Keywords

1 Introduction

Unlike the classical hydraulic jump accompanied by free surface breaking and rolling, the undular hydraulic jump has a coherent and smooth free surface with subtle fluctuations, and this fluctuation will persist downstream over an extended distance. Undular hydraulic jumps generally occur when Froude numbers approach 1, which is a transitional state from supercritical flow to subcritical flow. Therefore, this special phenomenon can often be observed on coasts and estuaries.

Presently, the predominant focus in undular hydraulic jump research revolves around the examination of factors such as Froude numbers, aspect ratios, Reynolds numbers, and their collective impact on the hydraulic jump characteristics. In a study by Chanson and Montes (1995) [1], experiments were conducted within smooth rectangular channels, varying Froude numbers from 1.05 to 3.0. The investigation delved into the velocity, pressure, and energy distribution along the centerline longitudinal profile and specific positions in the transverse profile. Notably, within the Froude number range of 1.5 to 2.9, the wavelength and amplitude of free surface waves were identified as functions of both the Froude number and the aspect ratio (yc/W). Subsequently, Dunbabin (1996) [2] extended the exploration by investigating the velocity and pressure distribution upstream of the first crest under three distinct conditions: Fr = 1.41, 1.52, and 1.63.

Ohtsu et al. (2003) [3] divided undular hydraulic jumps into two types, I and II, based on the ratio H1/B, representing the upstream depth to river width. The primary distinction lies in whether the shock wave on the sidewall crosses at the first crest. At the same time, the variation of flow conditions with Reynolds number was delineated, and an analysis was conducted on the influence of Reynolds number on undular hydraulic jumps. In a study by Montes and Chanson (1998) [4], pitot tubes were employed for velocity and pressure measurements in the first three crests and two troughs, elucidating certain three-dimensional flow characteristics resulting from the development of lateral shock waves. Ohtsu et al. (1996) [5] found that shock waves were always observed before the first crest when Fr ≥ 1.2. When the intersection point of the shock wave is upstream of the first crest, the impact of the shock wave on the undular hydraulic jump is significant. If the intersection point is downstream, the impact of the shock wave can be ignored.

Chanson (1995a) [6] conducted experiments maintaining consistent upstream inflow conditions but varying sidewall roughness compared to Chanson and Montes (1995) [6]. The findings revealed distinguishable flow patterns from those established by the latter on smooth sidewalls. Pasha and Tanaka (2017) [7] mentioned that when vegetation is inserted into a rectangular flume, there will be a proportional decrease in speed and energy with higher vegetation density.

This article conducted two series of experiments on undular hydraulic jumps generated by different types of bed roughness (including smooth, rough rubber-matted, and grated beds) in a 0.5m wide channel, such as Fig. 1. The investigations were dedicated to studying the influence of bed roughness on both the free surface characteristics and the evolution of the turbulent flow field associated with undular hydraulic jumps. Non-intrusive measurements of the free surface were carried out using an acoustic displacement meter (ADM) along the centerline of the sink, while invasive velocity measurements were conducted using an acoustic Doppler velocimeter (ADV). This study demonstrates the main locations and causes of different turbulence characteristics induced by bed roughness, providing a theoretical basis for solving sediment deposition issuse [8].

Fig. 1.
figure 1

Experimental phenomena of rough rubber-matted bed

Full size image

2 Experimental Setup and Instrumentation

2.1 Facility and Flow Conditions

Experiments were conducted in a rectangular channel equipped with a circulating water supply system. As shown in the Fig. 2(a), the entire system comprises a high-level tank, a horizontal channel, and a circulation pool. Water is pumped into a high-level tank with dimensions of 2 × 2 ×  2  m3 from the circulating pool. The flow is controlled by Siemens Micromaster 430 frequency converter and measured by LDG-200 electromagnetic flowmeter. At the same time, the flow is verified by a adjustable triangular overflow weir located at the tail of the flume, with a measurement accuracy of 2%. A movable baffle is positioned at the high-level tank's outlet to control the gate opening (inflow depth), while a triangular weir at the flume's tail regulates the toe position and tailwater depth. Rigorous checks are conducted on the tank's water level and gate opening to ensure uniform water flow into a horizontal channel measuring 12.5 m in length, 0.5 m in width, and 0.7 m in depth. Ultimately, the tail water is discharged into the circulating pool.

The experiment was conducted at Q = 241 m3/h, Fr = 1.75, and Re = 1.34 × 105, three sets of experiments were conducted on bed with different roughness. Nonintrusive measurements were performed using acoustic displacement meters on the free surface contour from the toe of the hydraulic jump to the end of the fourth wave, and invasive measurements were made using acoustic Doppler velocimeters on the velocity at the characteristic positions of the first two waves. The measurement cross-section selected for the experiment is the longitudinal section of the centerline of the flume. For recording convenience, a fixed point preceding the jump is designated as the starting point of the x-axis, with the bed denoted as y = 0. Figure 2(b) shows the geometric parameters measured in the experiments.

