The wedge-water-entry problem

Abstract

The wedge-entry problem can be formulated as an integral equation, involving a parameter $\beta$, $\beta$ small corresponding to wedges of semi-vertex angle just less than $\pi/2$. An existence proof for the wedge-entry problem is given in a Report (or unpublished notes) by McLeod and Fraenkel, henceforth denoted [LP].

G.Keady's 1997 computations

These are announced in a 1998 Conference Paper:
In "Engineering Mathematics and Applications Conference, EMAC98, July 1998, Adelaide". See my list of published papers. The EMAC paper is here, in postscript.

Numerics using a similar integral equation were first published in Dobrolovskaya (1969, Jnl of Fluid Mech.), and recently checked and extended to smaller $\beta$ in Zhao and Faltinsen (1993, Jnl of Fluid Mech.). This report available here contains the following.

• (I) Numerical coding and aspects of the existence work are brought closer together.
• (II)
  • Previously published numerical results are checked (and so far no serious error has been found).
  • The computations are developed as `small-machine' ones, using Matlab, in order to facilitate exploration of subproblems, of different iterations, etc..
  • The numerical solutions are checked against various bounds given in [LP].
• (III) Various items related to unsolved problems concerning thin wedges are collected together.

The separable and mixed-monotone operator structure within the problem is exposed. This was motivated by the last points on the list above - and the (long term, not presently realised) hope for a constructive existence proof.

The following are available from here.

• The draft report is given as a Postscript file. (There are times when this will have usepackage{showlabels} on, so that it is not the world's prettiest document on the Web. Some of the figures are missing as they are not yet in Postscript.)
• Some numerical codes are given:
  matlab.html
  (Left available until 2000. But removed Jan 2001 prior to revision/rewrite using Octave.)
• Various speculations about possible and worthwhile further work, cumulatively far more than I can do are given here.

G.Keady's 2001/2 Calculations for alpha small

A short announcement of this work has appeared in print in the BAIL 2002 Conference, Perth, July 2002. For details, see the list of my published papers.

This work concerns slender wedges. The outer flow has a nearly flat surface, and Mackie in the 1960s performed the linear potential theoretic calculations describing this. There are several (essentially equivalent) different formulations for integral equations describing the free boundary near the separation point. These are the main concern in the 2001/2 studies. More details.

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