Behavioir of Supercritical Nozzles .under Three-Dimensional ...AGARDograph 117 £ Behavioir of Supercritical Nozzles.under Three-Dimensional Oscillatory Conditions by L. Crocco and

AGARDograph 117

£

Behavioir of Supercritical Nozzles.under Three-Dimensional

Oscillatory Conditionsby L. Crocco and W. A. Sirignano

'A. AUG 21968

CLEA R G H1 0 U ' tE rhis docurnrnt . b r ., or." ,:edor pubki I I

IAGARD ograph 117

INORTH ATLANTIC TREATY ORGANIZATION4ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT1(ORGANISATION DU TRAITE DE L'ATLANTIQUE NORD)

BEHAVIOR OF SUPERCRITICAL NOZZLES UNDER

THREE-DIMENSIONAL OSCILLATORY CONDITIONS

by

Luigi Crocco* and William A.Sirignanot

"Professor of Aerospace Sciences

tAssistant Professor of Aerospaceand Mechanical Sciences

Princetan Uniiversity

Princeton, New Jersey, USA

1967

This AGARDograph was prepared at the request of thePropulsion and Energetics Panel of AGARD

!I

SUMMARY

A linearized treatment of three-dimensional oscillatory flow insupercritical nozzles has-be;in performed for the two cases where thesteady-state flow is axisymmetric or two-dimensional. In the axi-

symmetric case, perturbation series have-been employed to study the non-linear oscillations. In these analyses, variables have-been separated,reducing %%ie system of partial differential equations to a system of

ordinary differential equations. The variables describing the transversedependencies of the flow properties are governed by well-known differ-ential equations (for example, Bessel's equation is obtained). Theaxial dependencies, on the other hand, are governed by differential

equations which must be solved by numerical means. The nozzle admittancecoefficients are related tG the axial dependencies of the flow properties.Certain techniques have-been applied to reduce the order of the differ-

ential equations for the purpose of easier calculation of the admittancecoefficients.

These admittance coefficients are to be used in the boundary condition

applied at the exit of the chamber Joined to the nozzle and their cal-culation is the most important result of this research effort. ' 'The cal-culations have been performed for conical nozzles and are presented intabular form. Exiamples are presented which demonstrate the use of thetables in typical problems. Oscillatory pressures and velocities arealso calculated in a limited number of cases in order to provide physicalinsight to the oscillation. An asymptotic analysis has been performedwhich is valid in the low entrance-Mach-number regime. By predicting theadmittance coefficients and flow properties through closed-form solutions,the asymptotic analysis is an asset in the interpretation of results.

533.697.4

ii

'I SONNAIRE

Un traitement line'arise' d' un 4coulement oscillatoire tridiuensionneldana lea tuy~res suvercritiques a 4te' effectu4 pour lea deux cas o1' ecoulement permanent est axiaymetrique ou plan. Dana le cia axi-symtrique des sdries perturbatrices ont e'td utilis'es pour l'4tude desoscillations non lindaires. Dana lea analyses effectue'es on a s'par4lea variables en rdduisant i un systiue d dguations diffe'rentielles

I ordinsires le syatime d dquations diffdrentielles partielles. Leavariables gui ddcrivent lea dipendances tranaversales des propridtgsde l'e'coulement sont rigies par des dquations diffe'rentielles connues(par exemple, on obtient l'e'quation de Bessel). Par contre, leadedpendances axiales sont ddcrites par des 4quations diffe'rentiellesqu' il taut re'soudre par des moyena nume'riques. Lea coefficients d' entrdede tuyire soot lids aux ddpendances axiales des propridtes de 1' dcoulement.Certalnes techniques ont gt4 appliquies pour rdduire 1' ordre des iquationsdiffdrentielles en vue de faciliter le calcul des coefficients d'entr'e.

Ces coefficients d' entr6e seront employe's pour la condition limiteappliiae'e -- la sortie de la chainbre relide i I& tuye're et leur calculreprdsente le rdsultat le plus important de ces efforts de recherches.Lea calculs ont e'tg effectuis pour lea tuye'res coniques et sont pre'aentdAsous forme de tableaux. Des exemples sont, cite's pour dduontrer l utilisa-tion de ces tableaux pour I& solution de problimes types. Lea pressionset lea vitesses oscillatoires sont 4galeuent calculges dana tin nombreliuiti de ca pour obtenir tine connaissance physique de 1'oscillation.I Une analyse asymptotique a 4Ut re'aliado gui eat valable dana lea conditions

21 de faible nombre de Mach d' entrie. Elle s avere tria avantageuse pour1'interprdtation des rdsultats en peruettant de pre'voir lea coefficientsd' entrge au uoyen de solutions de forme ferude.

FOREWORD

The theory, on which the results presented in this monograph are based, was developedby the senior author over a decade ago and presented at a meeting at the Universityof Maryland. At that time, the complete formulation of the linear admittance co-

efficient for a supercritical nozzle had been derived in the general ease where bothvorticity and entropy oscillations exist at the nozzle entrance T the following

years, a small number of calculations were made with the purpose of providing thenecessary data for the determination of the combustion instsbility limits In a parti-

cular experimental rocket 1 . However, the publication of the theory was postponeduntil more complete calculations and interpretations would become available.

In recent years, the project became a cooperative venture and, primarily through

the efforts of the junior author, the computer calculations and other related analyseswere accomplished. Meanwhile, the need for the public availability has become pressing.due to the problem of combustion instability in rocket motors. One of the leading

rocket manufacturers in the United States has found it necessary to establish its own

computer program based on the above-mentioned theoretical developments'. Another

scientist has decided to attack the problem independently in a somewhat-differentL.anner 7 . We welcomed therefore the opportunity (made possible by AGARD) to publishthe theory, some of its extensions, the methods of calculation, and the numerical

results. It has been felt that they would fill a well-defined void In the existingtechnical literature and play a useful role.

In Part I, we present the theoretical background. The original theory of the senior

author is contained in Sections 1 through 12. Section 13 contains the discussion ofthe extension of the theory to the nonlinear shockless case performed by B.T. Zinn,under the supervision of the authors, as part of his Ph.D. thesis a . Part II containsthe discussion of the calculations, related analyses, interpretation of the results,and examples of applications and is mostly due to the junior author. Sectione 14. 15,and 16 present the method of calculation and a discussion of the results. Sections 17and 18 present an asymptotic theory useful for the purpose of interpretation. Finally,Section 19 contains a few sample applications of the results.

Wa wish to recognize the major technical support provided by the staff of the

Guggenheim Laboratories Computing Group at Princeton University. Also, we wish to

rpcognize the major financial support provided by the National Aeronautics and spaceAdministration. Additional support for the calculations was provided by the WilliamB.Reed. Jr. Fund for Engineering Research of Princeton University and (Indirectlythrough the support of the Princeton University Computing Center) by the NationalScience Foundation.

