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Page 5 ï~~Physical modeling of aeroacoustic sources in flute-like instruments Marc-Pierre Verge verge~ircam. fr Laboratoire d'Acoustique Musicale, (CNRS, Ministere de la Culture, Universit6 de Paris VI), 4, Place Jussieu, case 161, 75252, Paris Cedex 05, France Abstract The drive of the acoustic oscillations in the resonator of flute-like instruments is provided by the motion of an air jet on each side of a sharp edge. In addition to this mechanism, sound is also produced by two other aeroacoustic sources. Separation of the acoustic flow at the edge of the labium results in the formation of vortices in the mouth of the instrument. This phenomenon is crucial since it determines the playing amplitude and the timbre of the instrument. Turbulence in the mouth produces a broad band noise which is very characteristic of the tone of flute-like instruments. This paper describes how simple hydrodynamic models of these complex phenomena can be used for sound synthesis of flute-like instruments by physical modeling. 1 Introduction The geometry of a typical flute-like instrument is shown in Figure 1. The functioning of this type of instrument is based on the oscillation of an air jet on each side of the labium. This motion of the jet provides the "drive" that enables the maintenance of the auto-oscillations of the system. This jet-drive mechanism and its modeling have been discussed in detail by Verge et al. [Verge et al., 1994]. Visualizations of the flow in the mouth of the instrument show, however, that in addition to this jet motion, two other sources of sound exist in the mouth of the instrument [Verge, 1995]. First, one can observe the periodic shedding of vortices at the edge of the labium. These vortices interact with the acoustic field in the mouth of the instrument and produce acoustic work which strongly affects the behavior of the instrument. Furthermore, in normal playing conditions, the jet velocity is generally high enough for the jet flow to be turbulent. This produces a broad-band noise which is very typical of the tone of this kind of instrument. flue channel flue exit labium A Pipe foot mouth Figure 1: Typical geometry of a flute-like instrument. Ap Figure 2: One-dimensional representation of a flutelike instrument. One therefore must take into account these sound sources in order to obtain quantitatively and perceptively pertinent simulations. These phenomena are very complex, however, and their detailed description is only possible by the use of direct simulation techniques in order to solve the full Navier-Stokes equations. These techniques involve, however, a tremendous computing power [Skordos, 1995] and are certainly not adapted for real-time sound synthesis. Furthermore, they only work for low Reynolds number flow. The objective of this paper is to describe how simplified representations of these phenomena can be included in a one-dimensional representation of the instrument, shown in Figure 2, for efficient simulation of their main acoustic effects. 2 Vortex shedding Sound production by vorticity can be described, in free-field conditions, by Powell's theory of vortex sound [Powell, 1964]. This theory has been generalized by Howe [Howe, 1975, Howe, 1984] for ICMC Proceedings 1996 Verge
Page 6 ï~~internal flow. Within the framework of Howe's vortex sound theory, the vorticity C = V x vY of a flow velocity field v, can be interpreted as an aeroacoustic force density fc f= -p(Cx (1) where w = V x,v is the vorticity, and p is the density, corresponding to the Coriolis force experienced by an observer moving with the fluid particles. In the presence of an acoustic field 7, this force can perform a work W, given by Wa -I>dy, (2) Figure 3: Flow separation of the acoustic flow from the resonator, resulting in the formation of a jet. by Ingard and Ising (1967), the effects of vortices are assimilated to a fluctuating pressure jump/Apa across the mouth of the instrument of amplitude where V is the volume of integration and y the position vector of a particle in the source region where C # 0. The Coriolis force can either extract energy from the acoustic flow and therefore represent a loss mechanism or transfer energy to the acoustic field and produce sound. The direction of this energy transfer is determined by the relative orientation of the force ff and the acoustic velocity vector 17. Visualizations of the flow around the labium show that vortex shedding at the labium occurs as the result of the separation of the acoustic flow at the edge of the labium [Fabre et al., 1996]. This acoustically induced vortex is only present during a short fraction of the fundamental which implies that it can only extract energy from the fundamental of the acoustic field. In order to supply the fundamental with energy, it would have to remain at least until the opposite phase of the acoustic velocity which is not the case. This affirmation has been verified quantitatively by Fabre who showed that this vortex shedding represents the major non-linear amplitude limiting mechanism in flutelike instruments, much more important than the radiation and visco-thermic losses usually considered [Fletcher and Rossing, 1991].. The life span of this vortex is, however, long enough to transfer energy to the higher frequencies, which correlates with the presence of high harmonics in measured spectra of the acoustic field in the pipe. Note that the shedding of vorticity also occurs as a result