behavioir of supercritical nozzles .under three ... · agardograph 117 £ behavioir of...
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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 ORGANIZATION
4ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT
1(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 analyses
were 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 William
B.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
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]\
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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
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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 Coefficient
Sh =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
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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
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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
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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 bidimensional
n)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
functions defined in (136)
functions defined in (155)
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
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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 pIst (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 the
other 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 terms
of 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 the
elementary 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)eI t
p I w-- = pP() O(O)e i~
s' = s(Ob)W()ee i .
with !'(ql) =d/dlp , 8' (9) = dO/d . and w representing the nondimensional angular
frequency, 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 been
taken 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 the
fact that the amplitude of each of the perturbations defined by (44) vanishes r-n v
fixed diametricd planes of the nozzle. The second (56) leads to a rotating mode of
oscillations, 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 N27
2
-P-€ b -- = y- b2
* IWReplacing these values in (62), and writing for brevity
MITb N(TSm = ; N = (73)b
we obtainioR 1 + q2R2 + 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(9d4) 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 the
nozzle axis. Observe also that the subtractive term in the exponential represents thetravel time of a gas particle to any station 4 ; therefore the exponential exhibits
the 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 entropy
perturbation, 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 bidimensional
nozzle. 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 zero
the respective equation (64) or (65) is satisfied identically and the corresponding
dependent variable V, or W, disappears from the equations. All these cases can be
obtained 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 + (S2P 2q2 + 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)] 13
1 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 looking
for. 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~ ddOd 1 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 coefficients
are 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 divided
by 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 appears
from (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 vorticity
to 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 2In 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 represents
the 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 noticed between (104) and (147) through (151).
Substitution of (146) with C ~n -0 into (147), (148). (150), and (151) leads
to four equations which are linear in the th-ee constants 0n., 1 * mUav.Q - and
C2n.,1. . In similar fashion to the development in Section 10, the requirement of
compatibility leads to the following relationship between U, *. ' PU.8',q Vn.xV.Q
and Snv.q
U ,' ,(1) -n , q) -inc(o)] +un,mv.aq c n,MV. n.a l"Q
+V ii (O)- 1 ( q - _ 1n.mioq) -+Vn.atV.q-" "U .mv.q Pn.aVQ c fn.aV.q AM,
S c- FQt e() + n.w(O)g(2) fa. up, q 2 nIV.q - n. -V. 3n ] n
On n, MV, Q +-N2 ,vq-- Sn1mIq
+ ) 2 +-~fl (3 -1n +)
-. n ~ , 4 c~.a~.~n v Q n ,. -" 'n3 l , .2 n . v .q
+ inw 0 rt2 a.q 3n . ]
_ L
40
The following definitions have been made in order to obtain the simplif'ed version
(153)
1 dhnnmv, q dhnmQ (154)
g[ d d ]n.mV.q n.mv.q d ", hn.,mMq n. mJd, n.(155
fonthn.mi',qdoQdP a4
where J = 1.2,3 ,
1l(C1) = i 1- yv.q for n on n~mp~q Bn,.q -
\\94t't on
f d7 f n nfhy~~~2)~do do -f IIn.m
h 2 /
-- nawq =2 hdp" or qzf n:172() =F.. dE' .
-o, al. d fdonf' + on
Equation (153) applies everywhere along the convergent portion of the nozzle and.when applied at the nozzle entrance, is the nozzle admittance expression. The simi-
larity with (104) is immediately seen.
In the irrotationL case where 0 n, . and Cm.1 , are zero the nozzle admittanceexpression is simplilied. In that case (147), (148). nd (151) yield, after application
of the definitions (154) and (155),
Un.mq - n. mv, qVn. my. Q c fon n, my.Q + in o)Un,mV.Q +
+ n. m,,,q(Pa.myq + aUn.m..q)
T (2) (o)T-f; f(()on. my,', my. c onn mv. Q (156)
The problem remains to determine the functions nm'Q and !, J to substitureinto the relations (153) and (156). Reference 8 shows that in v q is governed bya complex Riccati equation in a similar manner to the function n the linear case
r
41
of Section 10. Also, if w were replaced by nzvo) , { by (J) by-LI) *fM is -
e(J) and "' , in (112). the governing euatio" for J iso'. nioed. So : ssu,,, ;l techniqies may be used to determine , and
Eij) as are used Lo determine and e(J) . These methods are dis"ed wth
respect to the linear theory in the next section.
There is one important difference between the linear case and the nonlinu, . case.
This difference appears on account of that inhomogeneous part of (144) proportional(3) c(3)
to F(3 ) and, consequently, on account of the solution of the functionn, ml'*q ,my
F(3 ) is a sum of terms each of which is proportional to the squares or double
products of the first order solutions for the flow properties as given in (102).
Therefore, in order to calculate the second admittance coefficients, it is necessary
to calculate the first order flow properties. Also. or , C1 . and C2 in (102)
should be known to completely determine the second order admittance coefficients. That
would require a coupled solution of the oscillatory phenomena in the nozzle and in the
chamber preceding the nozzle.
There is a prefeired method of solution which uncouples the numerical integration
of the nozzle equat-o.ns from the analysis of the chamber flow, although it was notia ur o p 3) ad f 10 ). F(3), edemployed in Reference 8. Due to the nature of F(3 Q and of (102). -n end
e(3) may be written in the convenient forms
a(3) 1(C)2 + 52(C)2 + 83(a)2 + S(CIC2) + 8S(a) + 8(C2)n.ml.q 1 22' '' 'V' s'1 6 '
(157)
' _= 1(c )2 + a(c2)' + c3(a)2 + ;(c~c2) + N(cla) + a6 (co)
where Si and 0 are indepen.ent of ,. C2 . and a. Since the coefficients of
S are constant throughout the nozzle, S1 may replace F(3) and cC1 may replnce3) V. in the modified version of (112). Of course, the a1' may be determined
by'numn(ical integration without a knowledge of the constants C1 , C2 * and a.N-e ha () (2) _ . 3) . . . .Note that - . v.q and a) may also be written in Identical form to
(157) so that the inhomogeneous parts of (153) and (156) may be presented in thatsame form, where C, . C2 . and o' remain to be determined.
Note that there are actually a double infinity (two n values and an infinity of q
values) of relations (153) and (156) for traveling waves. In the standing wave case.
there are a quadruple infinity (two n values, two m values, and an infinity of q
values) of those relations. Each of these infinite number of relations is to be applied
as the admittance relation for a particular eigenfunction. In a practical sense, con-
vergence will occur with a finite number of terms.
Qite similar relations to (153) and (157) are presented in Referen:e 8 for the
third order (in 6) quantities. Of course, the higher the order, the x re cumbersome
are the algebraic developments and the numerical integrations to be performed.
iii 4
29
1
,I
j V= ,
43
PART II. APPLICATIONS
14. CALCULATIONS FOR "CONICAL" NOZZLES
The vast majority of nozzles in practical devices are axisymetric. with thegeneratrix of the convergent portion (Fig. 17) consisting of a straight segen'. generatinga convergent cone, and circular arcs, generating the throat portion and possibly theentrance portion (which connects the conical convergent to the cylindrical combustionchamber).
The calculations were performed for a reference nozzle of this type, using theequations developed in Part I for the axisymmetric case. In order to include in asingle calculation nozzles of various contraztion ratios, the entrance portion of the
nozzle was assumed to be part of the conical convergent, so that the slope is dis-
continuous at the entrance (Fig. 17). Within the one-dimensional approximatton, thisallows choosing each station in the calculations to be the nozzle entrance station.
The effect of the presence of an entrance portion other than conical was studied
separately (Section 15) and found indeed to be generally small.
It must be observed that, according to Section 12. a whole family of nozzles canbe obtained by axially stretching or shrinking the reference nozzle. However, the
generatrix of the nozzles thus obtained is made of straight segments and elliptical
(rather than circular) arcs. This means that the correspondence between a practicallydesigned nozzle (different from the reference nozzle) and a nozzle of the family isonly approximate in the throat (and maybe the entrance) portions.
In Part I the physical quantities have been nondimensionalized with respect to the
steady-state stagnation properties. Indeed. this is the most appr, priate choice.because the ?Axe stagnation properties can be used as reference quantities in the
study of the combustion chamber behavior, which makes use of the admittance conditionat the nozzle entrance. For the same reason it is logical to choose the combustion
chamber radius (or, in the two-dimensional case. its height) to be the appropriate
reference length L* introduced in Section 2.
?-
44
On the other hand when this study was undertaken several years ago the referencequantities were taken to be the static properties at the throat and the throat radius(so that the nondimensional steady-state pressure, density, velocity and radius atthe throat took the value of unity), and the initial programming for the computer wasdeveloped on this basis. In subsequent calculations, rather than changing the program,it was decided to stick with this choice so far as the calculations were concerned,only switching to the stagnation reference properties when the final admittance co-efficients were calculated.
Although this may be a little confusing, the decisicn was made to keep in thisPart II the alternate choice of the reference system so as to avoid the laboriousconversion of numerical results or analytical developments besed on the throat refe-rence system to the stagnation reference system. However. we shall use for thevariables the same notations used in Part I, explicitly stating when the stagn.ationreference system is used instead of the throat reference system.
In view of the relation
Cth C
which immediately follows from (13) and from the definition (9), Equations (12) and(13) become
p ~ - -_I ___
P 2 2
Of course, since at the throat we have Z= =1= 1 =r?= 1 we obtain, from (43)I "i = .• Equation (88) becoses
(~) = -7 ~ ~(2 7+
QQ -iq dq
- c- dSP (2/-1) (158)
= _ (C F(:) + oa(2))cf o .
where, instead of (89). (77) and (85). we have
F")= s2.42/(/)
.--. df 3 s h ( 2 +s 1P q d-- +1w2io2Cf
45
f exp o
- 2 fo f
0
The appropriate solution of the equation for 4) is given by the equation corres-
ponding to (101).
4, = C 1 ) +o 4(2) +C2 h
where all the 4t s must be regular at the throat. The admittance relation can bewritten in the form corresponding to (121),
U + AP + BV + CS = 0 , (159)
with U , P .V and S still defined by (44) and A.B.C by (122), (123) and (124)with =1.
This, however, is not the most convenient form for use as a boundary condition onthe oscillatory flow in the combustton chamber. Indeed, the equations in the cylindri-cal chamber are usually separated by setting
U,= Ut(z)J,,(svhr)9(9)ei~tut = 13(z) jv-(shr)d()e1t
dr(160)
P,
s' = A(z)J,(s,,hr)4()ei~t
where r is nondiensionalized by the chamber radius, u' , v' by the stagnationsound velocity Z6 and p' by the stagnation pressure.
Comparing (160) with (44). and taking into account the difference of the referencesystems, we obtain
U = i + I + A
(v+1)v/(v1/ 2Y'I 1) '
After substitution the admittance condition becomes Ii
U+ P+ U+C = 0 . (161)
46
where
= c1CL) c 4( )
( 2 \i... I(i - 7)Wl) - i +
Certain (otherwise redundant) additional definitions are made necessary by the way
the original computer program %a written. Instead of 0 we introduce a new indepen-
dent variable
2- ,0 - Ot h)
The value P derives from the tollowing simple considerations. Since. by definition,dO = idz . we find d4 = 2- dz , with z representing again the physical axial co-
ordinate. Hence
dq1 I diK dZ
The particular choice
K = d
Vd z~h
results in dj-/d= 1 at 0 . This is the value of K that was adopted .n thefollowing calculations. For the throat geometry of Figure 17. K con be expressedexplicitly (using, for instance, the procedure outlined in Reference 9) as
2 2 1yK +R I
In conjunction with the independent variable the additional definitions were
made
=2K! 2Ke(1 ) , (a KWO . t K8&'.