Luigi CroccoWilliam A. Sirignano

iv

I,

CONTENTS

Page

SUMMARY ii

RESUME iii

FOREWORD iv

LIST OF TABLES vi

LIST OF FIGURES x

NOTATION xiii

PART I. THEORY

1. Introduction 1

2. The Equations 3

3. Linearization of the Equations 4

4. Choice of the Independent Variables: Axisymmetric Nozzle 6

5. Choice of the Independent Variables: Bidimensional Nozzle 8

6. Separation of the Variables for One-Dimensional UnperturbedFlow: Axisymmetric Nozzle 10

7. Separation of the Variables for One-Dimensional Unperturbed

Flow: Bidimensional Nozzle 14

8. Reduction of the System: Axisymmetric Nozzle 16

9. Reduction of the System: Bidimensional Nozzle 20

10. Admittance Condition at the Entrance of an Axisymmetric

Nozzle 2211. Admittance Condition at the Entrance of a Bidimensional

Nozzle 25

12. Similarity of Nozzles: Velocity Distribution for ReferenceNozzle 26

13. Nonlinear Analysis 29

PART II. APPLIPATIONS

14. Calculations for "Conical" Nozzles 43

15. Effect of Transition Region Between Cylindrical Chamberand Conical Convergent Nozzle 55

16. Flow Properties 58

17. Asymptotic Behavior of the Admittance Coefficients 63

18. Asymptotic Development of the Flow Properties 72

19. Results of the Nozzle Admittance Calculations and TheirApplications 78

REFERENCES 83

TABLES 84

FIGURES 124

DISTRIBUTION

LIST OF TABLES

Page

TABLE I Values of Svh 84

TABLE II 84

TABLE III Real Part of Pressure Admittance Coefficient sh = 0 85

TABLE IV Imaginary Part of Pressure Admittance Coefficient Svh = 0 85

TABLE V Real Part of Pressure Admittance Coefficient sh = 1 86

TABLE VI Imaginary Part of Pressure Admittance Coefficient svh = 1 86

TABLE VII Real Part of Pressure Admittance Coefficient Sh = 2 87

TABLE VIII Imaginary Part of Pressure Admittance Coefficient s.h = 2 87

TABLE IX Real Part of Pressure Admittance Coefficient Svh = 3 88

TABLE X Imaginary Pait of Pressure Admittance Coefficient Sh = 3 88

TABLE XI Real Part ot Pressure Admittance Coefficient svh. 4 89

TABLE XII Imaginary Part of Pressure Admittance Coefficient s., = 4 89

TABLE XIII Real Part of Pressure tdittance Coefficient S = 5 90

TABLE XIV Imaginary Part of Pressure Admittance Coefficient svh = 5 90

TABLE XV Real Part of Pressure Admittance Coefficient s = 7 91

TABLE XVI Imaginary Part of Pressure Admittance Coefficient sh = 7 91

TABLE XVII ReIl Part of Pressure Adzittance Coefficient a.,= 9 92

TABLE XVIII Ieinary Part of Pressure Admittance Coefficient Sh = 9 92

TABLE XIX Real Part of Radial Velocity Admittance Coefficient Sth = 1 93

TABLE XX Imaginary Part of Radial Velocity Admittance Coefficient

Svh - 1 93

TABLE XXI Real Part of Radial Velocity Admittance Coefficient a.,, = 2 94

TABLE XXII Imaginary Part of Radial Velocity Admittance Coefficient

h 2 94

vi

]\

Page

TABLE XXIII Real Part of Radial Velocity Admittance Coefficient Sh= 3 95

TABLE XXIV Imaginary Part of Radial Velocity Admittance Coefficient

Svh = 3 95

TABLE XXV Real Part of Radial Velocity Admi.ttance Coefficient Sh = 4 96

TABLE XXVI Imaginary Part of Radial Velocity Admittance CoefficientSvh = 4 96

TABLE XXVII Real Part of Radial Velocity Admittance Coefficient svh =5 9

TABLE XXVIII Imaginary Part of Radial Velocity Admittance CoefficientSvh = 5 97

TABLE XXIX Real Part of Radial Velocity Admittance Coefficient Sh = 7 98

TABLE XXX Imaginary Part of Radial Velocity Admittance CoefficientSvh = 7 98

TALE XXXI Real Part of Radial Velocity Admittance Coefficient sh 9 99

"E )XII Imaginary Part of Radial Velocity Admittance CoefficientSvh 9 99

TABLE XXv.III Real Part of Entropy Admittance Coefficient Svh - 0 100

TABLE XXXIV Imaginary Part of Entropy Admittance Coefficient sh= 0 100

TABLE XXXV Real Part of Entropy Admittance Coefficient sh = 1 101

TABLE XXXVI Imaginary Part of Entropy Admittance Coefficient svh = 1 101

TABLE XXXVII Real Part of Entropy Admittance Coefficient s = 2 102

TABLE XXXVIII Imagiiary Part of Entropy Admittance Coefficient Szh = 2 102

TABLE XXXIX Real Part of Entropy Admittance Coefficient svh = 3 103

TABLE XL Imaginary Part of Entropy Admittance Coefficient svh = 3 103

TABLE XLI Real Part of Entropy Admittance Coefficient sph = 4 104

TABLE XLII Imaginary Part of Entropy Admittance Coefficient svh 4 104

TABLE XLIII Real Part of Entropy Admittance Coefficient S = 5 105

TABLE XLIV Imaginary Part of Entropy Admittance Coefficient Svh =5 105

vii-

Page

TABLE XLV Real Part of Entropy Admittance Coefficient Sh 7 106

TABLE XLVI Imaginary Part of Entropy Admittance Coefficient S = 7 106

TABLE XLVII Real Fart of Entropy Admittance Coefficient Sh 9 107

TABLE XLVIII Imasinarv Part of Entropy Admittance Coefficient Sh = 9 107

TABLE XLIX Real Part of Irrotational Admittance Coefficient Svh = 0 108

TABLE L Imaginary Part of Irrotational Admittance Coefficient

svb = 0 108

TABLE LI Real Part of IrrotatJonal Admittance Coefficient Svh = 1 109

TABLE LII Imaginary Part of Irrotational Admittance Coefficient

Sh 1 109

TABLE LIII Real Part of Irratational Admittance Coefficient S., = 2 110

TABLE LIV Imaginary Part of Irrotational Admittance Coefficient= 2 110

TABLE LV Real Part of Irrotational Admittance Coefficient svh 3 111

TABLE LVI Imaginary Pt of Irrotational Admittance Coefficient

vh 3 11

TABLE LVII Real Part of Irrotational Admittance Coefficient S = 4 112

TABLE LVIII Imaginary Part of Irrotational Admittance CoefficientSIh = 4 112

TABLE LIX Real Part of Irrotational Admittance Coefficient s., 5 113

TABE LX Imaginary Part of IrrotatonL Admittance CoefficientSh =5 113

TPBLE LXI Real Part of Irrotational Admittance Coefficient Sh = 114

TABLE LXII Imaginary Part of Irrotational Admittance Coefficient

Svh = 7 114

TAB!,E LxIII Real Part of Irrotational Admittance Coefficient sVh = 9 115

TABLE LXIV Imaginary Part of Irrotational Admittance Coefficient9 115

TABLE LXV Real Part of Combined Admittance Coefficient S = 0 115

viii

Page

TABLE LXVI Imaginary Part of Combined Admittance Coefficient s = 0 116

TABLE LXVII Real Part of Combined Admittance Coefficient Svh = 1 117

TABLE LXVIII Imaginary Part of Combined Admittance Coefficient s = 1 117

TABLE LXIX Real Part of Combined Admittance Coefficient s;h 2 118

TABLE LXX Imaginary Part of Combined Admittance CoefficJent sh = 2 118

TABLE LXXI Real P.art of Combined Admittance Coefficient svh = 3 119

TABLE LXXII Imaginary Part of Combined Admittance Coefficient Sh = 3 119

TABLE LXXIII Real Part of Combined Admittance Coefficient slh = 4 120

TABLE LXXIV Imaginary Part of Combined Admittance Coefficient sh = 4 120

TABLE LXXV Real Part of Combined Admittance Coefficient 3 = 5 121

TABLE LXXVI Imaginary Part of Combined Admittance Coefficient Szh = 5 121

TABLE LXXVII Real Part of Combined Admittance Coefficient sth = 7 122

T.ABLE LXXVIII Imaginary Ppxt of Combined Admittance Coefficient sh = 7 122

TABLE LXXIX Real Part of Combined Admittance Coefficient S,,h = 9 123

TABLE LXXX Imaginary Psrt of Combined Alittance Coefficient sh = 9 123

LIST OF FIGURES

Page

Fig. 1 Geometry of convergent portion of nozzle 124

Fig.2 Scaling of admittance coefficients 124

Fig. 3 Real part of Z versus axial distance 125

Fig.4 Real part of 4:(2) versus axial distance 125

Fig. 5 Real part of pressure admittance coefficient versus axial distance 126

Fig. 6 Imaginary part of radial velocity admittance coefficient versusaxial distance 126

Fig.7(a) Real part of pressure admittance coefficient versus frequency 127

Fig. 7(b) Imaginary part of pressure admittance coefficient versus frequency 127

Fig.7(c) Real part of pressure admittance coefficient versus frequency 128

Fig.7(d) Imaginary part of pressure admittance coefficient versus frequency 128

Fig.8(a) Real part of radial velocity admittance coefficient versusfrequency 129

FAg.8(b) Imaginary part of radial velocity admittance coefficient versusfrequency 129

Fig.8(c) Real part of radial velocity admittance coefficient versusfrequency 130

Fig.8(d) Imaginary part of radial velocity admittance coefficient versusfrequency 130

Fig. 9(a) Real part of entropy admittance coefficient versus frequency 131

Fig. 9(b) Imaginary part of entropy admittance coefficient versus frequency 131

Fig.9(c) Real part of entropy admittance coefficient verbC&s frequency 132

Fig. 9(d) Imaginary part of entropy admittance coefficient versus frequemcy 132

Fig. 10(a) Real part of irrotational admittance coefficient versus frequency 133

Fig. 10(b) Imaginary part of irrotational admittance coefficient versus

frequency 133

x

Page

Fig. 10(c) Real part of irrotational admittance coefficient versus frequency 134

Fig. 10(d) Imaginary part of irrotational admittance coefficient versus

refrequency 134

Fig. 11(a) Real part of combined admittance coefficient versus frequency 135

Fig. 11(b) Imaginary part of combined admittance coefficient versus

frequency 135

Fig. 11(c) Real part of combined admittance coefficient versus frequency 136

Fig. 11(d) Imaginary part of combined admittance coefficient versus frequency 136

Fig. 12(a) Real part of pressure admittance coefficient versus frequency:Effect of throat wall curvature 137

Fig. 12(b) Imaginary part of pressure admittance coefficient versus frequency:Effect of throat wall curvature 137

Fig. 12(c) Real part of pressure admittance coefficient versus frequency:Effect of cone angle 138

Fig. 12(d) Imaginary part of pressure admittance coefficient versus frequency:Effect of cone angle 138

Fig. 13(a) Real part of radial velocity admittance coefficient versusfrequency: Effect of throat wall curvature 139

Fig. 13(b) Imaginary part of radial velocity admittance coefficient versusfrequency: Effect of throat wall curvature 139

Fig. 13(c) Real part of radial velocity admittance coefficient versusfrequency: Effect of cone angle 140

Fig. 13(d) Imaginary part of radial velocity admittance coefficient versusfrequency: Effect of cone angle 140

Fig. 14(a) Real part of entropy admittance coefficient versus frequency:Effect of throat wall curvature 141

Fig. 14(b) Imaginary part of entropy admittance coefficient versus frequency:Effect of throat wall curvature 141

Fig. 14(c) Real part of entropy admittance coefficient versus frequency:Effect of cone angle 142

Fig. 14(d) Imaginary port of entropy admittance coefficient versus frequency:Effect of cone angle 142

xi

Page

Fig. 15(a) Real part of irrotational admittance coefficient versus frequency:

Effect of throat wall curvature 143

Fig. 15(b) Imaginary part of irrotational admittance coefficient versusfrequency: Effect of throat wall curvature 143

Fig. 15(c) Real part of irrotational admittance coefficient versus frequency: -!Effect of cone angle 144

Fig. 15(d) Imaginary part of irrotational admittance coefficient versusfrequency: Effect of cone angle 144 i

Fig. 16(a) Real part of combined admittance coefficient versus frequency: JEffect of throat wall curvature 145

Fig. 16(b) Imaginary part of combined admittance coefficient versus frequency:Effect of throat wall curvature 145

Fig. 16(c) Real part of combined admittance coefficient versus frequency:Effect of cone angle 146

Fig. 16(d) Imaginary part of combined admittance coefficient versus frequency: IEffect of cone angle 146

Fig. 17 Nozzle geometry and comparison of entrance portions of approximateand actual nozzle contours 147

Fig. 18 Pressure perturbation versus axial distance from nozzle throat 147

Fig. 19 Axial velocity perturbation versus axial distance from nozzle throat 148

Fig.20 Radial velocity perturbation versus axial distance from nozzle

throat 148

Fig. 21 Irrotational admittance coefficient: Comparison between exact andasymtotic solutions 149

x:1i

NOTATION

A pressure admittance coefficient defined In (122) for axisym-metric nozzle

A1 pressure admittance coefficient for bidimensional nozzle

Qpressure admittance coefficient defir'd after (161) for axi-symmotric nozzle

A. An, An. 0q function defined in (130). its Fourier series coefficients,and its eigenfunction series coefficients, respectively

A,B,CD coefficients in (179)

constants in (186)

a constant defined after (177)

B radial velocity admittance coefficient defined in (123) for

axisymetric nozzle

B1 transverse velocity admittance coefficient for bidimensionaln)zzle

radial velocity admittance coefficient defined after (161) foraxisymmetric nozzle

