reed mouthpieces

Numerical simulations of fluid-structure interactions in single-reedmouthpieces

Da Silva, A. R., Scavone, G. P., & Van Walstijn, M. (2007). Numerical simulations of fluid-structure interactionsin single-reed mouthpieces. The Journal of the Acoustical Society of America, 122(3), 1798-1809. DOI:10.1121/1.2759166

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Numerical simulations of fluid-structure interactions insingle-reed mouthpieces

Andrey Ricardo da Silvaa� and Gary P. ScavoneComputational Acoustic Modeling Laboratory, Schulich School of Music, McGill University, Montreal,Quebec, Canada

Maarten van WalstijnSonic Arts Research Centre, School of Electronics, Electrical Engineering, and Computer Science, Queen’sUniversity Belfast, Belfast, United Kingdom

�Received 23 April 2007; revised 18 June 2007; accepted 21 June 2007�

Most single-reed woodwind instrument models rely on a quasistationary approximation to describethe relationship between the volume flow and the pressure difference across the reed channel.Semiempirical models based on the quasistationary approximation are very useful in explaining thefundamental characteristics of this family of instruments such as self-sustained oscillations andthreshold of blowing pressure. However, they fail at explaining more complex phenomenaassociated with the fluid-structure interaction during dynamic flow regimes, such as the transient andsteady-state behavior of the system as a function of the mouthpiece geometry. Previous studies havediscussed the accuracy of the quasistationary approximation but the amount of literature on thesubject is sparse, mainly due to the difficulties involved in the measurement of dynamic flows inchannels with an oscillating reed. In this paper, a numerical technique based on the latticeBoltzmann method and a finite difference scheme is proposed in order to investigate thecharacteristics of fully coupled fluid-structure interaction in single-reed mouthpieces with differentchannel configurations. Results obtained for a stationary simulation with a static reed agree verywell with those predicted by the literature based on the quasistationary approximation. However,simulations carried out for a dynamic regime with an oscillating reed show that the phenomenonassociated with flow detachment and reattachment diverges considerably from the theoreticalassumptions. Furthermore, in the case of long reed channels, the results obtained for the venacontracta factor are in significant disagreement with those predicted by theory. For short channels,the assumption of constant vena contracta was found to be valid for only 40% of the duty cycle.© 2007 Acoustical Society of America. �DOI: 10.1121/1.2759166�

PACS number�s�: 43.75.Pq, 43.75.Ef �NHF� Pages: 1798–1809

I. INTRODUCTION

The study of the acoustical properties of single-reed in-struments has followed a paradigm first proposed byHelmholtz,1 with these systems divided into linear and non-linear components representing the instrument’s bore andmouthpiece-reed, respectively.

Previous research on the resonator component has pro-vided an extensive list of experimental and theoretical stud-ies since the pioneering work of Bouasse.2 Much light hasbeen shed on the behavior of this system and, consequently,many satisfactory models have been proposed.

Conversely, the amount of available literature on themouthpiece-reed component is considerably smaller and themajority of models rely on the quasistationary approximationto describe the flow behavior. That is, the flow in a mouth-piece with an oscillating reed is assumed to be equal, at anyinstant, to the flow in a mouthpiece with a static reed havingthe same configuration.3 Moreover, the flow is considered tobe frictionless and incompressible. Consequently, the depen-dence of the volume flow U on the pressure difference across

the reed �p and on the reed opening h is normally describedby the Bernoulli obstruction theory based on the stationaryBernoulli equation, given by

UB = hw�2��p��

sgn��p� , �1�

where w is the channel’s width and � is the density of thefluid.

This approach was first presented by Backus,4 whosesemiempirical model was limited to low blowing pressureregimes. Years later, Worman5 presented a more complexmodel addressing, in further detail, phenomena such asthreshold of pressure and Bernoulli forces acting on the reed.Wilson and Beavers6 coupled the previous model to an ide-alized cylindrical resonator. More recent models involvingthe same approach were developed by Fletcher,7,8 Saneyoshiet al.9 Kergomard,10 and Olivier.11

The quasistationary approximation has also been used toderive a steady viscous flow representation by Hirschberget al.12 Their semiempirical model was based on the resultsobtained from the simulation of flow in a two-dimensional�Borda� tube based on the theory of potential flow. Theynoticed that, for Reynolds numbers Re�10, two patterns ofa�Electronic mail: [email protected]

1798 J. Acoust. Soc. Am. 122 �3�, September 2007 © 2007 Acoustical Society of America0001-4966/2007/122�3�/1798/12/$23.00

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flow may occur simultaneously, depending on the ratio l /h,where l is the length of the channel and h is its height. Theflow is fully detached along the channel, for short channels�l /h�1�, whereas for long channels �l /h�3� the flow isreattached at a roughly fixed point, lr, measured from thechannel’s entrance. They also observed that, in the case ofshort channels, the vena contracta factor �=Tj /h was ap-proximately constant with a value �0.6, where Tj is thethickness of the jet formed at the detached portion of theflow.

van Zon et al.13 provided an experimental validation ofHirschberg’s model using an idealized prototype of themouthpiece with a static reed and assuming the flow to betwo dimensional. They also derived a more sophisticatedflow model in which the transition between fully separated toattached flow is represented by a boundary layer solution.Other stationary measurements using realistic mouthpieceshave found the same flow behavior, such as those conductedby Valkering14 and by Dalmont et al.,15 in the case of clari-net, and by Maurin,16 in the case of saxophones.

