Supercavitating Flows–Small Perturbation Theory

Abstract

This paper contains analytical results which are both old and new. It also contains an historical introduction briefly outlining events leading up to the formulation of linearized theory and indicating the scope of its subsequent development and engineering applications. A large number of references to the extensive literature are presented. The new analytical results include: The logical development of the linearized theory of steady supercavitating flows; the presentation of a general solution for the linearized problem; introduction of a new nonlinear model for finite cavity flows; the formulation of a simple second-order theory; and the application of this theory to the case of flat plates and symmetric wedges. The logical development of the linearized theory involves a small perturbation expansion of the nonlinear problem in which the dependent variable is the complex function In Ψ' and the independent variable is the complex potential Ψ; the problem is formulated for a nonlinear finite-cavity model which features cavity termination in spiral vortices followed by a "smooth" wake which is closed at infinity. The resulting boundary-value problems are of identical form for both the first and second-order theories. The first-order theory is identical to the usual linearized theory, which has previously been formulated by more intuitive means. The simple second-order theory leads in the special cases of inclined flat plates and symmetric wedges in supercavitating flow to results for lift and drag in simple (no quadratures necessary) terms of the first-order results. The old analytical results contained herein refer particularly to: The hydrofoil-airfoil equivalence and results related thereto, which has been so useful in practice for designing hydrofoils at zero cavitation number; and also to problems involving hydrofoils operating at high speeds beneath (hydrofoil boats) and over (supercavitating propeller sections) free surfaces.