Kweon 2006 Computers & Fluids

8/7/2019 Kweon 2006 Computers & Fluids

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Computational study of an incident shock waveinto a Helmholtz resonator

Y.-H. Kweon a, T. Aoki a, Y. Miyazato a, H.-D. Kim b,*, T. Setoguchi c

a Department of Energy and Environmental Engineering, Kyushu University, 6-1, Kasuga kouen, Kasuga, Fukuoka 816-8580, Japanb

School of Mechanical Engineering, Andong National University, 388, Songchun-dong, Andong 760-749, Republic of Koreac Department of Mechanical Engineering, Saga University, 1, Honjo, Saga 840-8502, Japan

Received 14 July 2004; received in revised form 30 May 2005; accepted 10 September 2005Available online 23 November 2005

Abstract

The behavior of an incident shock wave into a Helmholtz resonator is very important from the acoustical point of view as well asthe fundamental researches of shock wave dynamics. When a shock wave propagates into a Helmholtz resonator, complicated wavephenomena are formed both inside and outside the resonator. Shock wave reflections, shock wave focusing phenomena, and shockvortex interactions cause strong pressure fluctuations inside the resonator, consequently leading to powerful sound emission. Thewave phenomena inside the resonator are influenced by detailed configuration of the resonator. It is well known that the gas insidethe resonator strongly oscillates at a resonance frequency, as the incident wavelength is larger, compared with the geometrical lengthscale of the resonator, but there are only a few works regarding a shock wave that has an extremely short wavelength. Meanwhile,the discharge process of the incident shock wave from the resonator is another interest with regard to an impulse wave generationthat is a source of serious noise and vibration problems of the resonator. In the present study, the wave phenomena inside and out-side the Helmholtz resonator are, in detail, investigated with a help of a computational fluid dynamics method. The incident shock

Mach number is varied below 2.0, and many different types of the resonators are explored to investigate the influence of the reso-nator geometry on the wave phenomena. A total variation diminishing (TVD) scheme is employed to solve two-dimensional,unsteady, compressible Euler equations. The computational results are compared with existing experimental data to ensure thatthe present computations are valid to predict the resonator wave phenomena. Based upon the results obtained, the shock wavefocusing and discharge processes, which are important in determining the resonator flow characteristics, are discussed in detail. 2005 Elsevier Ltd. All rights reserved.

1. Introduction

In 1860, Helmholtz [1] first established the mathemat-

ical formula to explain the wave phenomenon inside aresonator. His theory was strictly applicable only to aresonator, which has a circular opening in the resonatorwall, without having a neck. After that, Rayleigh [2,3]suggested a more simplified theory than the Helmholtzwork.

In general, Helmholtz resonator consists of a body tocontain gas and a hole or a neck, and it has long beenutilized as an effective acoustic attenuation device at

the low frequencies, which are the resonance dictatedby the combination of cavity and neck, and their relativeorientation. The characteristic frequency of the Helm-holtz resonator corresponds to the sound wave of thewavelength which greatly exceeds the size of the resona-tor, and is governed by its volume and the mass of thefluid [4,5]. When the acoustic wavelength notablyexceeds the resonators dimension, the gas in and nearthe neck moves compressing and expanding the gas vol-ume, and the vibration system, which is similar to a

0045-7930/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2005.09.001

* Corresponding author. Tel.: +82 54 820 5622; fax: +82 54 8235495/1630.

E-mail address: [email protected] (H.-D. Kim).

Computers & Fluids 35 (2006) 12521263

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spring-mass system is formed inside the resonator. Con-

sequently, the mass of the gas oscillates both back andforth in the neck, and inside the resonator with a reso-nance frequency.

Until now, a great deal of experimental and theoreti-cal researches has been carried out to investigate theacoustic characteristics of the Helmholtz resonator. Cha-naud [6] investigated the effects of geometry on the reso-nator frequency of Helmholtz resonators with a cavitythat is a rectangular parallel-piped and bounded byeither a circular, rectangular, and cross-shaped orifice,and developed mathematical formulae for the internaland external end corrections of Helmholtz resonator.He also developed theoretical equations for the internalend correction of a resonator with a cylindrical cavity [7].

Selamet and Lee [8] investigated theoretically andexperimentally the acoustic performance of a concentriccircular Helmholtz resonator with an extended neck,and showed that the resonance frequency can be con-trolled by the length, shape, and perforation porosityof the extended neck without changing the cavity.

