Waves in Flows 303067844X, 9783030678449

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Advances in Mathematical Fluid Mechanics

Tomáš Bodnár Giovanni P. Galdi Šárka Nečasová Editors

Waves in Flows

Advances in Mathematical Fluid Mechanics Series editors Giovanni P. Galdi, University of Pittsburgh, Pittsburgh, USA John G. Heywood, University of British Columbia, Vancouver, Canada Rolf Rannacher, Heidelberg University, Heidelberg, Germany

The Advances in Mathematical Fluid Mechanics series is a forum for the publication of high-quality, peer-reviewed research monographs and edited collections on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations and other significant viscous and inviscid fluid models. Titles in this series consider theoretical, numerical, and computational methods, as well as applications to science and engineering. Works in related areas of mathematics that have a direct bearing on fluid mechanics are also welcome. All manuscripts are peer-reviewed to meet the highest standards of scientific literature.

More information about this series at http://www.springer.com/series/5032

ˇ arka Neˇcasov´a Tom´asˇ Bodn´ar • Giovanni P. Galdi • S´ Editors

Waves in Flows

Editors Tom´asˇ Bodn´ar Department of Technical Mathematics Faculty of Mechanical Engineering Czech Technical University Prague, Czech Republic

Giovanni P. Galdi Department of Mechanical Engineering University of Pittsburgh Pittsburgh, PA, USA

ˇ arka Neˇcasov´a S´ Czech Academy of Sciences Prague, Czech Republic

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISBN 978-3-030-67844-9 ISBN 978-3-030-67845-6 (eBook) https://doi.org/10.1007/978-3-030-67845-6 Mathematics Subject Classification: 35Q30, 76D05, 76B15, 76B25, 76B55, 76B60, 76D33, 76B03, 76D03, 76N10 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkh¨auser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to the memory of Walter Craig, great person and our friend, who contributed to this volume, without having a chance to finally see it.

Preface This book is written as a monograph, covering the topic of Waves in Flows from different perspectives, with individual chapters written by world renowned experts in the corresponding fields. This volume may be seen as a contemporary summary of state-of-the-art knowledge in this area. The book is based on the lectures of invited speakers at a summer school “Waves in Flows”, held in Prague (Czech Republic) in August 27–31, 2018, and orˇarka Neˇcasov´ ganized by Tom´ aˇs Bodn´ ar (Czech Technical University in Prague), S´ a (Institute of Mathematics of the Czech Academy of Sciences) and Giovanni Paolo Galdi (University of Pittsburgh, USA). The summer school followed previous schools on various aspects of the mathematical fluid mechanics, held in Prague in the years 2011, 2012, 2014 and 2016, and it thus represented the fifth continuation of the series called the Prague-Sum. The aim of the summer school “Waves in Flows” was to offer the participants (i.e. graduate and postgraduate students, young scientists and other interested specialists) a comprehensive series of lectures on various themes and problems, connected with waves in fluids and their role in mathematical analysis and numerical simulation of fluid flows. The summer school was organized as a multidisciplinary event, and the presented lectures covered a wide range of topics, starting from mathematics up to physics and technical applications. A special attention was paid to models heading towards environmental, biomedical and industrial applications. The summer school as well as the present book was made possible due to generous direct support from Czech Academy of Sciences (special project to support research and educational activities for young people), Institute of Mathematics (institutional support Research Plan RVO 67985840), European Research Community on Flow, Turbulence and Combustion (ERCOFTAC) and Czech Science Foundation under the project no. P201-19-04243S. Prague, Czech Republic Pittsburgh, PA, USA Prague, Czech Republic

Tom´aˇs Bodn´ ar Giovanni Paolo Galdi ˇarka Neˇcasov´ S´ a

VII

Contents 1 A Priori Estimates from First Principles in Gas Dynamics 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Compensated Integrability . . . . . . . . . . . . . . . . . . . . 1.2.1 Evolution Problems . . . . . . . . . . . . . . . . . . . 1.3 Applications to Gas Dynamics (I): Euler Equations . . . . . . 1.3.1 Euler Equations for a Compressible Inviscid Fluid . . 1.3.2 Why Do We Care? . . . . . . . . . . . . . . . . . . . . 1.3.3 Estimating the Velocity Field . . . . . . . . . . . . . . 1.3.4 Flows in a Bounded Domain . . . . . . . . . . . . . . 1.3.5 Relativistic Gas . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Flows with External Force . . . . . . . . . . . . . . . . 1.4 Applications (II): Kinetic Models . . . . . . . . . . . . . . . . 1.4.1 Boltzmann-Like Models . . . . . . . . . . . . . . . . . 1.4.2 What Should a Dissipative Model Be? . . . . . . . . . 1.4.3 Discrete Velocity Models . . . . . . . . . . . . . . . . . 1.5 Applications (III): Mean-Field Models . . . . . . . . . . . . . 1.5.1 Vlasov-Type Models . . . . . . . . . . . . . . . . . . . 1.5.2 The DPT of a Single Vlasov-Type Equation . . . . . . 1.5.3 Genuine Plasmas . . . . . . . . . . . . . . . . . . . . . 1.6 Applications (IV): Molecular Dynamics . . . . . . . . . . . . 1.6.1 Mass–Momentum Tensor of a Single Particle . . . . . 1.6.2 Long-Range Forces . . . . . . . . . . . . . . . . . . . . 1.6.3 Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 7 10 12 13 16 18 19 20 22 23 23 28 31 34 34 36 38 40 40 41 43 46

2 Equatorial Wave–Current Interactions 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . 2.3 The Equatorial f -Plane Approximation 2.4 The Equatorial β-Plane Approximation

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49 49 51 57 58

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IX

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Contents 2.5

An Exact β-Plane Solution (Equatorially Trapped Wave) 2.5.1 Analysis of the Equatorially Trapped Wave Motion 2.5.2 Quantitative Aspects . . . . . . . . . . . . . . . . . 2.6 The Ocean Flow in the Equatorial Pacific . . . . . . . . . 2.6.1 The El Ni˜ no Phenomenon . . . . . . . . . . . . . . 2.6.2 A Model for the Equatorial Currents . . . . . . . . 2.6.3 Equatorial Wave–Current Interactions . . . . . . . 2.6.4 Linear Wave Theory . . . . . . . . . . . . . . . . . 2.6.5 Weakly Nonlinear Models . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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59 62 64 65 68 70 74 85 90 91

3 Linear and Nonlinear Equatorial Waves in a Simple Model of the Atmosphere 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Equatorial Waves . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Weakly Nonlinear Long Equatorial Waves . . . . . . . . . . . . . . 3.3.1 Long Linear Rossby and Kelvin Waves . . . . . . . . . . . . 3.3.2 Nonlinear Slow Dynamics of Long Waves . . . . . . . . . . 3.4 Equatorial Modons . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Equatorial Adjustment: Initial-Value Problem on the Equatorial Beta-Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Brief Summary and Discussion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 109 111

4 The Water Wave Problem and Hamiltonian Transformation Theory 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Water Waves and Hamiltonian PDEs . . . . . . . . . . . . . . . 4.2.1 Physical Derivation of the Governing Equations . . . . . 4.2.2 General Notions on Hamiltonian Systems . . . . . . . . 4.2.3 Examples of Hamiltonian PDEs . . . . . . . . . . . . . . 4.2.4 Zakharov’s Hamiltonian for Water Waves . . . . . . . . 4.3 Dirichlet–Neumann Operator and Its Analysis . . . . . . . . . . 4.3.1 Legendre Transform . . . . . . . . . . . . . . . . . . . . 4.3.2 Shape Derivative of H . . . . . . . . . . . . . . . . . . . 4.3.3 Invariants of Motion . . . . . . . . . . . . . . . . . . . . 4.3.4 Taylor Expansion of G . . . . . . . . . . . . . . . . . . . 4.4 Birkhoff Normal Forms . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Significance of the Normal Form . . . . . . . . . . . . . 4.4.2 Complex Symplectic Coordinates and Poisson Brackets 4.4.3 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Formal Transformation Theory and Birkhoff Normal Form 4.4.5 Solving the Third-Order Cohomological Equation . . . . 4.4.6 Normal Forms for Gravity Waves on Infinite Depth . . .

113 114 117 117 120 121 124 126 127 128 130 132 136 136 138 140 141 142 144

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93 93 96 99 101 102 104

Contents 4.5

XI

Model Equations for Water Waves . . . . . . . . . . . . . 4.5.1 Linearized Problem . . . . . . . . . . . . . . . . . . 4.5.2 Non-dimensionalization . . . . . . . . . . . . . . . 4.5.3 Canonical Transformation Theory . . . . . . . . . 4.5.4 Calculus of Transformations . . . . . . . . . . . . . 4.5.5 Boussinesq and KdV Scaling Limits . . . . . . . . 4.5.6 Modulational Scaling Limit and the NLS Equation 4.6 Initial Value Problems . . . . . . . . . . . . . . . . . . . . 4.6.1 Local Well-Posedness . . . . . . . . . . . . . . . . . 4.6.2 Recent Results on Global Well-Posedness for Small 4.6.3 Water Waves in a Periodic Geometry . . . . . . . . 4.7 Numerical Simulation of Surface Gravity Waves . . . . . . 4.7.1 Tanaka’s Method for Solitary Waves . . . . . . . . 4.7.2 High-Order Spectral Method . . . . . . . . . . . . 4.7.3 Collision of Solitary Waves . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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153 154 155 156 157 164 166 174 174 180 182 183 183 184 187 189

5 Gravity Wave Propagation in Inhomogeneous Media 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Propagation on Uneven Bottoms: First Order Stokes Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Second Order Stokes Waves . . . . . . . . . . . . . . . . . . 5.2.3 Propagation in the Presence of Current or Through Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Wave Scattering: 2D Case . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Standing Wave in a Tank: Resonance and Sloshing . . . . . 5.3.2 Case of Smooth Bathymetries: Sinusoidal Beds . . . . . . . 5.3.3 Case of Abrupt Bathymetries . . . . . . . . . . . . . . . . . 5.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Water Focusing: 3D Case . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Refraction—Snell—Descartes’ Law . . . . . . . . . . . . . . 5.4.2 Refraction-Diffraction . . . . . . . . . . . . . . . . . . . . . 5.4.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application to Wave Energy Device . . . . . . . . . . . . . . . . . 5.5.1 Oscillating Water Column . . . . . . . . . . . . . . . . . . . 5.5.2 Pressure Oscillation . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 198

6 Physical Models for Flow: Acoustic 6.1 Introduction . . . . . . . . . . . . . 6.2 Fluid Dynamics . . . . . . . . . . . 6.2.1 Conservation Equations . .

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198 202 205 208 209 210 217 220 229 230 231 234 240 252 252 256 260

Interaction 265 . . . . . . . . . . . . . . . . . . 265 . . . . . . . . . . . . . . . . . . 266 . . . . . . . . . . . . . . . . . . 267

XII

Contents 6.2.2 Constitutive Equations . . . . . . . . . . . . . . . . . . 6.2.3 Characterization of Flows by Dimensionless Numbers 6.2.4 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Towards Acoustics . . . . . . . . . . . . . . . . . . . . 6.3 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Simple Solutions: d’Alembert . . . . . . . . . . . . . . 6.3.3 Impulsive Sound Sources . . . . . . . . . . . . . . . . . 6.3.4 Free-Space Green’s Functions . . . . . . . . . . . . . . 6.3.5 Monopoles, Dipoles, and Quadrupoles . . . . . . . . . 6.3.6 Calculation of Acoustic Far Field . . . . . . . . . . . . 6.3.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . 6.3.8 Solution of Wave Equation Using Green’s Function . . 6.4 Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Lighthill’s Acoustic Analogy . . . . . . . . . . . . . . . 6.4.2 Curle’s Theory . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Vortex Sound . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Perturbation Equations . . . . . . . . . . . . . . . . . 6.4.5 Comparison of Different Formulations . . . . . . . . . 6.4.6 Acoustic Feedback Mechanisms . . . . . . . . . . . . . 6.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Human Phonation . . . . . . . . . . . . . . . . . . . . 6.5.2 Axial Fan . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Cavity at Low Mach Number [57] . . . . . . . . . . . . 6.5.4 Cavity at High Mach Number . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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270 274 274 279 284 284 286 289 290 292 293 297 299 302 302 308 312 317 319 320 323 326 332 338 342 346 348

Chapter 1

A Priori Estimates from First Principles in Gas Dynamics Denis Serre () Ecole Normale Sup´ erieure de Lyon Unit of Pure and Applied Mathematics Lyon France e-mail: [email protected]

In remembrance of Walter Craig

1.1

Introduction

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. Depending on the scale of the analysis, one distinguishes mainly three types of modelling. At the microscopic level, one is concerned with molecular dynamics, where the molecules are considered individually and we follow their trajectories through successive collisions. Although the mathematics behind it is extremely © Springer Nature Switzerland AG 2021 T. Bodn´ ar et al. (eds.), Waves in Flows, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-67845-6 1

1

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

simple, the global behaviour is usually extremely complicated because of the large number of particles. This is true even in the simplest case of spherical colliding particles. The state at given positive times is so much sensible to variations of the initial data that it is essentially impredictable, even by a powerful computer. One can only hope to capture a statistical information, giving up tracing individual particles. One goal is to estimate the number and the strength of the collisions in the whole history of the gas; this might give an information regarding the macroscopic pressure. The mesoscopic scale is the one where we reason in terms of statistical quantities only. It is described by a kinetic equation, where the unknown is the density f (t, y, ξ)dt dy dξ of particles, given the time t, the position y and the velocity ξ. The transport is modelled by the convective derivative ∂t f + ξ · ∇y f . In Boltzmannlike models, the interaction is accounted by an operator f (t, y, ·) → Q[f ], local in (t, y), but non-local in the velocity variable. It describes how the velocities are redistributed after collisions. The best known model involves the Boltzmann equation (1872). Its operator f → Q[f ] is quadratic since one takes into account only pairwise collisions; it relies upon the assumption that the velocities of colliding particles are uncorrelated (Maxwell 1867). In mathematical terms, the pairwise distribution f2 (t, y, ξ, ξ  ) is just given as the product f (t, y, ξ)f (t, y, ξ  ). This hypothesis is known as the molecular chaos hypothesis (Stosszahlansatz). We recall that, although molecular dynamics is a reversible physical process, the Boltzmann equation describes an irreversible evolution, in which Boltzmann’s H-theorem tells us that the total amount of an entropy is a monotonous non-decreasing function of time. At the last, macroscopic scale, one makes the assumption that the gas is locally at thermal equilibrium. By this, one means that in the H-theorem, the entropy production vanishes at the leading order. This assumption tells us that the density   3/2  ρ ρ|ξ − v|2 exp − f (t, y, ξ) = (1.1) 2πkB ϑ 2kB ϑ is a Maxwellian, where the fields (t, y) → (ρ, v, ϑ) are the local mass density, mean velocity and absolute temperature of the gas, and kB is Boltzmann’s constant. We warn the reader that the kinetic models are not rigorously compatible with molecular dynamics, except for extremely special initial data. It is instead meant to be a reasonable approximation when the number N of particles is large ; one may think of N ∼ 6, 02 · 1023 , the Avogadro number. This is why the chaos hypothesis is essential, and why thermodynamics is irreversible, unlike molecular dynamics. Likewise, the equilibrium assumption (1.1) is not compatible with space variability in kinetic models. However an asymptotic analysis can be carried out from the Boltzmann equation, which yields to thermodynamical models governing the macroscopic variables (ρ, v, ϑ). Depending on the context and the accuracy of the asymptotics, one ends with the Euler system or the Navier–Stokes system.

1.1. Introduction

3

One unifying feature of these three levels of description is that they all express the conservation of mass and momentum, where these quantities are defined in terms of a number of molecules or as integrals with respect to the space or space– velocity variables. These first principles provide us with a mathematical structure that seems to have been overlooked so far. The conservation laws are written in the abstract form Div T = 0, in terms of a mass/momentum tensor T . Here Div is the row-wise divergence in the time and space variables. Our tensor T takes values in the symmetric matrices, a property that encodes the compatibility with the conservation of angular momentum. This is the structure that we exploit in these notes. All the models listed above display, to some extent, the conservation of energy. This is a subtle concept, as energy may decompose into several forms and transform from one to the other. In thermodynamics, we sum an apparent kinetic energy, associated with the mean velocity, and a potential part that is expressed in terms of the pressure and the temperature. In a relativistic context, mass contributes to the total energy, by the celebrated amount 12 mc2 . Even at the level of molecular dynamics, the kinetic energy of each particle must be completed by some kind of internal energy, which can be rotational or vibrational; only for the mono-atomic case (say the Helium gas) may we neglect these contributions. Therefore an accurate model should describe precisely the way by which various forms of the energy are exchanged during a collision between particles, but this presupposes that we could measure at initial time the internal energy of each particle, something that is beyond our technological skills. This is why we will not base our analysis below upon a detailed balance of energy. Instead, we shall assume only that the total energy of our system is finite and conserved as time evolves. Since each part to the energy is non-negative, this will provide us with an a priori estimate of the total kinetic energy. This rather explicit information will be used together with the conservation of the total mass, to prove that T is integrable. Notice that we do not even need the exact conservation of energy, a global decay being a sufficient information; this allows us to consider incomplete models such as barotropic or isentropic gas dynamics. Our approach is twofold. On the pure mathematical side, we establish functional inequalities valid for positive semi-definite divergence-free tensors. We ask only that the tensor be integrable; or even less, we allow the entries to be bounded measures. In space dimension n, the size of T is the number d = 1 + n of space and time variables. Our main result is that the quantity (det T )1/n turns out to be integrable in space and time, where n is the space dimension. This represents a gain of integrability that we call Compensated Integrability, since the integrability 1 of T translates only into that of (det T ) n+1 . The second step of the analysis exploits the former by applying it to suitable mass–momentum tensor associated with the given modelling. As mentioned above,

4

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

the assumption of integrability is in general guaranteed by the conservation, or the decay, of the total energy, together with the conservation of mass. While the form of this tensor is pretty obvious in the case of the Euler system, its construction can be more involved for some other models, say in molecular dynamics, or for plasmas governed by the Vlasov–Poisson equation. In particular the crucial assumption of positiveness limits the applicability of the method to physical processes where the interaction between particles is repulsive. For instance, the analysis of the Euler system does not adapt to the Navier–Stokes system. Likewise, that for a plasma does not extend to galaxies, because the gravity, although given in a form similar to the electric force but with an opposite sign, is attractive. The deep reason for this limitation is that the results given by our abstract theorems are of dispersive nature; they resemble Strichartz estimates and are therefore incompatible with non-trivial equilibrium states that would result from some amount of self-attractivity. Instead, they are taylored for situations where mass and energy cannot concentrate too much in space and time. Warning These lecture notes do not pretend to be self-contained. Many details are left aside. We try to write only rigorous statements, but we skip most of their proofs. These can be found in the previous articles [22–26]. We choose to be as descriptive and clear as possible.

