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Citation: Elgamal, M.; Chaouachi, L.;
Farouk, M.; Helmi, A.M. Analysis of
Water-Surface Oscillations Upstream
of a Double-Right-Angled Bend with
Incoming Supercritical Flow. Water
2023,15, 3570. https://doi.org/
10.3390/w15203570
Academic Editor: Athanasios Loukas
Received: 28 July 2023
Revised: 8 October 2023
Accepted: 9 October 2023
Published: 12 October 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
water
Article
Analysis of Water-Surface Oscillations Upstream of a
Double-Right-Angled Bend with Incoming Supercritical Flow
Mohamed Elgamal 1, * , Lotfi Chaouachi 1, Mohamed Farouk 1,2 and Ahmed M. Helmi 3
1Civil Engineering Department, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318,
Saudi Arabia; lschaouachi@imamu.edu.sa (L.C.); miradi@imamu.edu.sa (M.F.)
2Faculty of Engineering, Ain Shams University, Cairo 11517, Egypt
3Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt;
ahmed.helmi@eng.cu.edu.eg
*Correspondence: mhelgamal@imamu.edu.sa; Tel.: +966-545-859-725
Abstract:
This study deals with the free-surface supercritical flow through a double-right-angled
bend (DRAB), which can be found in storm drainage networks in steep terrains. Laboratory experi-
ments showed that strong backwater effects and water-surface oscillations are generated upstream
of the DRAB, especially in supercritical flow conditions. This paper investigated the DRAB hy-
draulic behavior and water-surface heading up (backwater), and oscillations under supercritical flow
conditions. Thirty-four lab experiments were conducted with Froude numbers ranging between
1.03 and 2.63. Dye injection and video analysis were used to visually capture the flow structure and
to record water-surface oscillations. A tracker package was utilized to analyze the collected visual
data. Time series and spectral analysis were used to identify the statistical characteristics of recorded
water level time series and the dominant frequencies. It was found that the dominant frequencies of
water-surface oscillations upstream of the DRAB range between 1.6 and 4.6 Hz with an average value
of about 3 Hz. The Strouhal number of the water-surface oscillations is more sensitive to the Froude
number than to the Reynolds number. The Strouhal number ranged between 0.03 and 0.3 for Froude
numbers ranging from 2.63 to 1.03. The study confirms that near critical flow conditions exhibit the
highest water oscillation, and that the maximum nondimensional water depth upstream of the DRAB
is underestimated by both the Grashof formula and Knapp and Ippen (1939) model. A new formula
is proposed to estimate the maximum water depth upstream of the DRAB.
Keywords:
double-right-angled bends; bends in high-velocity channels; water oscillations; Strouhal
number; video tracking in hydraulics; backwater effect; water afflux
1. Introduction
High-velocity channels featuring steep gradients play a pivotal role in drainage sys-
tems, particularly in regions characterized by mountainous terrain, steep landscapes, or
urban environments where land costs are at a premium. In such scenarios, it is common
engineering practice to mitigate the erosive potential of high-velocity flows by employing
strategies such as the use of dissipators, cascades of weirs, or similar hydraulic structures to
reduce the energy slope of the channels. These approaches are widely adopted to safeguard
against erosion. However, despite their prevalence, there are exceptional cases where
engineers may opt for steeper channel slopes to fulfill specific project objectives. One such
exception arises when the design objective is the rapid discharge of water to a nearby dis-
posal point, necessitating higher flow velocities [
1
]. Another exception occurs in situations
where land acquisition costs are prohibitively high, and concrete-lined canals are utilized,
making higher-velocity channels a viable solution. This choice is driven by the inherent
capacity of concrete surfaces to endure high velocities and resist scouring, thereby reducing
land acquisition expenses through a reduction in the required channel bed width.
Water 2023,15, 3570. https://doi.org/10.3390/w15203570 https://www.mdpi.com/journal/water
Water 2023,15, 3570 2 of 26
In practical scenarios, channel layouts are often dictated by land use and topographical
constraints, leading to the inclusion of geometric features such as right-angled bends,
contractions, transitions, or bed steps. These features can result in flow constriction,
choked flow conditions, cross waves, standing waves, hydraulic jumps, and trans-critical
flows [1–4].
While it is generally advisable to avoid sharp right-angled bends in supercritical
flow channels to mitigate cross-waves and system oscillations, site-specific constraints may
necessitate their incorporation. Therefore, understanding flow conditions at such bends is
of paramount importance for practical applications.
Supercritical flow through channel bends is characterized by intricate phenomena,
including flow separation, secondary flows, energy losses, and variations in water-surface
elevation induced by the curvature of the bend [
5
]. Over the centuries, research on su-
percritical flow in curved channels has garnered significant attention, both in terms of
experimental and numerical investigations. Early contributions by Ippen and Knapp [
6
]
laid the foundation, involving experiments with varying relative radii of curvature and the
formulation of an initial equation for estimating maximum and minimum wave heights in
curved channels. Subsequent efforts by Reinauer and Hager [
7
–
10
] delved into theoretical
and experimental examinations of supercritical bend flow, elucidating the relationship
between flow characteristics and bend curvature. Numerically, numerous researchers
have explored the applicability of two-dimensional shallow water equations (2DSWEs) to
simulate supercritical free-surface flows in bends [
11
–
13
]. Valiani and Caleffi [
13
] noted
that 2DSWEs generally offer a qualitatively accurate representation of the flow field but
tend to underestimate maximum water depths with increasing relative curvature and
undisturbed Froude numbers. In alignment with these numerical endeavors, Soli et al. [
14
]
developed three-dimensional numerical models using a commercial CFD package (Flow-
3D Hydro) to simulate flow patterns in curved chutes and investigate the impact of adding
wall splitters on the separation zone. Frazao and Zech [
15
] conducted experimental and
numerical studies on the dam break problem in a channel with a 90-degree bend, proposing
a hybrid 1D/2D numerical model to overcome the limitations of 1D models, which tend to
underestimate water depths and overestimate wave propagation speeds.
The significance of these studies lies in the fact that the presence of these boundary
features in a supercritical flow regime can induce pronounced oscillations in water-surface
elevation and the flow field.
In recent decades, water-surface fluctuations in rivers and streams have garnered in-
creased attention. Investigations have encompassed a wide range of topics, from studying
water level fluctuations during the lock emptying process [
16
,
17
] and hydropower station
operations [
18
] to examining the impact of spur dikes on riverbanks [
19
] and the presence
of axisymmetric side cavities on water-surface oscillations in channels [
20
]. Additionally,
researchers have explored the influence of periodic cylinder arrays (similar to vegetation)
on flow dynamics and water-surface oscillations [
21
,
22
]. Furthermore, considerable ef-
fort has been dedicated to understanding free-surface fluctuations in hydraulic jumps
and the oscillatory behavior of these hydraulic phenomena [
23
–
28
]. Identifying water-
surface fluctuations holds practical applications, including enhancing navigation safety,
optimizing freeboard requirements for channel design, and determining submergence
depths for suction pipe intakes to prevent vortex formation. The prediction of oscillation
frequency assumes paramount importance for various reasons, including: (1) aiding in
pinpointing the primary source of oscillations within the system; (2) guiding the selection of
appropriate sensors (e.g., water depth sensors) and determining minimum sampling rates;
(3) facilitating the assessment of the likelihood of specific events, such as exceeding a
defined water-surface elevation threshold, which is critical for stochastic reliability analyses
through Monte Carlo simulation; and (4) supporting open-channel fluid dynamics by
providing experimental data to validate computational fluid dynamics (CFD) models.
The principal aim of the present research is to investigate the hydraulic performance
of high-velocity channels incorporating a double-right-angled bend (DRAB) configura-
Water 2023,15, 3570 3 of 26
tion under supercritical flow conditions. This study seeks to address the following key
questions:
•
What are the effects of altering the approaching flow and Froude number on the
upstream water elevation of the DRAB, and how can the maximum water depth
upstream of the DRAB be estimated?
•
Can the existing analytical/empirical equations used to predict the backwater effect
be applied to the DRAB configuration?
•
What are the flow conditions that induce water-surface oscillations upstream of
the DRAB?
•
What are the dominant frequencies characterizing these oscillations, and is there an
empirical formula for estimating these dominant frequencies?
To answer these questions, a series of laboratory experiments were conducted and
analyzed using visual inspection and a non-intrusive video tracking system comprising
five cameras to record spatial and temporal variations in water-surface elevation upstream,
within, and downstream of the DRAB.
The subsequent sections of this paper are organized as follows: Section 2introduces the
methodologies, including the experimental setup and measurement techniques. Section 3
outlines the visual inspection of the flow field and the analysis of free-surface oscillations.