For the engineering applications of Helmholtz reso-nator, it is known that a periodic array of Helmholtzresonator is very effective to annihilate a shock wavein propagation of nonlinear acoustic waves in an air-filled tube. Recently, Sugimoto et al. [9] examined exper-

imentally the effects of a periodic array of Helmholtz

resonators on forced longitudinal oscillations of an aircolumn in a closed tube. According to the results of theirexperiments, the array reduces the resonance frequencyand the peak value, while its dispersive effect (i.e., thedependence of sound speed on frequency) can effectivelyannihilate the shock wave. They also developed the non-linear cubic theory to obtain a frequency response ofshock-free, forced oscillations of an air column in aclosed tube with an array of Helmholtz resonators [10].

Moreover, the Helmholtz resonator is being employedas a means of noise control strategy, the wave phenom-ena generated in high-speed railway train, automobileand aerospace technologies, MEMS (micro-electro-mechanical systems) technologies, etc. In the high-speedrailway train/tunnel systems, the Helmholtz resonatorhas been employed as the noise control means of theimpulse noise which is generated by the discharge of acompression wave from the tunnel exit [11].

Vardy and Brown [12] investigated theoretically theinfluence of air pockets in ballast track that is linkedto a series of Helmholtz resonators. Nagaya et al. [13]investigated a new type of silencer to reduce the high-frequency noise generated in a blower, and suggesteda two-stage Helmholtz resonator with automaticallytuning control.

Nomenclature

a speed of sound, eigenvalue of flux Jacobianmatrix

C acoustic compliance

D diameter of resonatore total energy per unit volumefr resonance frequency of resonatorF numerical flux in the x directiong limiter functionG flux vectors in the y directionH neck height of resonatorL differential operator, neck length of resonatorM acoustic massMs Mach number of incident shock wavep static pressureR right eigenvector of the flux Jacobian, radius

of Helmholtz resonator

t timeTr resonance periodu velocity component in the x directionU vector of conservation variablesv velocity component in the y directionV volume of resonatorW flux vectors on the symmetric axis, neck

width of resonator

x longitudinal distance in Cartesian co-ordi-nate

y transverse distance in Cartesian co-ordinatec

ratio of specific heatsq densityDt time intervalDx grid space in the x directionw entropy correction function

Sub/superscripts

1 atmosphere state2 state behind the incident shock wave0 non-dimensional quantityd dischargef focusgeo geometrical focus

gas gas dynamic focush incident shock wave propagated into a reso-

natori space node in the x directionj space node in the y directionmax maximum peak valuen time step

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With regard to the development of high-impulsemomentum sources for application to micro-propulsionand flow control, such as micro-electro-mechanical sys-tems (MEMS) fabrication technique, the resonator isapplied at the walls inside the nozzle, leading to a betterthrust performance [14]. Yoda and Konishi [15] have

improved the Helmholtz resonator performance bymodifying the geometrical parameters to obtain effectivenoise reduction. They also tried to develop a compactand adaptive passive noise control system for theMEMS technologies, and suggested the noise controlsystem, being effective to control the acoustics in a duct.

Unfortunately, most of these works associated withtheHelmholtz resonator are mainly limited to sound waves orcompression waves, which have a comparatively longwavelength. There are only a few works with regard toshock waves, in which, the Helmholtz resonator gasdynamics can be different from that of the sound waves.In case that a shock wave propagates into the Helmholtz

resonator, the wave characteristics are influenced by verycomplicated shock wave dynamics, such as shock wavereflection and discharge [1622], shock wave focusing[23,24], shockvortex interaction [25], etc. At present,the associated wave phenomena are not known well.

Very recently, Matsuura et al. [26] carried out exper-imental work to investigate the behavior of a shockwave propagating into a Helmholtz resonator. Theirresearch was limited to a shock wave with a specificMach number. Further study is needed to understandthe wave characteristics of the shock wave into a Helm-holtz resonator.

The objective of the current work is numerically toinvestigate the propagation characteristics of a shockwave into a Helmholtz resonator in detail. Computa-tions were carried out to solve the unsteady, two-dimen-sional, compressible, Euler equations. The totalvariation diminishing (TVD) scheme of YeeRoeDavis[27] was used to discretize the governing equations. Forseveral configurations of the Helmholtz resonator, theMach number of the incident shock wave is varied inthe range from 1.1 to 2.0. The results obtained fromthe present computations are validated with the previ-ous experimental ones [23,26].

2. Computational analysis

2.1. Governing equations

In order to analyze the wave phenomena generated inthe Helmholtz resonator, two-dimensional, unsteady,compressible, Euler equations are used in the presentstudy. The governing conservation equations are given by,

oU

ot oF

ox oG

oy W 0; 1

where

Uq

qu

qv

e

26643775; F

qu

qu2 pquv

e p u

26643775;

Gqv

quv

qv2 pe pv

2664 3775; W 1yqv

quv

qv2

e pv

2664 3775;and x is the longitudinal distance, y is the transverse dis-tance, q is the density and u and v are the velocity com-ponents for the x and y directions, respectively. Thetotal energy e per unit volume of the gas is expressedas e = p/(c 1) + q(u2 + v2)/2.

