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Article

Analysis of Water-Surface Oscillations Upstream of a Double-Right-Angled Bend with Incoming Supercritical Flow

1
Civil Engineering Department, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318, Saudi Arabia
2
Faculty of Engineering, Ain Shams University, Cairo 11517, Egypt
3
Irrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt
*
Author to whom correspondence should be addressed.
Water 2023, 15(20), 3570; https://doi.org/10.3390/w15203570
Submission received: 28 July 2023 / Revised: 8 October 2023 / Accepted: 9 October 2023 / Published: 12 October 2023

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 experiments 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 hydraulic 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.

1. Introduction

High-velocity channels featuring steep gradients play a pivotal role in drainage systems, 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 disposal 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.
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 [1,2,3,4]. 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 supercritical 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 [7,8,9,10] 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 [11,12,13]. 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 increased 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 effort has been dedicated to understanding free-surface fluctuations in hydraulic jumps and the oscillatory behavior of these hydraulic phenomena [23,24,25,26,27,28]. 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) configuration 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 2 introduces 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 4 delves into the results, encompassing water oscillations upstream of the DRAB and maximum water depth. Furthermore, Section 4 discusses the challenges and limitations inherent to the current work. Lastly, Section 5 summarizes 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.

2.2. Experimental Runs

The main variables considered for experimental investigation were the flume bed slope and the water flow discharge. Table 1 provides the characteristics of the 48 experimental runs that have a Reynolds no. (Rn = Vo.Rh/n) ranging from 4622 to 55,513 and a Froude no. (Fn = Vo/ g y o ) 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.
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 supercritical 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.

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 2 provides the main characteristics of all the cameras used in the study [29,30,31].

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 software. 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 [32,33,34].
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 smartphones. 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 2 illustrates a typical example of auto-tracking the water surface just upstream of the DRAB using the Tracker package.

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 3 illustrates 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.

3.2. Visual Analysis

At the toe of the hydraulic drop structure, the flow undergoes a transition to supercritical 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. Additionally, 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 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.
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.
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.
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 8 presents 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.

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.
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 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.

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.

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).
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.
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.

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 (fo); normal flow depth (yo); ditch width (b); normal water velocity (vo) through the ditch (assuming uniform flow); channel bed slope (So); 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)
Since the normal depth is a function of So and n, Equation (1) can be reduced to:
fo = g(b,yo,vo,yt,ν,g)
Considering that the tail gate remained fully opened in all runs, this study does not account for the impact of changing yt 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)
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:
S t = f ( F n   ,   R n , b / R o )
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:
S t = f o . y o v o   o r ,   S t = f o . b v o   o r ,   S t = f o . R o v o   o r ,   S t = f o . y o g y o   o r ,   S t = f o . v o 2 g
Fn is the Froude no., which could be represented for a rectangular channel as:
F n = v o g y o
Rn is the Reynolds no., which could be represented for a rectangular channel as:
R n = v o . R o ν , where Ro is 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:
f o . b v o = 0.9586 . ϕ 1 0.9819   and   ϕ 1 = c 1 . R n c 2 . F n c 3
f o . R o v o = 1.3354 . ϕ 2 1.1111   and   ϕ 2 = c 1 . R n c 2 . F n c 3
f o . y o g y o = 1.0782 . ϕ 3 1.059   and   ϕ 3 = c 1 . R n c 2 . F n c 3
f o . y o v o = 1.1965 . ϕ 4 1.1031   and   ϕ 4 = c 1 . R n c 2 . F n c 3
f o . y o v o = 1.112 . ϕ 5 1.0656   and   ϕ 5 = c 1 . b R o c 2 . F n c 3
where F1 to F5 are 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 upstream 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.
Figure 14B,C depict the relationship between the Strouhal number (St) on the y-axis and the Froude number (Fn) and the Reynolds number (Rn) on the x-axis, respectively. Figure 14B reveals a substantial influence of the Froude number on the Strouhal number, where an increase in Fn leads to a decrease in St. On the other hand, Figure 14C illustrates a weak direct proportional relation between Rn 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 differences between the measured data and the proposed regression formulas. Table 3 presents the regression coefficients of Equations (5) to (9) along with their corresponding R2 values. Notably, regression equation no. 9 (Φ5) demonstrates the highest correlation with the measured data, as depicted in Figure 15.

