The Vox humana resonator

The design of theVox humana(“human voice”) resonator differs a lot from theCrumhornresonator regarding both its geometry and its purpose. TheVox humanaresonator consists of three sections:

(1) the shallot continues in a thin straight neck, which has a length of²/⁵of that of the complete resonator, (2) a flaring section, nearly as long as the neck, where the diameter increases greatly, and (3) a tapering section, which is open at the top. The resonator is cut to tune and can be tuned

8.5. VALIDATION 111

Parameter Value [mm] Parameter Value [mm]

Wall width at pipe foot 1.0 Outer Ø at open end 19.0 Wall width at open end 1.4 Lateral length of neck 56.5 Outer Ø at foot end 11.9 Lateral length of mid part 64.7 Outer Ø at middle 52.0 Lateral length of top part 35.5

Table 8.5.Geometry of theVox humanaresonator Mode Meas Z_R^(L&S) Err Z_R^(sim) Err

1^st 735 788 120.5 748 25.7

2^nd 2 319 2 339 14.9 2 317 −1.5 3^rd 3 192 3 239 25.3 3 212 10.8 4^th 4 605 4 695 33.5 4 650 16.8 5^th 5 425 5 566 44.4 5 502 24.4 6^th 6 469 6 695 59.4 6 619 39.7

Table 8.6.Measured and calculated eigenfrequencies of theVox humanaresonator.Z_R^(L&S):ZRcalculated by formulas of Levine & Schwinger,Z_R^(sim): ZRobtained from FE / IE models. Eigenfrequencies displayed in Hzunits, errors are in cents.

only in one direction by cutting from the top. While theCrumhorn resonator (with shallot) is tuned near the fundamental frequency, theVox humanais usually tuned to the third harmonic of the fundamental (196 Hzfor this pipe) in order to amplify the fifth above the octave in the steady state pipe sound. This resonator is only examined without its shallot here, for the sake of brevity.

The geometrical parameters of the resonator are listed in Table 8.5.

Table 8.6 displays radiation

impedance models the measurement and calculation results for the eigenfrequencies of theVox

humanaresonator. The calculation was performed with two different settings. First, the theory of Levine & Schwinger [93] was used for the calculation of the radiation impedanceZ_R, see eq.

(3.65). The corresponding results are shown in the columnZ_R^(L&S). Second, radiation impedance results from the finite element simulations presented in Section 8.3 were substituted into the one-dimensional model. The respective results are shown in the columnZ_R^(sim). As can be seen, applying the FEM results to determine the radiation impedance has a remarkable effect on the accuracy of the results, especially in the case of the first eigenfrequency. While the analytical formula leads to an error of120cents in this case, the FEM impedance model significantly re-duces this error to26cents. Apparently, the impact of the radiation impedance is smaller on the frequencies of higher modes and the errors given by the two approaches have the same order of magnitude here with the FEM errors being slightly smaller.

In order FEM / PML

simualation to get a picture of the sound pressure field in the resonator at the frequencies of

natural resonance, a finite element model of the resonator was assembled. The simulation ar-rangement imitated that of the transfer function measurements, as shown in Figure 8.8(a). Free field wave propagation conditions were ensured by applying the perfectly matched layer (PML) technique, as introduced in Section 4.4. The excitation was a point source located in the axis of the resonator, below the neck. The simulation was performed using the toolboxNiHu, see Appendix D.

The pressure fields inside the resonator are shown in Figure 8.8(b) for the first four modes.

The planar and spherical wavefronts are clearly observable in the cylindrical and conical parts of the resonator, respectively. Except for the first mode, the strongest pressure oscillations develop in the straight neck part of the resonator. The resulting eigenfrequencies show a good match with the measured ones with a slight underprediction of their frequencies, except for the first mode.

Resonator Domain extension

Source Perfectly

matched layers

(a) Simulation arrangement (b) First four acoustic modes of the resonator

Figure 8.8.Simulation of theVox humanaresonator

Based

evaluation on the results presented in this section, it can be assessed, that the two proposed supple-ments for the one-dimensional resonator model—i.e. calculation of the radiation impedance by means of the FEM and the low frequency shallot model—can both be successfully applied for the prediction of the natural resonance frequencies of axisymmetric resonators. Both the radiation impedance and the shallot models could be extended by taking other effects (such as viscother-mal losses) into account; however, these improvements are out of the scope of this chapter.

