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Nonlinear Analysis of Three-Dimensional Water-Wave Patterns via Exponential Asymptotics

通过指数渐近法对三维水波模式进行非线性分析

基本信息

批准号：
2004589
负责人：
Triantaphyllos Akylas
金额：
$ 34.2万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2004589&HistoricalAwards=false
关键词：
Nonlinear Analysis Three Dimensional Water

项目摘要

Three-dimensional water-wave patterns due to moving surface and submerged sources, such as ships and submarines, or due to currents flowing over seamounts and underwater ridges, are familiar from everyday experience. Apart from their aesthetic appeal, these wave phenomena also are fundamental to applications in various scientific fields, including geophysical fluid dynamics, ship hydrodynamics, oceanography, and meteorology. This research project will develop a novel mathematical methodology for analyzing three-dimensional water-wave patterns. In contrast to all previous analytical treatments, the new approach will not be restricted to waves of small steepness and aims to explain theoretically several striking features revealed by recent numerical simulations of water waves induced by currents over uneven ocean bottom topography. The project will also involve graduate students in the research.The generation of steady three-dimensional (3D) water-wave patterns by moving surface and submerged sources is a fundamental topic of fluid dynamics with applications to geophysical fluid modeling, oceanography, and ship hydrodynamics. The usual way of tackling these problems analytically is based on the linearized water-wave equations, assuming infinitesimally small disturbances. Departing from such linear analysis, this project will devise an analytical (asymptotic) technique that will enable nonlinear treatment of steady 3D free-surface gravity wave patterns due to various types of wave sources. Motivation comes from the recent discovery (by the PI's group) of a nonlinear mechanism that controls the amplitude of 2D forced gravity waves for a broad range of flow conditions, limiting severely the validity of linear analysis. This mechanism depends on all orders of the disturbance amplitude (nonlinearity parameter) and can be captured by perturbation expansions that go beyond all orders in the nonlinearity parameter. This project will develop such a beyond-all-orders (exponential asymptotics) technique suitable for 3D wave patterns. This new methodology will be used to advance understanding of nonlinear effects in 3D forced gravity water waves in two distinct flow regimes: (i) waves on water of finite depth due to a stream over fully localized topography or due to a localized pressure patch moving on the free surface; and (ii) deep-water waves due to flow past a point source or doublet submerged at finite depth from the free surface or due to a pressure patch moving on the free surface. These problems model (i) oceanic flows over seamounts and underwater ridges as well as ship wakes in shallow water; and (ii) ship wave patterns on deep water. The asymptotic analysis will aim to explain theoretically several striking nonlinear features revealed by recent numerical simulations of these mathematical models and to shed light on novel nonlinear aspects that as yet have not been captured numerically.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

由于移动的水面和潜水源，如船舶和潜艇，或由于流经海山和水下海脊的水流，三维水波图案在日常经验中很常见。除了它们的美学吸引力外，这些波浪现象也是各种科学领域应用的基础，包括地球物理流体动力学、船舶流体动力学、海洋学和气象学。这项研究项目将开发一种新的数学方法来分析三维水波模式。与所有以前的分析处理方法不同，新方法将不限于小陡度的海浪，其目的是从理论上解释最近对不均匀海底地形上的洋流诱导的水波进行的数值模拟所揭示的几个显著特征。该项目还将吸引研究生参与研究。通过移动的表面和水下源产生稳定的三维(3D)水波模式是流体力学的一个基本课题，应用于地球物理流体模拟、海洋学和船舶流体动力学。解析地处理这些问题的通常方法是基于线性化的水波方程，假定有无限小的扰动。从这种线性分析出发，这个项目将设计出一种分析(渐近)技术，它将能够非线性处理由各种类型的波源引起的稳定的3D自由表面重力波模式。动力来自于(由Pi的团队)最近发现的一种非线性机制，该机制可以在大范围的流动条件下控制2D强迫重力波的幅度，这严重限制了线性分析的有效性。这种机制依赖于所有阶次的扰动幅度(非线性参数)，并且可以通过超出非线性参数的所有阶数的微扰展开来捕获。该项目将开发一种适用于3D波形的超所有阶(指数渐近)技术。这种新的方法将被用来促进对两种不同流型下的三维强迫重力水波的非线性效应的理解：(I)由于完全局部化地形上的水流或由于在自由表面上移动的局部压力块而在有限深度的水中产生的波；(Ii)由于从自由表面经过有限深度处淹没的点源或二重流或由于压力块在自由表面上移动而产生的深水波。这些问题模拟(I)越过海山和水下海脊的海洋流动以及浅水中的船舶尾迹；以及(Ii)深水中的船舶波型。渐近分析旨在从理论上解释这些数学模型最近的数值模拟所揭示的几个显著的非线性特征，并阐明尚未从数字上捕捉到的新的非线性方面。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。