In terms of the variables the admittance coefficients may be written as follows:
a'/= - -' (2(V- ) i l).. 0 +)
LWK 1(Y-1) -Q(clem ) 0 -i' (162)
47
+ - - + (162)
The admittance coefficients are complex numbers since 0 AM f3 and iciare complex.
An interesting observation is that the second order differential equation (158)need not be solveJ for 'h • ') end (2) It Is sufficient indeed to solvedirectly the equations which are satisfied by (1) and (2) that is-
(d= (g + ib)Z - (j - ik) (103)
+ (164)
7'+1 4
i dO (Y, + 1)(I- €- )" +2 1Vh2(Y 2)ic
+ +----~~K +:f) (165)
where the following definitions apply:
b =- 7)
7+ I_'Td'
h -
4 4 Vh
Y - 1 7d7-,k = 1 d
4 cd~
Eq~utIou (163) Is the Riccati equatica obtained directly from (158) upon substitutiom offor (I/)(z/d.). Equatlocs (164) and (!65) are obtained, after the proper adJustaent, fromEquations (112).
48
The nonlinear, first order equation for is a complex Riccati equation and may onlybe solved by numerical integration. Once this is done, the linear, first order equationsfor ari (2) may be solved, obtaining the standard integral forms which areessentially identical to the form given by (111). However, rather than numericallyevaluating the integral solutions, it was more convenient to solve all three complex(or six real) equations simultaneously by numerical integration.
The numerical integration requires statements on the initial conditions. Theseare provided by the condition of regularity at the throat ( =0). After assuming a
Taylor series expansioL about the throat and evaluating the coefficients in the series,we find
U(O) c-: + iI38
I(0) = 0
g(2)(0) =0
where
+1A 2 -jh -
2 4 4
j2 + 1 22
8 4
c22 + (-7 )2
Since the Riccti equation is singular at the throat (7 - - 0), the numericalintegration cannot begin exactly there. So, in addition to the above conditions the
first derivative of t at the init~ia. point must be obtained from the regular expansionand supplied to the numerical integration scheme. This initial condition is
K4$)t h =~ +,
where 2
+yI)~ (y'+ I)2 (oC -/32) + (y + 1)c 0 o/ 0 +21: + (y'+ 1)2 2 0 i,8
+- (72 + y+ (y (+ 1)2 + 1o 3)
4 2 2 +4
- _+. _ - 'h)*'-jI .'b .
jY j
49
+ 1 ) 2 + i) °- a; + P 2 + 4 '0
8 2
Note that bis a coefficient in the power series expansion for q7- That is+ I + b2 + the (v a ; (i determine d by the nozle geometry in
the throat vicinity. The above-mentioned power series shows the convenience of thetransformation of the independent variable from 05 to qb;the coefficient of thefirst order term in the power series is unity.
Rather than numerically evaluating the integral for f3 , it is convenient to solvea first order differential equation for f3 simultaneously with the equations for
and (2) As directly checked, the proper equation is as follows:
d - _ i._ 1 dq-
7d-(C fl) - (C f 3 ) =--- ,(166)dc. 2i 2 d4)
subject to the initial condition that f3 (O) = 0
The steady-state velocity profile (6) must be determined for tnc given geometryof the convergent portion of the nozzle. This cannot be found in closed form butmust be determined either approximately or indirectly. It is difficult to find anapproximate form of (4) which is very accurate over a wide class of nozzle geometries.By indirect means the exact form of versus 4 may be determined and a table maybe constructed. This would proceed in the following manner:
z$=2K (0 - th)= 2K fo 4dz'
where z is the axial coordinate. Furthermore, for circular cross-sections, thecontinuity, isentropic, and isoenergetic relations may be used to relate the chamberradius r to the velocity q . The result is
r1
_2 2
This equation could be used directly for given nozzle geometry, to calculate l(4)) .
This procedure, however, is rather clumsy from the point of view of numerical calcula-tions, for several reasons that we shall not discuss.
50
Instead, a much simpler procedure was employed which results in a straightforwardcomputational procedure. By differentiation of the above relations, a first orderdifferential equation is found to describe ntato o The equation can merely be solvedsimultaneously with the equations for todescrband f3 * The equation is
of the form
d d dr dz d 1 dr
dO dr dz d0k 2K4 dr/d dz
so that it only remains to find an analytical form for dr/dz . For typical geometries,
this is a simple matter.
The major portion of the calculations has been performed for the nozzle which wehave already considered, with the generatrix shaped as a circular arc near the throatand with a smooth transition to a conical nozzle in the remainder of the convergentportion, as shown in Figure 1. The differential equation for this case is found to be:
(2R(r - 1) - (r - 1)2)1< - in circular region
2K(R + 1 - r) dr/d(^ = (167)
tan #1coia- in conical region
2R7 dr/d4
where all lengths are nondimensionalized with respect to the throat radius. Accord-ing to (167), and d/dck (but not d2 ,/dk2 ) are continuous at the matching point
(z = - R sino 1) between the circular and conical portions. This is necessary and
sufficient to perform the integration.
The initial condition is 1(0) =1 . Note that ' = 0 is a singular point sincer = 1 and dr/d = 0 there. One can, however, find a regular expansion which gives
di/d(0) = j (in agreement with the value (d--/d4)th = 1) thereby allowing the numericalintegration to be performed. From this same series expansion, the term b appearing
in the initial conditions for may be found. For the present geomentry, it is
2 - ^/
6( + 1)
The relationship between the nondimensional distance from the throat z and the meanvelocity & is
[ s (2R(r - 1) - (r - I)2) if 0 ) z > - R sin 0
z cos , -- Rsin (168)tar.s0 Cos 0
The system of differential equations (163) through (167) have been solved using aMilne variable-step-size technique by means of a Fortran IV program on an IBM 7094computer. The calculations are made with double precision and a predictor-corrector
51
scheme is employed which gives an accuracy of seven (or, in other words, the ratio of
the error in the integration to the correct value is of the order of 10-7). Theseare found to be the optimum precision and accuracy within the limits of the operating
range.
The iralue T = 1.2 was taken; w values were in the range 0 4 w 10 and Sphvalues in the range 0 4 svh 4 9 . For each combination of &) and Svh , the syztemwas integrated along the nozzle length beginning at the throat where the Mach number*is unity and extending to the point where the Mach number is 0.05. The solutionswere determined at certain specified values of the Mach number by interpolation and
then the admittance coefficients were calculated at these specified points by meansof (162).
In using these results, one would take the values of the admittance coefficientsat that Mach number which equals the entrance Mach number of the particular nozzle ofnterest. In this way, as already indicated at the beginning of this section, oneintration presents information for an infinity of contraction ratios. Of course.we are considering here a conical nozzle joined directly to a cylindrical chamber.
In practice, a smooth transition would occur between the conical and cylindricalportions. However, an exact calculation of this actual situation would require oneintegration for the contraction ratio. So the approximation employed in the calcula-
tions is a large-time-saving technique. The validity of this approximation was verifiedby calculations with the more realistic nozzle shape of Figure 17. These calculations
are discussed in Section 15.
The radius of curvature of the circular portion of the nozzle wall was equal tothe throat diameter (R = 2) while the semi-angle of the conical portion was 300 inthese calculatio's (Figure 1 is actually drawn with these specifications). It isimportant to remc.nber, however, that the admittance coefficients may be applied by ascaling procedure to a whole class of nozzle shapes. As shown in Section 13 andillustrated in Figure 2, any nozzle shape which can be obtained by "stretching" thewall of the reference nozzle uniformly in the axial direction by a factor 1/ is a
member of this class. "Shrinking" the nozzle shape is simply considered by lettingbe larger than unity. The scaling rules are given in the figure. Scaling in thetadial direction is trivial, since all lengths are nondimensionalized with respect tothe throat radius. Note that the radius of curvature at the throat R and the con-vergence angle 0, dto not vary independently in the scaling procedure. So all
"conical" nozzles cannot be obtained exactly by scaling from any one reference nozzle.However, for the range of shapes of practical interest, good approximations can usuallybe obtained by scaling from the reference nozzle. This is discussed in detail inSection 19.
Now that the possibility of scaling has been included, s~,h has been replaced by
Svhffi and this value will no longer necessarily correspond to an eigenvalue. Therefore,rather than taking the usual eigenvalues for Svh - it is more convenient to use
integral values. If results are desired for non-integral values of svh , interpolationsmay be made on certai- convenient cross-plots.
* Note that the mean flow ich number may replace the axial coordinate as the independentvariable for the purpose of presenting the results.
52
Figures 3 through 6 show results of the numerical integration for a sample case(0j= 0.5, svh = 1.0) where the frequency is not so high. In Figure 3, r is plotted
versus axial distance showing a gradual change in due to the relatively largepressure wavelength. Figure 4 shows r2) to be undulating* rapidly due to the
relatively small entropy and vorticity wavelengths. Note, of course, that pressurewaves propagate with the speed of sgund, while entropy and vorticity waves propagatewith the subsonic gas velocity. Figures 5 and 6 show the admittance coefficients Orand Bi (which are the most pertinent from a stability point of view) plotted versusaxial distance. Superimposed upon a gradual change due to the pressure waves, we seea rapid undulation due to the entropy and vorticity waves. It is apparent that the
admittance coefficients, and therefore the stability of the motor operation, can bequite sensitive to contraction ratio variations.
At higher frequencies the oscillations become more severe, since undulations in
the admittance coefficients occur due to pressure waves in that case. The undulationsdue to entropy and vorticity waves become still more rapid. It is worthwhile todiscuss the nature of the phenomenon at higher frequencies and its effect upon the
utility of the calculations.
An interesting phenomenon occurs in the solution of the second order equation forand, therefore, in t , M , e(2) . and the admittance coefficients. The real
part of the coefficient of in the second order Equation (158) is
*4 2 '-2/(Y-1) 2 = 2S-Svh = 4K
This may become negative for low values of the ratio W/s., , espotially in the highMach nur)er region (where & is large). Note that this never happens for longitudinaloscillations where sh = 0 . If, for a given oscillation (co and s,4 are fixed).this number changes sign as ! varies with , the equation has a turning point.In the region of low & , where this number is positive, the solution for '(') isundulatory, while in the region of high 1 where the number is negative. 1( 6) is
*not undulatory. Undulations in ( ) imply that longitudinal undulations as well astransverse undulations occur in the nozzle.
This turning point occurs in the range of physical interest for either transverse
mode undulations in configurations where the nozzle volume is large compared to the"1chaber t volume or for mixed mode undulations. The physical reason is that the same
frequency may be a low frequency in the small diameter, high . region near the
throat while it is at the same time a high frequency in the large diameter, lowregion away from the throat. In the case of transverse mode undulations in configura-tions where the nozzle volume is large compared to the chamber volume, the frequencyis determined by a characteristic dimension which is smaller than the chamber radius.Similarly, the characteristic dimension for mixed mode undulations is smaller than
the chamber radius. So, in both cases, the nozzle may have two separate regions.
One region, where the radius is large compared to the characteristic dimension, con-tains longitudinal undulations as well as transverse undulations. This regionobviously will always occur for mixed mode undulations, but even in that case need
J"'Undulation" pertains to 3pace-wise variations while *oscillation" pertains to time-w'sevariations.
t As already indicated, the chamber is that portion which is of constant diameter while thenozzle is the convergent portion.
1 _ _
-- - * -
53
not extend all the way to the throat. The other region, where the radius is smallcompared to the characteristic dimension, contains only transverse undulations. Thisregion always occurs for transverse mode undulations and, in that case, unless the
nozzle volume is quite large compared to the chamber volume, it extends over the wholevolume. It is seen, therefore, that in a given situation either or both regions may
occur.