of the separation of the jet flow at the edge of the labium. This effect will be neglected, however, since there is, at the present time, no available model of this complex phenomenon. In order to model the effects of the acoustically induced vortex shedding at the labium, a simple quasi-stationary free jet model is proposed. It is assumed that separation of the acoustic flow Qp from the pipe results into the formation of a free jet in the mouth of the instrument as shown in Figure 3. Vortices are then formed as a result of the roll-up of the shear layers bounding the jet. In this quasi-stationary free jet model, proposed APa = -j2Po(av Sm sign(Q) (3) where po is the density, a, is the vena-contracta factor of the jet, which, depending on the geometry, can vary within the range 0.6 < a, < 1. This pressure drop opposes the acoustical flow in the mouth and corresponds to the dissipation of the kinetic energy of the free jet by turbulence. Energy losses Ea induced by this model scale with the third power of the acoustic flow T Qp E o m (4) where T is the period of the acoustic signal. This crude model should constitute an upper limit approximation of the effects of the vortex shedding since it implicitly assumes that the vortex performs work during the entire period of the fundamental. Furthermore, the assumption of quasi-stationarity should only be valid when the Strouhal number Sr = wW2H/Qp <K 1, that is when the amplitude of the acoustic displacement is much larger than the characteristic dimensions of the mouth. Measurements indicate that the Strouhal number is rather of the order of 1. More sophisticated models [Peters, 1993], however, appear to yield equivalent power dissipation in this limit. Simulations performed with the simple one-dimensional model described by Verge [Verge, 1995] and that use this simple quasistationary free-jet model show that it enables one to correctly predict the playing amplitude. The fact that power losses vary with the third power of the acoustic flow, rather than with the second, as with radiation and visco-thermic losses, appears to be crucial. One of the drawbacks of this model, however, is that it does not account for the effects of the shape of the labium which strongly affects the tone of the instrument. Nolle Verge ICMC Proceedings 1996
Page 7 ï~~P P2 20 -15 -0 -5 -0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 a Figure 4: Ratio 13. of the amplitude of the fundamental and the second harmonic of the internal acoustic field in the pipe for different values of a. In the simulations, the pipe had a length of 0.28m, the driving pressure was 265 Pa, the labium was adjusted 0.2 mm below the jet centerline and the system oscillated on the second acoustic mode of the pipe (1085 Hz). [Nolle, 1983] has shown that rounding the edge of the labium reduces the harmonic content of the sound produced by the instrument. This is corroborated by instrument makers who say that the sharpness of the edge of the labium is crucial for the quality of the instrument. This can be understood by the fact that rounding the edge of the labium reduces the singularity of the potential flow around the labium and therefore the amount of vorticity shed in the flow. This effect can qualitatively be taken into account by modulating the amplitude of the pressure drop across the mouth associated with the formation of the free jet by a constant a. Spectra obtained from simulation show that a gradual inclusion of the vortex model while decreasing the amplitude of the fundamental, increases the amplitude of the second harmonic. A typical example is shown in Figure 4 where the ratio of the amplitude of the first and second harmonic of the internal acoustic field is shown for different values of the constant a. In these simulations, however, this vortex model did not seem to strongly affect the generation of higher harmonics. Another effect of an increase of the amplitude of the pressure drop is to lower the position of the transition, with respect to the blowing pressure, between operation in the first and second mode of the pipe. This results from a stronger damping of the fundamental and a transfer of energy to the second harmonic. 3 Turbulence noise Turbulence in the flow in the mouth of the instrument produces a noise which is very characteristic of the tone of flute-like instruments. This noise does not contribute fundamentally to the autooscillations of the instrument but it is nevertheless very important to include a turbulence noise model in order to obtain perceptively realistic simulations. Noise production by turbulence can be described by using the famous Lighthill theory describing sound production by aeroacoustic sources. Lighthill showed that for Mach numbers M Â~ 1 (M = U/co where U is the jet velocity and co the speed of sound), noise production by turbulence is a very inefficient process in free-field. In these conditions, the sound source can be shown to be dominated by quadrupoles which are weak acoustic sources and its intensity to scale with the 8-th power of the Mach number. Sound production can however be enhanced by placing an object in the flow which transforms the quadrupoles into stronger dipoles. The intensity of the noise produced by turbulence can then be shown to scale with the 4-th power of the Mach number. In a flute-like instrument, a sharp edge is placed right in the jet flow; noise production by turbulence can therefore be expected to have a dipolar nature and therefore to