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Experimental Thermal and Fluid Science

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On unsteady velocity measurements and profiling in compression waves inan asymmetrical trapezoidal channelRamith Fernando, Xinqian Leng, Hubert Chanson⁎

The University of Queensland, School of Civil Engineering, Brisbane, QLD 4072, Australia

A R T I C L E I N F O

Keywords:Compression wavesUnsteady turbulenceTrapezoidal canalSecondary currents

A B S T R A C T

A compression wave is a highly-unsteady rapidly-varied free-surface flow associated with a sudden rise in watersurface elevation and may travel over very long distances along flat prismatic canals. New experiments wereconducted to investigate the unsteady turbulent characteristics of compression waves, in a large-size channel.The focus was on the effects of the asymmetrical trapezoidal shape of the channel cross-section on the transientturbulence characteristics of the three-dimensional compression wave. The measurements showed a complicatedunsteady motion down the transverse slope, underneath the leading edge of the compression wave. A markedincrease in free-surface elevation was observed during the surge passage. A three-dimensional transient motionwas observed. An intense transient recirculation was seen next to the invert at the base of the transverse slopeand in the shallow flow zones. Visual observations and velocity measurements showed some recirculation pri-marily in the shallow water region. The results highlighted strong secondary currents on the transverse slope,albeit for short, transient periods.

1. Introduction

In an open channel, a compression wave is a highly-unsteady ra-pidly-varied free-surface flow associated with a sudden rise in watersurface elevation [27]. A compression wave can travel over very longdistances along flat prismatic canals, because it can absorb randomdisturbances on both sides of the front, making the compression wavestable and self-perpetuating [13,11]. Large-scale industrial applicationsencompass rejection and load acceptance surges in hydro-power canalsfollowing rapid turbine start or stoppage, surges in water supply sys-tems following rapid gate operation, and operation of lock gates innavigation canals [8,28]. Basic one-dimensional modelling may predictsuccessfully the propagation of a compression wave [24,21], but doesnot provide any information on the unsteady turbulence and three-di-mensional hydrodynamic transient flows, for example during incidentsleading to operational restrictions [9,30].

Recently, physical and numerical CFD studies of compression waveswere undertaken in rectangular channels. All the data showed a rapiddeceleration during the passage of the compression wave followed bysome transient recirculation underneath the wave front [12,6,17]. Theeffect of transient recirculation, and associated large vortical structures,were linked to secondary flows. Only a few investigations were con-ducted in non-rectangular flumes, including some free-surface

observations in symmetrical trapezoidal channels, and more recentlyvelocity measurements in an asymmetrical trapezoidal canal [15](Table 1).

In this study, new experiments were conducted to investigate theunsteady turbulent characteristics of compression waves in an asym-metrical trapezoidal canal. They were performed in a systematicmanner under controlled flow conditions, with detailed free-surfaceand velocity measurements in a large-size channel. The study focusedon the unsteady turbulence characteristics of the three-dimensionalcompression wave. Using acoustic Doppler velocimetry and profiling,the turbulent transient recirculation and secondary motion were char-acterised. The measurements showed a complicated unsteady motiondown the transverse slope during the passage of the compression wavefront.

2. Experimental methods, facility and instrumentation

2.1. Experimental channel and instrumentation

The experiments were conducted in a 19 m long, 0.7 m widechannel (Fig. 1A). The bed was made from PVC and the 0.52 m highsidewalls were made of glass. The trapezoidal cross section was mod-elled with a 0.14 m high 1V:3H transverse slope (Fig. 1B),

https://doi.org/10.1016/j.expthermflusci.2019.109986Received 20 April 2019; Received in revised form 29 October 2019; Accepted 5 November 2019

⁎ Corresponding author.E-mail address: h.chanson@uq.edu.au (H. Chanson).

Experimental Thermal and Fluid Science 112 (2020) 109986

Available online 07 November 20190894-1777/ © 2019 Elsevier Inc. All rights reserved.

T

manufactured in PVC and installed between 1.25 m < x < 17.2 m,with x the longitudinal distance from the upstream end of the glasssidewall channel. At the downstream end of the channel, a custommade Tainter gate was installed at x = 18.1 m (Fig. 1A). The rapid gateclosure was used to generate the compression wave. Fig. 1 shows asketch of the facility and Fig. 2 presents photographs of the compres-sion wave. Fig. 2A, B and C illustrate the surge propagation in the ex-perimental channel. The geometry of the channel was selected for twoapplications. The asymmetrical cross-sectional shape may correspondto natural systems affected by a compression wave: e.g., a river channelsubjected to a dam break wave. Such natural channels have typically anasymmetrical shape. Second, using a method of images, the currentconfiguration might be seen as a half section of a man-made trapezoidalchannel, with 1V:3H side slopes, e.g. used for water supply canals andhydropower systems.

