NASA/TM--2003-212019 AIAA-2003-0368

Nonlinear Resonant Oscillations of Gas in

Optimized Acoustical Resonators and the

Effect of Central Blockage

Xiaofan Li, Joshua Finkbeiner, and Ganesh Raman

Illin.ois In.stitute of Techn.ology, Chicago, I].linois

Christopher Daniels

Ohio Aerospace Insti.tute, Brook Park, Ohio

Bruce M. Steinetz

Glenn Research Center, Cleveland, Ohio

February 2003
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NASA / TM--2003-21_ 201_9 AIAA-2003-0368

Nonlinear Resonant Oscillations of Gas in

Optimized Acoustical Resonators and the

Effect of Central Blockage

Xiaofan Li, Joshua Finkbeiner, and Ganesh Raman

Illin.ois Institute of Technology, Chicago, I].linois

Christopher Daniels

Ohio Aerospace Insti.tute, Brook Park, Ohio

Bruce M. Steinetz

Glenn Research Center, Cleveland, Ohio

Prepared for the

41st Aerospace Scien.ces Meeting and Exh.i.bit

sponsored by the American Institute of Aeronautics and Astronautics

Reno, Nevada, January 6-!), 2003

National Aeronau.tics and

Space Administration

Gleru_ Research Center

February 2003
Trade names or manufacturers' names are used in this report for

identificatio_ only. This usage does not constitute a_ official

endorsement, either expressed or implied, by the National

Aeronautics and Space Administration.

The Propulsion and Power Program at

NASA Glenn Research Center sponsored this work°

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71121Standard Drive

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Available frorn

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5285 Port Royal Road

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Available electronically at bttp: //gltrs.grc.nasa.gov
NONLINEAR RESONANT OSCILLATIONS OF GAS IN

OPTIMIZED ACOUSTICAL RESONATORS AND THE

EFFECT OF CENTRAL BLOCKAGE

Xiaofan Li, Joshua Finkbeiner, and Ganesh Raman

Illinois Institute of Technology

Chicago, Illinois 60616

Christopher Daniels

Ohio Aerospace Institute

Brook Park, Ohio 44142

Bruce M. Steinetz

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

ABSTRACT

Optimizing resonator shapes for maximizing the ratio

of maximum to minimum gas pressure at an end of the

resonator is investigated numerically. It is well known

that the resonant frequencies and the nonlinear standing

waveform in an acoustical resonator strongly depend on

the resonator geometry. A quasi-Newton type scheme

was used to find optimized axisymmetric resonator

shapes achieving the maximum pressure compression

ratio with an acceleration of constant amplitude. The

acoustical field was solved using a one-dimensional

model, and the resonance frequency shift and hysteresis

effects were obtained through an automation scheme

based on continuation method. Results are presented for

optimizing three types of geometry: a cone, a horn-cone

and a half cosine-shape. For each type, different

optimized shapes were found when starting with

different initial guesses. Further, the one-dimensional

model was modified to study the effect of an

axisymmetric central blockage on the nonlinear

standing wave.

INTRODUCTION

The waveform of the standing wave in an oscillating

closed cavity is strongly influenced by the geometry of

the cavity (resonator). Lawrenson et al. 1 at MacroSonix

Corp first exploited the dependence for developing

high-amplitude acoustic pressures, referred to as

resonant macrosonic synthesis (RMS). Peak acoustic

pressures that exceed the ambient pressure three to four

times and ratios of maximum to minimum pressures of

27 were observed in RMS cavities. The size of the

demonstrated overpressure reached the level that

required by commercial applications such as acoustic

pump or compressor. They considered these types of

axisymmetric shape: cylinder, cone, hone-cone and

bulb, and appeared to find that the hone-cone shape

generates higher overpressure given an input power. A

companion paper by Ilinskii et al. 2 developed a onedimensional time domain model for studying the RMS

numerically. Their results confirmed the nonlinear

standing waveform and the related characteristics such

as shape-induced resonance hardening and softening

observed in the experiments by Lawrenson et al. 1Chun

and Kim 2 numerically investigated the cosine shapes in

addition to cylindrical and conical shapes using highorder finite-difference approximations. They claimed

that the half cosine-shape is more suitable to induce

high compression ratio than other shapes. To serve

commercial needs, such as in acoustic gas compressor

and acoustic liquid pump, the common goal in these

studies is to find an optimized shape for generating

higher overpressure. However, the optimization

procedure and result have not yet been discussed.