Notations We denote R+ = {s ∈ R ; s ≥ 0}. The Euclidian norm in Rd is | · |. We denote Sd−1 the unit sphere in Rd . Its (d − 1)-dimensional area is |Sd−1 |. If j ∈ [[1, d]] and ˆj = (. . . , xj−1 , xj+1 , . . .) ∈ Rd−1 (xj omitted). x ∈ Rd , we denote x Distributions and Measures Let Ω ⊂ Rd be an open domain. The space of distributions over Ω is D (Ω) and that of test functions is D(Ω). We denote by D+ (Ω) the cone of non-negative test functions. A distribution Γ is non-negative, and we write Γ ≥ 0, if Γ, φ ≥ 0 for every φ ∈ D+ (Ω). A non-negative distribution is actually a locally bounded measure: for every compact subset K ⊂ Ω, there exists a constant cK < ∞ such that   ∀φ ∈ D(Ω), (Supp φ ⊂ K) =⇒ | Γ, φ | ≤ cK sup |φ(x)| . x∈K

When there exists a constant c < ∞ such that ∀φ ∈ D(Ω),

| Γ, φ | ≤ c sup |φ(x)|, x∈K

Γ extends as a continuous linear form over the Banach space Cb (Ω) ; it is then a bounded measure (one also speaks of a finite measure). The space of bounded measures is M(Ω). The dual norm ΓM := sup{ Γ, φ ; φ ∈ Cb (Ω), φ = 1}

1.1. Introduction

5

is the total mass of Γ. A special case of bounded measures consists in integrable functions f ∈ L1 (Ω), in which case the total mass reduces to the L1 -norm. When M is a smooth manifold equipped with a volume form, we define as well the distributions, the test functions, the (locally) bounded measures and the total mass. We shall consider vector-valued distributions μ = (μ1 , . . . , μm ) ∈ D (Ω)m . They act on vector-valued test functions φ = (φ1 , . . . , φm ) by μ, φ =

m 

μj , φj .

j=1

We say that μ is a (locally) bounded measure if each μj has this property. When μ is a bounded measure, its total norm is defined by ⎧ ⎫ m ⎨ ⎬  μM := sup Γ, φ ; φ ∈ Cb (Ω)m , sup |φj (x)|2 = 1 . ⎩ ⎭ x j=1

Let q = (q1 , . . . , qd ) be a vector-valued bounded measure over Ω. Its divergence is defined as usual by div q =

∂qd ∂q1 + ··· + , ∂x1 ∂xd

where the derivatives are understood in the distributional sense. Let us assume that Ω has a Lipschitz boundary ∂Ω; we denote by ν (x) the unit outward normal vector at x ∈ ∂Ω. If both q and div q are square integrable, it is well-known that q admits a normal trace q · ν at the boundary, which is an element of the distribution space H −1/2 (∂Ω). It is characterized by the identity  ∀φ ∈ H 1 (Ω), (∇φ · q + φ div q) dx = q · ν , γ0 φ H −1/2 ,H 1/2 , (1.2) Ω

where γ0 : H 1 (Ω) → H 1/2 (∂Ω) is the usual trace operator. Likewise, we may define uniquely the normal trace q · ν under the assumption that both q and div q are bounded measures over Ω. This trace belongs to the dual of the space of restrictions to ∂Ω of elements of Cb1 (Ω). It is characterized by (1.2), in the following version: ∀φ ∈ Cb1 (Ω),

q, ∇φ + div q, φ = q · ν , φ|∂Ω .

(1.3)

be the extension of q by 0 outside of Ω. One verifies easily that div Q is a Let Q finite measure if and only if q · ν is a finite measure too. This suggests a definition. Definition 1.1.1. A vector field q over Ω is divergence-controlled if its entries qi , as well as div q and the normal trace q · ν , are finite measures. is the sum of the extension by 0 of div q, and of ( q · ν ) ◦ |∂Ω . Actually div Q

6

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

Linear Algebra If V, W ∈ Rd are vectors, then V ⊗ W is the d × d matrix whose entries are the vi wj ’s. This is a rank-1 matrix. . It satisfies M M = The cofactor matrix of M ∈ Md (k) is denoted by M M M = (det M )Id and = (det M )d−1 . det M (1.4) The space of d×d symmetric matrices with real entries is Symd . Its dimension d The convex cone of positive semi-definite matrices is Sym+ is d . If V ∈ R , + then V ⊗ V ∈ Symd . The open cone of positive definite matrices is SPDd . A 1 matrix Σ ∈ SPDd admits a unique positive square root, denoted by Σ 2 . If S ∈ + + Symd (resp., SPDd ), then S ∈ Symd (resp., SPDd ). If θ ∈ D (Ω), then the Hessian D2 θ is a symmetric matrix whose entries are distributions. If θ is actually a twice differentiable convex function, this Hessian takes positive semi-definite values. We recall a useful fact. n(n+1) . 2

Proposition 1.1.1. Let S ∈ Symd and Σ ∈ SPDd be given. Then the product SΣ (or ΣS as well) is diagonalisable with real eigenvalues. The list of signs of the eigenvalues is the same as that for the eigenvalues of S. Mind that SΣ is not symmetric in general, but it is conjugate to the sym1 1 metric matrix Σ 2 SΣ 2 . Passing to the limit, one obtains that if Σ is only positive semi-definite, then SΣ still has real eigenvalues; if moreover both matrices are positive semi-definite, then the spectrum of SΣ lies in R+ . Let us recall the Geometric-Arithmetic Means Inequality, which applies to a finite list of non-negative numbers: 1

∀a1 , . . . , ad ∈ R+ ,

(a1 · · · ad ) d ≤

1 (a1 + · · · + ad ). d

(1.5)

A direct application to the spectrum of a matrix gives ∀S ∈ Sym+ d,

1

(det S) d ≤

1 Tr S, d

(1.6)

where Tr M is the trace of a square matrix M . In light of Proposition 1.1.1, we also have 1 1 ∀S, Σ ∈ Sym+ (1.7) (det SΣ) d ≤ Tr (SΣ). d, d A classical consequence is Proposition 1.1.2. The map

is concave.

Sym+ d

−→

R+

S

−→

(det S) d

1

1.2. Compensated Integrability

7

When an n × n matrix M is given blockwise   A B M= , A ∈ GLp , C D the Schur complement formula tells us that det M = (det A) det(D − CA−1 B). In particular, M is non-singular if and only if the Schur complement D − CA−1 B is so. The real symmetric case (there, C = B T ) is especially interesting: if A ∈ SPDp , then M is positive semi-definite (resp., definite) if and only if the Schur complement D − B T A−1 B is so.

1.2

Compensated Integrability

Our abstract results concern symmetric matrices S of size d × d, whose entries are distributions: Sij ∈ D (Ω), where Ω is an open domain in Rd . If ξ ∈ Rd , the expression Sξ := ξ T Sξ is a distribution. We say that S is positive semi-definite if Sξ ≥ 0 for every ξ ∈ Rd , or equivalently for every ξ in the unit sphere Sd−1 . Positiveness implies that Sξ is a locally bounded measure over Ω, and we say that S itself is a locally bounded measure.  When S is a positive semi-definite tensor, then its trace γ = j Sjj is a non-negative measure. Taking ξ = a ei + b ej , we have Sξ = a2 Sii + b2 Sjj + 2abSij ≥ 0. For φ ∈ D+ (Ω), this gives a2 Sii , φ + b2 Sjj , φ + 2ab Sij , φ ≥ 0,

∀a, b ∈ R.

We infer the inequality 1

1

| Sij , φ | ≤ Sii , φ 2 Sjj , φ 2 ≤

1 Sii + Sjj , φ . 2

Thus every entry is absolutely continuous with respect to γ, and we may write Sij = fij γ for some function fij integrable against γ. If Φ : Sym+ d → R is a homogeneous function of degree 1, we define a locally bounded measure in the natural way Φ(S) := Φ(f )γ. In the sequel, we apply this 1 to Φ = Det d : 1 1 (det S) d := (det(fij )1≤i,j,≤d ) d γ ∈ M+ (Ω). 1

Applying (1.6) to the matrix F = (fij )1≤i,j,≤d , we have (det S) d ≤

1 d

γ.

8

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

Divergence-Free Tensors For a d × d tensor S, we define its row-wise divergence, which is vector-valued ⎞ ⎛ d  ∂j Sij ⎠ , Div S = ⎝ j=1

1≤i≤d

where the derivatives stand in the distributional sense. Notice the capital letter D, which we reserve for this context, to make it distinct from the divergence operator acting over vector fields. We say that S is divergence-free if Div S ≡ 0. More generally, we say that S is divergence-controlled if the rows of S are divergencecontrolled. We recall what it means: • each sij ∈ M(Ω), • for every ξ ∈ Rd , ξ T Sξ ∈ M+ (Ω), • each (Div S)i ∈ M(Ω), • the normal trace S ν ∈ M(∂Ω). Let us describe two explicit classes of divergence-free symmetric tensors, which do not exhaust this notion except if d = 2. (1) On the one hand we have diagonal tensors S = diag(a1 , . . . , ad ), where ∂j aj = 0 for each index j. In other ˆj . (2) On the other hand we have the class of spewords, aj depends only upon x 2 θ. Here θ is cial divergence-free symmetric tensors, which are of the form S = D 2,d−1 (Ω), and S is the cofactor matrix of the Hessian a scalar function of class W D2 θ. The fact that such a special tensor is divergence-free is classical, see [22]. Here is our basic result, from which all the other ones derive. Theorem 1.2.1 (d ≥ 2). Let S be a d × d positive semi-definite tensor over Rd , which is compactly supported. Assume that S is divergence-controlled. Then 1 the finite measure (det S) d is actually a measurable function, which belongs to d L d−1 (Rd ). Its norm is estimated by  d 1 1 d−1 (det S) d−1 dx ≤ Div SM . (1.8) 1 Rd d|Sd−1 | d−1 An immediate consequence is the following corollary. Corollary 1.2.2. Let S be a d × d positive semi-definite tensor over a smoothbounded domain Ω ⊂ Rd , d ≥ 2. Assume that S is divergence-controlled. Then 1 the bounded measure (det S) d is actually a measurable function, which belongs to d L d−1 (Ω). Its norm is estimated by  d 1 1 (det S) d−1 dx ≤ (S ν M + Div SM ) d−1 . (1.9) 1 Ω d|Sd−1 | d−1

1.2. Compensated Integrability

9

Comments • The positiveness is an essential ingredient in the theory. For if θ : Rd → R  2 θ of is a smooth, compactly supported function, then the cofactor matrix D the Hessian is a divergence-free symmetric tensor. Therefore it is not possible 1 in general to estimate | det S| d−1 in terms of Div S alone. • Decomposing γ = γa + γs into its absolutely continuous and singular parts, we have a decomposition S = Sa γa + Ss γs , from which it follows 1

1

1

(det S) d = (det Sa ) d γa + (det Ss ) d γs . The qualitative part of the theorem thus says that det Ss = 0, γs -almost everywhere. This is reminiscent to the main result of [11], which extended Alberti’s rank-one theorem; see also [14]. 1

L

d d−1

On the other hand the measurable function (det Sa ) d belongs to (γa ).

• Corollary 1.2.2 contains several classical results, such as the Isoperimetric Inequality, which we obtain by considering the tensor S ≡ Id . • The constant in (1.9) is sharp, as shown by the example of the Isoperimetric Inequality in the case of a ball. More generally, it is an equality whenever  2 θ for a smooth convex function θ such that ∇θ(Ω) is a ball. S=D • Likewise, Theorem 1.2.1 applied to S(x) = f (x)In yields the Gagliardo– Nirenberg inequality1 f 

d

L d−1

≤

1 1

d|Sd−1 | d−1

∇f L1 ,

(1.10)

where the constant is sharp. • Actually, our functional inequality is a far-reaching extension of a famous inequality due to Gagliardo for products of the form g(x) = g1 (ˆ x1 ) · · · gd (ˆ xd ), which writes gL1 (Rd ) ≤

d 

gj Ld−1 (Rd−1 ) .

j=1

To see this, consider a diagonal divergence-free tensor.

1 Personal

communication from Luis Silvestre.

10

1.2.1

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

Evolution Problems

The statement that we use for evolution problems involves an unbounded domain of the form Qτ = (0, τ )×Rn ; the overall dimension is d = 1+n, and it is meaningful to decompose x = (t, y) into a time and a space variable. In practice, the entries of S are labelled according to the indices 0, . . . , n, instead of 1, . . . , d. Theorem 1.2.3. Let S be a positive semi-definite tensor over Qτ , which is divergencecontrolled. In particular, we assume also that the normal traces S et at the top and bottom boundaries are finite measures. 1 Then the bounded measure (det S) d is actually an integrable function, which d belongs to L d−1 (Qτ ). Its norm is estimated by  τ  1 1 1+ 1 dt (det S) n dy ≤ et (0, ·)M + S et (τ, ·)M + Div SM ) n . 1 (S n n d|Sn | R 0 (1.11) Homogeneity Issue Despite its fundamental importance, Inequality (1.11) gives us seldom a significant estimate when applied to models of Mathematical Physics. The dependent and independent variables have their own dimensions, T for the time variable t, L for the space variables y and Σij for Sij . Each line of the control of the divergence of S expresses a conservation law (when it writes div q = 0) or more generally a balance law. In either case, this tells us that T −1 Σ0α = L−1 Σiα for i ≥ 1. There exists therefore a dimension Dα such that Σ0α = T Dα and Σiα = LDα otherwise. The symmetry of S tells us that actually D0 = T D and Dα = LD otherwise, for some dimension D. Now, although det S has a well-defined physical dimension T 2 L2n Dd , the situation is not so nice in the right-hand side of (1.11). On the one hand, the dimensions Dβ of the components (Div S)β do depend on β, and a compound quantity like Div SM does not make sense from the point of view of Physics. On the other hand, the same remark is valid for the first row S et = S0• , because S00 has dimension T 2 D, while the other S0α ’s have dimension T LD. A Homogeneous Estimate To fix the flaw quoted above, we first write our typical tensor blockwise   ρ mT S= , m A

(1.12)

where ρ = S00 is a scalar function (the mass density in our applications). The block A is still positive semi-definite. For the sake of simplicity, we assume that S is divergence-free. This implies, in particular, the conservation of total mass:   ρ(t, y) dy ≡ ρ(0, y) dy =: M0 . (1.13) Rn

Rn

1.2. Compensated Integrability

11

The physical dimensions of the blocks differ from each other. If R denotes that of ρ, then m has dimension RV and A has dimension RV 2 , where V = LT −1 is the dimension of a velocity. The expression  |S et | = ρ2 + |m|2 appearing in the right-hand side of (1.11) is therefore meaningless, from the physical point of view. To overcome this flaw, we apply Theorem 1.2.3 to a one-parameter family of positive semi-definite divergence-free tensors S λ . These are defined by a change of both dependent and independent variables: t = λt,

y  = y,

ρ = λ2 ρ,

m = λm,

A = A.

Applying (1.11) to S λ in the time interval (0, λτ ), and using |S et | ≤ ρ + |m|, we obtain the parameterized inequality  τ  1 1 1 λ− n 1+ n dt (det S) n dy ≤ . 1 (2λM0 + m(0, ·)M + m(τ, ·)M ) d|Sn | n Rn 0 To minimize the right-hand side, we choose λ=

m(0, ·)M + m(τ, ·)M 2nM0

and we obtain the following result. Theorem 1.2.4. Let S be a positive semi-definite divergence-free tensor over Qτ , whose total mass is finite. We assume also that the normal traces S et at the top and bottom boundaries are finite measures. 1 Then the bounded measure (det S) d is actually an integrable function that d belongs to L d−1 (Ω). Its norm is estimated, in terms of the blocks in (1.12), by 

τ 0



1 dt (det S) dy ≤ n n R 1 n



2dM0 |Sn |

 n1

(m(0, ·)M + m(τ, ·)M ) ,

(1.14)

with M0 given by (1.13). Remarks • One should not be troubled by the fact that if m ≡ 0, then (1.14) implies that S is singular. This extra assumption tells us that for almost every t, the positive semi-definite tensor A is divergence-free in the whole space Rn . Since it is also a bounded measure, (1.8) applied to A(t, ·) tells us that det A ≡ 0 and therefore det S = ρ det A ≡ 0. Actually, we can even prove that A ≡ 0, see Proposition 2.3 of [22]. • Both sides of (1.14) are homogeneous of dimension R1+ n Ln+2 T −1 . 1

12

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

Integrating in Time First Another subtle consequence of Theorem 1.2.3 deals with the same blockwise decomposition (1.12). Applying (1.11) to the modified tensor   ρ + φ mT Sφ := , m A where φ = φ(y) ≥ 0 is an arbitrary function, independent of the time variable, and using det Sφ = det S + φ det A ≥ φ det A,

Div Sφ = Div S,

we obtain this statement (see [25]): Proposition 1.2.5. We make the same assumptions as in Theorem 1.2.3. Then the bounded measure  τ 1 (det A) n dt Δ := 0

actually belongs to L  Δ(y)

n n−1

n n−1

 n−1 n dy

Rn

n

(R ), and its norm is estimated by

1 ≤ n



2d |Sn |

 n1

(S et (0, ·)M + S et (τ, ·)M + Div SM ) . (1.15)

Notice that  τ when n = 1, the statement above must be read as follows: the measure Δ = 0 A dt is actually a bounded function, whose norm is estimated by ΔL∞ (R) ≤

2 (S et (0, ·)M + S et (τ, ·)M + Div SM ) . π

If instead we apply Theorem 1.2.4 to Sφ , then we obtain a homogeneous estimate. Theorem 1.2.6. We make the same assumptions as in Theorem 1.2.4. Then the n bounded measure Δ actually belongs to L n−1 (Rn ), and its norm is estimated by  Δ(y) Rn

1.3

n n−1

 n−1 n dy

1 ≤ n



2d |Sn |

 n1

(m(0, ·)M + m(τ, ·)M ) .

