D-R74 652 SUPPRESSION OF FINITE-AMPLITUDE EFFECTS IN SLOSHING 1/1MODES IN CYLINDRICAL CAVITIES(U) NAVAL POSTGRADUATESCHOOL MONTEREY CA H Y SI DEC STUNCLASIFIE F/G12/1i W

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NAVAL POSTGRADUATE SCHOOLMonterey, California

DTI

BTHESISSUPPRESSION OF FINITE-AMPLITUDE EFFECTS

IN SLOSHING MODES IN CYLINDRICAL CAVITIES

'

by

C.-.Si Hwan Yum

December 1.983

C.2

Thesis Advisor: Alan B. Coppens

Approved for public release; distribution unlimited

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4. TI TLEI (and &ua*I. S. TYlet Of REPORT A PERIOD COVERED

Suppression of Finite-Amplitude Effects MatrsTeiDecember 1983

N?- in Sloshing Modes in Cylindrical Cavitiesi6. PERFORMING ORG. REPORT NUMBER

1AIJTH@NW 11. CONTRACT OR GRANT NUNUER(a)

Si Hwan Yum

SPERFORMUING ORGANIZATION N AMC AND ADDRESS W0. PROGRAM ELEMENT. PROJECT, TASK* . AREA A WOOK UNIT NUMBERS

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I&. SUPPI.EMENTARY NOTIES

Is. KEY WO01104 (CUo an ovovee ie Ifft ae pend sme Utlg , eek number)

non linear wave equation

)A pertubation expansion is formulated for the three dimension-

al, nonlinear, acoustic-wave equation with dissipative termdescribing the viscous and thermal energy losses encountered ina cylindrical cavity. The theoretical results show that non-linear effects in sloshing modes are strongly suppressed.

00 1473 gDo FINVO~ODSLT

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Suppression of Finite-Amplitude Effects in SloshingModes in Cylindrical Cavities

by

Si Hwan YumLieutenant Commander, Republic of Korea NavyB.S., Republic of Korea Naval Academy, 1973

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ENGINEERING ACOUSTICS

from the

NAVAL POSTGRADUATE SCHOOL-December 1983

Author:

Approved by:SThesis AdviTsor

Cha Egineering Acoustics Academic Committee

1 Dean of Science and Engineering

2

ABSTRACT

A pertubation expansion is formulated for the three-

dimensional, nonlinear, acoustic-wave equation with dissipa-

tive term describing the viscous and thermal energy losses

encountered in a cylindrical cavity. The theoretical results

show that nonlinear effects in sloshing modes are strongly

suppressed.

4%'T

AvilbliyCoe

AvAcein aaor

V Dist Aveil d

.3

TABLE OF CONTENTS

I. INTRODUCTION----------------------- ------------------ 6

II. THE NON-LINEAR WAVE EQUATION-------------------------- 7

A. GENERAL----------------------------------------- 7

B. APPLICATION TO THE CYLINDRICAL CAVITY--------------10

1. Symmetric Modes------------------------------- 11

2. Non-Symmetric Modes ---------------------- 14

JI. METHOD OF SOLUTION------------------------------------ 21

A. POWER SERIES METHOD------------------------------- 23

1. Coefficient --------------------- ------------ 23

2.------------------------------------------------ 27

3. Un(Y)----------------------------------------- 30

IV. CONCLUSIONS ------------------------------------- 314

LIST OF REFERENCES------------------------------------------ 36

INITIAL DISTRIBUTION LIST------------------------------------37

4'4e,

1%

LIST OF SYMBOLS

. phase velocity in the cavity

C1T~j~ C-674

CI effective phase speed associated with a standing wave

at resonance

Cspecific heat at constant pressure

k W/c, propagation constant associated with a standingwave

. peak Mach number of the driven standing wave

=P- P.=- oOIj acoustic pressure

.. instantaneous and equilibrium total pressure in the

field

_T absolute temperature in Kelvin

LA particle velocity

particle speed of the nth-order perturbation solution

infinitesimal-amplitude attenuation constant

=(Y + 1) /2 for a gas

=/?/Cv, ratio of specific heats for a gas

instantaneous and equilibrium densities

velocity potentialI;f, (angular) grequency at which the cavity is driven

(angular) frequency of a resonance

.4r

'"; .' 4 5<' - ..- ,,.'-- -. .. . . . .. . - . -.-- ..- . .