In this article, we introduce the numerical schemes and

present the results in searching for the optimal shape

parameters that give us the highest maximum-tominimum pressure ratio in each of the following types

of shapes: cone, horn-cone and half cosine-shape. In the

second part, we study the effect of a cylindrical central

blockage on the RMS.

I. GOVERNING EQUATIONS

In this section, we describe a one-dimensional model

for the acoustic wave field in the resonator that includes

a central blockage. We develop our model following

the work by Ilinksii et al. 2 for an axisymmetric

resonator in absence of the central blockage.

NASA/TM--2003-212019 1
Consider theacousticfieldinanoscillatingresonator

drivenbyanexternal force.AssketchedinFig.1,the

resonator isaxisymmetric withtheinneradiusgiven

byr r(x), O<x<l, where x is the coordinate along the

axis of symmetry. The central blockage is also

axisymmetric with the same axis of symmetry x, whose

shape is written as _ = r0(x ) .

The density of the gas p, the velocity u and the pressure

p satisfy the conservation of mass

1 0

(m-r:>.)--0

& r 2 _ r2 &

(1)

and the conservation of momentum

Ou Ou 1 @

& c_x p c_x

+(_+4r//3) lp _x0( r2-r_&O((r2-r_)u))

, (2)

where a(t) is the acceleration of the resonator enforced

by the external force; _ and r1 are coefficients of

viscosity. The state equation is specified by that of an

ideal gas

p=po(P/po) y, (3)

where P0 and P0 are the ambient pressure and density

respectively. The no-penetration boundary conditions at

the two ends require that the velocity vanish at x=O and

I.

Following Ilinksii et aL:, expressing the variables in

finite Fourier series in time, the equations (1) and (2)

can be reduced to a system of Ordinary Differential

Equations (ODEs) for the Fourier coefficients of the

velocity potential (p, defined as tt = Vq) :

#Ix r 2 - r2 ' (4)

where

v=(r 2- ,q)=._q)ke ,V= _)#e ,

^ ikt

a = 2_.,ace and

a_,= ak,(_,O_,x),L =L(_,O_,x). The

detailed expressions for akl andJ_ are the same as those

in Ilinksii et al. 2 except that the central blockage is

taken into account.

fk = -k2( r2 - ro2)(Pk^ + ik(/ - ro2)xak^

ik[v_L

r 2 -_

d(/-()

N N

^ ^ dx 2 ^ + Z Z ,: N+k ( r2 --r02) 3 Z: N+k

where c_ - _" + 4r//3

, c o is the small signal

P0

propagation speed, D' m is given by

D'm : -(/- l)xg_m - ira(7 - l)(_ m

7+1 v2

2(r 2_r_)2 [ ]m,

and

N

1= N+k

The no-slip boundary conditions at the two ends are

translated to the Fourier space: _)k = 0 at x=O,l.

After the Fourier coefficients { _k } and { _3k } are

obtained by solving the boundary-value problem Eq. (4),

the velocity potential (p and the modified velocity v are

computed from the inverse FFT. The density p is given

by the momentum equation

P

m=

P0

1

1 _ y_._.l_.2 Co1 1L_(,0 ot + 1 V2 c_ ____]) y_ 2(r2 _F02)2 +aX 1,2 __FO2

NASA/TM--2003-212019 2
andthepressurep can be obtained from the state

equation P/ Po = (P/ Po) y

II. NUMERICAL METHODS

In this section, we describe the procedures for finding

the optimal shape parameters so that the desired

standing waveform can be obtained.

To simplify our discussion, we introduce the following

dimensionless variables

x r r° A= a

x: 7, r:o,, R: 7, R0: 7, lo--7'

where co is the frequency of the periodic force acted on

2Z-C 0

the resonator, coo = -- is the fundamental frequency

1

of a cylindrical resonator of length I.