Any undulation in 'ph means that 4Phr -and %hi pass through zero, although, in
general, not simultaneously. This means can become large locally with the resultthat ) ( , and the admittance coefficients also exhibit wild behavior locally.
These regions of wild behavior do appear in the results of the calculations for certain
ranges of high w/5vh and low Mach number. The simple physical interpretation isthat nodal points occur in these regions and the reciprocal of the admittance co-
efficient (which is itself an admittance coefficient) is small. So, the wild behaviorin the mathematics is seen not to be a physical wild behavior by merely observing the
reciprocal of the results.
However, since wild behavior means large changes over a small range, it is not
feasible to interpolate between two calculated values in this range to obtain a thirdvalue*. It is desirable therefore to find another means for determining the admitcancevalues in this troLublesomerange. This has been accomplished by the development ofasymptotic soluti ,ns ft-r 9") , e(2) , and f3 which apply to the low Mach numberrange of the conical nozzle. This asymptotic theory is discussed in Section 17.
In addition to the admittance coefficients , , and C , two other complexadmittance coef~icients are useful and have been calculated. They are
l!~~ {'>'+ l
ic
where coc is the frequency nondimensionalized, as in Part I, by the ratio of thesteady-state stagnation speed of sound to the nozzle entrance radius.
c is the admittance coefficient to be used in the relation U = (W/y)P at thenozzle entrance in the absence of vorticity and entropy perturbations. This is essen-tially identical to (105) and (107), except that the separation of variables schemeand nondimensional scheme are slightly different. (See the discussion following (160)).Of course, when Svh = 0 . c is also the admittance coefficient corresponding to (107)for isentropic longitudinal oscillations.
is a combination of ( and 6 which becomes important in typical combustioninstability applications, as will be shown in a later section treating the applicationof the results. It can be shown that, for low Mach numbers, C and -8 are approxi-mately equal. This means that at low mean-flow Mach numbers E becomes approximatelyindependent of ) , even though a and 6 are dependent upon it.
° Due to time and expenses, neither is it feasible to perform the calculations with a sufficientlyfine mesh in the parameter space to allow interpolation.
54
The results of the calculations for the real and imaginar: parts of the five co-efficients Q 8 , * . E , and a are presented as functioki of frequency Co withSh and Mach number as parameters in Figures 7 to 11. Here the frequency is non-dimensionalized by the ratio of the steady-stage stagnation speed of sound to thenozzle throat radius. In order to obtain coc . oi must be multiplied by the squareroot of the area contraction ratio.
One of the most interesting results is that the nozzle muy have a destabilizingeffect upon the transverse modes of oscillation. This is indicated by negative valuesof the real part of OL and positive values of the real part of E . (The importanceof the signs of these quantities will be discussed in Section 19.) Negative Ur andpositive Er generally seem to occur in the range of "purely" transverse modes whereWc is close to Sh . So here the nozzle would have a destabilizing effect. Forlongitudinal modes and those mixed modes where the longitudinal dimensions are mostsignificant in determining the frequency (Oc >> S.h) * Xr is positive and er isnegative, so that the nozzle has a damping effect upon the oscillations.
It is seen from the figures that the wild behavior of the solutions at low Machnumbers makes interpolation of the results most difficult. For this reason, as alreadymentioned, the asymptotic solution has been employed at these low Mach numbers inorder to determine the admittance coefficients in that range.
The figures indicate that the value of the admittance coefficient C are generallyquite small compared to the coefficients C and 8 . This and the fact that theamplitude of the entropy oscillation is small compared to the amplitude of the pressureand velocity oscillations in most situations of physical interest ean that usually(159a vay be silified to the following form
U+QP+rU = 0. (159a)
As shown in Figure 8, - 0 whenever s = 0 . Furthermore I also equals zeroand it follows that (-a/y) and A. are identical in that case.
It should be noted that the results of the calculations for the standard throe-dimensional axisynmetric nozzle may be scaled for use with certain annular nozzles.The major restriction is that the inner wall of the annular nozzle must have the sameshape as a stream tube contour in the three-dimensional nozzle. This implies thatthe two nozzle flows are identical in the steady state (that is, of course, only inthe comon region where both flows exist). Also, under the long-nozzle, one-dimensionalsteady-state flow assumption, this means that. the ratio of the outer wall radius tothe inner wall radius is constant along the convergent section of the nozzle.
The equations for the annular nozzle may be separated in the same manner and thesame differential equations remain to be integrated as in the three-dimensional case.However, now sh is no longer the hth root of J,(x) = 0 but rather it is thehtb root of Jl(x)Y'(Ux) - J,(]Ox)Y,(x) = 0 . Here J and Y are Bessel functionsof the first and second kind, respectively, and 3 is the ratio of the inner to outerwall diameter. (v is an integer, here.) So using the proper value of s' , theresults of the three-dimensional nozzle calculations for both admittances and flowproperties may be used for the annular nozzle. The values of S,,h for various annulimay be found in Reference 100." In that Reference, 8 is defined as the reciprocal of our 6 , so that their value of suhmust be multiplied by their 6 to obtain our value of Svh
55
A limited number of calculations have been performed wherein the throat wall curva-ture, the cone angle, and the ratio of specific heats have been changed. It was foundthat changing the last parameter from the standard values of 7 = 1.2 to 7 = 1.4generally produced a change in the admittance coefficients of only a few percent. Theother two parameters affected the results more significantly, as shown in Figures 12through 16 where the results for different values of these parameters are compared.
Calculations were made with R = 3.0 (versus R = 2.0 in the standard cases) and01 = 300 and also with R = 2.0 and 0, = 150 (versus 0, = 300 in the standardcase-,). When R was changed and 0e left constant, the results changed most signi-ficantly in the high Mach number range near the throat. Further upstream in the lowMach number range, the difference between the R = 2. 0 and R = 3. 0 cases is smallerOn the basis of this small amount of evidence, it seems that far away from the throatthe resuits do not depend very strongly on the particulars of the nozzle shape nearthe throat. When 0, was changed and R left constant, the solution near the throatdid not change, of course. Only in the conical portion of the nozzle was a changeproduced.
15. EFFECT OF TRANSITION REGION BETWEEN CYLINDRICALCHAMBER AND CONICAL CONVERGENT NOZZLE
As mentioned in the previous section, it is expedient to disregard the actual shapeof the nozzle entrance portion, by means of the approximate assumption that the conicalportion and the chamber are directly connected. With this approximation the admittancecoefficients for an infinity of contraction ratios may be obtained from only onenumerical integration.
In order to ascertain whether this approximation does indeed produce the negligibleerror which would be expected, coefficients were calculated for a very limited numberof cases with more realistic nozzle entrance porticns. As Figure 17 indicates, thesecalculations have two phases. The first ohase involves the determination of theadmittance ccefficients at the entrance to the actual nozzle with contour ABCD. Forthe sake of fair comparison with the results of the calculations for the approximatenozzle shape with contour ABE. we must include the effect of the cyli!ndrical portionE2). So the second phase involves the calculation of the admittance coefficients atthe E-end of the cylinder given the coefficients at the D-end (which were the resultsof the first-phase calculations). It is the results of these second-phase calculations(that is, the coefficients at the E-end of the cylinder) which must be coupared tothe results of the calculations for the approximate contour. Obviously the entranceMach numbers for the two nozzles shown in the figure are identical.
In the calculation of the admittance coefficients for the actual nozzle, all thedifferential equations remain the same except the equation for the velocity 4 whichhas a different form in the transition region. In that regio:i '.he equation becomes
d 1 (2R2(re - r) - Cr- re)2)Z1
dx 2K! dr/d re - r - R2 (169)
Z Ze + (2R2 - r) - (re -
56
for the range z t z ze. where
Ze = Zt - R2(2(1 - cos 91) -(1- cos 1)2)2
R2 (I - cos 1) + R C1 G + 1 - r e
Zt = tan
Here the transition region wall contour is given by an arc of a circle with radius ofcurvature R. and the radius of the nozzle entrance cross-section is re . Thevelocity is still governed by (167) in the other regions of the convergent portion ofthe nozzle.
With the above simple modifications it is possible to find the admittance coefficients
for this nozzle by integration of the equations. Now it remains to calculate what theproper value of the admittance coefficient at the end of ".te approximate nozzle wouldhave to be. This value then must be compared to the results presented in the previoussection in order to determine the accuracy of using the approximate nozzle shape.
This second calculation involves the determination of the admittance coefficients
at one end of a cylinder given the coefficients at the other end. So, an analysis ofthree-dimensional oscillations with vorticity waves and entropy waves in a cylinderwith uniform mean flow is involved. This analysis is essentially a specialization of
that already performed for the variable cross-sectional area nozzle and is outlinedin the following discussion.
The linearized equations of motion (17) through (20) may be separated in a cylindricalcoordinate system by the scheme indicated by (160) with the addition that
Iw/ = W(z) - Jih(s~hr)e'(O)e i t
r
p, - R(z)Jh(suhr)(O)e wt
The reference length in the nondimensional scheme is now taken as the radius of thecylinder walls. z = 0 is one end of the cylinder and z = ' is the other end. Apositive uniform mean flo7 d in the axial direction exists in the cylinder.
The resulting system of ordinary differential equations may be solved to obtainthe following:
i%*z ix z
P(z) = ce +/8e 2
c 1 iX1z i X2 eixz ( )2U~z) e
2 2 6 e q)
-(Y) + V++ 2 1 + !Lh
L (170)
57
ixz e1)x2 z (1'70)O. e e3 -- z
U(z) = )J(z) -- i C + i5k- Y&+4k C2+ e Qyc+TlX1 ,w'+ 2 w2+ q Sh
-i= ZA(z) = -Ee q
R(z) = - a w
where a . / . S , and E are undetermined constants and
i= : (, 2 _ s h + -S1h) /(1 -
In the case of interest the admittance coefficients at z = 0 are known; that is.the complex numbers A , B , and C are known in the relation
(O) + P.0) + sU(0) + C*(o) = 0 (171)
by means of the first-phase calculation for the nozzle.
, first by applying (170) to determine U(o) , P(O) , U(O) , and A(O) in (171)
.2 'en using (170) to obtain four relations, one identity for each of U(t) . P(f)U() and A() . we have a system of five linear equations for the four unknownconstants a( , / , 8 . and C This means a certain relation must exist betweenthe flow properties at z = t . The relation is readily obtained by setting the
determinant of the system of five equations equal to zero. It is of the form
UM +QP( +BUM + CAM = 0
Since the formulas for a J B , and C are rather cumbersome, they are not presented
here. The reader may easily produce them solely by means of (170) and (171). The
formulas allow the calculation of a. 6 and C given ^ ,C, co t* , , ,6 , and C . (Note that O . P. , , and 6 have been eliminated.) Due to the fact thatdifferent reference lengths have been used in the two phases of the calculations, theabove value of &o is greater than the &j used in the calculations of (a, B , and Cby a factor equal to the chamber radius divided by the throat radius. Also since tmust correspond to the length ED in Figure 17, we see that t = (R2/re) tan (01/2)
The calculations were performed for various values of co , svb . , and & = eevaluated at the nozzle entrance (which determines the contraction ratio). Y = 1.2
01 = 300 , and R = 2.0 were taken in the calculations. Table II presents sampleresults which show the error produced in using the approximate nozzle shape.
