Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-18T16:15:02.015Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 March 2006

M. S. Howe
M. S. Howe
Affiliation:
Engineering Department, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 71 , Issue 4 , 28 October 1975 , pp. 625 - 673
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Atvars, J., Schubert, L. K. & Ribner, H. S. 1965 J. Acoust. Soc. Am. 37, 168.
Backus, J. 1970 The Acoustical Foundations of Music. London: John Murray.
Blokhintsev, D. I. 1946 Acoustics of a nonhomogeneous moving medium. N.A.C.A. Tech. Memo. no. 1399.Google Scholar
Brown, C. E. & Michael, W. M. 1955 On slender delta wings with leading edge separation. N.A.C.A. Tech. Note, no. 3430.Google Scholar
Brown, G. B. 1937 Proc. Phys. Soc. Lond. 49, 493.
Candel, S. M. 1972 Analytical studies of some acoustic problems of jet engines. Ph.D. thesis, Caltech.
Cannell, P. & Ffowcs williams, J. E. 1973 J. Fluid Mech. 58, 65.
Chernov, L. A. 1960 Wave Propagation in a Random Medium McGraw Hill.
Coltman, J. W. 1968 Phys. Today, 21 (11), 25.
Crighton, D. G. 1972 J. Fluid Mech. 51, 357.
Crighton, D. G. & Leppington, F. G. 1970 J. Fluid Mech. 43, 721.
Crighton, D. G. & Lesser, M. B. 1974 Physical acoustics and the method of matched asymptotic expansions. In Physical Acoustics(ed. R. N. Thurston & W. P. Mason), vol. 11, p. 75. Academic.
Crow, S. C. 1967 Phys. Fluids, 10, 1587.
Crow, S. C. 1970 Studies in Appl. Math. 49, 21.
Curle, N. 1953 Proc. Roy. Soc. A 216, 412.
Doak, P. E. 1973 J. Sound Vib. 26, 91.
Ffowcs Williams, J. E. 1963 Phil. Trans. A 255, 469.
Ffowcs Williams, J. E. 1969 Ann. Rev. Fluid. Mech. 1, 197.
Ffowcs Williams, J. E. & Hall, L. H. 1970 J. Fluid Mech. 40, 657.
Ffowcs Williams, J. E. & Hawkings, D. L. 1969 Phil. Trans. A 264, 321.
Ffowcs Williams, J. E. & Howe, M. S. 1975 J. Fluid Mech. 70, 605.
Ffowcs Williams, J. E. & Lovely, J. 1975 J. Fluid Mech. 71, 689.
Fletcher, N. H. 1975 J. Acoust. Soc. Am. 57, 233.
Howe, M. S. 1973 J. Sound Vib. 27, 455.
Howe, M. S. 1975 J. Fluid Mech. 67, 597.
Kraichnan, R. H. 1953 J. Acoust. Soc. Am. 25, 1096.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics Pergamon.
Laufer, J., Ffowcs williams, J. E. & Childress, S. 1964 Mechanism of noise generation in a turbulent boundary layer. AGARDograph, no. 90.Google Scholar
Lauvstad, V. R. 1968 J. Sound Vib. 7, 90.
Lauvstad, V. R. 1974 J. Acoust. Soc. Am. 56, 1407.
Liepmann, H. W. & Roshko, A. 1957 Elements of Gasdynamics. Wiley.
Lighthill, M. J. 1952 Proc. Roy. Soc. A 211, 564.
Lighthill, M. J. 1953 Proc. Camb. Phil. Soc. 49, 531.
Lighthill, M. J. 1954 Proc. Roy. Soc. A 222, 1.
Lilley, G. M. 1958 On noise from air jets. Aero. Res. Counc. 20, 376-N40-FM 2724.Google Scholar
Lilley, G. M. 1973 On noise from jets. AGARD CP 131, paper 13.Google Scholar
Marble, F. E. 1973 Acoustic disturbance from gas nonuniformities convected through a nozzle. Dept. Transportation Symp., Stanford.Google Scholar
Morfey, C. L. 1966 I.S.V.R. Rep., University of Southampton.
Morfey, C. L. 1973 J. Sound Vib. 31, 391.
Müller, E. A. & Matschat, K. R. 1959 The scattering of sound by a single vortex and by turbulence. Tech. Rep. Max-Planck-Inst. für Strömungsforschung, Göttingen.Google Scholar
Müller, E. A. & Obermeier, F. 1967 The spinning vortices as a source of sound. AGARD CP 22, paper 22.Google Scholar
Naumann, A. & Hermanns, E. 1973 On the interaction between a shock wave and a vortex field. AGARD CP131, paper 23.Google Scholar
Obermeier, F. 1967 Acustica, 18, 237.
O'Shea, J.1971 Scattering of sound by aerodynamic flows. Ph.D. thesis, University of London.
Pao, S. P. 1971 J. Sound Vib. 19, 401.
Pao, S. P. 1973 J. Fluid Mech. 59, 451.
Phillips, O. M. 1960 J. Fluid Mech. 9, 1.
Powell, A. 1961a Vortex sound. Engng Dept., University of California, Los Angeles, Rep. no. 61–70.Google Scholar
Powell, A. 1961b J. Acoust. Soc. Am. 33, 395.
Powell, A. 1964 J. Acoust. Soc. Am. 36, 177.
Rahman, S. 1971 Acustica, 24, 50.
Rayleigh, Lord 1880 Proc. Land. Math. Soc. 11, 57.
Rayleigh, Lord 1945 The Theory of Sound. Dover.
Richardson, E. G. 1931 Proc. Phys. Soc. Lond. 43, 394.
Smith, J. H. 1959 A theory of the separated flow from the curved leading edge of a slender wing. Aero. Res. Counc. R. & M. no. 3116.Google Scholar
Stüber, B. 1970 Acustica, 23, 82.
Tatarski, V. I. 1961 Wave Propagation in a Turbulent Medium. McGraw-Hill.