4.2. Water Depth Upstream of DRAB

Figure 16 plots the maximum nondimensional water depth (yDRAm/yo) upstream of the DRAB versus the far upstream Froude no. (Fn). 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 = ∞).
The best-fit trendline for data measurements of the maximum dimensionless water depth upstream of DRA bends is given in Equation (10).
y D R A m y o = 1.7558 e 0.3916 F n                            ( R 2   =   0.91 )
Figure 17 presents the measurements of the maximum water depth (upstream of DRABs) normalized by the critical depth (yDRAm/yc) and for different Froude numbers. It is interesting to note that (yDRAm/yc) 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.

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 supercritical flow conditions (1.03 ≤ Fn ≤ 2.63). In cases of very high supercritical flow conditions (Fn ≥ 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 understanding 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 capturing 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.
The recorded maximum nondimensional water depths (yDRAm/yo) 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 problem 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 measured 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.; software, 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.

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
CFDcomputational fluid dynamics
CHUconveyance heading up upstream DRAB [L]
DRABdouble-right-angled bend
fodominant frequency [1/T]
fpsnumber of recorded frames per second [1/T]
FnFroude 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]
Rohydraulic radius for normal water depth [L]
RnReynolds number [1]
R2coefficient of determination for a given regression [1]
Sochannel bed slope [1]
StStrouhal number [1]
Vocross-sectional averaged velocity assuming uniform flow (L/T)
Wwidth of the DRAB, Figure 1d
WSOwater-surface oscillation
WSPwater-surface profile
yonormal depth [L]
yttail water depth [L]
y D R A ¯ time averaged water depth upstream DRAB [L]
y D R A m maximum instantaneous water depth upstream DRAB [L]
νkinematic viscosity of water (L2/T)
Φ1 to 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(mm^2/Hz’)
xlabel(‘f (Hz)’),ylabel(‘Syy(mm^2/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));
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;