The first complex number given for the coefficients is the value calculated at theentranca of tho actual nozzle while the second number is the proper effective value tobe used at -:he entrance of the approximate nozzle. (This number resulted from the
second phase of the calculations.) The third number is the result of the calculations jfor the approximate nozzle discussed in Section 14. So the error is seen by comparingthe second and third numbers. Note that, instead of comparing the real and imaginary
parts separately, one should more properly compare magnitudes and phases.
58
In the first four examples the frequency co is large compared to the eigenvalueSvh * In those cases the frequency is primarily determined by the longitudinaldimen,,ion. The solutions are varying slowly Lbrough the transition region as indicatedby the small difference between the first two numbers. Most importantly, the smalldifference between the second and third numbers shows that the error introduced by theapproximate nozzle is small even when the radius of curvgture R is not-so-small.
In the next four cases &o and Sh are equal. Since . frequency is nondimension-alized with respect to the throat radius, the chamber frequency wc is actuallylarger than the eigenvalue. This is the region of mzi'ed mode oscillations where, forthe flow velocities indicated, the radial and longitudinal dimensions are equallyimportant in determining the frequency. Here the flow properties may vary more rapidlythrough the transition region, resulting in the possibility of significant errors byuse of the approximate nozzle. Still, as shown in Table II, the error generally issatisfactorily small for small radii of curvature at the nozzle entrance. This isexpected, since the actual nozzle approaches the approximate nozzle as R2 tends tczero.
The last example, when compared with the one immediately preceding it, indicatesthe improvement in the accuracy of the approximation as the frequency decreases belowthe eigenvalue. There, the range of "pure" transverse oscillations is approached.
In conclusion, the calculations show that the approximation usually introduceslittle error. However, in those ranges where the flow properties vary rapidly in theaxial direction, significant errors may be obtained for large radii of curvature atthe nozzle entrance.
16. FLOW PROPERTIES
It is interesting to examine the actual velocities, pressure, and entropy of theoscillatory nozzle flow, even though the knowledge of these quantities is not requiredfor the determination of the admittance coefficients. The determination of the flowproperties should lead to a better physical understanding of the nature of the oscilla-tion in the nozzle.
In order to detetmine these quantities one must solve (58) for and substitutethe solution into (102). In addition f0 , f1 , and f2 must be determined and -
C 1 , and C2 must be specified. In order to solve (158) it was convenient to firstchange the independent variable from k to 0 = 2K(q - qth) . The differentialequation becomes
d 24b d0b--- (g + ih) -=+ (J - ik) - (G + H) , (172)
where the coefficients b , g . h , j , and k are those defined immediately following(165) and G and H are given by
U = /(VY-1)fo 2 /4df - h f/ +
H = (f/2 f3fo + 2i2/ 1H ('1K)cf 0 [Q~ + ( f3 +
d 2 icZ qCY j
59
It is convenient for the purpose of numerical integration to reduce this second-order
complex differential equation to four first order equations by means of the following
definitions: yl+iY 2 d d Y3+iyM. G Gr+ iGi . and H Hr + iHiThen the system of equations becomes
S= Y3Y2 Y
Y = YII
(173)
y [g 3 J 1 by,, -ky G - Hr]/b
Y I gY - JY 2 + by3 + kY - G1 - H]/b."
Since b -0 at the initial point of the integration (as before), one must insurethat the regular solution is obtained by expanding the solution in a Taylor seriesabout the singular point at the throat in order to determine the value of the solution
and its first derivative at that point. These values are then used in the first step
of the numerical integration. Again. we use the steady-state velocity profileq2 = 1 4- q + b + . . So, by expanding the coefficients and inhosogeneous partsof (173) in a series .n 4 and determining the coefficients in the series solutionfor (173) in the vicinity of the throat, one finds the following initial conditionse:
y 1 (0) =1 , y(0) = Br
Y2 (0) = 0 y(O) = B
(174)y 3 (0) = Br. y*0) = 2C
y'(O) = B1 . y'(O) = 2C,
where the following definitions apply
+I
2 1
Br =+ 2
++ 1 ly12Y +&;I
2; 2 1B1 = 2 (2 j
BIIIn linearized problem dealing with periodic phenooenoa the amplitude and time-pbase arearbitrary In the final solutica. So with no loss of generality 0(0) = I was chosen.
60Cr 2o-(7)1
4[c)2 + (7 +1) 2
= 2wI 3 + 2(7 + 1)144l = 4[ 2 + (7 1 2
Defining further, we have
^2 .2 2Sv h - 0) S haIl - 4 4 Clr 2K(7 + 1)
iy 7- 1 h Or
- 5 ;4 CL)
4 4 2 2K(7 + 1)
+2 ___g213+ (7 + 1); Br + -v~ B2 ^ 13 \ r 4 Br 8 7 )--
3 +7Y, 8 2
4 -8 ? 11 - ( - + r
2- 6y -3 a CO
2(7 + 1)2 2K 7+ 14K
I.2 + (7 + 1); 4t +B
3 +7 7 2 YoBr - - cb_4 8 2
S 7-67)-3c7 W7 0*; - sff'-b'h (7 - 1)Cli- -CD--
8 Ir + 1)' 4 7 + 1 4K
When solving the system (173) and (1'74), fo , fj , f 2 ' and f3 are determinedsimultaneously. They are governed by the following system of equations which may be
obtained from (77), (81), (83, and (85):
cf) f 6()or f 0r6) -- 01 for(O) I
2/q2 r
f ( ) 2 q f0 i(O) 0
(175)
61
f/1 = ( (175)
I~(k - - -f -; f01(O) = 02 dk
f+f 2r (0) = 0 L
2r2q 2 2K q2'2r 0 0
f 1(k) 2 r6( ) _ f1 i2q 12---- -. f 21 (0) =0
(c~f3)r c f 31 _ 1 fdrq03r 2jF 2 dO frO
-IfOf wcf~r 3 1 (0) 0.
The numerical integration of Equations (175) is simpler th~ the numerical evaluationof the integrals appearing in their exact analytical solution. The flow propertieswere determined for the 300 conical nozzle described in Section 14. So (167) and (168)were solved simultaneously with Equations (175). The integrations were performed with
V an IBM 7094 computer in similar fashion to those previously mentioned.
(-2qy4 KYy +Ori +O2iir)
Um= 3 4 rfi
Uarg = ~ i ar tani +Kwy - 2q3y
Vmod = io [L(Yic Orfir0-f ii -C 1 2 ciY +cfi)2 +
r-I
+- (y2 -Cirfoi - Ciifor - 0rf21 -- aif2r) 2]
~rary Wag act - Cirfoi -Cii1for - '7rf21 - Oif2rY,- Cirf or + 11 f0 1 - rf2r + ojf2 i (176)
62
Smd=(o-, + a')z ( (176)
=arg arc tan rfoi + "1forrrf or Orif oiJ
where, for example, P P Pod exP (iParg)
The results given in (176) are nondimensionalized with respect to static throati conditions and are consistent with the variable separation scheme given by (44). It
was desired to have the final results nondimensionalized with respect to stagnationconditions and to be consistent with the variable separation scheme given by (160).
This was readily achiieved by multiplying, in (176).
2-/ Y "- 1 - /(-)
Pmod bymod + I T+ 1
Umod b1
and
Vod and Wmod by +
As already specified, r in (160) (as well as in the equation for w' ard p' givenin Section 15) must be interpreted as the radius divided by the local wall radius.
In order to readily compare the results for different cases, a phase was added tothe arguments of (176) such that Parg(0) always was equal to zero. Therefore the
phase angle A to be added was given by
A =- arc tan -(2Bi + )
- 2Br
Sample results of the flow properties calculations are given in Figures 18, 19,and 20 for the case of an oscillations with svh = 1.84129 and & = 1.00 . Here thetransverse variation corresponds to that of the first tangential mode. The samenozzle shape as discussed in Section 14 was considered. Both a and C1 are real
and equal to 0.5 for this case.
In Figure 18. it is seen that there is a gradual axial variation of the pressuredue to the acoustic wave phenomencn in the nozzle but there is essentially no variationdue to the presence of entropy and vorticity waves. The axial component of velocityhas a gradual variation due to the acoustic wave Plus a more rapid undulation of smaller
amplitude due to the entropy and vorticity waves, as shown in Figure 19. The trans-verse components of velocity have rather severe undulations, due to the entropy and
* vorticity waves, as indicated by the "artation of the radial velocity component shown
in Figure 20.
63
These entropy and vorticity waves travel at the particle velocity, which is subsonic,
so that their wavelength is shorter than the acoustic wavelength but increases as the
wave moves towards the throat. The values for o- and C1 in the figures were taken
to be considerably higher than those normally found in practice for the purpose ofdemonstration. The results are qualitatively similar for a wide range of values ofthese complex numbers. Even though the pressure does not exhibit the presence of
entropy waves, the density and temperature would.
Note that the admittance coefficients are independent of the values of tr and Cland therefore independent of the intensity of the entropy and vorticity waves. Of
course, as o" and C, go to zero, the general admittance relatio; ran be shown toreduce to the irrotational admittance relation.
17. ASYMPTOTIC BEHAVIOR OF THE ADMITTANCE COEFFICIENTS
Due to the wild behavior observed in the numerical solutions of Evations (163).
(164), (165), and (166) in the low Mach number range of the nozzle, it was desirableto obtain some analytical prediction of the solutions to those differential equations.
This was achieved by developing solutions which were asymptotic in the sense that
they apply in the limit as Mach number goes to zero. These solutions are satisfactory
for the present purpose since it is in the low Mach number region (away from thethroat location) that the behavior must be examined.
In the low Mach number region of the nozzle, we can say (using the continuityrelation and neglecting terms of order q1) that
(2/(y + I))1 A 2(Y 1 ) ) =
where the same nondimensional scheme as discussed in Section 14 is emplojed here.
Noting that for the conical portion of the nozzle dr = - dz tan8 dO(tan 0)/2K74
we find, after differentiation of the above relation, that
(+)2 (-) 2K
This may readily be integrated to obtain
= (a + c) 2 , (177)
where c is determined by specifying* ' at some value of P and
a I ~vi1112(-Y1)) ta2 2K
It is convenient to begin the development of the asymptotic solutions by a trans-
formation of the indp.dndent variable which is suggested by the form (177).
For the purposes here, specification of the value of c is unnecessary.
64
Setting y a + c , it readily follows that
-- y.2 2
(178)
d d-=-w- = adgb dy
will be determined by finding the homogeneous solution to (158) and then taking
the derivative of the logarithm of that solution. Using (178) to substitute into
(172), we have the equation with complex coefficients
@'+ (A- iC)@th + (B - iD)Lh = 0 , (179)
where
A = - 8y 3/(9/ + 1) + O(y7)
= C2 1 C2 svh - Y + 0(Y6)
B 2(y+)a 2 Y' 2( 7 + 1)a 2 4a 2 2 (;72 -0 +
2w 20oy~C = +0(y8 )(T+ )a ( + 1)a +
•-4(y7- 1) y3 + o )ii D- 2(,/ + 1)2 .(
The linear equation (179) with variable coefficients cannot be solved exactly;
however, there is the well-known WKBJ method for solving a second order linear
equation. The important criterion is that the solutions vary much more rapidly than
the coefficients in the differential equation. This is satisfied if /a is large,
which implies that the axial gradients in the unsteady state are larger than those in
the -t,:ady stnte, since a is proportional to tan 09 .
The IDXBJ method involves the determination of the two solutions in the form
= e4(Y) (180)
For equations with constant coefficients the derivative Pf is a constant. In thepresent situation, where the solution varies much more rapidly than the coefficients,
this quantity is "nearly" constant, so that tP"' is of higher order. Therefore, in
the first approximation we may neglect /il" so that, substituting (180) into (179),
we have A( IA tC- -+ ± i (B+- + OWy) .
