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THE PREDICTION OF THREE-DIMENSIONAL LIQUID-PROPELMT ROCKET NOZZLE ADMITTANCES

W. A. B e l l and B. T. Zinn

Georgia I n s t i t u t e of Technology A t l a n t a , Georgia 30332 NGL 11-002-085

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National Aeronautics and Space Administration Washington, D. C. 20546

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I 15. Supplementary Notes

Technical Officer, Richard J. Priem, NASA Lewis Research Center, 21000 Brookpark Road, Cleveland, Ohio 44135

116. Abstract

Croceo's three-dimensional nozzle admittance theory i s extended t o be applicable when the amplitudes of the combustor and nozzle o s c i l l a t i o n s increase o r decrease with time. a n a l y t i c a l procedure and a computer program f o r determining nozzle admittance values from the extended theory a r e presented and used t o compute t h e admittances of a family of l iqu id- propel lant rocket nozzles. Mach number and temporal decay coef f ic ien t s i g n i f i c a n t l y a f f e c t the nozzle admittance values. The theore t ica l predict ions a re shown t o be i n good agreement with ava i lab le experimental data.

An

The calculated r e s u l t s ind ica te t h a t the nozzle geometry, entrance

17. Key Words (Suggested by Authork))

combust i o n i n s t a b i l i t y Liquid rockets Nozzles

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Crocco's three-dimensional nozzle admittance theory is extended to be applicable when the amplitudes of the combustor and nozzle oscillations increase or decrease with time. program for determining nozzle admittance values from the extended theory are presented and used to compute the admittances of a family of liquid- propellant rocket nozzles. The calculated results indicate that the nozzle geometry, entrance Mach number and temporal decay coefficient significantly affect the nozzle admittance values. The theoretical predictions are shown to be in good agreement with available experimental data.

An analytical procedure and a computer

TABLE OF CONTEXVTS

ANALYSIS e e e e . e e e a . e 0 a e s e e e a e s e e e * e e m

Derivation of the Wave Equations e e . e . . . . . . . e e . , e . Method of Solution . . . e . . . . . . . e e . e . . . . e .

RESULTS AND DISCUSSION . . . . e . . . . . . . . . . . . . . . . . Admittances f o r Longitudinal Modes . . . . . . . . . . . e e e . Admittances f o r Mixed F i r s t Tangential-Longitudinal Modes . . e . Effect of Decay Coefficient upon Admittance Data . . . . . . . . .

SUMMARYAND'COUCLUSIONS . . o e e e . e . e e * e 0

F I G T m E S . . . . . . . . o e . . . . . . . . . . . . . . . . . . . . . . * ~

APPEXDIX e m e e . . e . e e e rn 0

REFERENCES e . . 9 . . e . e e e e 0

1

2

4 4 9 12

12

14 15 15 16 26

55

i

LIST OF ILLUSTRATIONS

Figure

1

2

3

4

5

6

7

8

9

10

A - 1

T i t l e

Typical Mathematical Model Used i n Combustion

I n s t a b i l i t y Analyses of Liquid Rocket Engines

Nozzle Contour

The Effect of Nozzle Half-Angle on the Theoretical

and Experimental Nozzle Admittance Values fo r

Longitudinal Modes

The Effect of Entrance Mach Number on the Theoretical

and Experimental Nozzle Admittance Values fo r Longitudinal Modes

The Effect of the Radii of Curvature on the

Theoretical and Experimental Nozzle Admittance

Values for Longitudinal Modes

The Effect of the Nozzle Half-Angle on the Theo-

r e t i c a l and Experimental Nozzle Admittance Values

fo r Mixed F i r s t Tangential-Longitudinal Modes

The Effect of Entrance Mach Number on the Theo-

r e t i c a l and Experimental Nozzle Admittance Values

f o r Mixed F i r s t Tangential-Longitudinal Modes

The Effect of the Radii of Curvature on the Theo-

r e t i c a l and Experimental Nozzle Admittance Values

fo r Mixed F i r s t Tangential-Longitudinal Modes

Effect of the Temporal Decay Coefficient on the

Theoretical Nozzle Admittance Values fo r Longitudinal

Modes

Effect o f the Temporal Decay Coefficient on the

Theoretical Nozzle Admittance Values for Mixed F i r s t

Tangential-Longitudinal Modes

Flow Chart for the Nozzle Admittance Computer Program

Page -

16 1.7

18

19

20

21

22

23

24

25

45

iii

LIST OF TABLE3

Table

1

A-1

A- 2

A-3 A-4 A-5

A-6

Ti t l e _.

smn Values of Transverse Mode Eigenvalues,

L i s t of Subroutines in the Computer Program Used t o Determine the I r ro t a t iona l Nozzle Admittance

Definition of FORllliAN Variables

Input Pwameters

Output Parameters

=s t ing of the Computer Program Used t o Determine

the I r ro t a t iona l Nozzle Admittance

Sample Output

13

27 28

32

33

34 44

i v

INTRODUCTION

The interact ion between the pressure osci l la t ions inside an unstable

rocket combustion chamber and the wave motion i n the convergent section of

the exhaust nozzle can have a s ignif icant e f fec t on the s t a b i l i t y charac-

t e r i s t i c s of the rocket motor and i s an important consideration i n analyt ical

studies concerned with the prediction of the s t a b i l i t y of liquid-propellant

rocket engines. This report i s concerned w i t h the investigation of t h i s

interact ion.

To determine the s t ab i l i t y of a liquid-propellant rocket engine, the

equations describing the behavior of the osc i l la tory flow f i e l d throughout

the rocket motor must be solved. To simplif'y the problem, i t i s convenient

t o analyze the osc i l la t ions i n the combustion chamber and the nozzle separately.

For such an analysis, the combustion chamber extends from the in jec tor face t o

the nozzle entrance as shown i n Fig. 1. All the combustion i s assumed t o take

place i n the combustion chamber where the mean flow Mach number i s generally

assumed t o be l o w , On the other hand, no combustion i s assumed to take place

i n the nozzle and i t s m e a n flow Mach number increases from a low value a t the

nozzle entrance t o uni ty a t the throat. Downstream of the throat the flow

i s supersonic and disturbances i n t h i s region cannot propagate upstream and

a f fec t the chamber conditions. Therefore, i n combustion i n s t a b i l i t y studies

it i s only necessary t o consider the behavior of t he osc i l la t ions i n the

converging section of the nozzle since only these osc i l la t ions can influence

the conditions i n the combustion chamber.