(1.16)

Applications to Gas Dynamics (I): Euler Equations

We now turn towards the applications of Theorems 1.2.3 and 1.2.4 to various models of gas dynamics.

1.3. Applications to Gas Dynamics (I): Euler Equations

1.3.1

13

Euler Equations for a Compressible Inviscid Fluid

In thermodynamics, the gas is described by a mass density ρ(t, y), a (mean) velocity field v(t, y) and another scalar field, which can be either the pressure p, the entropy s, the temperature ϑ or the specific internal energy e. Because our interest is focussed on the conservation of the mass and momentum, we choose the pressure. The equations write ∂t ρ + divy (ρv) = ∂t (ρv) + Divy (ρv ⊗ v) + ∇y p =

0, 0.

This system expresses that the mass–momentum tensor   ρ ρv T T := ρv ρv ⊗ v + p In

(1.17)

is divergence-free. This symmetric tensor is actually positive semi-definite because of   1 V := , T = ρV ⊗ V + p 0 ⊕ In , v and ρ, p are non-negative. Integrability Over Qτ In order to apply our abstract theorems, we check whether T is integrable in Qτ . Because of mass conservation,   ρ(t, y) dy ≡ M := ρ(0, y) dy, Rn

Rn

the upper-left entry is integrable whenever the total mass is finite, M < ∞. The integrability of the other entries is a consequence of the conservation (or decay) of the total energy. The Euler system is completed by another conservation law     1 1 ρ|v|2 + ρe + div ( ρ|v|2 + ρe + p)v = 0, (1.18) ∂t 2 2 from which we derive       1 1 2 2 ρ|v| + ρe (t, y) dy ≡ E := ρ|v| + ρe (0, y) dy. 2 2 Rn Rn For perfect gases, the pressure is given by an equation of state p = (γ − 1)ρe, where γ > 1 is the adiabatic constant. The energy estimate implies therefore that the lower-right block ρv ⊗ v + pIn of T is integrable (it actually belongs to L∞ (0, τ ; L1 (Rn ))), provided that the total energy is finite, E < ∞. Now, Cauchy– Schwarz inequality implies that the off-diagonal terms ρv share the same property.

14

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

Our analysis applies as well whenever we have an energy estimate and the pressure p is dominated by the internal energy ρe per unit volume and the mass density: p ≤ C(ρ + ρe) for some finite constant C. Actually, the energy estimate does not need a full conservation law as (1.18); we may content ourselves with a differential inequality     1 1 2 2 ∂t ρ|v| + ρe + div ( ρ|v| + ρe + p)v ≤ 0 (1.19) 2 2 instead, or something even weaker. What we need is only an explicit inequality    1 ∀t ∈ (0, τ ), ρ|v|2 + ρe (t, y) dy ≤ E (< ∞). (1.20) 2 Rn The latter situation actually happens when the gas is supposed isentropic: the Euler system is closed by an equation of state p = p(ρ) and cannot be compatible with one more differential identity. Instead, it is compatible with (1.19), which plays the role of a Lax entropy inequality. Here the specific internal energy is a de = p. The domination of p by ρe is therefore a function e = e(ρ), given by ρ2 dρ differential inequality ρe ≤ C(e + 1) for some finite constant C. Non-homogeneous Estimate In a first instance, we assume that the physical domain is the whole space Rn , and the gas has finite total mass and energy. The determinant of T equals ρpn . Applying (1.11) to T and using Div T = 0 and T et = (ρ, ρv), we obtain 



τ

1 n

dt 0

Rn

ρ p dy ≤



1

ρ

1

d|Sn | n



 1+

|v|2

dy +

t=0

ρ



1+

|v|2

t=τ

We treat the right-hand side with the help of Young’s inequality  a2 + 1 a 2 + |v| 1 + |v|2 ≤ 2a 2 for every parameter a > 0. Since  Rn

we infer





τ

1 n

dt 0

Rn

1 ρ|v|2 (t, y) dy ≤ E, 2

ρ p dy ≤



1 1

d|Sn | n

a2 + 1 M + 2aE a

1+ n1 .

1+ n1 dy

.

1.3. Applications to Gas Dynamics (I): Euler Equations Choosing the optimal

 a=

we obtain





τ

1

dt Rn

0

ρ n p dy ≤

15

M M + 2E

1 1

d|Sn | n

  1+ n1 2 M (M + 2E) .

(1.21)

We point out that the right-hand side of (1.21) does not depend on τ ; letting τ → +∞, we see that this estimate is still valid with τ = +∞. The Homogeneous Estimate We point out that (1.21) is not homogeneous from the physical point of view. On the one hand, the dimension of the expression n  n+1  ∞  1 dt ρ n p dy Rn

0

 is (V for velocity). On the other hand M (M + 2E) is a mix of M and √ MV ME ∼ MV . We can therefore optimize (1.21) with respect to the scaling: when (ρ, v, p) is a solution of the Euler system, we can construct a full one-parameter family of solutions and then apply (1.21) to each of them (see [22] for details). We therefore have a list of inequalities  ∞  1+ n1  n  1 1 dt ρ n p dy ≤ , 2λ− n+1 M (M + 2λ2 E) 1 d|Sn | n 0 Rn n n+1

where the scaling parameter λ > 0 is arbitrary. The optimal choice  nM λ= 2E yields the homogeneous inequality 



∞

dt 0

√ ρ p dy ≤ Cd M ME 1 n

Rn

1 n

 Cd :=

√  n1  2 2d d . |Sn | dn

(1.22)

Remark Of course, we could have applied (1.14) directly to the mass–momentum tensor and then obtain the same homogeneous estimates, up to the constant, which would be replaced by 1 3  2d n 22 Cd = . n |Sn | The more elaborated method described here above is slightly better than that used in Theorem 1.2.4 because we have Cd < Cd . We point out that this constant is moderate; for instance, C3 ∼ 0.56.

16

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

The Final Estimate The only flaw of (1.22) is that its left-hand side is Galileaninvariant, while the right-hand side is not: the total energy is modified in a new inertial frame. We may therefore optimize our estimate by choosing the “best” Galilean frame, that is the one in which the total energy is minimum. The latter is that in which the total momentum  ρv dy Rn

√ vanishes. This amounts to replacing (see again [22]) the factor M E by a Galileaninvariant expression in terms of the initial data. This yields our final result. Theorem 1.3.1. We suppose that there exists a finite constant C such that p ≤ C(ρ + ρe) for every state. For an admissible flow of the compressible Euler equation in the physical domain Rn , with finite mass M and (initial) energy E, there holds  ∞  1 1 √ dt ρ n p dy ≤ Cd M n D (1.23) 0

Rn

with Cd given as in (1.22). The expression D is given in terms of the initial data by    1 ρ0 (y)ρ0 (z)|v0 (z) − v0 (y)|2 dy dz + M ρ0 (y)e0 (y) dy. D := 4 Rn Rn Rn Comments The adjective admissible in the theorem refers to the assumption that the total energy obeys (1.20). Actually we may admit flows for which the total energy E[t] is not a monotone function of t, if it remains bounded. Then the result is given by (1.22), where E := supt>0 E[t]. Therefore it applies to some of the “non-standard” solutions constructed by De Lellis and Sz´ekelyhidi [12].

1.3.2

Why Do We Care?

The main consequence of this estimate is that the mass or the energy may not concentrate on a singular subset. To illustrate this fact, we shall assume either an ideal gas, p = (γ − 1)ρe, or an isentropic equation of state p = a2 ργ . The adiabatic constant γ lies within the realistic interval (1, γc ], where γc = 1 + n2 corresponds to a mono-atomic gas. The scaling constant a2 is irrelevant. In the first case above, the total energy is conserved for admissible flows and the previous analysis works out. We shall make the extra assumption that the entropy s = log e − (γ − 1) log ρ is, at initial time, bounded by below: s0 := inf s(0, y) > −∞. y

1.3. Applications to Gas Dynamics (I): Euler Equations

17

Since admissible flows do satisfy the second principle of thermodynamics, we infer s(t, y) ≥ s0 everywhere; there follows that p ≥ a2 ργ for some positive constant a2 . In both cases, (1.23) thus yields an estimate  ∞  1 dt ργ+ n dy ≤ C(s0 , M, D), (1.24) Rn

0

where the exponent γ + n1 is larger than that, γ, provided by the energy estimate. Remark however the price to pay: we have to integrate in time. This improved integrability can be combined with another information, given by a former study of the moment of inertia. We recall that a perfect gas satisfies an identity (see [9, 20, 21])     1 d ρ|tv − y|2 + t2 ρe dy ≤ t (2ρe − np) dy. (1.25) dt Rn 2 Rn In particular, a monoatomic gas has the property that     1 1 ρ|tv − y|2 + t2 ρe dy ≤ I := ρ0 |y|2 dy, 2 2 n n R R

∀ t ≥ 0.

More generally, combining (1.25) with a Gronwall argument, we derive the estimate  ρe dy = O(t−n(γ−1) ) as t → +∞, (1.26) Rn

provided that I is finite. This implies as above the estimate  ργ dy = O(t−n(γ−1) ). Rn

Combining this with (1.24), with the help of H¨ older inequality, we deduce  β  ∞ tα dt ρq dy ≤ C(M, E, I, s0 , q, γ) < ∞ (1.27) Rn

0 1 n

whenever γ < q < γ + and   1 − n = (γ − 1)(β − 1)n, α := (γ − 1) q−γ

β :=

1 , n(q − γ)

(1.28)

and we point out that β > 1. Let us summarize this analysis. Theorem 1.3.2. Consider an ideal gas with adiabatic constant 1 < γ ≤ 1 + 2 n . Consider an admissible flow with finite mass, energy and moment of inertia. Assume moreover that the entropy is initially bounded by below. Then for every exponent q ∈ (γ, γ + n1 ), we have an estimate (1.27), where α, β are given by (1.28).

18

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

1.3.3

Estimating the Velocity Field

In order to establish an estimate involving the velocity field, we may apply Theorem 1.2.4. The block A is the tensor ρv ⊗ v + pIn , whose determinant equals pn−1 (p + ρ|v|2 ). Under the same assumptions as in Theorem 1.3.1, we find that  τ 1 1 p1− n (p + ρ|v|2 ) n dy Δτ := 0 n

belongs to L n−1 (Rn ). The corresponding norm is bounded by 1 n



2d |Sn |

 n1

((ρv)(0, ·)M + (ρv)(τ, ·)M ) .

Cauchy–Schwarz inequality gives  (ρv)(t, ·)M ≤

 12

 Rn

ρ(t, y) dy ·

2

(ρ|v| )(t, y) dy Rn

≤

√ 2M E .

Because this bound does not depend upon τ , our estimate is actually valid when τ = +∞. Writing Δ for Δ∞ , we thus obtain the following statement. Theorem 1.3.3. We suppose that there exists a finite constant C such that p ≤ C(ρ + ρe) for every state. For an admissible flow of the compressible Euler equations in the physical domain Rn , with finite mass M and (initial) energy E, the bounded measure  ∞ 1 1 Δ := p1− n (p + ρ|v|2 ) n dt 0 n

actually belongs to L n−1 (Rn ) and there holds n ΔL n−1 ≤ (Rn )

2 n



2d |Sn |

 n1

√ 2M E .

(1.29)

Once again, the 1-dimensional case reads  ∞ Δ := (p + ρ|v|2 ) dt ∈ L∞ (R) 0

with ΔL∞ (R) ≤

4 √ 2M E . π

For an ideal gas, this yields the amazing remark that the energy density 12 ρv 2 + ρe not only belongs to L∞ (R+ ; L1 (R)) (conservation of energy) but also to the “mirror” space L∞ (R; L1 (R+ )) !

1.3. Applications to Gas Dynamics (I): Euler Equations

1.3.4

19

Flows in a Bounded Domain

When the physical domain O ⊂ Rn is bounded, with a rigid boundary, the Euler system is supplemented by the slip boundary condition v · ν = 0

on (0, τ ) × ∂O.

A direct application of Corollary 1.2.2 to our tensor T is not efficient, because we do not know how to bound the normal trace of T along the lateral boundary. Actually     0 0 T = . ν p v Instead, we apply (1.9) to a product φT , where φ = φ(y) ≥ 0 is a cut-off function, so that the normal trace of φT vanishes identically. The resulting tensor is no longer divergence-free, but its divergence Div (φT ) = T ∇φ is controlled, as a consequence of the a priori L1 -bound of T . A typical result is as follows. Theorem 1.3.4. We suppose that there exists a finite constant C such that p ≤ C(ρ + ρe) for every state. There exists an absolute constant cn < ∞ such that every admissible flow in a bounded domain O ⊂ Rd satisfies  τ  1 1 1 1 dt dist∂O (y)1+ n ρ n p dy ≤ cn (1 + τ )M n E diam(O) n . (1.30) 0

O

Comments • The inequality (1.30) does depend on the length τ of the time interval, contrary to the situation in the whole space Rd . One fundamental reason is that since the gas is confined, the density cannot decay to zero as t → +∞. The integral  1

1

dist∂O (y)1+ n ρ n p dy O

cannot tend to zero either. • The homogeneity √ must be worked out in a different way. This yields a factor E instead of M E in the right-hand side. • It is no longer possible to make a change of inertial frame. Thus the right-hand side of (1.30) involves the total energy E, instead of some Galilean-invariant expression.

20

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

1.3.5

Relativistic Gas

In the Minkowski space-time R1+n of special relativity, an isentropic gas is governed by the Euler system (see [18])   2   2 ρc + p ρc + p p − v = 0, + div ∂t y c2 − |v|2 c2 c2 − |v|2    2  2 ρc + p ρc + p ∂t v + Div v ⊗ v + ∇y p = 0, y c2 − |v|2 c2 − |v|2 where the constant c > 0 is the speed of light. Here ρ is the mass density at rest, and p is the pressure. The fluid velocity is constrained by |v| < c. The first equation above is actually the conservation of energy. It resembles a lot the conservation of mass, because the energy of the matter at rest equals mc2 , where m is the rest mass. The energy–momentum tensor   ρc2 +p p ρc2 +p T − v 2 2 2 2 2 c c −|v| T = c −|v| ρc2 +p ρc2 +p v 2 2 2 c −|v| c −|v|2 v ⊗ v + pIn is still divergence-free. We verify easily the formula det T = ρpn . This expression does not depend upon the fluid velocity. It is actually a relativistic invariant, which can be calculated by assuming v = 0, when T is diagonal. But of course a brute force calculation can be made, using the Schur complement formula. The facts that det T > 0 and that the lower-right block is obviously positive definite imply that T itself is positive definite (the semi-definite case is more subtle). Let us denote  E := Rn



ρc2 + p p − 2 c2 − |v|2 c





 dy = Rn

ρc4 + p|v|2 c2 (c2 − |v|2 )

 dy,

which is a constant for admissible flows of finite energy. Then Theorem 1.2.4 gives  1  ∞  1 1 2dE n n dt ρ p dy ≤ (m(0, ·)M + m(τ, ·)M ) , n |Sn | 0 Rn where

ρc2 + p v. c2 − |v|2 In order to make this estimate as explicit as possible, we make the physically relevant assumption that the pressure is given by a linear equation of state m :=

p = α2 ρc2

(1.31)

1.3. Applications to Gas Dynamics (I): Euler Equations

21

for some constant α > 0. For instance, the value α2 = 13 follows directly from the Stefan–Boltzmann law for a gas in thermodynamical equilibrium; see the discussion on page 12 of A. M. Anile’s book [2]. Then we have |m| = (α2 + 1) from which we infer m(t, ·)M ≤

ρc2 |v| , c2 − |v|2

α2 + 1 cE. 2α

We thus end with the following estimate. Proposition 1.3.5. Consider the relativistic Euler equations in R+ × Rn , with equation of state (1.31). Then every admissible flow with finite total energy E obeys the estimate  1  ∞  1 1 2d n α2 + 1 dt ρ n p dy ≤ cE 1+ n . (1.32) nα |S | n n 0 R Comments • There is a typo in [22], where the factor c is missing in the right-hand side of (1.32). • The remarkable point in the relativistic setting is that there is only one conserved quantity at stake in this analysis, namely the total energy. Of course, this is due to the principle of mass–energy equivalence. • One might wish to improve estimate (1.32) as we did in the classical case, by a clever choice of the reference frame. However this seems impossible, because both sides of the inequality are modified under a Lorentz transformation. The determinant ρpn of T is frame-invariant, but the domain t ≥ 0 is not preserved by a change of coordinate, because the new time variable    − 12 V ·y |V |2 t = γ t − 2 , γ = 1− 2 c c does depend on y. Thus the half-space {t > 0} does not contain and is not contained in the half-space {t > 0}. The way the left-hand side of (1.32) depends on the choice of the frame is thus unpredictable. Likewise, the total energy E is an integral over the hyperplane t = 0 and has little in common with the energy E  computed over t = 0. Likewise, we may apply Theorem 1.2.4, where now A=

ρc2 + p v ⊗ v + pIn , c2 − |v|2

det A =

c2 pn−1 (p + ρ|v|2 ). c2 − |v|2

We therefore obtain an estimate involving the velocity field.

