Numerical Investigations of Three and Two Dimensional Free Boundary Problems

这项工作是关于自由表面流动的理论。这里的自由表面是指两种流体之间的界面。这样一个曲面的位置是先验未知的，必须作为解的一部分来寻找。这导致了一个高度非线性的数学问题，PI将为此开发新的数值和分析方法。目的是发展三维非线性自由表面流动和二维界面流动的数值和解析理论。具体来说：(A)推导出三维非线性自由表面流动的有效而精确的数值方法。(B)将这些新方案与分析技术结合起来，进一步理解三维自由表面流动。这包括将线性和弱非线性理论扩展到完全非线性区域，并发现这些流动的极限构型。(C)检查现有船舶水动力近似的有效性并加以改进。(D)研究在两种等密度流体界面处传播的新型波的存在性。这些波应该具有介于“广义孤立波”和“经典锋面”之间的性质。因此，把它们称为“广义战线”是恰当的。它们的存在在以前的工作中得到了强烈的暗示，但它们并没有被明确地计算出来。(E)计算广义锋面并对其稳定性进行数值研究。自由表面流动在科学和日常生活的许多方面都很常见。例如海滩上的波浪，香槟中升起的气泡，融化的冰，从容器中流出的水流，以及在风中飘扬的帆。在这些例子中，自由表面是海洋表面、气体和香槟之间的界面、冰的表面、倾流的边界和帆的表面。PI将集中在流体力学方面的应用。然而，所开发的方法是通用的，在流体力学之外也有应用，PI也建议对它们进行研究。该研究将有利于研究非线性自由曲面流动的应用数学家和需要精确方案来求解三维自由曲面流动的应用科学家。所提出的应用于船舶产生的非线性波形是与船舶流体力学有关的。尖界面上波浪的研究在海洋学中有应用。在海洋、湖泊和大气中，由于温度、盐度或悬浮量的差异，密度不同的相邻质量之间形成了这种尖锐的界面。研究如此清晰的边界的原因之一是，它们出现在表面的锋线上，灰尘、泡沫、木材和其他物质聚集在那里。提出的许多问题都可以与研究生合作解决。

The work is concerned with the theory of free surface flows. Here a freesurface refers to an interface between two fluids. The position of such a surface is notknown a priori and has to be found as part of the solution. This leadsto a highly nonlinear mathematical problem, for which the PI willdevelop new numerical and analytical approaches. The objective is to develop numerically and analytically theories forthree-dimensional nonlinear free surface flows and for two-dimensionalinterfacial 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 furtherthe understanding of three-dimensional free surface flows. Thisincludes extending linear and weakly nonlinear theories to thefully nonlinear regime and discovering the limiting configurationsof these flows.(C) Check the validity of existing approximations in ship hydrodynamics and improve them.(D) Investigate the existence of new types of waves propagatingat the interface between two fluids of constant densities. These waves shouldhave properties intermediate between those of ``generalized solitary waves"and those of ``classical fronts". Therefore it is appropriate todescribe them as ``generalized fronts". Their existence is strongly suggested by previous work but theyhave not been calculated explicitly.(E) Calculate the generalized fronts and perform a numerical study oftheir 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 interfacebetween the gas and the champagne, the surface of the ice, the boundaryof the pouring flow and the surface of the sail. The PI will concentrateon applications arising from fluid mechanics. However the methods developed aregeneral and have applications outside fluid mechanics and the PI proposes toinvestigate them as well. The research will benefit applied mathematicians interested in nonlinearfree surface flows and applied scientists in need of accurate schemesto solve three-dimensional free surface flows. The proposed applicationto the nonlinear wave pattern generated by a ship is relevant toship hydrodynamics. The study of waves on sharp interfaces has applications in oceanography. Such sharp interfaces form in oceans, lakesand the atmosphere between adjacent masses of different density associated withdifferences in temperature, salinity or amount of suspension. One of thereasons for studying such sharp boundaries is that they appear at the surface asfrontal lines where dust, foam, timber and others accumulate.Many of the problems proposed can be done incollaboration with graduate students.