A rail mounted pointer gauge was used to measure the free-surfaceelevation in steady flows. Steady velocity measurements were per-formed using a Dwyer® 166 Series Prandtl Pitot tube (∅ = 3.18 mm),to check the validity of the acoustic Doppler velocimeter and Profilerdata, in the steady-state flow conditions. In steady flows, the Prandtl-Pitot tube was also calibrated to provide the boundary shear stress withthe tube lying on the boundary, acting as a Preston tube [2].

Ten non-intrusive Microsonic™ Mic +25/IU/TC acoustic displace-ment meters (ADMs) were used to measure the unsteady free-surfaceelevations. The ADMs were positioned at x = 6.96 m, 8.6 m, 10.96 m,17.41 m and 18.17 m. At x = 8.6 m, six ADMs recorded simultaneouslythe shape of the bore at six transverse locations (y = 0.1–0.6 m) as seenin Fig. 2D and 2E, with y the transverse distance from the right side-wall. The ADMs were sampled simultaneously and synchronously withand at the same sampling frequency as the velocimeter, i.e. 100 Hz and200 Hz with the Profiler and ADV respectively.

Unsteady velocity measurements were conducted using acousticDoppler velocimetry at x = 8.6 m. A Nortek™ acoustic Doppler veloci-meter Vectrino II Profiler (Profiler) (Serial number P27338, HardwareID VNO 1366) was used to record simultaneously 30 velocity mea-surements, from z-zbed = 0 (i.e. base of the bed) up to z-zbed = 0.029every 1 mm. The Profiler was sampled at 100 Hz. A Nortek™ acousticDoppler velocimeter (ADV) Vectrino+ (Serial No. VNO 0436) was usedfor point measurements, with a sampling volume 0.050 m away fromthe emitter and 0.0058 m above the base of the emitter. The side-looking ADV was sampled at 200 Hz and used to characterise accuratelythe unsteady Reynolds stresses.

While the Profiler signal output is known for the existence of outlierdata [7,19,29], neither the Profiler nor the ADV signal outputs weredespiked during unsteady flow conditions, and the raw data were useddirectly for analysis following Leng and Chanson [19]. Although anumber of studies detailed various ADV signal post-processing methods,all were developed and validated in steady flows only. In highly un-steady rapidly-varied flow conditions, these post-processing techniquesare not applicable (e.g. Nikora, 2004, Personal Communication) [16].

2.1.1. Experimental uncertainties and errorsIn steady flows, the water depths were measured with pointer

gauges with an accuracy of ± 0.5 mm. In unsteady flows, the waterdepths were recorded with acoustic displacement meters calibratedagainst the pointer gauges. The error on the unsteady water depth datawas expected to be within ± 0.5 mm. Three velocity measurementsystems were used: a Prandtl Pitot tube, an ADV Vectrino+, and aVectrino II Profiler. In steady flows, all the velocity measurements weresimilar within 2%. They were further checked in terms of conservationof mass against discharge measurements using a magneto flow meter.The results showed a deviation within 2–3% with the discharge mea-surements.

During the unsteady experiments, the high level of repeatability ofthe experiments and fast sampling rate enabled to synchronise the datawithin ± 0.05 s.Ta

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R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

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2.2. Steady flow conditions

Steady flow observations were conducted for discharges rangingfrom 0.006 m3/s to 0.100 m3/s. The open channel flow was at uniformequilibrium, i.e. normal flow conditions, along most of the flumelength. The free-surface flow was subcritical, i.e. the water depth wasgreater than the critical flow depth. Calculations of the critical flowdepth for a trapezoidal channel are detailed in Appendix A. Further thewater surface was horizontal in the transverse direction.

The velocity measurements indicated that the flow was three-dimen-sional, because of the asymmetrical cross sectional shape. The velocitiestypically decreased in shallow water depths and all the observations in-dicated a faster flow towards the deeper section of the channel (Fig. 3A).The boundary shear stress distribution was not uniform along the wettedperimeter. This is illustrated in Fig. 3C, with Fig. 3B presenting the de-finition of the perimetric coordinate Y, measured along the wetted peri-meter, with Y = 0 at the bottom right corner (y = 0, z = 0). Large var-iations were observed along both the sidewalls, bottom invert andinclined transverse slope. Minimum boundary shear stress was observedat the corners in steady flows. The minimum was linked to the occurrenceof secondary flow cells on each side of the corner bisector, and localturbulent losses [10,22]. Typical experimental results are shown inFig. 3C, where the horizontal axis is the dimensionless perimetric distanceY/W1, with W1 the horizontal invert width (Figs. 1B & 3B). In Fig. 3C, thevertical thick solid lines correspond to the bottom right corner Y = 0 at(y = 0, z = 0) and to the edges of the inclined transverse slope.

2.3. Unsteady flow conditions

This study focused on compression waves in translation, for whichthe mean characteristics were independent of the generation process.

The main sampling location (x = 8.6 m) was selected accordingly. Foreach experiment, the compression wave was generated by the rapidclosure of the downstream Tainter gate and the compression wave frontpropagated upstream (Fig. 2).

