MODELLING WAVE AND FLUID PROPAGATION IN WIND INSTRUMENTS - METHODS AND APPLICATIONS ¨ Wilfried K AUSEL, Helmut K UHNELT Department of Musical Acoustics, University of Music Singerstr. 26/a, A-1010, Vienna, Austria Phone: (43) 1-71155-4311, email:
[email protected] Phone: (43) 1-71155-4312, email:
[email protected]
ABSTRACT Current approaches to model wave propagation and fluid-acoustics in one-, two-, and threedimensional acoustical systems are covered and present ways to model sound production mechanisms in instruments are outlined. Emphasis has been put on the recently developed concept of the Lattice Boltzmann method in the context of flow-acoustics. It allows the study of the interaction between the jet and the acoustic field in flute-type instruments. Another branch of research is a time-domain simulation of trumpet sound generation including wave propagation on the lip surface.
1. INTRODUCTION When simulating wind instruments, a variety of acoustical and fluid mechanical phenomena have to be considered. Depending on the kind of instrument, apart from linear wave propagation, nonlinear effects like wave-steepening, flow-induced sound generation or sonic flow generation have to be included in the model. Sound production mechanisms in all musical instruments are based on highly non-linear elements as reeds, lips, jets, etc. In the most general case the Navier-Stokes equations for compressible, viscous, unsteady fluid have to be solved in order to simulate acoustic waves as well as flow effects like vortices directly. Especially tree-dimensional structures like vortices require three-dimensional models to develop naturally. Any computational grid has to be fine enough to resolve all scales of motion. This is a challenging task, which in many cases surpasses the today’s computational possibilities. Further more, low Mach number flow conditions, as observed in wind instruments, are causing numerical difficulties for most available flow solvers.
Therefore the problem often has to be simplified, reducing spatial dimensions or using underlying symmetries. This way simulations are restricted to idealized shapes, or sometimes results do not reflect the exact natural behavior. The most prominent simplification is the wave equation. It is obtained by neglecting mean flow, heat conduction, viscosity and gravity and postulating only small fluctuations of p and ρ around their atmospheric values. Often just plane wave solutions are considered. The three dimensional axial symmetrical case can be handled by modal decomposition. The wave equation does not cover the cases where significant air flow, high sound pressures or other non-linearities are present. A systematical classification of current modelling concepts goes beyond the scope of this paper. Therefore only a limited number of recent and promising modelling concepts are summarized and the related publications reviewed. Applications are presented and results are discussed. Original work is presented in the fields of distributed lip reed modelling. A time domain simulation of a real playing situation is outlined. The system simulation framework has been proposed by Kausel in [1], recent results have been published in [2]. Another branch of original work presented here, regards a three-dimensional Lattice-Boltzmann method applicable to flow-acoustics as proposed by K¨uhnelt [3]. Recent results of a flue pipe simulation including wave and fluid propagation in a viscous compressible medium are presented.
2. METHODS Impedance Analysis There are many ways and numerous publications on solving the wave-equation in time- or frequency-domain. Of special interest here is a recently proposed method of calculating the input impedance of wind instruments. Combining the efficiency of frequency domain wave guide modelling and the accuracy of a finite difference method in the time domain was elaborated by Noreland [4]. The rapidly flaring bell of a brass instrument together with a part of its environment was simulated in the time domain. A coordinate transformation was applied in order to obtain a coordinate grid better reflecting the bell’s geometry. A pulse response of the bell’s input impedance was computed and transformed into the frequency domain. It was used as terminating impedance for the one dimensional wave guide model of the rest of the instrument. Very accurate results have been obtained because the plane wave assumption was only made in slowly flaring and narrow parts of the instrument. Even the radiation impedance was not approximated by the usual infinite baffle model but contained the effect of the real environmental condition. A hybrid approach such as the one described above is especially useful when instruments are to be optimized. Usually the bell region is not allowed to be modified by the optimizer, so it does not need to be recalculated at each optimization step.