In this work, we assume the acceleration of the

resonator is harmonic, A(T) = _] cos(T). The

acoustic wave field in the resonator is determined by

the acceleration amplitude _], the ratio of specific

heats 7, the attenuation coefficient G, defined as

G = 7c(_" + 47] / 3)o o and the resonator oscillating

2

CoPo

frequency f2, defined as _ = o.

O0

Now, we briefly describe the method for finding the

optimal resonator shape.

In the optimization process, we fix the values of the

acceleration amplitude A = 5 x 10 4, the gas specific

heat ratio 7=1.2 and the viscosity-related parameter

G 0.01. Suppose the resonator shape R(X) is

determined by a number of shape parameters, So, S1,

.... Sn For example, a cone is given by

R(X) = S o + SIX. The compression ratio of a

resonator Rc, defined at the ratio of maximum pressure

to the minimum pressure at the narrow end of the

resonator

Rc(So,Sl,...,Sn) = P ......... atX:0, (6)

Pmin

is a function of the shape parameters, the dimensionless

frequency f2 and the history of f2 (due to the existence

of hysteresis effects). The method for obtaining I_ for

fixed resonator shape is explained later in the section.

We first outline the optimization schemes. A quasiNewton method, BFGS (Broyden-Fletcher-GoldfarbSharmo4), is used for maximizing the multi-variable

nonlinear function R ° (So, S1,...,S n ). Since the

evaluation of the objective function I_ itself involves

solving a nonlinear system of ODEs Eq. (4) for many

times, the gradient information of I_ required for the

BFGS method is not available analytically and is

derived by partial derivatives using a numerical

differentiation method via finite differences. This

entails perturbing each design variables, Si, in turn and

calculating the rate of change in the objective

function R ° (No, S 1,...,S n) . For two shape parameters,

the optimization takes 4 to 48 hours of CPU time

(depending on the type of the resonator shape and

initial guess for the shape parameters) on a 1.3GHz

Athlon T-Bird PC with the Lahey-Fujitsu FORTRAN

compiler.

For a given shape of the resonator and the central

blockage, the boundary value problem Eq. (4) is solved

numerically by a Multiple Shooting Method. Because

of the hysteresis effects, the solution is not unique near

the resonant frequency and the Multiple Shooting

Method will not converge unless a good initial guess of

the solution is provided. To circumvent the difficulty, a

continuation method is implemented: the system of

ODEs is solved starting from a frequency f2 that is far

away from the resonant frequency and the solution is

used as an initial guess for solving the ODEs for

increased or decreased f2; the steps are repeated until

all branches of the solution for all values of f2 near the

resonance is completed. The maximum ratio Ro among

different values of f2 is chosen as the compression ratio

for the resonator.

III. RESULTS

A. Characteristics of the standin_ waves

Before we present our results on finding the optimal

resonator's shape, we illustrate some of the important

properties of the pressure wave in a non-cylindrical

resonator, whose shape is described by that of a

horncone:

= IS0 cosh(S1X),0 _<X _<S 2

R(X) La + <_x <_l

(7)