Visual observations and free-surface measurements were under-taken for initial discharges Q ranging from 0.006 m3/s to 0.100 m3/s(Table 1). Different gate openings were used to alter the bore celerityand thereby the Froude number. Detailed velocity measurements wereconducted for three discharges (Q = 0.0026, 0.054 and 0.094 m3/s)(Table 2). Table 2 summarises the experimental flow conditions, whereQ is the initial steady water discharge, d1 is the initial water depth, h isthe Tainter gate opening after closure, U is the compression wave cel-erity positive upstream, y is the transverse distance from the rightsidewall, z is the vertical elevation, zbed is the local bed elevation(Fig. 1) and Fr1 is the compression wave Froude number defined as:

= +×

Fr U Vg

11

AB

11 (1)

with V1 the initial flow velocity, g the gravity acceleration, A1 and B1

the initial flow cross-section area and free-surface width respectively.Velocity data with the Profiler were obtained at various transverse lo-cations close to the bed: z-zbed = 0 to 0.029 m. These were com-plemented by data collected with the ADV.

All velocity measurement experiments were repeated 25 times andensemble-averaged, following Chanson and Docherty [5] and Leng andChanson [18]. The median free-surface elevations and velocity com-ponents were calculated from the total ensemble, as well as the fluc-tuating properties.

Finally the unsteady flow motion is a free-surface flow, for whcihthe gravity effects are important. Any upscaling of the results must bebased upon a Froude similitude [1,11].

(A) Side view of the compression wave propagating (from left to right) in the experimental channel

(B) Undistorted transverse cross-section looking downstream

Fig. 1. Definition sketch of a compression wave in an asymmetrical trapezoidal channel.

R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

3

(A) Q = 0.0063 m3/s, Fr1 = 1.3 - Details of wave front along transverse slope looking downstream (see Fig. 1B for coordinate system)

(B) Q = 0.026 m3/s, d1 = 0.105 m, U = 0.67 m/s, Fr1 = 1.34 - Sideview with compression wave propagating from left to right (arrow)

(C) Q = 0.054 m3/s, d1 = 0.15 m, U = 0.83 m/s Fr1 = 1.4, x = 8.6 m - Sideviews with compression wavepropagating from left to right (note the ADV stem and submerged ADV side-looking head)

Fig. 2. Photographs of compression wave propagation in the asymmetrical trapezoidal channel - Flow conditions are listed in the figure sub-caption, including theinitial discharge Q, initial water depth d1, compression wave celerity U, compression wave Froude number Fr1, and longitudinal distance x.

R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

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3. Basic flow patterns

Visual observations were undertaken for a range of initial flowconditions, with initial depths d1 measured next to the right sidewall(Table 1). Fig. 2 presents several photographs, with the details of flowconditions listed in the figure caption, while Appendix B lists a numberof video movies available in the Digital Appendix (Supplementarymaterial). Altogether, a number of unsteady compression wave patternswere observed at x = 8.6 m, depending upon the compression waveFroude number Fr1 and relative initial flow depth d1/D, with D beingthe maximum height of the transverse bed slope (D = 0.140 m, Fig. 1).For d1/D < 1, the initial free-surface width B1 was less than the fullcanal width B = 0.70 m, as sketched in Fig. 1. Such an initial flowcondition, i.e. d1/D < 1, is seen in Fig. 2A and B, and viewed in MovieCIMG5228.MOV (Appendix B). As the compression wave propagatedupstream, it expanded over the transverse slope as illustrated in Fig. 2Aand CIMG5228.MOV (Appendix B); then the new (conjugate) free-sur-face width was greater than the initial one. For d1/D > 1, the initialand conjugate free-surface breadths were identical(B1 = B2 = B = 0.70 m). This initial condition, i.e. d1/D > 1, is illu-strated in Fig. 2C, D and E, and in Movie CIMG5121.MOV (Appendix B).The propagation of the compression wave did not yield any change infree-surface width, as viewed in the Movie CIMG5121.MOV (AppendixB).

For all flow conditions, a key feature was the three-dimensional

shape of the compression wave front. For the present experimentalconditions (Table 1), the compression wave was typically breaking inshallow water next to the left side, while strong secondary waves fol-lowed the wave front in deep waters towards the right sidewall (MoviesCIMG5121.MOV and CIMG5228.MOV, Appendix B). The compressionwave front propagated upstream with an angle to the left sidewall lessthan 90° (Fig. 2 and Movies), although the celerity of the compressionwave was the same across the channel. The leading edge of the com-pression wave arrived first at the sampling location in the shallow watersection. Note that, for the present flow conditions, the compressionwave surface rose past the upper end of the transverse slope: i.e., theconjugate free-surface width was B2 = W2 = 0.70 m. Slow motion vi-deos (Appendix B) outlined an increased level of level of asymmetry ofthe wave front with reduced flow rates.

Dye-injection tests showed a transient recirculation zone about0.29 < y/B < 0.57. This transient recirculation zone was observedwith the dye travelling downstream towards the front of the compres-sion wave, and then changing direction immediately after the wavefront between 0.29 < y/W < 0.57. The compression wave front wasobserved to trap the dye for higher flow rates. After the dye reached thefront, the dye remained entrapped in the roller before slowly movingdownstream. Earlier dye injection tests by KIRI, et al. [15] showed adifferent, more contrasted dye dispersion pattern at the surge front,about y/W ≈ 0.5–0.6. Such an effect was captured, although not assignificantly, in the current study.

