Aerodynamic Sound Emission as a Singular Perturbation Problem
TL;DR: In this article, the authors examined the utility of Lighthill's and Ribner's results in the case M ≪ 1, Λ ˜ 1, and concluded that sound emission from large bodies of turbulence is an open problem.
Abstract: Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ≪ 1, Λ ˜ 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment Tij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ∇ × ψ{ω} + ∇φ The problem is solved for M ≪ 1, Λ ˜ 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if Tij is approximated by ρ0υiυj, with v = ∇ × ψ. The situation M ≪ 1, Λ ≫ 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.
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TL;DR: In this article, a reformulation of the Lighthill (1952) theory of aerodynamic sound is described, and 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.
Abstract: 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.
775 citations
TL;DR: In this article, the mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation and the results show that the phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed.
Abstract: The mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation. Details of the numerical method are briefly outlined and results are validated against an experiment at the same flow conditions (Stromberg, McLaughlin & Troutt 1980). Lighthill's theory is used to define a nominal acoustic source in the jet, and a numerical solution of Lighthill's equation is compared to the simulation to verify the computational procedures. The acoustic source is Fourier transformed in the axial coordinate and time and then filtered in order to identify and separate components capable of radiating to the far field. This procedure indicates that the peak radiating component of the source is coincident with neither the peak of the full unfiltered source nor that of the turbulent kinetic energy. The phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed which calls into question the commonly made assumption that the noise sources convect at a single velocity. Space–time correlations demonstrate that the sources are not acoustically compact in the streamwise direction and that the portion of the source that radiates at angles greater than 45° is stationary. Filtering non-radiating wavenumber components of the source at single frequencies reveals that a simple modulated wave forms for the source, as might be predicted by linear stability analysis. At small angles from the jet axis the noise from these modes is highly directional, better described by an exponential than a standard Doppler factor.
632 citations
Cites background from "Aerodynamic Sound Emission as a Sin..."
...9 jet 279 detailed theoretical criticisms of the analogy approach in general have appeared over the years (Doak 1972; Crow 1970; Fedorchenko 2000)....
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...We should also note that several detailed theoretical criticisms of the analogy approach in general have appeared over the years (Doak 1972; Crow 1970; Fedorchenko 2000)....
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TL;DR: A hierarchy of computational approaches that range from semi-empirical schemes that estimate the noise sources using mean-flow and turbulence statistics, to high-fidelity unsteady flow simulations that resolve the sound generation process by direct application of the fundamental conservation principles is discussed in this paper.
520 citations
Cites background from "Aerodynamic Sound Emission as a Sin..."
...…of the so-called Lighthill stress-tensor Tij : Simplification of the exact source term to that representing the momentum flux in a constant density, incompressible flow, Tij r0uiuj ; is asymptotically justified for a low-Mach number, acoustically compact noise-producing flow, see Crow (1970)....
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...The slow decay of dilatation away from the source region has raised serious criticism of this theory in the past [70,1]....
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TL;DR: In this paper, the authors give a simple, unified, analytical description of a wide range of mechanisms associated with the generation of sound by unsteady fluid motion, including radiation from compact and noncompact multipole sources, Lighthill's theory of sound emission from free turbulence, effects of source convection, sound generation from flow interaction with solid surfaces and inhomogeneities of the medium, and singular perturbation aspects of the aerodynamic sound problem.
430 citations
TL;DR: A critical review of computational techniques for flow-noise prediction and the underlying theories, in which the turbulent noise source field is computed and/or modeled separately from the far-field calculation, is provided.
Abstract: This article provides a critical review of computational techniques for flow-noise prediction and the underlying theories. Hybrid approaches, in which the turbulent noise source field is computed and/or modeled separately from the far-field calculation, are afforded particular attention. Numerical methods and modern flow simulation techniques are discussed in terms of their suitability and accuracy for flow-noise calculations. Other topics highlighted include some important formulation and computational issues in the application of aeroacoustic theories, generalized acoustic analogies with better accounts of flow-sound interaction, and recent computational investigations of noise-control strategies. The review ends with an analysis of major challenges and key areas for improvement in order to advance the state of the art of computational aeroacoustics.
399 citations
Cites background or methods from "Aerodynamic Sound Emission as a Sin..."
...In the limit of low Mach number and a compact vorticity region, the validity of Lighthill’s analogy has been shown using matched asymptotic expansions (Crow 1970)....
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...…extended into the acoustic far field, permitted the first direct application of several acoustic analogies, showing that despite certain concerns voiced about aspects of their formal derivation (Crow 1970, Doak 1972), their evaluation (by convolution integral in this study) gave the correct noise....
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TL;DR: In this paper, a theory for estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence is initiated, based on the equations of motion of a gas.
Abstract: A theory is initiated, based on the equations of motion of a gas, for the purpose of estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence. The sound field is that which would be produced by a static distribution of acoustic quadrupoles whose instantaneous strength per unit volume is ρv i v j + p ij - a 2 0 ρ δ ij , where ρ is the density, v i the velocity vector, p ij the compressive stress tensor, and a 0 the velocity of sound outside the flow. This quadrupole strength density may be approximated in many cases as ρ 0 v i v j . The radiation field is deduced by means of retarded potential solutions. In it, the intensity depends crucially on the frequency as well as on the strength of the quadrupoles, and as a result increases in proportion to a high power, near the eighth, of a typical velocity U in the flow. Physically, the mechanism of conversion of energy from kinetic to acoustic is based on fluctuations in the flow of momentum across fixed surfaces, and it is explained in § 2 how this accounts both for the relative inefficiency of the process and for the increase of efficiency with U . It is shown in § 7 how the efficiency is also increased, particularly for the sound emitted forwards, in the case of fluctuations convected at a not negligible Mach number.
