Aerodynamic Sound

The sound generated by vorticity in an unbounded fluid is called aerodynamic sound [60, 61]. Most unsteady flows of technological interest are of high Reynolds number and turbulent, and the acoustic radiation is a very small by-product of the motion. The turbulence is usually produced by fluid motion relative to solid boundaries or by the instability of free shear layers separating a high-speed flow (such as a jet) from a stationary environment. In this chapter the influence of boundaries on the production of sound as opposed to the production of vorticity will be ignored. The aerodynamic sound problem then reduces to the study of mechanisms that convert kinetic energy of rotational motions into acoustic waves involving longitudinal vibrations of fluid particles. There are two principal source types in free vortical flows: a quadrupole, whose strength is determined by the unsteady Reynolds stress, and a dipole, which is important when mean mass density variations occur within the source region.

Lighthill's Acoustic Analogy

The theory of aerodynamic sound was developed by Lighthill [60], who reformulated the Navier–Stokes equation into an exact, inhomogeneous wave equation whose source terms are important only within the turbulent (vortical) region. Sound is expected to be such a very small component of the whole motion that, once generated, its back-reaction on the main flow is usually negligible. In a first approximation the motion in the source region may then be determined by neglecting the production and propagation of the sound.

Influence of Rigid Boundaries on the Generation of Aerodynamic Sound

The Ffowcs Williams–Hawkings equation (2.2.3) enables aerodynamic sound to be represented as the sum of the sound produced by the aerodynamic sources in unbounded flow together with contributions from monopole and dipole sources distributed on boundaries. For turbulent flow near a fixed rigid surface, the direct sound from the quadrupoles Tij is augmented by radiation from surface dipoles whose strength is the force per unit surface area exerted on the fluid. If the surface is in accelerated motion, there are additional dipoles and quadrupoles, and neighboring surfaces in relative motion also experience “potential flow” interactions that generate sound. At low Mach numbers, M, the acoustic efficiency of the surface dipoles exceeds the efficiency of the volume quadrupoles by a large factor ∼O(1/M2) (Sections 1.8 and 2.1). Thus, the presence of solid surfaces within low Mach number turbulence can lead to substantial increases in aerodynamic sound levels. Many of these interactions are amenable to precise analytical modeling and will occupy much of the discussion in this chapter.

Acoustically Compact Bodies [70]

Consider the production of sound by turbulence near a compact, stationary rigid body. Let the fluid have uniform mean density p0 and sound speed c0, and assume the Mach number is sufficiently small that convection of the sound by the flow may be neglected. This particular situation arises frequently in applications. In particular, M rarely exceeds about 0.01 in water, and sound generation by turbulence is usually negligible except where the flow interacts with a solid boundary [111].

Jets and shear layers are frequently responsible for the generation of intense acoustic tones. Instability of the mean flow over of a wall cavity excites “self-sustained” resonant cavity modes or periodic “hydrodynamic” oscillations, which are maintained by the steady extraction of energy from the flow. Whistles and musical instruments such as the flute and organ pipe are driven by unstable air jets, and shear layer instabilities are responsible for tonal resonances excited in wind tunnels, branched ducting systems, and in exposed openings on ships and aircraft and other high-speed vehicles. These mechanisms are examined in this chapter, starting with very high Reynolds number flows, where a shear layer can be approximated by a vortex sheet. We shall also discuss resonances where thermal processes play a fundamental role, such as in the Rijke tube and pulsed combustor. The problems to be investigated are generally too complicated to be treated analytically with full generality, but much insight can be gained from exact treatments of linearized models and by approximate nonlinear analyses based on simplified, yet plausible representations of the flow.

Linear Theory of Wall Aperture and Cavity Resonances

Stability of Flow Over a Circular Wall Aperture

The sound produced by nominally steady, high Reynolds number flow over an opening in a thin wall is the simplest possible system to treat analytically. Our approach is applicable to all linearly excited systems involving an unstable shear layer, and it is an extension of the method used in Section 5.3.6 to determine the conductivity of a circular aperture in a mean grazing flow.