Transmission Line Matrix (TLM) method Wave propagation can be described by Huygens’ principle, which states that a wavefront can be decomposed into a number of point sources giving rise to spherical wavelets. The envelope of these wavelets forms a new wavefront which again gives rise to a new generation of spherical wavelets. A discrete equivalent to this principle is the Transmission Line Matrix (TLM) method were pressure pulses travel on a regular grid of acoustical tubes. The TLMmethod is valid for wavelengths large compared to the grid spacing, a condition, which is needed to minimize the discrete behavior of the model. The flexibility in the management of complex boundary conditions inherent to real problems is the crucial advantage of the TLM method, which can easily be extended in three dimensions. VirtualWaveTank [5], a small demonstration program for the TLM-method, was used for the following examples of wave propagation in realistic, musical-instrument-like geometries.
(a) Sound propagation in a horn
(b) Sound propagation in a rather deformed horn
Fig. 1: Screenshots of the VirtualWaveTank program [5] As an application Pelorson et al. [6] studied the wave propagation in the vocal tract, a problem with rather complicated boundary conditions, by a three-dimensional TLM model. Lattice Boltzmann Method The Lattice Boltzmann method is a quite recent approach for simulating fluid flow, which has proven as a valid and efficient tool in a variety of complex flow problems. In the field of aeroacoustics some fundamental studies have been undertaken showing its validity concerning linear [7] and non-linear wave propagation [8] and acoustical streaming [9]. There are only a few studies which dealt with applied problems [10], [11]. So the use of the Lattice Boltzmann Method as a tool for computational aeroacoustics can be seen as in its early stage. The Lattice Boltzmann method (LBM) is a specific discretization of the Boltzmann equation. It approximates the Navier-Stokes equations and thus incorporates fluid dynamics and acoustics intrinsically. LBM is numerically efficient on single and parallel computing environments. A major advantage of LB models is the possibility to implement realistic and irregular boundary conditions.
3. APPLICATIONS Trumpet Sound Synthesis, 2-D Lip Model An elaborate study of the operation of the lip reed in brass wind instruments has been published by Adachi in [12] and [13]. The first article describes a so called transversal model, where the lip valve is modelled by a spring-mass system activated only by the Bernoulli pressure between the lips. The pressure drop between mouth cavity and lip orifice has been derived from momentum and energy conservation laws assuming laminar flow. The pressure drop between the lip orifice and the mouthpiece layer, where a plane wave has already been established, has been derived from the momentum conservation law only. It was assumed that a jet is formed in this region and energy is dissipated by the jet. The second article extends the lip model by a second degree of freedom. While the stretching action of the lip is still controlled by the pressure in the lip orifice, a swinging motion of the lip is activated by the pressure difference between mouth and mouthpiece. In both cases results of a system simulation involving the pulse response of a real trumpet are presented. Trumpet sound was generated and claimed to be quite realistic. Some observations observed by trumpet players have been correctly reproduced. Simulated wave forms are in good agreement with measured quantities. Nevertheless, the fact that players can easily lip notes up and down from the center of their corresponding air column resonance is still an open modelling issue. Especially the idea of a ‘regime of oscillation’ [14] meaning that higher harmonics do contribute in establishing a self sustained oscillation, is very important. It deals with the fact that players can play tones even when there is no natural resonance of the air column present. A realistic model should explain the mechanisms behind this phenomenon. Wave steepening in Trombones Wave steepening and the resulting formation of shock waves at high playing pressures has already been suspected to be the cause of the so called ‘brassy’ sound of horns, especially noticeable in the trombone. Thompson and Strong presented a model [15] which was used to demonstrate wave steepening in a trombone sound production simulation. The interesting fact is that this kind of non-linearity has not been implemented in a time-domain model but in the frequency domain. Based on a one-dimensional lossy wave guide approach, the instrument was split into about 150 cylindrical slices propagating waves according to the wave equation. Losses were incorporated by allowing a complex valued wave number. The non-linear wave steepening correction is applied to outgoing pressure wave spectra whenever a slice has been passed. The input to the sequence of slices composing the trombone has been a measured pressure spectrum. The resulting pressure spectrum and wave form as shown in fig. 2 at an observers point somewhere in front of the bell has been compared with the data, recorded when the stimulus spectrum was taken. Inclusion of wave steepening in the model greatly reduces the error between predicted and measured spectra.
essure wave forms for the tone D4-1. ~a! Measured wave ~b! Measured wave on-axis 2.85 m in front of the bell. ~c! puted from the nonlinear model. ~d! Radiated wave comar model. DC offset has been neglected. The mouthpiece in Fig. 2. The radiated spectra are shown in Fig. 3.