Section 4delves into the results, encompassing water oscillations upstream of the DRAB
and maximum water depth. Furthermore, Section 4discusses the challenges and limitations
inherent to the current work. Lastly, Section 5summarizes the findings of this study and
presents future outlooks.
2. Methodology
The present study employs a combination of experimental and statistical approaches
to answer the above-mentioned questions.
This section is dedicated to describing the experimental setup and measurement
techniques. The statistical analysis, including time series and power spectral analysis, will
be detailed in Section 3.
2.1. Experimental Setup
The experimental investigation was conducted at the water resources engineering
laboratory of the College of Engineering at Imam Muhammad Ibn Saud University. An
8 m long tilting flume with a rectangular cross-section 310 mm wide and 600 mm deep was
used. The flume’s base is made of stainless steel and the sides of glass. The flume’s bed
slope is adjustable, and a tailwater gate at the end of the flume controls the downstream
water level. Water is recirculated using a pumping system equipped with a gate valve
for discharge control and a 100 mm electromagnetic flow meter for volumetric flow rate
measurements, as shown in Figure 1.
To create the DRAB, vertical plexiglass sheets were utilized and set perpendicular
using 210 mm (in length) plexiglass stiffener sticks. The resulting rectangular flume had
fixed walls, a width of 100 mm, a height of 600 mm, and a total length of approximately
8200 mm along the centerline. The DRAB was situated about 3 m downstream from
the channel drop inlet structure. The drop inlet structure, of a height of 295 mm and
laterally extended to the full width of the flume (310 mm), served two purposes: firstly,
to maintain a stationary water level in the upstream water reservoir to ensure steady and
non-oscillatory performance for the circulating pumping system; and secondly, to create
a waterfall generating the required supercritical flow conditions at the toe of the drop
structure and far upstream of the studied DRAB.
Water 2023,15, 3570 4 of 26
Water 2023, 15, x FOR PEER REVIEW 4 of 28
Figure 1. Experimental setup: (a) snapshot of the experimental setup (using camera 1); (b) schematic
of the experimental setup (plan view); (c) schematic of the experimental setup (front view); (d) top
view showing elements of DRAB (using camera 5): Legend of different elements pointed out by the
doed line arrows are given as follow: (1) span extent of the DRAB; (2) upstream inner edge of
DRAB; (3) downstream inner edge; (4) upstream reach of DRAB featured by conversion of super-
critical flow to subcritical flow via a hydraulic jump; (5) downstream reach of DRAB featured by
supercritical flow with cross waves; (6) location where water heading up and water oscillations are
Figure 1.
Experimental setup: (
a
) snapshot of the experimental setup (using camera 1); (
b
) schematic
of the experimental setup (plan view); (
c
) schematic of the experimental setup (front view); (
d
) top
view showing elements of DRAB (using camera 5): Legend of different elements pointed out by the
dotted line arrows are given as follow: (1) span extent of the DRAB; (2) upstream inner edge of DRAB;
(3) downstream inner edge; (4) upstream reach of DRAB featured by conversion of supercritical flow
to subcritical flow via a hydraulic jump; (5) downstream reach of DRAB featured by supercritical
flow with cross waves; (6) location where water heading up and water oscillations are measured;
(7) plexiglass sides of the channel; (8) stiffeners for supporting plexiglass sides. L is the length and
width throughout the bends.
Water 2023,15, 3570 5 of 26
2.2. Experimental Runs
The main variables considered for experimental investigation were the flume bed slope
and the water flow discharge. Table 1provides the characteristics of the 48 experimental
runs that have a Reynolds no. (R
n
= V
o
.R
h
/n) ranging from 4622 to 55,513 and a Froude
no. (F
n
= V
o
/
√gyo
) from 1.03 to 2.63. The runs can be classified into four sets aimed at
achieving different objectives. The first set (runs 1 to 11) aimed to measure the equivalent
Manning no. These runs were conducted on a straight rectangular flume with composite
material and a 100 mm bed width, with the DRAB removed from the setup.
Table 1. Characteristics of the conducted experimental runs.
Run No # Ditch Setup Experiment
Scope Slope (%) Water Flow
Q (L/s) Rn Fn Remarks
Straight DRAB
1XR 1 0.56 4622 1.56
2XR 1 1.94 12,296 1.49
3XR 1 3.06 18,088 1.45
4XR 1 3.89 21,825 1.53
5XR 1 5 25,208 1.44
6XR 1 6.39 30,028 1.48
7XR 1 9.17 37,278 1.50
8XR 1 16.11 52,219 1.53
9XR 0.2 1.39 8830 1.05
10 XR 0.2 2.78 15,449 1.14
11 XR 0.2 4.17 20,421 1.14
12 XDT 2 5.28 30,599 2.23 Right injection
13 XDT 2 5.28 30,599 2.23 Right & Left injection
14 XWSO 2.5 12.78 53,848 2.27
15 XWSO 2.5 9.44 45,225 2.38
16 XWSP/O 2.5 6.67 36,242 2.48
17 XWSP/O 2.5 3.89 24,766 2.58
18 XWSP/O 2.5 1.39 10,821 2.63
19 XWSO 2 15 55,513 1.93
20 XWSO 2 11.94 49,334 2.01
21 XWSO 2 10.56 46,079 2.06
22 XWSP 2 8.89 41,707 2.11
23 XWSP 2 7.5 37,593 2.16
24 XWSP/O 2 6.94 35,805 2.18
25 XWSP 2 5.83 31,945 2.22
26 XWSP/O 2 3.61 22,777 2.30
27 XWSP/O 2 1.67 12,395 2.35
28 XWSO 1.67 10.56 44,249 1.85
29 XWSO 1.67 8.61 39,433 1.91
30 XWSO 1.67 7.22 35,476 1.96
31 XWSO 1.67 5.83 30,973 2.01
32 XWSO 1.67 4.44 25,774 2.06
Water 2023,15, 3570 6 of 26
Table 1. Cont.
Run No # Ditch Setup Experiment
Scope Slope (%) Water Flow
Q (L/s) Rn Fn Remarks
Straight DRAB
33 XWSO 1.67 2.22 15,376 2.14
34 XWSO 1.25 12.22 44,545 1.52
35 XWSO 1.25 8.06 35,707 1.63
36 XWSO 1.25 5.28 27,598 1.73
37 XWSO 1.25 2.78 17,645 1.82
38 XWSO 1.25 0.83 6642 1.85
39 XWSO 1 13.33 43,735 1.3
40 XWSO 1 11.11 40,159 1.35
41 XWSO 1 8.33 34,632 1.43
42 XWSO 1 4.72 24,690 1.55
43 XWSO 1 1.67 11,610 1.66
44 XWSO 0.67 12.5 37,925 1.03
45 XWSO 0.67 10 34,347 1.08
46 XWSO 0.67 7.78 30,354 1.14
47 XWSO 0.67 4.44 22,060 1.24
48 XWSO 0.67 1.39 9634 1.35
Note: R: roughness, DT: dye test, WSP: water-surface profile analysis, WSO: water-surface oscillation analysis,
WSP/O: both water-surface profile and oscillation analysis upstream of the DRAB.
The second set of runs (runs 12 and 13) involved a visual study via dye injection to
visualize the flow structure resulting from the presence of the DRAB in the flume.
The third set of runs aimed to plot the spatial variation in the water-surface profile
upstream, through, and downstream of the DRAB.
The main objective of the last set of runs, totaling thirty-four experiments, was to track
the oscillation of the water-surface elevation and the backwater effect just upstream of
the DRAB.
2.2.1. Main Assumptions
The following principal assumptions were considered in this study:
•The channel is non-erodible with a fixed bed.
•The channel is a prismatic, constant-width rectangular section.
•
All runs were conducted with a constant width (W) of 100 mm and a constant DRAB
length (L) of 300 mm (i.e., bend length to width ratio L/W = 3).
•
The width through turns is equal to the bed width of the upstream and downstream
reaches (b/W = 1).
•
All turns within the DRAB are perfectly sharp turns (r = 0, b/2r =
∞
) (where r is the
radius of curvature of the turns).
•The slopes of all reaches remain constant for each run but vary across different runs.
•
All runs were conducted under approaching flow conditions classified as low super-
critical flow conditions (1.03 < Fn< 2.63).
•
The Manning n roughness varies with the water depth, and the depth-averaged value
is approximately 0.008 s/m1/3.
Water and flow measurements were conducted after reaching stationary conditions
for the measured variables. The assumption of stationarity is crucial for time series power
spectral analysis, as discussed later.
Water 2023,15, 3570 7 of 26
2.2.2. Work Flow
A brief overview of these workflow steps is as follows:
•
Adjust the flume bed slope, turn on the pump, and adjust the flow to the required value.