However, previous attempts to characterize flow in dy-namic regimes,13,17 i.e., flow in a mouthpiece with a movingreed, have suggested that the stationary behavior observed byvan Zon et al. is unrealistic. This is particularly evident inthe case of the steadiness associated with the detachment/reattachment phenomenon, which is strongly affected bysubtle modifications of the reed channel geometry as the reedmoves. The unsteadiness of the flow modulates the aerody-namic forces acting on the reed and plays an important rolein the reed’s behavior. In fact, this unsteadiness is respon-sible for the self-sustained oscillations in systems whoseacoustic coupling between the resonator and the exciter isweak or even absent. This is the case in the harmonium,18 inthe accordion,19 and in the human phonatory system.20–24

Moreover, the unsteadiness of the flow can explain whysmall modifications in a mouthpiece geometry can corre-spond to enormous changes in the transient behavior and thesteady-state sound of single-reed instruments.25,26

Unfortunately, the accurate quantification and visualiza-tion of a dynamic flow controlled by a moving boundary �inthis case, the reed� is a rather complicated task. For thisreason, previous attempts to do so are limited to qualitativeoutcomes.13,17 Similar difficulties are found when tacklingthe problem with unsteady numerical flow simulations usingtraditional computational fluid dynamic �CFD� techniquesbased on the continuum theory.27,28

The objectives of this paper are the presentation of anumerical model of fully coupled fluid-structure interactionin a single-reed mouthpiece in order to address the majoraspects of dynamic flow and its dependency on the reedchannel geometry and to verify the validity of the quasista-tionary theory in dynamic regimes. To accomplish that, weimplement a two-dimensional dynamic model of a single-reed mouthpiece based on a hybrid numerical approach in-volving the lattice Boltzmann method �LBM�, to representthe fluid and acoustic domains, and on a finite differencescheme to resolve the distributed model of the reed withvarying cross section, as proposed by Avanzini and vanWalstijn.29 The main advantage of this approach consists in

its simplicity in providing solutions of second-order accuracyto represent the fluid-structure interaction involving a mov-ing boundary.30 This simplicity is contrasted with the com-plexity and high computational demand associated with tra-ditional CFD techniques. Furthermore, the LBM can solvethe different scales associated with the flow and acousticfields in a single calculation, thus allowing the direct repre-sentation of the acoustic-flow interaction.31

The influence of the player’s lip and the coupling of theproposed system with the instrument’s bore and player’s vo-cal tract is not considered in this paper. Furthermore, thecontribution of aeroacoustic sources on the instrument’ssound content due to undisturbed flow will be left to futurework.

This paper is organized as follows: Section II describesthe model by presenting the lattice Boltzmann techniqueused, as well as the finite difference scheme to resolve thedistributed reed model and the coupling between both tech-niques. Section III compares the results obtained from a sta-tionary simulation with those provided by the quasistationarytheory. Section IV evaluates the characteristics of a dynamicflow in a reed-mouthpiece system without acoustic couplingfor three different reed channel geometries and compares theresults with those expected by the quasistationary theory. Fi-nally, the conclusions and remarks for future investigationsare presented in Sec. V.

II. THE REED-MOUTHPIECE MODEL

The following describes the implementation of the two-dimensional model of the mouthpiece-reed system. Themouthpiece is represented by the LBM, which includes solidstatic boundaries associated with the mouthpiece walls �face,rails, and cavity walls� and the fluid domain, described interms of acoustic and fluid fields. The moving boundary as-sociated with the reed is represented by a distributed modelof a clamped-free bar with varying cross section and re-solved with an implicit finite difference scheme, as proposedby Avanzini and van Walstijn.29

A. The lattice Boltzmann method

The LBM is classified as a particle or nonequilibriumtechnique. It simulates the space-temporal evolution of fluid-acoustic systems based on a time-space discretization of theBoltzmann equation, known as the lattice Boltzman equation�LBE� �see Eq. �2��.

Xe and Luo32 have demonstrated that the Navier-Stokesand continuity equations can be fully recovered from theLBE for low Mach numbers, namely �Ma�0.2�, by applyingthe Chapman-Enskog expansion, thus providing a physicalvalidity for the method. Detailed descriptions of the LBMare provided by Succi33 and Gladrow.34

The LBE controls two essential operations: advectionand collision of fluid particles. These particles are describedin terms of velocity distribution functions and can propagatein a discrete set of directions within the lattice.