i -#1, " "I ' _ , . , . . \ " '"" ""'

I. INTRODUCTION

The topic of finite amplitude acoustic standing waves in

* sloshing modes of a cylindrical cavity is interesting

theoretically, but development of the subject has not been

extensive. The purpose off this research is to present the

results of a power series perturbation approach to the

problem.

44*~6

II. THE NON-LINEAR WAVE EQUATION

A. GENERAL

It is well known [11 that for Z//4kand M<<, where

measures the fractional loss per wavelength and M is the peak

Mach number of the source, loss terms and nonlinear terms in*2.'-.

the constitutive equations can be separately approximated with

*.. the help of linear, lossless acoustic relations. The force

equation appropriate for acoustical processes in systems for

,',-- which gravitational effects are unimportant is

..%

whereU = U is the particle velocity, Cis the instantaneous

density of the fluid, P is the instantaneous total pressure in

the field, and the operator Lsymbolically describes those

physical processes leading to absorption and dispersion. We

used two additional equations. The first is the equation of

state for a perfect gas

p~CrTK (2-1-a)

where r is a constant whose value depends on the particular

gas involved and TK is the absolute temperature in Kelvin.

The second is the continuity equation

' , ' ' ' - :k.:... .:.;..., .. ,..: ..,. ,..,.v ... ..- ,... , .-.... .... ,7.

*k ."

+ V.L I+S)UJ (2-1-b)

where s = (-)/ois the condensation at any point and

is the equilibrium density of the fluid. If we ignore

rotational effects, then

U~ (2-2)

* where is the velocity potential. Combination of Eas.(2-1)-

(2-2) and the neglect of terms of orders higher then M2 , M(" /z),

and ( / )2 yields a quadratically nonlinear wave equation,

7-C)

where Co2=(.r/:j,) (adiabatic), p=acoustic pressure,(=U.(

-" is the ratio of heat capacities, and

C F1 Lo-7 (2-4)

The left-hand side of Eq.(2-3) is the classical, linear wave

equation with losses pertinent to the system under study.

The right-hand side can be interpreted as a forcing function

consisting of a three-dimensional spatial distribution of phase-

coherent sources.

In a second-order perturbation theory, this volume forcing

function is obtained from the classical(first-order) solution?l of the acoustical problem. The second-order perturbation

solution PI+P 2 describes the non-linearities P2 resulting from

the self interaction of the classical solution Pl.

Higher-order perturbation solutions consider the interaction

of the nonlinear solution with itself, and the forcing function

is composed of products of both classical and non-linearly

generated terms. Thus, if a system is driven at frequency w,

the non-linear term in Eq.(2-3) will force the existence of

all integer multiples 11W of the driving frequeney and the full

solution must contain all harmonics of the input frequency.

In a closed cavity, each of those nonlinearly generated waves

whose frequency lies near the resonance frequency of a standing

wave of the cavity and whose assciated spatial function matches

that of the standing wave can be strongly excited (2)

As far as the author has been able to determine, there has

been only one previous study of this system published in the

open literature. This was by Maslen and Moore in 1956 (3)

Their approach resulted in a series expansion which did not

converge if the relevant nomal mode frequencies were integerally

related. Their interpretation of the quenching of the nonlinear

effect was based on the "scattering effect of the wall".