(D) Q = 0.054 m3/s, d1 = 0.15 m, U = 0.71 m/s Fr1 = 1.24, x = 8.6 m, with dye injection, looking downstream

(E) Q = 0.094 m3/s, d1 = 0.205 m, U = 1.06 m/s, Fr1 = 1.35, x = 8.6 m, looking downstream - Note the ADV Profiler stem next to the left sidewall

Fig. 2. (continued)

R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

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Free-surface measurements showed the three-dimensional nature ofthe compression waves. Fig. 4 presents typical dimensionless time-series of ensemble-averaged free-surface data at x = 8.6 m and twotransverse locations y. The experiments were repeated 25 times andensemble-averaged in terms of the median surface elevation. Prior tothe arrival of the compression wave, i.e. t.(g.B1/A1)1/2 < 680 in Fig. 4,the water depth was constant in the initially steady flow. The arrival ofthe compression wave was associated with a rapid increase in waterdepth, seen in Fig. 4 about t·(g·B1/A1)1/2 < 700–710.

Three-dimensional free-surface profiles generated from the inter-polation of the six ADMs installed at x = 8.6 m between y = 0.1 m andy = 0.6 m, spaced 0.1 m apart. Fig. 5 presents data for two flow rateshighlighting the unsteady free-surface flow asymmetry. Note thatFig. 5A corresponds to the same flow conditions as for Fig. 2A and B.

Overall the present results showed a number of common features

with compression wave propagation in a compound flume [23] and inan asymmetrical canal [15], as well as with field observations of tidalbores in natural channels, e.g. in the Arcins Channel of the GaronneRiver, France [4,14,25].

4. Free-surface and velocity measurements

The compression wave was generated by the rapid gate closure atthe downstream end of the channel. The water piled up against theclosed gate leading to the wave formation and its upstream translation.Initially the shape of the compression wave evolved with time in re-sponse to the non-rectangular channel cross-section as well as the rapidchange in wetted area and channel shape during the generation andinitial acceleration phases. As the compression wave propagated furtherupstream, i.e. x < 14 m, its shape evolved more gradually with time,

(A) Longitudinal velocity contour map for Q = 0.0063 m3/s and d1 = 0.061 m

(B) Definition sketch of the perimetric coordinate Y (red axis), looking upstream

Y/W1 along wetted perimeter-1 -0.5 0 0.5 1 1.5 2 2.5 3

0

0.005

0.01

0.015

0.02

Q=0.00629 m3/sQ=0.026 m3/sQ=0.0545 m3/sQ=0.0945 m3/s

(C) Transverse distributions of dimensionless boundary shear stress as a function of the dimensionless

perimetric coordinate - Y = 0 at bottom left corner of channel and Y < 0 along the right sidewall (Fig. 3B),

with the bottom corners drawn as thick vertical lines - Data were offset vertically by +0.005 (Q = 0.026

m3/s), +0.010 (Q = 0.054 m3/s), and +0.015 (Q = 0.094 m3/s)

Fig. 3. Steady flow velocity and boundary shear stress in the asymmetrical trapezoidal channel under steady flow conditions.

R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

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and became nearly unchanged at the observation point x = 8.6 m.There, Fig. 5 presents typical time-variations of the two-dimensionalwater depth during the propagation of the surge. Fig. 5A corresponds toa mostly-undular bore for an initially-small discharge with d1/D < 1,showing a three-dimensional surface shape, consistent with the free-surface photographs and movies (e.g. Fig. 2B and movieCIMG5228.MOV (Appendix B)). Fig. 5B shows the variation with timeof the two-dimensional water depth during a breaking bore. The resultsillustrated a steep sharp front, similar to visual observations in Fig. 2Dand movie CIMG5141.MOV (Appendix B).

The free-surface elevation rose very rapidly during the wave passageat x = 8.6 m, and the free-surface became very turbulent. Instantaneous

free-surface elevation and velocity measurements were recorded syn-chronously at relatively high frequency (100 and 200 Hz for the Profilerand ADV respectively). Fig. 6 shows typical time-series of the waterdepth, instantaneous ensemble median longitudinal velocity compo-nent Vx and its instantaneous fluctuations (V75-V25) for a breaking surgewith undulations, in terms of the instantaneous water surface elevationz and longitudinal velocity component Vx, positive downstream, atdifferent transverse locations y. In Fig. 6, the data are ensemble-aver-aged and z is the free-surface elevation above the lowest bed elevation.The curves include the ensemble-median data and, in Fig. 6A, the dif-ferences between third and first quartiles. The latter may be consideredas a proxy of the instantaneous fluctuation. With the arrival of the

Table 2Experimental flow conditions for ensemble-averaged velocity measurements in a compression wave propagating upstream in an asymmetrical trapezoidal channel(Present study).