Fig. 2.
figure 2

Experimental facility and undular hydraulic jump flow: (a) experimental facility, (b) top view photo of flume and definition sketch of geometric parameters

Full size image

2.2 Equivalent Roughness Height

The velocity profile shape in the boundary layer above the rough bed in the measured subcritical clear-water flow conditions was compared with the Law of the Wall in the log region, and the equivalent roughness heights corresponding to the rubber mat surface and grille surface were calibrated. Choose the roughness that best fits the theoretical curve. Compare the measured centerline free surface profile with the theoretical backwater calculation to verify the accuracy of roughness calculation. It is ultimately believed that the results of using the log-law estimation to characterize the two types of roughness are reliable. The results show that the the roughness of smooth bed is ks = 0.5 mm, the rubber bed is ks = 11.2 mm, and the grated-bed is ks = 18.4 mm.

3 Results: Free-Surface Characteristics

Our observation of general flow patterns shows in Fig. 1 and Fig. 3 indicates that under smooth bed conditions, transverse shock waves intersect at the first crest and do not intersect when the bed is rough.

Fig. 3.
figure 3

Experiments under different bed roughness: (a) smooth bed (b) grated bed

Full size image

The waveform in Fig. 4 shows the average free surface water depth from the toe of the hydraulic jump to the end of the fourth wave within 1 min. The horizontal axis represents the distance from the starting point of the hydraulic jump, and the vertical axis represents the water depth, both of which have been dimensionless. It can be seen that the waveform characteristics of the first two waves are the most obvious, and the fluctuation of the subsequent waves gradually weakens.

Fig. 4.
figure 4

Time-averaged free-surface profiles of undular hydraulic jump at centerline.

Full size image

Figure 5 illustrates the water depths of the crests and troughs for the first four waves. Based on the waveform analysis, it is evident that the smooth bed exhibits the highest water depth at both the crest and trough of the wave. The grille and rough rubber mat show similar water depths at the trough, with slight variations at the crest. Specifically, at the first and second crests, the water depth of the grille is greater than that of the rough rubber mat. However, from the third crest onward, the water depth of the grille decreases sharply, only slightly surpassing that of the rough rubber mat, and eventually falling below the rough rubber mat at the fourth crest. This observation indicates that the waveform development of the grille is the most unstable, while the waveform of the rough rubber mat is the most stable. According to Fig. 5(a), the water depth of the second wave crest is the highest under all three operating conditions, and gradually decreases thereafter. In Fig. 5(b), the depth of the trough under smooth conditions is the most stable, while the grille is the least stable, showing a slow increasing trend overall. Across the first four waves, increased roughness corresponds to greater variation in trough depth as the waves develop. Simultaneously, as the depth of the wave crest gradually decreases, the trough depth gradually increases, and the undulating characteristics of the undular hydraulic jump gradually weaken. Comparing the Fig. 5(a) and(b), due to the difference in roughness, the difference in the dependent variable of the former is more significant, and the influence of roughness on the crest water depth is greater than that on the trough water depth.

Fig. 5.
figure 5

The water depth at the first four wave characteristic positions: (a) the water depth at crest, (b) the water depth at trough

Full size image

In Fig. 6, “L” represents the distance between two adjacent crests, for example, at first wave it represents the distance between the first and second crests, and the vertical axis represents the dimensionless crest distance value. In contrast to the pattern observed in crest and trough water depth, the distance between crests on a smooth bed is the smallest. Specifically, at the first crest distance, the rough rubber mat exceeds the grille, and in subsequent second and third crest distances, the grille exceeds the rubber mat. This pattern reaffirms that under grille conditions, the waveform characteristics of the first two waves are more pronounced. From the third wave onward, there is a significant increase in the distance between crests, a substantial decrease in crest depth, and a rapid weakening of waveform characteristics. This phenomenon may be attributed to the dissipation of most energy in the first two waves under grille conditions, leading to a notable attenuation of waveform characteristics at the onset of the third wave. Therefore, Fig. 6 also demonstrates that as the wave progresses downstream, the crest distance of the grille exhibits the most fluctuation. Additionally, the average value of crest distance is arranged in ascending order as follows: smooth, rough rubber mat, and grille.

Figure 7 illustrates the undulation of the first three waves, due to the significant difference in water depth between the upstream and downstream of the first wave crest, the calculation of the relief of the first wave crest has been improved. Across different roughness levels, the undulations of the first three waves follow a descending order: smooth, grille, and rough rubber mat. Notably, excessively small or large roughness intensifies the undulation of the first three waves, with the rough rubber mat exhibiting undulation between the other two and having the weakest undulation. The most significant difference in undulation occurs in the first wave, followed by a reduction in the disparity under various working conditions. The undulation of the rough rubber mat and grille becomes very close in the second and third waves.