In computations, Eq. (1) is rewritten in non-dimen-sional form by referring the quantities to atmosphericconditions, as follows:

p0 pp1

; q0 qq1;

u0 ua1ffiffiffic

p ; v0 va1ffiffiffic

p ; t0 tD=a1 ffiffifficp ; x0 xD ; y0 yD ;

2where the subscript 1 indicates the atmospheric condi-tion, D the diameter of Helmholtz resonator and a thespeed of sound. If the symbol ( 0), indicating the non-dimensional quantities, is omitted for the sake of sim-plicity, then the equation system is exactly equivalentto Eq. (1). For the present computational analysis, thetotal variation diminishing (TVD) scheme is applied tosolve the governing equations.

2.2. Numerical scheme

The TVD scheme has been known to be very effectivefor computing the phenomena of shock waves withoutpresenting the spurious oscillations, which were oftenencountered in the presence of strong discontinuities inconventional second-order schemes. The concept ofthe TVD scheme was first introduced by Harten [28],and has its origin in an important property of a scalarconservation law ut + fx = 0: the total variation (TV)

of any physically admissible solution

TV Z

ou

ox

dx;does not increase in time. The total variation in x (TVD)of a discrete solution to a scalar conservation law isdefined by

TVu Xi

ui1 ui. 3

A numerical solution is considered as a bounded totalvariation or a total variation if the total variation isuniformly bounded in t and Dx.

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Harten [28] proposed the total variation diminishingcondition as a monotonicity condition to be satisfied by

TVun1 6 TVun. 4A scheme satisfying the above condition is called a TVDscheme.

For the present computational analysis, the YeeRoeDavis total variation diminishing (TVD) scheme[27], as will be described later, is applied to discretizethe governing equations. For computation of the time-dependent flows, an operator splitting technique, whichwas suggested by Sod [29], is employed for temporal andspatial derivatives. Then Eq. (1) can be given by a set ofone-dimensional equations:

Lx :oU

ot oF

ox 0;

Ly :oU

ot oG

oy 0;

Lw :oU

ot W 0;

5

Un2i;j LwLxLyLyLxLwUnn;j; 6

LxUni;j Un1i;j Unj;j

Dt

DxbFni1=2 bFni1=2; 7

where subscripts i, j and superscript n indicate the spacenodes and time step, respectively, and Lx and Ly the dif-ferential operators for the x and y directions, respec-tively. In Eq. (7), Dt and Dx indicate the time intervaland the grid space in the x direction, respectively, and

bF denotes the numerical flux in the x direction, whichis expressed as,bFni1=2 12Fni Fni1 Rni1=2 Uni1=2; 8where Rni1=2 is a matrix whose column vectors are theright eigenvectors of the flux Jacobian oF/oU, evaluatedwith a symmetric average ofUi,jand Ui+1,j. The last termRni1=2 Uni1=2 represents the anti-diffusive flux contribu-tion that corrects the excessive dissipation of first-ordernumerical flux in a non-linear way. The numerical fluxbGnj1=2 in the y direction can be similarly expressed.

For the YeeRoeDavis second-order TVD scheme,the vector Un

i1=2is given as,

Ui1=2 DtDx

ai1=22gi1=2 Wai1=2ai1=2 gi1=2 !

;

9where ai+1/2 is the eigenvalue of the Jacobian matrix,ai+1/2 is the spatial difference of local characteristic vari-ables, and gi+1/2 is the limiter function. ai+1/2 and gi+1/2are defined respectively, as follows:

ai1=2 R1i1=2 Ui1;j Ui;j

; 10gi1=2 min modai1=2; ai1=2; ai3=2. 11

The function W(ai+1/2), called an entropy correctionfunction, is defined as,

Wai1=2 jai1=2j if jai1=2jP e;a2i1=2 e2=2e if jai1=2j < e;

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where e is a small positive number. This function cor-rects the entropy to prevent it from violating solutions,such as expansion shock waves.

2.3. Computational domain and boundary conditions

Fig. 1 shows the schematic description of the compu-tational domain, boundary conditions and initial condi-tions applied to the present computations. Thecomputational domain consists of the regions insideand outside the Helmholtz resonator. The upstreamdomain of the resonator is extended up to 3D from

the inlet of the resonator neck, where D and H meanthe resonator diameter and the neck height of the reso-nator, respectively. The computation was carried outonly in the half domain of the resonator because theflow field is assumed to be axisymmetric.

