Cavitation and Other Free Surface Phenomena

Abstract

This thesis develops a method of solving axisymmetric cavity flow problems using a relaxation or numerical technique. Chapter 1 contains a general review of the phenomenon of cavitation in fluids. Special reference is then made to fully developed cavities in an Euler or ideal fluid for both plane and axisymmetric flow. The basic theorems and equations are presented, with the various types of mathematical model which have been suggested. Details of the fundamental feature of this type of flow, namely the phenomenon of flow separation, are given. At the conclusion of the chapter the analytic methods of solution of plane cavitating flow and, in particular, those using the Riabouchinsky model, are outlined. The numerical results of a pertinant example of this type of flow are included in Appendix A with some additional comments on the phenomenon of choked cavity flow. Chapter 2 provides a brief account of the previous approaches to the problem of axisymmetric cavitating flow. These include; empirical results; theories based on source-sink and vortex sheet distributions; theories based on correlation with the corresponding plane flow solutions; previous applications of relaxation methods. Chapter 3 develops the basic equations for axisymmetric cavity flow in the transformed phi,psi plane in which it is proposed to solve for the dependent variable f (equal to r^2, where r is the radial variable in the physical plane). The equations prove to be of the non-linear elliptic type. Relations for the boundary conditions, and certain other relevant physical quantities, are then evolved in terms of the derivatives of f. The determinacy of the problem in this plane requires careful investigation. Special reference is made to two important phenomena; (i) that of the limiting condition of choked flow, for which certain important relations are developed and (ii) that of the two distinct types of separation in cavity flow. The derivation of expansions describing the singular behaviour of the flow in that region of the transformed plane is given in each case. Chapter 4 describes the adaptation of the results of chapter 3 to provide a numerical or relaxation method of solving axisymmetric cavity flows. The finite difference forms of the field equation and boundary conditions in the phi,psi plane are first derived. Their application is then discussed with special attention being paid to the separation point and to the free streamline, the treatment of which provides the crux of the problem. Details are then given of the treatment of the singular points, a subject which has commanded little attention in the literature for the case of nonlinear partial differential equations. Finally the application of the methods developed is summarized. Chapter 5 presents the results obtained by the author, both for the convergence of the methods and for the resulting cavity flows. Comparison is made with previous results with the corresponding plane flow solutions and with experiment. Special reference is made to the behaviour of cavity flows near the choked flow condition, results which have an added significance in view of the fact that most experiments are carried out in the restricted environment of a water tunnel. In the final section some analysis of the errors is given.

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