Ref. Q d1 h U Fr1 Velocimeter x y z-zbed

(m3/s) (m) (m) (m/s) (m) (m) (m)(1) (1) (1)

Q26a 0.026 0.102 00.025

0.670.50

1.311.11

Profiler 8.6 0.1, 0.2 & 0.3 0 to 0.029

Q54a 0.054 0.15 00.030

0.830.71

1.41.24

Profiler 0.1, 0.2, 0.3, 0.4 & 0.5

Q94a 0.094 0.205 0 1.06 1.35 Profiler 0.1, 0.2, 0.3, 0.4, 0.5 & 0.60.015 1.05 1.33 Profiler 0.60.065 0.84 1.11 Profiler 0.1, 0.2, 0.3, 0.4, 0.5 & 0.6

Q26b 0.026 0.103 0 0.67 1.32 ADV 8.6 0.2 0.0058–0.0780.26 0.0058–0.0550.28 0.0058–0.0550.3 0.0058–0.0710.4 0.0058–0.03740.5 0.0058

Q54b 0.054 0.0151 0 0.83 1.4 ADV 0.2 0.0058–0.11250.26 0.012–0.0240.28 0.012–0.0240.3 0.0058–0.10580.4 0.0058–0.07250.5 0.0058–0.03910.6 0.0058

Notes: d1: initial water depth; h: Tainter gate undershoot opening after closure; Fr1: Froude number of compression wave; Q: initially steady water discharge; U:compression wave celerity; y: transverse distance from right sidewall; zbed; bed elevation at the transverse location y; (1): measured at x = 8.6 m.

t.(g.B1/A1)1/2

d/d 1

680 690 700 710 720 730 740 7500.9

1.1

1.3

1.5

1.7

y = 0.2 m

y = 0.4 mQ=0.054 m3/s, h=0.015 m, x=8.6 my = 0.2 my = 0.4 m

Fig. 4. Dimensionless time-variations of ensemble-median water depth during the propagation of a compression wave - Flow conditions: Q = 0.054 m3/s,d1 = 0.151 m, Fr1 = 1.4, A1 = 0.0767 m2, B1 = 0.70 m, x = 8.6 m, transverse locations: y = 0.2 m and 0.4 m.

R. Fernando, et al. Experimental Thermal and Fluid Science 112 (2020) 109986

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compression wave, i.e. t·(g·B1/A1)1/2 < 805 (Fig. 6A) or 645 (Fig. 6B),the longitudinal velocity data showed a very brutal deceleration, fol-lowed by relatively large fluctuations of the velocity.

All the free-surface data showed that the compression wave wasthree-dimensional (Figs. 4–6). It was typically breaking in the shallowwater side and undular in the deep-water side. Prior to the compressionwave passage, the current velocity was positive downstream at alltransverse locations and vertical elevations. The wave propagation hada marked effect on the velocity field, as seen in Fig. 6A and B. Keyfeatures encompassed a brutal flow deceleration as well as fast fluc-tuations of each velocity component during and behind the compres-sion wave front. The present observations were consistent with earlierobservations in rectangular channels [16,18], although a key featurewas the three-dimensional nature of the compression wave front andassociated velocity field.

Video movies were produced from the contour plots of the in-stantaneous ensemble-median velocity components, measured with the

ADV and Profiler combined (Appendix C). In each movie, the in-stantaneous median free-surface elevation is shown with purple crossesand line, and the movie are replayed at 1/8th speed of real-time. Fig. 7presents instantaneous snap shots of three movies. During the com-pression wave passage, the longitudinal velocity data showed a veryrapid deceleration throughout the entire cross-section area, becomingnegative for 0.43 < y/W < 0.71 and a relative elevation above thebed (z-z0)/d1 < 0.1 (Appendix C, movies 26_0_Unsteady_Vx_Rev2.avi& 54_0_Unsteady_Vx_Rev3.avi). This transient recirculation zonechanged size and location, with the free-surface elevation. The trans-verse velocity component oscillated about zero for 0.43 < y/W < 0.71 and a relative elevation above the bed (z-z0)/d1 < 0.1(Appendix C, movies 26_0_Unsteady_Vy_Rev2.avi & 54_0_Un-steady_Vy_Rev3.avi). In the deep section near the right sidewall, thetransient secondary motion was larger in size, than in the shallowerzone, and varied in size, and it appeared to change at a faster rate thanin the shallow zone. The distinct secondary motion zones, either side of

(A) Q = 0.026 m3/s, d1 = 0.102 m, Fr1 = 1.3, A1 = 0.0466 m2, B1 = 0.598 m

(B) Q = 0.054 m3/s, d1 = 0.151 m, Fr1 = 1.3, A1 = 0.0767 m2, B1 = 0.70 m

Fig. 5. Dimensionless time-variations of the two-dimensional ensemble-median water depth during the propagation of a compression wave.

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the boundary between transverse slopes (y/W = 0.43), might have in-duced some transient vertical circulation, acting as some form of in-stantaneous vertical shear region (Appendix B, movies 26_0_Un-steady_Vz_Rev2.avi & 54_0_Unsteady_Vz_Rev3.avi).