The nozzle i s t he boundary condition t h a t must be

sa t i s f i ed b y the combustor flow osci l la t ions a t the nozzle entrance.

as the r a t i o of the axial velocity perturbation t o the pressure perturbation

a t the nozzle entrance, the nozzle admittance can also be used to determine

whether wave motion i n the nozzle under consideration adds or removes energy

from the combustor osci l la t ions. Furthermore, t h i s boundary condition

influences the s t ructures and resonant frequencies of the natural modes of

the combustor under investigation.

Defined

To theoret ical ly determine the nozzle admittance, the equations

which describe the behavior of the waves i n the convergent section of the

exhaust nozzle must be solved. These equations have been developed by

Crocco2 and were solved numerically t o obtain admittance values for one-

and three-dimensional osc i l la t ions , 'Ilhese values were tabulated over a

wide range of frequencies and entrance Mach numbers f o r a spec i f ic nozzle

geometry. By applying the scaling technique developed i n Ref, 2, the 2 admittances of re la ted nozzles can be determined.

however, t ha t interpolat ion of the tabulated values can r e su l t i n l a rge

e r rors i n the predicted nozzle admittances; f'urthermore, the accuracy of

the scal ing procedure i s open t o question. I n addition, Crocco's theory

i s only applicable t o constant amplitude per iodic wave motions, and i n i t s

present form it cannot be applied t o cases where the amplitude of the

osc i l la t ions var ies i n time.

It was pointed out,

I n t h i s report , the equations needed for computing the nozzle admit-

tance a re presented and the i r solutions are outlined.

extended t o account for wave-amplitude var ia t ion with time,

theore t ica l predictions are shown and compared with available experimental

data.

nozzle admittance are presented i n p lo ts showing frequency dependence of the

r e a l and imaginary par t s of the nozzle admittance. "he e f fec ts of the decay

coeff ic ient are a lso assessed.

program which calculates nozzle admittance values along w i t h a program

l i s t i n g i s presented i n the appendix.

Crocco's theory i s

Typical

The e f fec ts of the nozzle geometry and chamber Mach nwber on the

A manual describing the use of the computer

A9 B, c variable coefficients defined below Eq. (14) C

% Y % Y % un i t vectors

Jm

nondimensional speed of sound, c*//"g

i a Bessel function of the f irst kind of order m a f'unction having the following space and time dependence: Y e 9-Q

M Mach number a t the nozzle entrance

2

r

r cc

c t r

S

snm

Z

Y

5

0

h

P l-

Cp

m

4f w

number of mode diametral nodal l i nes

number o f mode tangential nodal l i nes

nondimensional pressure, p*/s;

nondimensional velocity, q*/E;

nondimensional radius r*/rE

nondimensional radius of curvature at the nozzle entrance,

r */r* nondimensional radius of curvature a t the nozzle throa t ,

4

cc c

.,& nondimensional frequency , uHr;/z*

the nth root of the equation

dJm(4 = o dx

nondimensional time, t*,*/r; nondimensional ax ia l veloci ty component > u*/E; nondimensional r ad ia l veloci ty corrponent , * / E ; noxdimensional tangential velocity component > w3c/c*

i r ro t a t iona l spec i f ic nozzle admittance defined i n Eq. (13)

0

0

I I U - - u * --

y = p*c* - = ypc -

nondimensional a x i a l coordinate, z*/r*

r a t i o of spec i f ic heats

a f’unction used to compute the nozzle admittance; defined below

Eq* (13)

PI* P I

C

tangent ia l coordinate, radians nozzle half-angle, degrees

nondimensional temporal decay coeff ic ient , h*r;/E;

nondimensional density, p*/p*

a flxnction used t o conpute the nozzle admittance; T = 1/5 nondiiensional steady s t a t e velocity potent ia l , Cps/E*r*

a function describing the v-dependence of the r a d i a l velocity

per turb a t i on

nondimensional steady s t a t e stream function, $c(cp) q(cp) r nondimensional frequency, o*r++/c;

0

o c

2

3

Subscripts:

C evaluated at the chamber w a l l i

0 stagnation value

r t h evaluated a t the nozzle t h r o a t

w evaluated at the nozzle w a l l 4 vector quantity

imaginary par t of a complex quantity

r e a l p a r t of a complex quantity

Superscripts : I

perturbation quantity

- steady s t a t e value

-E dimensional quantity

ANALYSIS

Derivation of the Wave Equations

The equations used by Crocco' t o compute the nozzle admittance will be developed from the conservation equations.

matically t rac tab le and yet physically meaningful, the following assumptions

were employed.

To keep the problem mathe-

(1) The nozzle f l o w i s a calor ical ly perfect gas consisting of

a s ingle species.

(2) Viscosity and heat conduction are negligible.

(3) The steady s t a t e flow i s one-dimensional.; t h i s a s s q t i o n implies 'chat the nozzle is slowly converging.

(4) The amplitudes of the waves are small so tha t only l i nea r

terms i n the perturbed quantit ies need t o be retained i n the

conservation equations.

(5) The osci l la t ions are assumed t o be i r ro ta t iona l . Using these assumptions, the equations of motion i n nondimensional

form become

Continuity

ap + v.(pq) = 0 at

4

Mome ntwn

-2 1 + -&q = - ;vp

2 and, from the isentropic conditions, c = p / p and p = py.