S2 2 4
65
This first approximation is differentiated to obtain
2P1, 2 Ya(.- 1 + O ( y 2 ) . ( 1 8 1 )
+ {2(7 + I)), y3 a{ 2 (y +i1)}i 7+ I1 4 +Z/ 2/
Now, in the second approximation, the qI" is kept when (180) is substituted into(179). with the following result:
A iC 2-- +- t i(B + + C ,/4) (182)
2 2
where 01" is given by (181).
For the third approximation, the result of the second approximation is differentiatedand substituted into (182), which then is solved for P' . Of course, this iterationmay be continued indefinitely. The result of the fifth approximation is convenientlyexpressed as follows (setting 1k = Or + ilki):
13O = - + Kly + K3y
3 + O(y5 )y
I = I a (+) y - 2 a (183)a2(/ + 1) 2 6a V 2
± + K2y2 + OW )
a(y + 1)
where the following definitions apply
IK i ( 7+iv/(..
K = {2( + 1)--27 3 C +la 2 7 3)} as ( Y j + 2+2T + 1 a 2(y + I)) }4 4 1-(
+ - y + 1)(y0 + 1))L G 8 R( j8 8 (2y + 1))}z 2,
3 as (2(y + 1)) 7i /(T-1) 3 {2(y + 1))i(7 + 1"a3K, h ..
.
4 \ 2 2
S+ 3 a2 h (v+ l (2^'-I)/(-I) a 2----- 11 -~--1(71). -
,+1 <4 \21 <)"d
1 Svh y+ 1
8 :Zr' ~2
I66
(183) shows that two solutions exist for P * which means that ' , and thereforeth , also have two solutions.
Integrating (183), we find that
br = b+log y + Kzy2 + K y )
2 4 (184)
q'P " q'f - c or - P(-) + d,
where b . c , and d are constants of integration and
"2 _ 1) 1)/{2(Y-I)}= a{2(y +1))y 2 2 a
+ CY + K2 y )
a(y + 1) 3(185)
_________ 1 ~ /7+ j~1 ){(I) yJ a2(/+ 1)i - j 2a ( '2)
N + L + oy5)a(Y + 1) 3
It can be shown that AP +) and APf- correspond to waves travelling in oppositedirections; P+) corresponds to g wave moving towards the nozzle entrance andcorresponds to a wave moving towards the nozzle throat.
Substitution of (184) into (180) yields the following results:
= jr e p {i(14 +) - c)} +Be'r exp ( C)) - d)) . (186)
where , . c and d remain unknown. Since exp (b) may be included in A andwith no loss of generality we will consider b = 0 in (184).
Noting that
(186) may be used to deduce that
a''-a(,Pf+)' +P-sin(Xq,) )m+ q,(- (c + d))
r (V-B)2 +~ I (. / cos (q14+) +'Pf - (c + d))
a(P ,+ '- ,) 2 (187)
1 2 o (( /+ ) _ )
2 + ~(TX/)2 1 2(V/B) cos ('I4+) + qPf- - (c + d,")
67
It is seen that the unknown constants appear in two combinations, (i/B) and (c + d)If and , were given at some initial value of y , the constants (VE) and(c + d) could be calculated, since qjr ' q+) and q'P) and their derivatives areknown as functions of y . Once these constants are known, (187) coula be used todetermine r and i at other values of y
If the relations given by (177) and (178) are used to transform (166), the resultis that
d i f) 2y 3-(C f3) -(Cf1I2ay
Upon integration we have the exact solution as follows:
c1f 3 = k exp (-i2/6ay3 ) + 2 exp (-i(/6exp (-i ) d
where Y is a dummy variable.
The definition is made that
In(y) = exp (-io5/6ay3) f'Y exp (-i5/6&) d7. (188)
Then ,e have the following form for the solution:
cf 3 = k exp (-i4 16ay3) + 213, (189)
where k is a constant which is determined when f 3 is known at a certain value of y
It remains to determine In(Y) ; an asymptotic series may be developed by successive
integration by parts. First, the following transformations are made
.at1/3 3
This means that (188) becomes
/-\(n+l)/3 -it CI = I eteir dIn Ta) 3 f (188a)
t i
Integration by parts generates the asymptotic series approximation
/(n+1)/3 Fr1 + ,Zbn a + i + b= (190)n W % t t 2
"' 88
where
&1 -7) %. =m -- a( + 2m)(-q + 2m - 1)
b - 7 -( 1 + 1), b+ I = - bn(7 + 2m + 1)(- + 2m)
The integral which would remain after 2M + 1 integrations represents the error inusing the series approximation in place of the exact function I. This error can beshown to be bounded above by the quantity
E (71 + 2M - 1) (7 + 2M- .2) -------- (7 +
( 3)7l,7+2K
For a given value of the argument t . therefore, an optimum value of the integer Mexists which minimizes the upper bound E . This optimum situation occurs when Mequals one plus the greatest integer in the quantity (t - 7)/2 . The smaller theMach number is. the larger t is, and the larger the optimum value for M is. Inthe limit of zero Mach number, t is infinite and the optimum value of M becomesinfinite.
I is not a rapidly undulating function of y for the values of in the rangeof interest. In particular. 13 is proportional to y' in its leading term. so that-f3 is essentially given by its homogeneous solution. That solution portrays theeffect of properties which propagate with the gas velocity rather than the soundvelocity since exp (- i4V6ay3) is an approximation to exp (-if(1/273)dx)
Asymptotic solutions may also be found for Equations (164) and (165). Equations(117) and (178) may be used to transform those differential equations to the followingforms*:
d 2()+ -i 1+MIdy M)+ L_ j (y + 1)(1 _ Y4 (' F(')(y) . (191)
where
M(1) = C1-y3) (1)M(2) = - (2)
'!h Y2)(1 2 / - 1 Y) (Vd
k2 / 2 3 -1
F () 3 1 71 \1/(v")[a +- -- -- yj~ (2i+2Y y1
Equations (189) and (190) may be employed to simplify these equations as follows:
F(") is not exactly the sae quantity here as it is in (88).
69
S+ (2" ^) /( 1 2
F ( ) y4 y (2 )
2iwkF(2 ) - exp (-icZ/6ay3) +
(y + 1)2yl
+ (2-)/O-1) r + (2 + 2k exp (-ij/6ay3 + 0(y2)41& y2
E(qiation (191) is readily integrated to obtain
= exp Eic/a) 1 2 - ]
[ica f (1 + (+ 1)(1-Y4 ]
Mei) =(Va ++
2y (y + 1)(1 -y)
h X
FJ()h exp i / + ( + 2 - 2) dy (192)
where is a complex constant of integration, Y is a dummy variable and it hasbeen noted that % = exp [(l/a) f dY)
The integrals appearing in (192) must now be approximated by an asymptotic solution.(185) and (186) will be used to evaluate Ph . One finds for the integrand
F( lb exp [i/a) I + I)(I - y
f (7~ +1 1(V) (1 - 4
2 4 2
ex---c-)+ - {2(y +
x( eic + exp [(i/a) (-1 + j)< ,. - + 1 ---- -
70
For our purposes, it will be proper and consistent to take exp (O(iy)) as suffici-ently-slowly varying to be replaced by unity. We see then that integrals of a certaintype sum to produce the integral in (192). This type of integrod is
Yn exp ic /a) ( y 2
In particular, for our desired accuracy we are interested only in the cases wheren = I and n = -1 . Note that the exp (O(i/y)) term represents wave propagationeffects and varies much more slowly than the exp (O(i/y 3)) term, whica representsparticle propagation effects. So there is a preferred way in which to integrate byparts. One finds that, to the desired accuracy.
exp [i(Va) (4 2( + 1) dy
exp i/a) ( 63- ] (y) -) IIy)
e+± [2y+ 1)} "({) ±y + 1)} ] d(
e exp [(uza) (i ± 2('y + 1) 1 dy)]
where (190) gives I1(y) and I.1 (y) . Now (192) yields
X3= (-y = e- t / 6 ay
MM (( 21 1 4./y K+ y 3 2K. + K+ yJ ((X/.) e(c+d) e'im/y +
4y ++ + K I +
2 e-I (c+d) e-21n/y _+K + i/~ ki)ei(cd) e-2im/y +(193)
where the quantity m has been defined as
a{2(y + I))-
I
71
In developing the asymptotic form of (192) for 2) integrals appear of the sameform as appeared in the relation for f(1) In addition integrals of the form
incY)!lex
exp a{2(- 1+ " d
appear. It can be shown that 4n ev F icda{2(y + 1)) 1 y can be put into a formsimilar to (188a) with t = + c/a{2(Y + 1)}iy aLd 7) = n + 2 . Then it follows that
()=3[3exp 1'+2(yn~~~y; = -+ I/ x a{2(y + 1)) Fy 'Y
where I3n,+,(y) is evaluated from (190) by using the above-mentioned relations fort and 77 . The "wiggly" sign over I has been used to denote the change in theargument as a function of n and y from the previous usage even though it remainsas the same function of t and 77 . Note that -, is special in that it is improper;its amplitude goes to infinity as 1/y . as y goes to zero. This really occursonly since we chose to set the lower limit of the inteiral at y = 0 . If we set thelower limit at some finite number, say E . the indefinite integral evaluated at thelower limit is merely a constant which my be incorporated with the constant whichappears before the homogeneous solution for M2 " This is done with no Tooss ofgenerality so that we may take
L3 + 2a2(7+ 1)( ir\ a{2(-/ + ) ' a{2(y+ 1))Iyj
Note that the above asymptotic solution which contains only two terms in the series
is in fact an exact solution for the integral.
It now follows that
M(2 ) = (1 y)(2)
2i k y 2 i k
k2 ~ e e.2 (-i/6y
-(+ )2 , ep~i~6~ (A~ -i(c+d) -2i-1n( ~ e (iILay)
(2 +K K r.+ K'\ .6 (A./Me~cue I Y +1K, Y
2 8/
X(2 exp (-i'6ay)++ K y3)(( /)e-i(c-d) pim l y + el T/ y )
U2 exp (-i6/6a 3 ), Y + 2K 2w j
(194)
72 (T./]) e-I (c+d) e2 1iM/Y I i) + I() 54 i exp (-i /6ay3 ) (194)
+e ( I/ (c) e e 2 1 /y + (.y + 1)y
2 e-+d) e-y/y i2-)/(-i) 2
x_ 2 3 1 K 21 +1 I
+ 4K y4 \ ,
I4
for t in (190).
The solutions given by the asymptotic theory provide a great amount of insight intothe oscillatory flew in nozzles. (186) shows that is the sum of two solutions,
one corresponding to upstream wnve propagation and the other to downstream wa .-propagatio. The equations following (185) show that 'pi goes as 2/y t. leadingorder. Noting that d = 2Ktd , dy = ad , and d4/dz = y2 we find 4'y is
linear in z , the axis coordinate, so that the solutions are undulatory with constantwavelength but their amlituden varyc. The radical behavior in is now easy toexplain. It is seen that behaves so a nearly-sinusoidal function, so thatone corr(d/di) behaves nearly as a cotangent which becomes quite large in c.rtainranges. The appearance of the cosine function in the denominators of (187) shows thatperiodically t may be.ome ,arg2.
It is especLtly interesting that to leading order does not depend pon stan
but only upon . The wavelength can be t a ve t happroxiatily that of a planarwave generated by a source of frequeny . In a cylndrical chamber, onn would have
foanes Th aearaengthe icoe fucto in th" eoiaosof(8)sosta
found a larger wavelength since ( s is the effective factor rather than c.
