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Basics of Aeroacoustics) Lecture notes Grundlagen der Aeroakustik (Basics of Aeroacoustics) WS 2016/2017 at Technische Universitat¨ Braunschweig Prof. Dr.-Ing. Jan Delfs Institut fur¨ Aerodynamik und Stromungstechnik¨ Abteilung Technische Akustik DLR – Deutsches Zentrum fur¨ Luft- und Raumfahrt e.V. Lilienthalplatz 7 38108 Braunschweig, Germany October 2016 Dieses von mir, Prof. Dr.-Ing. Jan Delfs, verfasste Skript unterliegt dem deutschen Urheberrecht. Es darf nur zum personlichen¨ Gebrauch zu Zwecken des Studiums verwendet werden. Insbesondere ist es nicht erlaubt, das Skript oder Teile daraus zu bearbeiten, zu ubersetzen,¨ zu kopieren oder in elektronischer Form zu speichern und an andere Personen weiterzugeben, weder in Kopie, noch auf elektronischem Wege per Email, auf Speichermedien (z. B. CD, USB-Stick usw.), uber¨ Datenbanken oder andere Medien und Systeme. Lediglich die Herstellung von Kopien und Downloads fur¨ den personlichen,¨ privaten und nicht kommerziellen Gebrauch ist erlaubt. i Introduction Among the various disciplines of classical mechanics, ”Aeroacoustics” is a comparatively young subject area. This is mainly because of the fact that it took until 1952, when Lighthill formulated his famous aeroacoustic wave equation and thus ”founded” the field of modern aeroacoustics. With the help of the source terms in Lighthills equation it became possible to physically under- stand for the first time the origin of sound produced by free turbulence. The derivation of this very equation initiated many developments extremely important for the understanding of aeroacoustic sound generation in general. These notes are an attempt to provide a very brief overview about these essential aspects of the theoretical foundations of aeroacoustics, at least from the author’s point of view. The aim is to convey to the reader the main ideas needed in order to understand the pertinent literature. The above selection of ”key-concepts” of aeroacoustics is necessarily non-unique, and many certainly interesting aspects had to be abandoned in order to fit the lecture into the given restricted time frame of one semester. Synopsis Some examples of ”noise generated aerodynamically” in section 1 are to make the reader aware of the variety of technical problems associated with aeroacoustics. Section 2 introduces to the subject area ”acoustics” in general, including various important definitions. Here the acoustic wave equation, describing the sound propagation in non-moving media is derived as a special case of the equations governing the dynamics of small perturbations in a flow field. Some of the very fundamental solutions of the wave equation are derived and discussed insofar as they illustrate the physics of sound generation and radiation as well as the motivation for the solution approaches in general cases. Then the general solution method of Green’s functions is introduced including some essential features of the theory of generalized functions, which are extremely useful for the mathematical handling of complex situations such as the description of the sound generation by moving bodies of arbitrary (and variable) shape. This section also contains the description of surface sound interaction. Moreover the Kirchhoff integral as basis for wave extrapolation is introduced. Next, the multipole-expansion of sound sources is briefly looked at. Then, the concept of acoustic nearfield, farfield and compactness along with the physical implications is introduced. Section 3 is devoted to describing sound waves in moving media and the motion of sound sources. The wave equation is generalized to account for various types of flow fields, such as potential flow and parallel shear flow. The sound intensity is generalized for sound propagating in a flow field. Some essential consequences on the propagation and radiation are discussed such as the effect of flow on the propagation of sound waves in ducts, convective amplification, sound refraction in shearlayers/temperature layers and Doppler-frequency shift. In Section 4 the various acoustic wave equations due to Lighthill, Mohring,¨ Lilley and Ffowcs Williams and Hawkings are derived and their source terms are discussed in view of a physical interpretation. Finally, secion 5 concludes the course with a discussion of some technical applications. Delfs 2016/17 CONTENTS ii List of Symbols f – scalar function f – vector function (bold face) x – spatial co-ordinates (usually position vector of observer) ξ – spatial co-ordinates (usually position vector of source) t – time (usually observer time) τ – time (usually source time or retarded time) ∇· – ”divergence of” r – ”gradient of” r× – ”curl of” · – simple contraction : – double contraction ∆ – Laplace operator ∇·∇ D – total (or material) derivative w.r.t. time (following a fluid particle) Dt δ – total variation of a thermodynamic variable Contents Synopsis i List of Symbols ii 1 Aeroacoustic Phenomena 1 2 Basic concepts of acoustics 1 2.1 Definitions . 