To obtain the l inear ized wave equations, the dependent variables

a re expressed i n the following form:

I I 4 i l = q + q , p = p + p f , p = p + p

Subst i tut ing these expressions i n t o Eqs. (1) and ( 2 ) , neglecting a l l non-

l i nea r terms involving primed quant i t ies , and separating the resu l t ing system

of equations in to a s e t of steady s t a t e equations and a s e t of unsteady

equations y i e ld the system o f steady s t a t e equations:

-2 -y - 1 y - 1 - 2 - -y V * ( q = 0; c = p =1-- 2 q ; P ' P

and the following system of unsteady l inear equations tha t describe the wave

motion :

I

+ V * ( q p ' -b 6q') = 0 3 3

I -2 I P = c P

To s i m p l i e the application of the boundary conditions at the nozzle

w a l l s , these wave equations are solved i n the orthogonal coordinate system

shown i n Fig. 1. I n t h i s coordinate system the steady s t a t e velocity

poten t ia l cp replaces the axial coordinate z, the steady s t a t e stream fbnction

$ replaces the r a d i a l coordinate r and the angle 8 i s used t o denote azimuthal

variations. Using t h i s coordinate system the veloci ty vectors can be expressec

as follows :

5

Using the def ini t ions of the steady s t a t e veloci ty po ten t i a l and stream

function for a one-dimensional mean flow, it can be shown 2 t h a t

Rewriting Eqs. (5) and (6) i n the (cp ,$,e) coordinate system yields the

following system of equations2:

Continuity

Momentum

cp - component

$-component

8 -component

-\- 1 f

YP + +G(%) = o - + F (9)

Equations (7) through (U) const i tute a system of f ive equations i n the f9ve

unknowns -- PI/;, U'/q, v'/rpq, rw', and p f / y p e by the method of separation of variables and the solutions are

These equations are solved

6

where

i ( u , - i h ) t cos m0e f o r standing waves

for spinning waves

These solutions ident ica l ly sa t i s fy the momentum and energy equations.

Substi tuting these solutions into Eq. (8) and eliminating variables give

the following d i f f e ren t i a l equation for the f'unction @ :

The f ine t ion @ can be related t o the specif ic acoustic admittance by 2 the formula

I y = y p c - = -- U - YPCS

P f i26 + i ( w - i h )

7

where 5 = - - '@ d i f f e r e n t i a l equation for 5 is derived:

Using the def in i t ion of 5 and Eq. (12) , the following @ dcp"

where

2- 2 2 smc

2 i ( w - i h ) - Y - 1: dc2] c = [(w - i h ) - - -

2 -2 acp C r

W

Equation (14) i s a complex Riccati equation which must be solved

numerically t o obtain 5 . Once the value of 5 i s determined a t the nozzle

entrance, the nozzle admittance can be computed d i r ec t ly from Eq. (13) . Inspection of Eq. (14) shows t h a t the value of 5 depends upon i t s coeff ic ients

A, B, and C which i n turn depend upon w, A , Sm, and the space dependence of

q and c i n the nozzle. The behavior of and

once the value of Y and the nozzle contour a re specified.

- i n the nozzle can be computed

To determine 5 for given values of w, A , Sm and y and a spec i f ic

nozzle contour, Eq. (14) must be integrated numerically. A major d i f f i c u l t y

which can occur during t h i s integrat ion is t h a t 5 becomes unbounded whenever

@ approaches zero, which causes numerical d i f f i c u l t i e s i n the integrat ion z scheme. Crocco and Sirignano

Mach numbers and high values of w/Sm.

they developed asymptotic solutions for 5 .

noted tha t t h i s phenomenon occurred for low A t these Mach nunibers and frequencies

Instead of using the asymptotic solut ion, an exact numerical solution

i s obtained i n t h i s study.

dependent var iable

The problem i s resolved by introducing a new

8

A s m approaches zero and the magnitude of 5 becomes large, T becomes s m a l l .

Introducing the def ini t ion of T i n to Eq. (14) gives the following Riccati

equation for T

A t those regions where 5 becomes unbounded, Eq. (15) i s integrated instead of Eq. (14) . Method of Solution

To obtain the nozzle admittance from Eq. (l3), values of 5 and T are

computed by numerically integrat ing Eq. (14) or (15). To evaluate the coeffi-

c ients A, 8 , and C , a d i f f e ren t i a l equation tha t describes the variations of

the steady s t a t e velocity i n the subsonic portion of the nozzle must be derived.

Differentiating the continuity equation

- 2- - 2 - P r q = pth rth qth = constant

- 2 - 2 where qth = cth = 2 / ( y + l), and using Eq. (4) y ie ld the following d i f f e r e n t i a l

equation

- dt2 dr

= -

Using Eq. (17) and the specified nozzle contour i n terms of r (z ) , the quantity

di/dcp can be obtained from the relat ionship

dq dr d i2 dq dr dz -=---=2-- dcp dr dz dcp dr dz

2

-2 Once q i s known the corresponding value of c2(cp) can be obtained by

use of Eq. (4) . To evaluate dr/dz i n Eq. (18), the nozzle contour shown i n Fig. 2 i s used. S ta r t ing a t the combustion chamber the contour i s generated

by a c i rcu lar arc of radius rcc turned through an angle el, the nozzle

half-angle. This arc connects smoothly t o a s t r a igh t l i n e which i s inclined

9

at an angle t o the nozzle axis. This s t r a i g h t l i n e then joins with another

c i r cu la r a rc of radius r th roa t . Using t h i s nozzle contour, i n regions I, I1 and I11 of Fig. 2

which turns through an angle and ends at the c t

r + r - r c t t h

Ut i l iz ing the appropriate e q r e s s i o n for dr/dz, Eq. (18) can now be solved

simultaneously with Eq. (14) o r (15) t o determine the nozzle admittance.

The numerical integrat ion of these equations must start a t some

i n i t i a l point where the i n i t i a l conditions are known. Since the equation

for 6 i s singular a t the throat2, the integrat ion is i n i t i a t e d a t a point

t h a t i s located a short distance upstream of the throat.

conditions are obtained by expanding the dependent variables i n a Tqrlor

s e r i e s about the throat . To obtain this TaJrlor se r ies , i t s coeff ic ients

The needed i n i t i a l

must be evaluated a t the throat where cp = 0.

These coeff ic ients a re evaluated by subs t i tu t ing the se r i e s

i n t o Eq. (14) and taking the l i m i t as cp 4 0. The r e su l t s a r e

cO 6, = 6(0) = - BO

where

10

i(w - ih) rt hr c t

- (6 + Y)]

The following relations are used in the evaluation of the above quantities:

Once 6, and 5 , are known, the initial condition at fp = cpl is obtained from

the expression C(cp1) = 5, + <,CP,.

