Eliminating the Nonlinear Kelvin Wake Behind a Moving Body on the Water Surface

Conceitos essenciais

It is possible to eliminate the nonlinear Kelvin wake behind a moving body on the water surface by carefully choosing the forcing distribution, and the corresponding wave-free solution is stable.

Resumo

The paper investigates the disturbance caused by a localized forcing moving at constant speed on the free surface of a liquid of finite depth, using the forced Kadometsev-Petviashvili (fKP) equation.
Key highlights:

The paper characterizes the v-shaped Kelvin wave pattern downstream of the forcing and the wedge angle as a function of the Froude number.
It is shown that the wake can be eliminated by a judicious choice of the forcing distribution, resulting in a wave-free steady state solution.
Numerical simulations demonstrate that the wave-free steady states are stable and are reached in the long-time limit of an initial value problem.

The authors first analyze the linear steady solutions of the fKP equation to determine the Kelvin wedge angle for different Froude numbers. They then construct nonlinear wave-free steady solutions by carefully choosing the forcing function to eliminate the pole responsible for the far-field wave pattern. Finally, they provide numerical evidence that these wave-free solutions are stable.
The results suggest the possibility of eliminating the Kelvin wake in real-world settings, which could lead to improvements in marine vessel design and the minimization of wake signatures.

Estatísticas

"The Kelvin wedge angle can be determined by examining the leading-order asymptotic behaviour of (7) far downstream, as r→∞, |θ| < π/2."
"When Fr= 1 the solutions satisfy tan φ = -1/2 cot θ, resulting in θk = π/2."
"When Fr< 1 there are two real solutions for φ provided that 0 < |θ| < θk(Fr), where 16(1 - Fr) tan^2 θk = 1."
"When Fr> 1 the permissible θ are bounded by the Kelvin wedge angle, θk, which satisfies 2(Fr-1) tan^2 θk = 1."

Citações

"Strikingly, we provide evidence that the wave-free solutions are stable, meaning that they are reached in the long-time limit of an initial value problem (IVP), leading to the possibility of observing them in a physical experiment."
"Viewed altogether, these calculations show the remarkable stability of the wave-free steady state; our third, and perhaps most significant result of the paper. A carefully constructed forcing term, as described in the previous section, will result in a wave-free profile that can be observed in a physical experiment."

Principais Insights Extraídos De

Towards eliminating the nonlinear Kelvin wake

by Jack Keeler,... às arxiv.org 10-03-2024

https://arxiv.org/pdf/2410.01255.pdf

Perguntas Mais Profundas

How could the insights from this study on the fKP equation be extended to the full Euler system to achieve wave-free solutions?

The insights gained from the study of the forced Kadometsev-Petviashvili (fKP) equation can be pivotal in extending the search for wave-free solutions to the full Euler system. The fKP equation serves as a simplified model that captures essential nonlinear dynamics of surface waves, particularly the formation of the Kelvin wake. To transition these insights to the full Euler system, several strategies can be considered:

Localized Forcing Functions: The study demonstrates that a carefully chosen localized forcing function can eliminate the Kelvin wake in the fKP framework. This principle can be applied to the Euler system by exploring various localized pressure distributions that could similarly suppress wave formation. For instance, using dipole-like forcing or multi-signed distributions may yield wave-free solutions.

Numerical Simulations: The numerical techniques employed in the fKP study, such as boundary-integral methods and Newton-Krylov approaches, can be adapted for the Euler system. By simulating the response of the Euler equations to different forcing distributions, researchers can identify configurations that lead to stable, wave-free states.

Asymptotic Analysis: The asymptotic methods used to analyze the fKP equation can be extended to the Euler system. By examining the behavior of solutions in the limit of small Froude numbers or other relevant parameters, it may be possible to derive conditions under which wave-free solutions exist.

Stability Analysis: The stability of wave-free solutions in the fKP equation suggests that similar stability properties could be investigated in the Euler system. By perturbing the wave-free states and analyzing the resulting dynamics, researchers can determine the robustness of these solutions against small disturbances.

By integrating these approaches, the transition from the fKP equation to the full Euler system could reveal new pathways to achieving wave-free solutions in practical scenarios.

What other types of forcing distributions, beyond the one considered here, could potentially lead to wave-free solutions in the fKP or Euler systems?

Beyond the Gaussian forcing distribution explored in the study, several other types of forcing distributions could be investigated for their potential to yield wave-free solutions in both the fKP and Euler systems:

Dirac Delta Function: A Dirac delta function as a forcing distribution represents an idealized point source. This type of forcing can be useful for studying localized disturbances and may lead to wave-free solutions under specific conditions, particularly when combined with appropriate boundary conditions.

Polynomial Distributions: Forcing distributions defined by polynomial functions could be explored. These distributions can be tailored to have specific properties, such as symmetry or decay rates, which may influence the resulting wave patterns and potentially lead to wave-free states.

Exponential Distributions: Exponential forcing functions, which decay rapidly away from a central point, could also be considered. Their rapid decay may help confine disturbances to a localized region, thereby minimizing the generation of waves in the far field.

Multi-Signed Forcing: As indicated in the study, multi-signed forcing distributions, which combine positive and negative pressure regions, could be effective in canceling out wave generation. This approach could be particularly useful in mimicking complex hull shapes or topographic features.

Randomized Forcing: Introducing randomness into the forcing distribution may also yield interesting results. By studying the effects of stochastic forcing, researchers could uncover new dynamics that lead to wave-free solutions, particularly in turbulent or chaotic flow regimes.

By exploring these diverse forcing distributions, researchers can expand the understanding of wave dynamics in fluid systems and potentially discover new configurations that lead to wave-free solutions.

What are the potential practical applications of wave-free solutions on the water surface, beyond improvements in marine vessel design?

The development of wave-free solutions on the water surface has several potential practical applications that extend beyond marine vessel design:

Environmental Protection: Wave-free solutions can significantly reduce the erosion of riverbanks and shorelines caused by boat wakes. By minimizing wave generation, these solutions can help protect fragile ecosystems and reduce sediment displacement in aquatic environments.

Aquaculture: In aquaculture settings, maintaining calm water conditions is crucial for the health of fish and other aquatic organisms. Wave-free solutions can create stable environments that promote growth and reduce stress on marine life, leading to more efficient aquaculture practices.

Underwater Structures: The principles of wave-free solutions can be applied to the design of underwater structures, such as pipelines, cables, and artificial reefs. By minimizing surface disturbances, these structures can be better integrated into their environments, reducing the risk of damage from wave action.

Hydrodynamic Testing: Wave-free conditions can provide a controlled environment for testing hydrodynamic models and experiments. Researchers can study the effects of various forces on submerged objects without the interference of surface waves, leading to more accurate data and insights.

Stealth Technology: In military applications, wave-free solutions can contribute to stealth technology for naval vessels. By minimizing the wake signature, vessels can operate with reduced detectability, enhancing operational security.

Recreational Activities: For recreational boating and water sports, wave-free solutions can enhance user experience by providing smoother water conditions. This can lead to safer and more enjoyable activities, such as kayaking, paddleboarding, and fishing.

By leveraging wave-free solutions, various industries can benefit from improved operational efficiency, environmental sustainability, and enhanced user experiences on the water.