplitude,length
lengthand andsteepness
steepness
theory solution of Lemoine, the Boussinesq equation solution of Anderson, earlier experimental
studies The
The andwave amplitude,
fieldamplitude,
observations length
length andsteepness
and
[21–23]. steepness areare shown
shown in Figure
in Figure 8 in8dimensionless
in dimensionless formsforms
(i.e.,
(i.e., a
aw/h0It /h
, wLcan
w/h , L
0 0, be /h
and , and a
0aw/Lfrom
w seen w). w /L
Both ).
w wave
Figure Both wave
8aamplitudeamplitude
that the wave and wave and wave length
length increases
amplitude were calculated
were calculated according
according
monotonically to
to the
with an
the definitions
definitions
increasing by by
surge KochKoch
Froudeandand Chanson
Chanson
number a [10];
to[10]; that
that
local is,is,the
maximum thewave
wave amplitude,aaww
amplitude,
for non-breaking , ,was
washalf
undular half of the
of
surges indifference
Stage 2b.
between the water depth at the first wave crest and at the first
between an abrupt decrease in wave amplitude is observed immediately after the appearance
However, wave trough, while the wave length,
of
LLw
w, was defined between the first and and second
second crests.
crests. The data are compared with the linear linear wave
wave
theory
theory solution
solutionofofLemoine,
Lemoine,thetheBoussinesq
Boussinesq equation
equation solution of Anderson,
solution earlier experimental
of Anderson, studies
earlier experimental
and field observations [21–23].
studies and field observations [21–23].
It can be seen from Figure 8a that the wave amplitude increases monotonically with an
increasing surge Froude number to a local maximum for non-breaking undular surges in Stage 2b.
However, an abrupt decrease in wave amplitude is observed immediately after the appearance of
numbers.
Figure 8c presents the dimensionless wave steepness data. It can be seen that the wave steepness
exhibits a similar trend to wave amplitude—aw/Lw shows an increasing and decreasing trend for non-
breaking and breaking undular surges, respectively. Additionally exhibited in this figure are the data
sets
Water obtained in undular hydraulic jumps [24,25]. For surge Froude numbers less than 1.29,7 ofthe
2018, 10, 1473 12
present data set is in excellent agreement with undular jump data.

(a)

Water 2018, 10, x FOR PEER REVIEW 8 of 12

(b)

(c)
Figure 8. Characteristics
Characteristics of undular surges as functions
functions of
of Fr
Fr00: (a) Dimensionless wave amplitude
w/h
aw /h0;0(b)
; (b)dimensionless
dimensionlesswave
wavelength
lengthLL
w/h 0; (c)
w /h wave
0 ; (c) steepness
wave aw/L
steepness . w.
aww/L

(3) It can be seen

Dispersion from Figure 8a that the wave amplitude increases monotonically with an increasing
characteristics
surge Froude number to a local maximum for non-breaking undular surges in Stage 2b. However,
The undular surges investigated in this study have the property of net mass transport, which is
in conjunction with the finite amplitude water waves [26]. For a finite amplitude wave, the linear
wave theory yields a dispersion relationship between the wave length Lw, wave period T and water
depth h:

gT 2  2π h 
Lw = tanh   (5)
Water 2018, 10, 1473 8 of 12

an abrupt decrease in wave amplitude is observed immediately after the appearance of wave breaking
at the first wave crest for breaking undular surges in Stage 3; subsequently, a gradual declining trend
was held before the disappearance of free-surface undulations. The relationship between aw /h0 and
Fr0 for undular surges is well described by the following equations
aw
= 1.39Fr0 − 1.49 for 1.07–1.10 < Fr0 < 1.26–1.28 (3)
h0
aw
= 7.50Fr0 −14.79 for 1.26–1.28 < Fr0 < 1.45–1.50 (4)
h0
Overall, the present data exhibit a qualitative agreement with all previous studies, as indicated in
Figure 8a. Moreover, although neither the linear wave theory nor the Boussinesq equation solution can
accurately predict the wave amplitude before surge breaking, the data tend to get close to Lemoine’s
solution for surge Froude numbers in the range of 1.2 to 1.28.
The dimensionless wave length data are shown in Figure 8b. Although the experimental trend is
close to that predicted by the theories of both Lemoine and Anderson and earlier experimental studies,
the data show consistently lower wave lengths, especially for very low or high Froude numbers.
Figure 8c presents the dimensionless wave steepness data. It can be seen that the wave steepness
exhibits a similar trend to wave amplitude—aw /Lw shows an increasing and decreasing trend for
non-breaking and breaking undular surges, respectively. Additionally exhibited in this figure are
the data sets obtained in undular hydraulic jumps [24,25]. For surge Froude numbers less than 1.29,
the present data set is in excellent agreement with undular jump data.
(3) Dispersion characteristics
The undular surges investigated in this study have the property of net mass transport, which is in
conjunction with the finite amplitude water waves [26]. For a finite amplitude wave, the linear wave
theory yields a dispersion relationship between the wave length Lw , wave period T and water depth h:

gT 2
 
2πh
Lw = tanh (5)
2π Lw

However, the undular surge is typical of a translation wave, i.e., water particles underneath the
water surface do not move backwards, indicating marked differences between undular surges and
finite amplitude waves. Therefore, a dimensionless dispersion parameter D can be defined for undular
surges as
 2
2πhconj
 
gT
D = Lw tanh (6)
2π Lw
Plots of D versus Fr0 for undular surges are presented in Figure 9. It is shown that a definite
correlation exists between the two dimensionless variables, with essentially no influence of undular
surge types. In the experimental range 1.07 < Fr0 < 1.50, the following empirical equation can be
proposed by fitting both data sets (i.e., for non-breaking and breaking undular surges)