The numerical solution is obtained by use of a modified Adams predictor-corrector scheme, and employing a Runge-Kutte scheme of order four to start the numerical integration. Initially, Eqs. (14) and (18) are integrated to determine 5; if the magnitude of 5 exceeds a specified value at which numerical difficulties can occur, the integration of Eq. (14) is terminated. Using the value of 5 at that point, T is computed and the

11

integrat ion proceeds using Eq, (15) e Simi la r ly , should the m@@&tude of T

become excessively la rge , the integrat ion of Eq, (15) is terminated, I: i s computed from the value of T at t h a t point, and the integrat ion 'proceeds

using Eq. (14) reached.

FORTRAN V for use on the UNIVAC 1108 computer and it is presented i n the

Appendix.

This process i s repeated u n t i l the nozzle entrance i s

A computer program u t i l i z i n g t h i s procedure has been wri t ten i n

RESULTS AND DISCUSSION

Using the previously mentioned computer program, theore t ica l values

of the r e a l and imaginary par t s of the nozzle admittance have been computed

for several nozzle configurations having contours similar t o the one presented

i n Fig. 2. In these computations the r a d i i of curvature, r assumed t o be equal, The admittance values are presented as f'unctions of the

nondimensional frequency S i n Figs. 3 through 9 where they are compared with

available experimental data obtained from Ref. 3. In these figures, the

frequency has been nondimensionalized by the r a t i o of t he steady s t a t e speed

of sound a t the nozzle entrance t o the chaniber radius r .

and ret, are cc

C

Admittances for Longitudinal Modes

Longitudinal-type i n s t a b i l i t i e s i n general occur i n the range of S

from 0 t o approximately 1.8 which i s i n the v i c in i ty of the cutoff frequency

of the first tangent ia l modes.

mode i s Sm' where Sm is the transverse mode eigenvalue and the sub-

s c r i p t s m and n respectively denote the number of diametral nodal l i n e s and

the number of tangent ia l nodal l ines . Values of S, are given i n Table 1 f o r

several values of m and n. For longitudinal modes good agreement ex is t s between the experimental

The cutoff frequency of a par t icu lar transverse

and theore t ica l values of the r e a l and imaginary par t s of the admittance as shown i n Figs. 3 through 5. presented i n Fig. 3 for a nozzle with an entrance Mach m b e r M of 0.,08 and

= 0.44.

The ef fec t of changing the nozzle half-angle i s

The data indicate tha t increasing €I1 increases the frequency ' c c k a t which the r e a l and imaginary par t s of the admittance a t t a in maximum values.

These data also indicate tha t t he assumption of a one-dimensional mean flow

12

Table 1, Values of Transverse Mode Eigenvalues; Sm

Transverse Wave Pattern ~

Longitudinal

F i r s t Tangential (1T)

Second Tangential (2T)

F i r s t Radial (lR) Third Tangential (3T) Fourth Tangential (4T) F i r s t Tangential, F i r s t Radial (lT,XR)

F i f th Tangential (5T) Second Tangential, F i r s t Radial (2T,lB)

Second Radial (2R)

m

0

1

2

0

3 4

-

1

5 2

0

n

0

0

0

1

0

0

1 0 1

2

I_

used i n the development of the theory appears to be valid. Even

smn 0

1 8413

3 00543

3 88317 4.2012

5.3175 5 e3313 6.4154 6.7060 7 -0156

for nozzles

with half-angles as high as 45 degrees, f o r which i t has been shown tha t the

mean flow i s two-dimensional, the experimental and theoret ical nozzle admit-

tance values are i n good agreement.

4

Examination of Fig. 4 shows tha t t he entrance Mach number M has a

s igni f icant e f fec t on the admittance values for el = 15 degrees and rcc/rc =

0.44.

influence of the entrance Mach number, and for = 45 degrees vaxiations i n

M has l i t t l e e f fec te3

of curvature for a nozzle with M = 0.16 and

The data presented i n Figs. 3 through 5 show t h a t f o r longitudinal

However, increasing the nozzle half-angle appears to decrease the

The dependence of t he nozzle admittance upon the radius

= 30 degrees i s shown i n Fig. 5.

modes the r e a l p a r t o f the nozzle admittance i s always posi t ive. A s indicated

by Crocco1’2 posi t ive values of the r e a l p a r t of the nozzle admittance imply

t h a t the nozzle removes acoustic energy from the combustor wave system which

implies t h a t the nozzle exerts a s tab i l iz ing influence upon the chamber

osc i l la t ions .

I n combustion in s t ab i l i t y analyses of liquid-propellant rocket motors, it i s of ten assumed tha t the nozzle i s short. This assumption implies t ha t

the nozzle length and throat diameter a re much smaller than the chamber length

and diameter s o tha t the wave t r ave l time i n the nozzle i s much shorter than

the wave t r a v e l time i n the chmber. For a short nozzle the r e a l and imaginary

p a r t s of the admittance are independent of frequency and are given by the

expressions 5

Yr = - ' " ' M ; y i = O 2

These theore t ica l short nozzle admittance r e s u l t s do not agree with the

r e s u l t s obtained for typ ica l l iqu id rocket nozzles presented i n Figs. 3 through 5. The disagreement i s especially evident for nozzles with low

values of el, which imply tha t the nozzle i s long, and for high values of

S where the wave length of the osc i l l a t ion becomes of the same order of

magnitude as a charac te r i s t ic nozzle dimension.

Admittances f o r Mixed F i r s t Tangential-LongLtudinal Modes

The mixed f i rs t tangential-longitudinal modes a r e those three-

dimensional modes which ex i s t 'between the cutoff frequencies of the f i rs t tangent ia l (S 2c 1.8) and second tangent ia l (S 21 3.0) modes. Theoretical

and experimental nozzle admittance data f o r these modes a re presented i n

Figs. 6 through 8. I n Fig. 6 the influence of the nozzle half-angle on the admittance

values i s shown.

agreement and they indicate tha t increasing el increases the frequency a t

which the r e a l and imaginary parts of the admittance reach maximum values.

The e f f ec t of Mach number on the admittance values i s presented i n

Fig. 7 for B1 = lfs degrees and rcc/rc = 0.44. especial ly s ign i f icant a t the higher frequencies. However, as shown i n Ref. 3, increasing the nozzle half-angle decreases the dependence of the

admittance values on the Mach nmiber.

curvature on the admittance values i s presented i n Fig. 8.

