arXiv subjects

Geoffrey Beck

Publications and source records attributed to Geoffrey Beck.

6 recordsLinked to original sources

arXiv2026

Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case

This paper deals with the interactions of waves governed by a non-linear dispersive Boussinesq type system with the vertical displacement of a cylindrical floating structure in an axisymmetric without swirl situation. The Boussinesq regime is a good approximation of free surface Euler's equations when the non-linear parameter and the shallowness parameter are small. The vertical motion of the floating body is governed by the Newton equation. The full coupled wave-structure interaction problem under consideration is reduced to a boundary problem. The boundary condition satisfied by the discharge is given in terms of the vertical displacement of the floating cylinder. The latter is calculated using an ODE, which requires knowledge of the trace of the surface elevation and its second-time derivative. We use the dispersion in order to exhibit a hidden second order ODE on the trace of the surface elevation. This finally allows us to rewrite the waves-structure interaction problem as a system of non-local conservative PDEs plus bounded radial terms with a dispersive boundary layer, combined with an ODE at the boundary. This is what we call the Augmented formulation. Afterwards we showed that this formulation is well-posed with two different methods. Finally, we study the return to equilibrium situation in the linear regime. In particular, we improved previous results on the explicit time decay. We showed that the center mass of the floating body cannot converge to its equilibrium faster than $\mathcal{O}(t^{-1/2})$ in 2D without viscosity and faster than $\mathcal{O}(t^{-3/2})$ with viscosity.

math.AP↗

arXiv2024

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: applications to fluid turbulence and generation of fractional Gaussian fields

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics has been recently proposed able to generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear Partial Differential Equation (PDE) stirred by an additive random forcing term which is delta-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space which aims at transferring energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We here present novel simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudo-spectral simulations. We show that the novel algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two and three spatial dimensions, and we display the solutions both in physical and Fourier spaces. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities.

physics.flu-dyn↗

arXiv2023

Chasing Gravitational Waves with the Cherenkov Telescope Array

The detection of gravitational waves from a binary neutron star merger by Advanced LIGO and Advanced Virgo (GW170817), along with the discovery of the electromagnetic counterparts of this gravitational wave event, ushered in a new era of multimessenger astronomy, providing the first direct evidence that BNS mergers are progenitors of short gamma-ray bursts (GRBs). Such events may also produce very-high-energy (VHE, > 100GeV) photons which have yet to be detected in coincidence with a gravitational wave signal. The Cherenkov Telescope Array (CTA) is a next-generation VHE observatory which aims to be indispensable in this search, with an unparalleled sensitivity and ability to slew anywhere on the sky within a few tens of seconds. New observing modes and follow-up strategies are being developed for CTA to rapidly cover localization areas of gravitational wave events that are typically larger than the CTA field of view. This work will evaluate and provide estimations on the expected number of of gravitational wave events that will be observable with CTA, considering both on- and off-axis emission. In addition, we will present and discuss the prospects of potential follow-up strategies with CTA.

astro-ph.HE↗

arXiv2023

A numerical method for wave-structure interactions in the Boussinesq regime

The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.

math.NA↗

arXiv2023

A Linear Stochastic Model of Turbulent Cascades and Fractional Fields

Turbulent cascades characterize the transfer of energy injected by a random force at large scales towards the small scales. In hydrodynamic turbulence, when the Reynolds number is large, the velocity field of the fluid becomes irregular and the rate of energy dissipation remains bounded from below even if the fluid viscosity tends to zero. A mathematical description of the turbulent cascade is a very active research topic since the pioneering work of Kolmogorov in hydrodynamic turbulence and that of Zakharov in wave turbulence. In both cases, these turbulent cascade mechanisms imply power-law behaviors of several statistical quantities such as power spectral densities. For a long time, these cascades were believed to be associated with nonlinear interactions, but recent works have shown that they can also take place in a dynamics governed by a linear equation with a differential operator of degree 0. In this spirit, we construct a linear equation that mimics the phenomenology of energy cascades when the external force is a statistically homogeneous and stationary stochastic process. In the Fourier variable, this equation can be seen as a linear transport equation, which corresponds to an operator of degree 0 in physical space. Our results give a complete characterization of the solution: it is smooth at any finite time, and, up to smaller order corrections, it converges to a fractional Gaussian field at infinite time.

math-ph↗

arXiv2021

Freely floating objects on a fluid governed by the Boussinesq equations

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis.

math.AP↗