JK 9

We feel our interpretation based upon nonlinearally generated

volume sources stimulating the allowed standing waves of the

cavity is more accessible and informative. Further, the

mathematical approach developed herein appear to avoid any

difficulties in convergence. Their conclusion that high

amplitude monofreqlzency transverse oscillations can exist is

consistent with our finding.

B. APPLICATION TO THE CYLINDRICAL CAVITY

The circular cylinder is provided with a "point" source of

*, sound. By properly positioning the cylinder with respect to

the sound source it is possible to effectively drive the

enclosed air into various modes of vibration. The rigid-

walled cylinder forces the component of particle velocity

perpendicular to each cavity surface to vanish at the surface.

The resulting steady-state solution to the linear wave equation

in cylindrical coordinates is

P ~ ~ ~ kcm~o a s ) cc65 (ttmtT(2-5)

where Jn are the cylindrical Bessel functions and applicationof the boundary condition to the sides yields

.5L

7'.

where a is the radius of cylinder and J are the argument7".t..

of the exterema of the Bessel function.

The nomal mode frequency is dependent on and

('vrn[' ,- (2- 5- b )

The standing wave well be identified by the ordered integers

(n, m, ) describing its spatial dependence.

1. Symmetric Modes

The simplest waves in cylindrical coordinates are those*. -"

- that depend only on the distance r from Z-axis, the gradient

takes the form

(2-6)

where

* .r (2-6-a)

and k=w/c is the wave number or propagation constant.

The Laplacian becomes

.3 \7 ~ I(2-6-b)

The suitable trial solution appropriate for symmetric modes is

the linear solution(O, m, 0) of frequency w

11

-. - - - .-.•*~- .

The velocity potential for this limit can be defined as

-Uand is

Lit M c ~ ,]J

Because we assume that the surfaces of the davity are rigid

then at r=a the appropriate boundary condition is J,'=O.

Thus, we have -k= j=0"

To generate the second order solution, we first note that

P - zZ o~t

and

Next, the second derivatives of Eqs.(2-10) and (2-11) with

respect to time are

12

T (2-12)

and

Li ~ )Z H ~t - -(2-13)

Substituting Eqs.(2-2) and (2-13) into the first term of the

right-hand side of Eq.(2-3) yields

(7W XS-~ (2-iL )

!'-/ co5, zt T,(-4and the second term of right-hand side of Eq.(2- 3 ) becomes

4-i 3 (2-15)

since

'J or

CO T,(2-17)

With the use of Eqs.(2-14) and (2-15) we can write the right-

13

hand side of Eq.(2-3) as

,~X,

-Cos Z4~ (2-18)

and the appropriate inhomogeneous wave equation for the second

order perturbation solution is

i e,. _ )' M = _z -oj L-Z.C: ]-t X (2u aj 4rT.+ z 2 1 ri

" - +.,. _(Y ",)JO CeSzr 7. (2-19)

2. Non-Symmetric Modes

The non planar waves in cylindrical coordinates are

those that depend on the distance r from Z axis and the angle

" from X-axis. The gradient takes the form

(2-20)7.=- . (± )o

where

14

. . .. .... *. . ... . . .... .. ... ;

I.P 17,

and the Laplacian becomes

V i< + w (2-20-a)

The suitable trial solution appropriate to the forcing function

of frequency w for exciting the (n, m, 0) standing wave is

P, D (2-21)

* where Jn(r) is the Bessel function of the first kind and

rder n. The velocity potential is approximated by

Ja

Tr =-r (L4,) NC TW~~~t (2-22)

W1%

- and t4, is

.?.

Thus,

or

F +t tVCts Z'A (2-25)

15

,...

and

(2-26)

and the second derivative of Eqs.(2-25) and (2-26) with respect

-. -to time is

,IM%]z_ --- - (2-27)

and

± ~ ~ ~ C [iZ2 2J } -/l }2-8)Substituting Eqs.(2-26) and (2-27) into the first term of

Eq.(2-3) yields

16

(2-29

i °. . . .

