Nonlinear hydroelastic waves on a linear shear current at finite depth

Abstract

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

Progressive flexural–gravity waves with constant vorticity

“…It is noted that the translating speed c and the Bernoulli constant B are also unknowns and need to be determined together with η(ξ ). The detailed derivation of (3.41) can be found, for example, in Gao et al (2019). The pseudo-differential operator T is defined as…”

“…Whether the middle layer will completely disappear depends on the bifurcation mechanism of hydroelastic solitary waves. More precisely, if the associated nonlinear Schrödinger equation is of focusing type at c min , indicating that hydroelastic solitary waves bifurcate from infinitesimal periodic waves (Gao et al 2019), then the vertical-transport layer will vanish as the wave speed reaches c min , but not vice versa.…”

“…Following Gao et al. (2019), the problem can be handled by using a conformal transformation that maps the physical fluid domain to a strip in a complex plane. For travelling waves, after the transformation the unknown free surface can be parametrised by , which satisfies an integro-differential equation where , is the Jacobian of the map, and the curvature in the new plane is of the form It is noted that the translating speed and the Bernoulli constant are also unknowns and need to be determined together with .…”

“…The validity and accuracy of such schemes have been checked by several groups, and the interested reader is referred to Vanden-Broeck (1994), Choi (2009) and Gao et al. (2019) and references therein for more details.…”

Hydroelastic solitary waves with constant vorticity

“…Thus the one parameter function γ(k * , Ω) ≡ γ * (Ω) defines whether the NLS is focussing type. From [48], it is found that γ * (0) ≈ −0.0067 and that γ * < 0 for Ω > Ω * and γ * > 0 for Ω < Ω * where Ω * ≈ −0.0076. From this we can conclude that for Ω < Ω * we expect solitary waves to bifurcate from the trivial state, whereas for Ω > Ω * they may bifurcate at finite amplitude (see the Ω = 0 case [47]).…”