B.BnBn,vvq function defined in (131), its Fourier series coefficients,

and its eigenfunction series coefficients, respectively

BrBi.CrCi parameters in initial conditions (174)

b.g,h,J,k coefficients in (163)

b,c.d iutegration constants in (184)

b nondimensional width if the bidlmensional nozzle

coefficient in power series expansion

C entropy admittance coefficient defined in (124) for axisymetricnozzle

C1 entropy admittance coefficient for bidimensional nozzle or

integration constant

C entropy admittance coefficient defined after (161) for axisym-metric nozzle

xiii

CO. C2,C 3 integration constants

SCO CY separation constants

C Cnen.my.q function defined in (132). its Fourier series coefficients.and its eigenfunction series coefficients, respectively

C in. W. q C2n. my. q integration constants

c speed of sound

Cp specific heat at constant preesure

D' spanwise velocity admittance coefficient for bidimensionalnozzle

DDn.Dn.w.q function defined in (133), its Fourier series coefficients, andits eigenfunction series coefficients, respectively

combined admittance coefficient for axisymetric nozzles

EEn.En.n.q function defined in (134). its Fourier series coefficients,

and its eigenfunction series coefficients, respectively

e unit vector

F (J ) functions defined in (89)

Fpj) functions defined in (97)

F(j) functions defined after (145)n~np.q

F.Fn.Fn,mvq function defined in (135), its Fourier series coefficients,and its eigenfunction series coefficients, respectively

fo function defined in (77)

fX function defined in (81)

f2 function defined in (83)

f3 function defined in (95)

ion functions defined after (139)

fIn f~2n fuLctions defined after (150)

f3n functions defined after (135)

G.H functions lefined after (172)

xiv

C ~ ~ ~ O - 0--'- ~ - - - ----

-. functions define after (139

Gn functions defined after (139)

Hn functions defined after (141)

ho.ho.h8.y defined in (23) and after (34)

inhomogeneous part defined in (141) or function defined by(188) and (190)

In.uvq inhomogeneous part defined in (144)

Il 112 143' parameters defined after (174)

i imaginary unit

Jv, Y Bessel functions of order v of the first kind and second

kind, respectively

integral defined after (193)

'J)Q functions defined after (155)

K1K K2' Ks 3constants defined after (183)

k related to separation constant in (54) or constant in (189)

characteristic length in nondimensional scheme

L combustion chamber length

-length El) of cylinder in Figure 17

MN integers describing m)de of transverse oscillation in bidimen-

sional nozzle

M integer in (190)

MM functions defined after (191)

m integer in Section 13, or parameter defined after (193)

Nn.m, q functions defined after (145)

n integer subs,'ript

P'P 1 separated pressure variables defined in (44) and (61)

P . coefficient in Bigenfunction series for pressure

P separated pressure variable defined in (160)

xv

p pressure

Q~h sepaated energy release per unit volume in combustion chamber

q velocity vector

RRt separated density variables defined in (44) and (61)

R.R2 nozzle wall radii of curvature at throat and entrance,respectively

Rn.1W.Q coefficient in eigenfunction series for density

R separated density variable defined in Section 15

r radial position or local wall radius in Section 14

SS1 separated entropy variables defined in (44) and (61)

Sn,Bv.q coefficient in eigenfunction series for entropy

Aseparated entropy variable defined in (160)

a entropy

Vh' lSv q eigenvalue corresponding to roots of the derivative of theBessel function

s8 parameter defined in (73)

t time or variable in (190)

UU t separated axial velocity variables defined in (44) and (61)

Una. q coefficient in eigenfunction series for axial velocity

separated axial velocity variable defined in (160)u,v,w 95, , and 0 or 4). 1, and y components of velocity,

respectively

VV separated radial velocity variables defined in (44) and (61)

Vnuy.q coefficient in eigenfunction series for radtal velocity

separated radial velocity variable defined in (160)

W.W1 separated azimuthal velocity variables defined in (44) and (61)

Wn'ms,.q coefficient in eigenfunction series for aziuthi velocity

xvi

Wseparated azimuthal velocity variable defined in Section 15

x coordinate for bidimensional nozzle

Y separated variable defined in (61)

y spanwise coordinate for bidimensional nozzle or normalizedtime in Section 13 or transformed velocity variable definee in

(178)

y1 'y2'y3'y; functions defined before (173)

z physical axial coordinate

OL phase constant in (56) or admittance coefficient defined in(105) or (107)

cy'1 coefficients in (157)

parameters appearing in initial conditions for (163)

L.A3 6. constants in (170)

function defined in (110) or scaling factor discussed inSection 12

Y ratio of specific heats

A phase angle in Section 16

81 elementary length

Sn elementary length in stremwise direction

Ss elementary length normal to streamline

e perturbation parameter

functions defined in (103) and (117). respectively

Ifunctions defined in (136)

functions defined in (154)

71 constant in (190) 24.4

716 functions defined in (136)

ftnctions defined after (141)

e separated variable defined in (44) and (61)

xvii

6 azimuthal position

01 semi-angle of convergent portion of conical nozzle

K transformation constant discussed in Section 14

Kn functions defined in (136)

2 constants defined after (170) or constants of integration in(192)

x 3 .X*M 5 constants defined in (179) and (200)

Vseparation constant defined in (53)

e),gJ Mfunctions defined in (103) and (117), respectively



7rn functions defined in (136)

p density

a integration constant related to initial entropy

parameter defined in (73)

an functions defined in (136)

n.miq integration constants

functions defined in (79) and (92), respectively

$J), J) particular solutions of (99) and (114), respectively

functions defined in (137)§n

n.mv. q functions defined in (143a) or (143b)

velocity potential function

separated variables defined in (44) and (61). respectively

'stream function or transformed variable in (180)

Co angular frequency nondimensionalized with respect to throatradius

cL angular frequency nondimensionalized with respect to chamberradius

xviii

Superscripts

star (*) dimensional quantity

bar steady-state variable or admittance uoefficient at end ofcylinder in Section 15

caret transformed variable discussed in Section 14

prime perturbation quantity

t) quantity pertaining to traveling wave

(s) quantity pertaining to standing wave

0 stagnation quantity

(o).(1).(2). etc. coefficient in perturbation series in Section 13

Subscripts

arg argument of a complex number

• quantity at nozzle entrance

h homogeneous svlut-on

iimaginary part of quantity

mod modulus of a complex number

p particular solution

r real part of quantity

ref quantity pertaining to reference nozzle

th quantity at nozzle throat

w quantity at nozzle wall

.4XIX

j -V.

I I ~iji~

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BLANK PAGE

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A

BEHAVIOR OF SUPERCRITICAL NOZZLES UNDERTHREE-DIMENSIONAL OSCILLATORY CONDITIONS

Luigi Crocco and William A. Sirignano

PART I. THEORY

1. INTRODUCTION

Maay propulsive devices are terminated by a nozzle through which se propulsivegases are discharged. Very often the nozzle operates In the supercritical range andis shaped as a classical Laval nozzle, converging up to a throat (where, in steadyoperation, the sonic ielocity is achieved) and Jiverging thereafter.

Unsteady condftionE are present in the nozzle iben the operation In the propulsivedevice Is unsteady. A particular type of unstez.y operation which has great importancein practice rest.lts from combustion instability in the propulsive device. In thiscase the operation is oscillatory, characterized by periodic variations of the flowparameters both in the combustio-a chamber and in the nozzle which foll;ss.

The study of combustion instability is of great importance for the safety of opera-tion of combustion devices, and has been already the subject of a great deal of theore-tical and experimental research. It is important in these studies to hnow the behaviorof the nozzle under oscillatory conditions. In particular, it in necessary to findout hqw a wave generated in the combustion chamber is partially reflected and partiallytransitted at the entrance of the nozzle. Mathematically, this is equivalent tosaying that it is necessary to know the bomdary conditions created by the nozzle tothe oscillatory flow present in the combustion chmber. For instance, on any solidwall of the chamber, the boundary condition is that the velocity component of thegases normal to the wall must be zero at every instant, and so must be. therefore, thecorresponding velocity perturbation, no matter what are the peiturbations of thetangential velocities, pressure. entropy. etc. Obviously If the entrance of the nozzleis considered to represent one of the bounaaries of the combustion chamber, the cor-respon6ing boundarr conditlons are more involved. No single perturbation can beassumed to vanish, and the boundary condition may be expected to be expressed by a

2

relation between the various perturbations which shall be called the admittancecondition*. In particular, if the perturbations are assumed to be of sufficientlysmall amplitude, so that the problem can be linearized, the aforesaid relation express-ing the boundary condition must be linear. Observe that a linearized treatment canbe only applied to the study of incipient instability, that is to the study of thecombustion stability limits of a given propulsive device. When the instability isfully developed, nonlinear effects become essential and must be taken into consideration.

If the unperturbed flow in the nozzle cap be assumed to be one-dimensional, a parti-cular!) simple case is obtained when also the perturbations are taken one-dimensional,that is the perturbations are assumed to be uniform, at each instant, on each sectionof the nozzle. Obviously in this case the conditions in the combustion chamber arealso one-dimensional, and any wave present in the chamber or in the nozzle is of an"axial" type. The behavior of the nozzle in the presence of axial waves has beenanalyzed by Tsien1 in a few simple cases, and by Crocco 2 , 3 for the most general typeof linear axial oscillation, in particular when the entropy is not constant. Thevariability of the entropy is in fact an unavoidable consequence of the combustiontaking place under oscillatory conditions, and may producL interesting effects on thecombustion instability 3. Ex,,iments carried out some time ago in nearly isentropicconditions show satisfactory agreement with the theoretical predictions".

However, the agial type of oscillations is only a particular case, and in actualconditions transverse perturbations are often present, this being particularly truein large propulsive devices. The problem is now complicated by the incre- ,ed numberof degrees of freedom, but under thc same assumptions that the perturt ,tioks aresmall and that the unperturbed flow in the nozzle is one-dimensional (hpnce, irrota-tional), isoenergetic and isentropic, it is amenable to a relatively simple analyticaltreatment. It is the purpose of this monograph to show how, in this simpler case,the admittance condition at the nozzle entrance can be expressed and to discuss thecorresponding numerical results. We shall treat both the case of axisymmetric nozzle(the most common in practical devices) and that of two-dimensional nozzle. We shallalso briefly discuss the nonlinear treatment of the nozzle admittance problem in theabsence of shock waves.

Observe that the results of the present study are applicable to any type of device,propulsive or not, involving combustion processes or not.

