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首页> 外文会议>IUTAM Symposium on Diffraction and Scattering in Fluid Mechanics and Elasticity Jul 16-20, 2000 Manchester, United Kingdom >NONLINEAR EFFECTS IN WAVE SCATTERING AND GENERATION: Flow interaction with topography
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NONLINEAR EFFECTS IN WAVE SCATTERING AND GENERATION: Flow interaction with topography

机译：波散射和生成中的非线性影响：与地形的流相互作用

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摘要

When a fluid flow interacts with a topographic feature, and the fluid can support wave propagation, there is the potential for waves to be generated upstream and/or downstream. In many cases when the topographic feature has a small amplitude the situation can be successfully described using a linearised theory, and any nonlinear effects are determined as a small perturbation on the linear theory. However, when the flow is critical, that is, the system supports a long wave whose group velocity is zero in the reference frame of the topographic feature, then typically the linear theory fails and it is necessary to develop an intrinsically nonlinear theory. It is now known that in many cases such a transcritical, weakly nonlinear and weakly dispersive theory leads to a forced Korteweg-de Vries (fKdV) equation. In canonical form, this is -u_t - Δu_x + 6uu_x + u_(xxx) + f_x = 0 where u(x,t) is the amplitude of the critical mode, t is the time coordinate, x is the spatial coordinate, Δ is the phase speed of the critical mode, and f(x) is a representation of the topographic feature. In this article we shall sketch the contexts where the fKdV equation is applicable, and describe some of the most relevant solutions. There are two main classes of solutions. In the first, the initial condition for the fKdV equation is u(x, 0) = 0 so that the waves are generated directly by the flow interaction with the topography. In this case the solutions are characterised by the generation of upstream solitary waves and an oscillatory downstream wavetrain, with the detailed structure being determined by Δ and the polarity of the topographic forcing term f(x). In the second class a solitary wave is incident on the topography, and depending on the system parameters may be repelled with a significant amplitude change, trapped with a change in amplitude, or allowed to pass by the topography with only a small change in amplitude.
机译：当流体流与地形特征相互作用并且流体可以支持波传播时，可能会在上游和/或下游产生波。在许多情况下，当地形特征的振幅较小时，可以使用线性化理论成功描述这种情况，并且可以将任何非线性效应确定为对线性理论的小扰动。但是，当流量很关键时，即系统在地形特征的参考系中支持群速度为零的长波时，通常线性理论会失败，因此有必要发展本征非线性理论。现在已知，在许多情况下，这种跨临界，弱非线性和弱色散理论导致了强迫Korteweg-de Vries（fKdV）方程。以规范形式，这是-u_t-Δu_x+ 6uu_x + u_（xxx）+ f_x = 0其中u（x，t）是临界模式的振幅，t是时间坐标，x是空间坐标，Δ是临界模式的相速度，f（x）代表地形特征。在本文中，我们将概述fKdV方程适用的上下文，并描述一些最相关的解决方案。解决方案主要有两类。首先，fKdV方程的初始条件为u（x，0）= 0，因此波是通过与地形的流动相互作用直接产生的。在这种情况下，解决方案的特征是产生上游孤立波和下游振荡波，其详细结构由Δ和地形强迫项f（x）的极性确定。在第二类中，孤立波入射在地形上，并且取决于系统参数，可能会因幅度明显变化而被排斥，由于幅度变化而被捕获，或者仅因幅度很小的变化而通过地形。

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