On the forming of sharp corners at a free surface

Abstract

By applying the technique for time-dependent irrotational flows proposed in the preceding paper, a new class of exact free-surface flows is derived. In these, the free surface has the form of a variable hyperbola, whose axes rotate in space. The angle γ between the asymptotes, and the angle δ of orientation of the axes, are found explicitly as functions of the time. The solutions fall into three groups. First there are those in which γ diminishes smoothly from 90° (a rectangular hyperbola) to zero (a slender hyperbola) while the angle of orientation 8 increases towards a finite limit. Secondly there are solutions in which γ diminishes to a positive minimum, and then returns again to 90°. Thirdly γ may begin from small values, increase to less than 45° and return again towards zero. In each case the total angle δ max remains finite. An exceptional but very interesting solution, in which the vertex angle γ remains constant at 45° and the free surface rotates with uniform angular velocity δ, is described in terms of elementary functions of the time t . It is suggested that these flows, which are generalizations of the symmetrical Dirichlet hyperbolae, are relevant to the flow near the tip of a breaking gravity wave. Since for large values of t the angle δ tends to its limit like t ~ 1 , the flows may be matched asymptotically to the parabolic arch of a plunging breaker. In other cases, the tip of the wave can curl over and appear to form a vortex. By the inclusion of terms cubic in the space coordinates it is also possible to represent a sharp crest pointing upwards and tending towards a cusp.

Singularities in the complex physical plane for deep water waves

“…The form of the solutions appears very similar to those observed in numerical simulations and in observations of actual breaking waves. We reproduce the results in Longuet-Higgins (1980) for Lagrangian motion and show clearly the consistency of the results with the presence of pole singularities in the complex arclength plane. By using a variant of the Taylor condition, a model for the inside curl of a breaking wave has been developed by Longuet-Higgins (1982) that agrees very well with observed surface profiles.…”

“…In Appendix B, we reproduce the solution found by Longuet-Higgins (1980) but in terms of Lagrangian motion. We show that…”

“…In addition, local solutions have been found that match the surface profile at the tip (Longuet-Higgins 1980) and the curling forward face of a breaking wave (LonguetHiggins 1982). In Appendix B, we reproduce the solution found by Longuet-Higgins (1980) but in terms of Lagrangian motion.…”

“…A family of solutions that take the form of hyperbolae in free fall has been proposed as candidates for the tip of a plunging breaker by Longuet-Higgins (1972, 1980. The solutions are exact, but should be viewed as the leading-order contribution to a local expansion near the tip.…”

“…In all cases R(p) > 0, even when the wave has clearly broken at t = 3.3 (as shown in figure 2). However, R(p) approaches zero at the tip of the plunging breaker, located near p = 3.2, indicating that the tip approaches free fall, one of the assumptions in Longuet-Higgins (1980). In Longuet-Higgins (1982), the assumption is made that r(p, t) = R(p, t)/s p (p, t) is only a function of time.…”