* The following considerations may help in understanding the nature of the admittance codition.It is clear that, if the flow in the nozzle is supercritical, for sufficiently small oscilla-'tions the stpersonic portion of the nozzle has no effect on the chamber conditions, becausedownstream of the throat the oscillations, no matter how distorted, must always propagatedownstream and cannot interfere wi'hh the upstream flow. Henze, the logical choice for thesurface on which boundary conditions must be prescribed would be the surface -Are "'te sonicvelocity is achieved, or, for small oscillations around an approximately one-ulmensional flow,the throet itself. It has been shown2.3 that the proper boundary condi" ion at the throat isthat the solution remains regular here (where. indeed, a singularity tends to result from theinability of the disturbances to propagate upstream from the supersonic into the subsonicregion). In practice, however, it is useful to divide the whole of the chamber plus thenozzle in two parts: the combustion chamber extending down to the no7zle entrance where theprocesses of combustion are taking place but the mean flow 1ach number is relatively low;and the nozzle where no combustion is assumed to take place but the mean Mach number grows upto unity. The result of this subdivision is to move the boundary of the combustion chamberfrom the throat up to the nozzle entrance, where the appropriate boundary conditions can beobtained by studying the oscillatory behavior of the nozzle per se and obtaining the properrelation between the perturbations at the nozzle entrance (the admittance condition) from th?condition of non-singularity at the throat.

3

2. THE EQUATIONS

Using stars to denote dimensional quantities and operators, the equations of motionfor an inviscid, non-heat-conducting gas are the following.

Continuity:

+ V*. (An = 0

Momentum:q 1 1

+ V(r*') + (V* x q*) x q* = -- VPt-2-p

where t* represents the time, p* and p* the density and pressure, and q* thevelocity vector.

A hen viscosity and heat conductivity are disregarded the energy equation in itssimplest form expresses the constancy of the entropy for a fluid particle after itenters the nozzle:

t7 + e*.V~s 0

where s*= cp I loge p- loge + constant M')

represents the entropy. The specific heat cp is assumed to be constant. With thisparticular form of the energy equation it is not necessary to introduce the equationof state, which is implicitly taken into account in Equation (1). and the four equations

just written are complete in the unknowns q* , p* , p s

These equations can be nondimensionalized using appropriate reference values.Assuming the gas entering the no7Lle in the unperturbed flow to be isoenergetic andisentropic (as well as irrotational), and hence to stay such in the following expansionthrough the nozzle, the corresponding stagnation quantities remain constant throughoutthe unperturbed flow and hence are suitable reference quantities. Hence we define

q p*q* p* p*q = --- ; P ; p = (2)

cp p

where c denotes the sonic velocity, the superscript 0 indicates stagnation valuesand the superposed bar unperturbed (steady) values. Observe that_-

0

4

The lengths can be nondimensionalized using a suitable characteristic length L* to

be further defined, and a nondimensional time is immediately obtained as

t = -- t (4)

Transforming also the operators to nondimensional coordinates we obtain the equationsof motion in the form

dp-- V. pq) = 0 (5)

q 1 1+ - V(q2 ) + (V x q) x q - Vp (6)

t 2 ^P

-+ q.Vs = 0, (7)

with

*- - log e p -log e p 4 constant (8)c0

representing the nondimensional entropy. Equation (3) is replaced by

c - (9)

representing the nondimensional sonic velocity.

3. LINEARIZATION OF THE SQUATIONS

We now introduce the ordinary ass,,mption that the (unsteady) perturbations around

the (steady) unperturbed quantities are of such small amplitude that caly linear termsin the perturbations need be considered. Hence we introduce

q + ; P = p +P' p = P+P s = i+s' (10)

in Equations (5) to (8) and, after neglecting all terms of order higher than first in

the primed quantities, we separate each equation into its unperturbed portion (containingonly the barred quantities) from the perturbation portion (.Inear in the primed

quantities). Clearly each portion of the equations must be satisfied separately. Theequations for the unperturbed flow are

div () 0; grad + (Vx ) x = - grad2 Y,

_ (11)

I5

1 rq.grad 0 - log e loge T + constant (11)

These equations can be replaced by the following simpler ones, obtained from (11) ina standard fashion when the flow is irrotational,

div(P) = 0; 5 - I--- . (12)

~2

The second expresses the constancy of the entropy s, the third that of the stagnationtemperature. Comparing this last equation to (9) we obtain the unperturbed sonicvelocity

= y-1 T = /(v-1) , (13)2

the last term, resulting from (12), providing the unperturbed density. When the nozzleis axisymmetric or two-dimensional the first (12) can be used to define a streamfunction. We may wrcite indeed

r = ea x Vqj (14)

in the axisymmetric case (r representing the nondimensional distance from the axis ofsymmetry and ea the unit vector in the tangential direction) and

b;5 = ey x q (15)

in the two-dimensional case (b representing the nondimensional width of the nozzle.which must be assumed to be finite if three-dimensional oscillations are to be con-sidered, and e, the spanwise unit vector.)

In the present assumption of irrotationality of the unperturbed flow, a potentialfunction can also be defined as

= ¢. (16)

The stream and potential functJP,-s introduced are nondimensional.

In what follows the unp,,turbed flow will be assumed to be in the meridional plane(axisymetric case) or pur,,.Ly two-dimensional (two-dimensional nozzle).

The perturbation equations can be written as

+V. (p +pq') =0 (17)Bt

q I 1 1- + V(4.q') + (V x q') x = --- =Vp' +. Vp (18)at PP

6

+ 0 (19)at

s' = P (20)

Equations (18) to (20) represent the system of equations to be solved. It is linear

in the perturbations, and has coefficients depending on the solution of the unperturbed

Equations (12).

It is useful to rewrite the second member of (18) in a different form, using the

following transformations:

I Vp + P' I p pl7 p I +p1 V

S-V -- -, - (21)

' - +(-)+ (V )s'

where use has been made of the second (12), of the second (11) and of (20).

4. CHOICE OF THE INDEPENDENT VARIABLES: A"ISYNNETRIC NOZZLE

Abandoning the vectorial representation, it is useful to choose the independent

spatial variables in a way appropriate to the introduction of thb boundary conditions

at the nozzle walls. In the case of axial symmetry a suitable choice is to take the

steady-state potential function to replace the axial variable, and the steady-state

stream function to replace the radial variable. Indicating with as and Sn ele-

mentary (nondimensional) lengths in the direction of the unperturbed streamlines and

of their normal in the meridlonal plane of Figure 1. Equations (14) and (16) can bewritten

= dq r'= - (22)

as Sn

Hence the _: .:-e of the elementary length dl can be written, in terms of do * d'I'

and dO (0 repiesenting the azimuthal variable) as

2 (1\2 2

d1 2 = d 2 + h2d4/P + h2de 2 - q-d 2 + d b + r d

from which we get1 1

h = h4,=-; h r. (23)Sq

7

These quantities are used to calculate the divergences, gradients, and rotors appearingin the equations. If f is any scalar and F = Foeo + Foe, + Feeo is any vector.e, . e4 . ee being unit vectors normal to the surfaces b constant .IJ= constant0 = constant , we have

e, + e# + eq (24)

1 1 f,= +h (rq0- e, .+(

r

Using these relations, the conservation equations can be written explicitly, noticingthat q = qe and defining the components of the velocity perturbation

V. F u'e + v'e + To •' (27)

We obtain:

'.'ntinuity (Equation (17) divided by p3):

) ,m( ' +(

8

Momentum Equation (18), taking into account (21):

46-component (divided by 1):

(q -n-s'.) (29)t2

qK-component (divided by rp5):

t + ~ ?.T rvq) (30)

e-component (multiplied by r) taking into account that, in view of its axial

symmetry, the unperturbed flow does not depend on 0

(rw') +- q7 (rw') +__ = 0 (31)

Entropy, Equation (19):

Bs' s- + 0 (32)-at

In view of (9), Equation (20) can be rewritten in the form

1 p1 pI

st (33)

The preceding equations have been purposely written in a way that shows thbt, insteadof just the perturbations, it is convenient to choose the dependent variables to be

the combinations p'f/ , u'/q , v'/rp , rw' , p'/Y , and s'

We see at once that, from the preceding six equations in these six dependent variables,one can obtain a system of four equations in the first four of the dependent variables

Just listed. In fact, a' can be obtained from Equation (32) independently of theother euations once the corresponding s'-distribution is preacribed at the nozzleentrance as a function of time. after which p'/y can be obtained from (33) in termsof p'/- . However, the resulting system is still way too complicated to be amenableto solution. A major step in the way to a solution would be accomplished if the vari-ables were separable, which they are not in the system as it is.

5. CHOICE OF THE INDEPENDENT VARIABLES: BIDINENSIONAL NOZZLE

If the nozzle, and the corresponding unperturbed flow, are two-dimensional, the

independent variables are chosen in a similar way. According to Equations (15) and

(16), Equations (22) can be replaced by

"q T ;s bpq = ' (34)s Sn

9

Ss and Sn are contained in the plane of the unperturbed flow and the square of theelementary length is given by

d2 d'f 2 +dy 2

where dy represents the (nondimensional) spanwise coordinate. Hence we obtain,instead of (23) (replacing the O-variable with the z-variable).

1 1h ( = -7; h = ''- ; hy=1q bp q

and, instead of (24), (25), and (26).

Vf = (e + e, + ey (35)

a+ (36). = b L (J q 1 "

V x = bpF1-- - e,& +

Applying these relations, noticing again that q = qe, and defining

Q I = ulek + vep + wey .

Equations (17) to (20) can be explicitly expressed as follows.

Continuity (divided by p):

+ , (L +b- - =0. (38)

Momentum:

0-component (divided by q):

+ 2+ (39)

.t -4) BO, Q) TO7-;5 2-

(

10

qi-component (divided by bT ):

-- + -- P (40)

y-component, taking into account that the unperturbed flow does not depend on y

"6w' w' ++ o- -0 (41)

Entropy:

s + s2p p' (42)+ 0; s (42)

These last two equations are the same as for the axisymmetric nozzle, and can be usedto reduce to four the number of dependent variables. However the same difficulty asbefore is found, due to the complexity of the equations which prevents the separationof the variables.

6. SEPARATION OF THE VARIABLES FOR ONE-DIMENSIONALUNPERTURBED FLOW: AXISYMMETRIC NOZZLE

The chief obstacle that prevents the separation of the variables in Equations (28)to (33) is the fact that ' and p depend on both k and 'P in a way determinedby the solution of (12). An additional obstacle Is due to the presence of the factorsr and r-' appearing in the two last terms of Equation (28). These factors shouldbe expressed in terms of the independent variables k , ' in a way depending againon the solution of (12).