•
Wait for a few minutes to reach stationary flow conditions, then start video recording
using sampling rate of 30 fps. Export recorded videos and open them in the Tracker
software. Adjust axes, scale, and apply suitable filters (if necessary) to ensure a clear
water interface. Then, perform auto-tracking for the water-surface upstream of the
DRAB. The recording duration should be at least 10 s to obtain sufficient time series
data for spectral power analysis. Export time series data of water-surface elevation and
vertical velocity and save them to CSV format. Perform power spectral analysis for the
water-surface elevation time series data and plot the corresponding power–spectral
density curve on a log–log scale using the signal processing toolbox in MATLAB. An
m-script was prepared to automate the analysis process (Appendix A).
•
Based on the obtained spectral power density curve, identify the dominant frequency
and the slope of the higher frequency data.
•
Conduct dimensional analysis to identify the relevant
π
-terms affecting the dominant
frequency of the water-surface oscillation upstream of the DRAB. Employ the least
squares approach to obtain the best fitting formula for data measurements.
2.3. Digital Cameras
To acquire data, five different digital cameras were used simultaneously. Camera 1, a
bridge-type Coolpix P600 Nikon digital camera-Japan, captured the temporal variation in
the water-surface just upstream of the DRAB and the full longitudinal water-surface profile
of the hydraulic jump generated upstream of the DRAB. The camera’s sampling rate was
adjusted to 30 fps for all runs. Camera 1 was positioned in two locations: 1A, just in front
of the DRAB, to capture the fluctuation of the maximum water surface at the first bend’s
upstream side using the video capturing mode; and 1B, to the right of position 1A and
slightly farther from the flume, providing a zoomed-out view to capture the extent of the
hydraulic jump formation upstream of the DRAB (Figure 1b).
Camera 2, the Sony Cyber-Shot DSC-RX100, was used to capture the spatial variation
in the water surface downstream from the DRAB (Figure 1b).
Webcams served as the third and fourth cameras, recording the lateral variation in
the water surface through the DRAB. The fifth camera, the Kiosk High Speed Webcam,
recorded the extent of the water features from the top view (Figure 1c). Table 2provides
the main characteristics of all the cameras used in the study [29–31].
Table 2. Characteristics of used cameras.
Camera No. Model Type Sensor Max Resolution
(MP)
Max Frame
Rate (fps)
Zoom
(Optical) Measured Parameter
1
Nikon, CoolPix
P600
Bridge-DSLR
styled
1/2.3”
BSI-CMOS 16 120 60X
Water surface upstream
of bends
2
Sony
Cyber-Shot
DSC-RX100
Point-and-
shoot 1” CMOS 20.1 1000 3.6X
Spatial variation in
water surface
downstream of bends
3 and 4 Microsoft-Life
Cam Studio Webcam CMOS 5 30 3X
Lateral water-surface
profiles throughout the
two bends
5
Kiosk High
Speed
Webcams
Webcam 1/3” CMOS 2 260 10X Top view for water
features extent
2.4. Video Tracking Packages
Video analysis and tracking methods can be applied either manually or automatically.
Manual tracking involves frame-by-frame analysis using specialized video analysis soft-
Water 2023,15, 3570 8 of 26
ware. Automatic tracking can be achieved through general image processing packages
like MATLAB/OCTAVE image processing toolbox/package or dedicated video analysis
packages developed specifically for object motion tracking, such as Vernier, Tracker, Logger
Pro, and others [32–34].
In this study, Tracker software (version 6.1.3) was utilized for video analysis. Tracker
is a freeware video analysis and modeling tool developed in 2010 as part of the Open-
Source Physics project (O.S.P.) [
34
]. It allows for automatic tracking of objects or features in
recorded video clips. To perform the auto tracking, a “template image” representing the
selected moving feature of interest is created by the user. The software then searches each
frame in the video for the best match to this template image. Two numerical parameters,
the “match score” and the “evolution rate percentage”, need to be set for this process. The
match score is used to identify the best match template image, while the evolution rate
percentage considers potential temporal changes in the shape and colors of the template
image. In this study, default values of 4 for the match score and a minimum value of 5% for
the evolution rate were adopted to prevent template image “drift” issues [35].
2.5. Measurement of Temporal Variation in Water-Surface Elevation
Video tracking techniques were employed to track the water surface and record its
temporal variations. This method offers several advantages; it is non-intrusive and cost-
effective, using inexpensive devices such as webcams, point-and-shot cameras, or smart-
phones. Moreover, it allows for capturing high water oscillations or temporal variations in
the water surface without additional equipment.
Video tracking has been widely applied in various studies to monitor surfaces over
time. For instance, video tracking was used to observe the evolution of scour holes due
to siphon action [
36
] and to track the falling rate of water surfaces in drainage tanks [
37
]
and at sluice gates [
38
]. In this study, the water surface was distinguished from the entire
water body using image processing techniques, which enabled the delineation of the water
surface from the water body through edge detection and delineation filters. Sufficient
lighting or adding color to the water aided in achieving clear delineation.
Figure 2illustrates a typical example of auto-tracking the water surface just upstream
of the DRAB using the Tracker package.
Water 2023, 15, x FOR PEER REVIEW 9 of 28
Figure 2. The process of auto-tracking the water surface in tank 1 via Tracker (refer to notes below
for the figure legend): (1) water-surface profile upstream of DRAB at a given time t = 1.635 sec (from
start of recording); (2) current elevation of water surface at location of oscillation measurements
upstream of DRAB; (3) current ploed elevation of the water surface measured from the bed; (4)
current vertical velocity v
y
; (5) data table showing current time and corresponding water-surface
elevation and vertical velocity; (6) tracking window, where water surface is tracked with time
course; (7) traces of tracking water-surface elevations at previous time steps; (8) upstream face of
DRAB; (9) upstream inner edge of DRAB. The green and blue colors shown represent colored plastic
sheets used to cover the external scenes outside the experiment.
3. Hydraulic Analysis of Supercritical Flow through DRAB
3.1. Estimation of the Equivalent Manning Coefficient
In this study, the composite section of the ditch, composed of stainless steel, glass,
and plexiglass materials, required an assessment of an equivalent roughness coefficient
and its variation with water depth. To achieve this, several runs were conducted on the
upstream straight part of the ditch before the DRAB was introduced. For these runs, the
drop inlet structure was omied, and additional deflector surfaces and turbulence sup-
pressors were temporarily added to promote nearly uniform flow conditions and mini-
mize cross waves at the ditch inlet. Figure 3 illustrates the vertical variation in the equiv-
alent Manning n as a function of water depth. It can be observed that the equivalent Man-
ning n tends to slightly decrease with increasing water depth. This trend is anticipated as
the weight of the less rough surfaces (glass and plexiglass) increases relative to the rela-
tively rougher stainless steel bed surface. The depth-averaged equivalent Manning n was
found to be approximately 0.008 s.m
−1/3
.
Figure 2.
The process of auto-tracking the water surface in tank 1 via Tracker (refer to notes below for
the figure legend): (1) water-surface profile upstream of DRAB at a given time t = 1.635 s (from start
Water 2023,15, 3570 9 of 26
of recording); (2) current elevation of water surface at location of oscillation measurements upstream
of DRAB; (3) current plotted elevation of the water surface measured from the bed; (4) current vertical
velocity v
y
; (5) data table showing current time and corresponding water-surface elevation and
vertical velocity; (6) tracking window, where water surface is tracked with time course; (7) traces of
tracking water-surface elevations at previous time steps; (8) upstream face of DRAB; (9) upstream
inner edge of DRAB. The green and blue colors shown represent colored plastic sheets used to cover
the external scenes outside the experiment.
3. Hydraulic Analysis of Supercritical Flow through DRAB
3.1. Estimation of the Equivalent Manning Coefficient
In this study, the composite section of the ditch, composed of stainless steel, glass, and
plexiglass materials, required an assessment of an equivalent roughness coefficient and its
variation with water depth. To achieve this, several runs were conducted on the upstream
straight part of the ditch before the DRAB was introduced. For these runs, the drop inlet
structure was omitted, and additional deflector surfaces and turbulence suppressors were
temporarily added to promote nearly uniform flow conditions and minimize cross waves
at the ditch inlet. Figure 3illustrates the vertical variation in the equivalent Manning n
as a function of water depth. It can be observed that the equivalent Manning n tends to
slightly decrease with increasing water depth. This trend is anticipated as the weight of
the less rough surfaces (glass and plexiglass) increases relative to the relatively rougher
stainless steel bed surface. The depth-averaged equivalent Manning n was found to be
approximately 0.008 s.m−1/3.
