Numerical Investigations of Three and Two Dimensional Free Boundary Problems

这项工作与自由表面流动理论有关。这里，自由曲面指的是两种流体之间的界面。这种曲面的位置是先验未知的，必须作为解决方案的一部分来找到。这导致了一个高度非线性的数学问题，PI将为此开发新的数值和分析方法。其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。具体地说：(A)推导出三维非线性自由面流动的有效和精确的数值方法。(B)将这些新格式与分析技术结合使用，以加深对三维自由表面流动的理解。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)检查船舶流体力学中现有近似的有效性并改进它们。(D)研究在两个恒定密度的流体之间的界面上传播的新型波的存在。这些波应该具有介于“广义孤立波”和“经典锋”之间的性质。因此，将它们描述为“广义前沿”是恰当的。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、从容器中倾泻出来的水流和在风中吹动的风帆。在这些例子中，自由表面是海面、气体和香槟的界面、冰的表面、倾倒的水流的边界和风帆的表面。PI将专注于流体力学的应用。然而，这些方法的发展是普遍的，并在流体力学之外有应用，PI建议也对它们进行研究。这项研究将有助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格式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与船舶水动力学有关。尖锐界面上的波浪研究在海洋学中具有应用价值。这种尖锐的界面在海洋、湖泊和大气中形成，相邻物质之间的密度不同，与温度、盐度或悬浮量的不同有关。研究如此尖锐的边界的原因之一是它们出现在表面作为积聚灰尘、泡沫、木材和其他物质的锋线。提出的许多问题都可以与研究生合作完成。

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.