The theore t ica l and experimental r e su l t s are i n good

Mach number e f fec ts a re

The e f f ec t of changing the r a d i i of

The r e s u l t s presented i n Figs. 6 through 8 show t h a t f o r mixed

first tangential-longitudinal modes the r e a l p a r t of the nozzle admittance

can be negative which means t h a t the nozzle radiates wave energy back i n t o

the combustor; t h i s process exerts a destabi l iz ing influence on the osc i l -

l a t ions i n the chamber.*

dimensional modes and, as shown by Crocco,2 t h e i r cause can be traced t o

the t e r m involving S, i n Eq. (12).

These negative values occur only fo r three-

For longitudinal modes, fo r which Sm

14

i s zero, the r e a l pa r t of the nozzle admittance i s always posi t ive, and fo r

those modes the nozzle always exerts a s tab i l iz ing influence upon the combustor

osc i l la t ions e

Effect of Decay Coefficient upon Admittance Data

The nozzle admittance theory has been modified t o include the e f fec ts

o f a temporal decay coeff ic ient , A . Ty-pical results are shown i n Figs. 9 and 10 for values of h of -0.05, 0 , and 0.05. These resu l t s indicate tha t

varying h af fec ts both the r e a l and imaginary parts of the admittance, There-

fore , the decay coeff ic ient should b e included i n the nozzle admittance

computations when the osc i l la t ions a re not neutral ly s table .

SUMMARY AND CONCLUSIONS

The equations necessary t o determine the nozzle admittance fo r one-

and three-dimensional osc i l la t ions have been developed. The ana ly t ica l

approach used i n solving the nozzle wave equations i s outlined and employed

t o obtain nozzle admittance data for typ ica l nozzle configurations. These

data show the dependence of the nozzle admittance values upon nozzle geometry,

nozzle Mach number, mode of osc i l la t ion , and the temporal damping coefficient.

The r e su l t s ea32 be summarized as follows f o r longitudinal and mixed

f i rs t tangential-longitudinal modes.

ing the nozzle half-angle and Mach number or b y decreasing the throat and

entrance rad i i of curvature decreases the frequency dependence of the nozzle

admittance, Good agreement exis ts between the theore t ica l predictions and

available experimental data . However, the nozzle admittance values fo r typical

l iqu id rocket nozzles are not i n agreement with the values obtained from short

nozzle theory.

Decreasing the nozzle length by increas-

Including the e f fec ts of a temporal damping coeff ic ient i n the nozzle admittance computations changes the admittance values. Therefore,

when the osc i l la t ions are not neutral ly s table , the temporal decay coefficient

should be accounted for i n the computations.

cd 4-1 0

16

I I I

I

I

Figure 2. Nozzle Contour

A 2

Y i

I

0

- I

-2

Figure 3. The Effect of Nozzle Half-Angle on the Theoretical and Experimental Nozzle Admittance Values for Longitudinal Modes

I

0 0 0. 0.4 0.6 1. 1.2 1.

Figure 4. The Effect of Entrance Mach Number on the Theoretical and Experimental Nozzle Admittance Values for Longitudinal Modes

Figure 5. The Effect of the Radii of Curvature on the Theoretical and Experimental Nozzle Admittance Values for Longitudinal Modes

20

2. 24 2.4 2.8 3.0 S

1 I I I I I

Figure 6. The Effect of the Nozzle Half-Angle on the Theoretical and Experimental Nozzle Admittance Values fo r fixed First Tangential- Longitudinal Modes

21

I I I I

1.8 2.0 2.2 3.0 - I

S

Figure 7. The Effect of Entrance Mach Number on the Theoretical and Experimental Nozzle Admittance Values for Mixed First Tangential-Longitudinal Modes

22

Y. I

S Figure 8. The Effect of the Radii of Curvature on the Theoretical and

Experimental Nozzle Admittance Values f o r Mixed First Tangenti al-Longitudinal Modes

23

2

'r

1

0

'i

2

1

0

-1

I

0.6 0.8 1.0 1.2 1.4 1.6 1.8 S

0.6 0.8 1.0 1.2 1.4 1.6 1.8 S

Figure 9 . Effect of t he Temporal Decay Coefficient on the Theoretical Nozzle Admittance Values f o r Longitudinal Modes

24

r Y

1

0

-1

S

2

1

0

-1

- - - -

1.8 2 .o 2.2 2.4 2.6

6 1.8 2 .o 2.2 2.4 2 S

Figure 10. Effect of the Temporal Decay Coefficient on the Theoretical Nozzle Admittance Values for Mixed F i r s t Tangential- Longitudinal Modes

25

APPENDIX

COMPUTER PROGRAM USED TO DETERMINE THE IRROTATIONAL NOZZLE ADMITTANCE

The computer program f o r calculat ing the i r ro t a t iona l nozzle a d m i t -

tance from Crocco's theory which is extended t o account f o r temporal damp-

ing is wri t ten i n FORTRAN V interpret ive language compatible with the

UNIVAC 1108 machine language compiler. This program consis ts of seven

routines - the main or control program and s i x subroutines.

the routines are l i s t e d i n Table A - 1 i n sequential order. The FORTRAN

symbols used i n these routines and t h e i r def in i t ions are presented i n Table

A-2 i n alphabetical order. The input parameters necessary f o r the admit-

tance computations must be specified i n the main program and a re l i s t ed i n

Table A-3. The output parameters and t h e i r def ini t ions are l i s t e d i n Table

A-4. A detai led flow chart of the computer program is shown i n Fig. A-1,

and the program l i s t i n g and sample output are presented i n Tables A-5 and

A- 6, re spec t ively . This computer program has been writ ten t o predict nozzle admittances

2

The names of

f o r nozzle contours shown i n F ig . 2 . The run time required depends upon

the number of admittance values desired and the nozzle length.

40 admittance values a t d i f fe ren t frequencies f o r the nozzles investigated

i n t h i s study, one t o two minutes of run time on the UNIVAC 1108 computer

are required.

