Royal Society Publishing

Experimental study of the particle paths in solitary water waves

Hung-Chu Hsu, Yang-Yih Chen, Hwung-Hweng Hwung

Abstract

We provide experiments that had been conducted to investigate the particle trajectory beneath a solitary water wave. Experimental results show that the surface drift is larger than the bottom drift. Meanwhile, the ratio of the net horizontal displacement to the total height of the trajectory at the free surface and subsurface will decrease with the initial vertical position and increase with the relative wave height.

1. Introduction

The steady finite-amplitude solitary waves were first noticed by Scott Russell in 1834. Forty years later, Boussinesq [1] and Rayleigh [2] provided an approximation solution of the wave. Korteweg & de Vries [3] derived a theory for irrotational solitary shallow water waves of small amplitude. Recently, Alvarez-Samaniego & Lannes [4] discussed the various regimes suitable for the modelling of water waves. In contrast to the case of periodic travelling waves, for solitary waves, it is known that the method of linearization is not appropriate, even for waves of small amplitude [5,6]. The existence theory for solitary waves (of small and large amplitude) propagating at the surface of water in irrotational flow and with a flat bed was developed in Amick & Toland [7], which extended earlier work [8,9]. Theoretical investigations of the flow beneath an irrotational solitary wave were undertaken recently by Constantin & Escher [10], Constantin [11] and Constantin et al. [12]. Because the quantity of the drift of a fluid particle plays an important role in the sediment transport for coastal engineering, it is worth carrying out some experiments to verify the analytical solution. Longuet-Higgins [13] observed the particle trajectories at the free surface in a solitary wave. However, the accuracy of the experiment could be improved. In the present paper, the solitary waves were generated by a piston-type wavemaker and the wave-board motions can be prescribed by the computer. The particle orbits beneath a solitary wave are observed using a high-speed camera that provides good-quality experimental data. The experimental data presented are in close agreement with the theoretical findings [1012].

2. Experimental facility

The experiments were carried out in a wave tank at the Tainan Hydraulics Laboratory (THL) of the National Cheng Kung University, Taiwan. This tank was 21 m long and 0.7 m deep, with a width of 0.5 m, and its glass side walls facilitate recording with a camera and permit the visual observation of the evolution of a wave. The target solitary waves were generated at one end of the flume by a programmable high-resolution wavemaker. A plane beach of slope 1/20 covered with a smooth layer of concrete starts 11 m from the wave paddle. A schematic of the experimental definition is shown in figure 1.

Figure 1.

Sketch of wave flume layouts and the definitions of the physical variables, with the exact positions of the wave gauges.

The arrangement of the measurement apparatus was deployed with wave gauges and a high-speed camera. Figure 2 shows photographs of the laboratory flume and of the measurement facilities employed in the reported experiments. The elevation of the local water surface was recorded by employing three capacitance-type wave gauges located between 3.56 and 7.63 m downstream of the wavemaker. It is noted that the wave probe located at x=3.56 m was referred to the reference gauge in all experiments. All wave gauges were calibrated through a standard method that concerns the change of water level to adjust the response voltage of each gauge before and after the experiments to ensure its linearity and stability. Figure 3ac shows the linear relationship between the water level and response voltage for three wave gauges. The linearity of the gauge response was given by a correlation coefficient of 0.9997–0.99991.

Figure 2.

View of the wave tank and the measurement apparatus in the present experiment: (a) wave gauge and (b) camera. (Online version in colour.)

Figure 3.

Calibration of the three wave gauges: (a) CH1, (b) CH2 and (c) CH3. The circles denote the measured data and the line is the linear regressive curve. The correlation coefficients for the three wave gauges are approximately 0.99991, 0.9997 and 0.99987, which mean there is a highly linear correlation between the water elevation and the measured voltage differences. (Online version in colour.)

3. Solitary wave generation

To push or withdraw a certain volume of water within a certain duration, the basic idea is to generate a solitary wave. In this study, the waves were generated by a hydraulically driven, dry back, piston-type wavemaker with a maximum stroke of 1 m. A programmable controller that can be accessed easily by a PC controlled the motion of the wave board at 25 Hz through a 16 bit analogue to digital/digital to analogue card. Different wave-board motions can be prescribed by the computer. Target solitary waves were generated at one end of the flume with a programmable, hydraulically driven, dry back, piston-type wavemaker. Of the two existing nonlinear algorithms developed, respectively, by Goring [14] and Synolakis [15], the former algorithm was employed here to generate solitary waves. Figure 4 shows measured wave-board trajectories of several trials and the computational results of Goring's [14] method are in good agreement, suggesting that the wavemaker is highly reliable. The measured solitary wave profiles also agreed with the first-order theory of Boussinesq [1] shown in figure 5, suggesting that the experimental results are highly repeatable and reliable. Note that Boussinesq theory is not applicable, as we do not deal with waves of small amplitude. However, the wave profile agrees to a great extent with the theoretical predictions by Amick & Toland [16]. The abscissa is a non-dimensional time variable Embedded Image (where d is water depth), as adopted by Synolakis [15]. The ordinate is the wave profile, normalized with the offshore constant water depth, η/H (where H is wave height). It is noted that the origin of normalized time is shifted to match the time when the wave crest passes the reference gauge. Clearly, small variations at the tail for higher wave nonlinearity are found and slight asymmetry of wave shape with respect to wave crest is observed. This phenomenon is due to the algorithm restriction of Goring [14]. Comparisons show that this facility validates the control of solitary wave generation.

Figure 4.