22

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

Proposition 1.3.6. Consider the relativistic Euler equations in R+ × Rn , with equation of state (1.31). Then every admissible flow with finite total energy E obeys the estimate  1 2d n α2 + 1 n ≤ cE, (1.33) ΔL n−1 (Rn ) nα |Sn | where



∞

Δ := 0

1.3.6



c2 pn−1 (p + ρ|v|2 ) c2 − |v|2

 n1 dt.

Flows with External Force

Suppose that the gas is subjected to an external force, proportional to its density. This may be gravity, or an electro-magnetic force. Then the balance law of linear momentum writes ∂t (ρv) + Divy (v ⊗ v) + ∇y p = ρF. Our mass–momentum tensor T , given by (1.17), is not any more divergence-free. We have instead   0 Divt,y T = . ρF It is however divergence-controlled over every slab (0, τ ) × Rn , provided that F is uniformly bounded, because of ρF M ≤ τ M F ∞ . Applying Theorem 1.2.3 to T gives  τ  √ 1 1 dt ρ n p dy ≤ cn (M + M E + τ M F ∞ )1+ n . Rn

0

Then a scaling trick, similar to pass from (1.21) to (1.22), yields the final estimate  τ  √  1 1 dt ρ n p dy ≤ cn M n M E + τ M F ∞ . (1.34) 0

Rn

We warn that (1.34) cannot be extended to the infinite time interval (0, +∞), because of the presence of τ in the right-hand side. Instead, it suggest that in average, the function  1 ρ n p dy t −→ Rn   1 1+ n behaves as an O M F ∞ as t → +∞. This is consistent with the expectation that the flow might tend to a non-trivial steady-state, for instance if the force field F is oriented towards a central point. A similar argument can be used in other models (e.g. Boltzmann, Vlasov) when an external force field is present.

1.4. Applications (II): Kinetic Models

1.4

23

Applications (II): Kinetic Models

In a mesoscopic formulation, the particles are still infinitesimally small and indistinguishable from each other. A gas is described by a density f (t, y, v) dt dy dv of particles, where now the velocity v becomes an independent variable, instead of a dependent one. The unknown f of the Cauchy problem for a kinetic equation is therefore a function over R+ × Rny × Rnv ; sometimes, the physical space Rny can be replaced by a subdomain Ω, with appropriate boundary conditions, or by a torus Tn . The macroscopic density ρ, momentum m and (kinetic) energy ε are defined in the obvious way by   ρ(t, y) = f (t, y, v) dv, m(t, y) = f (t, y, v)v dv, 

Rn

ε(t, y) = Rn

Rn

|v|2 f (t, y, v) dv. 2

(1.35)

The evolution of the density is governed by a kinetic equation (∂t + v · ∇y )f = Q[f ],

(1.36)

which results from two contributions. The left-hand side accounts for the transport, while the right-hand side represents the interaction between particles. The transport operator ∂t + v · ∇y describes how the particles would evolve in the absence of interaction: in free transport, f obeys the equation (∂t + v · ∇y )f = 0, which is equivalent to the formula f (t, y, v) = f0 (y − tv, v), where f0 = f (0, ·, ·) is the initial datum.

1.4.1

Boltzmann-Like Models

There exist several, qualitatively distinct, kinetic models. They differ by their interaction operator Q, but they share common features: Space-time locality. The restriction Q[f ](t, y, ·) depends only on the restriction f (t, y, ·). The operator Q therefore describes the way the velocities are redistributed after interaction. It can be viewed as acting over functions of the velocity variable only. Invariance. The operator Q is translation invariant: Q[g(· + w)] = Q[g](· + w). Likewise, it is rotationally invariant: if R ∈ SOn and h(v) := g(Rv), then Q[h](v) ≡ Q[g](Rv).

24

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

Consistency. The kinetic equation preserves the non-negativity of f as time increases. For this, we must have (g ≥ 0 and g(0) = 0) =⇒ (Q[g](0) ≥ 0). Conservations. The local mass, momentum and energy2 are conserved through the interaction. That is, for every reasonable distribution g ≥ 0 (say, bounded and compactly supported), we have    |v|2 dv = 0. Q[g](v) dv = 0, Q[g](v)v dv = 0, Q[g](v) 2 Rn Rn Rn (1.37) One says that the quantities 1, v and 12 |v|2 , as functions of the velocity, are the collision invariants. The latter property, together with a formal integration of (1.36) against the 2 measures dv, v dv and |v|2 dv yield the macroscopic conservation laws of mass, momentum and energy: ∂t ρ + divy m ∂t m + Divy S

= =

0, 0,

(1.38) (1.39)

∂t ε + divy q

=

0,

(1.40)

where the tensor S and the vector field q are given by   |v|2 v dv. f (t, y, v) v ⊗ v dv, q := f (t, y, v) S := 2 Rn Rn We point out the important relations ε=

1 Tr S, 2

m ⊗ m ≤ ρS,

(1.41)

the latter being a consequence of the Cauchy–Schwarz inequality. Another manipulation consists in multiplying (1.36) by a function h (f ), using the chain rule (mind that this needs some non-trivial justification) and then integrating over Rnv . One obtains    ∂t h(f (t, y, v)) dv + divy h(f (t, y, v))v dv = Q[f ](t, y, v)h (f (t, y, v)) dv. Rn

Rn

Rn

There is no reason why the right-hand side above should vanish, and the above equality does not look like a conservation law. The operator Q displays however 2 Mind that for the sake of simplicity, we consider only the kinetic part in the molecular energy. This applies to mono-atomic gases (e.g. Helium, Argon). For more general gases, we should consider the vibrational and rotational contributions.

1.4. Applications (II): Kinetic Models

25

the property that the corresponding integral is non-positive for the special choice h (f ) ≡ log f : for every reasonable function g ≥ 0, one has  Q[g](v) log g(v) dv ≤ 0. (1.42) Rn

Let us recall the so-called Boltzmann’s H functional, the opposite of the physical entropy  (f log f )(t, y, v) dv, H(t, y) := Rn

which satisfies therefore the differential inequality  ∂t H + divy (f log f )(t, y, v)v dv ≤ 0.

(1.43)

Rn

Inequation (1.43) is the mathematical translation of the second principle of thermodynamics. It reveals the irreversible nature of kinetic models. Integrating (1.43) in space, we obtain formally the decay of the total (mathematical) entropy  t −→ More precisely, we have   d H(t, y) dy+ R(t, y) dy = 0, dt Rn Rn

H(t, y) dy. Rn

 R(t, y) := −

Rn

Q[f ](v) log f (v) dv ≥ 0.

This identity suggests that as time increases, the gas tends to a local equilibrium; stated otherwise, the decay rate R(t, y) should tend to zero. A local equilibrium, or a Maxwellian, is therefore a distribution g(v) ≥ 0 that satisfies  Q[g](v) log g(v) dv = 0. (1.44) Rn

Let us point out that, because of (1.37), every Gaussian function g(y) = A exp −

|v − v¯|2 , B

(1.45)

with parameters A, B > 0 and v¯ ∈ R3 , is a Maxwellian. Conversely, for physically relevant models, every Maxwellian is a Gaussian. The formal asymptotics that, starting from a kinetic formulation, yields macroscopic models (Euler, Navier–Stokes equations) as the mean free path tends to zero is based on the fact that the set of Maxwellians is parametrized by the conserved quantities (ρ, m, ) defined in (1.35). The parameters are determined, say in space dimension n = 3, by v¯ =

m , ρ

B=

4ε 2 2 − |¯ v| , 3ρ 3

A = ρ(πB)−3/2 .

26

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

The quantity B is actually related to the temperature by B = 2kB ϑ, where kB is the Boltzmann constant. Finally, an elementary calculation gives the entropy  η= (f log f )(v) dv Rn

at equilibrium in terms of the macroscopic conserved quantities,   5 3 |m|2 η(ρ, m, ε) = ρ log ρ + cst. − log(2ε − ) . 2 2 ρ An important remark is that the thermodynamical limit does not depend upon the precise form of the interaction operator Q. What matters is only that Q satisfies (1.38), (1.39), (1.40) and (1.42) and that the only equilibria are local Gaussians. The Cauchy Problem Suppose that the physical domain is the whole space Rny (the theory is similar in the case of the torus Tn ). The Cauchy problem consists in finding the solution of (1.36) that fits with an initial datum f0 (y, v) ≥ 0. In order to benefit from the conservation laws (1.38)–(1.40) and of Boltzmann entropy inequality, it is natural to assume    1 + |v|2 f0 (y, v) dv dy < ∞, (1.46) n n  R ×R (f0 | log f0 |)(y, v) dv dy < ∞. (1.47) Rn ×Rn

According to (1.46), the total mass and energy are finite:   |v|2 f0 (y, v) dv dy. M := f0 (y, v) dv dy, E := Rn ×Rn Rn ×Rn 2 A priori estimates are that  f (t, y, v) dv dy ≡ M,



Rn ×Rn

sup t>0

together with



Rn

H(t, y) dy ≤

H(0, y) dy, Rn



+∞



dt 0

Rn ×Rn

|v|2 dv dy ≡ E 2





and

f (t, y, v)

Rn

R(t, y) dy ≤

H(0, y) dy. Rn

1.4. Applications (II): Kinetic Models

27

By a solution, one usually means a weak solution, in the sense of distributions. Alas, the theory has not been able so far to prove the existence of such solutions, except in space dimension one. For n ≥ 2, one only knows (DiPerna–Lions’ theory [13]) the existence of the so-called renormalized solutions, for appropriate operators Q. We refer, for instance, to the monograph by L. Saint-Raymond [19]. What matters for our purpose is that under reasonable assumptions, the conservation law (1.38) is valid, while (1.39) must be replaced by ∂t m + Divy (S + Σ) = 0,

(1.48)

where Σ is some symmetric positive semi-definite tensor, called a defect tensor. It is not known whether Σ really pops up in the limit, or if it vanishes identically. At least the energy estimate ensures that Σ has a finite mass in space, uniformly in time. We shall not elaborate on this, and we shall only use the fact that the tensor   ρ mT T := m S+Σ has finite mass on every band (0, τ ) × Rn , and that it is divergence-free. The inequality   ρ mT T ≥ T0 := ≥ 0d , m S implies

1

1

(det T ) d ≥ (det T0 ) d ,

(1.49)

while the Schur complement formula yields 1

1

(det T ) d ≥ (ρ det Σ) d .

(1.50)

An Extra Estimate for Boltzmann-Like Equations We do not consider a specific interaction operator Q for the analysis below. We only need that the Cauchy problem for (1.36) admits, for data that fulfil (1.46 and 1.47), a renormalized solution for which the macroscopic quantities (ρ, m, S) defined by (1.35) satisfy the conservation of mass (1.38) and the modified conservation of momentum (1.48), for some defect measure Σ ≥ 0n of finite mass. We may therefore apply Theorem 1.2.4 to the tensor T defined above. Using in addition (1.49), we infer an inequality 

τ 0



1 dt (det T0 ) dy ≤ n Rn 1 n



2dM |Sn |

 n1

(m(0, ·)M + m(τ, ·)M ) .

(1.51)

The right-hand side is as usual bounded in terms of the mass and energy, thanks to the Cauchy–Schwarz inequality: √ ∀t ∈ R+ , m(t, ·)M ≤ 2M E .

28

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

On the other hand, the tensor T0 is defined by a velocity integral:      1 1 T0 (t, y) = ⊗ f (t, y, v) dv. v v n R To evaluate its determinant, we apply Andreiev’s formula:    ⊗N 1 2 = (det((φi (ξj ) ))1≤i,j≤N ) dμ(ξ1 ) · · · dμ(ξN ), det φi φj dμ(ξ) N ! 1≤i,j≤N in which the symbol ⊗N means that we integrate over N independent copies of the original domain of the variable ξ. In the context of T0 , the variable ξ is v,  is 1 . The measure dμ is f (t, y, ·) dv, whose domain is Rn , and the vector field φ v and the matrix size N equals n + 1 ; for the sake of clarity, we shall use indices 0, 1, . . . , n instead of 1, . . . , n + 1. Finally the determinant inside the integral:    1 ··· 1    v 0 · · · v n  , is (up to the sign) n! times the volume of the simplex of Rn whose vertices are (v0 , . . . , vn ); we denote this volume by Δ(v0 , . . . , vn ). We have therefore the formula  ⊗n+1 n! Δ(v0 , . . . , vn )2 f (v0 )f (v1 ) · · · f (vn ) dv0 · · · dvn . det T0 = n + 1 Rn Hereabove, we discarded the dependence of f upon t and y for the sake of clarity. Eventually, we end up with the estimate 





τ

⊗n+1

dt Rn

0

Rn

Δ(v0 , . . . , vn )2 f (v0 )f (v1 ) · · · f (vn ) dv0 · · · dvn

 n1

√

2M E (1.52)

with 1 cn = n



2d2 n!|Sn |

 n1 .

If the defect tensor really exists, then (1.50) yields the estimate  τ  1 1 1 √ dt ρ n (det Σ) n ≤ cn M n 2M E . 0

1.4.2

1

dy≤cn M n

(1.53)

Rn

What Should a Dissipative Model Be?

Mathematical modelling for fluid dynamics can be carried out by following one among two strategies, depending on whether we adopt the point of view of continuum mechanics, or we start from a finer modelling, at meso- or microscopic scale.

1.4. Applications (II): Kinetic Models

29

On the one hand, we may ignore the molecules and treat the gas as a continuum, postulating that it is completely described by a few macroscopic fields. Then one establishes conservation laws from first principles. Finally, one closes the system of equations by making ad hoc assumptions, which must be consistent with the symmetries inherent to the problem. This is how the Euler equations and the Navier–Stokes equations were originally derived; a very nice presentation of this approach is given in C. Dafermos’ book [10], where the role of thermodynamics is highlighted. Even the Boltzmann equation can be established that way. Of course, each representation has its own domain of validity (dense gas, dilute gas), in which the assumptions are satisfied to a reasonable extent. On the other hand, one can view the various levels of description as different stages in a hierarchy, where a lower level (in terms of complexity) model must be obtain as a kind of singular limit of a higher level. At the top, one finds molecular dynamics, in which the molecules interact either through collisions or more realistically through a short-range force deriving from some potential. For a mono-atomic gas, this potential bears the name of Lennard-Jones. When the number of molecules becomes large, it makes sense to observe only statistical quantities instead of following each particle individually. The particle density obeys the Liouville equation, from which we extract balance laws for successive momenta, the so-called BBGKY hierarchy. This infinite list of differential equations is then truncated and completed by ad hoc closure laws; this yields various kinetic models. Contrary to the microscopic models, the kinetic ones are not time-reversible; instead, they display an increase of an entropy functional, which is reminiscent to the second principle of thermodynamics. These models admit a few special solutions for which the entropy remain constant. Most of them are homogeneous in space; they are referred to as equilibria. Eventually, when the mean free path becomes small with respect to the characteristic lengths of the apparatus, one expects that the gas is locally at, or close to, equilibrium and we derive fluid dynamics models, such as Euler or Navier–Stokes systems; this can be done by using a Hilbert or a Chapman–Enskog expansion around the local Maxwellian. It is remarkable that both approaches, the one by mathematical modelling or that following a hierarchy, produce the same systems of equations. This coincidence has been for a long time an argument in favour of the validity of these models, which are widely used in applications, such as weather forecast. Therefore every mathematical result about Euler, Navier–Stokes or Boltzmann is appreciated to a very high value. Ultimately, a mathematician who solved one of the central problems would be awarded a Fields medal or a 1 M$ cash prize. It might therefore seem foolish to keep discussing the relevance of these models. Nevertheless let us try to stand as the Devil’s advocate. In a first instance, let us admit that some kinetic model, say the Boltzmann equation, is relevant to such a large extent that we are allowed to start from it and perform the perturbative analysis, which leads to hydrodynamical equations as  → 0+. Given an initial datum f0 (x, v), or even a sequence of initial data f0 , which is bounded

30

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

in terms of total mass, energy and entropy, we are interested in the behaviour of the sequence of solutions f of the Cauchy problem, when  → 0+. The sequence (f0 )→0 is bounded in some functional space, as well as (f (t))→0 , uniformly in time. We may extract a sub-sequence that converges weakly-∗ in the appropriate topology. Still because of these a priori estimates, we may pass to the limit in the averaged Eqs. (1.38) and (1.39), in the distributional sense. We obtain therefore a limit divergence-free tensor T . Since T is the weak limit of positive semi-definite tensors T , it is itself positive semi-definite. But this property, though correct in the Euler model, is not fulfilled in the case of the Navier–Stokes equation, where the corresponding divergence-free tensor is 

ρ ρv

 ρv T . ρv ⊗ v + (p − λ div v)In − μ(∇v + ∇v T )

For ρ > 0, its positivity is equivalent to that of Schur’s complement (p−λ div v)In − μ(∇v + ∇v T ). Although it may occur here and then, there is no reason why it would occur ever and ever. In particular, this positivity is false when div v > p/λ. This paradox was discussed by Levermore [17] in the following terms: Fluid dynamics breaks down when the macroscopic gradient lengths are comparable to the mean free path of the flow. The viscous stress and heat conduction terms in the fluxes of the usual Navier–Stokes equations then become large enough to lead to unphysical effects. One such effect is the development of unrealizable values for the moments of the fluid particle distribution. This is clearly seen to arise in (3.2a) for the Navier–Stokes closure (3.21a), whenever the expression is no longer a non-negativedefinite matrix (the left side being manifestly so). Several strategies have been proposed in order to overcome this difficulty, starting with H. Grad’s celebrated 13-moment closure [15]. Levermore [17] pointed out a difficulty in Grad’s approach and elaborated an improved procedure. We shall not discuss the so-called moment closure problem here, and we refer to Section 4 of [17] instead. We only point out that Levermore’s criticism of the Navier–Stokes closure pins exactly the mathematical aspect that prevents us to apply Compensated Integrability to viscous fluid dynamics: the lack of negativity of the stress tensor. This observation leads us to question the validity either of the Navier–Stokes system for compressible fluids or that of the Boltzmann equation, because the latter cannot yield the former through a rigorous asymptotic procedure. Of course, we do not exclude that both these models are reasonably good approximations of the reality within some physical range; as Levermore’s discussion suggests, Navier–Stokes is not unreasonable when the velocity gradient is moderate enough. Thus the validity of viscous fluid dynamics, and not only its well-posedness or its predictive power, is intimately related to the question of whether this gradient remains bounded as time increases, the so-called regularity problem for the Navier–Stokes equations.