In this paper we use an isothermal two-dimensionalmodel known as D2Q9, after Qian et al.35 In this sense, thelattice grid is represented by squared two-dimensional lattice

J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al.: Dynamic flow in single-reed mouthpieces 1799


cells containing nine sites each �eight propagation directionsand one rest site�, as depicted in Fig. 1. Each cell connects toeight neighbor cells by the unity vectors ci, where i=1,2 , . . . ,8, indicates the propagation direction associatedwith each site. The null vector c0 is associated with a non-propagating site and plays an important role in improving theaccuracy of the model by removing the unphysical velocitydependency of pressure.36

The two main operations, namely, advection and colli-sion, are controlled by the LBE,

f i�x + ci�t,t + �t� − f i�x,t� = −1

��f i − f i

M� , �2�

where f i is the distribution function associated with thepropagation direction i at the site x� and time t�. � is therelaxation time or collision period, which acts to control thekinematic viscosity of the fluid, and f i

M is the equilibriumdistribution function for direction i, which depends on thelocal fluid velocity u��x� , t�� and local fluid density��x� , t��. Here and in the following, variables indicated witha prime are adimensional. The general expressions of theequilibrium function f i

M associated with the D2Q9 model are

f iM = ��i1 + 3ci · u� +

9

2�ci · u��2 −

3

2u�2 for i = 1,2, . . . ,8

��4

9−

2

3u�2 for i = 0 � �3�

with �1=�2=�3=�4=1/9 and �5=�6=�7=�8=1/36.The left-hand side of Eq. �2� represents the advection

operation and determines the diffusion of the distributionfunctions f i over the lattice grid. The right-hand term deter-mines the rate at which f i change due to intermolecular col-lisions between particles. This term is defined by a simplifiedcollision function, known as BGK, after Bhatnagar, Grass,and Krook,37 which is controlled by a single relaxation time� for all the advection directions i. This process, known asrelaxation, forces f i toward equilibrium and restitutes the vis-cosity of the fluid, recovering its nonlinear form whereby thecontinuity and Navier-Stokes equations are satisfied.

The local macroscopic variables �� and u� are obtainedin terms of moments of the local distribution functions f i by

��x,t� = �i

f i�x,t�, ��x,t�u��x,t� = �i

f i�x,t�ci. �4�

Other macroscopic parameters such as lattice pressurep�, lattice viscosity ��, and lattice speed of sound c0� areobtained by expanding the LBE into the Navier-Stokes equa-tion and are expressed by

p� =��

3, �� =

2� − 1

6, c0� =

1�3

. �5�

The adimensional lattice variables ��, p�, u�, x�, t�, and�� can be easily related to their respective physical counter-parts �, p, u, x, t, and by the following relations: �=��,p=�c0

2, u=u�c0 /c0�, x=x��x, t=�x�c0� /c0�t�, and �= �c0 /c0��x��, where c0 is the physical speed of sound.

B. The mouthpiece model

The mouthpiece model was implemented in a lattice gridcontaining 1002502 cells. The physical dimensions of thesystem are depicted in Fig. 2, as well as the dimensions ofthe grid. The lattice pitch was �x=410−5 m and the timestep �t=6.79210−8 s. As a matter of convenience, we haveopted to use an undisturbed fluid density �0�=�0=1.0 kg/m3.The relaxation time � was chosen to be 0.505, which impliesa lattice viscosity ��=1.6810−3 and a physical kinematicviscosity �=3.9510−5 m2/s, using c0=340 m/s as the ref-erence speed of sound.

Although the choice of values for �0 and � differ con-siderably from those of air in normal playing conditions, thedynamic similarity with the real system is obtained by forc-ing Re�1200 for a maximum Ma=0.1. These parametersalso allow the two essential criteria of the lattice BoltzmannBGK model to be met: the maximum compressibility �Ma

FIG. 2. Lattice grid representing the two-dimensional model of themouthpiece-reed system.

FIG. 1. The squared grid for the D2Q9 lattice Boltzmann model.

1800 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al.: Dynamic flow in single-reed mouthpieces


�0.1� before numerical instabilities34 and a minimum gridresolution �5.6 lattices per wavelength� to avoid spurious dis-persion and dissipation effects associated with the numericalbulk viscosity, as described by Crouse et al.38

C. The reed model

The reed is represented as a clamped-free bar with non-uniform thickness b�x�, constant width w, and driven by aforce F�x , t�. The partial differential equation describing thevertical displacement y�x , t� as a function of F�x , t� is givenby

�rA�x��2y

�t2 �x,t� +�2

�x2YI�x� 1 + ��

�t� �2y

�x2 �x,t� = F�x,t� ,

�6�

where x� �0,L� is the horizontal position, A�x�=wb�x� is thecross section, �r is the material density, Y is the Young’smodulus, I�x� is the moment of area about the longitudinalaxis, and � is the viscoelastic damping coefficient. Table Ishows the values used in the simulation, obtained experimen-tally by Avanzini and van Walstijn.29 Equation �6� considersonly reed motion associated with flexural waves in the ver-tical direction and, thus, torsional and longitudinal modes areneglected. This is similar to the approach used by Chaigneand Doutaut39 to simulate xylophone bars. In our model, aterm associated with the energy dissipation of the reed due tothe work exerted on the surrounding fluid is neglected in Eq.�6�. However, this is taken into account implicitly by thefully coupled fluid-structure interaction scheme presented inSec. II E.