Fluid motion in the immediate vicinity of a solid surface is usually controlled by viscous stresses that cause an adjustment in the velocity to comply with the no-slip condition and by thermal gradients that similarly bring the temperatures of the solid and fluid to equality at the surface. At high Reynolds numbers, these adjustments occur across boundary layers whose thicknesses are much smaller than the other governing length scales of the motion. We have seen how the forced production of vorticity in boundary layers during convection of a “gust” past the edge of an airfoil can be modeled by application of the Kutta condition (Section 3.3). In this chapter, similar problems are discussed involving the generation of vorticity by sound impinging on both smooth surfaces and surfaces with sharp edges in the presence of flow. The aerodynamic sound generated by this vorticity augments the sound diffracted in the usual way by the surface. However, the near field kinetic energy of the vorticity is frequently derived wholly from the incident sound, so that unless the subsequent vortex motion is unstably coupled to the mean flow (acquiring additional kinetic energy from the mean stream as it evolves) there will usually be an overall decrease in the acoustic energy: The sound will be damped. General problems of this kind, including the influence of surface vibrations, are the subject of this chapter. We start with the simplest case of sound impinging on a plane wall.

Damping of Sound at a Smooth Wall

Dissipation in the Absence of Flow

The thermo-viscous attenuation of sound is greatly increased in the neighborhood of a solid boundary where temperature and velocity gradients are large.

This book deals with that branch of fluid mechanics concerned with the production and absorption of sound occuring when unsteady flow interacts with solid bodies. Problems of this kind are commonly known under the heading of aerodynamic sound but often include more conventional areas of acoustics and structural vibration. Acoustics is here regarded as a branch of fluid mechanics, and an attempt has therefore been made in Chapter 1 to provide the necessary background material in this subject. Elementary concepts of classical acoustics and structural vibrations are also reviewed in this chapter. Constraints of space and time have required the omission or the curtailed discussion of several important subareas of the acoustics of fluid-structure interactions, including in particular many problems involving supersonic flow. The book should be of value in one or more of the following ways: (i) as a reference for analytical methods for modeling acoustic problems; (ii) as a repository of known results and methods in the theory of aerodynamic sound and vibration, which have tended to become scattered throughout many journal and review articles over the past forty or so years; and (iii) as a graduate level textbook. Chapter 1 and selected topics from Chapters 2 and 3 have been used for several years in teaching an advanced graduate level course on the theory of acoustics and aerodynamic sound.

Theoretical concepts are illustrated and sometimes extended by numerous examples, many of which include complete worked solutions. Every effort has been made to ensure the accuracy of formulae, both in the main text and in the examples. The author would welcome notification of errors detected by the reader and more general suggestions for improvements.

Each of Chapters III–V and VIII, which are devoted, respectively, to birods, beamshells, axishells, and general shells, begins with a derivation of the equations of motion by a descent from the equations of balance of translational and rotational momentum of a three-dimensional material continuum. These equations are written in integral-impulse form, which allows us to account for discontinuities in internal and external variables. Later, appropriate forms of the Second Law of Thermodynamics are derived by descent from the three-dimensional, integral form of the Clausius-Duhem inequality.

The present chapter summarizes those laws of three-dimensional continuum mechanics relevant to our approach to shell theory. For a detailed discussion of these laws, we refer the reader to books and articles by Truesdell & Toupin (1960), Truesdell & Noll (1965), Chadwick (1976), Truesdell (1977), Spencer (1980), and Gurtin (1981), among others.

The Integral Equations of Motion

A body B is represented by a set of points called particles that move through three-dimensional Euclidean space. This set of points at time t, St, is called the shape (or image) of B at t; S0 ≡ S is called the reference shape and is assumed to be a connected region.

In classical mechanics, a particular particle pk in a set of n may be identified (i.e., indexed) by a positive integer k = 1,2, …, n. In continuum mechanics a particle in B may be identified by its positionx in S relative to a given Cartesian frame Oxyz represented by the standard orthonormal triad of basis vectors {ex, ey, ez}.

