Numerical and Analytical Studies of Wave Phenomena in Fluid Mechanics

Milewski 9704606我们建议使用数值和渐近方法来研究流体力学和波传播中的几个物理相关问题。特别是，我们建议与几位合作者一起研究：赤道开尔文波和层流中某些波的共振现象；海滩上孤立波的强相互作用；自由表面流中旋转-无旋分量的强相互作用；侧边界突变引起的非线性长波的绕射。这些项目对于发展有效的拟谱数值方法和理解一大类地球物理波传播问题中的共振效应都是有意义的。我们还继续就二维和三维自由表面重力流动的奇点和撞击刚性墙的射流的相似解、反应扩散方程中六边形图案的演变以及应用于稳定火焰锋面的动力学进行持续合作。在这项建议中，我们使用数学和计算方法来研究流体力学中具有物理和地球物理重要性的问题。具体地说，我们研究大气中的赤道和中纬度波。这些海浪被认为对全球气候有影响，因此，从数学上了解它们与其他海浪和地形特征的相互作用具有重要的实际意义。2)波浪靠近海滩时的绕射和放大。在这里，我们试图从数学上预测海滩附近的海浪的大小和模式。然后，可以将数学预测与风暴期间海浪的真实数据进行比较。3)流体自由面的形状。我们研究了不同类型的奇异自由面流动(如尖波海岸)和冲击物体的流体射流的形状。4)火焰锋面花样的演变。在这里，我们研究了火焰锋面在燃料-空气混合物中的传播和稳定性。本文所用的数学形式和高效的数值方法适用于广泛的物理问题。

Milewski 9704606 We propose to apply numerical and asymptotic methods to study several problems of physical relevance in fluid mechanics and wave propagation. In particular we propose to study, together with several collaborators: The resonance phenomena of equatorial Kelvin waves and of certain waves in stratified flows; The strong interactions of solitary waves on a beach, and of the rotational-irrotational components in free surface flows; The diffraction of nonlinear long waves due to abrupt changes in the lateral boundaries. These projects are relevant both in developing efficient psudo--spectral numerical methods and in understanding resonance effects in a large class of geophysical wave propagation problems. We also continue ongoing collaboration on Singularities in two- and three-dimensional free surface gravity flows and similarity solutions of jets impinging on rigid walls; The evolution of hexagonal patterns in reaction-diffusion equations with applications to the dynamics of stabilized flame fronts. In this proposal we use mathematical and computational methods to study problems of physical and geophysical importance in fluid mechanics. Specifically we work on 1) Equatorial and mid-latitude waves in the atmosphere. These waves are thought have an impact on the global climate and a mathematical understanding of their interaction with other waves and with topographical features is therefore of practical importance. 2) the diffraction and amplification of waves approaching a beach. Here we attempt to predict mathematically the size and pattern of ocean waves near a beach. The mathematical predictions can then be compared to real data from ocean waves during a storm. 3) Shapes of the free surfaces of fluids. We study different types of singular free surface flow (such as sharp wave coasts) and the shapes of fluid jets impinging on objects. 4) The evolution of patterns in flame-fronts. Here, we study the propagation and stability of flame fronts in a fuel-air mixture. The mathematical formulation and efficient numerical method used here is applicable to a wide class of physical problems.