Equation �6� is solved by performing a space-temporaldiscretization based on an implicit finite difference schemedescribed by Chaigne and Doutaut.39 This results in a matri-cial difference equation in which the spatial coordinate isvectorialized,

y�n + 1� = A0 · y�n� + A1 · y�n − 1� + AF · F�n� , �7�

where y�n+1�, y�n�, and y�n−1� represent the displacementvector at successive time instants and A0, A1, and AF arecoefficient matrices. F�n� is a vector representing the longi-tudinally distributed force on the reed. The interaction be-tween the reed and the mouthpiece lay is considered to beinelastic. This is achieved by nullifying the kinetic energy ofthose reed sections that collide with the mouthpiece side rail,which presents an upper boundary to the reed. The inelasticassumption for the reed/lay interaction is discussed and jus-tified in Ref. 29.

D. Initial and boundary conditions

The algorithm assumes a no-slip condition of flow at thewalls by implementing a bounce-back scheme proposed byBouzidi et al.40 The bounce-back scheme works to invert thedirection of propagation of a distribution function f i just be-fore it reaches a solid boundary. This procedure creates a nullfluid velocity at the walls and provides second-order accu-racy to represent viscous boundary layer phenomena.

The problem of a moving boundary, the reed, within thelattice is tackled by using an interpolation scheme proposedby Lallemand and Luo.30 This technique preserves second-order accuracy in representing the no-slip condition and thetransfer of momentum from the boundary to the flow. Oneconstraint of this approach is the velocity limit defined byMa�0.5, Ma=ub /c0, ub being the velocity of the boundary.However, such a limitation does not represent a problem inour simulation because it corresponds to values of velocitymuch higher than those found for reeds at normal playingconditions.

The mean flow is initiated by using a fairly well knowntechnique in CFD called absorbing boundary conditions�ABC�. This technique has been adapted to LBM by Kamet al.41 and consists in using a buffer between the fluid regionand the open boundary to create an asymptotic transitiontoward a target flow defined in terms of target distributionfunctions f i

T. This is done by adding an extra term to Eq. �2�to represent the transition region,

f i�x + ci�t,t + �t� − f i�x,t� = −1

��f i − f i

M� − ��fM − f iT� ,

�8�

where �=�m� /D�2 is the absorption coefficient, �m is aconstant, normally equal to 0.3, is the distance measuredfrom the beginning of the buffer zone, and D is the width ofthe buffer. The f i

T is constant and can be obtained in the samemanner as f i

M using Eq. �3�, where the local velocity u� andlocal density � are replaced by the desired target flow uT andtarget density �T, respectively. Another desired feature of thistechnique is the anechoic characteristic that avoids any re-flection or generation of spurious waves at the open bound-aries.

E. Numerical procedures

Seven different operations are executed at every timestep in order to couple the lattice Boltzmann model with thefinite difference scheme. The sequence of operations is de-picted as a flowchart in Fig. 3. Before the simulation begins,the initial conditions associated with the fluid and reed vari-ables are set and the definition of solid boundaries within thelattice are defined. The reed variables such as displacementy�x ,0�, velocity y�x ,0�, and force F�x ,0� are set to zero, aswell as the variables associated with the fluid domain, suchas the local fluid velocities u. The initial fluid variables areused to define the initial distribution functions based on Eq.�3�, so that, in the first time step f i= f i

M. The flow is started byprescribing a target pressure difference at the ABC layer,defined as �pT= ��in

T −�outT �c0

2, where the indexes “in” and

TABLE I. Characteristics of a plastic reed �Plasticover� obtained by Avan-zini and van Walstijn �Ref. 29�.

Length Lreed=3410−3 mWidth w=1010−3 mDensity �reed=500 kg/m3

Young’s modulus Y =5.6109 N/m2

Viscoelastic const. �=6.010−7 sFluid damping coef. �air=100 s−1



“out” indicate inlet and outlet, respectively. The values of�pT depend on the type of simulation being conducted andare described in the next sections of this paper.

With respect to the flowchart in Fig. 3, the followingoperations take place after the initial conditions are set: �a�calculate the relaxation functions f i

M’s using Eq. �3�; �b�propagate f i to all directions, ignoring the presence of pre-defined solid boundaries and perform their relaxation basedon Eq. �2�; �c� find the lattice positions of f i that have crossedsolid boundaries during the propagation step in �b�; �d� re-place f i found by the previous operator with new valuesbased on two different interpolation strategies: f i at crossedstatic boundaries are replaced by values calculated using the

simple bounce-back scheme, proposed by Bouzidi et al.40

Otherwise, f i are replaced by values calculated using themoving boundary scheme proposed by Lallemand and Luo.30

In this case, the calculation of the new f i requires the actualvalues of y�x , t� in order to take into account the transfer ofmomentum from the reed to the flow; �e� determine newvalues of u� and �� using Eq. �5�; �f� evaluate the new dis-tributed force F�x , t� on the reed model based on local latticepressures across the reed boundary; and �g� calculate thereed’s new position y�x , t� and velocity y�x , t�.

