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About this book

This volume offers an overview of the area of waves in fluids and the role they play in the mathematical analysis and numerical simulation of fluid flows. Based on lectures given at the summer school “Waves in Flows”, held in Prague from August 27-31, 2018, chapters are written by renowned experts in their respective fields. Featuring an accessible and flexible presentation, readers will be motivated to broaden their perspectives on the interconnectedness of mathematics and physics. A wide range of topics are presented, working from mathematical modelling to environmental, biomedical, and industrial applications. Specific topics covered include:

Equatorial wave–current interactionsWater–wave problemsGravity wave propagationFlow–acoustic interactions

Waves in Flows will appeal to graduate students and researchers in both mathematics and physics. Because of the applications presented, it will also be of interest to engineers working on environmental and industrial issues.

Table of Contents


Chapter 1. A Priori Estimates from First Principles in Gas Dynamics

The topic that one calls Gas Dynamics studies the motion of a very large number of molecules when there is enough room between them so that each one can move, and the motion is dominant in the overall behaviour, contrary to the case of liquids. The molecules interact in one way or another. At first glance, one often identifies interactions to collisions, but we may also consider medium-range interaction through some potential such as that of Lennard-Jones. One considers only pairwise interactions, either because the collisions involving three or more particles are extremely rare, or because forces involve only pairs of particles.
Denis Serre

Chapter 2. Equatorial Wave–Current Interactions

The ocean covers more than 70% of the Earth’s surface, and about 97% of the Earth’s water can be found in the ocean. The water’s ability to absorb large amounts of heat without a large increase in temperature, in combination with the transport of warmed or cooled water by ocean currents and waves over several hundred or thousand kilometres, regulates the global climate by reducing the uneven distribution of solar radiation reaching the Earth’s surface. Despite being central to the Earth’s climate, we still struggle to understand the fine details of ocean dynamics (and even at the level of the large-scale circulation, a consistent global description remains elusive).
Adrian Constantin

Chapter 3. Linear and Nonlinear Equatorial Waves in a Simple Model of the Atmosphere

Atmospheric dynamics in the equatorial region significantly differs from that in higher latitudes due to the simple fact that the rotation axis of the Earth lies in the tangent plane to the equator. As is well known starting from the pioneering paper by Matsuno [11], see also [4], the dynamics is dominated by equatorial waves. These waves are specific, due to orthogonality of the gravity acceleration and rotation axis, and are well-identified in meteorological in situ and satellite observations, e.g., [18]. It is important that equatorial waves are well-captured already in the simplest rotating shallow water (RSW) model of the atmosphere that was used in the classical works of Matsuno and Gill. The exposition below will be based on this model, with an important addition of the effects of moist convection, which are primordial in the tropical atmosphere. After having recalled the construction of the RSW model and its moist-convective generalization, the mcRSW model [1], we will analyze the linear wave spectrum, with an accent on its long-wave part, the most important for weather and climate, and then analyze the nonlinear effects in the equatorial wave dynamics, which lead to formation of specific coherent structures.
Vladimir Zeitlin

Chapter 4. The Water Wave Problem and Hamiltonian Transformation Theory

About 70% of the Earth’s surface is covered by water and about 40% of the world’s population lives within 100 km of the coast and estuaries. It is thus not surprising that ocean waves, directly or indirectly, affect human activities and have been a constant subject of interest.
Walter Craig, Philippe Guyenne, Catherine Sulem

Chapter 5. Gravity Wave Propagation in Inhomogeneous Media: Wave Scattering and Interference Process

This lecture aims to give an overview of water waves and their propagation in inhomogeneous media. Effects of varying bathymetries, varying currents, or structures including porous media are then considered. After some generalities on water waves, governing equations for regular plane waves for the study of wave scattering including reflection, refraction and diffraction, due to either varying bathymetry and currents or structures are presented for both 2D and 3D cases. Analytical and numerical solutions are then presented and compared to experiments. For 2-D cases, examples are given for wave reflection through interference process including Bragg resonance. For 3-D cases, various examples including wave scattering due to a shoal, a structure, periodic structures or varying currents are given. Applications to both shore protection solutions and wave energy are also presented.
Vincent Rey

Chapter 6. Physical Models for Flow: Acoustic Interaction

Aeroacoustic is a quite young scientific discipline, which focuses on flow-induced sound. In general, solving the full set of compressible flow dynamics equations results in both, the flow structures and the radiated sound. However, for practical applications, this approach is infeasible, and we have to investigate the physics of the sound generation mechanism. The book chapter will provide an overview of aeroacoustic modeling approaches and will discuss applications in technical and medical science: (1) Human phonation; (2) Axial fan; (3) Cavity flow at low Mach number; (4) Cavity flow at high Mach number.
Manfred Kaltenbacher, Stefan Schoder
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