Botn obstacles are removed if the unperturbed flow is one-dimensional. This meansthat the dependence of 4 and p on 'P can be practically disregarded, so that theycan be considered practically uniform on each surface 4 = constant . It means alsothat the angle of obliquity of the streamlines with respect to the axis of symmetryis sufficiently small so that its cosine is practically 1 and the element of normalSn along the surface 4' = constant can be'identified with dr . Hence Eouation (22)can be integrated, providing

r2

'P r22

or

22r 2 --- , ('.3)pq

which provides a sirdple expression for r2 , to be introduced in (28). An additional

advantage of dropping the dependence of on 'P is 'hat ',wo terms of Equation (30)vanish identically.

11

Without writing down explicitly the equations obtained assuming .( , =;05PM

and introducing (43). we observe that the variables are now separable. In fact, if

we assume a harmonic time dependence, expressed in the complex form, and express the

dependent variables as follows':

p' -- = R(O )T(t)()(O)ei &Ot

P

Ul-- - U( ) (p)e(9)ei~t

v1

= V() ' (41,)E)()ei (44

rwi = W(O)T(,)E)'(O)eIt

p I w-- = pP() O(O)ei~

s' = s(Ob)W()ee i .

with !'(ql) =d/dlp , 8' (9) = dO/d . and w representing the nondimensional angularfrequency, related to the dimensional co* by

-c=o* (45)

The equations, divided by 'T6 exp (icot) , take the following form:

icoR4+q R' + 2pqV -+ +- 0 (46)

d - 1_ d- "iwU+-(qU) + P' - 2 .- S = 0 (47)

2 0 0 (48)(io V + Q V' + P) -=0 (48)

',(IW + 7W' +P)- = 0 (49)

The various functions appearing in (44) may be complex functions of the corresponding variable.

12

aS + QTS' 0 (50)

1S =y -P -R .(51)

where, again. primes represent differentiations with respect to the correspondingindependent variable.

It is interesting to express the components of the flow vorticity using (26) and(44). The result Is

V x q = ;5 (W - V)u'e'eite, + i- (U - W')-O'eitt4, + rpq2 (V' - U)IEei~tee . (52)r "

We see that the variables are separated in Equations (47) to (51). which are made upof factors depending on a single independent variable. Thus the only requirement forcomplete separation is that the variables qj and e disappear from (46). For 0to disappear. 0"/0 must be a constant; more precisely, in view of the necessaryperiodicity of 0 , we must have

- 2 (53)

E)

v representing an integer.

Introducing (53) in Equatiov (46). it appears that the dependence on ' disappearsonly if V = W and WO) satisfies the equation

"P + -- 4 constant. (54)

with the value of the constant to be determined from the boundary conditions.

This result shows that the requirement for the seprtation of the variables introducesa certain restriction on the solution of our equations. In fact; the condition

V = .W (55)

is equivalent, as (52) shows, to the condition that the vorticity component of theperturbed flow along the streamlines must be identically zero. The resulting loss ofgenerality does not appear too important, since in most practical cases no mechanimis present to generate axial vorticity in combustion chambers.

The general solution of (53) is*

e = cos V(e- c) ; 8 = e~i"(6 -a * (56) I

where OL represents a constant, and an unessential multiplicative constant has beentaken to be unity.

The follwing discussion concerning the 0 and T functione proceeds along the well knownlines used for acoustic oscillations in cyllnders . lie repeat it here for the purpoae ofcompleteness.

13

The first (56) leads to a standing mode of oscillation, as clearly seen from thefact that the amplitude of each of the perturbations defined by (44) vanishes r-n vfixed diametricd planes of the nozzle. The second (56) leads to a rotating mode ofoscillations, where the amplitude of the perturbations remains constant for fixed

values of ± v6 +cot . The anguiar speea of the rotating modes is do/dt = . Inboth cases : represents the number of the diametral nodal planes (standing or rotating)

for the mode under consideration.

Next. c3nsider (54). Taking () as Independent variable, this is transformed

into a Bessel equation of order Y . The general solution, if the constant appearingon the right-hand side is negative*. and set equal to - k 2/4 , is given by

r = C3JR(k(4,11) + CY,¥(k[idi)

where J. and Y. are Bessel functions of the first and second kind. respectively.Since Y. does not remain finite as 1P approaches zero. C, must vanish identicallyif the perturbations are to be finite on the nozzle axis. Hence, setting the un-essential multiplicative constant C3 to be unity, we have

j -- J(k[] ) (57)

The constant k is determined by considering the boundary condition v' = 0 atthe nozzle wall. which is a %P = constant surface. Calling 'Pr the value of thestream function at the wall t. and recalling (43). the condition v' = 0 at the wall

is satisfied ifJ;"(kf[plf ) = .3. (58)

The prime here indicates differentiation with respect to the argument. Hence. if

is the hth zero of J'(x) . in order to satisfy (58) we rst have k =and (57) becomes":

VI) = JR,(s'jhb PkW]+) = .3, (s-h P) (the last term being obtained from (43). Observe that the last form of T(0) and theforms (56) of 0(8) are exactly the same as for the acoustic oscillations of a gasin a cylindrical chamber, the only difference being that in (59) r w is the variableradius. rw(0') . of the nozzle sections, instead of a constant. Hence within the

validity of the one-dimensional assumption for the unperturbed flow, the perturbations

are distributed on each section of the nozzle in the same way as they are in the acoustic

oscillations in a cylindrical chamber. Corresponding to the lowest values of z, and

h we have for s,,h the values given in Table I on p.84 (Ref. 5)tf.

" Positive values of the constant, leadlng to the Bessel functions of imaginary argument, wouldnot allow the boundary condition at the nozzle wall to be satisfied.

f The value of 0. Is imediatcly obtaincd from (43) evaluated at the throat or at tte nozzleentrance, once the reference length L* has been chosen. Appropriate values for L* areeither tbo radiud of the throat section or the radius of the entrance section of the nozzle.

The index h has a meaning simflar to that of z : h - I represents the number of nodalcircles for the pressure perturbations, that is. the number of circles on each section onwhich the pressure does not oscillate.

tt Ths argument of the Bessel function in Reference 5 is va,,tr/rw so that their elgenva!ues3must be multiplied by ,v in order to obtain the values in Table I.

14

Inserting in Equation (46) the relations (55), (53), and (54) with the value of theconstant in the last relation given by - k2/4 - S~h/4 /w , we obtain

- - ,2 h2o +qR+ 2U1 - h -icoR +qR' +=q 2 0 . (60)

The separation of the variables is now complete.

7. SEPARATION OF THE VARIABLES FOR ONE-DIMENSIONALUNPERTURBED FLOW: BIDIIMENSIONAL NOZZLE

When ! and p are taken to be functions only of k . Equations (38) to (42)becomo separable too. We take

bp'

- = R,((P)T!(O)Y(y)eiwt

p1

ut

a = S1(kY)1(v)Y(y)eti t ,

and replace these expressions in the equations. Dividing all equations by) exp (it) , we obtain

iwn,+ a 1+ u 1 1+ b qv 1 = 0 (62)

u +qu+-qu2 - d-s = 0 (63)

T'P

(ico V + q- V + P1) - 0 (64)

(ico W + q' + P ) 0 (65)

L

x (I =q b, 0 (66

11

Sl , P -Ri" (67)

The flow vorticity is given by (37) after substJtution of (61):

V x qI = bq. I Vdl)yleicuteo + q(UI - WI)TY 'eiwteo +

+ -j(V' - U,)T'ye ~ez (6

We see that in Equations (63) to (67) the variables are separated. In order that the

separation may be complete we must have

I , Cy (69)

C. and Cy representing constants to be determined through the boundary conditions.

We notice that, contrary to the axisymmetric nozzle, in the two-dimensional caseno restriction is placed on the solution by the requirement of variable separation,

and that, according to (68), the vorticity of the flow may have non-vanishing componentsin any direction. The values of Co and Cy are determined by the boundary conditionsof vanishing normal velocity at the nozzle walls, that is v' = 0 at q/ = 0 and

P =w and w' = 0 at y = 0 and y = b *. We obser that, since in the assump-tion of small obliquity Sn may be identified with dx in the second Equation (34),x representing the nondimensional coordinate along the nozzle height, we obtain

'= bpjx, qw = bpqa , (70)

where a = a(qb) is the nondimensional height of the nozzle section under consideration.The value of Ow (corresponding, as for the axisymmetric nozzle, to the total fluxof mass through the nozzle) can be determined once the reference length L* has beenchosen, According to (61) the boundary conditions can be written

1(o) = (~) : 0; Y'(0) = Y'(b) = 0. (71)

The proper solutions of (69) satisfying (71) are

=Cos(M7T ~ =Cos(U7T.I

(72)

Y(y) = cos(N77) ,

* For the sake of symtry, the *axis" of the two-dimensional nozzle should be taken as theline 0 = 0 , y = 0 . It is more c.avenient, however, to take 1 = 0 on one of the curvedwalls of the nozzle, and y = 0 on one of the plane walls. The other curved wall is then a,= constant = Ow surface, and the other plane wall is the y = b surface.

16

M and N representing integral numbers defining the mode of transverse oscillation.

Their meaning is immediately found to be the number of nodal surfaces for the pressureperturbation in the corresponding direction. Equations (72) are the same as for theacoustic oscillations of a gas in a chamber of rectangular cross-section, so that againwe find that under the assumption of one-dimensional unperturbed flow the perturbationsare distributed in a similar fashion on each cross-section of the actual duct as theyare in the acoustic oscillations in a rectangular chamber.

The comparisor, of (72) and (69) gives

M* _

M 2 /y1 N

272

-P-€ b -- = y- b2

* IWReplacing these values in (62), and writing for brevity

MITb N(TSm = ; N = (73)b

we obtainioR 1 + q

2R2 + q2U1 - s2pq 2 V1 - 2W1 = 0 . (74)

The separation of the variables is now complete.

8. REDUCTION OF THE SYSTEM: AXIS'MMETRIC NOZZLE

We first observe that the case v = 0 , h =1, resulting in svh =0 correspondsto purely axial oscill&tions, since 0 and T become constants. In this case (48)and (49) are automatically satisfied, and Equations (46), (47). (50), and (51) containonly the variables R , U P , and S . The solution for this case has been discussedin References 2 and 3, but it will again be included as a particular case in the follow-ing treatment.