III. QUASISTATIC MODEL

The following compares the results obtained for a sta-tionary simulation �static reed� using the model described inSec. II with the results provided by the quasistationary ap-proximation for two-dimensional flows in single-reed mouth-pieces with constant channel cross section, as proposed byvan Zon et al.13 The model was based on the results from astationary measurement involving an idealized two-dimensional prototype of the mouthpiece-reed system. Simi-lar measurements using real mouthpieces have found identi-cal results and were conducted by Valkering14 andDalmont,15 in the case of a clarinet, and by Maurin,16 for thesaxophone.

A. Overview of the analytical model

Hirschberg et al.12 derived a semiempirical analyticalmodel for the viscous steady flow in a two-dimensionalsingle-reed mouthpiece channel with constant height. Themodel was based on the numerical study of a two-dimensional channel �Borda tube� using a potential flowscheme.

They observed two types of flow for Reynolds numbersRe�10�Re=U /w�, depending on the ratio between thechannel height h and its length l. In both cases, a jet isformed at the sharp edges of the channel’s entrance. Forsmall ratios �l /h�1�, the jet does not reattach along thechannel walls, whereas, for high ratios �l /h�3�, the jet re-attaches at a fixed point lr�h measured from the channel’sentrance.

Thus, in the case of short channels, the flow is describedby the Bernoulli equation �Eg. �1�� scaled with a constantvena contracta factor �, whereas, in the case of long chan-nels, the detached segment is represented by the Bernoulliflow and the reattached part is represented by the Poisseuilleflow.

These results were confirmed experimentally by van Zonet al.13 who derived a more accurate steady flow model inwhich the transition between fully separated to Poisseuilleflow is described by a boundary layer flow. In this case, thevelocity profile u�x ,y� within the boundary layer of thickness �x� is assumed to increase linearly with the distance y fromthe wall.

Similar to the model proposed by Hirschberg et al., theflow in short channels �l /h�1� is given by

FIG. 3. Flowchart of the integrated algorithm.



U = �UB, �9�

where UB is the Bernoulli flow given by Eq. �1� and �, 0.5��0.61 is the constant vena contracta factor whose valuedepends on the external geometry of the mouthpiece.

For long channels �l /h�4� and �l�� c, where c is thecritical boundary layer thickness, van Zon et al. describe thevolume flow by

U =�w

ch�lc − lr� . �10�

The term �lc− lr� is the length of the transition betweenfully separated flow to Poisseuille flow, given by

lc − lr

l − lr=

12c�1 − *�2

24c − 11 −�1 −

h4�24c − 1��p

72��2�l − lr�2�1 − *�2 ,

�11�

where * is the generalization of the critical boundary layerthickness c for a channel of arbitrary height h, expressed by

* = c

h=

4

9 1 −� 5

32� = 0.2688 �12�

and

c =1

64 * + 9 ln�1 − *� +

5 *

1 − * = 0.01594. �13�

B. Stationary results

The stationary simulations were conducted for differentcases involving geometries with the same characteristics asthat shown in Fig. 2, but with different channel profiles asdepicted in Fig. 4. For each geometry, different steady-statevalues of U are achieved by prescribing different target pres-sure values �pT from 0 to 9 kPa. The simulations used thesame characteristics described in Sec. II in terms of initialand boundary conditions, lattice discretization, and fluidproperties. However, in this case the reed is maintained fixed�or static� throughout the simulations.

Figure 5�a� presents the numerical results obtained forthe three cases in terms of vena contracta factor �=U /UB asa function of the modified Reynolds number proposed byvan Zon et al.13 These results are compared with those pre-dicted by the quasistationary model presented in the previoussection. For short channels, the values of � were chosen torepresent two geometry cases, namely, a slit in an infinitewall and a tube with sharp edges �Borda tube�. According topotential flow theory,42 � is determined by the turning angleof the upstream flow into the channel, which depends on thecharacteristics of the external geometry. For the slit in aninfinite wall, one finds �=0.61, whereas for the Borda tube�=0.5. Therefore, in the case of a short single-reed mouth-piece channel �l /h�1�, one should expect an intermediatevalue between the two extreme cases, i.e., 0.5��0.61.Figure 5�b� plots the same simulation results in terms ofvolume flow U as a function of the pressure difference �pand compares that with the theory provided for short andlong channels as presented in Sec. III A.