D = 0.046 exp(2.952Fr0 ) (7)

It is clearly demonstrated that for a surge Froude number Fr0 less than 1.1, the dispersion
parameter D is close to 1, and hence the dispersion relationship based on the linear wave theory is
assumed to hold.

3.2.3. Energy and Momentum Fluxes Properties

A positive surge is seen as a quasi-steady flow situation by an observer travelling at the surge
speed S. In a rectangular, horizontal channel, if both the rate of energy dissipation and friction loss are
negligible, the equations of conservation of energy and momentum can be rewritten as [26]
Water 2018, 10, x FOR PEER REVIEW 9 of 12

Water 2018, 10, 1473 9 of 12

2
E h 1h 
E∗ = = +  c  = const (8)
hc hc 2  h 
1 hc 2
 
E h
E∗ = = + = const (8)
hM
c hhc 21  hh 2
M ∗ = 2 = c +   2 = const (9)
M∗ = h2c = h + 2  hc  = const
M hc 1 h
(9)
hc h 2 hc
where
where E is
is the
the specific
specific energy;
energy; M is
is the
the specific
specific momentum;
momentum; h is is the
the flow
flow depth;
depth; E*
E* and
and M*
M* correspond
correspond
to the dimensionless specific energy and specific momentum, respectively; and hc is the
the dimensionless the critical
critical depth
depth
in the system of coordinates in translation with the undular surge front, and is defined as
in the system of coordinates in translation with the undular surge front, and is defined as
q
= 3 SS22hh20 /g
hhc = 3 2
c g 0
(10)
(10)

This pair of equations may be be viewed

viewed as as the
the parametric representationofofM*
parametricrepresentation M*==f f(E*), withh/h
(E*), with as
h/hc as
a parameter. The function E* −−M* M*has
hastwo
twobranches
branches(the(thered
red dash
dash line
line in
in Figure
Figure 10), intersecting
intersecting at
(1.5, 1.5). The lower branch of the curve E* − M*corresponds
− M* correspondsto toaa supercritical
supercritical flow
flow while
while the
the upper
upper
branch represents a subcritical flow. flow. Figure
Figure 1010 shows
shows aa comparison
comparison of of M* f (E*) and
M* == f(E*) and the
the entire
entire dataset
dataset
in this
in this study.
study. ItItisisexpected
expectedthat
thatthe
the initial
initial flow
flow data
data areare located
located on onthethe supercritical
supercritical branch
branch whilewhile
the
the corresponding
corresponding conjugate
conjugate flow flow
data data are
are on theonsubcritical
the subcritical branch.
branch. It can
It can also bealso be concluded
concluded from
from Figure
Figure
10, that10,atthat
the atsubcritical
the subcritical
branchbranch of the
of the curve E* E*
curve − M*,
− M*, someoverlaps
some overlapsexist
existbetween
between regions
corresponding to
corresponding to different
different flow stages, indicating marked differences between travelling positive
surges and stationary hydraulic jumps [27].

Variation of D with Fr00 for

Figure 9. Variation
Figure for undular
undular surges.
surges.

Figure 11
Figure 11 shows
shows aa comparison
comparison of of the
the dimensionless
dimensionless specific
specific energy
energy in in the
the initial
initial flow
flow (E *) and
(E00*) and
the corresponding conjugate flow (E *). It can be found that the values
the corresponding conjugate flow (Econj*). It can be found that the values of Econj* are basically larger
conj of E conj * are basically
largerthat
than than
of that of E
E0* for 0 * for non-breaking
non-breaking undularundular
surges in surges
Stagein 2b.Stage
This 2b. This is probably
is probably due to thedue to the
pressure
pressure distribution
distribution beneath an beneath
undularansurge
undular
that surge
is less that
thanishydrostatic
less than hydrostatic
beneath thebeneathwave crest theand
wave crest
greater
and greater than hydrostatic beneath the wave trough. With the ensuing
than hydrostatic beneath the wave trough. With the ensuing development of undular surges, Econj* development of undular
surges, EconjE* 0approaches
approaches * for breaking E0 *undular
for breaking
surges undular
in Stagesurges
3, and inisStage 3, and is
essentially essentially
less than E0* for than E0 *
lessbreaking
for breaking
surges surges
in Stage 4. This in may
Stagebe4.primarily
This mayattributed
be primarily attributed
to energy to energy
dissipation due dissipation due to wave
to wave breaking, and
breaking, and hence a smaller specific energy in the conjugated flow than in the
hence a smaller specific energy in the conjugated flow than in the initial flow is obtained in breaking initial flow is obtained
in breaking surges.
surges.
In Figure
In Figure 12,
12, aa comparison
comparison of of the
the dimensionless
dimensionless specific
specific momentum
momentum in in the
the initial
initial flow
flow (M(M00*)*) and
and
the corresponding conjugate
the corresponding conjugate flow (Mconjflow (M *) is presented. The values
conj*) is presented. The values of Mconj of M * are consistently
conj* are consistently larger larger
Water 2018, 10, 1473 10 of 12
Water 2018, 10, x FOR PEER REVIEW 10 of 12
Water 2018, 10, x FOR PEER REVIEW 10 of 12
than that
than that of M00* for the entire present test range. The main reason for this may be the deviation from
of M from
than
the that of M
hydrostatic0*and
for the
the entire present
complicated test range.
velocity The main
distribution inreason for
undular this
and may be
breaking the
the hydrostatic and the complicated velocity distribution in undular and breaking surges. deviation
surges. from
the hydrostatic and the complicated velocity distribution in undular and breaking surges.