1.4. Applications (II): Kinetic Models

31

Perhaps the most obvious suspect in the whole family of models should be the Boltzmann equation, which is based upon the unrealistic assumptions that particles collide and that their shape is spherical. The latter hypothesis might be valid for mono-atomic gases (the so-called noble gases) but is clearly false for all other molecules, including the simplest ones such as H2 , O2 or N2 . At first approximation, these can be viewed as ellipsoids, and even a simple process by elastic collisions yields an intricate dynamics; a reasonable model of the statistical evolution seems hard to elaborate. On another hand, pairs of particles do not really collide but interact through a short-range potential, the simplest one being that of Lennard-Jones. Although it is true that particles repel each other quite strongly at very short distance, a Van der Waals contribution lets them attract each other at larger distances, when the electronic clouds do not intersect. Because of this, rather weak, attractive feature, the mass–momentum tensor T is not positive semidefinite in general. Now, it is clear that if we start from a realistic kinetic model, in which the divergence-free tensor T is not necessarily positive, then Levermore’s objection above does not stand, and we should not exclude a priori the possibility of deriving a Navier–Stokes-like system in the hydrodynamic limit. One such model is the Vlasov equation where the force derives from a Lennard-Jones potential, or a similar one. As a last remark, we do not exclude at this stage the following scenario, which already occurs in the context of the Vlasov–Poisson equation for a plasma [23], see Paragraph 1.5.2. Although the viscous-stress tensor in Navier–Stokes is not positive in general, it might happen that its row-wise divergence be identical to that of another tensor S, this one positive semi-definite. Here S would be the weak limit of the tensor arising in the Boltzmann equation. This possibility, which we leave as an open question, would also provide a satisfactory answer to Levermore’s concern.

1.4.3

Discrete Velocity Models

The study of discrete velocity models is quite an old topic, which deals with kinetic equations where the velocity variable is constrained to take values in a finite set V ⊂ Rn . The simplest, non-trivial example is the Broadwell system, where V = {± ej | 1 ≤ j ≤ n}. The density function becomes a map f : R+ × Rn × V → R+ . It can be viewed as a vector of functions fv (t, y) indexed by v ∈ V . This vector is denoted f (t, y) ∈ RV ; in the case above, f (t, y) ∈ R2n and the components are denoted by fi± . A component fv obeys an equation (∂t + v · ∇y )fv = Qv (f )  ! " transport

32

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

where Qv is a numerical function over RV+ , often a quadratic form. The exchange must be consistent with the usual physical rules, such as conservation of term Q mass, momentum and energy, and with the symmetries of the set V . We point out that discrete velocity models are never consistent with Galilean invariance, because the finite set V cannot be translation invariant. For the Broadwell system, the equations are (∂t ± ∂i )fi± = λ

n 

(fj− fj+ − fi− fi+ ),

1 ≤ i ≤ n,

j=1

where λ > 0 is a parameter whose dimension is the inverse of a length. Since the characteristic length is a mean-free path, λ is expected to be a large number. A thermodynamical limit occurs as λ → +∞. The macroscopic mass and momentum   ρ= (fj− + fj+ ), m = (fj+ − fj− ) ej j

j

obey the conservation laws = 0, ∂t ρ + div m

∂t mi + ∂i (fi− + fi+ ) = 0.

(1.54)

Because V is contained in the unit sphere, the macroscopic energy coincides with the mass. More elaborate models, where V is contained in several spheres, display an extra conservation law, that of energy. Nevertheless, the solutions are always subjected to |m| ≤ vmax , where vmax is the maximal modulus of the elements of V. The mass–momentum (divergence-free) tensor is thus   ρ m T T = m diag(fi− + fi+ ) and its determinant equals ρ

n 

(fi− + fi+ ) −

i=1

n  j=1

m2j



(fi− + fi+ ).

i=j

A local equilibrium is a vector f for which fi+ fi− =: h2 does not depend on the index i. It can be expressed in closed form in terms of (ρ, m), where |m| ≤ ρ, in the following way. On the one hand, h > 0 is the unique root of # ρ= 4h2 + m2j . j

Then fj+ = aj h and fj− = h/aj , where aj is the positive root of a2 −

mj a − 1 = 0. h

1.4. Applications (II): Kinetic Models

33

The thermodynamical limit, which occurs in the limit λ → +∞, is therefore made of the equations (1.54), where the flux of momentum is computed at equilibrium, that is # fi+ + fi− = 4h2 + m2i . Because the Broadwell system implies ∂t +





fi± log fi±

i,±

∂i (fi+ log fi+ − fi− log fi− ) = λ



i

(fj− fj+ − fi− fi+ )(log(fj− fj+ ) − log(fi− fi+ ))

i 0.3) and incompressible (Ma ≤ 0.3), and is defined by v (6.51) Ma = c with c the speed of sound. In unsteady problems, periodic oscillating flow structures may occur, e.g. the K´ arm´an vortex street in the wake of a cylinder. The dimensionless frequency of such an oscillation is denoted as the Strouhal number, and is defined by l (6.52) St = f v with f the shedding frequency. Characteristic numbers are relevant for measurement setups, comparability of physical effects, simulation validation, and modeling considerations.

6.2.4

Vorticity

The motion of a fluid particle is a superposition [3] of • translational motion • rotation • distortion of shape, i.e. strain . The velocity gradient tensor ∂vi /∂xj can be expressed as the sum of its symmetric and anti-symmetric part, i.e.     ∂vi 1 ∂vi ∂vj ∂vj 1 ∂vi = + − + . (6.53) ∂xj 2 ∂xj ∂xi 2 ∂xj ∂xi The symmetric part defines the rate of strain tensor  (see (6.41)) and corresponds to a stretching of fluid particles along the principle axes. The anti-symmetric part   1 ∂vi ∂vj − (6.54) ωij = 2 ∂xj ∂xi

6.2. Fluid Dynamics

275

has only three independent components, since ωii = 0, ωij = −ωji . These components can be combined into a vector ω, which is called the vorticity and computes by ω = ∇×v. (6.55) The vorticity gives a measure of the angular rotation of fluid particles. For example, a fluid particle with the angular velocity Ω about the origin, i.e. the fluid velocity v is given by Ω × r, has the vorticity ω = 2Ω . Vorticity lines (cut through a vortex tube) are always tangential to the vorticity vector and form closed field lines. These lines pass through every point of a simple closed curve and define the boundary of a vortex tube. For a tube of small crosssectional area dΓ the product ω · dΓ is called the tube strength, which has to be constant because of the following vector identity: ∇ · ω = ∇ · ∇ × v = 0. Therefore the divergence theorem (also known as Gauss integral theorem) leads to =  ∇ · ω dx = ω · dΓ = 0 . (6.56) Ω

∂Ω

We will now derive the partial differential equation for vorticity and show that vorticity is transported by convection and molecular diffusion. Therefore, an initially confined region of vortex loops can frequently be assumed to remain within a bounded domain. We assume a Stokesian fluid (see (6.43)) and a homentropic flow, where the density ρ is only a function of pressure p. We start at the conservation of momentum (divided by the density) and use the vector identity (6.278) to arrive at   1 4 ∂v + v · ∇v + ∇p = −ν ∇ × ω − ∇∇ · v . (6.57) ∂t ρ 3 In the next step, we use for the second term on the left hand side the vector identity according to (6.281) and for the third term the relation (6.265) to obtain Crocco’s form of momentum conservation   4 ∂v + ω × v + ∇B = −ν ∇ × ω − ∇∇ · v . (6.58) ∂t 3 Thereby, B denotes the total enthalpy in a homentropic flow  dp 1 2 B= + v ρ 2

(6.59)

and the vector ω × v is called the Lamb vector. When the flow is incompressible (denoted by the subscript ic), Crocco’s equation reduces to ∂vic + ω × vic + ∇B = −ν∇ × ω , ∂t

(6.60)

276

Chapter 6. Physical Models for Flow: Acoustic Interaction

in which case dissipation occurs only for ω = 0. Applying the curl to (6.60) and considering the vector identity for ω ∇ × ∇ × ω = ∇∇ · ω − ∇ · ∇ω = ∇ · ∇ω results in

  ∂ω + ∇ × ω × vic = ν∇ · ∇ω . ∂t

(6.61)

Finally, applying (6.279) and exploring the relation ∇ · vic = 0, we arrive at the vorticity equation Dω = ω · ∇vic + ν∇ · ∇ω . (6.62) Dt The terms on the right hand side determine the mechanisms describing the change of vorticity: • Term 1: ω · ∇vic In the absence of viscosity, vortex lines move with the fluid. They are rotated and stretched in a manner determined by the flow. When a vortex tube is stretched, the cross-sectional area is decreased and therefore the amplitude of vorticity has to increase in order to preserve the strength of the tube. • Term 2: ν∇ · ∇ω This term is only important in regions of high shear, in particular near boundaries. Near walls the velocity becomes very small, so that (6.62) reduces to a diffusion equation ∂ω = ν∇ · ∇ω . (6.63) ∂t Vorticity is generated at solid boundaries and the viscosity is responsible for its diffusion into the fluid domain, where it may subsequently be convected by the flow. On the other hand, if the vorticity within an incompressible flow is zero, we arrive at the potential flow. Then, we can describe the flow by a scalar potential φ (also called flow potential) (6.64) vic = ∇φ . According to the incompressibility condition, we obtain the describing partial differential equation (6.65) ∇ · vic = ∇ · ∇φ = 0 . Please note that a potential flow is both, divergence- and curl-free. Therefore, according to the Helmholtz decomposition, the velocity field vic for an incompressible flow can be split into two vector fields vic = ∇φ + ∇ × ψ = ∇φ + vv ,

(6.66)

6.2. Fluid Dynamics

277

where φ describes the potential flow (boundary effect) and vv = ∇×ψ the vortical flow (vorticity driven) with ω = ∇ × vic = ∇ × vv .

(6.67)

By applying the curl to (6.66), we obtain for the vector potential ∇ × ∇ × ψ = ∇ × vic = ∇ × vv = ω .

(6.68)

Furthermore, by using the vector identity ∇ · ∇ψ = ∇∇ · ψ − ∇ × ∇ × ψ we may write ∇ · ∇ψ = −ω .

(6.69)

To obtain ψ, we can use Green’s function for the Laplace equation and arrive at  ω(y) dy . (6.70) ψ(x) = 4π|x − y| Ω

On the other hand, knowing the vorticity ω, we may compute vv by   ω(y) (y − x) × ω(y) dy = vv = ∇x × dy , 4π|x − y| 4π|x − y|3 Ω

(6.71)

Ω

which is a pure kinematic relation. Now, because vorticity is transported by convection and diffusion, an initially confined region of vorticity will tend to remain within a bounded body, so that it may be assumed that the fluid dynamic field vv → 0 as |x| → ∞ with O(1/|x|3 ) [26]. To conclude, a fluid perturbation will gradually decay by O(1/|x|3 ) as we move away from the perturbation origin. Finally, let us consider a pulsating sphere as displayed in Fig. 6.2. Since there are no sources in the fluid and we assume the fluid to be incompressible, we can model it by the Laplacian of the scalar velocity potential ∇ · ∇φ = ∇2 φ = 0 .

(6.72)

Since the setup is radially symmetric, we obtain (using spherical coordinates)   ∂ 1 ∂ ∇2 φ = 2 r2 φ = 0; r > a r ∂r ∂r and hence

A +B. r We assume that φ vanishes at ∞, so B can be set to zero. Furthermore, with the boundary condition ∂φ/∂r = vn at r = a we get φ=

φ(t) = −

a2 vn (t) for r > a . r

(6.73)

278

Chapter 6. Physical Models for Flow: Acoustic Interaction

Γ

r

vn

0

a

Figure 6.2: Pulsating sphere Assuming a non-viscous fluid ([τ ] = 0), no external forces (f = 0), and neglecting the convective term, we may write the momentum conservation (see (6.15)) for an incompressible flow by ∂vic + ∇pic = 0 . ρ0 (6.74) ∂t Using the scalar potential φ, we arrive at the following linearized relation: ∂φ . ∂t With (6.73) we can compute the resulting pressure pic to pic = −ρ0

∂φ a2 ∂vn = ρ0 . ∂t r ∂t The volume flux qΩ (t) at any time computes as = = qΩ (t) = ∇φ · dΓ = , ∇φ · er dΓ = 4πa2 vn (t) ,  ! " pic (t) = −ρ0

∂Ω

(6.75)

(6.76)

∂Ω ∂φ/∂r=vn

and so we can rewrite (6.73) by qΩ (t) for r > a . (6.77) 4πr This solution also holds for r → 0 (see [26]). To conclude, a potential perturbation will gradually decay by O(1/|x|) as we move away from the perturbation origin. φ(t) = −

6.2. Fluid Dynamics

6.2.5

279

Towards Acoustics

In the previous section, we have demonstrated the decomposition of an incompressible flow into its potential flow described by the scalar potential φ and its vortical flow described by the vector potential ψ. Now, we consider conservation of mass for a compressible fluid 1 Dρ ρ Dt

=

−∇ · v

(6.78)

and again applying a Helmholtz decomposition according to v = ∇φ˜ + ∇ × ψ˜ .

(6.79)

Thereby, we obtain 1 Dρ ρ Dt

=

−∇ · ∇φ˜ − ∇ · ∇ × ψ˜  ! "

=

−∇ · ∇φ˜ .

=0

(6.80)

In this case, the scalar potential φ˜ also includes compressible effects, which may include wave propagation, since this is just possible in a compressible fluid. Indeed, as already described in [3], the overall compressible flow may be decomposed into three parts: • Irrotational deformation without volume change (see Fig. 6.3a): The classical theory of potential flow equations describes this velocity field and is well known in fluid dynamics. The resulting flow field is both, divergencefree and curl-free. • A rigid-body rotation at an angular velocity (see Fig. 6.3b): Typical vortical flow structures can be described by the vorticity ω and its dynamics. • Isotropic expansion (see Fig. 6.3c): This part is proportional to the volumetric rate of expansion ∇ · v. The field component can be described by a scalar potential associated with the compressibility of the fluid. The extension of the classical Helmholtz decomposition is the Helmholtz– Hodge decomposition, which also considers the topology of a domain, and in special considers the decomposition on a bounded domain Ωr (bounded by ∂Ωr ) with nonzero velocity field on the boundary. In such a case, every square integrable vector field3 u ∈ [L2 (Ω)]3 , C 1 smooth, on a simply connected, Lipschitz domain 3 Note that we changed the variable (for the flow velocity) here intentionally to express that this decomposition is not exclusive for the flow velocity and can also be applied for any vector field.

280

Chapter 6. Physical Models for Flow: Acoustic Interaction

(a) Irrotational deformation without volume change.

(b) A rigid-body rotation.

(c) Isotropic expansion.

Figure 6.3: General decomposition of a flow field. (a) Irrotational deformation without volume change. (b) A rigid-body rotation. (c) Isotropic expansion Ω ⊆ R3 has an L2 -orthogonal decomposition u = uv + uc + uh = ∇ × Av + ∇φc + uh ,

(6.81)

with the vector potential Av ∈ H(curl, Ω), the scalar potential φc ∈ H 1 (Ω), and the harmonic component uh ∈ [L2 (Ω)]3 . If the decomposition is either computed in the scalar potential formulation or in the vector potential formulation, the obtained components include the harmonic component due to the boundary values of the partial differential equation. Therefore, we mark both potentials with a star superscript. u u∗c

= =

uv + uc + uh = ∇ × A∗v + ∇φ∗c = u∗v + u∗c uc + αuh

(6.82) (6.83)

u∗v α+β

= =

uv + βuh 1.

(6.84) (6.85)

For any decomposition that is possible, we like to separate the harmonic part into the compressible or the vortical part by choosing the boundary conditions. The L2 -orthogonality of the decomposed components implies that the L2 -inner product is zero  u∗v · u∗c dx = 0 . (6.86) (u∗v , u∗c ) = Ω

Inserting the definition of the joint potentials and applying the orthogonality of the three components yield   (u∗v , u∗c ) = u∗v · u∗c dx = αβuh · uh dx = 0 , (6.87) Ω

Ω

and therefore a condition for α and β. In order to be orthogonal, the harmonic part must be either separated into the compressible component (α = 1 and β = 0) or in the vortical component (α = 0 and β = 1). We prefer α = 0 and β = 1,

6.2. Fluid Dynamics

281

since for low Mach numbers the obtained vortical field recovers the incompressible flow solution. Using the orthogonality condition, we derive a unique boundary that leads to a distinct separation of the harmonic field (which represents the exterior influence of the decomposed vector field u) into the vortical part with α = 0 and β = 1. Formulation for Scalar Potential In this section, we derive the equation for the scalar potential φ∗c ∈ H 1 (Ω), which is associated with the compressible part and the property ∇ × u∗c = 0. Physically motivated by low Mach number flows, we aim to separate the harmonic part from the compressible part (α = 0 and β = 1). In doing so, the obtained compressible part will be the acoustic field. Helmholtz’s decomposition is defined by u = ∇φ∗c

= ∇φc + ∇φh =

u∗c

=

∇ × A∗v + ∇φ∗c ,

(6.88)

uc + αuh |α=0 = uc .