'4,

- a or

.

IT I±tl -1+C- (2-30)

since

::r ' < % 7- Tr._, r--, )

In 6rder to solve the second term of Eq.(2-3), we can use

C V =x v, (1 I z .. (233

.4.:

mft~ =j; + o (2-35)

17

n .'.e."to solve the.second term of.,'3-" : "..we

can use-.-.:1 2.

j7.

"* since

T(2-36)

andiXTr I'p It,(2-37)

Thus, Substituting Eqs.(2-33) through(2-35) into the secondI

term of Eq.(2-3) yields

C os;C- 7TTtt .., v -7.v IJ + Tl , - TjO ±A) (2-38)

and

CC. + -T14;;5/0-14 TMj4)-D 1 (-9

Ci 18

I ° . o . • -

--C ,L-' .m ',.' " ', " -'" ' " " "d" "" " " ' " " " " " " """ "" " ' """"" " " " "" " "" "" . .

With Eqs.( 2-30), (2-38) and (2-39) we can get the inhomogeeneous

wave equation for the second order perturbat ion as

8X

-5..:i.~~~~ -;r -/ -T l- )

V'..-o

c, El+Tltll-

z +

7- +

/,0 ')T3 i.+ - 1 T1

- FT~Jt TT +(Ni

l,

..w % t, € .. .. . . .. .. .... .... . ..... ... ....... .. .-... .- '. .-

71 - . -7 .- -

If we let n=O in Eq(2-40), it reduces to Eq.(2-1 9 ), as it must.

Substituting n=l into Eq.(2-40) yields the equation with

forcing term resulting from the (1, m, 0) sloshing mode,

4%,

+ M

4.. CL T, ]7

(2-41)

' " ' "2 0

))" o,

" ,' '.'.I - J ._ LV ... ,". . . . '.ZY' .' '' . ) )C, & $'"" . j " - %h ", % "- ' "-,ij

m4.

III. METHOD OF SOLUTION

Recall that the equation with forcing term resulting from

the(l, m, 0) sloshing mode can be written as a function of

frequency,2'.

-- 4 z 0.

Co-_) (3-1)

and the left-hand side of Eq.(2-3) is

-4 V~ +- (3-1-a)

If the harmonics of the frequency at whichi the cavity is driven

are not close to any resonant frequencies, we can ignore the

lossy term. So, Eq.(3-1-A) can be written as

Let us assume that the solution of Eq.(3-1) can be expressed as

P:C -- w UR()+' I '57_ (3-3)

21

, o .21

:."' 'i++ - .

• , . - . . .° . - . . .- .- . *, , . * *- - -- v-,.. .. J , w * .- ~ -,

-. ' - and define

= ~V1 -~)CO C-(n (3-..4)

and

PV' M ,Co t 35

Combination of Eqs.(3-2) and (3-3) yields

I,,

-.. +

and

-p..°'

-- -C (- ' - ) TJ (3-7)

22-.

A. POWER SERIES METHOD

In order to solve the right-hand side of Eq.(3-7), we can

use the definition [4 ]

* ~ ~ ~ T(9T4V K+)~ (-I )9*(4& ztt('~

i ( -Z (3-8)

where3

1. Coefficient

a. Substituting V =L: n+2 into Eq.(3-8) yields

4+:4

or

> (3-10)

where

4 (3-11)

23

'. *, ". . " , - . - - - ,- . --. . - *- , . - . . . .. . . . . . . . .

b. Substituting ' = n+2,,t=n-2 into Eq.(3-8) vyelds

,-:7 :-0: 4:

"E ( . .. .T-~K) = (+i). - 1' Z -.a!!(31

(3-12)

