Numerical Investigations of Three and Two Dimensional Free Boundary Problems

这项工作涉及自由表面流动的理论。 这里，自由表面指的是两种流体之间的界面。 这样一个表面的位置不是先验已知的，必须作为解的一部分来找到。这导致了一个高度非线性的数学问题，PI将开发新的数值和分析方法。目标是发展三维非线性自由表面流和二维界面流的数值和解析理论。具体而言：（A）推导出三维非线性自由表面流动的有效和精确的数值方法。(B)使用这些新的方案与分析技术，以进一步了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)检查船舶水动力学中现有近似的有效性并改进它们。(D)研究在两种恒定密度流体的界面上传播的新型波的存在性。这些波的性质应该介于“广义孤立波”和“经典波前”之间。因此，将其描述为"广义战线”是恰当的。他们的存在是强烈建议由以前的工作，但他们还没有被明确计算。(E)计算广义锋并对其稳定性进行数值研究。自由表面流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，香槟酒中升起的气泡，融化的冰，容器中的倾倒流和风中吹的帆。在这些例子中，自由表面是海洋表面，气体和香槟酒之间的界面，冰的表面，the boundary边界of the pouring倾注flow流and the surface表面of the sail帆. PI将专注于流体力学的应用。然而，开发的方法是通用的，并有流体力学以外的应用和PI建议调查他们以及. 这一研究将有利于对非线性自由表面流动感兴趣的应用数学家和需要精确方案来求解三维自由表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型是与船舶水动力学有关的。研究尖锐界面上的波在海洋学中有应用。在海洋、湖泊和大气中，由于温度、盐度或悬浮量的不同，密度不同的相邻物质之间形成了这种尖锐的界面。研究这种尖锐边界的原因之一是，它们在表面上看起来像锋线，灰尘、泡沫、木材和其他东西都在那里堆积。

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.