Similarly, the cases in which either v = 0 or h = 1 (purely radial or purelytangential oscillations) can be included in the following treatment as particularcases. We shall, therefore, discuss the case where v and h are arbitrary integers.

In this case, in order to satisfy (48) and (49), the corresponding expressions inparentheses must vanish. Subtracting one from the other we eliminate P and obtain

dic(W-V) + - (W-V) = 0 (75)

The general solution of this can be written in the form

W-V = Cof o , (76)

40

17

where, for brevity, we have written

fr =) e-iW do (77)

q5' being an integration variable and the lower integration limit being arbitrarilyset at the throat. Clearly the integration constant CI has to be taken zero inorder to satisfy (55), with the aforementioned loss of one degree of freedom in thesolution.

Equations (50) and (51) result in

s(5 = . = rf0 (78)

a= S(Oth) being an integration constant.

Introducing a function 4(P() such that

U = (79)

(47) can be integrated to

iwP+ q' +P = af 1 (80)

with

f1 (5 fo0(0') !2-Q dO' (81)

2fth

The integration constant has been incorporated in , which is defined anyway withinan arbitrary constant.

Since V satisfies the equation

iV + QV' +P = 0

we can eliminate P from this equation and (80), obtaining

d

which, upon integration, provides

V -f2 + Cif0 (82)

18

with

f _ f (€')f2( ) = f0(4) J __.. _- .-----f(-d'. (83)

Using (78). Equation (80) can be rewritten as

iwco + q2 + C 2R = o(f 1 -c 2f0 )

Equation (60) can be written in the same dependent variables making use of (82), asS2 2

+ "-Sv ' _!, - - (~f _ f )iWR + R 1 a 2 + C0fo)

Elimination of R from these two equations and use of (13) produce the followingequation

S21l76, d -Q),-+2 4 , ,:f 2 WWd hP

i~j (Ify +- If1 2h 2] vh -- jf-f _ 7 -f + -, o.y

This is the fundamental equation to which our system can be reduced; its solution canonly be obtained by numerical integration*, using the appropriate boundary conditions.

The right-hand side of (84) can be written in a somewhat simpler form by introducing

the function

f3 f (85)

0

and observing that the first two terzis within the brackets can be written as

d -dd f . fdkqd (f -IN) = jd df ficafo(f 3 - 1) + - (f 3 - 1) : { d fo(f 3 - )] - J (f3 1 f

Use has been made of the equation

dfoq" + ifo = 0, (86)

An exception is represented by the case of axial oscillations (s. = 0) and q linear with* in which case (84) reduces to the hypergeocetric equation

2,3

19

which fo satisfies. The last term within the brackets can also be expressed through

f3 , observing that with the help of (86) we get from (83)

f . fd If - f f2 1 1 f df

I'of~th (1 fth f/

= fo= f (87)

Hence (84) can be written in the form

L() = -cf 0 (CF(') + OT(2)) (88)

where L(O) represents the left-hand side of Equation (84) and the functions F areexpressed in terms of f3 alone by

22 df3 + .hP (F() _ q F(2) = qL -df-- 3- +---(2 Q(9

d4) 2q/jw 2iL ~ "t

The flow rotation can also be calculated from (52). Beczuse of (55) the >-componentvanishes, while the other two components are proportional to U - V1 . In view of(79) and (82) this can be written as

V/ = 0-df+ dfo

From the third expression (87) for f and (81) one obtains

df 2 1 df1 - _

d 21oio d (q

Lwhere [2- ( = (Y + I)) is the value at the throat, obtained from (13) by setting

= . Hence, making use of (86), one has

- -- [a( - )- -i C

2q2 1

and the vorticity* is given by

Vx exp i t- - .~e2 - q

tbt

Taking Into account that the variables are different we have

V d1/dP (S/2tk,))J(sh[k///Ii)

,, = N,,h

20

Observe that both TE)'/r and rT'@ vanish at r = 0 for v 1 . Only for v 1

are they finite at r = 0 and only for this v can there exist a vorticity on thenozzle axis. Observe also that the subtractive term in the exponential represents thetravel time of a gas particle to any station 4 ; therefore the exponential exhibitsthe transport of vorticity with the fluid elements. Clearly for Isentropic oscillations

(o = 0) and C, = 0 the oscillations are also irrotational, but, if there is an entropyperturbation, vorticity must be present no matter how C1 is chosen, except for axial

oscillation, when ' " = 0

9. REDUCTION OF THE SYSTEM: BIDIMENSIONAL NOZZLE

The developments of Section 8 can be repeated almost identically for the bidimensionalnozzle. The case of purely axial oscillations (obtained for H = N = 0) provides equa-tions which coincide with those of the axisymmetric nozzle. The difference between

the two only appears when transverse oscillations are present. If M or N is zerothe respective equation (64) or (65) is satisfied identically and the corresponding

dependent variable V, or W, disappears from the equations. All these cases can beobtained as particular cases from the case when neither M nor N vanish; in which

case we obtain, from (64) and (65),

W1 - V1 = Cofo , (90)

C. being an integration constant which (contrary to the case of circular section)

can be taken non-vanishing.

From (66) and (67) we get

s = P,-c , f (S1)

o= S(O) being again an arbitrary integration constant.

Taking

U = (92)

one obtains, from (63),

iwu 1 + QT + P 0of 1 (.93)

Combining this with (64) we obtain, as before,

V, = cf 2 + C1 f o . (94)

Hence we obtain, from (91) and (93).

ip + ' + 7R = 0-(f 1 - 7f 0 )1

.1a

21

and, from (74). making use of (90), (92), and (94).

iazR + Q2R' + -)(s2 2q2 + or = - (0-2 + C fo)(s q 2 + -) + 2Co f

Eliminating R from the last two equations we obtain

dO 2 c]

q2(c2C f(~ 2c ) + w 0~d +S~ (0-N)C+ ON

- Cc(s:pq + oN)fo + COC 2 oafO (95)

which can also be written

L 0( ( = o1 + F +rF( 2 ) ) (96)

where L1 (dQ) represents the left-hand side of (95) and

F( °) = 2

F!)= S2-__

" su q +N ;(97)

= df_3 + (S2P2q2 + o 2 + q

d M N' 21&j

From (68) we obtain the vorticity. Following a line similar to that of Section 8we obtain

b°'-- 2 - 2iwC1 hiyej e f

g e V eexpressing the vorticity property of being transportedtthe flud, and again we notice that the vorticity cannot vanish identically with

noniseutropic oscillations, except for purely axial oscillations, when ' Z= 0

22

10. ADMITTANCE CONDITION AT ,HE ENTRANCE OF

AN AXISYMMETRIC NOZZLE

The general solution of (88) is made of the general solution of the homogeneous

equation and of a particular solution of (88). In other words, if we know the solutionsof the equations

.(oh ) = 0 (98)

L(V~J ) ) = _-efo0F ( j ) (j = 1, 2) o(99)

the general solution is

1 = C ) +o$( 2 ) + C2'% + C h Al (100)

where 4 h are two independent solutions of (98) and C2 C3 arbitrary constants.Now observe that (98) has the singular points ! = 0 , = F= 1th = {2/(y + 1)} andA= co . For a supercritical nozzle with a finite entrance section. only the sonicsingularity at the throat is of concern. If the solution must be regular at the throat.nuthe solution, say , which is regular there can appear in (100); and the parti-

cular solutions V J ) must also be regular at the sonic point. Since all the singu-larity appears now in the independent solution h , the condition of regularity at

the throat is simply expressed by C3 = 0

Hence the rrp er solution of (88) is

O= c 1x ) ,-o ( ) +Clh, (101)

with V'Z ) 2 ) and h regular at the sonic point.

From (101) one obtains, making use of (79), (82). (55), (80). and (78). theexpressions

CU = (Pd +" a q + c 2 d=dq d d(dP

=C f) (102)! +P + q'U = - C Iio.)4 (Z) _ or(la;D( 2 1 _ f 1) - C2't4 b

( ,S = o"fo 0

If this is considered to be a system of linear equations for the determination ofC1 I a , and C2 with given values of U , V = W , P , and S for a given qbwesee that only three of these values can be arbitrarily prescribed. In other words, arelation must ex. st between any such set of four values, a relation that can be obtained

from the relatic of compatibility of the four linear equations (102). that is fromthe vanishing of the determinant

.L ____ _

r

23

d0 2) dd'h0~ 00d

V fo 0 (2) _ f 2 'Ph

P + q2U -1 01) f - if (2 ) iC1h

S 0 f0 0

Developing the determinant, dividing byf 2 h, writing for simplicity

- 1 d~ ; . :(i)(q) = 1 I j )d'b d4 (J)] 131 dph 'j(103)

and applying the last relation (37) we obtain

U" ) - - i&0] + Viwjc (1 ' + P(c 200') - -

ScP (Y + =2 - (104)

This relation (104) holds for any value of i ; in particular, it holds for the entrancesection of the nozzle. where it represents the admittance condition we have been lookingfor. The adittance condition simplifies in the isentropic case by taking S = 0identically. If the perturbed flow is irrotational according to the discussion ofSection 9 on the flow rotation, it must be isentropic as well, so that C1 = a = 0The corresponding admittance condition can be found directly from (102) which reduce to

U %V C~h. P + QfU = C2icL h.

j Elininatint C2 oue obtains

u = v - P = -P (105)

which represents two &&ittanca conditions when applied to the entrance of thb nozzle.Here . represents the irrotational admittance coefficient.

For purely axial oscillations, aince (48) and (49) are identically satisfied, onehas to disregard Equation (82), which makes use of (48). The corresponding equationscan be obtained from (102) disregarding the second one, and all terms in Z.

............... : -

ij 24

With the same procedure used above one finds from these equations the corresponding

admittance condition

U(; + iCo) + P + Sc(io (2) - f3 ) 0 (106)

For isentropic oscillations this condition can be used in the form

U P = -P = R (107)

In this case oL corresponds to the admittance ratio of the acousticians. Observethat, contrary to the c of (i05). this ci has to be calculated for s,, = 0 .