To obtain

26

Table A-1. L i s t of Subroutines i n the Computer Program Used t o Determine the I r ro t a t iona l Nozzle Admittance

Subroutine De s c r i p t ion

MAIN

NOZADM

RETZ

RKZDIF

RKTDIF

ZADAMS

TADAMS

Specif ies the nozzle geometry and operating conditions i n the converging sec t ion of the nozzle

Specifies i n i t i a l conditions a t the throat, computes the f i n a l nozzle a d m i t - tance values, and contains a l l output formats

Uses the Runge-Kutta of order four t o obtain i n i t i a l values f o r the modified Adams integrat ion routine

Computes the d i f f e r e n t i a l element i n the converging sect ion of the nozzle used t o solve Eq . (14) Computes the d i f f e r e n t i a l element i n the converging sect ion of the nozzle used t o solve Eq . (15) Numerically in tegra tes Eq. ( 14) using the modified Ad- numerical integrat ion scheme

Numerically in tegra tes E q . (15) using the modified Adams numerical integrat ion scheme

Table A-2. Definit ion of FORTRAN Variables (Page 1 of 4)

Variable Definit ion

A

A ( 5 ) AF ANGL8

APR

B I

BR

BO1

BOR

B 1 1

BIR

C

C I

CM

COR( 5

CR

C O I

C OR

C 1 I

C l R

DP

( 5)

DR

DU

R e a l coeff ic ient A of Eqs . (14) and (15) Coefficients of the Runge-Kutta formulas of order four

Nondimensional temporal damping coeff ic ient h

Nozzle half-angle, degrees

Derivative of the coeff ic ient A evaluated a t the throa t

Imaginaq p a r t of the coeff ic ient B i n E q s . (14) and (15) R e a l par t of the coeff ic ient B i n E q s . (14) and (1-5) Value of B I a t the throat

Value of BR a t the throa t

Derivative of B I evaluated a t the throat

Derivative of BR evaluated a t the throat

Nondimensional speed of sound squared, c

Imaginary p a r t of the coeff ic ient C i n Eqs . (14) and ( 15) Mach number a t the nozzle entrance

Formula f o r the corrector i n the modified Adams in te - gra t ion routine

Real par t of the coeff ic ient C i n E q s . (14) and (15) Value of C I at the throat

Value of CR a t the throat

Derivative of C I evaluated a t the throa t

Derivative of CR evaluated a t the throa t

Integration stepsize

Derivative used i n the corrector formula i n the modified Adams integrat ion routine

Derivative of t he loca l w a l l radius with respect t o axial distance

Derivative of the nondimensicnal veloci ty i2 with respect t o the w a l l radius r

Increment of the nondimensional frequency w

2

DWC

28

Table A-2, Definition of FORTRAN Variables (Page 2 of 4)

V a r i abie De f i n i ti on

F2

GAM

G ( 5 )

H

I

IP

J

JOPT

K N Nu

NWC

P

PARG

PHI1

PHIR

Derivative used i n the modified Adams integrat ion scheme

Constant given as v y p evaluated a t the nozzle entrance

Derivative used i n the Runge-Kutta method

Lutnped parameter determined by the conditions a t the throat

Lumped parameter determined by the conditions a t the throat

Ratio of spec i f ic heats y

Dependent variable i n the Runge-Kutta integrat ion routine

Integrat ion s tepsize

Integer counter

Integer constant. If IP = 0 the nozzle admittance i s output. pressure osc i l la t ion m e output along the length of the nozzle

If I& = 2, the integrat ion of Eq. (15) for T i s complete

= 1: Eq. (15) fo r T i s integrated = 2: Eq. (14) f o r 5 i s integrated

Integer variable

= 1: Eq. (15) for T i s integrated = 2: Eq. (14) fo r 5 i s integrated

Integer variable

Integer variable

Number of d i f f e ren t i a l equations t o be solved by the Runge-Kutta or the modified Adams integrat ion routine

Number of frequency points

Value of the steady s t a t e veloci ty poten t ia l

Phase of the pressure osc i l l a t ion i n the nozzle

Imaginary par t of m Real p w t of @

I f IP f 0 the amplitude and phase of the

Table A-2, Definit ion of FORTRAN Variables (Page 3 of 4)

Variable De f i n i t ion

Imaginary pa r t of the pressure osc i l l a t ion

Magnitude of the pressure osc i l l a t ion

Real par t of the pressure osc i l l a t ion

Predictor formula for the modified Adams integrat ion routine

Q

&BAR R

RCC

RCT

RHO RT

R 1

R2

SRTR

SVD

SVNR

SYI

SYR

T

TDN

T I

TMAG

Nondimensional steady s t a t e veloci ty 6 Local wall radius r

Ratio of the radius of curvature a t the nozzle entrance t o the radius a t the nozzle entrance

Ratio of the radius of curvature at the throat t o the radius a t the nozzle entrance

Nondimensional, steady-state density

Nondimensional throat radius

Nondimensional radius a t the entrance to Section 2 of the converging portion of the nozzle

Nondimensional radius a t the entrance to Section 3 of the converging portion of the nozzle

Constant give as =/re

Imaginary pa r t of the spec i f ic admittance y

Real pa r t of the spec i f ic admittance y

Nozzle half-angle, i n radians

Inverse of the square of the magnitude of 5 Imaginary p a r t of T

Magnitude of 7

Derivative of T I with respect to cp TPI

Table A-2, Definition of FORTRAN Variables (Page 4 of 4)

Variable Definition

TPR

TR TZ

T2

U

uz W

wc X

y(5)

Y I

YR

ZDN

Z I

ZMAG

ZPI

ZPR

ZR

Z O I

ZOR

Z 1 I

ZLR

22

Derivative of TR with respect t o cp

Real p a r t of T

Value of cp a t the nth integrat ion point

Square of the magnitude of T

Steady s t a t e veloci ty squared, c2 Dependent variable i n the Runge-Kutta integrat ion scheme

Nondimensional frequency S

Nondimensional frequency w

Value of cp a t the nth integrat ion point

Dependent variable used i n the modified Adams integrat ion scheme

Imaginary par t of the i r ro t a t iona l nozzle admittance defined by Crocco i n Ref. 2