- or

T Z i (3-13)

where

C~= ~-< ~ I(3-14)

c. Substituting Vhn+l,.(t~n-1 into Eq.(3-8) yields

-.. 1

(315

or

[-z. (3 -16 ):. .2(--o

24

,:: . ...::?::.""- ".:'-' " :-::.-,--.:-.': . --. ---.. -:.- .- .- . -- v ..- . .. -.. -..... -. . - . -. .- .. -.-

"-" where

( + (3-17)

d. Substituting V=?t=n into Eq.(3-8) yields

II

" )(V-z (3-18)

or(CC

where

A0m-.2')! (3-20)

e. Substituting V't:n+l into Eq.(3-8) yields

~44

7--A+Z-k - (7-)) -2-k

x (3-21)

-- k

25

%:,

or

where

f. SubstitutingV==n-l into Eq.(3-8) yields

00

.!..'.:4z Iez,.., (-, ka -2&) ! (-kiD

or.t .

where

26

* . * * . . . . . .°- . , . .

g. Substituting V--U-=n-2 into Eq.(3-8) yields

ZT-4!

or

I , where

2. Vn(y)

With the help of Eqs.(3-9) through(3-29) Eq.(3- 7 ) can

be written as

z IJ F\tw&~ j> dl J (3-30)

27

a,."

I--'. J

where! 5*4

* (3-30-a)

and

(3-30-b)

Now, let us assume that

'I0- L Q~1)(3-31).- C- t z

The first derivative with respect to y=:r can be expressed as

_______(3-32)

and the second derivative with respect to y=ir can be expressed

as

_____ V _ (3-33)

Substituting Eqs.(3-31) through(3-33) into the left-hand side

of Eq.(3-30) yields

4V.

28

* ° ~ '- . .° • • . .

-04 z?) (3-34)a

Eq.(3-34) must be equal the right-hand side of Eq.(3-30). Thus,

"- ±-i16"±+ - -

where k 0, 1, 2, 3 .(3-3)

If Eq.(3-35) is substituted into Eq.(3-36), then Eq.(3-35) can

be expressed as

.o,

ctA) t(-L) (iZ-I ) 0C;= - -G 4i DtI + AEbni)

(3-37)

29

"P;-"

"

The normal component of particle velocity is equal to zero on

the boundaries. From the application of the boundary condition

at the rigid boundary at r=a,

I"]" +* -Zj(3-38)

where

Substituting Eq.(3-36) into Eq.(3-38) yields

011 (-L ) (r+ 1 *""r+y -. ) 1 . . - .

0 (3-39)

3. Un(y)

With Eqs.(3-9) through(3-29), Eq.(3-6) can be expressed as

(3-40O)

30

k ' ";' -> t;,,,',.,':,.i:.-;.':'.:'.,,,-.-,;" -.:::.'::,-.><:; .-. :',-,.-,.-:-,- ' .". - ' . - •"- '-. " -." "-".-"'-

. ... ~~~~ ~ ~~ ~ ~~~ , . .. ."E-.' -,,, . -7,0 . . -.-. , ,,..,, . ,

W.4-3 . .7T -. 4 -7 -77.j -;6 - 7 -77,2- 7 .7 7 7

,4 where, as before, A Y /2-1/8 and a =

Now let us assume that

The first derivative with respect to y:4r can be expressed as

U10a DO kj -~ (3-'2

and the second derivative with respect to y=-.rcan be expressed

as

*" Substituting Eqs.(3-41) through(3-43) into the left-hand side of

Eq.(3-40) yields Q!fl4'>1z z( _ )JK i)4=0

Eq.(3-44) must equal the right-hand side of Eq.(3-40).

Thus,

314,:a

S

77:*

'P %

where -:1, 2, 3 .....

If Eq.(3-45) is substituted into Eq.(3-36), then Eq.(3-45) can

be expressed as

16

---z .,+ H,, E,,

,'. '

4.2.

(3-46)

32

.,. . . ....

. . . .