have the character of a force. This assumption has been verified from spectra measured on an experimental flue organ pipe [Verge, 1995] where the amplitude of the noise source was shown to scale with the square of the jet velocity which is proportional with the force provided by the labium and acting on the jet flow. A realistic turbulence noise source model can therefore be obtained by placing a broad-band dipolar noise source, equivalent to a pressure jump across the mouth of the instrument (as in Figure 2), and modulating its amplitude with the square of the jet velocity. This simple representation used in the one-dimensional model of the instrument described by Verge [Verge and Hirschberg, 1995] appeared to correctly reproduce the filtering effect of the noise source by the pipe and the background noise level measured on a real instrument with and without auto-oscillations of the jet [Verge, 1995]. A perceptively significant effect can be obtained by modulating the amplitude of the noise source with the jet displacement at the labium. This is justified by flow visualizations which show that the jet oscillations result in the periodical injection of puffs of turbulence on each side of the labium and that therefore the noise source can be expected to be intermittent. Similar effects were obtained in speech synthesis by modeling the modulation of the jet flow by the closing of the vocal folds [Hermes, 1991] and are consistent with the observations of Chafe [Chafe, 1.995] who detected period-synchronous spectral changes in flute tones. This effect should however vanish at high blowing pressures (Reynolds numbers ICMC Proceedings 1996 Verge
Page 8 ï~~> 3000) when the amplitude of the jet motion, as shown by flow visualization, becomes smaller than the half width of the jet. Furthermore, a phase delay between this modulation and the jet oscillations is perceptible and could account for the fact that, for low jet velocities, the flow only becomes turbulent downstream of the labium. It is however difficult to quantitatively relate this effect to the jet motion. 4 Conclusion Complex hydrodynamic phenomena occurring in the mouth of flute-like instruments have important consequences on the playing amplitude and tone produced by the instrument. A precise description of these phenomena is not adapted for sound synthesis and therefore one has to rely on simplified representations. Simple and efficient models of acoustically induced vortex shedding at the edge of the labium and of the production of noise by turbulence have been presented. The free jet model used to represent vortex shedding at the labium enables one to correctly predict the playing amplitude and to qualitatively account for the effects of the rounding of the labium, at least for the first and second harmonics. The turbulence noise source can be represented by a pressure jump across the mouth of the instrument having an amplitude scaling with the square of the jet velocity. A perceptively significant effect is obtained by modulating the amplitude of the noise source with the jet position at the labium, indicating that a thorough study of the behavior of the noise source should include a description of the dynamic of the phenomenon. References [Chafe, 1995] Chafe, C. (1995). Adding vortex noise to wind instrument physical models. In International Computer Music Conference, pages 57-60, Banff, Canada. [Fabre et al., 1996] Fabre, B., Hirschberg, A., and Wijnands, A. (1996). Vortex shedding in steady oscillations of a flue organ pipe. to be published in Acta Acustica. [Fletcher and Rossing, 1991] Fletcher, N. and Rossing, T. (1991). The Physics of Musical Instruments. Springer-Verlag, New-York. [Hermes, 1991] Hermes, D. J. (1991). Synthesis of breathy vowels: Some research methods. Speech Communication, 10:497-502. [Howe, 1975] Howe, M. (1975). Contribution to the theory of aerodynamic sound, with applica tion to excess jet noise and the theory of the flute. J. Fluid Mech., 71:625-673. [Howe, 1984] Howe, M. (1984). On the absorption of sound by turbulence and other hydrodynamic flows. Institute of Mathematics and its Applications Journal of Applied Mathematics, 32:187-209. [Ingard and Ising, 1967] Ingard, U. and Ising, H. (1967). Acoustic nonlinearity of an orifice. J. Acoust. Soc. Am., 42(1):6-17. [Nolle, 1983] Nolle, A. (1983). Flue organ pipes: Adjustments affecting steady waveform. J. Acoust. Soc. Am., 73(5):1821-1832. [Peters, 1993] Peters, M. (1993). Aeroacoustic Sources in Internal Flows. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands. [Powell, 1964] Powell, A. (1964). Theory of vortex sound. J. Acoust. Soc. Am., 36:177-195. [Skordos, 1995] Skordos, P. (1995). Modeling Flue Pipes: Subsonic Flow Lattice Boltzmann, and Parallel Distibuted Computers. PhD thesis, Department of Electrical Engineering, Massachussetts Institute of Technology, USA. [Verge, 1995] Verge, M. (1995). Aeroacoustics of Confined jets, with Applications to the Physical Modeling of Recorder-Like Instruments. PhD thesis, Eindhoven University of Technology, The Netherlands. [Verge et al., 1994] Verge, M., Causs6, R., Fabre, B., Hirschberg, A., Wijnands, A., and van Steenbergen, A. (1994). Jet oscillations and jet drive in recorder-like instruments. Acta Acustica, 2:403-419. [Verge and Hirschberg, 1995] Verge, M. and Hirschberg, A. (1995). A physical model of recorder-like instruments. In International Computer Music Conference, pages 37-44, Banff, Canada. Verge ICMC Proceedings 1996