Real part of the nozzle admittance defined by Crocco in Ref. 2

Inverse of the square of the magnitude

Imaginary par t of 5 Magnitude of 5 Derivative of Z I with respect t o cp

Derivative of ZR with respect t o cp

Real par t of 6 Value of Z I a t the throat

Value of ZR a t the throat

Value of ZPI a t the throat

Value of ZPR at the throa t

Square of the magnitude of f :

of 6

Table A-3, Input Parameters

Var i ab l e Definition

GAM

CM

SVN

Ratio of spec i f i c heats, y

Mach number a t the nozzle entrance

Nth root of the equation dx = 0. Corresponds t o

are given i n Table 1 fo r various

dJ@

. Values of S smn mn acoustic modes

WC I n i t i a l value of tu

DWC Increment of frequency

?XWC Number of frequency points desired

ANGLE Nozzle half-angle, degrees

RCT Radius of curvature a t the throat nondimensionalized

RCC Radius of curvature a t the nozzle entrance nondimens ional-

with respect t o the chamber radius

ized with respect t o the chamber radius

IP

AF

= 0: nozzle admittances are printed # 0:

Temporal damping coeff ic ient h

presswe magnitude and phase are printed a t each point along the nozzle

32

Table A-4. Output Parameters

Variable Definition

wc YR

Y I

W SYR

SYI

Nondimensional frequency, w

Real pa r t of the admittance as defined by Crocco i n Ref. 2

Imaginary par t of the admittance as defined by Crocco i n Ref. 2

Nondimensional frequency

Real p a r t of the spec i f ic admittance y

Imaginary par t of the specif ic admittance y

33

Table A-5* Listing of the Computer Program Used to Determine the Irrotational nozzle Admittance (Page 1 of 10)

24+ 25+

28*

34

Table A-5, Continued (Psge 2 of LO)

2* 3+ 4* 5* 6* 7* 8* 9*

lO* 11s 12* 13* IS* 15* 16* 17* le* l9* 20* 21* 22* 23* 2Q* 25s 26* 21* 28* 29, 30* 31* 32* 33+ 34* 35* 36*

38* 39* 40* Itl* 42* 43* Yb* 45* Q6* 47* U8* 49* 50 * 51 * 52* 53* 5S* 55* 56* 57* 58* 59* BO+

. 3 r +

35

Table A-5, Continued (Page 3 of 10)

69r 76 * 71* 7 2 * 73+ 7LI 9 7 5 * 7 5 , 7 7 * 7t. * 79* BG + 6l* L12 * 85* 84* 85 * 8b* 67+ dS* B O * 9f, 4 91* 9;* 93, 94 4

95s 9 5 e 97* 9 b + 99*

1004 I91* 1022

os+ 1rJQ 1u5* 1 O b *

Table A-5* Continued (Page 4 of 10)

3* 5+ 5s 6* 7s 8* 9*

10* 11* 12* 13* 19* 15* 16* 17* 10* 19* 2G* 21* 22* 23* 24. 25+ 2 5 * 27* 2 H * 24* 3G * 31* 32* 33* J U * 35* Jh* 37* 3 0 + 39* u c + Ul* 42* b3* 4 b * r)5*

C I I Y )

37

Table A-5, Continued (Page 5 of‘ 10)

81 9*

12r 13* 14* 15* 16* 17* 10* 1 9* 20 * 21+ 22* 23* 2%* 25* 26* 27* 26* 29s 3 c 4

31, 32* 35* 3Q* 35* 3h 37 3tr* 34+

38

Table A-5, Continued (Page 6 of 10)

4* 5* b* 7* 8* 9* 10* 11* 12* 13* ]Lo* 15* lb* 17. 18* 19* 20* 21* 22* 23* 2L)* 251 ' 26* 27* 20* 29+ 30* 31* 32* 33 + 3 b * 35* 36+

39

Table A-5- Continued (Page 7 of 10)

lo*

13* 14r 15* 16* 1 7 * 16+ 19* 2G+ 21+ 22+ 234 24* 2 5 r 264 2 7 * 2t* 2% 30 r 31* 32 * 33 Jb+ 3S* 3 5 4 370 36 L 39* 40 * 4l* 424 4:r rc4c rc5* 4464 u7+ rbe e 45s 50 @

51* 52 53 54 s5* 56* 514 56 59+ 6 C * 61 4

40

Table A-5, Continued (Page 8 of ID)

62*

64 * 668

68* 69* 70 * 71* 72* 73* IS* 75* 76* 77. 78* 79* 80* 01* 82, 83* BY* B5* 86* 87* 88* 89* 90* 91. 92* 93* 9rc * 95* 96* 97* 98* 99*

l o o + 101* 1U2* 103* lo&* 105* l o b * 107* 108* 109* 110* 111+ 112* 113* 114* 115* 116* 117* i i a * 119* 12O* 121* 122*

639

65*

67*

41

T a b l e A-5, Continued (Page 9 of 10)

3*

5* 6* 7* 8* 9* O * %* 12* 13+ t u * 15* 16* 174 18+ 194 2 3 , 21* 2 2 + 2 3 e 24+ 2>* 25* 27* 26+ 29* 30 * 31* 32+ 33+ 3I(* 35* 3 b * 37* J B + 392 4 3 * 4 1 * 421 434 444 45* 46* 474 4&+ Y 7 * ba* 51* 52+ 53+ 54* 554 5b* 57* 50s 59* 60 * 61* b2*

42

Tab le A-5, Continued (Page ID of 10)

6.3 64 4

65* 66*

6b* 69* 70 * 71* 72* 73* 7Q* 75* 76* 77* 78* 79* 80 * 81* 82* 83* 84* 85* 86* 87* 88* 89* 90* 91* 92 * 93* 94* 95* 96* 97* 98* 9.9 *

l o o * 101* 102+ 103* loo* 105* 106* 107* 10e* 109* 110* 111* 112* 113+ 114* 115* 116* 11% 118* 119* 120* 121* 122* 123*

67+

43

a, rl

u1

M ('. t-

44

INPUT WC, DWC, NWC, CM, RCC,

Figure A-1. Flow Chart for the Nozzle Admittance Computer Program (Page 1 of u))

45

DP = -0.001

CM, SVN, RCT,

t COMPUTE INITIAL

VALUES OF PHIR, PHII, U, A,B,C ,Cm,C1I,

--b

ZR = ZOR ZI = ZOI G(1) = U G ( 2 ) = ZR G(3) = ZI G(4) = PHIR G(5) = PHII

f P = P + D P U = G(1) ZR = G ( 2 ) ZI = G ( 3 ) PHIR = G(4) PHII = G ( 5) COMPUTE DERIVATIVES

+ NEW VALUES OF

DY(1,I) = GP(1) DY(2,I) = GP(2)