NASA/TM--2003-212019 3
where

& =0.028333,S1=5.7264,S2=0.25,a=&cosh(S1S2),

and/3=SoS 1 sinh(S1S 2). The shape parameters for

the homcone are obtained from those used in the

experiments by Lawrenson et al. 1

In Fig. 2, we show the pressure waveforms at the ends

of cylindrical resonator and the horncone resonator at

their corresponding resonance frequencies for the

effective viscosity G=0.01, the gas specific heat ratio

7=1.2, the acceleration A = 10 3. Throughout this

paper, we will keep the effective viscosity G and the

gas specific constant at these values, which are close to

the values used in the experiments 1 and our numerical

simulations have suggested that the pressure waveform

is not a strong function of these parameters. At the

same level of the acceleration, the waveform at the

narrow end of the horncone shows large variation in

which the ratio of the maximum and minimum

pressure, Rc, exceeds the value of 30.5. On the other

hand, the waveform for the cylinder shows the

formation of shocks at the resonance and the

compression ratio Rc is below 1.32. The difference in

the waveforms shows the strong dependence of the

acoustic field on the resonator shapes. The graph also

shows that the variation in pressure at the wide end of

the horncone is much milder than that of the narrow

end, oscillating within 23% above or below the value of

the reference pressure P0

In Fig. 3, the ratio of the amplitudes of the second and

the first harmonics of the pressure, P2 / Pl, are plotted

against the reduced amplitude of the first harmonic,

Pl / P0, calculated at one end of the cylinder and at

the narrow end of the homcone with increasing

frequency. The figure shows that the second harmonic

reaches its maximum at small amplitude of the first

harmonic for the consonant (cylinder) resonator and the

second harmonic increases slowly as the first harmonic

increases for the dissonant (homcone) resonator. In

other words, the energy is pumped into the first

harmonic more for the horncone than that for the

cylinder, creating the large-amplitude pressure wave in

the horncone resonator. In contrast, the energy is

efficiently absorbed by all harmonics of the pressure

wave in a cylindrical resonator and the amplitudes of

the harmonics reach their maximums at small amplitude

of the pressure.

In Fig. 4, the amplitude of the first harmonic is plotted

against the frequency of the oscillating resonator for

two different levels of acceleration: A = 2 x 10 4 and

= 10 3. For the smaller acceleration, the pressure is

uniquely determined at each frequency; for the larger

drive amplitude, the pressure takes one of the multiple

values near the resonance, depending on the direction

of the change in frequency. The existence of hysteresis

and hardening resonance in homcone resonators

requires that the largest pressure compression ratio be

obtained through an upward frequency sweep. Our

numerical simulations show that the frequency

increment size near the resonance must be small for the

convergence of the Multiple Shooting method.

B. Optimal conical shapes

In following subsections, we present results on

optimizing resonator shapes to achieve maximum

compression ratio Rc at one end of the resonators. For

simplicity and reducing computing cost, we assume the

central blockage is absent, i.e., R0=0, and keep ratio of

specific heats ?= 1.2, the attenuation coefficient G=0.01

and the acceleration i] = 5 x 10 4 constant.

A conical resonator can be written as

R(X)= So+ S,X, for 0 < X < 1. (8)

Using the optimization procedure described in Sec. II

and starting with the conical shape given in Lawrenson

et al., 1 we found that the compression ratio Rc of a

conical resonator reaches a maximum when the cone

has the shape constants: 3'o=0.032945 and 5'1=0.26800.

The achieved compression ratio is R_ =5.048 for the

fixed values of the parameters given earlier. Recall that,

due to hysteresis, in order to find R_, one has to trace

out the entire branch by incrementing the frequency f2.

The compression ratio reaches the value when the

frequency is increased to 1.3135. The conical resonator

studied in Lawrenson et al. 1 and Ilinksii et al. 2 has

almost identical shape 3'o=0.032941 and 5'1=0.26800.

The adopted optimization scheme BFGS is designed for

finding a local extreme of a multivariable function. Our

numerical simulations indicate that the compression

ratio, as a function of the shape parameters, have many

local extrema. Starting with a different initial shape, we

obtained another locally optimized conical shape with

So=O. 010413 and SI=0.19572. When the frequency is

raised to the value of 1.3862 by small increments, the

compression ratio reaches the value of 5.343, which is

about 6% higher than that for the first optimal conical

resonator. In the current optimization, the slope of the

cone is restricted to be less than 0.268 and we require

that the radius at the small end be larger than 0.01. The

NASA/TM--2003-212019 4
twooptimal conical resonators areshowninFig.5.The

secondcone(Fig.5(b))hasthesmallernarrowendand

smallerslopethanthefirstone(Fig.5(a)).

C. Optimal horn-cone shapes

Horn-cone geometry is described by

= IRh cosh(DhX ) for 0 <_X <_X o

R(X) _a + fl(X- X_) for X_ < X < I

where

a = R h cosh(DhX C) and/_ = RhD h sinh(DhX C).

