Quasi-periodic Water Waves and Their Stability

准周期水波及其稳定性

• 批准号: 1716560
• 负责人: Jon Wilkening
• 金额: $ 27.5万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-15 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1716560&HistoricalAwards=false
• 关键词: Quasi; periodic; Water; Waves; Their

Understanding the complex dynamics of water waves is essential in many engineering applications, including power generation from ocean waves, early detection of tsunamis, and safeguarding coastal power plants. While pure traveling waves and pure standing waves have been studied extensively in the past, real ocean waves generally contain more than two dominant frequencies and wavelengths. This project will develop mathematical and numerical techniques for studying quasi-periodic water waves. Traveling and standing waves are special cases with one and two quasi-periods, respectively. The stability of these waves will also be investigated, including stability transitions. Of course, only stable solutions will be seen in the ocean or laboratory, but unstable solutions can be computed numerically, often completing the picture of how the stable solutions fit together. These methods will also be used to compute microseisms, which are geophysical elastic waves in the sea bed generated by nearly coherent standing waves at the ocean surface, and Faraday waves, which are surface waves on water or oil in a wave tank that self-organize into various standing wave patterns when the container is driven to oscillate with a prescribed motion. The numerical results of this project will be compared with wave tank experiments done by Diane Henderson's group at Penn State in the William G. Pritchard Fluid Mechanics Laboratory. Additional broader impacts include course and curriculum development, organization of seminars and minisymposia, advising of graduate students, and development of new computational tools with many applications beyond water waves.The first technical goal of the project is to devise and implement a generalized shooting method for computing quasi-periodic solutions of differential equations. Several new types of solutions of the free-surface Euler equations are expected to be found, including traveling-standing waves, KdV-like elastic collisions, and NLS-like breathers. Harmonic and subharmonic stability of standing waves and other relative periodic solutions will also be determined, with the goal of studying orbital stability, long-time dynamics, and quasi-periodic perturbations of relative-periodic solutions. Subharmonic stability will be investigated using Bloch theory in space and Floquet theory in time, with solutions of the linearized Euler equations computed in parallel batches to compute the monodromy operator. Standing-wave analogues of the Benjamin-Feir instability will also be studied. A new approach to computing cyclic steady states and stability transitions of parametrically driven Faraday wave systems will also be developed, along with a new algorithm for computing the Dirichlet-Neumann operator in a cylindrical geometry using tensor products of orthogonal polynomials tailored to the cylindrical geometry and a variant of the Transformed Field Expansion technique. The Arnoldi algorithm will be used to compute the eigenvalues of the monodromy operator for the Faraday wave problem to obtain the largest Floquet multipliers. The method can track families of solutions through unstable branches to yield a more complete picture of the stability transitions that affect pattern formation. Quasi-periodic forcing of Faraday waves will also be investigated.

了解水波的复杂动力学在许多工程应用中至关重要，包括海浪发电，海啸的早期检测和保护沿海发电厂。虽然过去已经广泛地研究了纯行波和纯驻波，但是真实的海浪通常包含两个以上的主频率和波长。该项目将开发用于研究准周期水波的数学和数值技术。行波和驻波分别是具有一个和两个准周期的特殊情况。 这些波的稳定性也将被调查，包括稳定性过渡。当然，在海洋或实验室中只能看到稳定的解，但不稳定的解可以通过数值计算得到，通常可以完成稳定解如何组合在一起的画面。这些方法还将用于计算微地震和法拉第波，微地震是由海洋表面的几乎相干的驻波在海床中产生的地球物理弹性波，法拉第波是波浪槽中的水或油上的表面波，当容器被驱动以规定的运动振荡时，法拉第波自组织成各种驻波图案。本计画的数值结果将与宾州州立大学的戴安亨德森小组在威廉G。普里查德流体力学实验室。其他更广泛的影响包括课程和课程开发，组织研讨会和minisymopsia，研究生的建议，并开发新的计算工具与许多应用超越waterwaves.该项目的第一个技术目标是设计和实施一个广义射击方法计算准周期解微分方程。自由表面欧拉方程的几种新的类型的解决方案，预计将被发现，包括行驻波，KdV类弹性碰撞，和NLS类呼吸。驻波和其他相对周期解的谐波和次谐波稳定性也将被确定，目的是研究轨道稳定性，长期动力学和相对周期解的准周期扰动。 分谐波稳定性将使用布洛赫理论在空间和Floquet理论在时间上进行研究，与线性化欧拉方程的解决方案计算在并行批次计算的monodromy运营商。还将研究Benjamin-Feir不稳定性的驻波类似物。一种新的方法来计算循环稳定状态和稳定性转换的参数驱动的法拉第波系统也将开发，沿着一个新的算法计算的Dirichlet-Neumann算子在一个圆柱形的几何形状，使用张量产品的正交多项式的圆柱形几何形状和一个变种的变换场扩展技术。Arnoldi算法将用于计算法拉第波问题的monodromy算子的本征值，以获得最大的Floquet乘子。 该方法可以跟踪家庭的解决方案，通过不稳定的分支，以产生一个更完整的图片的稳定性转变，影响图案的形成。还将研究法拉第波的准周期强迫。

项目成果

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

周期性行进水波的空间准周期分岔以及使用有符号奇异值检测分岔的方法

• DOI: 10.1016/j.jcp.2023.111954
• 发表时间: 2023
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Wilkening, Jon;Zhao, Xinyu
• 通讯作者: Zhao, Xinyu

Numerical algorithms for water waves with background flow over obstacles and topography

具有越过障碍物和地形的背景流的水波的数值算法

• DOI: 10.1007/s10444-022-09957-z
• 发表时间: 2022
• 期刊: Advances in Computational Mathematics
• 影响因子: 1.7
• 作者: Ambrose, David M.;Camassa, Roberto;Marzuola, Jeremy L.;McLaughlin, Richard M.;Robinson, Quentin;Wilkening, Jon
• 通讯作者: Wilkening, Jon

Traveling-Standing Water Waves

行进的水波

• DOI: 10.3390/fluids6050187
• 发表时间: 2021
• 期刊: Fluids
• 影响因子: 1.9
• 作者: Wilkening, Jon

Computing the Dirichlet--Neumann Operator on a Cylinder

计算圆柱上的狄利克雷-诺依曼算子

• DOI: 10.1137/18m1204796
• 发表时间: 2019
• 期刊: SIAM Journal on Numerical Analysis
• 影响因子: 2.9
• 作者: Qadeer, Saad;Wilkening, Jon A.
• 通讯作者: Wilkening, Jon A.

Spatially quasi-periodic water waves of finite depth

• DOI: 10.1098/rspa.2023.0019
• 发表时间: 2023-01
• 期刊: Proceedings of the Royal Society A
• 作者: J. Wilkening;Xinyu Zhao