An incident shock wave with Mach number Ms is ini-tially located at x/D = 0.5 away from the inlet of theresonator neck, and at the instant of the start of compu-tation, it is assumed to propagate into the resonator. Asshown in Fig. 1(b), p1 and p2 indicate the atmospheric

H

L

D=2R

2D

H=D

(Helmholtz resonator) (Half circular reflector)

(a) Helmholtz resonator and half circular reflector

pp

2

1

Incidentshock wave

Initial conditions

0-3 x/D

Inflow/outflowconditions

Symmetric conditions

Slip-wall conditions

y/D

(b) Boundary conditions and initial conditions

Fig. 1. Computational domain and boundary conditions.

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pressure and the pressure just behind the incident shockwave, respectively, in which its Mach number Ms is var-ied between 1.1 and 2.0 in the present computations.

As the inflow and outflow boundary conditions, thezeroth-order extrapolation is used for the conservativevariables. The symmetric conditions are applied on thecentre-line of the resonator. The boundary conditionson all the surfaces inside and outside of the resonatorare the slip-wall conditions.

A square grid system is employed in the present com-putation. The fineness of computational grid required toobtain grid independent solutions was first examined

with the experimental data [23]. A grid density overDx = Dy = R/200 seemed to no longer change the accu-racy of the obtained solutions, as can be seen in Fig. 2. Agrid size ofDx = Dy = R/250 was employed so that thesolutions obtained were independent of the grid density.The dependence of the numerical solutions on the griddensity employed is found in Fig. 2, where the maximumpeak pressure pmax is defined as the peak value achievedby the shock wave focusing. The maximum peak pres-sure is no longer changed, when the number of thenumerical grid n is larger than 250.

Several different types of the resonator are employedin the present study. In order to change the configura-tion of the resonator, the value of H/D is varied between0.1 and 0.6 at fixed D. The length of the resonator neckL is held constant at 0.23D. In addition, computationsare carried out for the half circular reflector whose His the same as the resonator diameter (i.e., H/D = 1.0),as schematically shown in Fig. 1(a).

3. Acoustic theory of Helmholtz resonator

The resonance frequency of the Helmholtz resonatoris obtainable using the acoustic theory of sound wave. It

is based upon acoustic mass and acoustic compliance,where the acoustic mass M can be expressed by

M qSL; 13where Sis the cross sectional area of the neck and is givenas S= HW, q is the density of the gas inside the resona-

tor, L and Hare the neck length and neck height of theresonator, respectively, and W is the neck width of theresonator. In the present study, W is equal to 2.27D forthe purpose of comparison with Matsuura et al.s exper-iment (D = 22 mm, H= 10 mm, L = 5 mm, and W=50 mm) [26]. The acoustic compliance C is related tothe volume of the resonator and the speed of sound, asfollows:

C V=qa2; 14where Vmeans the volume of the resonator, and a is thespeed of sound. From two equations above, the reso-nance period Tr and frequency fr are obtained as

Tr 2pffiffiffiffiffiffiffiffiMCp ; 15fr 1=Tr. 16For instance, the resonance frequency of the resonator,which is H/D = 0.46, is calculated to be 3.89 kHz. Thisresonance frequency increases with an increase in H/D,as shown in Fig. 3.

According to the linear acoustic theories which aregiven in Eqs. (13)(16), the resonance frequency of aresonator is derived using the assumption that for

0x105

1x105

2x105

3x105

4x105

5x105

6x105

Grid number

11.0

11.5

12.0

12.5

13.0

pmax

/p1

x=y=R/n

n=100

n=150

n=200 n=250 n=300

Fig. 2. Dependence of the numerical grid density on the predictedmaximum peak pressure (H/D = 0.46, Ms = 1.7).

0.1 0.2 0.3 0.4 0.5 0.6H/D

0.00

0.25

0.50

0.75

1.00

Tr

(ms)

L/D=0.23

(a) Resonance period

0.1 0.2 0.3 0.4 0.5 0.6H/D

1.0

2.0

3.0

4.0

5.0

fr(kHz)

L/D=0.23

(b) Resonance frequency

Fig. 3. Relationship between resonance frequency and H/D.



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oscillation, mass is concentrated on the neck of the res-onator, and that spring constant is given by the resona-tor volume. The present study is related to the shockwaves that are strong nonlinear phenomenon with highfrequency. When a shock wave propagates into the res-onator, very complicated wave phenomenon take place

inside outside of the resonator, such as shock reflection,shock wave focusing, shockvortex interaction, and theshock wave discharge from the resonator. Due to thecomplicated shock wave reflections that occur insidethe resonator, the pressure continues to fluctuate afterthe wave discharge. Such wave phenomena are differentfrom the simple sinuous wave-motion of mass-springsystem.