1 2.1.1 General terms . 1 2.1.2 Acoustic quantities . 2 2.1.3 Characterisation of acoustic events . 4 2.1.4 Physiology of the ear . 4 2.1.5 Acoustic quantities adapted to hearing . 6 2.2 Basic equations . 15 2.3 Linear equations of gas dynamics . 19 2.4 Acoustics in stagnant homogeneous media . 20 2.4.1 Intensity and power . 20 Delfs 2016/17 CONTENTS iii 2.4.2 Acoustic wave equations . 22 2.4.3 Plane-, spherical- and cylindrical free acoustic waves . 27 2.4.4 More general elementary solutions to the homogenous wave equation . 34 2.4.5 From a source to its sound field in free space . 34 2.5 Mathematical tools . 39 2.5.1 Useful relations from generalized functions theory . 39 2.5.2 Green’s function method . 41 2.6 Acoustics in stagnant homogeneous media – part 2 – . 43 2.6.1 Pressure field in the presence of obstacles . 43 2.6.2 Acoustic surface properties . 47 2.6.3 Kirchhoff integral . 50 2.6.4 Expansion of sources into multipoles . 50 2.6.5 Compactness, nearfield and farfield . 59 3 Moving medium / moving source 62 3.1 Sound propagation in steady parallel flows . 62 3.1.1 Uniform flow . 64 3.1.2 Kirchhoff integral for uniform flow . 77 3.1.3 Parallel shear flows . 79 3.2 Sound propagation in steady potential flows . 91 3.2.1 Generalization of sound intensity for potential flows . 91 3.2.2 Acoustic wave equation in potential flow . 93 3.3 Motion of sound sources . 95 3.3.1 Moving mass- or heat source . 96 3.3.2 Moving point force . 98 3.3.3 Moving multipole . 99 3.3.4 Doppler effect . 99 3.3.5 Kirchhoff integral for moving surface . 100 4 Description of aerodynamic sources of sound 103 Delfs 2016/17 CONTENTS iv 4.1 The quiecent-fluid view: Lighthill’s acoustic analogy . 103 4.1.1 Solution of Lighthill’s equation for free field problems . 105 4.1.2 Solution of Lighthill’s equation for free field problems in the farfield . 106 4.2 Lilley’s equation: Analogy for shear flows . 107 4.3 Mohring’s¨ wave equation: Analogy for potential flows . 110 4.4 Curle’s equation: noise from steady objects in low speed flows . 112 4.5 Ffowcs-Williams and Hawkings equation: noise from objects in motion . 114 5 Technical application 120 5.1 Jet noise . 120 5.2 Noise from compact objects . 121 5.3 Noise from non-compact objects . 122 6 Further reading 123 References 123 A Example Problems A1 A.1 Sound of a bursting bubble . A1 A.1.1 Approach . A1 A.1.2 Boundary conditions . A1 A.1.3 Initial conditions . A2 A.1.4 General solution . A3 A.2 Sound of a harmonically pulsating (”breathing”) sphere . A5 A.3 Sound of a harmonically oscillating sphere . A7 A.4 Scattering of plane wave at cylinder . A9 B Free field Green’s functions B1 C Solutions for point mass or heat source in uniform flow C1 C.1 Particle velocity for 3D point mass or heat source in uniform flow . C1 C.2 2D acoustic field of line mass or heat source in uniform flow . C1 Delfs 2016/17 CONTENTS v C.3 1D acoustic field of plane mass or heat source in uniform flow . D1 D Nearfield terms in FW-H integral D1 Delfs 2016/17 1. Aeroacoustic Phenomena 1 1 Aeroacoustic Phenomena Aeroacoustics deals with the sound generated aerodynamically. Certainly, there are many coupled problems, in which unsteady fluid loading on structures causes them to vibrate and radiate sound. In a way, one may think of this also as ”sound generated aerodynamically”, because the loading then is of aerodynamic origin. But what we really mean, is the noise generated in the unsteadily moving fluid itself, and the influence of (non-vibrating, stiff) bodies on this very process. There is a variety of applied problems where aeroacoustic phenomena are the major contributors to noise, i.e. aircraft engine noise, propeller noise, rotor noise, noise of fans (ducted and unducted) in whatsoever application, cavity noise, tones of wires and cylindrically shaped bodies in transverse wind, airframe noise at aircraft, high speed trains and cars, noise in valves and nozzles, etc. In the following we will try to explain the nature of all these phenomena theoretically employing the well established theory of aeroacoustics, which was initiated by Lighthill in 1952 [2]. 2 Basic concepts of acoustics 2.1 Definitions 2.1.1 General terms The field of acoustics contains numerous parts of which important ones are listed below: • physical acoustics: Dynamics of continua - building acoustics (vibrational excitation of noise from the floor, sound transmission through walls,...) - room acoustics (design of concert halls, tone studios, ...) - aero-/flow acoustics (flow induced noise) • electro acoustics: generation, amplification, measurement • musical acoustics: music, instruments • physiological acoustics: functioning of the ear • psycho acoustics: perception investigations with respect to different sound events • technical acoustics: generic term for all acoustic issues, which are connected to technical prod- ucts Delfs 2016/17 2.1 Definitions 2 2.1.2 Acoustic quantities The perception of sound is the response to a physical stimulus to the ear.
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