In general, Figs. 5�a� and 5�b� show that the results ob-tained for geometries 1, 2, and 3 agree very well with thetheory presented in Sec. III A. However, Fig. 5�a� shows thatthe result for geometry 1 is in considerable disagreement forh Re/ �l−s��60 when compared with the limits provided bythe theory for short channels and fully detached flow �0.5��0.61�. This type of disagreement is commonly re-ported in the literature and is attributed to the influence ofviscous effects at low Reynolds numbers, as described byDurrieu et al.43 and Blevins.44 Curiously, the results obtainedfor geometry 2 are very similar to those found for geometry3 and agree very well with those predicted by the theory forlong channels �Eq. �10��, despite the fact that geometry 3 hasa rather diverging channel profile due to the presence of thechamfer.

The flow profiles were found to be roughly constant forall geometries. In the first geometry, the flow remained fullyseparated for Re�30, whereas in geometries 2 and 3 theflow separated at the beginning of the channel and reattachedat lr�2h for Re�60.

IV. DYNAMIC RESULTS

The goal of the following is to investigate the main as-pects of the fluid-structure interaction in dynamic regimes byusing the same geometries investigated in Sec. III �Fig. 4�.We also intend to substantiate the validity of quasistationarytheory by evaluating the main assumptions associated with

FIG. 4. Different reed channel profiles used in the simulation: �a� L /h=1,�b� L /h=4, and �c� L /h=4 with a chamfer.



the steadiness of the flow reattachment point and steadinessof the vena contracta factor when the oscillation of the reedis taken into account.

For all three cases, we use the same initial and boundaryconditions described in Sec. II. The flow is initiated by pre-scribing �pT=5 kPa. This value corresponds to a middlepoint between the threshold of oscillation and the maximumpressure found for a clarinet mouthpiece.15 It must bestressed that the ABC scheme used at the inlet and outlet ofthe system �Fig. 2� provides a complete anechoic behavior,which avoids any sort of acoustic coupling between the reedand the upstream and downstream chambers. Therefore, thereed can only move if an aerodynamic force FB exists due toflow detachment with an ensuing reattachment.25 This aero-dynamic force can explain the movement of the reed duringthe transient state of the flow but its existence alone is, how-ever, insufficient to explain a self-sustained oscillatory re-gime. This can only happen when the net energy exchangedbetween the flow and the reed during one duty cycle is posi-tive: E=�0

TFB · ytip�0, where FB is the space averaged aero-dynamic force on the reed and ytip is the velocity of the reedmeasured at its tip. In other words, the amount of energyabsorbed by the reed from the flow during one duty cycle hasto be greater than the energy imparted to the flow by thereed. As explained by Hirschberg,25 in the absence of acous-tic coupling, a positive net energy after one duty cycle ispossible due to several reasons: �a� the difference in the reedchannel geometry between opening and closing phase; �b�the inertia of the flow in the channel;18 and �c� variability ofthe separation/reattachment point behavior.21

A. General results

Figure 6 depicts the time histories associated with dis-placement of the reeds measured at their tips for all geom-etries. The self-sustained oscillation regime is achieved forall geometries in Fig. 4. The long channel geometries de-picted in Figs. 4�b� and 4�c� present very similar behaviorwith high oscillation amplitudes, which forces the tip of thereed to close the channel completely. For the geometry withthe short channel �Fig. 4�a��, the reed oscillation is roughly

FIG. 5. Comparison between theory and numerical results for a stationaryreed: �a� Vena contracta factor as a function of the modified Reynolds num-ber and �b� pressure difference across the reed channel as a function of thevolume flow.

FIG. 6. Time histories of the reed dis-placement measured at the tip for dif-ferent channel geometries.



sinusoidal and the average value of the tip displacementytip�8.010−4 m. Long-channel geometries present similaroscillation periods that are �6.5% shorter than those foundfor the short-channel geometry. This vibratory behavior at afrequency close to the reed’s first natural frequency f0 wasexpected, given the absence of acoustic coupling between thereed and the downstream and upstream cavities.

Further analysis was carried out by investigating the dy-namic characteristics of one single oscillation period. Theselected duty cycles are related to the sixth oscillation periodof each case and are indicated between dashed lines in Fig. 6.

Figure 7�b� shows the normalized energy flows E=FBytip as a function of time in terms of fraction of one dutycycle. The negative areas indicate transfer of energy to theflow due to the work of the reed. They take place during thephases associated with the opening of the reed, as shown inFig. 7�a�. Conversely, the positive areas in Fig. 7�b� takeplace when the reed is closing and represent the energy ab-sorption by the reed due to flow work. In the regions ofnegative energy flow, ytip and FB are out of phase but becomein phase as the reed starts to close again. In all cases, theshift from negative to positive energy flow also coincideswith the maximum volume flow U, shown in Fig. 7�c�. Theseresults present the same behavior found in the experimentsconducted by Thomson23 for an idealized model of the hu-man larynx.