Application of the boundary condition at r=a yields

4 L .'' (3-4.L7)

where

Substituting Eq.(3-36) into Eq.(3-47) yields

+

(3-48)

33

'*--*.r*-..3 -3

t7,..,

.17-7

IV. CONCLUSIONS

Recall that we can compute from Eqs.(3-37) through(3-39)

and Eqs.(3-14), (3-17), (3-20). Thus, with the use of Eq.(3-31),

we can get Vn . From Eqs.(3-46) through(3-48) and Eqs.(3-11),

(3-20), (3-23), (3-26), (3-29), we can compute 11. Thus, with

the use of Eq.(3-14), we can get Un . Therefore we can compute

Vn(Y) and Un(y). Let us calculate the finite amplitude effects

resulting from the nonlinear distortion of a forced radial mode.

If we excite a(O, m, 0) mode and obtain the pressure at the

circumference, then Given this value of use

of Eqs.(3-14), (3-17), (3-20) give the quantities C, D, E.

From Eqs.(3-35) and (3-36) we can compute O . Thus, we can get

Vn(Y) from Eq.(3-31). Let us calculate the finite amplitude

effects resulting from the nonlinear distortion of a forced

sloshing mode. If we excite a(l, m, 0) mode and obtain the pre-

ssure at the circumference, then :-1'4, Given this value

of use of Eqs.(3-ll), (3-20), (3-23), (3-26), (3-29) give the

quantities B. E, G, H, 0. From Eqs.(3-45) we can compute

Thus, we can get Un(y) from Eq.(3-41). Substituting n=0 atradial modes into Vn(Y) and Un(y) yields V0 (y):U0 (y). Therefore,

with the use of Eq.(3-3), we can get the solution of Eq.(3-2).

A. RADIAL MODES(0, m, 0)

0L % MT 0 , (~1 kWt

.34

then

-" so that

B. SLOSHING MODESUl, m, 0)

'..< then

P2 -2 c s zt

so that

",V2

)/ I

..-.-

LIST OF REFERENCES'a..

1. Blackstock, D. T., "Convergence of the Keck-BeyerPerturbation Solution for Plane Waves of Finite Amplitudein a Viscous Fluid", J. Acoust. Soc. Am., Vol. 39, No. 2,pp. 411-413, February 1966.

2. Coppens, A. B., and Sanders, J. V., "Finite-AmplitudeStanding Waves Within Real Cavity", J. Acouft. Soc. Am.,Vol. 58, No. 6, pp. 1133-1140, December 1975.

3. Maslen, S. H., and Moore, F. K., "On Strong TransverseWaves Wi±hout Shocks in a Circular Cylinder", Journal ofthe Aeronautical Sciences, Vol. 23, No. 6, pp. 583-593,June 1956.

4. Abramowitz, M. and Stegun, I. A., "Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables",National Bureau of Standards Applied Mathematics Series,No. 55, pp. 255, 360, December 1965.

'...

-.'

,a-

36

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No. Copies

1. Defense Technical Information Center 2Cameron Station

Alexandria, VA 22314

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, CA 93943

3. Department Library, Code 61Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943

4. Professor Alan B. Coppens, Code 6lDz 5Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943

5. Professor James V. Sanders, Code 61Sd 5

Department of PhysicsNaval Postgraduate SchoolMonterey, CA 93943

6. Chief of Naval Research

800 Quincy St.Arlington, VA 22217

7. LCDR Charles L. Burmaster, Code 6lZrDepartment of PhysicsNaval Postgraduate SchoolMonterey, CA 93943

8. LCDR Chil-Ki Baek 5Republic of Korea Naval AcademyDepartment of PhysicsChin-Hae City, Seoul, Korea

9. The University of Texas at Austin 1Applied Research LaboratoriesAttn: D. T. BlackstockAustin, TX 78712

37

z

010

II

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00.

II WiIAv

& ;,

j,. -1,,. .1 t , it %0 D ; 1w7

*74

'A-