We see that in every case the knowledge of three functions. . and e(2)

defined in (103) is sufficient to determine the admittance condition. The Pumerical

procedure to calculate these functions is explained in Section 14. Here, however. itis useful to obtain a more explicit expression for the f(J) . From (98) and '99).recalling that L() is given by the first member of (84). we obtain

{ oT~) -%L (J))= ( - Thd) .(fh ( S) -(1 -_ 2i(C f0\ 5

=~ 0 hF (J (108)

Q2

This first order equstion has the general solution

b c--fo h()e ( '5 d(19

cfolc : 0 ( 0) onstant + f (c _2) Fe d)'] (109)

4 th

where

and '2s1aain-v - loge - (110) I

and 46' is again an integration variable, of which all quantities in the integrandsare functions. The lower limit in the integral of (110) is essential. The presenceof the second tcrm on the right-hand side of (110) causes exp 0(0)) to become

infinite at the sonic point. Clearly the only change for f(J) to remain finite

there (which is necessary if the solution must be regular) 'R the vanishing of theintegration constant in (109), since the lower limit of the corresponding integral

has been taken at the throat. Hence the proper form for the solution of (108) is

xpexp ( -Cd

_______ P_____ 21w

! A:.M el) - + c __ d,' .llp, d o'.

.h

25

where f0 has been replaced by its expression (77) and Ph by

Ph = exp {f p') d .immediately obtained from (103).

We see that VJ) can, in principle, be obtained from (111) once is known, usingonly integration processes, and that the solution of the Equations (99) is not necessary

if our scope is limited to the admittance condition. However, in practice (111) presentsnasty problems of evaluation, since - as the form of the exponents shows - both theintegrand and the factor in front of it tend tc oscillate faster and faster when thesonic point is approached, or when 'Z becomes small, toward the nozzle eutrance.

These oscillations are artificial and tend to cancel each other, but they make itdifficult to carry on the last integration which would require absurdly small integra-tion intervals. A better way of calculating the () is to make use of the equation

dio -W 2 . (112)

(0 - ql]ifJ)] + - (( x j - _

the validity of which -an 1e directly checked from (111).

11. ADMITTANCE CONDITION AT THE ENTRANCE OF A

BIDIMENSIONAL NOZZLE

For the bidimensional nozzle one has to consider one constant and one equation morethan in the previous case. That is. if the solutions of the equations

L =(',h) 0 (113)

L ((i)) = - fo1) (j = 0. 1.2) (114)

are known, the general solution of (96) remaining regular at = = (2f(Y + I)} -is

ti= COCO) + CIC') + ort 2 ) tll

where all partial solutions must be regular at

Hence we obtain, from (92). (94). (90). (93). and (91).

U1 d 0 d~ ddOd1 d~d~

V, =Co c (I(4' -o ' - ) 1 ) - C2) +

V1 -W S Co0 (115)

- -

26

p + -- - - + - - i (115)

Again the compatibility condition of these five equations between the four constants,

requires the vanishing of the corresponding determinant. After development, and divi-sion by f0Ph , the condition becomes

q 7e -) ico] + V icoc 2(ep) - + W iwc~0) +

PI(ce( )- S~c q2 h ) i ) (116)

1ne ::::ah~ ___ . j) + UV) - f3)

1~ l dJ) ( 117)1 ~1 c i.;P C- dO6 d ] -dki 0 1 h d

'uen applied at the nozzle entrance, (116) represents the admittance condition.

We notice that if the axial component of the vorticity vanishes (W, = V1) Equation(116) becomes identical to (104). However, the functioLs appearing in tle coefficientsare different, depending on the solution of different equations. The ispntropic caseagain can be derived imnediately from (116) taking S, = 0 , and the isentropic,

.irrotational case leads to two admittance conditions identical to (1M,), the only

difference being the subscript 1 on the various quantities. Finally, the case of

axial oscillations leads to equations identical with (106) and '107). Evidently in

this case there Is no difference between the quantities with the subscript 1 and thosewithout subscript; indeed Equations (84) and (95) coincide when s uh -= S¥ -= N = 0

About the functions , and eJ , which are the only ones to be determined in everycase for the purpose of obtaining the admittance condition, the same observationsdeveloped at the end of the preceding section for the corresponding quantities andc(J) apply with identical conclusions.

12. SIMILARITY OF NOZZLES: VELOCITY DISTRIBUTION

FOR REFERENCE NOZZLE

The solutioa of Equation (88), or of (98) and (99), depenis on the pczameters oand Sh (pw has a fixed value once the reference length is chosen; see footnote,p. 13); it depends moreover on the noz?.le geometry through the function (4)(fromwhich U( ) and (kO) are derived). The results obtained for a given nozzle that we

shall call the reference nozzle can, however, be immediately applied to a wholefamily of nozzles obtained by linear deformation of the axial scale. If z is theaxial lengt'. coordinat.e measured from the throat and positive in the flow direction,

i., the proper scale factor and r(z) is the axial distribution of velocity for

the :eference nozzle, &(z) = ,?re-jz) will be the general velocity distribution

...-... .. ..-- --- .... -- .- - . -

I27

for the derived family of nozzles. From (22), since, in view of the one-dimensional

assumption, Ss = dx , we obtain, indicating with Oref the potential for the reference

nozzle,

3dO = 83(z)dz = T-Jf(8z)d(8z) = d'ref

It appears that 1(&k- kth) = cref is also a function only of 83z = Zref , and thatas a result can be considered to be the same function of r for the whole family.

It is immediately seen that if L(k) , that is the first member of (84), is dividedby 32 , its form remains unaltered if the independent variable is taken to be efand the parameters w and sh are replaced by Wref = w/3 and svh ref sso that

s (O '-h) = e(e , rf SAre

where href is the solution of (98) obtained for the reference nozzle. It appearsfrom (103) chat

= 1 1 d _ 1 d~ret _ ( 191 S h) 1 dk re dOref ref (4>ref Wref Svh ref (119)

We conclude from (118) or (119) that it is sufficient to solve (98V only for the

reference nozzle. Howcver, the solution should be computed not only for the discreteset of values s~h of Table 1 (p.84), but for S~h continuously variable in a certainrange, in order to cover the possible range of values of the scale factor /3 . We seealso that if the nozzle entrance velocity is the same for all the nozzles of the given

family (that is, if the area contraction ratio of these nozzles is prescribed), thevalue of Pref corresponding to the entrance section is also the same so that ref

is only a function of ref and Svh ref For given scale factor 83 and mode (sh)

ref , as well as , is only a function of ao.

Similar observations can be .made on the quantities (J . Inspection of (89)

ahows that

P() (0; sh /ref\ref s h refF F (2 () (

F(02 6; 6o, sA ) = 8F \ @re f O ref ; Svh ref).

Hence we obtain at once from (111) that

1 ( ' ) (0 ; , s v I : = P -r e f 'r e f r f

g (2) . o Sh) = (2 ) o .1 (120)'W Svb ref (ref' ref' S h refd

We are particularly interested in the repercussion of these relztions on the admittancecondition (11)4). We can rewrite this condition in the form

U + AP + BV + CS = 0 , (121)

28

where the admittance coefficients are

A (122)

B- - (123)

c7(7 Th l + -~l () f

C -(124)

Because of (119) and (120) we see that

A(C;w. Svh) = Aref (ref ;'ref Svh refd

B(q; CO. sph) = 8Bref(kref ;' rer' Svh refd (125a)

SCU; Co. Svh) = Cref (4ref ; Wref * Svh ref)

Hence, from the admittance coefficients calculated for the reference nozzle in theappropriate range of the independent variable (or of ) and of the parameters, one

can obtain the admittance coefficients for any nozzle of the family.

The case of the two-dimensional nozzle presents the same property of nozzle

similarity. In fact, it is eauily seen from equations (113) and (117) that, defining

sM ref = SM/N3 and cTN ref = ON1 , we obtain

Plh ( k ; C' SM '7N) - ih ref (Oref Coref SM ref C N ref)

1 ( k ; o , SM a N) - 1 ref(kref Wref, Smref ONref)

and. from (97) and (111).

I . . = 1 ref((kref: c"ref' sM ref' 07t refd

I 5 M 0SN) = 89')ef(ref ;'ref' SMref ON ref)

Here again the subscript ref denotes solutions relative to the reference nozzle. As

a consequence if the 3dmittance coudition (116) is written in the form

U1 t AlPi + B1V1 + C18 + DW = 0

- _ _ ... ... .... .. ... . ..

Z M

where the expression for admittance coefficients A1 , Bi , C1 , Di is immediatelyobtained by comparison witn (116). Then it followc that

AI(O; w, s, crN) = A1 ref(ore"' ;oref. SMref I N refd

Bl(k;co, s m crN) Bi ref(Oref':oref, SMref, Nref)(125b)

Ci (O; co, sM ' N ) = Ci ref (Oref; ref I SM ref' N refd

D105; W, sMCIN) = /Dref(dref; Wref. SM ref,ONref) I/

Hence from the admittance coefficients for the reference nozzle calculated in a

sufficient range of variation of the independent variable and of the parameters onecan obtain directly those for any nozzle of the family.

It is interesting to notice that, as (125a) and (125b) show clearly, the scale changefrom the reference nozzle to any nozzle of the family affects in a different fashion

the relation between scalar or axial quantities (pressure, entropy, and axial velocitycomponents) and those including the transveise components of velocity.

13. NONLINEAR ANALYSIS

The linearized analysis which has been performed applies to small.-amplitudeoscillations and is most useful in the treatment of spontaneous inst-ibilities. It canbe used in the prediction of the stability of the steaay-state nozzle operation and,if the regime oscillation in the unstable situation has a small amplitude, it can beused to predict some characteristics of the oscillation. However, if the oscillationdoes not initiate spontaneously but instead requires a finite-size disturbance to thesteady-state operation in order to excite an oscillation, the linearized analysis isnot sufficient. Also, if the regine oscillation does not have a small amplitude (asis often the case), the linearized analysis does not accurately predict all of thecharacteristics of the oscillations. In these situations, a nonlinear analysis isbetter suited on the basis of accuracy.

The analysis of the axisymmetric nozzle was extended to include nonlinear effectsby Zinna. A perturbation series was employed where the perturbation quantity was anamplitude parameter. Of course, the first order solution is identical to the linearizedsolution discussed in previous sections. The second and higher order solitions re-present the nonlinear effects. In his work, Zinn completed the analysis up to andincluding third order; however, for the sake of brevity only the second order resultswill be presented here and the reader is referred to the original work for third orderresults.