Water 2023, 15, x FOR PEER REVIEW 10 of 28
Figure 3. Vertical variation in equivalent Manning n for the experimental ditch (runs 1 to 8).
3.2. Visual Analysis
At the toe of the hydraulic drop structure, the flow undergoes a transition to super-
critical conditions, and due to the presence of the DRAB, a hydraulic jump is formed up-
stream of the bends, leading to subcritical flow. Within the DRAB, the flow exhibits a gen-
erally three-dimensional behavior with subcritical, followed by transcritical, characteris-
tics. Additionally, the presence of two free-surface vortex structures is evident just down-
stream of the inner side of each bend forming the DRAB. Upon exiting the bends, the flow
reverts to supercritical conditions and forms a clear paern of positive (shock) and nega-
tive standing waves downstream of the last bend.
To visually investigate the flow structure upstream of the DRAB, two dye injection
runs were conducted. The injection location was set at a distance of 300 mm (3W) up-
stream of the DRAB. Two thin needles were used for injection, with their tips positioned
at the same elevation just below the water surface. The syringe pistons of both needles
were connected to ensure equal injection speed and exerted stress. The first needle, loaded
with blue dye, was placed closer to the right side of the ditch, while the second needle,
loaded with red dye, was positioned closer to the left side of the channel. Figure 4 depicts
a plan view of the setup apparatus used for dye injection.
0
20
40
60
80
100
120
0.0065 0.007 0.0075 0.008 0.0085 0.009
water depth (mm)
Equivalent Manning n (metric)
Figure 3. Vertical variation in equivalent Manning n for the experimental ditch (runs 1 to 8).
3.2. Visual Analysis
At the toe of the hydraulic drop structure, the flow undergoes a transition to supercrit-
ical conditions, and due to the presence of the DRAB, a hydraulic jump is formed upstream
of the bends, leading to subcritical flow. Within the DRAB, the flow exhibits a generally
three-dimensional behavior with subcritical, followed by transcritical, characteristics. Ad-
ditionally, the presence of two free-surface vortex structures is evident just downstream of
the inner side of each bend forming the DRAB. Upon exiting the bends, the flow reverts to
supercritical conditions and forms a clear pattern of positive (shock) and negative standing
waves downstream of the last bend.
To visually investigate the flow structure upstream of the DRAB, two dye injection
runs were conducted. The injection location was set at a distance of 300 mm (3W) upstream
of the DRAB. Two thin needles were used for injection, with their tips positioned at the
same elevation just below the water surface. The syringe pistons of both needles were
connected to ensure equal injection speed and exerted stress. The first needle, loaded with
Water 2023,15, 3570 10 of 26
blue dye, was placed closer to the right side of the ditch, while the second needle, loaded
with red dye, was positioned closer to the left side of the channel. Figure 4depicts a plan
view of the setup apparatus used for dye injection.
Water 2023, 15, x FOR PEER REVIEW 10 of 28
Figure 3. Vertical variation in equivalent Manning n for the experimental ditch (runs 1 to 8).
3.2. Visual Analysis
At the toe of the hydraulic drop structure, the flow undergoes a transition to super-
critical conditions, and due to the presence of the DRAB, a hydraulic jump is formed up-
stream of the bends, leading to subcritical flow. Within the DRAB, the flow exhibits a gen-
erally three-dimensional behavior with subcritical, followed by transcritical, characteris-
tics. Additionally, the presence of two free-surface vortex structures is evident just down-
stream of the inner side of each bend forming the DRAB. Upon exiting the bends, the flow
reverts to supercritical conditions and forms a clear paern of positive (shock) and nega-
tive standing waves downstream of the last bend.
To visually investigate the flow structure upstream of the DRAB, two dye injection
runs were conducted. The injection location was set at a distance of 300 mm (3W) up-
stream of the DRAB. Two thin needles were used for injection, with their tips positioned
at the same elevation just below the water surface. The syringe pistons of both needles
were connected to ensure equal injection speed and exerted stress. The first needle, loaded
with blue dye, was placed closer to the right side of the ditch, while the second needle,
loaded with red dye, was positioned closer to the left side of the channel. Figure 4 depicts
a plan view of the setup apparatus used for dye injection.
0
20
40
60
80
100
120
0.0065 0.007 0.0075 0.008 0.0085 0.009
water depth (mm)
Equivalent Manning n (metric)
Figure 4. Plan view of the setup apparatus for dye injection (run 12).
In the first dye-injection run, only the blue dye was injected into the right needle at
an average injection speed of about 1.5–2 times the flow speed. Figure 5A–H display the
spreading of the dye from the injection point over time. Notably, the injected dye near the
right side of the channel experiences slight upward vertical drift.
Water 2023, 15, x FOR PEER REVIEW 11 of 28
Figure 4. Plan view of the setup apparatus for dye injection (run 12).
In the first dye-injection run, only the blue dye was injected into the right needle at
an average injection speed of about 1.5–2 times the flow speed. Figure 5A–H display the
spreading of the dye from the injection point over time. Notably, the injected dye near the
right side of the channel experiences slight upward vertical drift.
Figure 5. Side views of the dye injection from right (inner) needle with time course (run 12) ((A–D)
are from camera 1 whereas (E–H) are from camera 3).
In the second dye-injection run, both blue and red dyes were injected simultaneously
from the right and left needles, respectively. Figure 6A–H demonstrate the spread of the
dyes away from both needles over time. It is evident that the injected red dye (from the
left side) experiences vertical downward drift, while the injected blue dye (from the right
side) does not.
Figure 5.
Side views of the dye injection from right (inner) needle with time course (run 12) ((
A
–
D
) are
from camera 1 whereas (E–H) are from camera 3).
Water 2023,15, 3570 11 of 26
In the second dye-injection run, both blue and red dyes were injected simultaneously
from the right and left needles, respectively. Figure 6A–H demonstrate the spread of the
dyes away from both needles over time. It is evident that the injected red dye (from the left
side) experiences vertical downward drift, while the injected blue dye (from the right side)
does not.
Water 2023, 15, x FOR PEER REVIEW 11 of 28
Figure 4. Plan view of the setup apparatus for dye injection (run 12).
In the first dye-injection run, only the blue dye was injected into the right needle at
an average injection speed of about 1.5–2 times the flow speed. Figure 5A–H display the
spreading of the dye from the injection point over time. Notably, the injected dye near the
right side of the channel experiences slight upward vertical drift.
Figure 5. Side views of the dye injection from right (inner) needle with time course (run 12) ((A–D)
are from camera 1 whereas (E–H) are from camera 3).
In the second dye-injection run, both blue and red dyes were injected simultaneously
from the right and left needles, respectively. Figure 6A–H demonstrate the spread of the
dyes away from both needles over time. It is evident that the injected red dye (from the
left side) experiences vertical downward drift, while the injected blue dye (from the right
side) does not.
Figure 6. Dye injection of the blue and red inks from right (inner) needle and the left (outer) needle,
respectively, with time course (run 13). (
A
–
D
) are from camera 1 whereas (
E
–
H
) are taken with
camera 3.
This phenomenon can be attributed to the centrifugal forces generated due to the
curvature of the streamlines throughout the DRAB and formation of a weak helical flow
typical of sharp bends [
39
,
40
], resulting in upward vertical velocity near the right (inner)
side and downward vertical velocity near the left (outer) side of the ditch. This helical flow
leads to the vertical downward drift of the injected red dye observed in Figure 6.
The steep 90-degree alignment of the bend results in the formation of two vortex
structures. The first free vortex structure (shown as point 5 in Figure 7A) is located along
the inner side of the bend downstream of the inner edge of the first bend (point 1). This
vortex is relatively shallow (less than 30% of the local water depth) and rotates clockwise.
In contrast, the second vortex (shown as point 6 in Figure 7A) lies just downstream of
the inner edge of the second bend along its inner side. This vortex is relatively deep and
rotates counterclockwise. Figure 7B provides a typical example of the second free-surface
vortex, offering more details about its size and depth. Based on Figure 7B, the water-surface
elevation on the left side accelerates and contributes to the angular momentum flux for
the lower part of the vortex, while the water-surface elevation on the right side seems to
decelerate and contribute to the angular momentum flux of the upper surficial part of
the vortex.
Figure 8presents the typical spatial variation in water surface (as recorded with camera
2) downstream of the second bend. It is clear to notice that the flow is supercritical with
strong cross waves.
Water 2023,15, 3570 12 of 26
New Figure 7A
Figure 7.
Flow structure through the DRAB: (
A
) plan view of the full DRAB (using camera 5), (
B
) side
view of the second free-surface vortex (taken with camera 4).
Water 2023,15, 3570 13 of 26
Water 2023, 15, x FOR PEER REVIEW 14 of 28
Figure 8. Spatial variation in supercritical water surface downstream of DRAB (using camera 2).