Equations (103) and (194) show that and e(2) are affected not only by
acoustic oscillations but by oscillations of entropy and vorticity as well. The
acoustic behavior appears through the exp (± Iraly) terms, xhile the entropy and
vorticity behavior appears through the exp (-i'/6&y 3 ) terms. Bth '(:) .d 2)
have sinusoidal and cosinusoidal terms appearing in denomJnators of their exkre.;sions
(193) and (194) so they become large periodically for the same reason that becomes
large.
18. ASYMPTOTIC DEVELOPMENT OF THE FLOW PROPERTIES
In addition to the development of the asymptot.ic theory for the admittance co-
efficients, it is convenient to develop an asymptotic theory to determine the pressure,
0h 'e variation is, of courae, due to convergence of the nozzle.
I
73
donsity, and velocity undulations. This theory describes the effects of the acoustic
waves and the entropy and vorticity waves upon the oscillatory flow properties, at
least for the low Mach rumber region if the nozzle. Of course, it provides information
that could be found qualitatively in the results of Section 16 only by means of hind-
sight. Quantitatively it gives the orders of magnitude of various interesting effects.
The asymptotic solution of (172) for low Mach number is required. The relationsk (177) and (178) are employed again and the coefficients* of (172) may be represented
by their asymptotes as follows:
7+1 Ub 2. Y y
g 4ay' + o(y')
h = y
(195)
( - 2 y6 + O(y1 0)
4 4 2 Y +1I( \ 2/lX y7-1/~y
k = 2 +-o( .
The transformations discusFed oi: Section 14 are used in conjunction with (77),
(177) and (178) to yield
-o= exp (iL/6ay3)
where
3= exp (-iO/6ay 3) exp "- 4 " (196)
y Iaid € correspond to the same axial position somewhere in the low Mach number
region of the nozzle. Pow, using the definitions folowing (172), together with (177),
(178), and (196), we find that the inhomogeneous part of (172) has the following
asymptote:
+ .4 2 (Ye + -)/
Cr ( 2 a: i~fY3
+ 9,-h +: 2-v )-? - :+ O(yi) + 0(y i o ei /6ay3) (197)
•ctually, these coefficients are defined immediately after (165).
74
Inspecting the coefficients (195) and the inhomogeneous part (197), we expect to
find a particular solution* p of the form
^ 3 CD=e e y :E Any n Z. Bn y (198)n=-wo nu=-D
Substitution of (195), (197), and (198) into (172) allows the determination of thecoefficients An and Bn . The particular zolution is found to be
(3 1 2 ( l 1) eiZ/6aY3 6 20 iaKW 2, , 2 / (
+X 3 i f-- -F 2a 2 Sh -- l(2-01)/(O-1) (4)
2 glh 6/ + i~~E2 )
-4)+ O(y') + O(y ° ei/yl6+y 3 ).
I - (199)
The solution is given to higher order in the undulating terms than the others, since
their gradients are large and consistent accuracy in the gradients is required. Theabove solution when added to C2
4 h , where h is given by (186), represents theasymptotic solution of (172), so we write
Use of (81), (83), (177), (178), (196), and the transformations discussed in
Section 14 yields= -f 2X3 ei /6ay3 = + O(Y7 eiZ/6ay3
f = eiC/6aY3 3ik (200)
where
X Ql- f W-2 fo -d
X 3 +f 6ik - ,/6ylAS - L - o -- d --- j2f 32 d
110
Cle rly, thAI.particular solution must essentially equal the particular solutionC* + 04A of (88): the only difference is in the nondimensionalization scheme, asexplained at the beginning of Section 14.
'I 75
The relations (102) give the formulas for the axial dependence of the flow properties
(of course, now the nondimensionalization scheme differs). Using the transformations
given in Sections 14 and 17, we replace & by KiL) and dW/do by 2aKdIPdy in those
relations. Also, C1 4,( 1) + G(2) is replaced by Pr as given in (199) and 'ph ' f0
f, and f2 are given by (186), (196), and (200), respectively. Under this procedure,
the relation (102) for the axial component of the velocity perturbation becomes, taking
note of the form of (44), as follows:
2aK(BeidC2 )iy [e-i(c+d) e i + - ' j x
+ 2 _K Ijl h Y (7+t)I{ ' )}
+y_2K 3 +- K, s + Ji +a{2(y + WL)} 8
4 +K)i 2. I - e-(c+d) eiTl(+ i<"
+2a (9 eid C.)y2 e+ e ]
i2'x - + 1) y 2 + - K, .y 3 + Y0-, 5KI-
a( +1) 2 2 2a(y + 1) 8
2 3
- (,Kc+ "+ 1) y 4 ei(/6aY 4-
-16(&k)i \- )- 0 + 3b
(201)
Realizing that y is of the order of the square root of M (the Mach number along
the nozzle) we see that the axial component of the perturbation velocity goes to zero
as (M)f . This component of velocity is primarily of an acoustic nature with vorticity
and entropy effects of the order of crM2 and C1M2
. As discussed in Reference 12,whenever the entropy and vorticity waves are generated by the combustion process theyare typically of O(M) . In that case, 0, and C1 would therefore by of O(M) ,
causing the effect of vorticity and entropy waves upon the axial velocity componentto be of O(M3)
Ii
76 4
The relations (102) for the transverse components become
V= = -E (i KC + (--) - (O 3 + CX] eic/6aY3 -
1 ( o a2a 2 1)'!-h-/27 + Iy)/(VI
g,2 4F +B 1t
Y-1) e
2 8 2
fy (y+ 1)2a2 +1- .(Ji
+(eidC2)(A ei(c+d) ei ei )
__,+21K 2 (202)
(44) shows that the radial velocity perturbation v' behaves as pqrV and thetangential velocity perturbation w' behaves as (1/r)W = (1/r)V . From the steady-state continuity equaticn. we have that ;r2 is a constant along the nozzle. So,combining r,/r with the radial dependence, we obtain tha; bot4 v' and wl haveaxial dependencies which behave as V/rw or, therefore, rs WNt at low Mach numbers.So. each of the two transverse velocity components are primarily affected by theentropy and vorticity waves rather than the acoustic waves. The rotational effectshave both an undulatory and a non-undulatory portion. These effects appear to beO(Mi&) while the acoustic effect is O(M)
The thermodynamic properties now remain to be determined. Following the same pro-cedure as above, (102) gives the following relation for the axial dependence of thepressure perturbation:
YP Y + 1' ( (C2 ed)iK ei(c+d) eto + e i X2/
K 2ai 3 K3 + KI 21 31aK+ .y y y02 6) 8 7+1
2aKy t------ -) (C2 eid)y3 e-(c+d) ei~ l + e - i X-- 2
{2(" + 1)), 2a{2(-y + 1)}) 2 ,f-j +a
2 CO~ J 203)
77
/ I 2as1( + \(27-I)/(Y-1) _4 y uo+
+ Y( +' (i y ++ 2 Ik) k 2c
9h 4 / + 1\ '/ (Y' (1 + 1)2a2
+ T2 -2 X
t xl6a2(7 +1) 2) ]Y}+Oh 7
The pressure perturbation has an acoustic behavior which is of O(M ) . To O(M3)there is no undulatory effect due to the entropy and vorticity waves. There is,
however, a non-undulatory effect of O(a) due to the entropy waves.
Finally, we find that the entropy and density are given as
S = cfo = crX 3 elZ/6ay3
R = - <.x 3 y+ ) ( eiZ/6ay3 + 2(7P) + (204)
-Y'+ I '/(OT+ 1) (I + I
It is seen that the entropy wave has a constant amplitude and the density has an
undulating effect (due to entropy) which is of O((a)
The results of this section agree qualitatively with the results of Section 16.Figures 18. 19, and 20 show that pressure has the slightest effect due to entropy and
vorticity, axial velocity has a small effect, and transverse velocities have a largeeffect.
It should be noted that there is a slight inconsistency throughout the last two
sections. The steady-state density was considered as a constant of 0(1) but,
actually, compressibility causes variations of 0(yu) . This metnas that &ny termsin the previous results which are of O(y) (or 0(N2)) higher than the leading terms
cannot be considered as quantitatively accurate. The reason, however, that the
analyses were continued to higher orders, was to establish the orders of magnitude
of the acoustic, entropy, and vorticity effects. While the steady-state compres-sibility would modify the solution to higher orders, it would not change the orders
of magnitudes of these effects. In principle, the steady-state density variation
could have been considered but the simplicity of (177) would have been destroyed.
T ,e solutions contain several groups of parameters: (C2 geid) (R/9 e (c+d))
(ioKCI + 0/2) , (ak) , (a\3) . (aX1 + CI3) , and (X\/k) . The first two parametersplus o- and C1 are given by inJtial conditions on the problem. Then the other
five parameters may be calculated. Actually, one of the parameters should be con-sidered as unit' and the other parameters should be referenced tc it since the problemis linear.
78
Instead of calculating the asymptotes for ( and (2) as described in Section17. one could take the results for the particular solutions of (172) and substitute
them into the defining relations following (162).
19. RESULTS OF THE NOZZLE ADMITTANCE CALCULATIONSAND THEIR APPLICATIONS
The results of the nozzle admittance calculations are presented in Tables III through
LXXX. The values of the real and imaginary parts of the coefficients G , , C* and 8 are given for various values of the nozzle entrance Mach number M , the
angular frequency w , and the eigenvaiue s,, . As previously noted, the frequencyw must be multiplied by the square root of the contraction ratio in order to obtain
W. (the frequency nondimensionalized by the chamber radius). The coefficients werecalculated at constant intervals of the variable which do not correspond to con-stant intervals of the Mach number.
Of course, if one were interested in values of Svh , o) . and/or M other thanthose presented it would be necessary to interpolate or extrapolate. This can readilybe done, _s will be shown in some of the examples given later in this section.
The most important admittance coefficients in the combustion instability applicationare CL and & . The coefficient C( depends solely upon the function and therefore
does not contain the effects of vorticity and entropy which are contained in6(2) , and f3 • is a combination of C and 8 and therefore contains neither6(2) nor f 3 Furthermore, C and 8 combine in such a way so as to suppress the
effect of (1) at low Mach numbers. So contains only slight effects of the
vorticity and no effects of the entropy waves. For these reasons cL and 8 do not
behave as widly as the coefficients G , , and C and interpolationL are moreaccurate for C( and 8 than for Q. and C.
Certain relationships have been developed which provide necessary and sufficient
conditions for the neutral stability of small perturbations in a rocket combustion
chamber*. The derivations of these relationships may be found in References 1, 3 and12. A typical relationship for purely transverse oscillations which is found in
Reference 12 is as followst:
Sf 2 (205)
where Qvh is the axial dependence of the energy release per unit volume, L is the
combustion chamber length, k is the gas velocity to stagnation speed of sound ratio
at the nozzle entrance, and 60 is the difference between the angular frequency Ocand the eigenvalue s. A similar relationship for longitudinal oscillations whichemploys Ox instead of & is also developed in Reference 12.
• Presently. the science of combustion instability has practical use only for the admittancecoefficients OL and &; a and 6 have been used only in the combined form given as 8and C have been neglected. Here a more general viewpoint was adopted, so that all of theadmittance coefficients have been calculated at little extra expense.
f Actually, in the reference, another term due to droplet drag has been included. However, itis not important for the purpose of the example.
l,
• #
79
In (205) the term on the left-hand side represents the integral effect over tile
chamber of the perturbation of energy release divided by the pressure perturbation.In general, it will be a complex number and have a destabilizing effect. On the right-
hand side, (y + 1)/y has a stabilizing effect and is due to various terms in thedifferential equations of the gas motion. The second term on the right-hand siderepresents the effect of the nozzle upon the stability and upon the frequency. Whenever
the real part of 8 is positive it has a destabilizing effect, while whenever the realpart of 8 is negative it has a stabilizing effect. The last term represents thechange in frequency from the acoustic frequency. It has no effect upon the stability
since it is imaginary.