The resonance frequency, as given in Eq. (16), is notadequate for the shock wave discharge from the resona-tor. The present study yields the shock focusing time,which is associated with the mean-velocity of shockwave in the resonator. Thus, the shock wave focusing

time can be an important parameter in predicting theshock wave discharge, as will be described later.

4. Results and discussion

Figs. 4 and 5 show the behavior of the shock wavepropagating into a Helmholtz resonator, where Ms =1.7 and H/D = 0.46, and non-dimensional time t0 isassumed to be zero for the instant that the incidentshock wave arrives at the inlet of the resonator neck(x/D = 0). At t 0 = 0.18, just after the incident shock

wave entered the inlet of the resonator neck, a part ofthe shock wave propagates into the resonator throughthe neck while the other part reflects from the wall ofthe resonator neck and propagates back toward the inletof the resonator. The expansion waves are generatedbehind the reflected shock waves, causing shock wavediffractions near the edges of the resonator neck. It isalso found that the vortices are formed at the inlet ofthe resonator neck. At t 0 = 0.37, the shock wave propa-gates into the resonator and the resulting wave patternsare changed from a normal to a curved shock wave. Astime increases, the curved shock wave propagatestoward the vertex of the resonator, and in this case,the regular reflections are formed on the upper andlower walls of the resonator (see Fig. 4(c)). Due to thepressure difference in the region between the curvedshock wave and the expansion waves generated fromthe edges of the resonator neck, a secondary shock waveis formed and interacting with two vortices, as seen inFig. 4(c) and (d).

When the curved shock wave completely reflects fromthe vertex of the resonator, the resulting shock wave isconverging. The reflected shock wave is convex to the res-onator. Thus, the compression waves are formed alongthe resonator wall surface. The triple point is generated

by the two reflected shock waves and the compressionwaves. Fig. 5 shows the formation and movement of thetriple point and the symbol T indicates the triple point.The triple point moves toward the centre-line of the reso-nator, consequently coalescing at a certainlocation on thecentre-line. Shock wave focusing is obtained at this loca-tion, where the location pressure reaches a maximumvalue. After the shock wave focusing, the reflected shockwaves are crossed and propagate toward the neck of theresonator with time. As the reflected shock waves stronglyinteract with the vortices and the secondary shock waves,the complicated wave structures are formed, and dis-charged from the resonator, as shown in Fig. 4(e). Froma comparison of the present computations and the previ-ous experimental results [26], it is found that there is qual-itative agreement.

Fig. 4. Experimental and computed Schlieren images (H/D = 0.46,Ms = 1.7).

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Fig. 6 shows the propagation process of the shockwave inside the resonator, where the Mach number Msof the incident shock wave is 1.7, and H/D is 0.1. Att 0 = 0.51, the shock wave inside the resonator is dif-fracted more significantly due to the smaller neck height

of the resonator, compared with the case of H/D = 0.46in Fig. 4. With an increase in time, the vortices generatednear the edges of the resonator neck grow graduallyand move into the resonator. At t0 = 1.59, the shockwave focusing is formed. The shock waves reflectedinside the resonator are discharged through the neck of

the resonator as an impulse wave, as shown in Fig. 6(g)and (h).For Ms = 1.5, Fig. 7 shows the detailed pressure

distributions before and after shock wave focusing atH/D = 0.6. For reference, the pressure distributionsalong the centre-line of the resonator are also presented.Fig. 7(a) shows the pressure distribution before theshock wave focusing, and the triple points and the vor-tices are obviously found inside the resonator. Outsidethe resonator, the shock waves reflected from the wallof the resonator neck propagate back toward theupstream boundary, and the regular reflection is formedon the centre-line. With an increase in time, the regular

reflection is changed to the Mach reflection, as can beseen in Fig. 7(c). At the instant of the shock wave focus-ing, as shown in Fig. 7(b), the local pressure reaches amaximum value at x/D = about 0.9. After the shockwave focusing, the shock wave propagates toward theneck of the resonator, and strongly interacts with thevortices. This gives rise to the complicated wave struc-tures inside the resonator, causing large pressurefluctuations.

At the same conditions, the shock waves dischargedfrom the resonator are shown in Fig. 8. At t 0 = 2.19,the strong vortices are observed inside the resonator,

and the shock waves propagate toward the neck of theresonator, interacting with the vortices. At t 0 = 2.78,the shock waves begin to discharge from the neck ofthe resonator, leading to the impulse waves. The strongvortices are indicated as the concave parts in the pre-dicted pressure contours. The wave phenomenon, whichis generated outside the Helmholtz resonator, at thevicinity of resonance condition, can be important withregard to such engineering applications as mentionedpreviously. In order to make clear such wave phenom-ena, a tremendous computing time would be required.The present study is concentrated on the flow phenom-ena inside the Helmholtz resonator.

