Scientists observe and explain the oscillations of circular hydraulic jumps

In a new study published in Physical Examination LettersScientists are exploring how small jets of water can create stable periodic oscillations on a solid disk, discovering a link between these movements and the waves they generate and providing insight into the dynamic interplay of fluid behavior.

A hydraulic jump is a phenomenon that occurs when a fast-flowing liquid abruptly encounters a slower-flowing or stagnant region. This sudden transition results in a change in the flow characteristics, causing a jump to form or a visible increase in liquid height.

In this process, the kinetic energy of the fast-flowing liquid is converted into potential energy, resulting in changes in flow velocity and depth. This phenomenon is commonly observed in various contexts, for example when a jet of liquid hits a surface, for example in rivers or downstream of dams.

French researchers studied a scenario in which a circular hydraulic jump undergoes stable periodic oscillations on a solid disk.

Explaining the team’s motivation behind the study, lead author Aurélien Goerlinger told Phys.org: “The hydraulic jump is a ubiquitous phenomenon that seems simple. However, it is counterintuitive because nature prefers smooth transitions to abrupt ones.

“Therefore, the hydraulic jump is difficult to model, although it has been studied since the time of Leonardo da Vinci. As many fundamental aspects remain to be understood, or even discovered, the hydraulic jump remains an area of ​​study active for our team.”

Circular hydraulic jumps and water jets

The experimental setup of the study involved generating circular hydraulic jumps on a solid disk using a submillimeter water jet.

The researchers fired a submillimeter jet of water, with an internal diameter of 0.84 mm, aimed at a plexiglass disk with an edged surface at a 90-degree angle positioned 1 cm below the point of impact.

This process resulted in the formation of a circular pattern of discontinuity where the liquid established a thin film around the point of impact. The thin film suddenly thickened at a certain radial distance, giving rise to the characteristic circular shape of the hydraulic jump.

To help visualize this phenomenon, Goerlinger provided an analogy by stating: “When you turn on your kitchen faucet and look at the bottom of the sink near the impact of the liquid jet, you can observe a approximately circular liquid wall separating two distinct zones. .

“The inner zone, near the jet, is shallow but the flow is fast, while the outer zone is much deeper but the flow is also much slower. This liquid wall is called a circular hydraulic jump.”

The researchers then varied the experimental parameters, including flow rate (2 to 3 mL/s) and disk radius (1 to 6 cm). They observed different behaviors based on these parameters, such as stationary jumps, transient states with oscillations, bistable states with periodic oscillations, and systematic stable periodic oscillations.

The analysis revealed that the oscillation period did not depend on the flow rate but showed a linear dependence on the disk radius.

Interestingly, for disk radii greater than 5 cm, the data points showed two distinct linear trends with different slopes, indicating two distinct oscillation modes, which the researchers call fundamental and harmonic modes.

Interaction between hydraulic jumps and gravity waves

The researchers developed a theoretical model to explain the observed stable spontaneous oscillations, suggesting that they arise from the interaction between the hydraulic jump and surface gravity waves formed in the disc cavity.

Surface gravity waves propagate along the liquid surface and reflect at the edge of the circular hydraulic jump. This reflection contributes to the establishment and maintenance of oscillations. Additionally, these waves would be amplified when they align with one of the disk cavity modes.

Remarkably, the researchers’ theoretical model not only explains the observed oscillations, but also offers predictive capabilities. It predicted the coupling of distant jets to induce oscillations in opposite phases, a phenomenon confirmed by experimental observation.

In practical terms, this means that the rhythmic ebb and flow of one water jet could influence the oscillations of the other, creating a synchronized dance where the peaks and valleys of one jet inversely correspond to those of the other.

Goerlinger emphasized the importance of their work: “Despite extensive research into this phenomenon, the circular hydraulic jump was found to remain stationary in most cases. However, our study is the first to report stable spontaneous oscillations of the hydraulic jump occurring when the impact jet “is stable. Furthermore, we successfully built a model that predicts the behavior of these oscillations.”

Potential applications and future work

By successfully modeling stable periodic oscillations, the theoretical framework contributes to a deeper understanding of the complex dynamics involved in hydraulic jumps.

This understanding may have implications in various fields, including fluid dynamics and related engineering applications.

“Hydraulic jumps are of great interest in areas where cooling and cleaning of surfaces are necessary. They can also find their interest in high-speed or 3D printers,” explained Goerlinger.

Goerlinger believes they are only scratching the surface of this research and says they plan to do further research in this area.

“We have only partially explored the rich physics of this new phenomenon. The effects of many experimental parameters remain to be studied, such as fluid properties or substrate geometry.

“Moreover, our work opens the way to study the interactions between multiple oscillating jumps and the interactions between hydraulic jumps and waves in general,” he concluded.

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