Applications of variational theory in certain optimum shape problems in hydrodynamics.

PART I In a recent paper Wu, T.Y. & Whitney, A.K., the authors studied optimum shape problems in hydrodynamics. These problems are stated in the form of a singular integral equation depending on the unknown shape and an unknown singularity distribution; the shape is then to be determined so that some given performance criterion has to be {lcub}maximized/minimized{rcub} In the optimum problem to be studied in this part we continue to assume that the state equation is a linear integral equation but we generalize the Wu & Whitney theory in two different ways. This method is applied to functional of quadratic form and a necessary condition for the extremum to be a minimum is derived. PART II The purpose of this part is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces {lcub}maximum lift/minimum drag{rcub} The problem is discussed in three cases when there is a {lcub}full/partial/zero{rcub} cavity flow past the hydrofoil. The liquid flow is assumed to be two-dimensional steady, irrotational and incompressible and a linearized theory is assumed. Two-dimensional vortex and source distributions are used to simulate the two dimensional {lcub}full/partial/zero{rcub} cavity flow past a thin hydrofoil. This method leads to a system of integral equations and these are solved exactly using the Carleman-Muskhelishvili technique. This method is similar to that used by Davies, T.V. We use variational calculus techniques to obtain the optimum shape of the hydrofoil in order to {lcub}maximized/minimized{rcub} the {lcub}lift/drag{rcub} coefficient subject to constraints on curvature and given length. The mathematical problem is that of extremizing a functional depending on {lcub}? vortex strength/ ? source strength{rcub} these three functions are related by singular integral equations. The analytical solution for the unknown shape z and the unknown singularity distribution y has branch-type singularities at the two ends of the hydrofoil. Analytical solution by singular integral equations is discussed and the approximate solution by the Rayleigh-Ritz method is derived. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. PART III The purpose of this work is to evaluate the optimum shape of a two-dimensional hydrofoil of given length and prescribed mean curvature which produces minimum drag. A thin hydrofoil of arbitrary shape is in steady, rectilinear, horizontal motion at a depth h beneath the free surface of a liquid. The usual assumptions in problems of this kind are taken as a basis, namely, the liquid is non-viscous and moving two-dimensionally, steadily and without vorticity, the only force acting on it is gravity. With these assumptions together with a linearization assumption we determine the forces, due to the hydrofoil beneath a free surface of the liquid. We use variational calculus techniques similar to those used in Part II to obtain the optimum shape so that the drag is minimized. A sufficient condition for the extremum to be a minimum is derived from consideration of the second variation. In this part some general expressions are established concerning the forces acting on a submerged vortex and source element beneath a free surface using Blasius theorem.