(6.89)

A harmonic part uh can be further split into parts accounting for closed boundary curves [63]. The homogeneous Neumann boundary is intrinsically satisfied by the finite element formulation. However, a general penetrating wall implies an inhomogeneous Neumann boundary. By taking the divergence of (6.88), we obtain a scalar valued Poisson equation with the dilatation ∇ · u as forcing ∇ · ∇φ∗c = ∇ · u .

(6.90)

This scalar Poisson equation can be solved computationally efficient compared to the more involved curl–curl equation (number of unknowns is about a factor of three lower compared to the equivalent curl–curl equation). Since this equation has the same structure as the pressure correction equation, one may compute (6.90) inside the flow solver. Furthermore, the boundary conditions of (6.90) must obey orthogonality. Integration by parts leads to a general orthogonality condition for the boundaries of the scalar potential  φ∗c u∗v ·n dΓ (u∗v , u∗c ) = (u∗v , ∇φ∗c ) = −(∇ · u∗v , φ∗c ) +  ! "  !" ∂Ω =0



φ∗c (u − ∇φ∗c ) · n dΓ = 0 .

=

=u−∇φ∗ c

(6.91)

∂Ω

This results in an orthogonal boundary integral for the scalar potential; either the scalar potential has a homogeneous Dirichlet boundary or the normal component of the velocity is entirely captured by the scalar potential. If the normal component of the velocity is entirely captured by the scalar potential, the scalar potential would have a harmonic part and α = 0. Thus, the function space of the scalar potential φ∗c has to obey (6.91) to enforce an orthogonal decomposition and has to be adjusted for different flow boundaries in order to have α = 0.

282

Chapter 6. Physical Models for Flow: Acoustic Interaction

Wall At perfectly smooth, no-slip, non-penetrated wall boundaries, the flow velocity in normal direction is equal to the compressible wall penetration movement (u · n = 0 ⇒ u∗v · n = −u∗c · n). Assuming a non-penetrating wall for the vortical component implies ∂φ∗c = u∗c · n = 0 . (6.92) ∂n The condition represents a sound hard wall, if the compressible component is interpreted as acoustics. Outlet and Inlet Since the condition of a velocity inlet is in general nonzero (u∗v · n = 0), the orthogonality condition (6.91) leads to φ∗c = 0 .

(6.93)

Having set the boundary conditions, we ensure that α = 0 for a given flow configuration. Formulation for Vector Potential The Helmholtz decomposition, formulated by the vector potential A∗v ∈ H(curl, Ω), is associated with the vortical part and the property ∇ · u∗v = 0. We aim to decompose the velocity field u = ∗ ∇ × Av = u∗v =

∇ × A∗v + ∇φ∗c , uv + βuh |β=1 = uv + uh

(6.94) (6.95)

such that the harmonic part is united with the vortical part (α = 0 and β = 1). If we have β = 1 the harmonic solution is incorporated into the joint vector potential A∗v . In the case of holes inside the domain, these holes enrich the function space. For example, imagine a flow around a cylinder at very low Mach number and then decompose this flow. As a result the vortical part of the flow simulation will converge to the incompressible flow solution as the Mach number approaches zero. Furthermore, if we decompose an incompressible flow solution we would recalculate this incompressible flow field by obtaining the vortical part of the decomposition. As it applies for the scalar potential formulation, the harmonic part accounts for the exterior and interior cutouts [63]. The function space of the vector potential has to enforce an orthogonal and unique decomposition (6.97). By taking the curl of Eq. (6.94), the curl–curl equation for the vector potential A∗v is obtained ∇ × ∇ × A∗v = ∇ × u = ω ,

(6.96)

forced by the vorticity ω = ∇ × u. Any numerical scheme can be used to solve this equation as long as the appropriate function space is used. This function space

6.2. Fluid Dynamics

283

must also ensure the orthogonality of the decomposed components. Applying the integration by parts to the orthogonality leads to  A∗v · ( u∗c ×n) dΓ (u∗v , u∗c ) = (∇ × A∗v , u∗c ) = (A∗v , ∇ × u∗c ) +  ! "  !" ∂Ω =0



u−∇×A∗ v

A∗v · (u − ∇ × A∗v ) × n dΓ = 0 ,

=

(6.97)

∂Ω

as uniqueness condition at the boundary. We ensure that the harmonic part is united with the vortical part by assuming (u − ∇ × A∗v ) × n = 0. Based on the orthogonality condition (6.97), typical flow boundaries in combination with the curl–curl problem can be identified. Wall For a rigid and perfectly smooth wall, the compressible velocity component in tangential direction uc × n = 0 is zero if the no-slip holds since a rigid wall will not deform per definition. Therefore, a no-slip and non-penetrating wall boundary requires that the overall tangential flow velocity is equal to the wall movement uwall in tangential direction u∗v × n = (∇ × A∗v ) × n = uwall × n .

(6.98)

Stationary, rigid walls, with a no-slip condition uwall = 0, enforce a homogeneous Neumann boundary for the vector potential (∇ × A∗v ) × n = 0). Inlet and Outlet by

At the inlet and the outlet the tangential velocity is described u∗v × n = (∇ × A∗v ) × n = uinlet/outlet × n .

(6.99)

Here, we assume that the tangential velocity component of the total field is dominated by the vortical part u∗v to a sufficient extent u∗v × n|∂Ω := u × n|∂Ω . Firstly, the amplitudes of the acoustic perturbation are small and if the numerical setup is wave dissipative the waves do not travel until the free boundaries (e.g. sponge zone). Secondly, the acoustic perturbations are a longitudinal process and typical boundaries are arranged mostly orthogonal to the radiation direction (since absorbing boundary conditions are usually designed for orthogonal wave impingement and work best if the domain is designed according to that fact) and therefore the compressible part in the tangential velocity component is weighted by the sine of the relative angle. This reduces the amplitude further. Thirdly, if the numerical setup is not dissipative the prescribed radiation condition at flow boundaries is optimal for normal wave impingement. Consequently, the domains are designed to satisfy normal wave impingement. Summarizing, the formulation of the scalar potential, with its algorithmic simplicity and computational efficiency (compared to the vector potential formulation), has two computational drawbacks:

284

Chapter 6. Physical Models for Flow: Acoustic Interaction

• The computational domain of the scalar potential should capture compressible effects e.g. acoustics. Acoustic radiation, in a free field configuration, reaches far into the surrounding since the acoustic field only decays by O(1/||x||2 ). This slow decay results either in a huge domain or in an involved boundary condition that accurately fulfills free field behavior. A possible solution strategy for elliptic differential equations is the infinite mapping layer [52]. • The second issue arises for non-convex domains with a C 0 smooth boundary, like reentrant corners [51]. For such flow domains, the computation of the scalar potential leads to a singular point at reentrant corners and corrupts the solution [56]. Using the finite element method, a graded mesh can treat this singularity. However, in most cases, these reentrant corners are locations where aeroacoustic sources are present and therefore these regions have to be treated carefully. Caused by these two drawbacks, the application of the computationally efficient scalar potential formulation is limited and we suggest the application of the vector potential formulation.

6.3 6.3.1

Acoustics Wave Equation

We assume an isentropic state, where the total variation of the entropy is zero and the pressure is only a function of the density. For linear acoustics, this results in the well known relation between the acoustic pressure pa and density ρa pa = c20 ρa

(6.100)

with a constant isentropic speed of sound c0 . Furthermore, the acoustic field can be seen as a perturbation of the mean flow field p

= p0 + pa ; ρ = ρ0 + ρa ; v = v0 + va

(6.101)

with the following relations: pa  p0 ; ρa  ρ0 .

(6.102)

In addition, we assume the viscosity to be zero, so that the viscous stress tensor [τ ] can be neglected, and the force density f is zero. We call ρa the acoustic density and va the acoustic particle velocity.

6.3. Acoustics

285

For a quiescent fluid, the mean velocity v0 is zero, and furthermore we assume a spatial and temporal constant mean density ρ0 and pressure p0 . Using the perturbation ansatz (6.101) and substituting it into (6.8) and (6.15), result in   ∂(ρ0 + ρa ) + ∇ · (ρ0 + ρa )va ∂t  ∂va  + (ρ0 + ρa )va · ∇va (ρ0 + ρa ) ∂t

=

0

(6.103)

=

 −∇ p0 + pa ) .

(6.104)

In the next step, since we derive linear acoustic conservation equations, we are allowed to cancel second order terms (e.g., such as ρa va ), and arrive at conservation of mass and momentum ∂ρa + ρ0 ∇va ∂t ∂va + ∇pa ρ0 ∂t

=

0

(6.105)

=

0.

(6.106)

Applying the curl-operation to (6.106) results in ∇×

∂va = 0, ∂t

(6.107)

which allows us to introduce the scalar acoustic potential ψa via va = −∇ψa .

(6.108)

Substituting (6.108) into (6.106) results in the well known relation between acoustic pressure and scalar potential pa = ρ0

∂ψa . ∂t

(6.109)

Now, we substitute this relation into (6.105), use (6.100) and arrive at the well known acoustic wave equation 1 ∂ 2 ψa − Δψa = 0 . c20 ∂t2

(6.110)

On the other hand, we also obtain the wave equation for the acoustic pressure pa exploring (6.105), (6.106), and (6.100) 1 ∂ 2 pa − Δpa = 0 . c20 ∂t2

(6.111)

286

Chapter 6. Physical Models for Flow: Acoustic Interaction

pa = pa (x, t) Figure 6.4: Propagation of a plane wave

6.3.2

Simple Solutions: d’Alembert

In order to get some physical insight in the propagation of acoustic sound, we will consider two special cases: plane and spherical waves. Let us start with the simpler case, the propagation of a plane wave as displayed in Fig. 6.4. Thus, we can express the acoustic pressure by pa = pa (x, t) and the particle velocity by va = va (x, t)ex . Using these relations together with the linear pressure-density law (assuming constant mean density, see (6.100)), we arrive at the following 1D linear wave equation ∂ 2 pa 1 ∂ 2 pa − = 0, (6.112) ∂x2 c20 ∂t2 which can be rewritten in factorized version as    ∂ 1 ∂ ∂ 1 ∂ − + pa = 0 . ∂x c0 ∂t ∂x c0 ∂t

(6.113)

This version of the linearized, 1D wave equation motivates us to introduce the following two functions (solution according to d’Alembert): ξ

=

t − x/c0 ; η = t + x/c0

with properties ∂ ∂t ∂ ∂x

= =

∂ ∂ξ ∂ ∂ξ

∂ξ ∂ + ∂t ∂η ∂ξ ∂ + ∂x ∂η

∂η ∂ ∂ = + ∂t ∂ξ ∂η   ∂ ∂η 1 ∂ = − . ∂x c0 ∂η ∂ξ

Therewith, we obtain for the factorized operator 1 ∂ ∂ − ∂x c0 ∂t

=

−

2 ∂ ; c0 ∂ξ

∂ 1 ∂ 2 ∂ + = ∂x c0 ∂t c0 ∂η

and the linear, 1D wave equation transfers to −

4 ∂ ∂ pa = 0 . c20 ∂ξ ∂η

6.3. Acoustics

287

The general solution computes as a superposition of arbitrary functions of ξ and η (6.114) pa = f (ξ) + f (η) = f (t − x/c0 ) + g(t + x/c0 ) . This solution describes waves moving with the speed of sound c0 in +x and −x direction, respectively. In the next step, we use the linearized conservation of momentum according to (6.104), and rewrite it for the 1D case (assuming zero source term) ∂pa ∂va + = 0. (6.115) ρ0 ∂t ∂x Now, we just consider a forward propagating wave, i.e. g(t) = 0, substitute (6.114) into (6.115) and obtain  1 ∂pa dt va = − ρ0 ∂x  1 ∂f (t − x/c0 ) = − dt ρ0 ∂x  1 ∂f (t − x/c0 ) ∂(t − x/c0 ) = − dt ρ0 ∂t ∂x  1 1 pa ∂f (t − x/c0 ) = dt f (t − x/c0 ) = . (6.116) ρ0 c0 ∂t ρ0 c0 ρ0 c0 Therewith, the value of the acoustic pressure over acoustic particle velocity for a plane wave is constant. To allow for a general orientation of the coordinate system, a free field plane wave may be expressed by pa = f (n · x − c0 t) ;

va =

n f (n · x − c0 t) , ρ0 c0

(6.117)

where the direction of propagation is given by the unit vector n. A time-harmonic plane wave of angular frequency ω = 2πf is usually written as pa , va ∼ ej(ωt−k·x)

(6.118)

with the wave number (also called wave vector) k, which computes by k = kn =

ω n. c0

(6.119)

The second case of investigation will be a spherical wave, where we assume a point source located at the origin. In the first step, we rewrite the linearized wave equation in spherical coordinates and consider that the pressure pa just depends on the radius r. Therewith, the Laplace operator reads as ∇ · ∇pa (r, t) =

1 ∂ 2 rpa ∂ 2 pa 2 ∂pa = + 2 ∂r r ∂r r ∂r2

288

Chapter 6. Physical Models for Flow: Acoustic Interaction

and we obtain 1 ∂ 2 rpa 1 ∂ 2 pa − = 0. r ∂r2 c20 ∂t!2"

(6.120)

1 ∂ 2 rpa r ∂t2

A multiplication of (6.120) with r results in the same wave equation as obtained for the plane case (see (6.112)), just instead of pa we have rpa . Therefore, the solution of (6.120) reads as pa (r, t) =

1 (f (t − r/c0 ) + g(t + r/c0 )) , r

(6.121)

which means that the pressure amplitude will decrease according to the distance r from the source. The acoustic intensity is defined by the product of the two primary acoustic quantities I a = pa v a . (6.122) The assumed symmetry requires that all quantities will just exhibit a radial component. Therewith, we can express the time averaged acoustic intensity Iaav in normal direction n by a scalar value just depending on r Iaav · n = Irav and as a function of the time averaged acoustic power Paav of our source Irav =

Paav . 4πr2

(6.123)

According to (6.123), the acoustic intensity decreases with the squared distance from the source. This relation is known as the spherical spreading law. In order to obtain the acoustic velocity va = va (r, t)er as a function of the acoustic pressure pa , we substitute the general solution for pa (see (6.121), in which we set without loss of generality g = 0) into the linear momentum equation (see (6.106)) ∂va ∂t va

1 = − ρ0 1 = − ρ0

  ∂pa 1 ∂ f (t − r/c0 ) =− ∂r ρ0 ∂r r   ∂ F (t − r/c0 ) ∂r r

with f (t) = ∂F (t)/∂t. Using the relation 1 ∂F (t − r/c0 ) ∂F (t − r/c0 ) =− ∂r c0 ∂t

(6.124)

6.3. Acoustics

289

and performing the differentiation with respect to r result in va (r, t)

= =

1 ρ0 1 ρ0 c0

−

1 ∂F (t − r/c0 ) F (t − r/c0 ) + r ∂r ρ0 r2 1 ∂F (t − r/c0 ) F (t − r/c0 ) + ∂t ρ0 r2 ! " r

(6.125) (6.126)

f /r=pa

=

pa F (t − r/c0 ) + . ρ0 c0 ρ0 r2

(6.127)

Therewith, spherical waves show in the limit r → ∞ the same acoustic behavior as plane waves. Now with this acoustic velocity-pressure relation, we may rewrite the acoustic intensity for spherical waves as Ir =

pa 2 pa + F (t − r/c0 ). ρ0 c0 ρ0 r2

With the relation (just outgoing waves) pa = we obtain Ir =

1 ∂F f = r r ∂t

pa 2 1 ∂F 2 (t − r/c0 ) , + ρ0 c0 2ρ0 r3 ∂t

which results for the time averaged quantity (assuming F (t − r/c) is a periodic function) in the same expression as for the plane wave Irav =

6.3.3

(pa 2 )av . ρ0 c0

Impulsive Sound Sources

The sound being generated by a unit, impulsive point source δ(x)δ(t) is the solution of 1 ∂ 2 ψa − ∇ · ∇ψa = δ(x)δ(t) (6.128) c20 ∂t2 with ψa the scalar acoustic potential. Now, since the source exists only for an infinitesimal instant of time t = 0, the scalar potential ψa will be zero for t < 0. Due to the radial symmetry, we may rewrite (6.128) in cylindrical coordinates for r = |x| > 0 by   1 ∂ 1 ∂ 2 ψa 2 ∂ − 2 (6.129) r ψa = 0 for r > 0 . c20 ∂t2 r ∂r ∂r

290

Chapter 6. Physical Models for Flow: Acoustic Interaction

According to Sect. 6.3.2 (see (6.121)) the solution is f (t − r/c0 ) g(t + r/c0 ) + . (6.130) r r The first term represents a spherically symmetric wave propagating in the direction of increasing values of r (outgoing wave) and the second term describes an incoming wave. Physically, we have to set g to zero, since according to causality (also known as the radiation condition) sound produced by a source must radiate away from this source. To complete the solution, we have to determine the function f , which results in (see [26]) 1 δ(t − r/c0 ) (6.131) f (t − r/c0 ) = 4π and the solution becomes 1 1 δ(t − r/c0 ) = δ(t − |x|/c0 ) . ψa (x, t) = (6.132) 4πr 4π|x| ψa =

This represents a spherical pulse that is nonzero only on the surface of the sphere with r = c0 t > 0, whose radius increases with the speed of sound c0 . It clearly vanishes everywhere for t < 0. Compared to the solution of a potential flow generated by a pulsating sphere (see Sect. 6.2.5) we have as an argument the retarded time.