Wave-board displacement time histories. Solid line represents Goring [14] and dashed line represents ramp trajectory.

Figure 5.

Comparison of the time history of the water elevation displacement for repeated experiments (triangles) measured at the wave gauge. The dashed line denotes the theoretical profile, whereas the solid line is the experimental data (Boussinesq). The free-surface displacements of several trials at the wave gauge and the theoretical predictions are in good agreement. (ad=15 cm, H=8.64 cm and (b) d=20 cm, H=7.07 cm.

4. Experimental procedure

Solitary waves of relative incident wave heights (H/d) ranging from 0.182 to 0.576 were generated, where H is the wave height and d is the constant water depth. The test conditions are listed in table 1. Time series of local water surface elevation were recorded by capacitance-type wave gauges distributed at three stations along the tank. Each sensor of the gauges was mounted at 0.25 m away from the side wall. The resolution of the gauges was 12 bit on a dynamic range of 0.6 m. As these gauges were entirely out of the water before the wave passed by, their outputs were carefully examined to avoid incorrect signals. In this laboratory, signal synchronization from numerous parallel inputs and signal decay owing to the long-distance transmission were two important data-acquisition problems. To cope with these challenges, measuring data and wave-paddle motion were all recorded simultaneously at a 50 Hz sampling rate for 60 s using a multi-node data-acquisition system, developed by THL. The still-water depth was also measured after each test to maintain the same initial conditions. The repeatability of all experiments was satisfactory, with a maximum derivation in reference wave height of approximately 3 per cent between repetitions. The experimental data were synchronized before analyses.

View this table:
Table 1.

The test conditions.

The water particles were simulated with fluorescent spherical polystyrene beads (PS) with a diameter of about 0.1 cm, as shown in figure 6. The density of primitive PS is about 1.05 g cm−3 heavier than the water. After boiling with water, the PS density is approximately equal to the water density 1 g cm−3, and will remain neutrally stable in a fixed position in water. Images were captured using a high-speed camera (A301fc-type, Basler Company) that can take 80 frames per second. The camera was controlled using the BCamGraber program and linked to a 1394 card to collect and analyse the data. Four powerful lamps (110 V and 500 W) were set up to reinforce the brightness of the images of PS motion in the water waves for easier identification. A transparent acrylic-plastic sheet (1 m×45 cm×2 mm) plotted with 2×2 cm square grids, as shown in figure 7, was placed in the still water centred along the width of the tank. It was first photographed before being removed from the tank. The network grids in the photograph were programmed into the computer and used to analyse the continuous images of particle trajectories captured by the high-speed camera. A copper pole (150 cm long with a diameter of 0.5 cm), calibrated at 0.1 cm intervals and perforated below 70 cm with 20 holes having a diameter of 0.3 cm, was erected vertically in front of the viewing glass in the still water tank. The PS was pushed out horizontally from the holes of the copper pole at different water levels into the still water. Then, the copper pole was slowly removed from the tank before the waves were generated to avoid interfering with the incident waves and PS motion.

Figure 6.

The fluorescent spherical polystyrene beads with a diameter of about 0.1 cm for simulating the water particle. (Online version in colour.)

Figure 7.

The transparent acrylic-plastic sheet was plotted with 2 cm interval network grid points. (Online version in colour.)

5. Discussion and conclusions

The particle motions beneath the solitary waves are shown in figure 8. In the case of periodic irrotational travelling waves, it is known that the particles perform a forward–backward motion, with no closed particle paths in the absence of an underlying current, cf. Constantin [17] and Constantin & Strauss [18] for theoretical considerations and Chen et al. [19] for experimental data. It is obvious that for solitary waves, the particle orbit does not comprise any backward motion because of the existence of a displacement that persists with it along the wave direction; this phenomenon is shown in figure 8. Figure 8 shows the trajectories that would occur in a succession of widely separated solitary waves for surface and subsurface particles. It can be seen that a particle advances horizontally after each period through a distance known as the drift or mass transport in the direction of wave propagation. The vertical excursion is less than its horizontal displacement and diminishes rapidly with the depth of the trajectory below the free surface. At the bottom, the trajectory becomes a straight line because the vertical movement of the particle is zero, and only a horizontal displacement exits. The net forward horizontal displacement during one complete wave cycle is largest at the free surface, decreases at deeper levels, and is smallest at the bottom. Therefore, the surface drift is larger than the bottom drift. The ratios r=Y/X of the net horizontal displacement X to the total height Y of the trajectory at the free surface and subsurface for the six cases are shown in table 2. It is clearly shown that the ratio r decreases with the initial vertical position and increases with the relative wave height H/d. The experimental data presented are in close agreement with theoretical findings [1012].

Figure 8.

The orbits of water particles obtained from the experimental measurements of the polystyrene beads motions at different water levels b in the four experimental wave cases. (a) d=20 cm, H=7.07 cm; (b) d=20 cm, H=8.56 cm; (c) d=30 cm, H=5.46 cm and (d) d= 30 cm, H=7.56 cm.

View this table:
Table 2.

The total horizontal displacement X versus the maximum vertical displacement Y at the surface and subsurface of a solitary wave in a water of undisturbed depth d.

Acknowledgements

The authors thank Mr C.Y. Lin who performed the experimental work. The authors also acknowledge the insightful critiquing of Prof. Adrian Constantin and two referees for some helpful comments. Support under grant numbers NSC 99-2915-I-006-046 and NSC 99-2923-E-110-001-MY3 from the National Science Council, Taiwan is gratefully acknowledged.

Footnotes

References

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