5. Velocity fluctuations and Reynolds stresses

The instantaneous velocity fluctuations were defined as the differ-ence between the 75th and 25th percentiles of the ensemble. The ve-locity fluctuations were related to the turbulence and energy levelswithin flow. Maximum fluctuations would characterise times and lo-cations where the greatest levels of turbulent mixing occurred. Typicalresults are presented in Fig. 8 next to the bed. All the velocity com-ponents presented large fluctuations after the passage of the compres-sion waves, with fluctuation magnitudes comparable to the residualvelocity magnitude after the wave passage. The data showed also amarked increase in longitudinal velocity fluctuations during the flowdeceleration, corresponding to the passage of the compression wave(Fig. 8A).

The instantaneous Reynolds stress tensor is defined as:

= × × =vvv

[v v v ]xy

z

x y z

xx xy xz

yx yy yz

zx zy zz (2)

where v is the instantaneous velocity component fluctuation, calculatedas:

=v V V̄ (3)

with V the instantaneous velocity measurement and V̄ the ensemble-median velocity. In the following paragraphs, the ensemble-medianReynolds stress data are presented. Since the instantaneous velocitydata presented large fluctuations (Fig. 8), the instantaneous turbulentstresses similarly showed large and rapid fluctuations during andshortly after the passage of the compression wave. This is illustrated inFig. 9, presenting the time-variation of ensemble-median turbulentstress data. The passage of the compression wave was characterised bya sudden increase in median normal stresses (Fig. 9A) and rapid oscil-lations of the tangential stresses (Fig. 9B). The data showed further that

Fig. 6. Dimensionless time-variations of ensemble-median longitudinal velocity Vx and water surface elevation - ADV data at x = 8.6 m.

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the time lags between wave passage and large turbulent stresses wasabout the same irrespective of the position within the cross section.These findings were consistently observed within the entire cross-sec-tion area, although it is acknowledged that the measurements wereconducted in the initially clear-water column, i.e. z/d1 < 1.

6. Conclusion

This investigation assessed the instantaneous free-surface elevationsand velocity fields in a compression wave propagating in an asymme-trical trapezoidal channel, to understand the three-dimensional turbu-lent flow and mixing processes. Physical experiments were conducted ina relatively large laboratory flume (19 m long, 0.7 m wide), modelling ahalf trapezoidal channel, with a 1V:3H transverse slope, alongside a flattransverse bed (0.28 m wide). The experiments were repeated 25 timesand ensemble averaged.

Undular bores and breaking bores with secondary undulations wereobserved. A marked increase in free-surface elevation was observedduring the surge passage. This was associated with a drastic decelera-tion of the flow. Some three-dimensional transient flow motion wasobserved in this investigation, linked to the asymmetry of the cross-

section. There was a close correlation between velocity oscillations andfree-surface undulations, in line with the potential flow theory andstreamline patterns. In the deep water region, the velocities decreasedduring the compression wave passage and remained positive after thesurge passage. In contrast, a transient recirculation occurred in theshallow water zone, being the strongest next to the base of the trans-verse slope. Dye-injection tests showed a sizeable region where atransient recirculation occurred. The unsteady transverse velocitycomponent was larger next to the bed, in particular close to the jointbetween the sloped and flat bed, compared to near the free-surface. Themagnitudes of local transverse velocities were larger than local long-itudinal velocities, at this location in particular. Rapid and large fluc-tuations of the velocities were observed as the compression wavepassed.

Overall, this study shows a complicated three-dimensional unsteadyturbulent motion in the trapezoidal channel. Strong recirculation andsecondary currents were observed on the transverse slope, albeit forshort, transient periods. Further investigations should focus on the ef-fects of different transverse slopes and assess the effects at the transitionbetween different transverse gradients.

Fig. 7. Instantaneous velocity contour maps before and during a compression wave in the trapezoidal channel - Flow conditions: Q = 0.054 m3/s, h = 0.0 m,Fr1 = 1.4, d1 = 0.151 m. (a) Arrival of compression wave (b) Deceleration phase (c) Shortly after compression wave passage, (LEFT) Vx (MIDDLE) Vy (RIGHT) Vz.

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t·(g.B1/A1)1/2

V/V

1

d/d 1

80 90 100 110 120 130 140 1500 0.9

0.2 1

0.4 1.1

0.6 1.2

0.8 1.3

1 1.4

1.2 1.5

1.4 1.6Vx(50)Vx(75-25)d y=0.3m_(50)

(A) Longitudinal velocity data Vx

t·(g.B1/A1)1/2

V/V

1

d/d 1

80 90 100 110 120 130 140 150-0.1 0.9

0 1.2

0.1 1.5

0.2 1.8Vy(50)Vy(75-25)d y=0.3m_(50)

(B) Transverse velocity data Vy

t·(g.B1/A1)1/2

V/V

1

d/d 1

80 90 100 110 120 130 140 150-0.2 0.9

-0.1 1

0 1.1

0.1 1.2

0.2 1.3

0.3 1.4

0.4 1.5

0.5 1.6Vz(50)Vz(75-25)d y=0.3m_(50)

(C) Vertical velocity data Vz

Fig. 8. Time-variations of instantaneous median velocity V50, velocity fluctuations V75-25 = V75 - V25 and water depth d during the passage of the compression wave -Flow conditions: Q = 0.054 m3/s, h = 0.0 m, Fr1 = 1.4, d1 = 0.151 m, y = 0.3 m, z-zbed = 0.006 m.