Rh, Xo and Dh are positive constants. We consider that

the connection point of horn and cone in the horncone

is fixed atXo =0.25. We optimize the remaining two

shape parameters X 0 --- R h and X 1 --- D h . We found

flaat the dimensions of the horn-cone given in

Lawrenson et al. 1, Rh =0.028333, Dh =5.7264, are

nearly locally optimized if we start the optimization

with these values. For these parameters, the maximum

compression ratio 13.10 is achieved when the frequency

is increased to 1.4671. In searching the optimal

horncone design, the parameters, such as acceleration

and specific heat ratio, are fixed at the values given in

the beginning of Sec. III B. Starting with a different

initial geometry of the horncone, we can obtain a higher

value of the compression ratio Ro = 17.12 when Rh

=0.024488, Dh =5.1434. For the compression ratio, the

frequency was increased to 1.4110 gradually with small

increments. Comparing with the compression ratio for

the homcone in Lawrenson et al. 1 (shown in Fig. 6(a)),

flae second optimal horncone (shown in Fig. 6(b))

improved the ratio by more than 30% at the same value

of acceleration. Shown in Fig. 6(c), the pressure

waveform at the narrow end for new homcone design is

a little more complicated than that for the original

horncone given Lawrenson et al. 1

D. Optimal 1⁄2-cosine shape

We define a 1/2-cosine shape as

R(x) = s0 + sl (1- cos( X), for 0 _<X _<1.

The parameters of the resulting optimal cosine shape

are 3'o=0.016307, S1=0.093041, shown in Fig. 7(b),

compared with S0=0.025, S1=0.095, the 1/2-cosine

dimensions reported in Chun and Kim) For the optimal

shape, we obtain the compression ratio Rc =10.67 at the

frequency f2=1.5683, which is about 11% better than

that of Chun and Kim 3 at the same level of acceleration.

Under the specified conditions, we found that horncone shape is better than the cosine shape in generating

higher compression ratio at the narrow end, which is

opposite to the findings of Chun and Kim) Shown

in Fig. 7(c), the pressure waveform of the optimal

1/2-cosine design is smoother than that of Chun and

Kim)

E. The effect of the central blockage

In this section, we discuss the influence of the central

blockage on the pressure waveforms in a resonator of

fixed shape. In this work, we limit ourselves to

cylindrical central blockage, though our equations

derived in Sec. I apply to arbitrary axisymmetric shape

of central blockage.

Figure 8 shows the pressure waves at the narrow end of

a conical resonator described by Eq. (8) with

So=0.032945 and S1=0.26800 for different central

blockage radius sizes: Ro = 0, 0.01 and 0.02. These

pressure profiles were obtained at the fixed frequency

f2 = 1.3132 and for _] = 5 × 10 4, G = 0.01, 7 = 1.2.

The central blockage in this case was a simple cylinder.

It appears that the presence of a central blockage

reduces the amplitude of the pressure wave. However,

Fig. 9 shows the resonant frequency of the conical

resonator with a central blockage is shifted. Keeping

other parameters constant, the pressure waveforms for

the conical resonator with the cylindrical blockage of

radius R0=0.02, at different oscillating frequencies f2 =

1.2812, 1.3068 and 1.3291, are shown in Fig. 9.

Comparing to the pressure wave shown in Fig. 8 for a

resonator without a central blockage, the amplitude of

the pressure is nearly the same when the central

blockage of radius 0.02 is included. The results indicate

that a cylindrical central blockage in a conical resonator

would affect the resonant frequency but would have

negligible effect on the amplitude and overall shape of

the pressure wave at the resonant frequency of the

resonator/blockage system.

IV. CONCLUSION

We have performed local optimization schemes for

finding the resonator shapes that maximize the pressure

compression ratio at one end of the resonators. The

dimensions for several types of optimal resonators are

reported: cone, homcone and 1/2-cosine shapes. For each

type, we found there are many different designs that

achieve local extrema. Trying with different initial

guesses for the optimal design, we show that one can

NASA/TM--2003-212019 5
getasmuchas30%improvement onhecompression

ratiowithafixedlevelofacceleration. Fortheshapes

weconsidered, it appearsthathehornconeshape

generateshehighestcompression ratio.Strategies for

searchinggloballyoptimalshapesareunder

investigation.

