Title:

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 {maximized/minimized} 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 twodimensional hydrofoil of given length and prescribed mean curvature which produces {maximum lift/minimum drag} The problem is discussed in three cases when there is a {full/partial/zero} cavity flow past the hydrofoil. The liquid flow is assumed to be twodimensional steady, irrotational and incompressible and a linearized theory is assumed. Twodimensional vortex and source distributions are used to simulate the two dimensional {full/partial/zero} cavity flow past a thin hydrofoil. This method leads to a system of integral equations and these are solved exactly using the CarlemanMuskhelishvili 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 {maximized/minimized} the {lift/drag} coefficient subject to constraints on curvature and given length. The mathematical problem is that of extremizing a functional depending on {? vortex strength/ ? source strength} these three functions are related by singular integral equations. The analytical solution for the unknown shape z and the unknown singularity distribution y has branchtype singularities at the two ends of the hydrofoil. Analytical solution by singular integral equations is discussed and the approximate solution by the RayleighRitz 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 twodimensional 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 nonviscous and moving twodimensionally, 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.