Fig. 6.
figure 6

Dimensionless crest distance of the first three waves: The abscissa represents the number of waves. L/d2 means the distance between the crests.

Full size image
Fig. 7.
figure 7

Undulation of the first three waves. The horizontal axis represents the first, second, and third waves, and the vertical axis relief is defined as: 2Ai/Li. Furthermore, the undulation of the first crest is (A1 + A0)/2Lw

Full size image

4 Results: Turbulent Velocity Characteristics

In the study of free surface wave characteristics, it was found that the waveform characteristics of the first two waves were the most prominent, and the turbulent flow field evolved more violently. After the second wave, the waves gradually weakened. In the examination of turbulent flow field evolution characteristics, the turbulent flow field information of the first two waves was explored, encompassing velocity, turbulence intensity, and turbulent kinetic energy at the crest and trough of the waves. Figure 8 depicts the distribution of velocity along the water depth at the crests and troughs of the first two waves. The smooth bed exhibits the smallest flow velocity and negative flow velocity appears near the bottom of the first wave crest. In contrast, the other two rough beds do not manifest negative flow velocity. This phenomenon is attributed to the minimal shear force and turbulence of the water flow at the bottom of the smooth condition, resulting in the emergence of a ‘static water zone’ at the bottom. With the oscillation of the jump toe, negative velocity appears at the bottom of the first wave crest. As the bottom roughness increases, the shear force of the bottom water flow intensifies, and turbulent kinetic energy rises (as depicted in Fig. 9), leading to the formation of numerous vortices in the static water zone. The streamline in the static water zone is no longer parallel to the free surface, and a small amount of mainstream may traverse the static water zone, disrupting the stability of the first crest bottom water flow. Consequently, the static water zone disappears, preventing the occurrence of negative velocity.

Fig. 8.
figure 8

Dimensionless longitudinal velocity distributions at different characteristic positions: c1, the first crest. t1, the first trough. c2, the second crest. t2, the second trough.

Full size image

At the first crest, the velocity gradient near the surface of the rough bed is greater than that of the smooth bed. This observation may be attributed to the fact that under rough conditions, the shock wave intersects at the first crest, resulting in significant turbulence in the water flow near the free surface, as depicted in Fig. 9, with pronounced shear effects. Conversely, under smooth conditions, the shock wave does not intersect at the first crest. The flow velocities at the first and second crests exhibit a trend of increasing and then decreasing, and as the hydraulic jump develops, the gradient of flow velocity changes becomes smaller. This pattern aligns with observations made by Montes and Chanson (1998) in the velocity profile measured at the first crest of the undular hydraulic jump centerline. The decrease in speed near the water surface is attributed to surface rolling and breaking.

Figure 9 shows the distribution of turbulent intensity along the water depth. As the height increases, the turbulence intensity first increases and then decreases. Furthermore, as the hydraulic jump develops, this change diminishes gradually until it eventually disappears.

Roughness exerts a substantial impact on the first wave, with the most pronounced influence observed at the first crest. However, as it progresses to the second crest, the influence of roughness on turbulence intensity diminishes significantly, and the three curves closely approach overlapping. Across different roughness levels, the maximum turbulence intensity of the first two waves is situated in the lower half of the first wave crest. Specifically, at the bottom of the first crest (y/d1 < 0.5), the smooth bed exhibits the lowest turbulence intensity, while the grille displays the highest, indicating that greater bed roughness corresponds to increased velocity fluctuation at the bottom of the first crest.

Fig. 9.
figure 9

Distribution of turbulent intensity in the mainstream direction along the water depth at the characteristic positions of the first two waves

Full size image

5 Conclusion

An experimental study was conducted on a undular hydraulic jump in a 0.5m wide flume. The surface undulations of four standing waves were recorded. Under three different roughness conditions of smooth bed, rough rubber mat, and grille, different wave characteristics are displayed.

In the study of turbulent flow characteristics, the influence of roughness is mainly on the first wave, with the greatest impact on the first crest. The greater the roughness of the bed, the greater the fluctuation of the velocity at the bottom of the first crest.

At the first crest of the smooth bed, negative velocity was observed near the bottom, while the other two rough beds did not exhibit negative velocity. This phenomenon is attributed to the small shear force and turbulence of the water flow at the bottom of the smooth bed, resulting in the formation of a “static water zone” at the bottom. Simultaneously, with the oscillation of the toe, negative velocity emerges at the bottom of the first wave crest.

Under various roughness levels, the undulations of the first three waves follow a descending order: smooth, grille, and rough rubber mat. Excessive small or large roughness intensifies the undulation of the first three waves. The most substantial difference in undulation occurs in the first wave.