Now, suppose for example that one vishes to determine the stability characteristicsof a combustion chamber with a nozzle of the shape employed in the calculations (300
half-angle in the conical section and wall radius of curvature of the throat equal tothe throat diameter). The nozzle entrance Mach number in this example is 0. 113 andthen the contraction ratio is 5.28. The first tangential mode is examined io that
S = 1.84 and, taking the frequency as wc = 1.72 , one has w = wc/(5.28)Z = 0.750
In order to find the values of Sr and 8i . interpolations of the results presentedin the tables must be made. The following summarizes the information extracted fromthe tables of results for the purpose of this example:
CO Svh M 8r Si
0.5 1 0.099 0.074 0.114
1.0 1 0.099 -1.080 -1.538
0.5 2 0.099 0.223 1.132
1.0 2 0.099 0.097 0.063
0.5 1 0.124 0.100 0.216
1.0 1 0.124 -0.383 -1.008
0.5 2 0.124 0.390 1.319
1.0 2 0.124 0.130 0.230
By a linear interpolation over the frequency parameter we obtain, for &j 0.75
M er 2i1 0.099 -0.503 -0.712
2 0.099 0.160 0.598
1 0. 124 -0. 142 -0. 396
2 0.124 0.260 0.775
By another linear interpolation over the eigenvalue parameter, we obtain the
following, for C 0.75 and sh = 1.84
Ci
0.099 0.054 0.3880.124 0.195 0.588
80
'1 Now, finally interpolating over the Mach number, we have for co 0.75 svh 1.84and M 0.113
er = 0.133; i 0.500
For the purpose of comparison, the equations discussed in Section 14 were integratedand the admittance coefficients were calculated for the values co = 0.75 and
Svh = 1.84 . With M = 0.113 that exact solution was
Er = 0.136 ; 2, = 0.481.
This compares favorably with the results obtained by linear interpolations; there is2% error in Er and 4% error in Si
There are many cases where significant errors are introduced by the linear inter-polation. Often the situation may be improved by employing a more sophisticated inter-polation scheme. For example, if the value of the admittance coefficient at three
values of a parameter were known, a parabola could be fitted to determine the admittance
coefficient at neighboring values of the parameters. If f represents the particular
admittance coefficient and x represents a particular parameter, the following formularesults from such a parabolic fit to the points x0 . x1 , and x2
f(x) (x - x,)(x - X2) f(x )(X0 - x1)(Xo - X2 )
(x - x2)(x - x 0)x + x " - ) f(x 2)- (206)
(X - Xo)(X2 - x)
The use of (206) shall be demonstrated in an example where interpolations are per-
formed only over the Sh parameter. Of course, it may be used successively to inter-
polate over as many parameters as desired. Consider coc = 2.36 . Sph = I. 8 4 . and
M = 0.250 . The contraction ratio is 2.45 so thtt co - 1.50 . The following summarizesthe pertinent results from the tables:
S&'h er E1 -0.601 -0.625
2 0.116 -0.060
3 0.406 0.689
Using the results for svh 1 and svh = 2 . a linear interpolation gives, forSvh -1.84.
r= 0.001; = -0.154.
I
81
On the other hand. (206) may be used with x representing s., and f representing
Er at first and secondly &i Then xo =I . x1 = 2 . and x2 = 3. The result isthat, for S~,h = 1.84 ,
Er 0.030; 81 = -0.163
An exact solution for co= 1.50 . s*h = 1.84 . and M = 0.250 yields
0.036 ; i= -0.198
So it is seen that the more sophisticated interpolation scheme is more accurate in
this case, although about 20% error -til1 exists*.
In both cases considered above 8, is positive so that the nozzle has a de-
stabilizing effect. For low M , , so that in the first case er/yqe = 1.00while in the second case it equals 0. 120. For 7 = 1.2 .(y' + ")/Y = 1.83 in (205),so that the destabilizing effect is significant by comparison in the first case
although small in the second case.
Suppose nowv that we did not have a 300 but instead a 150 nozzle. The scalingfactor 3 ictroduced rr Section 12 is chosen as (tan 150)/(tan 300) = 0.465 . Ingeneral the scaled nozzle will not be identical to the actual nozzle in the throat
portion, although it has been made identical in the conical portion. So. whilethere is some error here in replacing the actual nozzle by the scaled nozzle, thiserror is expected to be small whenever the conical portion of the nozzle is con-siderably larger than the throat portion.
In the present example consider the throat wall radius of curvature R = 2
=2.97 . Szh = 1.0 , and the entrance Mach number M = 0.152 . Then the contractionratio is 3.94 and co = 1.50 . c4/0 equals 3.22 and Svh/' = 2. 15 . Since
O(U. . Svh) = Gref(M. W/A. s~h/)
and
8(M.co, Sh) = 3b8ref(M, -&/, Sh1)
it follows that
8(M. - .& Sh) = ere f( ( h
It remains now tc determine e for M = 0.152 o= 3.22 , and S = 2.151refFrom the tables of results, one obtains the following information for M = 0. 152
3.0 2 -0.694 -0. 184
3.5 2 -0.982 -0. 1453.0 3 -0.669 -0.0963.5 3 -0.883 -0. 193
* The errors are considerably smaller when the more proper comparison is made with the differencesbetween the values of 4r (and 8j) at the points sp = 1, 2. and 3
82
By means of successive linear interpolations over the frequency and eigenvalue para-meters. it is found that. for M = 0.152 . &- 3.22 . and Sh= 2. 15.
= = -0.908 ; - -0.163
1 An exact c'alculation for these values of co and s.h with 81 = 150 and R 2yields
Er = -1.050; - -0. 129.
which favorably agrees with the approximete results obtained by using the scalednozzle. The difference results because the throat wall radius of curvature R forthe scaled nozzle is greater than that for the reference nozzle by a factor(11,)2 = 4.62 while the actual and reference nozzles have the same value of R . Ofcourse, if the actual nozzle had an identical throat contour to the scaled nozzlethere would be no difference. It also follows that. if the elliptical arc which isthe generatrix of the throat wall of the scaled nozzle (obtained by scaling the cir-cular arc of the reference nozzle) has identical curvature at the throat to the curva-ture of the actual nozzle, the difference in the results would be quite small.
In certain cases, the asymptotic analysis may provide a better procedure for thedetermination of the admittance coefficients than interpolation. This is especiallyso in the regions of wild behavior as indicated in Figure 21. There the irrotationaladittance coefficient O' is plotted versus Mach number along the noTzle. It isseen that the asymptotic solution compares favorably with the exact comuter solution.Wiereas interpolation with the exact results over Mach number increments of 0.025would obviously produce serious errors in the regton of wild behavior, the asymptoticsolution more accurately approxisates the exact solution.
As noted in Section 17. the constants of integration in the asymptotic solutionmust be determined by matching the exact solution to the aeMtotic solution at somepoint in the low Mach number range. In Figure 21, the matching point is M = 0.099The comparison tetween the two tvlutions is most favorable when the matching point isin or near the region of wild behavior. Uhenever the matching point is away from thisregion the comparison becomes unfavorable.
Furthermore. thc coavirison for 4 is better than for (') and (2) Notethat C depend. solely upon , while the other admittance coefficients depend upon2(1) and sometimes (2) as well. Therefore the comparison is more favorable forOL thin for the other coefficients. It is believed that the unfavorable comparisonsfor -(1) and f(2) are due to computational difficulties rather than theoreticaldifficulties. At this time. the asymptotic solutions do provide a substantial amountof insight to the nature of the oscillatory flow but caution must be used wheneverthey are employed to actually calculate admittance coefficients.
83
REFERENCES
1. Tsien. H.S. The Transfcr Functions of tiocket Nozzles. American
Rocket Society Journal. Vcl. 22, 1959, P. 139. p. 162.
2. Crocco. L. Supercrirical Gaseous Discharge zsith High Frequency
Oscillations. Aerotecnica. Rona. Vol.33, 1953, p.46.
3. Crocco. L. Theory of Conbustion Instability in Liquid Propellant
Cheng, S.I. Rocket Motors. AG.ARD-graph No.8, Butterworths ScientificPublications, London. 1956.
4. Crocco, L. Verification of Nozzle Admittance 77;zory by Direct Measure-et al. nent of the Admittance Parameter. American Rocket Society
Journal. Vol.31. 1961. p.6.
5. Morse. P.M. Vibration and Sound. McGraw-Hill, New York. 1948. p.399.
6. An Experimental Investigation of Combustion StabilityChcracteristics at High Chamber Pressure: High FrequencyComputer Program. Aerojet-General Report 11741/SA6-P,Vol.2, 31 August 1966.
7. Culick. P.E.C. Stability of High Frequency Pressure Oscillations in Gasand Liquid Rocket Combustion Chambers. MassachusettsInstitute of Technology. Aerophysics Laboratory Report480. June 1961.
8. Zinn. B.T. A Theoretical Study of Nonlinear Transverse CombustionInstability in Liquid Propellant Rocket Motors. PrincetonUniversity Department of Aerospace and Mechanical SciencesReport 732. May 1966.
9. Crocco, L High Speed Aerodynamics and Jet Propulsion, Fundamentalsof Gas Dynamics. Vol.III, Section B.6. Princeton Univer-
sity Press, 1958.
10. Bridge, J.F. An Extended Table of Roots of Jn(x)Yn(Bx) - Jn(x)Y=(x)Angrist, S.W. 0 . Mathematics of Computation. Vol.16. 1962. 78, P.198.
11. Reardon. P.1H. An Investigation of Transverse Mode Combustion Instabilityin Liquid Propellant Rocket Motors. Princeton University
Aeronautfcal Engineering Report 550. 1961.
12. Crocco, L. Theoretical Studies on Liquid Propellant Rocket Instability.Tenth Symposium (International) on Combustion. CombustionInstitute. June 1965. p. 1101.
84
TABLE I
Values of sh
Ii 12 3
0 0 3.8311 7.0156t1 1.8413 5.3313 8. 5263
2 3.0543 6.14060 9.9695
TABLE II
vh R 2 je Tc
A5.0 2.0 2.0 0.1 -0. 830 +0. 0001 0.000 -0. 0301 0.000 +0. 0001-0. 827 +0.0201 0. 002 -0. 0321 0.000 +0. 0001-0.831 -0.0581 0.000 -0.0321 0.000 +0.0001
5.0 2.0 1.5 0.1 -0.841 -0. 0101 0.000 -0. 0301 0.000 +0.0001-0.817 +0.0201 0.001 -0.0331 0.000 +0.0001
j-0. 831 -0. 0581 0.000 -0. 0321 0.000 +0.0001
5.0 2.0 2.5 0. 1 -0.844 +0.004i 0 000 +0.0001 0.000 +0.0001-0.843 -0.005i 0.001 -0. 0311 0.000 +0001f-0.'831 -0.0581 0.000 -0.0321 0.000 +0.0001
A5.0 2.0 2.0 0.2 -0.827 -0.0021 0.000 -0.0851 0.000 +0.0001-0.824 +0.0091 0.004 -0.0921 0.000 +0.0001-0.835 -0.0981 -0.001 -0.0931 0.000 +0.0011
2.0 2.0 0.5 0.1 -0.619 -0. 3531 -0.017 -0. 0261 0 -0. 0031-0. 524 -0. 2421 0.010 -0. 1201 0 +0.0031I-0.506 -0.2251 0.033 -0. 1081 -0.001 +0. 0031
2.0 2.0 1.0 0.1 -0.849 -0.4191 -0.002 -0. 1381 0.001 +0.0031-0.542 -0. 2441 -0. 011 -0. 1241 0.001 +0. 002&1-0. 506 -0. 2251 0.033 -0. 1081 -0. 001 +0. 0031
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Ri
Conicalregion *
dfh=,rth Nozzle axis-SThrot t
Flow portion
Direction onic flowat throat
1 Geometry of convergent portion of nozzle
(ZwSh) = Aref ( Z, , - )
z,,,,s,,, = cef ( , , )
Reference nozzle
R- -Scoled nozzle
Fig. 2 Scaling of admittance coefficients
125
2.5
1.5 Sh/O2.0
.5
-.5
-1.0-5 -4 -3 -2 -/ 0Z
Fig. 3 Real part of versus axial distance
3
-I
€ow = .5-2 S $ =.OO
-3 i-5 -4 -3 -2 - 0 0
z
Fig.4 Real part of (2) yersus axial distance
126
.2 -.
