# Generation of impulse surface waves by a moving piston

Wladimir Sarlin, Zhaodong Niu, Alban Sauret, Philippe Gondret, Cyprien Morize

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## To cite this version:

Wladimir Sarlin, Zhaodong Niu, Alban Sauret, Philippe Gondret, Cyprien Morize. Generation of impulse surface waves by a moving piston. International Conference of Theoretical and Applied Mechanics (ICTAM) 2024, Aug 2024, Daegu, South Korea. [hal-05222879](https://hal.science/hal-05222879v1)

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# GENERATION OF IMPULSE SURFACE WAVES BY A MOVING PISTON

Wladimir Sarlin<sup>1,2</sup>, Zhaodong Niu<sup>1,2</sup>, Alban Sauret<sup>3</sup>, Philippe Gondret<sup>1</sup> & Cyprien Morize<sup>1*</sup>

<sup>1</sup>Université Paris-Saclay, CNRS, Laboratoire FAST, F-91405 Orsay, France

<sup>2</sup>Laboratoire d'Hydrodynamique (LadHyX), UMR 7646 CNRS-Ecole Polytechnique, IP Paris, 91128

Palaiseau, France

<sup>3</sup>University of California, Santa Barbara, Department of Mechanical Engineering, CA 93106, USA

**Summary** The generation of impulse waves is still a significant challenge in environmental fluid mechanics. In this study, water waves are produced at the laboratory scale by the sudden translation of a rigid piston. The velocity and total stroke of the piston, along with the initial water depth, are independently adjusted. We report a wide range of wave regimes, and these are plotted on a phase diagram based on the Froude number and the dimensionless stroke of the piston. The experimental outcomes during the generation stage are successfully compared with existing theoretical results when the relative acceleration of the piston is low. Contrariwise, for high Froude numbers, a quasi-static hydraulic jump theory is developed and shown to describe finely the experiments.

## CONTEXT

Although the evolution of surface waves has been a subject of studies for centuries, the physical mechanisms involved in their generation continue to raise numerous questions due to the complexity of the phenomenon [1]. Recently, there has been significant attention from the fluid mechanics community in understanding the initial stages of impulse surface wave formation. Reaching a better description is crucial for improving the modeling of large-scale phenomena like tsunamis caused by landslides, pyroclastic flows, or iceberg calving. The present study reports experiments of impulse wave generated by the motion of a piston, where the velocity and stroke of the latter was varied, alongside with the initial water depth.

## EXPERIMENTAL SETUP

The experimental apparatus, shown in figure 1 (a), consists of a rectangular glass tank of 2 m long, 30 cm high and 15 cm wide, which is filled with water over an initial height $h$. The controlled horizontal motion of the vertical piston, at one side of the tank, generates a wave by displacing a certain volume of water. The flow and the wave elevation are assumed to be invariant in the $y$ transverse direction given the dimensions of the tank and the wave generator. The motion of the vertical wall is controlled by a brushless servo-motor for a reproducible regulation of its displacement and velocity. The law of motion of the piston is chosen to be quadratic in time at constant acceleration $\gamma$ on the first half of the total piston stroke $L$ followed by a constant slow down stage at $-\gamma$ on the second part, so that the forcing of the wave, i.e. its phases of acceleration and deceleration, is symmetrical. We note $U$ the maximum velocity of the piston at mid-stroke. This wall motion is very simple to implement and allows independent control of the total stroke length $L$ and the maximum velocity $U$.

The dynamics of the wave generation is imaged with a video camera through the transparent sidewall of the tank. Image processing is performed using a custom-made Matlab routine based on a thresholding method that allows to extract both the instantaneous amplitude $A(t)$ and the mid-height width $\lambda(t)$ of the impulse wave. More than 280 experiments have been performed varying the maximum velocity and the total stroke of the piston in the range $0.13 \le U \le 1.2 \text{ m s}^{-1}$ and $2 \le L \le 30 \text{ cm}$, respectively, while the water depth was also varied in the range $1 \le h \le 23 \text{ cm}$. In these conditions, the maximal Froude number $Fr = U/\sqrt{gh}$, defined as the ratio of the maximal velocity of the moving wall to the typical velocity of gravity waves in shallow water conditions, has been varied in the range $0.1 \le Fr \le 2.3$. The total stroke length relative to the water depth have been varied over two decades in the range $0.1 \le L/h \le 10$.

