This paper describes a reformulation of the Lighthill (1952) theory of aerodynamic sound. A revised approach to the subject is necessary in order to unify the various ad hoc procedures which have been developed for dealing with aerodynamic noise problems since the original appearance of Lighthill's work. First, Powell's (1961 a) concept of vortex sound is difficult to justify convincingly on the basis of Lighthill's acoustic analogy, although it is consistent with model problems which have been treated by the method of matched asymptotic expansions. Second, Candel (1972), Marble (1973) and Morfey (1973) have demonstrated the importance of entropy inhomogeneities, which generate sound when accelerated in a mean flow pressure gradient. This is arguably a more significant source of acoustic radiation in hot subsonic jets than pure jet noise. Third, the analysis of Ffowcs Williams & Howe (1975) of model problems involving the convection of an entropy ‘slug’ in an engine nozzle indicates that the whole question of excess jet noise may be intimately related to the convection of flow inhomogeneities through mean flow pressure gradients. Such problems are difficult to formulate precisely in terms of Lighthill's theory because of the presence of an extensive, non-acoustic, non-uniform mean flow. The convected-entropy source mechanism is actually absent from the alternative Phillips (1960) formulation of the aerodynamic sound problem.In this paper the form of the acoustic propagation operator is established for a non-uniform mean flow in the absence of vortical or entropy-gradient source terms. The natural thermodynamic variable for dealing with such problems is the stagnation enthalpy. This provides a basis for a new acoustic analogy, and it is deduced that the corresponding acoustic source terms are associated solely with regions of the flow where the vorticity vector and entropy-gradient vector are non-vanishing. The theory is illustrated by detailed applications to problems which, in the appropriate limit, justify Powell's theory of vortex sound, and to the important question of noise generation during the unsteady convection of flow inhomogeneities in ducts and past rigid bodies in free space. At low Mach numbers wave propagation is described by a convected wave equation, for which powerful analytical techniques, discussed in the appendix, are available and are exploited.Fluctuating heat sources are examined: a model problem is considered and provides a positive comparison with an alternative analysis undertaken elsewhere. The difficult question of the scattering of a plane sound wave by a cylindrical vortex filament is also discussed, the effect of dissipation at the vortex core being taken into account.Finally an approximate aerodynamic theory of the operation of musical instruments characterized by the flute is described. This involves an investigation of the properties of a vortex shedding mechanism which is coupled in a nonlinear manner to the acoustic oscillations within the instrument. The theory furnishes results which are consistent with the playing technique of the flautist and with simple acoustic measurements undertaken by the author.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075002777

Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute

https://www.win.tue.nl/~sjoerdr/papers/boek.pdf

An introduction to acoustics

The Stewartson-Warn-Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat’s eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale Sq, i.e. the initial absolute vorticity gradient dq,/dy times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate du,/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat’s-eye pattern, probably leading to chaotic motion there, with very large ‘eddy-viscosity ’ values. Those values correspond to critical-layer Reynolds numbers A-’ Q 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen. The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection) might be prolonged indefinitely is ruled out, in many cases of interest (even if the ‘eddy viscosity’ is large), by the existence of a rigorous, general upper bound on the magnitude of the time-integrated absorptivity a(t). The bound is uniformly valid for all time t. The absorptivity a(t) is defined aa the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, no matter how large t may become, I a(t) I cannot greatly exceed the rate of absorption predicted by linear theory multiplied by the timescale on which linear theory breaks down, say the time for the cat’s-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that I a(t) I cannot greatly exceed the initial absolute vorticity gradient dq,/dy times the cube of the

Do Rossby-wave critical layers absorb, reflect, or over-reflect?

Over the past twenty-five years , nonlinear acoustics has developed into a vigorous and distinctive branch of science. The subject covers a wide range of topics, including steady finite-amplitude waves in supersonic aerodynamics, weak-shock theory, unsteady finite-amplitude waves in gases, liquids, and solids, bubble dynamics and cavitation in liquids, and phonon interactions and the quantum acoustics of solids. Practical applications are on the increase: the steady wave systems of supersonic projectiles; the nonlinear parametric sonar array devised by Westervelt (1963) for the production of a highly directional low-frequency beam in water or air; the damaging effects of cavitation and bubble implosion on ship structures in water and on nuclear reactors cooled by liquid sodium; and the use of ultrasonics in biomedical and engineering non­ destructive testing. These are all well-known examples, though they hardly exhaust the possibilities . A series of eight (as of 1978) international symposia in the USA, Europe, and the USSR has greatly stimulated work in many aspects of nonlinear acoustics

Model equations of nonlinear acoustics

List of Figures List of Tables Notation Preface 1. Differential and difference equations 2. Characterization and classification 3. Fourier theory 4. Distributions and fundamental solutions 5. Approximation by finite differences 6. The Equations of continuum mechanics and electromagnetics 7. The art of modeling 8. The analysis of elliptic equations 9. Numerical methods for elliptic equations 10. Analysis of parabolic equations 11. Numerical methods for parabolic equations 12. Analysis of hyperbolic equations 13. Numerical methods for scalar hyperbolic equations 14. Numerical methods for hyperbolic systems 15. Perturbation methods 16. Modeling, analyzing, and simulating problems from practice Appendices. Useful definitions and properties Bibliography Index.

D.G. Crighton

Papers

Physical Acoustics and the Method of Matched Asymptotic Expansions