Numerical Investigations of Three and Two Dimensional Free Boundary Problems

New Jersey Institute Of Technology, Newark NJ

Investigators

Abstract

The work is concerned with the theory of free surface flows. Here a free surface refers to an interface between two fluids. The position of such a surface is not known a priori and has to be found as part of the solution. This leads to a highly nonlinear mathematical problem, for which the PI will develop new numerical and analytical approaches. The objective is to develop numerically and analytically theories for three-dimensional nonlinear free surface flows and for two-dimensional interfacial flows. Specifically: (A) Derive efficient and accurate numerical approaches for three-dimensional nonlinear free surface flows. (B) Use these new schemes together with analytical techniques to further the understanding of three-dimensional free surface flows. This includes extending linear and weakly nonlinear theories to the fully nonlinear regime and discovering the limiting configurations of these flows. (C) Check the validity of existing approximations in ship hydrodynamics and improve them. (D) Investigate the existence of new types of waves propagating at the interface between two fluids of constant densities. These waves should have properties intermediate between those of ``generalized solitary waves" and those of ``classical fronts". Therefore it is appropriate to describe them as ``generalized fronts". Their existence is strongly suggested by previous work but they have not been calculated explicitly. (E) Calculate the generalized fronts and perform a numerical study of their stability. Free surface flows are common in many aspects of science and everyday life. Examples are waves on a beach, bubbles rising in a glass of champagne, melting ice, pouring flows from a container and sails blowing in the wind. In these examples the free surface is the surface of the sea, the interface between the gas and the champagne, the surface of the ice, the boundary of the pouring flow and the surface of the sail. The PI will concentrate on applications arising from fluid mechanics. However the methods developed are general and have applications outside fluid mechanics and the PI proposes to investigate them as well. The research will benefit applied mathematicians interested in nonlinear free surface flows and applied scientists in need of accurate schemes to solve three-dimensional free surface flows. The proposed application to the nonlinear wave pattern generated by a ship is relevant to ship hydrodynamics. The study of waves on sharp interfaces has applications in oceanography. Such sharp interfaces form in oceans, lakes and the atmosphere between adjacent masses of different density associated with differences in temperature, salinity or amount of suspension. One of the reasons for studying such sharp boundaries is that they appear at the surface as frontal lines where dust, foam, timber and others accumulate. Many of the problems proposed can be done in collaboration with graduate students.