## RESULTS

The experiments reveal a variety of wave shapes, showcasing a broad spectrum of forms and behaviors, as illustrated in Fig. 1(b-e). This mirrors the distinct regimes previously documented in the context of wave generation by a rigid wall [2] or the collapse of granular material into water [3, 4]. In situations where both the piston stroke and velocity are sufficiently small ($L < h$ and $U < \sqrt{gh}$), dispersive waves emerge, as depicted in Fig. 1(b), resembling damped oscillations with amplitude decay over time. The dispersive behavior gets balanced by non-linearity when the piston stroke increases ($L \sim h$ and $U < \sqrt{gh}$), resulting in the formation of solitary waves, as shown in Fig. 1(c). The existence of hydrodynamic solitons indicates that the motion law governing the piston corresponds to specific scenarios explored in previous studies. With a substantial relative piston stroke ($L > h$), bore waves are produced, exhibiting a

<footnote>* Corresponding author. E-mail: cyprien.morize@universite-paris-saclay.fr</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1. (a) Schematic view of the experimental setup. (b-e) Examples of the different regimes of impulse waves observed: (b) dispersive wave, (c) solitary wave, (d) plunging breaking bore and (e) jet-splash. (f) Diagram of the impulse wave regimes in the (Fr,L/h) plane. Full symbols correspond to breaking bore, while empty symbols correspond to non breaking bore. Stars correspond to experiments at the boundary between two regimes.</figcaption></figure>

characteristic nonlinear steepening of the wavefront. As the values of $U$ approach $\sqrt{gh}$, the spilling breaking of the bore occurs progressively closer to the piston, with nonlinear effects intensifying when the initial impulse increases. This ultimately leads to the development of plunging breakers for a piston velocity sufficiently large ($U > \sqrt{gh}$), accompanied by air entrainment (Fig. 1(d)). Another experimental scenario arises when the acceleration of the piston is high enough. In this case, and as illustrated in Fig. 1(e), a thin filament of fluid is propelled upstream from the initial water bump situated near the advancing piston. Once the jet separates from the vicinity of the piston, it seems to undergo a ballistic trajectory before striking the undisturbed water free surface, dissipating a lot of energy in the process. This jet-splash is reproducible and occurs when $U^2/L \ge 0.45g$, where $g$ is the gravity acceleration.

Based on these empirical observations, the parameters $h$, $U$, and $L$ were systematically varied to create a comprehensive diagram in the $(Fr, L/h)$ plane, which is displayed in Fig. 1(e). The different wave regimes are represented with distinct symbols for (○) dispersive waves, (□, ■) nonbreaking and breaking solitary waves, (▷, ►) spilling and plunging breaking bores, and (●) water jets, respectively. In the diagram, empty and filled symbols represent non-breaking and breaking waves respectively, while the stars correspond to waves at the boundary between several identified regimes. The full line separates the region where water jets develop from the rest of the dataset. Overall, we observe that the value of the relative piston stroke $L/h$ selects the generated wave's shape while the Froude number $Fr$ controls the transition between stable and breaking waves. Additionally, successful comparisons between experimental outcomes in the generation stage and existing theoretical results [5] are made when the piston's relative acceleration is low. Conversely, a quasi-static hydraulic jump theory is developed which describe accurately the experiments for high Froude numbers.

## CONCLUSION

In this study, we investigate experimentally the generation and early propagation of impulse surface waves induced by the motion of a rigid piston. Several wave regimes are obtained, showing the richness of the physics at play. At least, five typical waveforms could be identified from the experiments, ranging from dispersive waves obtained at low $Fr$ and low relative stroke of the piston $L/h$ to unstable spilling or plunging breaking bores at high values of $Fr$ and $L/h$.

## References

[1] Dauxois T. et al.: Confronting grand challenges in environmental fluid mechanics. *Phys. Rev. Fluids* **6** (2): 020501, 2021.

[2] Noda E.: Water waves generated by landslides. *Journal of the Waterways, Harbors and Coastal Engineering* **96** (4): 835-855, 1970.

[3] Fritz H., Hager W., Minor H.: Near field characteristics of landslide generated impulse waves. *Journal of the Waterways, Harbors and Coastal Engineering* **130**: 287-302, 2004.

[4] Sarlin W., Morize C., Sauret A., Gondret P.: Nonlinear regimes of tsunami waves generated by a granular collapse. *J. Fluid Mech* **919**: R6, 2021.
[5] Joo S., Schultz W.W., Messiter A.F.: An analysis of the initial-value wavemaker problem. *J. Fluid Mech* **214**: 161–183, 1990.