I Srh: 1.0
V
ar-. 3
-. 3-- -
-. 4 - __ -
-55 -4 -3 -2 -I 0z
Pig. 5 Real part of pressure admittance coefficient versus axial distance
./
-.1-A2 -
-.3
-. 5
-4 -3 -2 -/ 0z
Fig. 6 Imaginary part of radial velocity admittance coefficient versus axial distance
.5
-.
-'.5
0 3
R -2.0
Me .294
.5 1.0 0. 2.0 2.5 3.0 3.5 4.0 4.5 1.0 5.5Frequency W
Fig. 7 (a) Real part of pressure admittance coefficient versus frequency
2.5-
~'1.22.0- R- 2.0
51S"JM- .294
1.5-
I0
S.5-
.5 1.0 4.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 7(b) Imaginary part of pressure admittance coefficient versus frequency
128
0
U 2' '1.2
2 *8 3001.6 3R'&2.0I
(3 Mu .501
.5 1.0 1.5 2.0 2.5 30 3.5 4.0 4.5 5.0 5.5Frequoncy wd
Fig.7(c) Real part of pressure admittance coefficient versus frequency
2.0
1.2-2 I
.8- yx 1.2O's 3,0Rx 2.0M' .501
-. 4
/5 .0 1.5 2.0 Z 2 . 0 3.5 40 4.5 5A0 5.5
Fig. 7(d) Im~aginary Vnrt of pressure admittance coefficient versus frequency
A 129
7.5
yxl.25.5 - , =30"
R :2.0MP .294
'3.5
2 501 h
1.5 oJ
-. 5
-. 5 n
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 55Frequency w
Fig.8(a) Real part of radial velocity admittance coefficient versus frequency
0
-2
"3 Y-.
,z3o"
'-4R2.
M z .294
-55
-6
-7.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
PF-.8(b) Imaginary part of radial velocity admittance coefficient versus frequency
--
130
" 16
12 5'Y- 1.22t Sh ,:30 °
R= 2.0S8 .501
4
0
-4I I I.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Fig. 8(c) Real part of radial velocity admittance coefficient versus frequency
0
-2
%' 4-
C-) M= .501
- / I I I I.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
i Fig. 8(d) Imaginary part of radial velocity admittance coefficient versus frequency
1'
131
.04
.02
-02 A 1.22Srh 8-"=30-3 R -2.0
5 M- .294
-. 04
-. 06
.5 /.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency .'
Fig. 9(a) Real part of entropy admittaDce coefficient versus frequency
.16
Yz 1.2
.12 ' = 30"
3 R-2.0
S.08h M - .294
04
0
.04-.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5,0 5.5
Frequency w
Fig. 9(b) Imaginary part of entropy admittance coefficient versus frequency
132
.04
0
~04-SY h -J2 R =2.0
-. 08
-. /6
- .2 I I I I I.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Pig.9(c) Real part of entropy admittance coefficient versus frequency
: .20
.16 5
2 S~h r1. 2
1 4:IS30.12 JR r2.0
M .501
.08
° .04 -
0o 1 1 ..5 /.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Fig. 9(d) Imaginary part of entropy admittance coefficient versus frequency
133
2.00-_______ __-______
1.25
- -. 25
-3.25z .294I
.5 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 10(a) Real part of irrotational admittance coefficient versus frequency
- .45
-85
-1.25Srh y = 1.2
-9 30,
-1.65 M = .294
.5 /.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 10(b) Imaginary part of irrotational admittance coefficient versus frequency
134
2.00
1.25-
.50-
.~.25
-1.75
.5 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 10(c) Real part of irrotational admittance coefficient versus frequency
.25-
-. 50
-. 75 M.0
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 10(d) imaginary part of irrotational admittance coefficient versus frequency
135
7.5
5.5 5 y :123 0, =300
2 S;,hR =:2.0M = .294
-21.5
.5 h.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 11(a) Real Part of combined admittance coefficient versus frequency
16
12 rl 2
L 9 : 30-
21Sj~hM = .294
U0'
0
-4.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency wo
Fig. 11(b) Imaginary part of combined admittance coefficient versu~s frequency
~ 7.553 A1.
55 4 R =2.0.0 M: .501
.5 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency wd
Fig. 11(c) Real Part of combined admittance coefficient versus frequency
211.
a/c 300
M: .50117
-3
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5,5Frequency w
Fig. 11(d) Imaginary part of combined admittance coefficient versus frequency
137
1 0
r1.2M = .704R -2.0
-2. R--3.0--
.5 h.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
I Frequency w~
Fig. 12(a) Real part of pressure admittance coefficient versus frequency: Effect of
* throat wall curvature
2.C0 ____________________________
1.6-
1.2 - -
S.4 N. R z 3.0 ---
138_ _ _ _ _ _ _ _ __
- 2.5
t5 %. 1. .0 25..5 5 -. 0 4. 50 55
2.5 .29
-R 2..5 1002.0 Z5 3. . .245 . 55
1.5
1~J .5 -
-.5
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 55
* Frequency w
Fig. 12(d) Imaginary part of pressure admittance coefficient versus frequency:i
Efec _ _ _ _ _eanl
139
12 20 2 F1.2y
0= 30, .. .M = .704"%) R --2.0
4 I I
.5 h. 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency ow
Fig. 13(a) Real part of radial velocity admittance coefficient versus frequency:Effect of throat wall curvature
R =3.0---8
-/0-
- I I ! I ! I 1 ,I!
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 13(b) Imaginary part of radial velocity admittance coefficient versus frequency:Effect of throat wall curvature
140
7.5
Rz 2.0iM M= .294
4 3.5 \a _9_r1 _
-2.5
.5 1.0 1.5 2.0 2.5 3.0 4.5 4.0 4.5 5.0 5.5FroquencyC J
Fig. 13(c) Real Part of radial velocity admittanice coefficient versus frequency:Effect of cone angle
0:.
Me .294
-6-/
-7.5 1h0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency wa
Fig. 13(d) Imagiary part of radial velocity admittance coefficient versus freque., y:* Effect of cone angle
* 141
0
-. 04 I
-. 2.2
-. 201.
.0
.5 .0 1.5 20 2.5 3. 3. 4.. 2 . .Frequency00
Fig.~~~~~~~~~~~~~~~ &4b .7g0r4at fetoyamitnecefcin essfe~e:Efc
of thrat wal cur.ture0
R - 3.0--
142
.. 0
.02
.02
R=2.0M= .294
- .04
6=0
-. 06
-08.5 1.0 1.5- 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Fig. 14(c) Real part of entropy admittance coefficient versus f~equency: E1fiect ofcone angle
.16
r=1.2R =2.0
. 12 M = .294
(.08
0
15 .0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 14(d) Imaginary part of entropy admittance coefficient versus frequency: Effectof cone angle
2-OOF-143
.*25
0
r01.249 =30-
M = .704
R=3.0-1.75
.5 /.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 15(a) Real part of irrotational admittance coefficient versus frequency: Effectof thruat wall curvature
.35
-.05
Freqenc =1w
F~~~~~~~~~~~g. ~ ~ ~ ~ ~ ~ ~ ~ 9 =5b Imgnr3ato0roat0a ditnecefcin essfeunyEffcct~~~ of thoa7al0cr4tr
144
-.25 -/M. -- M .294
-3. 251II I.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
V . Frequency w
Fig. 15(c) Real part of irrotational admittance coefficient versus frequency: Effectof cone angle
.75- 1
3.5-
-. 05
-. 6.5Y 1.
-2.05 L.5 /.0 '.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Fig. 15(d) Imaginary part of irrotational admittance coefficient versus frequency:Effect of cone angle
145
8 YI.0, = 300
M=.704R =2.0-
6 ~R =3.0 ---
0
-2
4 i II.5 40 1.5 2.0 2.5 3.0 3,5 4.0 4.5 5.0 5.5
Frequency w
F'ig, 16(a) Real part if combined admittance coefficient versus frequency: Effect ofthroat wall curvature
32.5
25.0 y =/. 2
Mz .704Sph7. R z2.0-
i~ = R 3.0-- -
0.0
2.5
5.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Frequency w
Fig. 16(b) Imaginary part of combined admittance coefficient versus frequency: Effectof throat wall curvature
146
7.5 __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
y .25.5 R 2.0
M -. 294
3. 0IN
.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 16(c) Real part of combined admittance coefficient versus frequency: Effect ofcone angle
20
12 ~ 12R =2.0
51ShM: .2949,= 150---
04
0-4
.5 1.0 0. 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5Frequency w
Fig. 16(d) Imgtners part of combined admittance coefficient Versus frequency: Effectof cone angle
147
Entrance portionof
approximate nozzle D
R*:=Rr,h*
Entrance portion of_ A actual nozzle
rth Nozzle throat Nozzle axis Nozzle entrance
Fig. 17 Nozzle geometry and comparison of entrance portions of approximate and actualnozzle contours
5.2r .2
4r "\_ _ __
b4.8[ 0
2.4 -1.2 Cone srmi-angle -30°
S, 1 . 841292.0 -. 4 o" :.50
3.6 6
2.6 "5.0 -4.5 -4.0 3.5 -3.0 -2.5-2.0-I.5 -1.0 -. 5 0 .5 1.0
Fig. 18 Pressure perturbation versus axial distance from nozzle throatILVTw=/
148
2.2F 4-52.0 4.o0 i
I I \
1 1.6F 3.0 , -Z r2.O
0
/.0 11,5 Cone semi-angle S 300
~I. 2.5 .. . . . . . . . . . . . . . . . .
I. - - /. 8. 1-
.4 L o- JI"
II
-5.0 -4.5 -4.0 -3.5 -3.0 -Z5 -1. -. 0 - .5 0 .5 .0
4xioi distance
Fig.19 Axial velo-Cty perturbation versus axial distance from nozzle throat
5 .6 1.6
4.5j 1.- ' :.42
i 5- -2 .0 - 5
--85
Axiol distnce
ig. 19 Raial velocity perturbation versus axial distance from nozzle throatI. 4-* .2 ~ \.~ I ----
.8.50
0 - 4 5. - 3. -- 25-20-5 / .5 0 5 10
, 5 L .2 - xi Conei setanle
Fig.20 adial elocityperturatinvruxia l distance fo ozetra
149
/0 ScJ 1 .5
ISvh= 2.05 1 Asymptotic solution r =1.2
Iir /jjExoct solution 61, = 30oL/i ~-5, 1iI
6
Asymptotic solution
IExact solution2-
-2 \
1 I I I ! I I
.05 .10 .15 .20 .25 .30 .35 .40Moch number
Fig. 21 Irrotational admittance coefficient: Comparison between exact and asymtotirsolutions
[:4