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Declaration of Competing Interest

The authors have no conflict of interest nor vested interest.

Acknowledgements

The authors thank Ms. Urvisha Kiri for her helpful advice, Ms.

Ebony Bell for her assistance with some experiments, and Mr. JasonVan Der Gevel and Stewart Matthews (The University of Queensland)for technical assistance.

Appendix A. Application of the momentum equations to a compression wave in non-rectangular asymmetrical channel

A compression wave is as a sudden rise in free-surface elevation constituting a discontinuity of the pressure and velocity fields. In the system ofreference following the wave front, the integral form of the equations of conservation of mass and momentum gives a series of relationships betweenthe conjugate flow properties, i.e. in front of and behind the surge front [11,20]:

+ × = + ×(V U) A (V U) A1 1 2 2 (I-1)

× + × × × + × + = × × + ×(V U) A ( (V U) (V U)) P dA P dA F W sin1 1 1 1 2 2A A

fric f

2 1 (I-2)

where V is the flow velocity and U is the surge celerity for an observer standing on the bank, ρ is the water density, g is the gravity acceleration, A isthe channel cross-sectional area measured perpendicular to the main flow direction, β is a momentum correction coefficient, P is the pressure, Ffric isthe flow resistance force, Wf is the weight force, θ is the angle between the bed slope and horizontal, and the subscripts 1 and 2 refer respectively tothe initial flow conditions and the flow conditions immediately after the tidal bore. Note that Eqs. (I-1) and (I-2), as well as the following equations,were developed for a compression wave propagating upstream. A similar reasoning may be expanded for a compression wave travelling downstream.

The above system of equations may provide some analytical solutions within some basic assumptions. Neglecting the flow resistance and theeffect of velocity distribution (β1 = β2 = 1), and assuming a horizontal channel (sinθ ≈ 0), the momentum equation (I-2) becomes:

× + × × = × ×(V U) A (V V ) P dA P dA1 1 1 2A A2 1 (I-3)

Assuming a hydrostatic pressure distribution in front of and behind the compression wave, the net pressure force resultant consists of the increase

Fig. 9. Time-variations of instantaneous ensemble-median Reynolds stress tensor components τxx and τyz - - Flow conditions: Q = 0.054 m3/s, h = 0.0 m, Fr1 = 1.4,d1 = 0.151 m, y = 0.3 m, z-zbed = 0.070 m.

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of pressure ρ × g × (d2 − d1) applied to the initial flow cross-section area A1 plus the pressure force on the cross-section area ΔA = A2 − A1. Thislatter term equals:

× × × = × × × ×zg (d ) dA 12

g (d d ) B'A

A

2 2 12

1

2

(I-4)

where z is the distance normal to the bed, d1 and d2 are the flow depths in front of and behind the wave front, and B′ is a characteristic free-surfacewidth. It may be noted that B1 < B′ < B2 where B1 and B2 are the upstream and downstream free-surface widths. Another characteristic free-surface width B is defined as:

=B A Ad d

2 1

2 1 (I-5)

The equation of conservation of mass (I-1) may be expressed as:

= + ×V V (V U) A AA1 2 1

2 1

2 (I-6)

The combination of the equations of conservation of mass and momentum yields [3]:

= × × × + ×Fr 12

AA

BB

2 B'B

B'B

AA1

2

1

1 2

1 (I-7)

where the Froude number of the compression wave Fr1 is defined as:

= +×

Fr U Vg

11

AB

11 (I-8)

Eq. (I-10) defines the compression wave Froude number for an irregular channel based upon momentum considerations.Eq. (I-7) gives an analytical solution of the surge Froude number as a function of the ratios of cross-sectional areas A2/A1, and dimensionless

characteristic widths B'/B and B1/B. It may be rewritten to express the ratio of conjugate cross-section areas A2/A1 as a function of the upstreamFroude number [3]:

= ×+ × ×( ) ( )

AA

12

2 8 Fr 22

1

B'B

21

2 B'B

B'B

B'BB1B

(I-9)

which is valid for any positive in an irregular flat channel. The effects of channel cross-sectional shape are taken into account through the ratios B'/Band B1/B.

In some particular situations, the cross-sectional shape satisfies the approximation B2 ≈ B ≈ B′ ≈ B1: for example, a channel cross-sectional shapewith parallel walls next to the waterline or a rectangular channel. A well-know solution of the momentum equation is the Bélanger equationdeveloped for a rectangular horizontal prismatic channel in absence of friction.