References

  1. Chong, N.B. Numerical simulation of supercritical flow in open channel. Comput. Methods Appl. Mech. Eng. 2006, 3, 269–289. [Google Scholar]
  2. Stockstill, R.L. Hydraulic Design of Channels Conveying Supercritical Flow. Urban Flood Damage Reduction and Channel Restoration Demonstration Program for Arid and Semi-Arid Regions. 2006. Available online: https://www.researchgate.net/publication/235103810_Hydraulic_Design_of_Channels_Conveying_Supercritical_Flow/related (accessed on 26 July 2023).
  3. Defina, A.; Viero, D.P. Open channel flow through a linear contraction. Phys. Fluids 2010, 22, 036602. [Google Scholar] [CrossRef]
  4. Viero, D.P.; Lazzarin, T.; Peruzzo, P.; Defina, A. Supercritical flow overpassing forward-or backward-facing steps non-orthogonal to the flow direction. Phys. Fluids 2023, 35, 036604. [Google Scholar] [CrossRef]
  5. Han, S.S.; Ramamurthy, A.S.; Biron, P.M. Characteristics of flow around open channel 90 bends with vanes. J. Irrig. Drain. Eng. 2011, 137, 668–676. [Google Scholar] [CrossRef]
  6. Ippen, A.T.; Knapp, R.T. A study of high-velocity flow in curved channels of rectangular cross-section. Eos Trans. Am. Geophys. Union 1936, 17, 516–521. [Google Scholar] [CrossRef]
  7. Reinauer, R.; Hager, W.H. Supercritical bend flow. J. Hydraul. Eng. 1997, 123, 208–218. [Google Scholar] [CrossRef]
  8. Reinauer, R.; Hager, W.H. Supercritical flow in chute contraction. J. Hydraul. Eng. 1998, 124, 55–64. [Google Scholar] [CrossRef]
  9. Reinauer, R.; Hager, W.H. Supercritical flow behind chute piers. J. Hydraul. Eng. 1994, 120, 1292–1308. [Google Scholar] [CrossRef]
  10. Reinauer, R.; Hager, W.H. Shockwave in air-water flows. Int. J. Multiph. Flow 1996, 22, 1255–1263. [Google Scholar] [CrossRef]
  11. Ghaeini-Hessaroeyeh, M.; Tahershamsi, A.; Namin, M.M. Numerical modelling of supercritical flow in rectangular chute bends. J. Hydraul. Res. 2011, 49, 685–688. [Google Scholar] [CrossRef]
  12. Brufau, P.; Garcia-Navarro, P. Two-dimensional dam break flow simulation. Int. J. Numer. Methods Fluids 2000, 33, 35–57. [Google Scholar] [CrossRef]
  13. Valiani, A.; Caleffi, V. Brief analysis of shallow water equations suitability to numerically simulate supercritical flow in sharp bends. J. Hydraul. Eng. 2005, 131, 912–916. [Google Scholar] [CrossRef]
  14. Souli, H.; Ahattab, J.; Agoumi, A. Investigating Supercritical Bended Flow Using Physical Model and CFD. Model. Simul. Eng. 2023, 2023, 5542589. [Google Scholar] [CrossRef]
  15. Soares Frazao, S.; Zech, Y. Dam break in channels with 90 degrees bend. J. Hydraul. Eng. 2002, 128, 956. [Google Scholar] [CrossRef]
  16. Dian-Guang, M.A.; Lei, W.; Jun-yi, Y. Water Level Fluctuation Patterns in Restricted Intermediate Navigable Channels with Different Lock Emptying Processes. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2021; Volume 826, p. 012044. [Google Scholar]
  17. Wan, Z.; Li, Y.; An, J.; Wang, X.; Cheng, L.; Liao, Y. Exploring water-level fluctuation amplitude in an approach channel under the regulation of a dual cascade hydro-plant in the dry season. Water Supply 2022, 22, 4159–4175. [Google Scholar] [CrossRef]
  18. Yang, Z.; Zhu, Y.; Ji, D.; Yang, Z.; Tan, J.; Hu, H.; Lorke, A. Discharge and water level fluctuations in response to flow regulation in impounded rivers: An analytical study. J. Hydrol. 2020, 590, 125519. [Google Scholar] [CrossRef]
  19. Ohomoto, T.; Nirakawa, R.; Watanabe, K. Interaction between Water Surface Oscillations and Large Eddies in an Open Channel with Spur Dikes. In Proceedings of the 31st IAHR Congress, Seoul, Republic of Korea, 1 January 2005; CD-Rom. Volume 1, pp. 347–348. [Google Scholar]
  20. Meile, T.; Boillat, J.L.; Schleiss, A.J. Water-surface oscillations in channels with axi-symmetric cavities. J. Hydraul. Res. 2011, 49, 73–81. [Google Scholar] [CrossRef]
  21. Zhao, K.; Cheng, N.S.; Huang, Z. Experimental study of free-surface fluctuations in open-channel flow in the presence of periodic cylinder arrays. J. Hydraul. Res. 2014, 52, 465–475. [Google Scholar] [CrossRef]
  22. Viero, D.P.; Pradella, I.; Defina, A. Free surface waves induced by vortex shedding in cylinder arrays. J. Hydraul. Res. 2017, 55, 16–26. [Google Scholar] [CrossRef]
  23. Hager, W.H. Energy Dissipators and Hydraulic Jump; Kluwer Academic Publishers, Water Science and Technology Library: Dordrecht, The Netherlands, 1982. [Google Scholar]
  24. Jesudhas, V.; Murzyn, F.; Balachandar, R. IDDES Evaluation of Oscillating Hydraulic Jumps. In E3S Web of Conferences; EDP Sciences: Paris, France, 2018; Volume 40, p. 05067. [Google Scholar]
  25. De Leo, A.; Ruffini, A.; Postacchini, M.; Colombini, M.; Stocchino, A. The effects of hydraulic jumps instability on a natural river confluence: The case study of the Chiaravagna River (Italy). Water 2020, 12, 2027. [Google Scholar] [CrossRef]
  26. Wang, H.; Chanson, H. Experimental study of turbulent fluctuations in hydraulic jumps. J. Hydraul. Eng. 2015, 141, 04015010. [Google Scholar] [CrossRef]