The high amplitudes of oscillation found in the case oflong-channel geometries are explained by the higher ratiosbetween absorbed E+ and lost energy E− during one cycle, asshown in Table II. The excess of energy given to the reed isdissipated internally by the viscous damping predicted by thethird term on the right-hand side of Eq. �6� and by the in-elastic collision of the reed against the side lays of themouthpiece. Furthermore, Fig. 7�b� shows that the reeds inthe long-channel geometries start to receive energy from theflow at 0.6 T of the duty cycle, which represents a delay of0.13 T compared to the short channel geometry. This is dueto a higher flow inertia caused by larger fluid volume withinlong channels and due to the effect of flow driven by themoving reed, as will be discussed later in this paper. Table IIpresents some aspects related to the oscillation frequenciesachieved by each geometry, as well as aspects related to theenergy exchange between the flow and the reed.

Figure 8 provides a better understanding of the resultspresented in Figs. 7�a�–7�c� by depicting snapshots of thenormalized velocity field unorm= �ux

2+uy2�1/2 /max�ux� in the

mouthpiece models, taken at four different instants withinthe same duty cycle.

In all cases, a jet is formed at the channel’s entrance asthe reed starts to open. At this point, the jet rapidly adheresto the rail tip but remains detached elsewhere. This situationcontinues until the gradient of pressures between the jet andthe reed is enough to force the jet to attach to the reed’ssurface. The gradient is originated by the entrainment of flowbetween the jet and the reed wall due to viscous momentumtransfer and it is proportional to the downstream volumeflow. This phenomenon, known as the Coanda effect, plays

an important role in the self-sustained oscillations in vocalfolds20–24 and in reed instruments such as the accordion19 andthe harmonium.18

During the opening stage the volume flow U in the shortchannel accelerates earlier into the mouthpiece chamber. Infact, for the same channel aperture ytip, the volume flow intothe short channel is much higher than that into the long-

FIG. 7. Oscillation characteristics as function of time in terms of fraction ofone duty cycle: �a� channel aperture, �b� normalized energy flow, and �c�volume flow.



channel geometries, as shown in Figs. 7�a� and 7�c�. Theearly acceleration provides the necessary pressure gradientfor the jet to detach from the rail tip and adhere on the reedat �0.5 T, in contrast with the long channel geometries inwhich the same phenomenon happens at �0.7 T, as depictedin Fig. 8. The separation/adhesion phenomenon is confirmedby the determination of the skin friction based on the shearstress on the reed surface.

As already mentioned, there are two explanations for theearly volume acceleration in the case of the short channel.First, the fluid volume within the channel has a reduced in-ertia. The second reason is attributed to the effect of the flowdriven by the reed Uwall. This is because, in the case of adynamic regime, the effective volume flow can be expressedby U=U�p+Uwall, where U�p is the flow driven by the pres-sure difference �p across the reed channel. Thus, during the

opening stage the reed exerts work on the flow by pulling itout of the mouthpiece chamber in the upstream direction,which means that U�p and Uwall are out of phase. In shortchannels, the influence of Uwall on the effective flow U ismuch smaller than in the case of long channels, which ex-plains the early acceleration.

The effect of Uwall also becomes significant at instantsnear the complete closure of the channel �0.9T� t�1�. Dur-ing this period, U�p and Uwall are in phase and Uwall maybecome higher than U�p, which could explain the consider-able unsteadiness of the flow at this fraction of the dutycycle. This phenomenon has been reported by Devergeet al.45 in the case of experiments involving prototypes of thehuman glottis. In their observations, however, the effect ofUwall seems to be more evident in channels with constant

TABLE II. Aspects of dynamic flow in the different channel profiles.

L /h f�Hz� f / f0 E �E+ /E−�

Geometry 1 1 1760.3 1.00 120.28 1.10Geometry 2 4 1877.2 1.07 188.59 1.24Geometry 3 4 1855.7 1.06 179.30 1.22

FIG. 8. �Color online� Snapshots of the velocity field for different instants within the same duty cycle: �a� L /h=1, �b� L /h=4, and �c� L /h=4 chamfered.



height. This fact contrasts with our case in which the reedchannel becomes divergent near the closure stage.

Furthermore, the aerodynamic force FB caused by thepressure gradient increases when the jet attaches to the reedand becomes proportional to the attachment length. This ex-plains the higher oscillation amplitudes in geometries 2 and3. The increase in FB acts to decelerate the reed until it stops.At this point, ytip and FB become in phase and the reed startsto receive energy from the flow. The stronger FB in longchannels also explains the positive pitch shift in these geom-etries, because a stronger FB forces the reed to close morerapidly.

B. Discrepancy from the quasistationary predictions

The snapshots of flow during one duty cycle depicted inFig. 8 show some fundamental deviations between the qua-sistationary assumptions and the numerical results regardingthe detachment/adhesion phenomenon. In the case of theshort channel geometry, Fig. 8�a�, the constant fully sepa-rated flow assumed in the quasistationary theory has not beenobserved. In fact, for the first half of the duty cycle the flowis detached from the reed but remains attached to the rail tip.For the second half of the duty cycle the flow attaches toboth reed and rail tip.