A true shell has double curvature: If undeformed, its Gaussian curvature vanishes nowhere, except possibly along certain curves (such as the crown of a toroid) or at isolated points (such as the apex of a very flat dome). Shells of revolution, with the exception of conical and cylindrical shells, are the simplest, useful class of true shells. It is a corollary of Gauss’ Theorema egregium (Struik, 1961) that if a surface is bent, its Gaussian curvature will change only if the surface is simultaneously stretched. Thus, a shell will be relatively stiff if, by virtue of symmetry, loading, or boundary conditions, inextensional deformation is impossible. Such is the case with a shell of revolution that undergoes torsionless, axisymmetric deformation without rigid body motion. This feature of axishells is in contrast to the behavior of beamshells where near inextensional bending is typical.

In a beamshell there is one extensional strain, one shearing strain, and one bending strain; an arbitrary prescription of these three fields always yields some displacement field. However, in an axishell there are six strains: two extensional strains, one shearing strain, two bending strains, and one normal bending strain; these fields cannot be prescribed independently but must satisfy compatibility conditions if an associated displacement field is to exist. There is one differential compatibility condition if the rotation is introduced as a dependent variable, but three—two are differential and one is algebraic—if the strains are the basic kinematic unknowns.

In the preceding chapters on birods, beamshells, and axishells, we studied three important classes of structures that, under special loads and boundary/initial conditions, suffer strains, rotations, and displacements that are functions of one spatial variable only. The resulting simplifications of the general governing equations are enormous and in static problems, partial differential equations reduce to ordinary ones. Because strains, unlike rotations or displacements, are unaffected by rigid-body motions, it seems natural and possibly fruitful to seek other shells that, subject to suitable loads and boundary/initial conditions, suffer deformations in which the strains depend on one spatial variable only but in which the rotations and displacements depend (possibly) on a second one. The simplest and most important example is the bending of a pressurized tube of circular planform and arbitrary cross section—the subject of Sections A and B.

Specifically, in Section A we consider tubes subjected to a net end moment and, if pressurized, to an equilibrating net end force as well. We formulate the governing equations, discuss various approximations, and survey the literature on the collapse of such tubes. In Section B, we develop variational principles for curved tubes of either closed or open cross section, allowing for loads that act on the boundaries of the latter.

Beginning with Section C, we extend our analysis to unishells, i.e., to any shell with a set of loads and boundary/initial conditions (which must include the trivial ones that produce no distortion) such that there is a system of coordinates (σ, τ) on the reference surface in which the components of the deformed metric and curvature tensors are independent of τ.

We can only obtain analytic solutions to the equations describing nonlinear waves for a very limited set of geometries, constitutive equations, and boundary conditions. For example, the Riemann integrals yield a solution for a one-dimensional simple wave in an infinite volume of nonlinear-elastic material; however, obtaining analytic solutions for the interaction of two simple waves ranges from difficult to impossible depending upon the constitutive equations of the medium. Without the advent of the digital computer and sophisticated methods of numerical analysis, advances in the field of nonlinear wave motion and the development of constitutive theories for dynamic loading of materials would have stagnated over the past 40 years.

What Is a Shell?

A shell is a curved, thin-walled structure. A quantitative definition will be given in later chapters. Two important degenerate classes of shells are plates (shells which are flat when undeformed) and membranes (shells whose walls offer no resistance to bending). Shells may be made of a single inhomogeneous or anisotropic material or may be made of layers of different materials. The primary function of a shell may be to transfer loads from one of its edges to another, to support a surface load, to provide a covering, to contain a fluid, to please the eye or ear, or a combination of these. Shells occur in nature and as artifacts and include aortic valves, automobile hoods, balloons, beer cans, bellows, bells, bladders, bowls, contact lenses, crab carapaces, diaphragms, domes, ducts, egg coverings, footballs, funnels, inner tubes, light bulb casings, loudspeaker cones, manhole covers, parachutes, peanut hulls, Ping-Pong balls, panels, pipes, pressure vessels, silos, skulls, straws, tents, tires, trumpets, umbrellas, vaults, wine glasses, and woks. The aim of shell theory is to describe the static or dynamic behavior of such structures by equations that involve no more than one or two spatial variables.

Elastic Shells and Nonlinear Behavior

If a shell is prevented from moving as a rigid body, then it is elastic if, upon application and removal of a sufficiently small load, it tends to return to its initial shape.