  27. Lennon, J.M.; Hill, D. Particle image velocity measurements of undular and hydraulic jumps. J. Hydraul. Eng. 2006, 132, 1283–1294. [Google Scholar] [CrossRef]
  28. Murzyn, F.; Chanson, H. Free-surface fluctuations in hydraulic jumps: Experimental observations. Exp. Therm. Fluid Sci. 2009, 33, 1055–1064. [Google Scholar] [CrossRef]
  29. Nikon, CoolPix P600 Digital Camera, User Manual. Available online: https://www.bhphotovideo.com/lit_files/104252.pdf (accessed on 27 January 2023).
  30. Sony Cyber-Shot DSC-RX100 Digital Camera, User Manual. Available online: https://manuals.plus/wp-content/sideloads/cyber-shot-dsc-rx100-manual-original.pdf (accessed on 27 January 2023).
  31. Microsoft LifeCam Studio WebCam, Data Sheet, Rev.1606A. 2016. Available online: https://download.microsoft.com/download/0/9/5/0952776D-7A26-40E1-80C4-76D73FC729DF/TDS_LifeCamStudio.pdf (accessed on 27 January 2023).
  32. Eadkhong, T.; Rajsadorn, R.; Jannual, P.; Danworaphong, S. Rotational dynamics with Tracker. Eur. J. Phys. 2012, 33, 615. [Google Scholar] [CrossRef]
  33. Elgamal, M.; Abdel-Mageed, N.; Helmy, A.; Ghanem, A. Hydraulic performance of sluice gate with unloaded upstream rotor. Water SA 2017, 43, 563–572. [Google Scholar] [CrossRef]
  34. Brown, D. Tracker Introduction to Video Modeling (AAPT 2010). Portland, Oregon. 2010. Available online: http://www.compadre.org/Repository/document/ServeFile.cfm?ID=10188&DocID=1749 (accessed on 24 December 2022).
  35. Brown, D. Autotracker. 2010. Available online: https://physlets.org/tracker/help/autotracker.html (accessed on 24 December 2022).
  36. Elgamal, M.; Fouli, H. Sediment removal from dam reservoirs using syphon suction action. Arab. J. Geosci. 2020, 13, 943. [Google Scholar] [CrossRef]
  37. Elgamal, M.; Kriaa, K.; Farouk, M. Drainage of a water tank with pipe outlet loaded by a passive rotor. Water 2021, 13, 1872. [Google Scholar] [CrossRef]
  38. Lazzarin, T.; Viero, D.P.; Defina, A.; Cozzolino, L. Flow under vertical sluice gates: Flow stability at large gate opening and disambiguation of partial dam-break multiple solutions. Phys. Fluids 2023, 35, 024114. [Google Scholar] [CrossRef]
  39. Blanckaert, K.D.V.H.; De Vriend, H.J. Secondary flow in sharp open-channel bends. J. Fluid Mech. 2004, 498, 353–380. [Google Scholar] [CrossRef]
  40. Lazzarin, T.; Viero, D.P. Curvature-induced secondary flow in 2D depth-averaged hydro-morphodynamic models: An assessment of different approaches and key factors. Adv. Water Resour. 2023, 171, 104355. [Google Scholar] [CrossRef]
  41. Yunus, A.C. Fluid Mechanics: Fundamentals and Applications (Si Units); Tata McGraw Hill Education Private Limited: Noida, India, 2010. [Google Scholar]
  42. Chow, V.T. Open-Channel Hydraulics, Classical Textbook Reissue; McGraw-Hill: New York, NY, USA, 1988. [Google Scholar]
  43. Liggett, J.A. Fluid Mechanics; McGraw-Hill Inc.: New York, NY, USA, 1994. [Google Scholar]
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.
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.
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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 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 vy; (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.
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 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 vy; (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.
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Figure 3. Vertical variation in equivalent Manning n for the experimental ditch (runs 1 to 8).
Figure 3. Vertical variation in equivalent Manning n for the experimental ditch (runs 1 to 8).
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Figure 4. Plan view of the setup apparatus for dye injection (run 12).
Figure 4. Plan view of the setup apparatus for dye injection (run 12).
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Figure 5. Side views of the dye injection from right (inner) needle with time course (run 12) ((AD) are from camera 1 whereas (EH) are from camera 3).
Figure 5. Side views of the dye injection from right (inner) needle with time course (run 12) ((AD) are from camera 1 whereas (EH) are from camera 3).
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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). (AD) are from camera 1 whereas (EH) are taken with camera 3.
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). (AD) are from camera 1 whereas (EH) are taken with camera 3.
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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).
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).
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Figure 8. Spatial variation in supercritical water surface downstream of DRAB (using camera 2).
Figure 8. Spatial variation in supercritical water surface downstream of DRAB (using camera 2).