DY(4,I) = GP(4) DY(5,I) = GP(5)

DY(3,I) = GP(3)

Figure A-1, Continued (Page 2 of 10)

46

SUBROUTIHJE

UZ(J) = U ( J ) + A(I + l)*H* FZ(I - 1,J)

S UBROUTI: NE

Figure A-1. Continued (Page 3 of 10)

47

CALL [ SUBROUTINE I

RKZDIF

Figure A-1. Continued (Page 4 of 10)

48

FOR SECTION I

FOR SECTION

Figure A-L. Continued (Page 5 o f 10)

49

FOR SECTION I SECTION I1 SECTION rrr

Figure A-1. Continued (Page 6 of 10)

X = X + H

ZR = PRED(2) ZI = PRED(3) PHIR = PRED(4) PHII = PRED( 5) COMPUTE C,R

u = PRED(~)

COMPUTE DR

COMPUTE DU ,A ,BR , BI,CR,CI ,PHIR, PHII

DO THROUGH I = 1,N

COMPUTE DR FOR

?

DY(I,l) = DY(I,2) DY(I,2) = DY(I,3) DY(1,3) = DY(1,4)

C ,DU,A ,BR,BI , CR GI ,PHIR ,PHI1 ,

Figure A-1. Continued (Page 7 of 10)

Figure A-1. Continued (Page 8 of 10)

2

X = X + H

TR = PRED(2) TI = PRED( 3) PHIR = PRED(4) PHI1 = PRED(5) COMPUTE C,R

u = PRED(~)

COMPUTE DU, A,

1 k g

DY(I,1) = DY(I,2) DY(I,2) = DY(I,3) DY(I,3) = DY(I,4)

53

+ CONVERT T TO

f

COMPUTE I N I T I A L r VALUES

Figure A-1. Concluded (Page 10 of 10)

54

REFERENCES

1, Crocco, Le, and Cheng, S. I-,

Liquid Propellant Rocket Motors

t ions Limited, London, 1956. Crocco, L., and Sirignano, W. A., Behavior of Supercr i t ica l Nozzles

AGARDograph 8, Butterworth Publica-

2.

, AGARDograph 117, Butterworth Publications Limited, London, 1967.

3. Bel l , W. A., "Experimental Determination of Three-Dimensional Liquid

Rocket Nozzle Admittances," PH.D. Thesis, School of Aerospace Engineer-

ing, Georgia I n s t i t u t e of Technology, Atlanta, Georgia, July 1972.

Serra, R. A , , "Determination of In te rna l Gas Flows by a Transient

Numerical Technique," AIAA Journal, Vol. 10, May 1972, p. 603. Zinn, B. T., "Longitudinal Mode Acoustic Losses i n Short Nozzles,"

J. of Sound and Vibration, 22(1), pp. 93-10?, 1972.

4.

5.

55

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L i b r a r y Stanford Research Ills t i t u t e 333 Ravenswood Avenue Menlo Park, Califol*,Lj.a 132'7

Rudy Reichel Aerophysics Research So 'p e

P 3 s t o f f icc BOX 187 Belleme, Washin :ton C,d( J C ~

Princeton Univerlii L,, Junes Forrestal ,:m.ps Li l i r . a !y Attn: D. Harrje Post Office Box 710 Princeton, New ~ I e i - ~ t , J>154C

U. S. Naval Weapons Center Attn: T. Inouyt 'Xdc 4581 China Lake, Gal2 f'oJn?ia 93555

Office of Naval ;;e;czi~ch Navy Departmen - Attn: R. D. Jackel, 1173 Washington, D. C . 20360

A i r Force Aero Propulsion Laboratory Attn: APTC Lt. F'i. Johnson Wright-Patterson Am, Ohio 45433

Naval Underwater Systems Center Energy Conversion Department Attn: D r . R . S . Lazar., Code TB 131 Newport, %ode Island 02840

HASA Langley Research Center Attn: R . S. Levint.. JW 213 Hampton, Virginia 25365

Aerojet General Corporation Attn: David A. Fajrchild, Mech.

P m t Office Box 15'347 (Sect. 9732) Sacramento, Cal i fornia 35809

Design

NASA Lewis Research C e ~ l t e ~ , MS 500-313 Rockets & Spac :craft Prceuremsnt S tc t icn 21000 Brookpa ck Road Cleveland, Ohic &135

NASA L e w i s Research :-ate;* Attn: H. W. Douglass:? ivE ' k ~ 0 - 2 O 1 , 21000 Brookpark Fcatl Cleveland, 13hj' '-'!135

Univer it^ of Ca.lifornia Depart, ,lent of' Chemical Engineering Attn: A. I(. 3;penheim 6161 E t che vcr I-; I I a l l Berkeley, Chlifcrnia 914720

Army B d . i ; s t i es Laboratories Attn: cJG R . Osborn Aberdeen Proving Ground, Marylan$i -5

Sacrarntxto State Coli-ege School or' Engineering Attn: F. H. Reardon 6000 J. Street Sacramento, Cal i fornia 95819

Purdue University School of Mechanical Engireering Attn: B. A. Reese Lafayette, Indiana 47907

NASA George C. Idarshall Space Flighi, :;c nte=r Attn: R. J. Richnond, SNE-ASTL-I-I' Huntsville , Alabaraa 35812

Colorado S ta t e University Mechanical Ehgineering Department Attn: C. E. Mitchell Fort Collins, Colorado 80521

University of Wisconsin Meck.,ani cal. Engine er ing Department Attn: PI S1 Myers 1513 University Avenue Madison, Wisconsin 53706

North American Rockwell Corporation Rocketdyne Division Attn: J. A. Nestlerode,

6633 Canoga Avenue Canoga Park, California 91304

~ ~ 4 6 D/596-121

J e t Propulsion Laboratory Cal i fornia I n s t i t u t e of Technology Attn: J. HI Rupe 4800 Oak Grove Drive Pasadena, California 91103

University of Cal i fornia Mechanical Ehgineering Thermal Systems Attn: Professor R. Sawyer Berkeley, California 94720

ARL (ARC) Attn: K. Scheller Wright-Patterson AIB, Ohio 45433