Application to an asymmetrical trapezoidal channel

In the present study, the propagation of compression waves was investigated in an asymmetrical non-rectangular channel (Fig. 1). The initial flowcross-section area and free-surface width are:

= × × + × = × + <A 12

d W d & B d W d D1 12

1 1 1 1 1 1 (I-10a)

= × × = >A d W D2

& B W d D1 1 22

1 2 1 (I-10b)

where λ−1 is transverse slope of the bed (Fig. 1). For this geometry, three boundary conditions may occur depending upon the initial and new(conjugate) flow depths: (a) d1 < d2 < D, (b) d1 < D < d2, and (c) D < d1 < d2. The last case, D < d1 < d2, is trivial, yielding B1 = B = B' =B2.

Case d1 < d2 < D

The analytical solutions of the characteristic widths B and B′ are:

= + × × +B W 12

(d d )1 1 2 (I-11)

=× × + × × × × + × × + × × + × ×

×( ) ( ) ( )

B'd W d d d W d d W d

(d d )2

2 12 1

12 2 2 1 1

12 1 1

2 12 1

13 1

12 2 1

2 (I-12)

Case d1 < D < d2

The analytical solutions of the characteristic widths B and B′ are:

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13

=× × × + ×( )

BW d D d W d

d d2 2 2

21 1 2 1

2 1 (I-13)

=× + × × × + × × × × + ×

×B'

D d D (d D) W (d d ) d d d

(d d )6

32 2 2

12 1 2 1

22 2 1

23 1

3

12 2 1

2 (I-14)

Using the method of images, the above solution may be applied to symmetrical trapezoidal channel. In Fig. 1, the vertical line y = 0 would be thesymmetry line. The resulting symmetrical trapezoidal channel would have a bottom width 2 × W1 with 1V:λH sideslopes.

Appendix B. Movies of compression wave experiments in an asymmetrical trapezoidal channel (digital appendix)

Video-movies of the compression wave experiments were taken using a digital camera Casio™ Exilim Ex10 (120 fps, resolution: 640 px × 480 px).Table A1. describes each video movie available in the digital appendix. The movies were recorded at 120 fps and replayed at 30 fps.

Appendix C. Movies of ensemble-median unsteady velocity field beneath compression wave in an asymmetrical trapezoidal channel(digital appendix)

Video movies were developed based upon the instantaneous ensemble-median ADV and Profiler velocity data at x = 8.6 m. At each time step, a y-z cross-sectional contour profile was generated, including the no-slip condition at the fixed boundaries. The resultant images were played back inslow motion movies. The ensemble-median velocity data were combined with the ADM data to include the instantaneous transverse surface profile.It is acknowledged that the spacing between sampling points might yield some error in the spatial interpolation throughout the cross section.Practically, the data showed the highest velocity gradient next to the bed in the steady-state and unsteady velocities at the bed. With velocityprofiling every millimetre next to the bed, from z = zbed up to 30 mm, the main effects were believed to be captured, including a marked effect ofcompression wave passage on the vertical velocity gradient, as well as the transverse secondary motion along the inclined (1V:3H) slope.

Each movie started at the steady state, with the solid purple line and cross symbols showing the water surface. The arrival of the compressionwave may be seen with the initial raise and following oscillations of the transverse water level. The legend gives the colour coding of the di-mensionless velocity component Vi/V1, with V1 the initial flow velocity and i = x, y or z. Finally the video movies are played at a 1/8th speed of real-time (see Table A2).

Appendix D. Supplementary material

Supplementary data to this article can be found online at https://doi.org/10.1016/j.expthermflusci.2019.109986.

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Table A2Video movies of unsteady velocity data beneath compression wave in an asymmetrical trapezoidal channel.

Filename Velocitycomponent

Description

26_0_Unsteady_Vx_Rev2.avi Vx Flow conditions: Q = 0.026 m3/s,So = 0.002216, Fr1 = 1.3,d1/D = 0.74

26_0_Unsteady_Vy_Rev2.avi Vy

26_0_Unsteady_Vz_Rev2.avi Vz

54_0_Unsteady_Vx_Rev3.avi Vx Flow conditions: Q = 0.054 m3/s,So = 0.002216, Fr1 = 1.4,d1/D = 1.07

54_0_Unsteady_Vy_Rev3.avi Vy

54_0_Unsteady_Vz_Rev3.avi Vz

Table A1Video movies of the compression wave propagation in an asymmetrical trapezoidal channel.

Filename Camera Description

CIMG5121.MOV Casio™ Exilim Ex10Frame rate: 120 fps

Three-dimensional compression wave propagation at x = 8.6 m, viewed looking downstream. Flow conditions: Q = 0.054 m3/s,So = 0.002216, Fr1 = 1.4, d1/D = 1.07. Movie duration: 17 s

CIMG5228.MOV Casio™ Exilim Ex10Frame rate: 120 fps

Three-dimensional compression propagation looking downstream from x = 7 m. Flow conditions: Q = 0.026 m3/s, So = 0.002216,Fr1 = 1.32, d1/D = 0.74. Movie duration: 36 s

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