10. Dimensionless
Figure 10. Dimensionless relationship
relationship between
between the
the momentum
momentum and both undular
and energy fluxes for both undular
Figure
and 10. Dimensionless
breaking surges.
and breaking surges. relationship between the momentum and energy fluxes for both undular
and breaking surges.

Figure 11. Comparison

Figure11. of EE000* and E
Comparison of conj**for
Econj
conj forboth
bothundular
undularand
and breaking
breaking surges.
surges.
Figure 11. Comparison of E0* and Econj* for both undular and breaking surges.

Figure 12. Comparison of M00* and Mconj

conj* for both undular and breaking surges.
Figure 12. Comparison
Figure12. ofM
Comparisonof and M
M00* and conj**for
Mconj forboth
bothundular
undular and
and breaking
breaking surges.
surges.
4. Conclusions
4. Conclusions
The free-surface properties of a positive surge induced by the progressive opening of an
The free-surface
upstream properties
plate gate were of a positive
investigated surge induced
experimentally by the channel
in a rectangular progressive
with opening
a smoothofbed.
an
upstream plate gate were investigated experimentally in a rectangular channel with a smooth bed.
Water 2018, 10, 1473 11 of 12

4. Conclusions
The free-surface properties of a positive surge induced by the progressive opening of an upstream
plate gate were investigated experimentally in a rectangular channel with a smooth bed. Both undular
and breaking surges were studied with a wide range of surge Froude numbers Fr0 ranging from 1.07
to 1.57. The occurrence of non-breaking undular surges was observed for 1.07–1.10 < Fr0 < 1.26–1.28,
breaking undular surges for 1.26–1.28 < Fr0 < 1.45–1.50, and breaking surges for Fr0 > 1.45–1.50.
The range of Froude numbers corresponding to each type of surge is consistent with previous
findings [11,13,15], although previous studies were concerned with positive surges induced by the
rapid closure of a downstream sluice gate.
A detailed analysis of undular surge characteristics was conducted. First, during the propagation
process of undular surges, the wave height of the first wave crest increases with the surge Froude
number at a much greater rate for non-breaking undular surges than for breaking undular surges.
Second, the maximum wave height of the first wave crest is in very close agreement with the McCowan
theory. Third, neither the linear wave theory nor the Boussinesq equation solution can accurately
predict the wave amplitude and steepness for small surge Froude numbers (Fr0 < 1.26–1.28), and the
wave length data are consistently lower than the values predicted from the above theories. Based on
the experimental results, two empirical equations in terms of the surge Froude number were proposed
to estimate wave amplitude. Fourth, the dispersion of undular surges is consistent with the linear wave
theory only for surge Froude numbers close to unity (Fr0 < 1.1), demonstrating marked differences with
undular surges investigated in previous studies [11,12]. Therefore, a novel dimensionless parameter
defined as Lw /[(gT2 /2π) × tanh(2πhconj /Lw )] was introduced to characterize the dispersion of undular
surges induced by a progressive increase of discharge and it was found to solely depend on the surge
Froude number.
Furthermore, on the subcritical branch of the function E* − M* proposed by Benjamin and
Lighthill [27], the large overlaps of different data regions, corresponding to three different surge types,
indicate some key differences between travelling positive surges and stationary hydraulic jumps.
This study provides important data for the development and verification of numerical schemes for
computing such positive surges. In the case of navigation canals, the present results may be applicable
in assisting engineers to predict the characteristics of positive surges and further estimate the sudden
surge loads on lock gates and navigation ships during the emptying operation of lock chambers.

Author Contributions: F.Z. performed experiments and wrote the paper; Y.L. and G.X. led the work performance
and edited the manuscript; Z.L. and L.Z. collected data through the review of relevant literature.
Funding: This research was funded by the National Key R&D Program of China (Grant No. 2016YFC0402003).
Acknowledgments: The authors would like to express their gratitude to Yue Huang and Xiujun Yan for their help
and support in this experiment.
Conflicts of Interest: The authors declare no conflict of interest.

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