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

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. Characteristics of the conducted experimental runs.

Run No #	Ditch Setup		Experiment Scope	Slope (%)	Water Flow Q (L/s)	Rn	Fn	Remarks
	Straight	DRAB
1	✓		R	1	0.56	4622	1.56
2	✓		R	1	1.94	12,296	1.49
3	✓		R	1	3.06	18,088	1.45
4	✓		R	1	3.89	21,825	1.53
5	✓		R	1	5	25,208	1.44
6	✓		R	1	6.39	30,028	1.48
7	✓		R	1	9.17	37,278	1.50
8	✓		R	1	16.11	52,219	1.53
9	✓		R	0.2	1.39	8830	1.05
10	✓		R	0.2	2.78	15,449	1.14
11	✓		R	0.2	4.17	20,421	1.14
12		✓	DT	2	5.28	30,599	2.23	Right injection
13		✓	DT	2	5.28	30,599	2.23	Right & Left injection
14		✓	WSO	2.5	12.78	53,848	2.27
15		✓	WSO	2.5	9.44	45,225	2.38
16		✓	WSP/O	2.5	6.67	36,242	2.48
17		✓	WSP/O	2.5	3.89	24,766	2.58
18		✓	WSP/O	2.5	1.39	10,821	2.63
19		✓	WSO	2	15	55,513	1.93
20		✓	WSO	2	11.94	49,334	2.01
21		✓	WSO	2	10.56	46,079	2.06
22		✓	WSP	2	8.89	41,707	2.11
23		✓	WSP	2	7.5	37,593	2.16
24		✓	WSP/O	2	6.94	35,805	2.18
25		✓	WSP	2	5.83	31,945	2.22
26		✓	WSP/O	2	3.61	22,777	2.30
27		✓	WSP/O	2	1.67	12,395	2.35
28		✓	WSO	1.67	10.56	44,249	1.85
29		✓	WSO	1.67	8.61	39,433	1.91
30		✓	WSO	1.67	7.22	35,476	1.96
31		✓	WSO	1.67	5.83	30,973	2.01
32		✓	WSO	1.67	4.44	25,774	2.06
33		✓	WSO	1.67	2.22	15,376	2.14
34		✓	WSO	1.25	12.22	44,545	1.52
35		✓	WSO	1.25	8.06	35,707	1.63
36		✓	WSO	1.25	5.28	27,598	1.73
37		✓	WSO	1.25	2.78	17,645	1.82
38		✓	WSO	1.25	0.83	6642	1.85
39		✓	WSO	1	13.33	43,735	1.3
40		✓	WSO	1	11.11	40,159	1.35
41		✓	WSO	1	8.33	34,632	1.43
42		✓	WSO	1	4.72	24,690	1.55
43		✓	WSO	1	1.67	11,610	1.66
44		✓	WSO	0.67	12.5	37,925	1.03
45		✓	WSO	0.67	10	34,347	1.08
46		✓	WSO	0.67	7.78	30,354	1.14
47		✓	WSO	0.67	4.44	22,060	1.24
48		✓	WSO	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.

provides 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/$\sqrt{g{y}_o}$) 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 < F_n < 2.63).
• The Manning n roughness varies with the water depth, and the depth-averaged value is approximately 0.008 s/m^1/3.

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. 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

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 (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 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:

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:

$$St=\mathrm{f}({\mathrm{F}}_n,{\mathrm{R}}_n,\mathrm{b}/{\mathrm{R}}_o)$$

(4)

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:

$$St=\frac{f_o.y_o}{v_o}or,St=\frac{f_o.b}{v_o}or,St=\frac{f_o.R_o}{v_o}or,St=\frac{f_o.y_o}{\sqrt{g{y}_o}}or,St=\frac{f_o.v_o}{2g}$$

F_n is the Froude no., which could be represented for a rectangular channel as:

$$F_n=\left(\frac{v_o}{\sqrt{g{y}_o}}\right)$$

R_n is the Reynolds no., which could be represented for a rectangular channel as:

${\mathrm{R}}_n=\left(\frac{v_o.R_o}{\nu }\right)$, where R_o 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:

$$\frac{f_o.b}{v_o}=0.9586.{\left({\varphi }_1\right)}^{0.9819}\mathrm{and}{\varphi }_1=c_1.{\left(R_n\right)}^{c_2}.{\left(F_n\right)}^{c_3}$$

(5)

$$\frac{f_o.R_o}{v_o}=1.3354.{\left({\varphi }_2\right)}^{1.1111}\mathrm{and}{\varphi }_2=c_1.{\left(R_n\right)}^{c_2}.{\left(F_n\right)}^{c_3}$$

(6)

$$\frac{f_o.y_o}{\sqrt{g{y}_o}}=1.0782.{\left({\varphi }_3\right)}^{1.059}\mathrm{and}{\varphi }_3=c_1.{\left(R_n\right)}^{c_2}.{\left(F_n\right)}^{c_3}$$

(7)

$$\frac{f_o.y_o}{v_o}=1.1965.{\left({\varphi }_4\right)}^{1.1031}\mathrm{and}{\varphi }_4=c_1.{\left(R_n\right)}^{c_2}.{\left(F_n\right)}^{c_3}$$

(8)

$$\frac{f_o.y_o}{v_o}=1.112.{\left({\varphi }_5\right)}^{1.0656}\mathrm{and}{\varphi }_5=c_1.{\left(\frac{b}{R_o}\right)}^{c_2}.{\left(F_n\right)}^{c_3}$$

(9)

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 (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 differences between the measured data and the proposed regression formulas.

Table 3. Coefficients of regression equations for Strouhal no.

Regression Formula	Function	Equation No	Coefficients			R2
			c1	c2	c3
1	Φ1	5	42.22809	−0.4635	−0.71255	0.784
2	Φ2	6	0.081352	0.026588	−1.08519	0.743
3	Φ3	7	0.003221	0.441993	−0.66306	0.74
4	Φ4	8	0.003678	0.424442	−1.54091	0.897
5	Φ5	9	0.777935	0.85343	1.13919	0.9001

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 (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 = ∞).

The best-fit trendline for data measurements of the maximum dimensionless water depth upstream of DRA bends is given in Equation (10).

$$\frac{{y_{DRA}}_m}{y_o}=1.7558e^{0.3916F_n}({\mathrm{R}}^2=0.91)$$

(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.

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 ≤ 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 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 ≤ F_n ≤ 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 (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 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.