Fig. 9 shows the pressure-time histories at the inlet Aof the resonator and its gas-dynamic focus [24], wherexgas indicates the distance between the gas dynamicfocus B and the vertex C of the resonator. At the loca-tion A, the pressure sharply rises up to a certain overpressure level (p2/p1 = 3.2), as the incident shock wavewith Ms = 1.7 reaches the point A at t

0 = 0.28, and itmaintains constant for a short time. Then the pressuresuddenly decreases, as the expansion waves generatedfrom the edges of the resonator neck reaches the pointA. At t0 = 2.5, the pressure rises again due to the inci-dent shock waves reflected back from the resonator.

Fig. 6. Computed iso-pressure contours (H/D = 0.1, Ms = 1.7).

Fig. 5. Formation and movement of triple point (H/D = 0.46, Ms = 1.7).



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At t0 = 3.0, the pressure rise is due to the discharge ofthe complicated wave structures from the resonator.

At point B, the pressure rises to a certain value ph/p1,which corresponds to the incident shock wave propa-gated into the resonator. At t 0 = 1.4, the pressure shar-ply rises to a maximum peak value (pmax,f/p1), which isdue to the shock wave focusing. This peak pressure isextremely high and very sharp, being about 12 timesatmospheric pressure. After the shock wave focusing,the pressure decreases with time, and fluctuating dueto the complicated wave structures inside the resonator.

Fig. 10 represents the relationship between the maxi-mum peak pressure achieved by the shock wave focusingand the Mach number of the incident shock wave, wherepmax,f/p1 means the maximum peak pressure due to theshock wave focusing, divided by atmospheric pressure.At fixed values ofL/D and H/D, pmax,f/p1 increases withan increase in Ms. This tendency seems remarkable asH/D increases. It is also found that for a given Ms,pmax,f/p1 increases with an increase in H/D. For instance,

for the resonator ofH/D = 0.6 and Ms = 2.0, the maxi-mum peak pressure is about 150% higher than that ofH/D = 0.10. From a comparison of a half-circularreflector and the Helmholtz resonator, it is found thatpmax,f/p1 for the half circular reflector is much higher,compared with the Helmholtz resonator.

The variation of the maximum peak pressure with theneck height of the resonator is shown in Fig. 11. For aweak incident shock wave (Ms 6 1.3), the value of(pmax,f ph)/p1 increases slightly with an increase inH/D. However, for strong shock waves, it increases withH/D and becomes almost constant for further increasein H/D. It is, thus, believed that for strong incidentshock waves, the value of (pmax,f ph)/p1 is not influ-enced by the neck height of the resonator which hasH/D larger than a certain value.

Fig. 12 shows the relationship betweenph/p2 and Ms, forvarious values ofH/D, where p2 is the pressure just behindthe incident shock wave and ph the pressure of the incidentshock wave passing through the point B (see Fig. 9). For a

Fig. 7. Iso-pressure contours before and after shock wave focusing (H/D = 0.6, Ms = 1.5).



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given H/D, the value ofph/p2 decreases with an increase inMs. This tendency is rather weak as H/D increases.

For various values of H/D, the relationship betweenpd/p1 and Ms is represented in Fig. 13, where pd is thepeak pressure at the point A, which is due to the shockwave discharged from the resonator (see Fig. 9). Thevalue of pd/p1 increases with an increase in Ms. It isinteresting to note that for H/D > 0.1, the value ofpd/p1 has an inflection point at Ms = about 1.5, whilefor H/D = 0.1, it increases monotonously with Ms. Thismay be because the shock wave propagated into theresonator is diffracted more significantly due to thesmaller neck height of the resonator and its strengthbecomes very weak, compared with a larger neck height.

It is difficult to obtain the resonance frequency for theshock wave propagating into the Helmholtz resonator,because the complicated wave structures are formedinside the resonator due to the shock wave reflection,shockvortex interaction, etc. However, it may beobtained if the computing time is not limited, under a

very big computational domain enough to contain thewaves discharged from the resonator. In the presentstudy, the wave phenomena inside the Helmholtz reso-nator are analyzed using the shock wave focusing timeand the shock wave discharge time.

Fig. 14 shows the shock wave focusing and dischargetime, where t0f is the shock wave focusing time, and t

0d the

shock wave discharge time in which both are taken fromthe instant that the incident shock wave arrives at theinlet of the resonator neck (see Fig. 9). It is found thatt0f and t

0d slightly decrease with an increase in both

H/D and Ms. From a comparison with the previousexperimental data [26], the present computation predictwell the experimental results of t0f and t

0d.