3.3. Spatial Variations in Water Surface
In Figure 9, we present longitudinal water-surface profiles obtained during runs 16,
17, and 18 in the upstream section of the DRA bends. These profiles were recorded with
camera 1. As expected, a reduction in water flow rate leads to a corresponding decrease
in both the length of the hydraulic jump and the upstream water elevation just before the
DRA bends. It is noteworthy that all experimental runs consistently exhibited the for-
mation of classic hydraulic jumps in the upstream reach (i.e., choked flow conditions),
with no instances of oblique jumps observed. Further insights regarding this observation
will be elaborated upon in the forthcoming “Limitations and Outlook” section.
Figure 8. Spatial variation in supercritical water surface downstream of DRAB (using camera 2).
3.3. Spatial Variations in Water Surface
In Figure 9, we present longitudinal water-surface profiles obtained during runs 16,
17, and 18 in the upstream section of the DRA bends. These profiles were recorded with
camera 1. As expected, a reduction in water flow rate leads to a corresponding decrease in
both the length of the hydraulic jump and the upstream water elevation just before the DRA
bends. It is noteworthy that all experimental runs consistently exhibited the formation
of classic hydraulic jumps in the upstream reach (i.e., choked flow conditions), with no
instances of oblique jumps observed. Further insights regarding this observation will be
elaborated upon in the forthcoming “Limitations and Outlook” section.
Water 2023, 15, x FOR PEER REVIEW 15 of 28
Figure 9. Water-surface profile upstream of the DRAB (as recorded with camera 1) for runs 16, 17
and 18, and water-surface profile along the DRAB (as recorded with camera 3) for runs 22, 23 and
25.
Figure 9. Cont.
Water 2023,15, 3570 14 of 26
Water 2023, 15, x FOR PEER REVIEW 15 of 28
Figure 9. Water-surface profile upstream of the DRAB (as recorded with camera 1) for runs 16, 17
and 18, and water-surface profile along the DRAB (as recorded with camera 3) for runs 22, 23 and
25.
Figure 9.
Water-surface profile upstream of the DRAB (as recorded with camera 1) for runs 16, 17 and
18, and water-surface profile along the DRAB (as recorded with camera 3) for runs 22, 23 and 25.
Figure 9also offers a visual representation of time-averaged spatial variations in the water
surface, as observed along the left and right sides of the DRA bends. These observations were
captured with camera 3 during runs 22, 23, and 25. Within the DRA bends, a consistent pattern
emerges, with the water surface on the left side being conspicuously elevated compared to
the right side. Furthermore, the flow dynamics near the left side of the DRA bends reveal
post-inner-edge acceleration at x = 0, while the right-side experiences localized acceleration
both before and within the region of the first free vortex, followed by a subsequent mild
deceleration. The longitudinal water-surface profiles downstream of the DRA bends are
skillfully recorded with camera 2 for runs 20, 21, and 24 and presented in Figure 10. The
spatial variations in the water-surface profiles unequivocally display the presence of cross
waves, and it is intriguing to note that as the flow rate decreases, the amplitudes and phase
shifts between the right and left sides of these cross waves exhibit a diminishing trend.
Water 2023, 15, x FOR PEER REVIEW 16 of 28
Figure 9 also offers a visual representation of time-averaged spatial variations in the
water surface, as observed along the left and right sides of the DRA bends. These obser-
vations were captured with camera 3 during runs 22, 23, and 25. Within the DRA bends,
a consistent paern emerges, with the water surface on the left side being conspicuously
elevated compared to the right side. Furthermore, the flow dynamics near the left side of
the DRA bends reveal post-inner-edge acceleration at x = 0, while the right-side experi-
ences localized acceleration both before and within the region of the first free vortex, fol-
lowed by a subsequent mild deceleration. The longitudinal water-surface profiles down-
stream of the DRA bends are skillfully recorded with camera 2 for runs 20, 21, and 24 and
presented in Figure 10. The spatial variations in the water-surface profiles unequivocally
display the presence of cross waves, and it is intriguing to note that as the flow rate de-
creases, the amplitudes and phase shifts between the right and left sides of these cross
waves exhibit a diminishing trend.
Figure 10. Water-surface profile downstream of the second turn of the DRAB (as recorded with
camera 2): (A) run 20; (B) run 21; (C) run 24.
3.4. Water Oscillation Analysis
3.4.1. Time Series Analysis
The dominant frequency of water-surface oscillations upstream of the DRAB was de-
termined through spectral analysis applied to the time series data obtained from Tracker.
Illustrations of the acquired time series data are presented in Figure 11. Subsequently, the
Figure 10. Cont.
Water 2023,15, 3570 15 of 26
Water 2023, 15, x FOR PEER REVIEW 16 of 28
Figure 9 also offers a visual representation of time-averaged spatial variations in the
water surface, as observed along the left and right sides of the DRA bends. These obser-
vations were captured with camera 3 during runs 22, 23, and 25. Within the DRA bends,
a consistent paern emerges, with the water surface on the left side being conspicuously
elevated compared to the right side. Furthermore, the flow dynamics near the left side of
the DRA bends reveal post-inner-edge acceleration at x = 0, while the right-side experi-
ences localized acceleration both before and within the region of the first free vortex, fol-
lowed by a subsequent mild deceleration. The longitudinal water-surface profiles down-
stream of the DRA bends are skillfully recorded with camera 2 for runs 20, 21, and 24 and
presented in Figure 10. The spatial variations in the water-surface profiles unequivocally
display the presence of cross waves, and it is intriguing to note that as the flow rate de-
creases, the amplitudes and phase shifts between the right and left sides of these cross
waves exhibit a diminishing trend.
Figure 10. Water-surface profile downstream of the second turn of the DRAB (as recorded with
camera 2): (A) run 20; (B) run 21; (C) run 24.
3.4. Water Oscillation Analysis
3.4.1. Time Series Analysis
The dominant frequency of water-surface oscillations upstream of the DRAB was de-
termined through spectral analysis applied to the time series data obtained from Tracker.
Illustrations of the acquired time series data are presented in Figure 11. Subsequently, the
Figure 10.
Water-surface profile downstream of the second turn of the DRAB (as recorded with
camera 2): (A) run 20; (B) run 21; (C) run 24.
3.4. Water Oscillation Analysis
3.4.1. Time Series Analysis
The dominant frequency of water-surface oscillations upstream of the DRAB was
determined through spectral analysis applied to the time series data obtained from Tracker.
Illustrations of the acquired time series data are presented in Figure 11. Subsequently, the
stationary nature of the water elevation time series was verified, indicating the absence of
any seasonal trends.
Water 2023, 15, x FOR PEER REVIEW 17 of 28
stationary nature of the water elevation time series was verified, indicating the absence of
any seasonal trends.
Figure 11. Time series of water surface upstream of DRAB (runs 31 to 33).
3.4.2. Power Spectral Analysis
Spectral analysis was employed on the time series data obtained from Tracker to as-
certain the dominant frequency of water-surface oscillations. To achieve this, a custom
Matlab script was developed (see Appendix A).
Figure 12 shows a sample of power spectral density (psd) curves produced by Matlab
script for runs no. 18, 26, 31, 36, 41 and 44 respectively. It is noted that all psd curves have
single-peak values that identify the dominant frequency.
Figure 11. Time series of water surface upstream of DRAB (runs 31 to 33).
3.4.2. Power Spectral Analysis
Spectral analysis was employed on the time series data obtained from Tracker to
ascertain the dominant frequency of water-surface oscillations. To achieve this, a custom
Matlab script was developed (see Appendix A).
Water 2023,15, 3570 16 of 26
Figure 12 shows a sample of power spectral density (psd) curves produced by Matlab
script for runs no. 18, 26, 31, 36, 41 and 44 respectively. It is noted that all psd curves have
single-peak values that identify the dominant frequency.
Water 2023, 15, x FOR PEER REVIEW 18 of 28
Figure 12. Sample of power spectral density curves for: (A) run 18; (B) run 26; (C) run 31; (D) run
36; (E) run 41; (F) run 44. Square symbols denote the points of dominant frequency.
In Figure 13, the dominant frequencies of water level oscillations are presented for all
the conducted runs. It is noted that the dominant frequency ranges from 1.4 Hz to 4.4 Hz
with an average of 3 Hz.
Figure 12.
Sample of power spectral density curves for: (
A
) run 18; (
B
) run 26; (
C
) run 31; (
D
) run 36;
(E) run 41; (F) run 44. Square symbols denote the points of dominant frequency.
