Three-wave interactions among surface gravity waves in a cylindrical container

Abstract

The motion of a container filled with fluid perturbs the free surface and may result in spilling. In practice, most of the energy is localized in the modes of lowest frequencies (the gravest modes), and sloshing can be predicted once the dynamics of these modes is known. In this Rapid Communication, we investigate the nonlinear interactions between such grave modes in a cylindrical container. We first show that energy can be transferred from modes to modes with three-wave interactions: we derive the resonance conditions and characterize the early stage of this interaction. This result strongly contrasts with resonant interactions between surface gravity waves in extended domains such as the ocean, which involve at least four waves and are thus less efficient. An experiment is then performed to provide evidence of these nonlinear interactions.PACS numbers: 46.40.Ff, 47.35.Bb Introduction. Although the study of sloshing is almost two centuries old [1], it is still extremely challenging to predict the surface deformation of a container undergoing back-and-forth oscillations. The main source of difficulties is nonlinearities, which must be taken into account starting from very small amplitudes: The experiments of Cocciaro et al. [2,3] have proved that the first nonlinearity to occur corrects the dissipation, and it has been recently demonstrated that, in perfect wetting, this effect becomes sizable when the sloshing amplitude compares to the thickness of the boundary layers (a fraction of millimetre in everyday life containers) [4]. This explains why, whereas natural frequencies are experimentally found in good agreement with linear sloshing theory [2, 3, 5-7], damping rates involve larger discrepancies [2,[5][6][7].As the forcing amplitude is further increased, other nonlinear phenomena arise. The natural frequencies evolve with the oscillation amplitude [8,9], and spatial and temporal harmonics appear, referred to as bound waves in oceanography and first studied by Stokes for plane waves [10]. The dynamics of the different modes get coupled by nonlinearities: for a cylindrical container, back-and-forth displacement at the gravest eigenfrequency may, for instance, result in a pattern rotating with a periodic or a chaotic dynamics [11,12]. Energy can also be transferred between modes of different eigenfrequencies by resonant interactions. Such energy transfers have been mainly studied in the context of oceanography, i.e., for surface gravity waves in a fluid of infinite depth and without lateral boundaries. In that case, resonant interactions involve at least four waves [13], although three-wave interactions have been recently reported in the presence of a turbulent wave background [14,15]. The associated growth rate and phase locking have been first computed by Longuet-Higgins [16], and accurately describe experimental results for a large range of parameters (see [17] and references therein).In this Rapid Communication, we investigate how these resonant energy transfers are affected by the presence of solid boun...

Experiments in Surface Gravity–Capillary Wave Turbulence

“…Laboratory observations of the inverse cascade in gravity wave turbulence were limited to a narrow inertial range due to the small size of the container used (Deike et al 2011), while recent attempts have been inconclusive using a larger cylindrical basin (Campagne et al 2019a) or another type of forcing (Nazarenko & Lukaschuk 2016). Michel (2019) showed that in a cylindrical container, gravity waves may sustain three-wave resonant interactions due to confinement as a consequence of a new conserved quantity (angular pseudo-momentum). These interactions have been evidenced in a gravity wave turbulence experiment in a high-gravity environment (Cazaubiel et al 2019b).…”

Wave Turbulence on the Surface of a Fluid in a High-Gravity Environment

“…Although forbidden by weak turbulence theory for plane gravity waves, 3-wave interactions are authorized theoretically in cylindrical containers where axisymmetric eigenmodes are important [28]. Indeed, axisymmetric modes imply a new conserved quantity (angular pseudomomentum) [28]. The large-scale axisymmetric modes being important in our study, they thus modify the type of interaction mechanism for gravity waves, and probably change the one for capillary waves.…”

“…In the gravity regime, we find that 3-wave interactions occur (c ≈ 0, b = 0). Although forbidden by weak turbulence theory for plane gravity waves, 3-wave interactions are authorized theoretically in cylindrical containers where axisymmetric eigenmodes are important [28]. Indeed, axisymmetric modes imply a new conserved quantity (angular pseudomomentum) [28].…”

Forced three-wave interactions of capillary-gravity surface waves

“…For laboratory generated waves, the size and the shape of the container can also play a role. Recently, an experimental and theoretical study [33] has indeed demonstrated that in a confined cylindrical geometry the large modes of gravity surface waves interact with each other by a three-wave resonant process, whereas plane gravity waves are subjected only to the four-wave resonant interaction mechanism [32]. Here, despite the cylindrical shape of the container, the mother waves are forced as plane waves and at frequencies significantly larger than those corresponding to the first tank eigen-modes.…”