Fig. 15 shows the relationship between Ms and xgas,where the geometrical focus of the resonator is givenby xgeo, and defined as a location where linear acousticwaves reflected from the concave solid wall are focused,and for a circular reflector it is 0.25D, regardless ofH/Dand Ms. All of the present computational data show that

Fig. 8. Shock wave discharged from Helmholtz resonator (H/D = 0.6, Ms = 1.5).



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xgas decreases, as Ms increases, and that it is considerablydifferent from the geometrical focus of the resonator. Forthe same Mach number Ms, xgas becomes shorter as H/Dincreases. For instance, in the case of Ms = 1.7 andH/D = 0.46, the present computation somewhat over-

predicts the location of the gas dynamic focus, comparedwith the experimental data [26]. The shock wave focusingis a typical nonlinear phenomenon, and the presentcomputational analysis was made using unsteady, com-pressible Euler equations. The discrepancy between theexperimental and computed results can be due to the vis-cous effects which are essentially involved in the shockreflection and shock/shock interaction phenomena.However, for the half circular reflector (i.e., H/D =1.0), the present computation predicts the experimentalresults [23] with quite good accuracy.

A

(Inlet of Helmholtz resonator)

(Gasdynamic focus) B C

Vertex of Helmholtz

resonator

xgas

0 1 2 3 4 5t

0

2

4

6

8

10

12

14

p/p

1

td

tf

pmax, f/p1

pd/p1p2/p1 ph /p1

at A

at B

Fig. 9. Pressure-time histories (H/D = 0.6, Ms = 1.7).

0.1 0.2 0.3 0.4 0.5 0.6

H/D

0

1

2

3

4

5

6

7

(pmax,

f-ph

)/p

1

L/D=0.23

Ms=2.0

Ms=1.7

Ms=1.5

Ms=1.3

Ms=1.1

Fig. 11. Variation of maximum peak pressure with H/D.

1.0 1.2 1.4 1.6 1.8 2.0

Ms

0

5

10

15

20

25

30

pmax,

f/p1

H/D=0.10H/D=0.20

H/D=0.30

H/D=0.46

H/D=0.60

Half circular

L/D=0.23

Fig. 10. Variation of maximum peak pressure with Ms.

1.0 1.2 1.4 1.6 1.8 2.0

Ms

0.6

0.7

0.8

0.9

1.0

1.1

ph

/p2

L/D=0.23

H/D=0.10

H/D=0.20

H/D=0.30

H/D=0.46

H/D=0.60

Fig. 12. Magnitude of the incident shock wave propagated intoHelmholtz resonator.



11/12

5. Conclusion

In the present study, the wave phenomena inside theHelmholtz resonator are analyzed with a help of compu-tational method. Two-dimensional, unsteady, compress-ible, Euler equations are numerically solved using the

YeeRoeDavis total variation diminishing (TVD)scheme. Several kinds of resonators are employed toinvestigate the effect of the resonator configuration onthe complicated wave phenomena inside the Helmholtzresonator. The present computational results are vali-dated with the previous experimental data available.The obtained results show that the maximum peak pres-sure at the shock wave focusing location increases andthe focusing location is closer to the resonator wall, asthe Mach number of the incident shock wave and theneck height of the resonator increase. For a given inci-dent shock wave, an increase in the neck height of theresonator leads to a stronger shock wave, which is prop-

agated into the Helmholtz resonator, resulting in astronger wave discharge from the resonator. It is alsofound that an increase in the neck height of the resona-tor decreases the shock wave focusing and dischargetime. Because the complicated wave structures areformed inside the resonator due to the shock wavereflection and discharge, shockvortex interaction, etc.,the wave phenomena inside the Helmholtz resonatorcan more effectively analyzed using the shock wavefocusing time and discharge time than theoretical reso-nance frequency of the resonator.

References

[1] Helmholtz H. Theorie der Luftschwingungen in Rohren mitoffenen Enden. Crelle 1860;62:172.

[2] Rayleigh L. On the theory of resonance. Philos Trans1870;161:77118.

[3] Rayleigh L. Theory of sound. New York: Dover Publications;1945.

[4] Howe MS. On the Helmholtz resonator. J Sound Vib 1976;45(3):42740.

[5] Vinokur R. Infrasonic sound pressure in dwellings at theHelmholtz resonance actuated by environmental noise andvibration. Appl Acoust 2004;65:14351.

[6] Chanaud RC. Effects of geometry on the resonance frequency of

Helmholtz resonators. J Sound Vib 1994;178(3):33748.[7] Chanaud RC. Effects of geometry on the resonance frequency ofHelmholtz resonators, Part II. J Sound Vib 1997;204(5):82934.

[8] Selamet A, Lee I. Helmholtz resonator with extended neck.J Acoust Soc Am 2003;113(4):197585.