In Figure 13, the dominant frequencies of water level oscillations are presented for all
the conducted runs. It is noted that the dominant frequency ranges from 1.4 Hz to 4.4 Hz
with an average of 3 Hz.
Water 2023,15, 3570 17 of 26
Water 2023, 15, x FOR PEER REVIEW 19 of 28
Figure 13. Dominant frequency of water level oscillation. Doed blue line represents the average
value.
3.5. Regression Frequency Formulas Based on Dimensional Analysis
In this section, we apply a dimensional analysis to develop a regression formula for
the dominant frequency of water level oscillations upstream of the DRAB. The proposed
formula establishes a relationship between the Strouhal number, representing the domi-
nant frequency, and other relevant key parameters. The Strouhal number is a dimension-
less quantity that characterizes the oscillating mechanisms by relating the characteristic
flow time to the period of oscillation [41]. It also serves to describe the ratio of inertial
forces resulting from the local acceleration of the flow to those due to convective acceler-
ation.
To investigate the factors influencing the dominant water frequency upstream of the
DRAB, we applied dimensional analysis to the following set of variables: dominant fre-
quency (f
o
); normal flow depth (y
o
); ditch width (b); normal water velocity (v
o
) through
the ditch (assuming uniform flow); channel bed slope (S
o
); channel roughness (Manning
n); tailwater depth (y
t
); kinematic viscosity of water (ν); and gravitational acceleration (g).
f
o
= f(b,y
o
,v
o
,S
o
,n,y
t
,ν,g) (1)
Since the normal depth is a function of S
o
and n, Equation (1) can be reduced to:
f
o
= g(b,y
o
,v
o
,y
t
,ν,g) (2)
Considering that the tail gate remained fully opened in all runs, this study does not
account for the impact of changing y
t
in the results. Nevertheless, it should be noted that
slight changes in the tailwater level would not significantly affect the flow oscillation up-
stream of the DRAB. This can be justified by visually inspecting the flow upstream and
downstream of the DRAB, where it is observed that in all cases, the flow far upstream is
supercritical and transitions to subcritical after the formation of the hydraulic jump up-
stream of the DRAB. Within the DRAB, the flow starts as subcritical, then becomes trans-
critical and supercritical again just downstream of the second bend. Consequently, Equa-
tion (2) can be further reduced to:
f
o
= h(b,y
o
,v
o
,ν,g) (3)
The next step involves formulating the relevant π-terms. Following a dimensional
analysis approach, we obtained the following equation that relates the main dependent
Strouhal number to the following π-terms:
Figure 13.
Dominant frequency of water level oscillation. Dotted blue line represents the aver-
age value.
3.5. Regression Frequency Formulas Based on Dimensional Analysis
In this section, we apply a dimensional analysis to develop a regression formula for
the dominant frequency of water level oscillations upstream of the DRAB. The proposed
formula establishes a relationship between the Strouhal number, representing the dominant
frequency, and other relevant key parameters. The Strouhal number is a dimensionless
quantity that characterizes the oscillating mechanisms by relating the characteristic flow
time to the period of oscillation [
41
]. It also serves to describe the ratio of inertial forces
resulting from the local acceleration of the flow to those due to convective acceleration.
To investigate the factors influencing the dominant water frequency upstream of
the DRAB, we applied dimensional analysis to the following set of variables: dominant
frequency (f
o
); normal flow depth (y
o
); ditch width (b); normal water velocity (v
o
) through
the ditch (assuming uniform flow); channel bed slope (S
o
); channel roughness (Manning n);
tailwater depth (yt); kinematic viscosity of water (ν); and gravitational acceleration (g).
fo= f(b,yo,vo,So,n,yt,ν,g) (1)
Since the normal depth is a function of Soand n, Equation (1) can be reduced to:
fo= g(b,yo,vo,yt,ν,g) (2)
Considering that the tail gate remained fully opened in all runs, this study does not
account for the impact of changing y
t
in the results. Nevertheless, it should be noted
that slight changes in the tailwater level would not significantly affect the flow oscillation
upstream of the DRAB. This can be justified by visually inspecting the flow upstream and
downstream of the DRAB, where it is observed that in all cases, the flow far upstream
is supercritical and transitions to subcritical after the formation of the hydraulic jump
upstream of the DRAB. Within the DRAB, the flow starts as subcritical, then becomes
trans-critical and supercritical again just downstream of the second bend. Consequently,
Equation (2) can be further reduced to:
fo= h(b,yo,vo,ν,g) (3)
The next step involves formulating the relevant
π
-terms. Following a dimensional
analysis approach, we obtained the following equation that relates the main dependent
Strouhal number to the following π-terms:
St =f(Fn, Rn, b/Ro)(4)
Water 2023,15, 3570 18 of 26
where St represents the Strouhal number for the oscillated water depth upstream of the
DRAB. It should be noted that there may exist different formulations for the Strouhal
number, and all these formulations will be examined in this study to identify the most
relevant one based on measured data. Examples of different St formulations include:
St =fo.yo
vo
or,St =fo.b
vo
or,St =fo.Ro
vo
or,St =fo.yo
√gyo
or,St =fo.vo
2g
Fnis the Froude no., which could be represented for a rectangular channel as:
Fn=vo
√gyo
Rnis the Reynolds no., which could be represented for a rectangular channel as:
Rn=vo.Ro
ν, where Rois the normal hydraulic radius for a rectangular channel.
Based on the dimensional analysis and the generated
π
-terms, the following regression
formulas could be proposed for further examinations:
fo.b
vo
=0.9586.(φ1)0.9819 and φ1=c1.(Rn)c2.(Fn)c3(5)
fo.Ro
vo
=1.3354.(φ2)1.1111 and φ2=c1.(Rn)c2.(Fn)c3(6)
fo.yo
√gyo
=1.0782.(φ3)1.059 and φ3=c1.(Rn)c2.(Fn)c3(7)
fo.yo
vo
=1.1965.(φ4)1.1031 and φ4=c1.(Rn)c2.(Fn)c3(8)
fo.yo
vo
=1.112.(φ5)1.0656 and φ5=c1.b
Roc2
.(Fn)c3(9)
where F1to F5are regression functions and c1, c2, and c3 are regression coefficients.
4. Results and Discussion
This section presents the outcomes of the experimental runs and provides an in-depth
analysis of the key findings.
4.1. Oscillation of Water Level Upstream DRAB
Figure 14A illustrates the predominant frequencies observed in the water-surface
elevation upstream of the DRAB. The analysis reveals that these dominant frequencies span
a range of 1.4 to 4.4 Hz, with an average value of approximately 3 Hz. Dashed and dotted
lines on the graph delineate the measured frequency ranges corresponding to different flow
conditions, specifically oscillatory/stable/strong jumps and undular jumps, as reported in
references [28] and [27], respectively.
Notably, our findings indicate a marginal increase in the measured frequencies up-
stream of the DRAB when compared to the measurements obtained at the end of the
jump roller as reported in prior studies. This discrepancy might suggest the presence of
dual sources contributing to the water-surface oscillations upstream of the DRAB. The
primary source is attributed to the water-surface oscillations of hydraulic jump formation
upstream of the DRAB, while a secondary source stems from the oscillations generated at
the channel’s asymmetric contraction inlet and the additional instability induced by the
sharp upstream bend of the DRAB, leading to the development of a secondary spiral flow.
Water 2023,15, 3570 19 of 26
Water 2023, 15, x FOR PEER REVIEW 21 of 28
Figure 14. (A) Dominant frequency of water fluctuation upstream of DRAB. (B) St. no. as a function
of Froude No. (C) St. no as a function of Reynolds No.
Notably, our findings indicate a marginal increase in the measured frequencies up-
stream of the DRAB when compared to the measurements obtained at the end of the jump
roller as reported in prior studies. This discrepancy might suggest the presence of dual
sources contributing to the water-surface oscillations upstream of the DRAB. The primary
source is aributed to the water-surface oscillations of hydraulic jump formation up-
stream of the DRAB, while a secondary source stems from the oscillations generated at
Figure 14.
(
A
) Dominant frequency of water fluctuation upstream of DRAB. (
B
) St. no. as a function
of Froude No. (C) St. no as a function of Reynolds No.
Figure 14B,C depict the relationship between the Strouhal number (St) on the y-axis
and the Froude number (F
n
) and the Reynolds number (R
n
) on the x-axis, respectively.
Figure 14B reveals a substantial influence of the Froude number on the Strouhal number,
where an increase in F
n
leads to a decrease in St. On the other hand, Figure 14C illustrates
a weak direct proportional relation between R
n
and St. The Strouhal number values for
Water 2023,15, 3570 20 of 26
the conducted experimental runs range from a minimum of 0.03 (at higher Froude number
flows) to a maximum slightly higher than 0.3 (near critical conditions).