Theeffectofincludingacylindrical central blockagein

aresonator isstudiedinthisworkaswell.Foraconical

resonator, it seemsthatheresonancefrquencyis

increasedwhenacentral blockageispresent andthe

compression ratioatresonance isnotsignificantly

affected.

REFERENCES

1Lawrenson, C.C., Lipkens, B., Lucas, T.S., Perkins,

D.K., and Van Doren, T.W., "Measurements of

macrosonic standing waves in oscillating closed

cavities," J. Acoust. Soc. Am. Vol. 104(2), Pt. 1,

pp. 623-636, August 1998.

2Ilinskii, Y.A., Lipkens, B., Lucas, T.S., Van Doren,

T.W., and Zabolotskaya, E.A., "Nonlinear standing

waves in an acoustical resonator," J. Acoust. Soc. Am.

Vol. 104(5), pp. 2664-2674, November 1998.

3Chun, Y.-D. and Kim Y.-H., "Numerical analysis for

nonlinear resonant oscillations of gas in axisymmetric

closed tubes," J. Acoust. Soc. Am. Vol. 108(6),

pp. 2765-2774, December 2000.

4Gill, P.E., Murray, W., and Wright, M.H., Practical

Optimization (Academic Press, New York, 1981).

r=r (x)

Figure 1. Sketch of an axisymmetric resonator and a central blockage.

homconei narrow end

hom¢one, wide end

10 t2

Figure 2. The reduced pressure p / P0 at the ends of a homcone and acylinder at their corresponding

resonance frequencies for the same acceleration A = 10 3.

NASA/TM--2003-212019 6
OA

03

02

0;1

8

©

÷ horncone

cylinder

o,0 0.1, &2' 014 0:5.... 0.6 0:7 o,8 0,9

PilPo

Figure 3. The ratio between the amplitudes of the second and the first harmonics of the pressure wave, P2 / P],

is plotted as a function of the dimensionless amplitude of the first harmonic reduced pressure p] / P0 With

increasing frequency, these amplitudes are calculated for the narrow end of the horncone and at one end

of the cylinder.

0:9

OL7

0 ̧. ̧6  ̧

0:5

0.4

O:S

0_2

O,T

u i

A=O 0002

A--OUil,..,pw rd

l _ A_O:OOLd°wnwardsweeP J

i i

o

©

El

[]

[3

0 []

@ f Z]

o ++

1:35 i,4 i.45 lr 5

Figure 4. The effect of the frequency and the amplitude of acceleration on the dimensionless amplitude of the

first harmonic reduced pressure p] /P0 The pressure is calculated for the narrow end of the homcone for the

acceleration A = 2 x 10 4(crosses) and A = 10 3 (circles and squares).

NASA/TM--2003-212019 7
cc

(a) (b)

0.5 0:5

rr 0

O 02 0.4 016 0:8 1 0 0.2 0.4 0.6 0.8 1

X X

Figure 5. Optimized conical shapes: the compression ratios corresponding to the shapes in (a) and (b) reach the

values of 5.048 and 5.343 respectively. For shape parameters, see the text.

0.4

0.2

n.- 0

-0.2

-0.4

2ii5

_ 2

1.5

1

(a) (b)

0.4

02

rr- C

-O.f,

-0.4 = •

0.2 0,4 0,6 0.8 1 0 0.2 0.4 0.6 0.8 1

X X

(c)

0 5 1O

T

Figure 6. (a) The homcone shape in Lawrenson et al. 1 (b) Optimized homcone shape. For shape parameters,

see the text. (c) The pressure waveform at the narrow end for the homcone in Lawrenson et al. 1 (the solid line)

and that for the second optimized homcone (the dashed line) are shown.

NASA/TM--2003-212019 8
0.4

0.2

tax 0

-0.2

-0.4

(a)

0.4

0,2

rr 0

-0.2

-0.4

0 0.2 0,4 0,6 0.8

X

(c)

(b)

0 0,2 0.4 0,6 0,8

X

0 5 10

T

Figure 7. (a) The 1A-cosine shape as in Chun and Kim 3. (b) The optimized 1A-cosine shape. (c) The pressure

waveform at the narrow end for the 1A-cosine resonator in Chun and Kim 3 (the solid line) and that for the

optimized 1A-cosine shape (the dashed line) are shown.

o!5

0 ! _ !