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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.
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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.
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.
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Figure 11. Time series of water surface upstream of DRAB (runs 31 to 33).
Figure 11. Time series of water surface upstream of DRAB (runs 31 to 33).
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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.
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.
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Figure 13. Dominant frequency of water level oscillation. Dotted blue line represents the average value.
Figure 13. Dominant frequency of water level oscillation. Dotted blue line represents the average value.
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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 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.
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Figure 15. Best-fit regression equation for the Strouhal number upstream of DRA bends.
Figure 15. Best-fit regression equation for the Strouhal number upstream of DRA bends.
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Figure 16. Maximum nondimensional water depth (normalized by yo) upstream of DRABs.
Figure 16. Maximum nondimensional water depth (normalized by yo) upstream of DRABs.
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Figure 17. Maximum nondimensional water depth (normalized by yc) upstream of DRABs. The doted red line represents the average value of the measurements.
Figure 17. Maximum nondimensional water depth (normalized by yc) upstream of DRABs. The doted red line represents the average value of the measurements.
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Table 1. Characteristics of the conducted experimental runs.
Table 1. Characteristics of the conducted experimental runs.
Run No #Ditch SetupExperiment ScopeSlope (%)Water Flow Q (L/s)RnFnRemarks
StraightDRAB
1 R10.564622 1.56
2 R11.9412,296 1.49
3 R13.0618,088 1.45
4 R13.8921,825 1.53
5 R1525,208 1.44
6 R16.3930,028 1.48
7 R19.1737,278 1.50
8 R116.1152,219 1.53
9 R0.21.398830 1.05
10 R0.22.7815,449 1.14
11 R0.24.1720,421 1.14
12 DT25.2830,599 2.23Right injection
13 DT25.2830,599 2.23Right & Left injection
14 WSO2.512.7853,848 2.27
15 WSO2.59.4445,225 2.38
16 WSP/O2.56.6736,242 2.48
17 WSP/O2.53.8924,766 2.58
18 WSP/O2.51.3910,821 2.63
19 WSO21555,513 1.93
20 WSO211.9449,334 2.01
21 WSO210.5646,079 2.06
22 WSP28.8941,707 2.11
23 WSP27.537,593 2.16
24 WSP/O26.9435,805 2.18
25 WSP25.8331,945 2.22
26 WSP/O23.6122,777 2.30
27 WSP/O21.6712,395 2.35
28 WSO1.6710.5644,249 1.85
29 WSO1.678.6139,433 1.91
30 WSO1.677.2235,476 1.96
31 WSO1.675.8330,973 2.01
32 WSO1.674.4425,774 2.06
33 WSO1.672.2215,376 2.14
34 WSO1.2512.2244,545 1.52
35 WSO1.258.0635,707 1.63
36 WSO1.255.2827,598 1.73
37 WSO1.252.7817,645 1.82
38 WSO1.250.836642 1.85
39 WSO113.3343,735 1.3
40 WSO111.1140,159 1.35
41 WSO18.3334,632 1.43
42 WSO14.7224,690 1.55
43 WSO11.6711,610 1.66
44 WSO0.6712.537,925 1.03
45 WSO0.671034,347 1.08
46 WSO0.677.7830,354 1.14
47 WSO0.674.4422,060 1.24
48 WSO0.671.399634 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.
Table 2. Characteristics of used cameras.
Table 2. Characteristics of used cameras.
Camera No.ModelTypeSensorMax Resolution (MP)Max Frame Rate (fps)Zoom (Optical)Measured Parameter
1Nikon, CoolPix P600Bridge-DSLR styled1/2.3” BSI-CMOS1612060XWater surface upstream of bends
2Sony Cyber-Shot DSC-RX100 Point-and-shoot1” CMOS20.110003.6XSpatial variation in water surface downstream of bends
3 and 4Microsoft-Life Cam StudioWebcamCMOS5303XLateral water-surface profiles throughout the two bends
5Kiosk High Speed WebcamsWebcam1/3” CMOS226010XTop view for water features extent
Table 3. Coefficients of regression equations for Strouhal no.
Table 3. Coefficients of regression equations for Strouhal no.
Regression
Formula
FunctionEquation NoCoefficientsR2
c1c2c3
1Φ1542.22809−0.4635−0.712550.784
2Φ260.0813520.026588−1.085190.743
3Φ370.0032210.441993−0.663060.74
4Φ480.0036780.424442−1.540910.897
5Φ590.7779350.853431.139190.9001
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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

AMA Style

Elgamal M, Chaouachi L, Farouk M, Helmi AM. Analysis of Water-Surface Oscillations Upstream of a Double-Right-Angled Bend with Incoming Supercritical Flow. Water. 2023; 15(20):3570. https://doi.org/10.3390/w15203570

Chicago/Turabian Style

Elgamal, Mohamed, Lotfi Chaouachi, Mohamed Farouk, and Ahmed M. Helmi. 2023. "Analysis of Water-Surface Oscillations Upstream of a Double-Right-Angled Bend with Incoming Supercritical Flow" Water 15, no. 20: 3570. https://doi.org/10.3390/w15203570

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