No abstract

No abstract

The Lagally theorem for unsteady flow expresses the forces and moments acting on a rigid body moving in an inviscid and incompressible fluid in terms of the singularities of the analytically continued flow within the body. Previous generalizations of the Lagally theorem, originally given by Lagally (1922) for steady flows, are due to Cummins (1957) and Landweber & Yih (1956), who consider the effect of flow unsteadiness on the forces and moments. In these, the system of image singularities within the body was assumed to consist of isolated or continuous (surface or volume) distributions of sources and doublets. A further extension of Lagally's theorem ie due to Landweber (1967), who derived expressions for the steady forces and moments acting on a rigid body generated by isolated or a continuous distribution of multipoles. The purpose of the present paper is to generalize the Lagally theorem so as to include the effects of multipoles in unsteady flow, and deformability of the body, as well as to present a briefer derivation of the resulting formulae. Two examples, illustrating the application of the force and moment formulae, will be presented.

The dynamical theory of the motion of a body through an inviscid and incompressible fluid has yielded three relations: a first, due to Kirchhoff, which expresses the force and moment acting on the body in terms of added masses; a second, initiated by Taylor, which expresses added masses in terms of singularities within the bòdy; and a third, initiated by Lagally, which expresses the forces and moments in terms of these singularities. The present investigation is concerned with generalizations of the Taylor and Lagally theorems to include unsteady flow and arbitrary translational and rotational motion of the body, to present new and simple derivations of these theorems, and to compare the Kirchhoff and Lagally methods for obtaining forces and moments. In contrast with previous generalizations, the Taylor theorem is derived when other boundaries are present; for the added-mass coefficients due to rotation alone, for which no relations were known, it is shown that these relations do not exist in general, although approximate ones are found for elongated bodies. The derivation of the Lagally theorem leads to new terms, compact expressions for the force and moment, and the complete expressions of the forces and moments in terms of singularities for elongated bodies.

An iteration formula for Fredholm integral equations of the first kind is applied in two new methods for obtaining the steady, irrotational, axisymmetric flow of an inviscid, incompressible fluid about a body of revolution. In the first method a continuous, axial distribution of doublets is sought as a solution of an integral equation of the first kind. A method of determining the end points and the initial trends of the distribution, and a first approximation to a solution of the integral equation are given. This approximation is then used to obtain a sequence of successive approximations whose successive differences furnish a geometric measure of the accuracy of an approximation. When a doublet distribution has been assumed, the velocity and pressure can be computed by means of formulas which are also given. In the second method the velocity is given directly as the solution of an integral equation of the first kind. Here also a first approximation is derived and applied to obtain a sequence of successive approximations. In contrast with the first method, which, in general, can give only an approximate solution, the integral equation of the second method has an exact solution. Both methods are illustrated in detail by an example. The results are compared with those obtained by other well-known methods.