Surface gravity waves in deep fluid at vertical shear flows

Abstract

Special features of surface gravity waves in deep fluid flow with constant vertical shear of velocity is studied. It is found that the mean flow velocity shear leads to non-trivial modification of surface gravity wave modes dispersive characteristics. Moreover, the shear induces generation of surface gravity waves by internal vortex mode perturbations. The performed analytical and numerical study provides, that surface gravity waves are effectively generated by the internal perturbations at high shear rates. The generation is different for the waves propagating in the different directions. Generation of surface gravity waves propagating along the main flow considerably exceeds the generation of surface gravity waves in the opposite direction for relatively small shear rates, whereas the later wave is generated more effectively for the high shear rates. From the mathematical point of view the wave generation is caused by non self-adjointness of the linear operators that describe the shear flow.

Disturbance growth in two-fluid channel flow: The role of capillarity

“…4 was not seen in previous studies in which interfacial tension was neglected (South andHooper, 1999, 2007) but ''double-peak" behavior has been previously pointed out by Olsson and Henningson (1995) for watertable flow. Oscillatory growth in other non-normal systems has been linked to global instability (Coppola and DeLuca, 2006) and appears to be a generic feature of systems which support different types of modes (here capillary and shear modes) between which energy can be exchanged (Chagelishvili et al, 1997(Chagelishvili et al, , 2005.…”

10.1007/s11447-008-1012-5

“…Taking the divergence of the first three equations in (6), we obtain the equation for pressure (9) where Δ is the Laplace operator. Using (9) to eliminate P from the second equation in (6), we find that the transverse velocity component is governed by the equation (10) which implies, in particular, that if Δv = 0 at t = 0, then Δv = 0 at any t > 0.…”

“…It should be noted that the shear direction plays a crucial role in instability development. It was shown in a recent study [9] that only gravity wave dispersion is modified (though nontrivially) in a vertically sheared flow.…”

Wave generation on an interface by vortex disturbances in a shear flow

“…We note that in the framework of the so-called non-modal approach, a number of general features of the interaction between the wave and vortex disturbances in shear flows were studied in [6][7][8][9][10][11]. As applied to the problem of excitation of surface gravity waves (the upper layer is vacuum), this interaction was considered in [10]; however, the resonant nature of the interaction has not been investigated in [10].…”

“…As applied to the problem of excitation of surface gravity waves (the upper layer is vacuum), this interaction was considered in [10]; however, the resonant nature of the interaction has not been investigated in [10].…”