6.3.4

Free-Space Green’s Functions

The free-space Green’s function G(x, y, t − τ ) is the causal solution of the wave equation by an impulsive point source with strength δ(x − y)δ(t − τ ) located at x = y at time t = τ . The expression for G is simply obtained from (6.132), when we replace the source position x = 0 at time t = 0 by x − y at t − τ . This substitutions result in   1 ∂2 − ∇ · ∇ G = δ(x − y)δ(t − τ ) where G = 0 for t < τ (6.133) c20 ∂t2   |x − y| 1 δ t−τ − . (6.134) 4π|x − y| c0 This describes an impulsive, spherical symmetric wave expanding from the source at y (therefore y are called the source coordinates) with the speed of sound c0 . The wave amplitude decreases inversely with the distance to the observation point x. Now, Green’s function is the fundamental building block for the computation of the inhomogeneous wave equation with any generalized source distribution F(x, t)   1 ∂2 − ∇ · ∇ pa = F(x, t) . (6.135) c20 ∂t2 with

G(x, y, t) =

6.3. Acoustics

291

The key idea is that the source distribution is regarded as a distribution of impulsive point sources4 T  F(x, t) =

F(y, τ )δ(x − y)δ(t − τ ) dy dτ . 0

Rn

Therefore, the outgoing wave solution for each constituent source strength F(y, τ )δ(x − y)δ(t − τ ) is given by F(y, τ )G(x, y, t − τ ) . Therefore, the overall solution is obtained by adding up all the individual contributions T  pa (x, t) = F(y, τ )G(x, y, t − τ ) dy dτ 0 Rn

  F(y, τ ) |x − y| δ t−τ − dy dτ |x − y| c0 n 0 R    F y, t − |x−y| c0 1 dy . 4π |x − y| 1 4π

=

=

T 

(6.136)

Rn

This integral formula is called a retarded formula, since it represents the pressure at position x (observation point) and time t as a linear superposition of sources at y radiated at earlier times t − |x − y|/c0 . Thereby, the time of travel for the sound waves from the source point y to the observer point x is |x − y|/c0 . In general, finding a (tailored) Green’s function of given configuration (including, e.g., scatterer) is only marginally easier than the full solution of the inhomogeneous wave equation. Therefore, it is not possible to give a general recipe. However, it is important to note that often we can simplify a problem already by the corresponding integral formulation (as done above) using free field Green’s function. Furthermore, the delta-function source may be rendered into a more easily treated from by spatial Fourier transform. Thereby, (6.134) leads to the free field Green’s function in the frequency domain (setting τ = 0) ∞ ˆ G(x, ω)

= −∞

  |x − y| 1 δ t− e−jωt dt 4π|x − y| c0

e−jkr 4πr with r = |x − y| and k = ω/c0 . =

4 Symbol

Rn denotes the Euclidean space.

(6.137)

292

6.3.5

Chapter 6. Physical Models for Flow: Acoustic Interaction

Monopoles, Dipoles, and Quadrupoles

A volume point source q(t)δ(x) as a model of a pulsating sphere (as considered in Sect. 6.2.5) is called a monopole point source. Now, we consider a compressible fluid and the corresponding wave equation   1 ∂2 − ∇ · ∇ ψa = q(t)δ(x) . c0 ∂t2 The solution can be simply obtained by using (6.136), replacing pa by ψa and setting F(y, τ ) = q(τ )δ(y) ψa (x, t) =

q(t − r/c0 ) q(t − |x|/c0 ) = . 4π|x| 4πr

(6.138)

This differs from the solution obtained within an incompressible fluid (see (6.77)) by the dependence on the retarded time t − r/c0 . Any change at the source is now communicated to a fluid element at distance r after an appropriate estimated delay r/c0 required for sound to travel outward from the source. In the next step, we will investigate in a point dipole. Then, a source on the right hand side of the wave equation (6.135) of the following type: F(x, t) = ∇ · (f (t)δ(x)) =

∂ (fj (t)δ(x)) ∂xj

(6.139)

is called a point dipole located at the origin. The sound generated by such a source computes according to (6.136)    δ t − |x−y| c0 ∂ 1 pa (x, t) = dy . (6.140) (fj (t − |x − y|/c0 ) δ(y)) 4π ∂yj |x − y| Rn

In a first step, we perform an integration by parts and arrive at ⎞ ⎛  |x−y|  1 ∂ ⎝ δ t − c0 ⎠ dy pa (x, t) = − fj (t − |x − y|/c0 ) δ(y) 4π ∂yj |x − y| Rn   |x−y|  δ t − c0 1 + n · ej dΓ . (6.141) (fj (t − |x − y|/c0 ) δ(y)) 4π |x − y| Γ

Thereby the second integral has to be evaluated at a surface for which yj = ±∞, so that due to the property of the delta function δ(y) = 0 at yj = ±∞, it vanishes. Furthermore, we explore the relation ⎞ ⎞ ⎛  ⎛  |x−y| |x−y| δ t − δ t − c0 c0 ∂ ⎝ ⎠=− ∂ ⎝ ⎠, (6.142) ∂yj |x − y| ∂xj |x − y|

6.3. Acoustics

293

and arrive at

⎞ ⎛  |x−y| δ t − c0 1 ∂ ⎝ ⎠ dy fj (t − |x − y|/c0 ) δ(y) 4π ∂xj |x − y| Rn    δ t − |x−y| c0 1 ∂ dy . (6.143) fj (t − |x − y|/c0 ) δ(y) 4π ∂xj |x − y| 

pa (x, t)

=

=

Rn

Due to the property of the delta function, we can directly obtain the solution for the acoustic pressure by   fj (t − |x|/c0 ) ∂ pa (x, t) = . (6.144) ∂xj 4π|x| Therefore, a distributed dipole source F(x, t) = ∇ · f (x, t) results in the following expression for the acoustic pressure:  fj (y, t − |x − y|/c0 ) 1 ∂ dy . (6.145) pa (x, t) = 4π ∂xj |x − y| Rn

A point dipole at the origin oriented in the direction of the unit vector n is entirely equivalent to two point monopoles of equal but opposite strengths placed a short distance apart (much smaller as the wavelength). Furthermore, a combination of four monopole sources, whose net volume source strength is zero, is called a quadrupole. A general quadrupole is a source distribution being characterized by a second space derivative of the form F(x, t) =

∂ 2 Lij . ∂xi ∂xj

(6.146)

Here, Lij are the components of an arbitrary tensor. In the context of aeroacoustics, [L] will denote the Lighthill tensor (see Sect. 6.4.1). Applying the procedure as in the case of the dipole source two times results in the corresponding acoustic pressure  ∂2 Lij (y, t − |x − y|/c0 ) 1 dy . (6.147) pa (x, t) = 4π ∂xi ∂xj |x − y| Rn

6.3.6

Calculation of Acoustic Far Field

We will now discuss useful approximations for the evaluation of    F y, t − |x−y| c 1 0 dy , pa (x, t) = 4π |x − y| Rn

(6.148)

294

Chapter 6. Physical Models for Flow: Acoustic Interaction

when computing the sound in the far field. Thereby, as mostly true for practical applications, we assume that F(x, t) is nonzero only in a finite source region, as displayed in Fig. 6.5. Furthermore, the source region contains the origin O of the x

path from y y path from 0

A 0 x.y x

source region

Figure 6.5: Acoustic far-field calculation coordinate system. In a first step, we assume |x|  |y|, so that the following approximation will hold |x − y|

= = ≈ ≈

1 |x|2 − 2x · y + |y|2 2  1 2x · y |y|2 2 |x| 1 − + |x|2 |x|2   12 2x · y |x| 1 − |x|2 |y| x·y for  1. |x| − |x| |x|



(6.149)

In a second step, we investigate in the term 1/|x − y| using the above result   1 1 1 1 ≈ = . (6.150) x·y |x − y| |x| − x·y |x| 1 − |x| 2 |x| Now, we develop the term in the parenthesis in a Taylor series up to first order and arrive at   1 x·y 1 1 x·y ≈ + . = 1+ 2 |x − y| |x| |x| |x| |x|3 This approximation demonstrates that in order to obtain the far-field approximation of (6.148), which solution behaves like 1/r = 1/|x| as |x| → ∞, it is sufficient to replace |x − y| in the denominator of the integrand by |x|. However, in the argument of the source strength F it is important to retain possible phase differences between the sound waves generated by the source distribution at location

6.3. Acoustics

295

y. Therefore, we replace |x − y| in the source argument by the approximation obtained in (6.149) and arrive at    1 |x| x·y pa (x, t) ≈ F y, t − + dy , |x| → ∞ . (6.151) 4π|x| c0 c0 |x| Rn

This approximation when computing the acoustic far field is known as Fraunhofer approximation. The source region may extend over many characteristic wavelengths of the sound. By retaining the contribution x · y/(c0 |x|) to the retarded time, we ensure that the interference between waves generated at different positions within the source region is correctly described by this far-field approximation. Let us consider the setup as displayed in Fig. 6.5. The acoustic travel time from a source point y to a far-field point x is equal to that from the point labeled by A to x when x goes to infinity. The travel time over the distance OA computes by tOA =

1 1 x . y · ex = y· c0 c0 |x|

Therefore, the time obtained by |x|/c0 − x · y/(c0 |x|) is the correct value of the retarded time when x goes to infinity. Let us apply the above approximation for a dipole source distribution. In doing so, we use the far-field formula according to (6.151) to a dipole source F(x, t) = ∇ · f (x, t) and obtain ⎛ ⎞    ∂ ⎝ 1 x·y |x| fj y, t − + dy ⎠ . (6.152) pa (x, t) ≈ 4π|x| ∂xj c0 c0 |x| Rn

In the next step we replace the space derivative with a time derivative, which is usually more easily estimated in practical applications. This operation is done as follows:     x·y x·y |x| |x| ∂fj ∂ ∂ fj y, t − + + t− = . ∂xj c0 c0 |x| ∂t ∂xj c0 c0 |x| Now, the second term evaluates as   x·y |x| ∂ + t− = ∂xj c0 c0 |x| = = ≈

1 ∂ 1 ∂|x| + − c0 ∂xj c0 ∂xj



x·y |x|



1 yj |x| − x · y xj |x|−1 1 xj + c0 |x| c0 |x|2 yj x · y xj 1 xj + − − c0 |x| c0 |x| |x|3 1 xj for |y|  |x| . − c0 |x| −

296

Chapter 6. Physical Models for Flow: Acoustic Interaction

Collecting these results, we can provide the far-field approximation for a source dipole F = ∇ · f (x, t) as follows (cancelling all terms which are proportional to 1/|x|2 as well as x · y xj /|x|4 ) ⎛ ⎞    −xj ∂ ⎝ x·y |x| fj y, t − + pa (x, t) ≈ dy ⎠ . (6.153) 4πc0 |x|2 ∂t c0 c0 |x| Rn

Please note that the term xj xj 1 xj 1 = = |x|2 |x| |x| |x| r is not changing the rate of the amplitude decay, which is still given by 1/r. The first term xj /|x| is the jth component of the unit vector x/|x| and so it does just influence the directivity pattern (see Fig. 6.6 for the directivity of a dipole). x2

r j

x1

Figure 6.6: Directivity of a dipole source. The plotted directivity is ∝ p2 (proportional to the intensity) Furthermore, it is necessary to realize that the rule of interchanging a space derivative with a time derivative is given by ∂ 1 xj ∂ . ≈− ∂xj c0 |x| ∂t

(6.154)

We will now explore this relation, when deriving the far-field approximation for a quadrupole source given by F(x, t) =

∂ 2 Lij (x, t) . ∂xi ∂xj

According to (6.154) we directly arrive at ⎛ ⎞    2 xi xj ∂ ⎝ x·y |x| pa (x, t) ≈ Lij y, t − + dy ⎠ . 4πc20 |x|3 ∂t2 c0 c0 |x| Rn

(6.155)

(6.156)

6.3. Acoustics

297 x2 r

(1, 2)-quadrupole directivity

x1

Figure 6.7: Directivity of a quadrupole source: radiation in the x1 -x2 plane (ϕ = 0, π) Now, for a point quadrupole in the x1 -x2 plane F(x, t) =

∂2 (L(t)δ(x)) ∂x1 ∂x2

we obtain by exploring (6.156) the following far-field pressure:   x1 x2 ∂ 2 |x| pa (x, t) ≈ L t− x → ∞. 4πc20 |x|3 ∂t2 c0

(6.157)

If we use spherical coordinates, such that x1 = r cos ϑ ; x2 = r sin ϑ cos ϕ ; x3 = r sin ϑ cos ϕ we may rewrite the pressure by pa (x, t) ≈

  sin 2ϑ cos ϕ ∂ 2 |x| L t − 8πc20 |x| ∂t2 c0

x → ∞.

(6.158) 2

The directivity pattern of the sound intensity, which is given by ∝ (pa ) , is therefore represented by sin2 2ϑ cos2 ϕ. Its shape is displayed in Fig. 6.7.

6.3.7

Compactness

We consider a rigid sphere of radius a oscillating at a small amplitude of velocity U ex1 , as displayed in Fig. 6.8. Assuming that a is very small, we may model the source as a point dipole of amplitude 2πa3 U (t), so that the acoustic potential ψa computes by solving    1 ∂2 ∂  2πa3 U (t)δ(x) . − ∇ · ∇ ψa = (6.159) 2 2 c0 ∂t ∂x1

298

Chapter 6. Physical Models for Flow: Acoustic Interaction x2 r

Θ

x1

x3

Figure 6.8: Oscillating sphere According to (6.144), the solution is given by   2πa3 U (t − |x|/c0 ) ∂ ψa (x, t) = . ∂x1 4π|x| With the following two relations:   1 ∂ ∂x1 |x| ∂ (U (t − |x|/c0 ) ∂x1

= =

(6.160)

x1 |x|3 ∂U (t) 1 x1 − ∂t c0 |x| −

and r = |x|, x1 = r cos Θ, we arrive at ψa (x, t) = − 

a3 cos Θ ∂U (t − |x|/c0 ) a3 cos Θ − . U (t − |x|/c ) 0 2r2 ∂t ! "  2c0 r ! " near field far field

(6.161)

Thereby, we observe that the near-field term is dominant at sufficiently small distances r from the origin 1 1 ∂U f 1 1  ∼ = . r c0 U ∂t c0 λ

(6.162)

Hence, the near-field term is dominated when r  λ.

(6.163)

The motion becomes incompressible when c0 → ∞. In this limit, the solution reduces entirely to the near-field term, which we also call the fluid dynamic nearfield and its amplitude decreases like 1/r2 as r → ∞. Thereby, the retarded time can be neglected and the near-field coincides with that of the incompressible potential flow. In regions, e.g. at boundaries, where the acoustic potential ψa varies significantly over a distance l, which is short compared to the wavelength λ, the acoustic

6.3. Acoustics

299

field can be approximated by the incompressible potential flow. We call such a region compact, and a source size much smaller than λ is a compact source. For a precise definition, we define a typical time scale τ (or angular frequency ω) and a length scale l. Then, the dimensionless form of the wave equation reads ∂ 2 ψa ∂ 2 ψa = He2 2 ∂x ˜i ∂ t˜2

(6.164)

with t˜ = t/τ = ωt and x ˜i = xi /l. In (6.164) He denotes the Helmholtz number and computes by l ωl 2πl He = =  1. = c0 τ c0 λ Note that the time derivative term in (6.164) is multiplied by the square of a Helmholtz number. Therefore, if He is small, we may neglect this term and the wave equation reduces to (6.165) ∇ · ∇ψa = 0 . Hence, we can describe the acoustic field by the incompressible potential flow, which allows us to use incompressible potential flow theory to derive the local behaves of acoustic fields in compact regions. To conclude, acoustic compactness can be used to justify acoustic assumptions, simulation model simplifications, and introduce evidence based defeaturing.

6.3.8

Solution of Wave Equation Using Green’s Function

We consider a stationary medium, in which the acoustic field is computed by the linear wave equation. The domain may include surfaces generating sound and surfaces, where the sound waves are scattered, as displayed in Fig. 6.9. Our goal is to compute the acoustic pressure pa at observer position x and time t due to

Radiating sound waves

Scattering surface

vn

Radiating surface

Figure 6.9: Radiating and scattering surfaces in a sound field

300

Chapter 6. Physical Models for Flow: Acoustic Interaction

some sources in space y and time τ . In doing so, we write the wave equation in terms of y and τ 1 ∂ 2 pa (y, τ ) ∂ 2 pa (y, τ ) − = 0. c20 ∂τ 2 ∂yi2

(6.166)

In the next step, we introduce the Green’s function G being the solution of the inhomogeneous wave equation 1 ∂2G ∂2G − = δ(x − y) δ(t − τ ) . c20 ∂τ 2 ∂yi2

(6.167)

Thereby, the Green’s function is distributed in space as a function of y and τ , but also depends on the observer position x and time t, and so we write G = G(x, t|y, τ ). Now, we multiply (6.167) by pa (y, τ ), (6.166) by G(x, t|y, τ ) and subtract the so obtained equations to achieve at     1 ∂2G ∂2G ∂ 2 pa ∂ 2 pa − G − G p − p = δ(x − y) δ(t − τ ) pa (y, τ ) . a a c20 ∂τ 2 ∂τ 2 ∂yi2 ∂yi2 (6.168) Next, we integrate over τ and the volume Ω(y) and explore the property of the delta function  T pa (x, t) = Ω

0

1 c20

 pa

∂2G ∂ 2 pa −G 2 ∂τ ∂τ 2



  ∂2G ∂ 2 pa − G − pa dτ dy . (6.169) ∂yi2 ∂yi2

The first integrand in (6.169) may be rearranged as follows:     ∂pa ∂2G ∂G ∂ 2 pa ∂ −G −G = pa pa ∂τ 2 ∂τ 2 ∂τ ∂τ ∂τ

(6.170)

from which follows:  T Ω

∂ ∂τ

 pa

∂pa ∂G −G ∂τ ∂τ

 



pa

dτ dy =

0

Ω

∂pa ∂G −G ∂τ ∂τ

τ =T dy .