The results for geometries 2 and 3, Figs. 8�b� and 8�c�,are very self-similar. The presence of a chamfer in geometry3 did not play a significant role on the stability of the attach-ment phenomenon. In those cases, the flow remains detachedfrom the reed for nearly �70% of the duty cycle. At �0.7 T,the flow adheres to the reed and gradually detaches from therail tip until the complete channel closure. This pattern con-trasts with the theory, which assumes a constant separationregion between the channel’s entrance and lr=2h and fullattachment of the flow afterwards.

As expected, the numerical results for the vena contractafactor also diverge considerably from the theoretical predic-tions. Figure 9 depicts the comparison between numericaland theoretical values for � along one duty as a function ofthe modified Reynolds number proposed by van Zon et al.The hysteresis observed for all cases in Fig. 9 agrees quali-tatively with that found in a dynamic flow measurement con-ducted by van Zon et al.13 The hysteresis observed in theshort channel geometry �Fig. 9�a�� is much smaller than thatobserved in the remaining cases �Figs. 9�b� and 9�b��. This isprobably due to a less significant influence of the flow drivenby the reed Uwall, as previously discussed. Furthermore, theflow adhesion segment is much shorter in the case of geom-etry 1, which minimizes the contribution of shear dissipationon the hysteresis. Figure 10 depicts the numerical values of �as a function of time in terms of fraction of a duty cycle. Forthe short channel, the values of � remain constant for only35% of the duty cycle, namely 0.50 T� t�0.85 T. The val-ues of � become very unstable as the reed approaches theclosed position. As already discussed, this characteristic isattributed to the effect of Uwall, which becomes higher thanthe flow driven by the pressure difference across the reedchannel.

V. CONCLUSIONS

We propose a numerical technique based on the latticeBoltzmann and finite difference methods to represent theproblem of fully coupled fluid-structure interaction in singlereed mouthpieces. The model provides second-order accu-racy at representing boundary layer phenomena and was usedto evaluate the behavior of three different reed channel ge-ometries in two types of regimes, namely, stationary and dy-namic. The stationary results agree very well with those pre-dicted by the quasistationary theory, in terms of volume flow

FIG. 9. Numerical and theoretical results for the vena contracta factor asfunction of the modified Reynolds number: �a� geometry 1, �b� geometry 2,and �c� geometry 3.



and vena contracta factor. Furthermore, we observed thesame behavior found experimentally by van Zon et al.,13

associated with the steadiness of the vena contracta factor fordifferent Reynolds numbers, in the case of short channels,and with the steadiness of the detachment / reattachmentphenomenon in long channels.

However, the results obtained during the dynamic simu-lations are very different from those predicted by the quasis-tationary theory. For the short channel geometry, � wasfound to be constant for only �40% of the duty cycle, andfor long channels, the values of � were in stark disagreementwith the quasistationary predictions. Moreover, the patternsobserved in stationary measurements such as fully detachedflow, in the case of short reed channels, and the twofoldpattern, in the case of long channels, were not observed inthe dynamic simulations. The main difference in the flowbehavior between short and long channels was found to bethe time taken by the flow to adhere on the reed wall withinone duty cycle. This characteristic was attributed to the effectof inertia associated with different fluid volumes within thereed channel and to the flow driven by the reed. The resultsalso show that different levels of self-sustained oscillationscan be achieved in the absence of acoustic feedback due tothe complexity of hydrodynamic forces acting on the reed,which supports the hypothesis proposed by Hirschberget al.12,25 in the case of single reed mouthpieces.

The two-dimensional nature of our numerical approachrestricts the results to a qualitative analysis. Another limita-tion is associated with the lack of acoustic feedback, whichneglects eventual influences of the acoustic field on the flowwithin the reed channel. Nevertheless, we feel it is worth-while to focus on the aerodynamicaly oscillating situationpresented in this paper. The widespread assumptions made inmodeling wind instrument reeds that have been reportedmany times in this journal and others are based on a quasis-tationary assumption that itself does not take into account theinfluence of the acoustic field on the flow behavior. Thesimulations reported in this paper show that there are signifi-cant deviations from these long held assumptions that callinto question the validity of the currently accepted model.These deviations might easily be obscured in the presence ofacoustic feedback. Furthermore, the approach presented inthis paper contributes to our understanding of the behavior of

dynamic flow in single-reed mouthpieces and its dependencyon the characteristics of the reed channel geometry.

More investigations are needed in order to understandthe behavior of the dynamic flow when the acoustic couplingbetween mouthpiece-reed system and resonator is taken intoaccount. Another interesting step could be taken in order toinvestigate the mechanisms of energy transfer between flowand the acoustic field, as well as the characterization ofaeroacoustic sources in the mouthpiece and its contributionto the instrument’s sound content.

ACKNOWLEDGMENTS

The authors would like to thank Professor Luc Mogeauand the anonymous reviewers of this paper for their helpfulsuggestions. A.R.d.S wishes to thank CAPES �FundingCouncil of the Brazilian Ministry of Education� for support-ing his doctoral research.

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