Lip Surface
FIG. 6. Acoustic pressure wave forms for the tone D4-3. ~a! Measured wave in the 2: mouthpiece. Measured wave of on-axis 2.85 m in front of the bell. ~c! Fig. Wave~b!Steepening Trombone Sound [15] Radiated wave computed from the nonlinear model. ~d! Radiated wave com-
puted from the linear model. DC offset and frequencies above 20 kHz have been neglected. The mouthpiece spectrum is shown in Fig. 2. The radiated spectra are shown in Fig. 4. Playing Activity during Trumpet
IONS
It was noticed by observations that lips do not oscillate like swinging or sliding doors. Stroboscopic studies revealed complex multi-dimensional wave patterns travelling on the surface are indeed encouraging. For small-amplitude ofmodels the lips which linear and the nonlinear predict wellhave been recently studied by Yoshikawa and Muto in great detail [16].
ata. At large amplitudes, the nonlinear model Kausel proposed accurate in its predictions, while the lineara two dimensional distributed lip model [2] which is able to exhibit surface waves travelling underestimates the amplitudes of the high-between the teeth and the mouth piece rim. Surface waves are interacting onics. However, at shock formation with forcesdistances originating from the pressures in the mouth, in the mouthpiece and in the lip ss than the length of the instrument, the non- depends on the tension of the lip. orifice. Phase speed ends to overestimate the amplitudes of the harmonics. The simulation was carried out in the time domain using the electrical circuit simulator may be improved upon by using a greater . The equivalent circuit was presented in [1]. Figure 3(a) shows pressure signals S PICE nders in the approximation, which would efand lip displacements at several points along the lip surface cross-section between teeth and bute the losses at the walls of the cylinders mouthpiece rim while playing a glissando over the first four natural tones. y. Including more harmonics in the computap to reduce numerical error in the wave steepFigure 3(b) shows enlarged curves of pressures, velocities, lip displacements and crosson. Using mouthpiece data with a lower noise sectional areas when the pedal tone is sounding. Figure 4 illustrates the surface waves on the rmit evaluation of the model at higher frelipbeshowing animation of the upper and lower lip cross-section. In the real animation the ore accurate model could formulatedanusing uous cones instead of cylinders. pressure is color coded and the velocity is represented by the arrow. direct comparisons between the present rework of Msallam et al. ~1997!, cannot be vely, this research buildsComputational upon their work byClarinet Analysis ternate method for including wave steepening FIG. 7. Comparison of the average errors between measured and computed harmonic SPLs from both models. Each line ranges from the mean error in onal model of trombone production: An sound extensive computational analysis of the clarinet was recently done by Facchinetti et al. SPLs minus the standard deviation of the error to the mean plus the standard wall losses are handled in a more realistic [17]. Coupled fluid and solid three-dimensional finitesignal element models for the reed and the deviation. Only frequencies below the mouthpiece cutoff frequencies teepening is applied systematically to the en~see Table I! were included. The different dynamics at each pitch are numair model load in first 10bered cm inofincreasing the pipe and a lumped elements model for the main part of the and the accuracy of the is the analyzed order, from soft to loud.
pipe were used.
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M. W. Thompson and W. J. Strong: Modeling of wave steepening
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First the eigenmodes of an isolated reed rigidly clamped on the section corresponding to the ligature and having a stress-free boundary elsewhere were computed. As a second step the dynamics of the reed influenced by air loading was studied using a coupled fluid-solid model. The system then was composed of the reed, the mouthpiece and the barrel. It was found that the coupling to air changes the normal modes of the isolated reed. Therefore, the modes of the whole system have to be taken as source for the acoustic field in the mouthpiece.