4,697 citations
TL;DR: In this paper, an extension to Lighthill's general theory of aerodynamic sound was made to incorporate the influence of solid boundaries upon the sound field, and it was shown that these effects are exactly equivalent to a distribution of dipoles, each representing the force with which unit area of solid boundary acts upon the fluid.
Abstract: An extension is made to Lighthill's general theory of aerodynamic sound, so as to incorporate the influence of solid boundaries upon the sound field. This influence is twofold, namely (i) reflexion and diffraction of the sound waves at the solid boundaries, and (ii) a resultant dipole field at the solid boundaries which are the limits of Lighthill's quadrupole distribution. It is shown that these effects are exactly equivalent to a distribution of dipoles, each representing the force with which unit area of solid boundary acts upon the fluid. A dimensional analysis shows that the intensity of the sound generated by the dipoles should at large distances x be of the general form I$\propto $ $\rho \_{0}$ U$\_{0}^{6}$a$\_{0}^{-3}$ L$^{2}$x$^{-2}$, where U$\_{0}$ is a typical velocity of the flow, L is a typical length of the body, a$\_{0}$ is the velocity of sound in fluid at rest and $\rho \_{0}$ is the density of the fluid at rest. Accordingly, these dipoles should be more efficient generators of sound than the quadrupoles of Lighthill's theory if the Mach number is small enough. It is shown that the fundamental frequency of the dipole sound is one half of the frequency of the quadrupole sound.
1,760 citations
TL;DR: In this article, the acoustic properties of the system are studied in the special case of decaying isotropic turbulence, and it is shown that the intensity of sound at large distances from the turbulence is the same as that due to a volume distribution of simple acoustic sources occupying the turbulent region.
Abstract: A finite region, with fixed boundaries, of an infinite expanse of compressible fluid is in turbulent motion. This motion generates noise and radiates it into the surrounding fluid. The acoustic properties of the system are studied in the special case in which the turbulent region consists of decaying isotropic turbulence. It is assumed that the Reynolds number of the turbulence is large, and that the Mach number is small. The noise appears to be generated mainly by those eddies of the turbulence whose contribution to the rate of dissipation of kinetic energy by viscosity is negligible. It is shown that the intensity of sound at large distances from the turbulence is the same as that due to a volume distribution of simple acoustic sources occupying the turbulent region. In this analogy, the whole fluid is to be regarded as a stationary and uniform acoustic medium. The local value of the acoustic power output P per mass of turbulent fluid is given approximately by the formula P = ─3/2 α d u 2 ¯ /d t ( u 2 ¯ / c 2 ) 5/2 , where α is a numerical constant, u 2 ¯ is the mean-square velocity fluctuation, t is the time, and c is the velocity of sound in the fluid. The constant α is expressed in terms of the well-known velocity correlation function f(r) by assuming the joint probability distribution of the turbulent velocities and their first two time-derivatives at two points in space to be Gaussian. The numerical value α ~ 38 is then obtained by substituting the form of f(r) corresponding to Heisenberg’s theoretical spectrum of isotropic turbulence. It is found that the effects of decay make only a small contribution to the value of α, and that the order of magnitude of α is not changed when widely differing forms of the function f(r) are used.
329 citations
TL;DR: The theory of sound generated aerodynamically, that is, of sound radiation fields which are by-products of airflows, has been extended and improved by Curie and Fowcs Williams as mentioned in this paper.
Abstract: The author’s original theory of sound generated aerodynamically, that is, of sound radiation fields which are by-products of airflows, has been extended and improved by Curie and Ffowcs Williams. It is explained in this lecture fully but simply, and used as a framework for short analyses of our experimental knowledge on pulse-jet noise, hydrodynamic sound generation, aeolian tones, propeller noise, and boundary-layer noise, as well as for a somewhat extensive discussion of the noise of jets, both stationary and in flight. Improved knowledge of space-time correlations in turbulent flow is used to throw new light on the noise radiated by turbulent boundary layers, as well as by jets at the higher Mach numbers. Supersonic bangs and the scattering of both sound and shock waves by turbulence are briefly touched upon. The lecture ends with a discussion of the methods used for the reduction of jet aircraft noise, in the light of our knowledge of its physical basis.
183 citations
TL;DR: In this article, the scattering of a sound wave in a medium undergoing shear flow confined to a finite region is investigated under the assumption that the total velocity field is everywhere small compared to the velocity of sound.
Abstract: The scattering of a sound wave in a medium undergoing shear flow confined to a finite region is investigated under the assumption that the total velocity field is everywhere small compared to the velocity of sound. Formulas are obtained for the angular distribution and frequency distribution of the scattered wave in terms of the four‐dimensional Fourier transform of the shear velocity field. The cross sections for the scattering of a plane wave of frequency ω by a shear flow of given scale and spatial structure go typically as ω4M2, where M is a characteristic Mach number of the flow. The coupling between the shear and longitudinal velocity fields has a tensor character such that the scattering vanishes at 180° and at 90°. The spectrum of the scattered sound wave is very sharp in the forward direction and becomes broader at larger scattering angles. Explicit expressions for the cross sections are obtained for the case of scattering from a region of isotropic turbulence. When the frequencies of importance ...
146 citations