The same nondimensional scheme given in Section 2 has been employed so that thenonlinear system of equations describing the oscillatory phenomenon is given byEquations (5) through (9). It is convenient to transform the time variable by settingy =cot , where co is the angular frequency, so that the solution is 27-periodic iny . In accordance with the standard proced'tre in nonlinear mechanics, the followingseries are substituted into the nonlinear system of equations:

LI

30

q = + + q() + + 0(e 3 )

p = +Ep ( + e 2 p (2 + O(e3 )

p = p +Ep(!) + E2p(2) + o(63)

s = S + Es 1 ) + e2 s(2 ) + O(e 3 )

_ O((O) +ej 1 ) +E 2 O(2) + 0(e 3 )

where E is the amplitude parameter and barred quantities, as before, denote steady-

state solutions. Separation of the equations according to powers of 6 yields the

sme equations for the steady-state quantities as (11), (12), and (13). The equations

for the first order coefficients q(l) . pO) . (1) , and s(1) are the same as

Equations (17) through (21) for q' . pl , p' , and s' , except that the partial-

differential operator B( )/t is replaced by (,(O)a( )/-ay

The second order system of equations becomes

Continuity:

By .O(2) (126)

Momentum:

Bq (2) _ 1 p (2)1a(0) 5y + V(4.q(2)) + (V x q(2)) x q + -- (.) ) +___p(2)

2 p ^P

(1) Zq(1) ) p") aq(') p (1)0) _ (.q())L a y By p

p(1) 2 )]

+- (V x q(l)) x + - V(q(l).q ( 1) ) + ( X q(1)) q(' (127)p 2

Entropy Equation:

s(2) s (1)

(o) By + .Vs = - q(l).vs(1) -((I)-B (128)

Equation of State:

(22 Y2+ (2) = ~ ( 1) 2 (~y (19(2) +

.. .. .. .. (1..

..29.. ..

31

As done in the linear analysis, a stream function and a potential are defined forthe axisymmetric case according to Equations (14) and (16). Furthermore, they arestill used as independent variables, as discussed in Section 4. Then we have (con-sistent with (27)) u , v , and w such that q = ueo + veo + we6 . Now the equationsfor p(1) , P(1) , s ( *) , u (1 ) , v (' ) , and w( l ) become identical to Equations (28)through (33) for the primed quantities except that again the time derivative has beentransformed. The equations for the second order quantities become the following:

Continuity:

_ ~ (_2+1 ( ( 2p)(0 + r p +

( ... .P1 u3

() f; 1

(r27 V1) ( I 1 rW~) PI) A. (130)

'k -Component of Momentum:

4(0)~ su +2 - + ~ ~ (2)

= (1 (( \ 1 dq2 y p( 2 p1 2-I- 4---- u (WI -

(1) I q -; I

2 BO ___ 2

2p 'k 2 BOk r (3

-B V (V(1)--.

32ii i b-Component of Momentum:

9-Component of Momentum:

TY(o) (rw( 2 )) "0 (r) + :-- - cL ) 2- - -

Iy Z - Ur(2 ) + 0 ..~ 'a (1 y)'a PL

To --- ( r( 1 ) + -- :-- - -- (132)

4~I~O q zia~p/ 5-P

Entropy:

c - q- 2 __ - .. =E. 14

Equation of State:

-+ = - =2 1) 7 (15

p 7,0 2(_ c \Y /

The first order condition given by (52) and (55) (zero axial vorticity) has been used

in deriving the above equations.

The time dependence of the first order solutions as skoun by" (44) and the form ofthe inhomogeneous parts of (130) through (135) indicate that

the y dependence of

those inhonogeneous terms is readily expressed as a Fourier series. This indicates

that second order solutions are of the form

n

(2)-- einY (.O.)Or n (133)

I,.

I

Entropy

33

rw (2 einy (h , 0n (136)

p(2)-- = Z enYKn((, qE . 0)

n

p(2)"- einyn(O. ,. 0)

7,0 n

8s2) E e e'n , 0,) .

n

As shown later, the solutions contain only terms corresponding to n = 0 and n = 2

Substitution of (136) into (130) through (135) and separation of the Fourier com-ponents yields

n % Kn Y- n h Pni 0 y+ + I'g'K +-+ 2pq--( L) + - = An (130a)

'a7T I d7 aBainoiOIw + 4 1 + :- -a - (131a)

in(°)71n + q 1- + T Z= (132a)

__a (133a)

inco(o)VA + q-Z.t En (124a)

(=n + Kn) - = Fn, (135a)

where An. etc.. are the coefficients in the Fourier expansion of A

ZB/-, etc.

In analogy with (79) we define

- - (137)

Then (131a), (132a), (133a). and (137) yield, upon combination and integration,

1n= fOn [ Cn a d d + In-_ -U (dth. .0)] fon' n0.7 thh 6.) (138)

oth

~34I _nG ~ +IO 7 nfo O h -P +' fo:(Oth'Vk'

0 ) ,(139)

where the definitions are made that

Gn = 2 0 -n V dP + Bn

fon = e-nwo do

th

In addition, (134a) is readily integrated to obtain

' f d0' +o"n(th. 0) (140)

Ith o

Using (131a). (135a). and (137) through (140) to substitute into (130a), we find thst

q- (7+ Gn) -2Pfo +2 I , (141)

uthere i is defined as

Z0 2-n-Zdb +-Dt..O n to.

H. = Pon

tb

and is a potential function defined by the relation

The inhomo(eneous part IW in (141) has been siaplified by assuing zero axial vorticityto second order.Kt

I35

The form of Equations (137) through (141) and the form of the first order solution(as given by (44)). together with the boundary condition that the radial velocityvanishes at the nozzle wall, indicate how the radial and tangential dependencies ofthe second order solution may be obtained; the solution may conveniently be found inthe form of an expansion in eigenfunctions. Following standard procedures. Zinn obtainsthe following for a wave traveling* in the negative 0 direction

'n( t) = In(tin)v. q(Jnv 9nv e (142a)

while for a standing wave he obtains

Its) = ( ) J.~ (0i) jCos . (142b)If n A 0.2 In(s) (t) 0 . =e symbol means that m = 0 and m 2

In 0h=0,2

droduce the only nonzero values of In.av, . Note that w( ) was properly set equalto zero in the above relations f3r the cowbustion instability problem considered inReference P. Otherwise, the first harmonic would appear above. Note that sa. Qis the q root of J',(x) = 0 . Solutions are found in the following form fortraveling waves

4I--

- -.

77nt) = ni V ( t ) (0) J de n0 nsa.n.nv.q d n

~t) = niv[ W t)(~n(si~ (1P ) Ielae((143a)

Since transverse standing waves are the sun of two transverse traveling waves in tb2 linearcase. no distinction was necessary there. However. here in the ntnlinear case. it isnecessary since no siaxle relation applies between the tic types of waves.

36

7TI~) ~ q(;{)] ~" 9 (143a)

a 0. , q ()Jm1 , ) osv, Q

o . U e I) o

t~) (~t V j ( ( 'ii Co

m=o.2 1 Lq=0 * \c''* ~*

n( SiD 21-'9 (14(3b).

2,, 2 (s) (o

0= 1

Vrs (S) ill (SU.

n~1[ n. m., Wm'Q dip qCos msv&

am=0,2 q= 0

-2 / ( ~ i 19(4b

7 nT(S) = P(S) (~n,3U ~ o v

K cos ove .

Cos MVu

Again. Lbe only aloal ausfor m are 0 and 2.

37

Substitution of the proper forms of (142) and (143) into (141) and separation of

the elgenfunctions leads to the following ordinary differential equation for both

traveling and standing waves

, - 21 - -+ 2ir.'o d nmv. +

7 17c d95 --

- d I QC2i n a'. (144)2 c 2qw .

Reference 8 shows that the inhomogeneous part cay be organized into a m-ore convenientform. For both traveling and standing waves. (141) and (142) yield

_ (f (C F F F( 3 ) (145)cfon iD.3'.q n.O.q fl.D.Q Omq D. n.Q-.

wheres2

F(2) - -, s-. a'.q q--d-'(b 2 iw . -m 2 ( 2cf.n qi

FM, (A.,,q + E..n. y. q fon n v

fn-' d Fn n ' " Bn ' : ' "v q n = ' qd

MU ,

in, d(o F

+ i n(n[F.. q v Bn.nu.q Nn.n . -

C B

_ P n. W..q - B.,,.9 a 2P' de'

th

ln~ - I - 7 f dc'

2cf. d c'3 ,Q ~ N12

CiD.a . 3'. h .D D ath)

38 0 '"a.q "- n"'VQ(h)

Nn'v' -2 fond 1 Q'fon

fth ,th/

An~mvq, Bn. a q , etc., are coefficients in the eigenfunction expansions of A.Bn aec. The details of their evaluation are presented in Reference 8. Of course,

the coefficients are different for traveling waves and standing waves.

The similarity between (145) and (75) is instructive; it indicates a similar method!of solution should be pursued for the nonlinear problem as has been pursued for the

linear problem. The solution of (144) is of the form

myQ n.my qn.my + On. my. n. my. Q+

+ b(3)v~ + C2n.em %i'n.mv, q + 03n~mv.q~hn~ay~q •(146)

Each of the terms above except one corresponds to a term of (100). The first twoterms n vqand qn v are particular solutions due to the vorticity andentropy, respectively. The third term 1I(my is a particular solution due to non-

linear t:wrotational effects and does not correspond to anything in (100). The lasttwo terms are homogeneous solutions. One of these hn,nv, q is regular at the throat;in order to eliminate the singular solution thnmv,q , C3n~mv.q = 0 is imposed.

Now (143) may be substituted into (137) through (140) and those equations ma beseparated to yield

= dn'mv'q (147)Un. myq dq6

7f:lth

+ tbn, my, In C, my, qfon - 'n, my. qf211(48

Wn. my~ Vnmq (149)

Snmu q= ~n En -- 'QdV +a-,mv, (150)qon

Lh

JJ

39

where

f2n =,n for on

h

in 2 f ond4d/th

Integration of (131a) and combination with (137), (143a), and (150) determines

Pn mV~q •Then this result is combined with (135). (143). and (150) to determine

RA.m1IQ • The results are

1 2 do' - d + .q - -n, .q I .v.q (151)

th

Rn. V (Fn.v.q +Pnsj.q)-Snup~q (152)

Obviously, since the coefficients A , Bn,mv q etc., are different for

traveling and standing, so will the solutions given by (146) through (152) be different.

Note further that the co in the linear solutions given by (102) actually representsthe 001 of (144) through (152). In the linearized problem it is not necessary

to distinguish between the two, since the difference is of higher order. Similarity

is notice

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