(a) Glissando over first 4 Resonances.
(b) Pedal Tone Curves. Fig. 3: Time Domain Simulation of Lip Oscillator An experimental modal analysis on reeds by means of holographic interferometry showed the validity of the numerical model of the reed coupled to air within 10 − 20% of the measured resonance frequencies. In the cylindrical part of the mouthpiece acoustic waves can be considered already plane within a very good approximation. Therefore the finite element model of the first 10 cm of the pipe was connected with a lumped-element model of the remaining part of the instrument in order to simulate the modal behavior. Flow-acoustic Simulations Acoustically Deflected Jet. Adachi [18] used a two-dimensional finite element method to simulate the acoustical deflection of a jet in an external sound field perpendicular to the jet. It was shown that the method used is able to track the sound-flow interaction. Edge-Tone. Bamberger et al. [19] simulated edge tones using a two-dimensional simulation of the incompressible, isothermal Navier-Stokes equations using an adaptive finite element method and compared the results with measurements. They examined edge tones in the fundamental fluid dynamic mode varying the height of the nozzle, the edge distance and the
Fig. 4: Moving Lip Cross-Section jet velocity. A good proportionality of the frequency with the flow velocity was found, both in experiment and simulation. Experiment and simulation differed in the oscillation conditions. For a wide nozzle, oscillations started at lower velocities than experimentally observed. The authors consider the missing noise in the simulation to be the cause for this behavior. Glottis. Zhao et al. [20] simulated the flow and the acoustic field in an idealized vocal tract with time-varying geometry. The compressible Navier-Stokes equations in a twodimensional axisymetric moving coordinate system were solved, spatially discretized by a sixth-order compact finite-difference method and temporally by a fourth-order Runge-Kutta method. The sound sources within the flow were identified by means of an acoustic analogy. Three types could be assigned: An unsteady volume velocity monopole source associated with the motion of the vocal fold, a dipole source due to the oscillating force, which arises from the interaction between vortical structures shed from the glottis and the velocity field, and a quadrupole source due to net unsteady flow inside the vocal tract. The assumption of two-dimensional axisymmetric geometry prevents the generation of turbulence, which is expected to dramatically change the flow pattern inside and downstream of the glottis, as well as the subsequent acoustic radiation. Therefore the computed contribution from the monopole source, which is believed to be negligible in speech production, was overestimated. The quadrupole source was underestimated about one order of magnitude compared to large eddy simulations of the sound radiation from subsonic turbulent jets done by the same authors. Side Branch Resonators. In engineering, coaxial side branches attached to a main duct acting as quarter-wave resonator are commonly used as a silencer in ducts. At certain flow conditions large wave amplitudes are excited in the side branches and high noise levels in the main duct. The mechanism of vortex induced sound excitation in low Mach number flow (M a < 0.1) is to a certain degree comparable to the sound generation in flutes and flue pipes. Radavich et al. [21] computed such a system, solving the two-dimensional, unsteady compressible Navier-Stokes equations with the k − turbulence closure model using an implicit, non-iterative, operator splitting algorithm. The region of oscillations measured in experiments was successfully reproduced. Although the simulations were not fully comparable
with experiments due to differences in the boundary conditions, the sound-pressure level differed about 6 dB at maximum. The comparison of computed and experimental visualizations of the vortical structures in the flow in the vicinity of the side branches suggests that the flow-acoustic coupling is captured well in the computations. A large net acoustic source caused by the interaction between the vortex, the velocity field and the acoustical velocity was localized over the junction between the main duct and the side branch. In her thesis, Dequand [22], [23] covered a larger area of flow-acoustic interactions, from the aeroacoustic response of sharp bends in ducts, self-sustained oscillations in a cross-junction, to the excitation of a Helmholtz resonator by a low Mach number flow, all studied experimentally and computationally. The numerical method used here is based on the Euler equations for two-dimensional inviscid compressible flows. To simulate