To determine the values of the regression coefficients and identify the formula that
best fits the data, a least square error analysis was performed using the GRG nonlinear
gradient Excel solver. The objective was to minimize the summation of squared differences
between the measured data and the proposed regression formulas. Table 3presents the
regression coefficients of Equations (5) to (9) along with their corresponding R
2
values.
Notably, regression equation no. 9 (
Φ5
) demonstrates the highest correlation with the
measured data, as depicted in Figure 15.
Table 3. Coefficients of regression equations for Strouhal no.
Regression
Formula Function Equation
No
Coefficients
R2
c1c2c3
1Φ15 42.22809 −0.4635 −0.71255 0.784
2Φ26 0.081352 0.026588 −1.08519 0.743
3Φ37 0.003221 0.441993 −0.66306 0.74
4Φ48 0.003678 0.424442 −1.54091 0.897
5Φ59 0.777935 0.85343 1.13919 0.9001
Water 2023, 15, x FOR PEER REVIEW 22 of 28
the channel’s asymmetric contraction inlet and the additional instability induced by the
sharp upstream bend of the DRAB, leading to the development of a secondary spiral flow.
Figure 14B,C depict the relationship between the Strouhal number (St) on the y-axis
and the Froude number (F
n
) and the Reynolds number (R
n
) on the x-axis, respectively.
Figure 14B reveals a substantial influence of the Froude number on the Strouhal number,
where an increase in F
n
leads to a decrease in St. On the other hand, Figure 14C illustrates
a weak direct proportional relation between R
n
and St. The Strouhal number values for
the conducted experimental runs range from a minimum of 0.03 (at higher Froude number
flows) to a maximum slightly higher than 0.3 (near critical conditions).
To determine the values of the regression coefficients and identify the formula that
best fits the data, a least square error analysis was performed using the GRG nonlinear
gradient Excel solver. The objective was to minimize the summation of squared differ-
ences between the measured data and the proposed regression formulas. Table 3 presents
the regression coefficients of Equations (5) to (9) along with their corresponding R
2
values.
Notably, regression equation no. 9 (Φ
5
) demonstrates the highest correlation with the
measured data, as depicted in Figure 15.
Table 3. Coefficients of regression equations for Strouhal no.
Regression
Formula Function Equation No Coefficients R
2
c
1
c
2
c
3
1 Φ
1
5 42.22809 −0.4635 −0.71255 0.784
2 Φ
2
6 0.081352 0.026588 −1.08519 0.743
3 Φ
3
7 0.003221 0.441993 −0.66306 0.74
4 Φ
4
8 0.003678 0.424442 −1.54091 0.897
5 Φ
5
9 0.777935 0.85343 1.13919 0.9001
Figure 15. Best-fit regression equation for the Strouhal number upstream of DRA bends.
4.2. Water Depth Upstream of DRAB
Figure 16 plots the maximum nondimensional water depth (y
DRAm
/y
o
) upstream of
the DRAB versus the far upstream Froude no. (F
n
). The wide dashed green line refers to
the Knapp and Ippen 1939 model; the doed black line refers to the Grashof formula
[42,43] for subcritical flow, which was found to match with our study conditions due to
the formation of the hydraulic jump upstream of DRA bends. The solid continuous blue
line represents the subsequent depth relationship between supercritical and subcritical
flow in the classical hydraulic jump and the thin continuous red line refers to the best-fit
trendline of the DRA bend data measurements. It should be noted that both the Knapp
and Ippen model and the Grashof formula are calculated for 90-degree sharp bends
Figure 15. Best-fit regression equation for the Strouhal number upstream of DRA bends.
4.2. Water Depth Upstream of DRAB
Figure 16 plots the maximum nondimensional water depth (y
DRAm
/y
o
) upstream of
the DRAB versus the far upstream Froude no. (F
n
). The wide dashed green line refers to the
Knapp and Ippen 1939 model; the dotted black line refers to the Grashof formula [
42
,
43
]
for subcritical flow, which was found to match with our study conditions due to the
formation of the hydraulic jump upstream of DRA bends. The solid continuous blue
line represents the subsequent depth relationship between supercritical and subcritical
flow in the classical hydraulic jump and the thin continuous red line refers to the best-fit
trendline of the DRA bend data measurements. It should be noted that both the Knapp and
Ippen model and the Grashof formula are calculated for 90-degree sharp bends assuming a
relative curvature (b/2r = 1) (where b is the bend width and r is the average bend radius of
curvature). It is interesting to note that both the Knapp and Ippen model and the Grashof
formula underestimated the DRA bend measurements and this is justified because our case
study represents an ideally sharp bend with relative curvature of infinity (i.e., r = 0 and
b/2r = ∞).
Water 2023,15, 3570 21 of 26
Water 2023, 15, x FOR PEER REVIEW 23 of 28
assuming a relative curvature (b/2r = 1) (where b is the bend width and r is the average
bend radius of curvature). It is interesting to note that both the Knapp and Ippen model
and the Grashof formula underestimated the DRA bend measurements and this is justi-
fied because our case study represents an ideally sharp bend with relative curvature of
infinity (i.e., r = 0 and b/2r = ∞).
The best-fit trendline for data measurements of the maximum dimensionless water
depth upstream of DRA bends is given in Equation (10).
= 1.7558𝑒.
(R
2
= 0.91) (10)
Figure 16. Maximum nondimensional water depth (normalized by y
o
) upstream of DRABs.
Figure 17 presents the measurements of the maximum water depth (upstream of
DRABs) normalized by the critical depth (y
DRAm
/y
c
) and for different Froude numbers. It is
interesting to note that (y
DRAm
/y
c
) is almost constant and equals 2.5. In other words, the
maximum water depth is found to be approximately 2.5 times the critical depth for all the
runs and for different Froude numbers.
Figure 16. Maximum nondimensional water depth (normalized by yo) upstream of DRABs.
The best-fit trendline for data measurements of the maximum dimensionless water
depth upstream of DRA bends is given in Equation (10).
yDR Am
yo
=1.7558e0.3916Fn(R2=0.91)(10)
Figure 17 presents the measurements of the maximum water depth (upstream of
DRABs) normalized by the critical depth (y
DRAm
/y
c
) and for different Froude numbers. It
is interesting to note that (y
DRAm
/y
c
) is almost constant and equals 2.5. In other words, the
maximum water depth is found to be approximately 2.5 times the critical depth for all the
runs and for different Froude numbers.
Water 2023, 15, x FOR PEER REVIEW 23 of 28
assuming a relative curvature (b/2r = 1) (where b is the bend width and r is the average
bend radius of curvature). It is interesting to note that both the Knapp and Ippen model
and the Grashof formula underestimated the DRA bend measurements and this is justi-
fied because our case study represents an ideally sharp bend with relative curvature of
infinity (i.e., r = 0 and b/2r = ∞).
The best-fit trendline for data measurements of the maximum dimensionless water
depth upstream of DRA bends is given in Equation (10).
= 1.7558𝑒.
(R
2
= 0.91) (10)
Figure 16. Maximum nondimensional water depth (normalized by y
o
) upstream of DRABs.
Figure 17 presents the measurements of the maximum water depth (upstream of
DRABs) normalized by the critical depth (y
DRAm
/y
c
) and for different Froude numbers. It is
interesting to note that (y
DRAm
/y
c
) is almost constant and equals 2.5. In other words, the
maximum water depth is found to be approximately 2.5 times the critical depth for all the
runs and for different Froude numbers.
Figure 17.
Maximum nondimensional water depth (normalized by y
c
) upstream of DRABs. The
doted red line represents the average value of the measurements.
Water 2023,15, 3570 22 of 26
4.3. Limitations and Outlook
The findings of this study provide crucial insights into the calculations of freeboard
for channels with DRABs. However, certain limitations should be acknowledged when
applying these findings:
•
The experimental runs were conducted for a specific geometry of the DRAB, where the
spacing length between the two bends is three times its width (L/W = 3). It is important
to note that altering the L/W ratio could lead to the occurrence of “trans-critical
flow conversion” earlier within the distance between the two bends. Consequently,
supercritical cross waves might be present not only in the downstream reach but also
within the gap between the two bends.
•
Another limitation is that all the runs were conducted with a channel bed bend–width
ratio (b/W) equal to unity. By reducing the b/W ratio, oblique hydraulic jumps may
potentially form inside the gap between the two bends, contrary to the observations
made in this study. The influence of different L/W or b/W ratios warrants further
investigation in future work.
•
The study focused solely on DRABs in channels with a rectangular section. However,
in trapezoidal channels, the local flow regime can exhibit a mixture of subcritical
and supercritical flow at the same cross-section, introducing additional complexities,
three-dimensionality, and water depth oscillations. The investigation of DRABs in
trapezoidal channels presents a challenging yet important task for future research.