0 2 4 6 8 10 12

T

Figure 8. Effect of the center blockage. The pressure waveforms at the narrow end of a cone resonator are

plotted for the different radii of the central blockage: Ro = 0 (solid line), Ro =0.01 (dashed line) and R0=0.02

(dotted line). The parameters corresponding to the plot are given by if2= 1.3132, A = 5 x 10 4, G = 0.01, 1 = 1.2.

NASA/TM--2003-212019 9
S

i

&5i

i i i i i ̧ i

0 2 4 6 8. 10 12

t

Figure 9. Pressure waveform at different frequencies: f2 = 1.2812 (the dotted line), f2 = 1.3068 (the dashed line)

and f2 = 1.3291 (the solid line), for a conical resonator with the cylindrical central blockage of radius Ro=0.02.

The rest of the parameters corresponding to the plot are given by _] = 5 × 10 4, G = 0.01, _ = 1.2.

NASA/TM--2003-212019 10
Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3, REPORT TYPE AND DATES COVERED

February 2003 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Nonlinear Resonant Oscillations of Gas in (_timized Acoustical Resonators and

the Effect of Central Blockage

& AUTHOR(S)

Xiaofan Li, Joshua Finkbeiner, Ganesh Raman, Christopher Daniels,

and Bruce M. Steinetz

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Resem'ch Center at Lewis Field

Cleveland, Ohio 44135 - 3191

WBS-22-708-28-15

8. PERFORMING ORGANIZATION

REPORT NUMBER

E----13723

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

Washington, DC 20546-0(101 NASA TM------2003-212019

AIAA-2003---0368

11. SUPPLEMENTARY NOTES

Prepared for the 4Ist Aerospace Sciences Meeting and Exhibit sponsored by the American Institute of Aeronautics and

Astronautics, Reno, Nevada, January 6-9, 2.003. Xiaofan Li, Joshua Finkbeiner, and Ganesh Raman, Illinois Institute of

Technology, Chicago, Illinois 60616; Christopher Daniels, Ohio Aerospace Institute, Brook Pro'k, Ohio 44142; and

Bruce M. Steinetz, NASA Glenn Research Center. Responsible person, Bruce M. Steinetz, organization code 5950,

216-433-3302.

12a, DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 71 Distribution: Nonstandard

Available electronicaJly at ['..ttp://_lt:.-s._rc.nasa.aov

_lttis publication is available from the NASA Center for AeroSpace Im'brmadon, 301-621-0390.

12b. DISTRNBUTIONCODE

13. ABSTRACT (Maximum 200 words)

Optimizing resonator shapes *or maximizing the ratio of maximum to minimum gas pressure at an end of the resonator

is investigated numerically_ It is well known that the resonant frequencies and the nonlinear stranding waveform in an

acoustical resonator strongly depend on the resonator geometry. A quasi-Newton type scheme was used to find

optimized axisymmetric resonator shapes achieving the maximum pressure compression ratio with an acceleration of

constant amplitude. The acoustical field was solved using a one-dimensional model, and the resonance t>equency shift

and hysteresis effects were obtained through an automation scheme based on continuation method. Results are presented

for optimizing three types of geometry: a cone, a horn-cone, and a half cosine-shape. For each type, different optimized

shapes were found when starting with different initial guesses. Further, the one-dimensional model was modified to

study the efl;cct of an axisymmetric central blockage on the nonlinear standing wave.

14. SUBJECT TERMS

Acoustic resonance; Cavity resonators; Oscillators; Sound generators; Standing waves

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

NSN 7540-01-280-5500

15. NUMBER OF PAGES

16

16. PRICE CODE

18, SECURITY CLASSiFiCATiON 19. SECURITYCLASSiFiCATION 20. LiMiTATiON OF ABSTRACT

OF THIS PAGE OFABSTRACT

Unclassified Uncl assifi ed

Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102