the generation of vorticity in the wall boundary layer the Kutta-condition was imposed. The Euler equations were spatial discretisized using a second-order accurate cell-centered finite-volume technique. The time-integration was based on a second-order accurate four-stage Runge-Kutta method. The simulations of self-sustained oscillations in a cross-junction could reproduce the region of oscillation. Comparison with measurements, others than those mentioned above, showed that the oscillation frequency was predicted within 2%, but the calculated pulsation amplitude was overestimated by 40%. Since previous calculations by Hofmans [24] of a similar pipe system based on a method including visco-thermal losses did also overestimate the same measurements by 30%, the author concluded that only a minor part of this deviation should have been cased by the absence of visco-thermal losses in her model, the major part could be due to experimental problems such as wall vibrations. Flow Driven Helmholtz Resonator. Dequand [22] also studied self-sustained oscillation in a Helmholtz-like resonator by simulations and experiments. The influence of the shape of the neck of the resonator on the flow behavior and the generation of sound by vortex shedding was extensively examined in a series of different configurations. The computations made by the above mentioned Euler solver predicted the resonance frequency within 5%. A comparison to experiments showed, that the computations performed best in a configuration having a neck with a 90◦ angle and both the upstream and the inner edge chamfered. Here the predicted pulsation amplitude was about 20% too low. In the configurations with a 60◦ neck and having a sharp and a chamfered upstream edge respectively the predicted amplitudes were 40% too low, and in the configuration with a sharp upstream edge and a chamfered inner edge the predicted pressure amplitude was at least a factor 4 too low. Flute Simulation with the LBM The sound generation in a small stopped flue pipe with recorder-like proportions was simulated by K¨uhnelt [3] using a three dimensional Lattice Boltzmann Method. Here some results of recent simulations are presented. The dimensions of the stopped pipe are 60 mm × 7 mm × 7 mm for the resonator, 15 mm × 4.5 mm × 1.2 mm for the flue. The ratio of the distance from flue exit to the labium to the height of the flue exit is about 2.8 and the angle of the labium is approx. 14◦ . The whole pipe is embedded into a volume of 75 mm × 7 mm × 19 mm, which was simulated only partially. Due to the limited computational resources the lattice spacing had to be chosen rather large, δx = 0.22 mm. The speed of sound was cs = 340 m/s used, resulting in a time step of δt = 3.74 · 10−7 s. The time for rising the flow in the flue was 5000 δt = 1.87 ms, which is rather short compared to the rising time found in
Fig. 5: Simulated stopped organ pipe
Fig. 6: Cross section of the simulated pipe. Dimensions in lattice units. flue instruments. A slightly higher viscosity than air was assumed to enhance the stability of the calculation. Formation of the jet at startup. When the pressure rises at the inlet and a flow through the flue develops, a jet is formed at the flue exit due to the effect of viscosity, as shown in fig. 7. The flow separates and vortices above and below the flue exit are clearly visible. This shows that with the LBM effects of viscosity emerge naturally, with no needs of special boundary conditions. Starting transients and steady-state oscillations. Fig. 8 shows the density inside the resonator near the stopped end. During startup, the first mode is excited first but not supported, so it decays – visible in fig. 9. The second pipe mode then grows until it reaches a saturation. The frequency of the fundamental mode is 1197 Hz. The frequency of the second mode is 3235 Hz and is about 10% deeper than three times the fundamental frequency. Fig. 10(a) shows the spectrum of the decaying first mode and the growing second mode, whereas fig. 10(b) shows the spectrum during steady state oscillations. The velocity in the mouth of the flute during a oscillation period is shown in fig. 11. Complicated flow and vortex shedding at the labium occur during the operation of the flute. Fig. 12 shows the complete simulated domain.
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Fig. 7: Formation of the jet during start-up. The value of the velocity ranges from blue (u = 0) to red (max. velocity).
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Fig. 11: Velocity of the jet in the mouth of the pipe during one oscillation period
Fig. 12: Velocity field in the stopped flue pipe and the surrounding during steady state oscillations
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