8.6 Concluding remarks

In this chapter a one-dimensional modeling approach for composite resonators having axial sym-metry was introduced. Such resonators are used in a great number of reed pipe stops in the organ, such as theCrumhorn, theTrumpet, theOboe, theClarinet, and theVox humanaamong others. The general transmission line model was supplemented by two novel components: (1) the finite el-ement simulation of the radiation impedance, and (2) a low frequency shallot model. By means of comparison to measurement results it was shown, that the one-dimensional model with the proposed extensions is capable of the more accurate prediction of the natural resonance frequen-cies of axisymmetric resonators and shallots of reed organ pipes. The results

application presented in this

section have been incorporated into the software toolReedResonatorSim, introduced later in Appendix C.

Chapter 9

Air jet and edge tone simulation in an organ pipe foot model

Different computational fluid dynamic (CFD) models for the simulations of the air jet emerging in the windway of a labial organ pipe are reported and evaluated in this chapter. The numer-ical examination is carried out using 2D and 3D geometries with a laminar and a Large Eddy Simulation (LES) model. Two different configurations of the air jet are investigated here. First, the free jet is examined, without the interacion with the upper lip. The quality of the simulation results for this model is assessed by means of comparison to reproducible hot wire anemometer measurements carried out on a high precision pipe foot model. In the second arrangement, the upper lip is included in the model and the jet shows oscillations around this edge. The modal frequencies of the resulting edge tone are compared to sound recordings of the edge tone on the same pipe foot model. It is shown, that the extension of the simulation model into three dimen-sions has a remarkable impact on the quality of the simulation results in both configurations.

This chapter is a revised and extended version of the conference paper [C13].

9.1 Introduction

Examinationof the air jet and edge tone generation in flue organ pipes by means of computa- motivation tional fluid dynamics aims at gaining knowledge about the effect of the geometry of the pipe

mouth and the parameters adjusted in various voicing steps on the sound generation of the pipe, especially in the attack phase. While these effects can also be investigated by means of experiments [12, 119], the numerical framework provides more flexibility and is cheaper than measuerements on prototypes requiring special equipment.

In particular, the aim of the study presented here is to extend two-dimensional air jet and edge tone simulations published recently by Vaik & Paál [140] into three dimensions. The reason for this extension is twofold. On the one hand, the effect of the 3D extension on the simulation results—which was claimed to be negligible by the aforementioned authors—can be examined.

On the other hand, 3D simulations are necessary to model geometries that are inherently three-dimensional. One specific example is the simulation ofnicking, a voicing step often applied in labial organ pipes, which can be an interesting target of future examinations.

Beside the edge

tone phe-nomenon playing an important role in the sound generation of air reed wind instruments, the

edge tone phenomenon is also present in other industrial applications, such as the wind noise generated by pantographs of high-speed electric trains, for example. It was found earlier (see e.g. [44, 73]) that the edge tone has tonal components with clearly distinguishable frequencies, also called modes or stages [121]. In case of labial organ pipes the resulting characteristics of the

113

Foot hole

Figure 9.1.Außerlechner’s pipe foot model [18, 19]. Left: geometry and adjustable parameters. Right: Air jet and edge tone generation in the foot model and the corresponding aeroacoustic source types.

pipe sound are attained by the coupling of the excitation mechanism and the acoustic resonator, as it was discussed in Chapter 2. In the process of voicing the organ builders tune the excitation (i.e. the air jet and the edge tone) to arrive at the desired speech of the pipe. This is traditionally done by modifying the flow geometry in fine steps during the voicing phase, see Section 2.1.4.

The edge tone is also an important component of the pipe sound in the attack transient, it can often be heard in the initial phase of the sound. In the steady state pipe sound, the edge tone can also appear in some special cases, resulting in a “rough” sound. The latter phenomenon was studied recently by Trommeret al.[139].

The edge tone recent

results

and its effect on the sound generation of flue organ pipes have already been investigated by a number of researchers, as it was discussed in Chapter 2. In a recent paper by Vaik & Paál [140] flow simulation results in an organ pipe foot model using different turbulence models have been published. Good agreement with the measurements of Außerlechneret al.[19]

has been found. However, the investigations were limited to two-dimensional models and thus three-dimensional effects inside the flow field were not considered.

The objectives

objectives of this contribution are to extend the flow simulation model into three dimen-sions and to assess the quality of the results obtained using 2D and 3D models by means of comparison with previous CFD results and measurements. Two main flow configurations are examined. In the first case a free jet without upper lip is discussed and the resulting flow profiles are evaluated. The second case focuses on the edge tone with the analysis of its tonal components.