•
The experimental runs were limited to approaching flows categorized as low super-
critical flow conditions (1.03
≤
F
n≤
2.63). In cases of very high supercritical flow
conditions (F
n≥
6), the formation of humped non-stationary waves with sustained
supercritical flow throughout the bends, without the formation of hydraulic jumps,
might be expected.
•
Furthermore, the present analysis did not consider the effect of an erodible channel
with a movable bed, which could be a significant aspect for future exploration.
Notwithstanding these limitations, this work represents a crucial step in understand-
ing supercritical flows through bends. The findings serve as a foundation for future
investigations, not only for estimating suitable freeboards in similar applications but also
for assessing the accuracy of current computational fluid dynamics (CFD) models in cap-
turing the dynamics of water-surface oscillations at bends and hydraulic structures. The
research presented here contributes valuable knowledge to the field and sets the stage for
further advancements in the study of supercritical flow phenomena in channel bends.
5. Conclusions
This experimental study investigated the influence of having a double-right-angled
bend (DRAB) on supercritical free-surface flow in a rectangular channel. The investigation
involved visual examinations through dye injection, video analysis using the Tracker
package, and power spectral analysis to determine the dominant frequency of water-surface
fluctuations and the maximum water depth upstream of the DRAB.
The research findings have revealed several important observations:
•
The flow through the DRAB is characterized by high complexity and three-dimensionality.
The approaching supercritical flow in the upstream reach undergoes a conversion to
subcritical flow through the formation of a hydraulic jump upstream of the DRAB. This
hydraulic jump was consistently observed in all experimental runs
(1.03 ≤Fn≤2.63).
•
The dye injection experiments provided valuable insights, showing the formation of
a secondary anticlockwise swirl flow just upstream of the DRAB. This flow pattern
contributes to water set-up (superelevation) along the left (outer) side of the water
surface compared to the right (inner) side of the upstream reach.
•
A non-intrusive video tracking system via a set of 5 cameras was used to record the
spatial and temporal variations in water surface upstream, within, and downstream of
the DRAB.
Water 2023,15, 3570 23 of 26
The recorded maximum nondimensional water depths (y
DRAm
/y
o
) upstream of a
DRAB are consistently underestimated by both the Grashof formula and Knapp and Ippen
1939 model because the DRAB bends are perfectly sharp turns (r = 0, b/2r =
∞
). Therefore,
a new empirical equation (Equation (10)) was proposed for the case of perfectly sharp
DRABs. This study found that, regardless of Froude number, the maximum water depth
upstream of the DRAB was approximately 2.5 times the critical depth for all runs. This
is an interesting finding that warrants further investigation, especially for DRABs with
different geometries.
•
Spectral analysis was used to identify the dominant frequencies of water-surface
fluctuations upstream of the DRAB. It is noted that the dominant frequencies span a
range of 1.4 to 4.4 Hz (with an average of 3 Hz). This range is slightly higher than the
recorded values (by previous researchers) at the end of different hydraulic jump types.
The marginal increase in the measured frequencies upstream of the DRAB suggests
the presence of dual sources contributing to the water-surface oscillations at the DRAB.
The primary source is the hydraulic jump, while the secondary source probably stems
from the additional instability induced by the secondary spiral flow that is developed
by the action of the centrifugal force just upstream of the first bend and the crosswaves
generated far upstream at the asymmetric contraction of the channel inlet.
•
Due to the sharp 90-degree bends in the DRAB, two distinct free vortex structures
were observed. The first vortex is shallow in depth, rotates clockwise, and exists along
the inner side of the junction and downstream of the upstream inner edge. The second
vortex is comparatively deeper, rotates anticlockwise, and lies just downstream of the
downstream inner edge along the inner side of the second bend. As the flow progresses
through the DRAB, the subcritical flow is influenced by the formation of these two
free vortices, resulting in a transition to trans-critical flow and eventually supercritical
flow, with pronounced cross waves at the junction outlet in the downstream reach.
•
The Strouhal number corresponding to the water-surface oscillations upstream of the
DRAB is found to be strongly dependent on the Froude number and weakly dependent
on the Reynolds number. A decrease in the supercritical Froude number leads to an
increase in the Strouhal number, indicating that the highest water-surface oscillations
are associated with critical flow conditions.
•
The recorded water surface and dominant frequencies data set for the DRAB prob-
lem could be used for calibration and verification of CFD models. These data not
only enable a rigorous comparison between the CFD model’s predictions and mea-
sured time-averaged values, but they could also provide a new basis for a higher
level of model calibrations in which the measurements of the dominant frequency of
fluctuations are compared against CFD model outputs.
In summary, this study has provided valuable insights into the behavior of supercritical
flow in a rectangular channel with a double-right-angled bends. The findings contribute
to a better understanding of flow characteristics and oscillations, which are essential for
hydraulic design and management of such systems.
Author Contributions:
Conceptualization, M.E. and M.F.; methodology, M.E., M.F. and L.C.; soft-
ware, M.E.; validation, M.E. and M.F.; formal analysis, M.E. and M.F.; investigation, M.E., L.C. and
M.F.; resources, L.C.; data curation, M.E.; writing—original draft preparation, M.E. and A.M.H.;
writing—review and editing, M.E. and M.F.; visualization, M.E. and L.C.; supervision, M.E.; project
administration, M.E.; funding acquisition, M.E. All authors have read and agreed to the published
version of the manuscript.
Funding:
This research was supported and funded by the Deanship of Scientific Research, Imam
Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, Grant No. (221414007).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data are available on request from the authors.
Water 2023,15, 3570 24 of 26
Acknowledgments:
The authors extend their appreciation to the Deanship of Scientific Research,
Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research
work through Grant No. (221414007).
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
bchannel bed width [L]
c1,c2,c3coefficients of regression Equations (5)–(9), Table 3
CFD computational fluid dynamics
CHU conveyance heading up upstream DRAB [L]
DRAB double-right-angled bend
fo dominant frequency [1/T]
fps number of recorded frames per second [1/T]
Fn Froude number [1]
gthe acceleration of gravity (L/T2)
Llength of DRAB as per Figure 1d
n Manning roughness coefficient [T/L1/3]
O.S.P. open-source physics project
Qwater flow rate [L3/T]
Ro hydraulic radius for normal water depth [L]
Rn Reynolds number [1]
R2coefficient of determination for a given regression [1]
So channel bed slope [1]
St Strouhal number [1]
Vocross-sectional averaged velocity assuming uniform flow (L/T)
Wwidth of the DRAB, Figure 1d
WSO water-surface oscillation
WSP water-surface profile
yonormal depth [L]
yttail water depth [L]
yDR A time averaged water depth upstream DRAB [L]
yDR Ammaximum instantaneous water depth upstream DRAB [L]
νkinematic viscosity of water (L2/T)
Φ1to Φ5regression functions, Equations (5)–(9), Table 3
Appendix A. MATLAB Code
% remove the mean and trends (if any) from time series data using detrend function
y = detrend(X);
fs = 30;
% using pwelch function from signal processing toolbox to carry out power
% spectral analysis
[Syy,f] = pwelch(y,[],[],[],fs);
loglog(f,Syy)
xlabel(‘f (Hz)’),ylabel(‘Syy(mmˆ2/Hz’)
xlabel(‘f (Hz)’),ylabel(‘Syy(mmˆ2/Hz)’)
grid
% identifying the dominant frequency corresponding to maximum density
Syymax = max(Syy);
K = find(Syy == Syymax);
K5 = find(f > 5&f < 5.1);
K10 = find(f > 10&f < 10.1);
Kfull = length(f);
Fo = f(K);
logSyy = log10(Syy(K:Kfull,1));
logf = log10(f(K:Kfull,1));
Water 2023,15, 3570 25 of 26
logSyy5 = log10(Syy(K:K5,1));
logf5 = log10(f(K:K5,1));
logSyy10 = log10(Syy(K:K10,1));
logf10 = log10(f(K:K10,1));
b_full = polyfit(logf, logSyy, 1);
Slp_full = b_full(1);
b_5 = polyfit(logf5, logSyy5, 1);
Slp_5 = b_5(1);
b_10 = polyfit(logf10, logSyy10, 1);
Slp_10 = b_10(1);
mdl_full = fitlm(logf,logSyy);
RSQ_full = mdl_full.Rsquared.Ordinary;
mdl_5 = fitlm(logf5,logSyy5);
RSQ_5 = mdl_5.Rsquared.Ordinary;
mdl_10 = fitlm(logf10,logSyy10);
RSQ_10 = mdl_10.Rsquared.Ordinary;
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