AI 中文总结
本文提出一种仅依赖自由表面的二维闭合模型，用于一般深水流动上的线性波，显著简化计算，并首次发现足够陡峭的二维Gerstner波不稳定。
AI 中文摘要
我们提出了一组新颖的空间二维（2D）方程组，用于研究在一般稳态三维（3D）背景自由表面流动上传播的线性深水表面波。我们未假设背景自由表面为平面，也未假设背景流动无旋。所得模型不仅在理论上显著简化，而且计算速度提高了数个数量级。线性化欧拉方程在基础流动的自由表面上进行评估，并提出一个闭合条件来解释自由表面处的垂直导数。这推广了近期针对纯旋转背景流动的结果（Zuccoli, Brambley & Barkley, 2025, arXiv:2405.12078）。最终模型由五个耦合的一阶偏微分方程（PDEs）组成，在二维自由表面上求解，涉及五个未知量：水平和垂直扰动速度；自由表面扰动高度；以及出乎意料地，表面处扰动压力随深度的梯度。我们使用两个测试问题来验证模型：一个具有平坦自由表面的单向涡量随深度变化的基础流动；以及行进的Gerstner波解的扰动。我们计算了线性化欧拉方程的特征值和特征函数，并与模型的相应结果进行比较。结果显示两者之间具有显著的一致性。据我们所知，我们的研究首次发现，足够陡峭的二维Gerstner波是不稳定的。
英文摘要
We present a novel, spatially two-dimensional (2D) set of equations to study the propagation of linear deep-water surface waves over a general steady three-dimensional (3D) background free surface flow. No assumptions of a flat background free surface, nor of an irrotational background flow, are made. The resulting model is not only a significant theoretical simplification, but also results in orders of magnitude faster computations. The linearized Euler equations are evaluated on the free surface of the base flow, and a closure condition is proposed to account for the vertical derivatives at the free surface. This generalizes a recent result for purely rotating background flows (Zuccoli, Brambley & Barkley, 2025, arXiv:2405.12078). The final model consists of five coupled first order partial differential equations (PDEs) to be solved on the 2D free surface, involving five unknowns: the horizontal and vertical perturbation velocities; the free surface perturbation height; and unexpectedly the gradient of perturbation pressure with depth at the surface. Two test problems are used to validate the model: a uni-directional vortical depth-varying base flow with a flat free surface; and perturbations to a travelling Gerstner wave solution. Eigenvalues and eigenfunctions of the linearized Euler equations are computed and compared with those of the model. Results show remarkable agreement between the two. Our study finds for the first time, to the best of our knowledge, that sufficiently steep two-dimensional Gerstner waves are unstable.
Comments28 pages, 13 figures