(6.171)

τ =0

The integrand is zero at the lower limit τ = 0, when we specify pa and ∂pa /∂τ to be zero at τ = 0. Furthermore, we explore the causality condition that sound heard at time t must be generated at time τ < t. This implies that the Green’s function G(x, t|y, τ ) and its time derivative ∂G(x, t|y, τ )/∂τ are zero for τ ≥ t, so that the integrand at the upper limit is also zero for t < T . A similar expansion is done for the second term in (6.169) obtaining       ∂2G ∂ ∂G ∂ 2 pa ∂pa −G pa 2 − G 2 dy = pa dy . (6.172) ∂yi ∂yi ∂yi ∂yi ∂yi Ω

Ω

6.3. Acoustics

301

Here, we can use the divergence theorem to transform the volume integral to the enclosing surface integral with normal vector pointing into the volume  T  pa (x, t) = Γ

∂G(x, t|y, τ ) ∂pa (y, τ ) − G(x, t|y, τ ) pa (y, τ ) ∂yi ∂yi

 ni dΓ(y) dτ .

0

(6.173) In free field application, the Sommerfeld radiation condition eliminates the need to include the exterior boundary in the surface integral. The resulting integral equation (6.173) solves the linear acoustic wave equation and can be evaluated from knowledge of the pressure and the pressure gradient on surfaces that bound the region of interest as well as other surfaces, i.e. scattering surfaces or sound generating surfaces. Furthermore, we need to know the Green’s function, which must satisfy the inhomogeneous wave equation (6.167) and a causality condition, e.g. free field Green’s function according to (6.134). Then we can use (6.173) for both sound radiation and scattering problems involving prescribed surface motion and/or surface pressure as displayed in Fig. 6.9. For radiating surfaces, the pressure gradient term can be obtained by the acoustic momentum equation5 (6.106) as ∂pa ∂va,i ∂vn ni = −ρ0 . ni = −ρ0 ∂yi ∂τ ∂τ

(6.174)

Having the relation between the normal pressure gradient and the normal acceleration at the surface, (6.173) may be rewritten as pa (x, t)

=

  T  ∂G(x, t|y, τ ) pa (y, τ ) ni dΓ(y) dτ ∂yi Γ

0

 T  +

ρ0 Γ

∂vn G(x, t|y, τ ) ∂τ

 dΓ(y) dτ .

(6.175)

0

Since for many cases, we perform computations in the frequency domain, we also provide (6.173) in the frequency domain via a Fourier transform. Thereby, we obtain    ˆ ∂ G(x|y) ∂ pˆa (y, ω) ˆ − G(x|) (6.176) pˆa (x, ω) = pˆa (y, ω) ni dΓ(y) ∂yi ∂yi Γ

and for our special case of a radiating surface according to (6.175)       ˆ ∂ G(x|y) ˆ jωρ0 vˆn G(x|y) dΓ(y) pˆa (x, ω) = pˆa (y, ω) ni dΓ(y) + ∂yi Γ

Γ

(6.177) 5 In the case of a non-homogeneous acoustic momentum equation, this equality looks different. This is particularly important for flow-induced sound.

302

Chapter 6. Physical Models for Flow: Acoustic Interaction

ˆ i.e. using free field Green’s function according to (6.137). with G, As an example, let us compute the acoustic pressure generated by an pulsating sphere with radius a and vibration velocity vˆn . Assuming the radius a to be ˆ and ∂ G/∂y ˆ very small, the spatial dependency of G i becomes negligible, and we may write for the far-field solution 2 3   ˆ ∂G ˆ pˆa (x, ω) = pa ni dΓ(y) + [G]yi =0 jωρ0 vˆn dΓ(y) . (6.178) ∂yi yi =0 Γ

Γ

The first term represents the net force exerted on the fluid by the sphere, and is zero because the pressure is constant on the surface. Therefore, we obtain pˆa (x, ω) =

jωρ0 a2 vˆn jk|x| e . |x|

(6.179)

For the general case, we often do not know pa (y) on the surfaces and (6.176) or (6.177) cannot be directly solved. So, a numerical method, i.e. the boundary element method has to be applied. However, in cases that the surface is a rigid ˆ scatterer, we get rid of the first term, when we force ni ∂ G/∂y i to be zero. Such Green’s functions are called tailored (modified) Green’s functions [19, 26].

6.4 6.4.1

Aeroacoustics Lighthill’s Acoustic Analogy

The sound generated by a flow in an unbounded fluid is usually called aerodynamic sound. Most unsteady flows in technical applications are of high Reynolds number, and the acoustic radiation is a very small by-product of the motion. Thereby, the turbulence is usually produced by fluid motion over a solid body and/or by flow instabilities. Lighthill transformed the general equations of mass and momentum conservation to an exact inhomogeneous wave equation whose source terms are important only within the turbulent region [37]. Lighthill was initially interested in solving the problem, illustrated in Fig. 6.10, of the sound produced by a turbulent nozzle and arrived at the inhomogeneous wave equation. However, at this time a volume discretization by numerical schemes was not feasible and so a transformation of the partial differential equation into an integral representation was performed, which can just be achieved for a free field setup, for which Green’s function is available. Therefore, Lighthill’s theory in its integral formulation just applies to the simple situation as given in Fig. 6.10. This avoids complications caused by the presence of the nozzle. The fluid is assumed to be at rest at the observer position, where a mean pressure, density, and speed of sound are respectively equal to p0 , ρ0 , and c0 . So Lighthill compared the equations for the production of density fluctuations in the real flow with those in an ideal linear acoustic medium (quiescent fluid).

6.4. Aeroacoustics

303 sound

sound

v turbulent nozzle flow (a) Turbulent nozzle flow.

(b) Isolated turbulent region.

Figure 6.10: Sound generation by turbulent flows. (a) Turbulent nozzle flow. (b) Isolated turbulent region For the derivation, we start at Reynolds form of the momentum equation, as given by (6.12) neglecting any force density f ∂ρv + ∇ · [π] ∂t

=

0,

(6.180)

with the momentum flux tensor πij = ρvi vj + (p − p0 )δij − τij , where the constant pressure p0 is inserted for convenience. In an ideal, linear acoustic medium, the momentum flux tensor contains only the pressure perturbation 0 πij = (p − p0 )δij = c20 (ρ − ρ0 )δij

(6.181)

and Reynolds momentum equation yields  ∂ρvi ∂  2 c0 (ρ − ρ0 ) = 0 . + ∂t ∂xi

(6.182)

Rewriting the conservation of mass in the form ∂ ∂ρvi (ρ − ρ0 ) + =0 ∂t ∂xi

(6.183)

allows us to eliminate the momentum density ρvi in (6.182). Therefore, we perform a time derivative on (6.183), a spatial derivative on (6.182), and subtract the two resulting equations. These operations lead to the equation of linear acoustics satisfied by the perturbation density     1 ∂2 − ∇ · ∇ c20 (ρ − ρ0 ) = 0 . (6.184) c20 ∂t2 Because flow is neglected, the unique solution of this equation satisfying the radiation condition6 and we obtain ρ − ρ0 = 0. 6 This is an ordinary acoustic wave equation that can possibly radiate sound. Regarding the assumptions, no surfaces, boundary conditions, or sources are incorporated in this mathematical model.

304

Chapter 6. Physical Models for Flow: Acoustic Interaction

Now, it can be asserted that the sound generated by the turbulence in the real fluid is exactly equivalent to that produced in the ideal, stationary acoustic medium forced by the stress distribution in addition to the state of rest   0 = ρvi vj + (p − p0 ) − c20 (ρ − ρ0 ) δij − τij , (6.185) Lij = πij − πij where [L] is called the Lighthill stress tensor. Indeed, we can rewrite (6.180) as the momentum equation for an ideal, stationary acoustic medium of mean density ρ0 and speed of sound c0 subjected to the externally applied stress Lij 0 ∂πij ∂ρvi + ∂t ∂xj

=

−

 ∂  0 πij − πij , ∂xj

(6.186)

or equivalent  ∂ρvi ∂  2 + c0 (ρ − ρ0 ) ∂t ∂xj

=

−

∂Lij . ∂xj

(6.187)

By eliminating the momentum density ρvi using (6.183) we arrive at Lighthill’s equation    2  1 ∂2 ∂ 2 Lij c − ∇ · ∇ (ρ − ρ ) = . (6.188) 0 0 c20 ∂t2 ∂xi ∂xj It has to be noted that (ρ − ρ0 ) = ρ is a fluctuating density not being equal to the acoustic density ρa , but a superposition of flow and acoustic parts within flow regions.7 In the definition of the Lighthill tensor according to (6.185) the term ρvi vj is called the Reynolds stress [69]. It is a nonlinear term  except  and can be neglected where the motion is turbulent. The second term (p − p0 ) − c20 (ρ − ρ0 ) δij represents the excess of moment transfer by the pressure over that in the ideal fluid of density ρ0 and speed of sound c0 . This is produced by wave amplitude nonlinearity, and by mean density variations in the source flow. The viscous stress tensor τij properly accounts for the attenuation of the sound. In most applications the Reynolds number in the source region is high and we can neglect this contribution. Please note that the terms in Lij account not only for the generation of sound, but also includes acoustic self-modulation caused by • acoustic nonlinearity, • the convection of sound waves by the turbulent flow velocity, • refraction caused by sound speed variations, • and attenuation due to thermal and viscous actions. 7 Region, where aeroacoustic sources occur and where the flow perturbation is not decayed to zero.

6.4. Aeroacoustics

305

The influence of acoustic nonlinearity and thermoviscous dissipation is usually sufficiently small to be neglected within the source region. Convection and refraction of sound within the flow region can be important, e.g., in the presence of a mean shear layer (when the Reynolds stress will include terms like ρv0i vj , where v0 and v  respectively denote the mean and fluctuating components of v). Such effects are described by the presence of unsteady linear terms in Lij . Therefore, we can summarize that Lighthills’ inhomogeneous wave equation is a general model to describe flow-induced sound. Solving this partial differential equation by the correct boundary conditions, i.e. ∂p /∂n = 0 at the surface of solid bodies, using a volume discretization method includes all sources of the sound. The main challenge in doing so is that the whole set of compressible flow dynamics equations have to be solved in order to be able to calculate Lighthill’s tensor. However, this means that we have to resolve both the flow structures and acoustic waves, which is an enormous challenge for any numerical scheme and the computational noise itself may strongly disturb the physical radiating wave components [8]. Therefore, in the theories of Phillips and Lilley interaction effects have been, at least to some extend, moved to the wave operator [38, 40]. These equations predict certain aspects of the sound field surrounding a jet quite accurately, which are not accounted for Lighthill’s equation due to the restricted numerical resolution of the source term need in (6.188) [22]. The solution of (6.188) for free field radiation condition with outgoing wave behavior can be rewritten in integral form as follows (see Sect. 6.3.5):  ∂2 Lij (y, t − |x − y|/c0 ) 1 dy . (6.189) c20 (ρ − ρ0 )(x, t) = 4π ∂xi ∂xj |x − y| Rn

Thereby, y defines the source coordinate and x the coordinate at which we compute the acoustic density fluctuation. This provides a useful prediction of the sound, if Lij is known. Now, let us consider the situation for which the mean density and speed of sound are uniform throughout the fluid. The variations in the density ρ within a 2 low Mach number, high Reynolds  number source flow are then of order O(ρ0 Ma ). 2 Thus, ρvi vj = ρ0 1 + O(Ma ) vi vj ≈ ρ0 vi vj . Furthermore, if c(x, t) is the local speed of sound in the source region, it can be shown that c20 /c2 = 1 + O(Ma2 ), so that we obtain p − p0 − c20 (ρ − ρ0 ) = (p − p0 )(1 − c20

ρ − ρ0 ) ≈ (p − p0 )(1 − c20 /c2 ) ∼ O(ρ0 v 2 Ma2 ) . p − p0  ! " 1/c2

(6.190) As a consequence and if viscous dissipation is neglected, we may approximate the Lighthill tensor by (6.191) Lij ≈ ρ0 vi vj for Ma2  1 .

306

Chapter 6. Physical Models for Flow: Acoustic Interaction

With this result we obtain for the pressure fluctuation using the isentropic pressuredensity relation the following integral representation:  ρ0 vi vj (y, t − |x − y|/c0 ) ∂2  dy (6.192) p (x, t) ≈ ∂xi ∂xj 4π|x − y| Rn    ∂2 xi xj x·y |x| ≈ ρ v v + y, t − dy . (6.193) 0 i j 4πc20 |x|3 ∂t2 c0 c0 |x| Rn

To obtain (6.193), we have used the far-field approximation, which allows the following substitution (see Sect. 6.3.6): 1 xj ∂ ∂ . ≈− ∂xj c0 |x| ∂t

(6.194)

Now, we want to derive the order of the magnitude of the acoustic pressure as a function of the flow velocity v. In doing so, we introduce a characteristic velocity v and length scale l of a single vortex as displayed in Fig. 6.11. Fluctuations in vi vj x

vortex

l

turbulent source region Ωf

Figure 6.11: Single vortex in a turbulent flow region at its acoustic radiation towards the far field occurring in different turbulent regions by distances larger than O(l) will be treated to be statistically independent. So the sound may be considered to be generated by a collection of Ω f /l3 independent vortices. The characteristic frequency of the turbulent fluctuations can be estimated by f ∼ v/l so that the wavelength λ of sound will result in λ=

c0 l l c0 ∼ =  l for Ma = v/c0  1 . f v Ma

6.4. Aeroacoustics

307

Hence, we arrive at the quite important conclusion for low Mach number flows that the turbulent vortices are all acoustically compact. This means that in the relation (6.193) the retarded time variation x · y/(c0 |x|) can be neglected. Therefore, the value of the integral over one source vortex in (6.193) can be estimated to be of order ρ0 v 2 l3 . The order of the magnitude for the time derivative in (6.193) is estimated to be v ∂ ∼ . ∂t l Collecting all these estimates, we may now state that the acoustic pressure in the far-field, generated by one vortex, satisfies pa ∼

l ρ0 v 4 l ρ0 v 2 Ma2 . = |x| c20 |x|

The acoustic power defined by = = Pa = pa va · dΓ = pa va · n dΓ Γ

(6.195)

(6.196)

Γ

can be computed in the far-field with the relation va · n = pa /(ρ0 c0 ) as follows: = pa 2 Pa = dΓ . (6.197) ρ0 c0 Γ

This formula allows us to estimate the acoustic power generated by one vortex Pa ∼ 4π|x|2

pa 2 l2 ρ0 v 8 ∼ = ρ0 l2 v 3 Ma5 . ρ0 c0 c50

(6.198)

This is the famous eighth power law. To conclude, the eighth power law describes the scaling of the acoustic sound in the far-field with respect to the characteristic velocity. A rigorous analysis of the leading order term of Lighthill’s tensor for low Mach number flows in an isentropic medium applying the method of matched asymptotic expansion (see, e.g., [8]) has been done in [9]. Sound emission from an eddy region involves three length scales: the eddy size l, the wavelength λ of the sound, and a dimension L of the region. The problem is solved for Ma  1 and L/l ∼ 1 by matching the compressible eddy core scaled by l to a surrounding acoustic field scaled by λ. Thereby, Lighthill’s solution is shown to be adequate in both regions, if Lij is approximated by Lij ≈ ρ0 vic,i vic,j

(6.199)

with vic = ∇ × ψ(ω) and vorticity ω = ∇ × vic . Such a flow field is described by solving the incompressible flow dynamics equations. Thereby, we obtain an

308

Chapter 6. Physical Models for Flow: Acoustic Interaction

incompressible flow velocity vic and pressure pic . For an incompressible flow, the divergence of vic is zero, which allows to rewrite the second spatial derivative of (6.199) by  ∂2  ∂vic,j ∂vic,i ρ0 vic,i vic,j = ρ0 . (6.200) ∂xi xj ∂xi ∂xj Furthermore, applying the divergence to (6.12) provides the following equivalence (using ∇ · vic = 0 and assuming f = 0): ∇ · ∇pic = −ρ0

∂ 2 vic,i vic,j . ∂xi ∂xj

(6.201)

With such an approach, we totally separate the flow from the acoustic field, which also means that any influence of the acoustic field on the flow field is neglected.

6.4.2

Curle’s Theory

The main restriction of Lighthill’s integral formulation8 is that it can just consider free radiation. Therewith, it cannot consider situations where there is any solid body within the region. In [10] this problem was solved by deriving an integral formulation for the sound generated by turbulence in the vicinity of an arbitrary, fixed surface Γs as displayed in Fig. 6.12. Thereby the surface Γs is defined by the function f (x), which has the following property (see Fig. 6.12): ⎧ for x on Γs ⎨ 0 < 0 for x within the surface f (x) = ⎩ > 0 for x in Ω . This surface may either be a solid body, or just an artificial control surface used to isolate a fixed region of space containing both solid bodies and fluid or just fluid. To derive Curle’s equation we start with the momentum equation according to (6.187) and multiply it with the Heaviside function H(f ) H(f )

 ∂  2 ∂ρvi + H(f ) c0 (ρ − ρ0 ) ∂t ∂xi

=

−H(f )

∂Lij . ∂xj

(6.202)

8 Here, we are explicitly pointing towards the integral formulation with the use of a freefield Green’s function. There are available a lot of numerical methods that use tailored Green’s functions. Using tailored Green’s function in combination with Lighthill’s analogy will result at the same solution as Curle’s theory.

6.4. Aeroacoustics

309 f >0 n

Lij = 0 Ω

dΓ Ω f