[9] Sugimoto N, Masuda M, Hashiguchi T, Doi T. Annihilation ofshocks in forced oscillations of an air column in a closed tube.J Acoust Soc Am 2001;110(5):22636.

[10] Sugimoto N, Masuda M, Hashiguchi T. Frequency response ofnonlinear oscillations of air column in a tube with an array ofHelmholtz resonator. J Acoust Soc Am 2003;114(4):177284.

[11] Raghunathan RS, Kim HD, Setoguchi T. Impulse noise and itscontrol. Prog Aerospace Sci 1998;34(1):144.

[12] Vardy AE, Brown JMB. Influence of ballast on wave steepening intunnels. J Sound Vib 2000;238(4):595615.

0.1 0.2 0.3 0.4 0.5 0.6

H/D

0.0

0.5

1.0

1.5

2.0

tf,

td

tf

td

tf

td

L/D=0.23

Ms=1.1

Experiment - Ref.26

(Ms=1.7,H/D=0.46)

Present CFD

Ms=1.3 Ms=1.5 Ms=1.7

Ms=2.0

Ms=2.0 Ms=1.7 Ms=1.5Ms=1.3 Ms=1.1

Fig. 14. Effect of H/D on shock wave focusing time and dischargetime.

1.0 1.2 1.4 1.6 1.8 2.0

Ms

0

0.1

0.2

0.3

0.4

0.5

xgas

/D

H/D=0.10

H/D=0.20

H/D=0.30

H/D=0.46

H/D=0.60

Half circular

H/D=0.46(Ref.26)

Half circular (Ref.23)

L/D=0.23

xgeo/D=0.25

Fig. 15. Gas dynamic focus vs Ms.

1.0 1.2 1.4 1.6 1.8 2.0

Ms

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

pd/p1

H/D=0.10

H/D=0.20

H/D=0.30

H/D=0.46

H/D=0.60

L/D=0.23

Fig. 13. Peak pressure of the shock wave discharged from Helmholtzresonator.



12/12

[13] Nagaya K, Hano Y, Suda A. Silencer consisting of two-stageHelmholtz resonator with auto-tuning control. J Acoust Soc Am2001;110(1):28995.

[14] Muller MO, Bernal LP, Moran RP, Washabaugh PD, Parviz BA,Allen Chou TK, et al. Thrust performance of micromachinedsynthetic jets. AIAA paper 2000-2404, Fluids 2000, 1922 June2000, Denver, USA.

[15] Yoda M, Konishi S. Estimation of an open-loop compactadaptive passive noise control system with microstructures.J Acoust Soc Am 2001;110(1):63841.

[16] Bazhenova TV, Fokeev VP, Gvozdeva LG. Regions of variousforms of Mach reflection and its transition to regular reflection.Acta Astronaut 1976;3:13140.

[17] Kim HD, Kweon YH, Setoguchi T. A study of the impulsive wavedischarged from the inclined exit of a tube. IMechE Part C2003;217(2):2719.

[18] Kim HD, Kweon YH, Setoguchi T, Aoki T. Weak shockreflection from an open end of a tube with a baffle plate. IMechEPart C 2003;217(6):6519.

[19] Kim HD, Setoguchi T. Study of the discharge of weak shock froman open end of a duct. J Sound Vib 1999;226(5):101128.

[20] Setoguchi T, Kim HD, Kashimura H. Study of the impingementof impulse wave upon a flat plate. J Sound Vib 2002;256(2):197211.

[21] Kim HD, Lee DH, Setoguchi T. Study of the impulse wavedischarged from the exit of a right-angle pipe bend. J Sound Vib2003;259(5):114761.

[22] Kim HD, Kweon YH, Setoguchi T. A study of the weak shockwave propagating through an engine exhaust silencer system.J Sound Vib 2004;275(4):893915.

[23] Hashiba Y, Saida N. Reflection of plane shock wave from concavewalls. Proc Jpn Shock Wave Symp 1990:6058.

[24] Kim HD, Kweon YH, Setoguchi T, Matsuo S. A study on thefocusing phenomenon of a weak shock wave. IMechE Part C2003;217(11):120919.

[25] Sun M, Takayama K. The formation of a secondary shock wavebehind a shock wave diffracting at a convex corner. Shock Waves1997;7:28795.

[26] Matsuura K, Funabiki K, Abe T. Behavior of an incident shockwave into a Helmholtz resonator. Proc Jpn Shock Wave Symp1997:758.

[27] Yee HC. Upwind and symmetric shock capturing schemes. NASATM-89464; 1987.

[28] Harten A. A high resolution scheme for the computation of weaksolutions of hyperbolic conservation laws. J Comput Phys1983;49:35793.

[29] Sod GA. A numerical study of a converging cylindrical shock.J Fluid Mech 1977;83:78594.


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