Vorticity effects on nonlinear wave–current interactions in deep water

Abstract

The effects of uniform vorticity on a train of 'gentle' and 'steep' deep-water waves interacting with underlying flows are investigated through a fully nonlinear boundary integral method. It is shown that wave blocking and breaking can be more prominent depending on the magnitude and direction of the shear flow. Reflection continues to occur when sufficiently strong adverse currents are imposed on 'gentle' deep-water waves, though now affected by vorticity. For increasingly positive values of vorticity, the induced shear flow reduces the speed of right-going progressive waves, introducing significant changes to the free-surface profile until waves are completely blocked by the underlying current. A plunging breaker is formed at the blocking point when 'steep' deep-water waves interact with strong adverse currents. Conversely negative vorticities augment the speed of right-going progressive waves, with wave breaking being detected for strong opposing currents. The time of breaking is sensitive to the vorticity's sign and magnitude, with wave breaking occurring later for negative values of vorticity. Stopping velocities according to nonlinear wave theory proved to be sufficient to cause wave blocking and breaking.

Stability analysis of deep-water waves on a linear shear current using unsteady conformal mapping

“…It should be noted that, although the waves of interest are irrotational even for Ω = 0, in this paper, only the waves without a shear current are referred to as irrotational waves (Ω = 0). On the basis of this formulation, the full Euler system has been numerically solved by Simmen & Saffman (1985), Teles da Silva & Peregrine (1988) and Vanden-Broeck (1996) for steady waves, and by Choi (2009) and Moreira & Chacaltana (2015) for unsteady waves. It has been known that the limiting form of steady waves of symmetric profile can have a corner at the wave crest with an inner angle of 120 • which is independent of the shear strength Ω (Milne-Thomson 1968, p. 403, §14.50), similarly to the case of irrotational waves (Ω = 0).…”

Approximate Dispersion Relations for Waves on Arbitrary Shear Flows

“…Key nonlinear treatments are those of Simmen and Saffman () and Teles da Silva and Peregrine (). Of particular interest is the numerical scheme for fully nonlinear stationary and solitary waves on arbitrary 2D shear currents due to Dalrymple (), and followed by further numerical work on this question (Ko & Strauss, ; Moreira & Chacaltana, ; Nwogu, ). The present effort is restricted to linear waves; a comparison of the effect on wave speed of shear versus that of nonlinearity is an important topic for the future; refer to Swan and James () for some results up to second order in wave steepness.…”

Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

“…For a review on the recent rigorous results, the reader can refer to Constantin & Varvaruca (2011) and Kozlov & Kuznetsov (2014). Among the authors using asymptotic methods or purely numerical methods, on can cite Tsao (1959), Dalrymple (1974), Brevik (1979), Simmen & Saffman (1985), Teles da Silva & Peregrine (1988) , Kishida & Sobey (1988), Vanden-Broeck (1996), Swan & James (2001), Ko & Strauss (2008), Pak & Chow (2009), Cheng, Cang & Liao (2009 , Moreira & Chacaltana (2015), Hsu, Francius, Montalvo & Kharif (2016), Ribeiro-Jr, Milewski & Nachbin (2017). Although the recent important theoretical developments have confirmed that periodic waves can exist over flows with arbitrary vorticity, it appears that their stability to infinitesimal disturbances and their subsequent nonlinear evolution have not been studied extensively so far.…”