SCALING P R O B L E M S ASSOCIATED WITH UNSTABLE B U R N I N G IN SOLID PROPELLANT ROCKETS R. W. HART AND J. F. BIRD The general problem of acoustic instability was surveyed at a panel session of the Eighth International Symposium on Combustion. At that time emphasis was placed on the kinds of experimental measurements which would be necessary before our understanding of the phenomena could be put on a quantitative and practically useful basis. The nature of the role of theory in reaching for this goal was also considered. Since that time much new research has been undertaken and much progress has been made. Accordingly, it seems to us that a reexamination of our status visdvis the objective of a practical understanding could be valuable, particularly in focusing attention on essential unresolved problems, but also partly in enhancing a general awareness of recent progress and present activity. This reexamination is undertaken here within the restricted framework associated with scaling oscillatory instability. It is not possible, in general, to determine the stability of a full scale motor by merely noting the stability of a scaleddown version. Nevertheless, it is possible to analyze the acoustic gains and losses via small scale tests, and one may hope thereby to evaluate the degree of stability of full scale motors. Of Course it is possible to carry out this evaluation only to the extent that the important gainloss parameters are known. Sample analyses are presented for such cases, and the limitations imposed by our present lack of knowledge are discussed.
Introduction Unstable burning of solid propellant rocket motors has been a rather commonly occurring malfunction. One of the most troublesome kinds of unstable burning is characterized by the generation of acoustic fields within the motor cavity. Various undesirable effects may be produced by these acoustic fields. For example, the burning rate of the propellant is often sufficiently enhanced that severe overpressure occurs and the motor explodes. Although much research has been devoted to acquiring a detailed understanding of this unstable burning,14 the details are far from complete, and the translation of this understanding into engineering precepts useful in motor design has hardly begun. In fact, as the phenomena become better understood, one becomes increasingly impressed with the difficulty of that translation. Since full scale testing tends to be prohibitively costly, the desirability of assuring the acoustic stability of a motor by means of small scale laboratory testing is quite apparent. But examination of the factors affecting this stability indicates that the possibility of rigorous scaling of motors as entities seems remote, although limited 993
success has occasionally been achieved over limited regions with some motors? Some of the difficulties are similar to those affecting the stability of liquid fuel motors while other problems arise which are unique to the solid motors. I n spite of the general infeasibility of a gross scaling of the overall motor, it turns out that one may still hope to infer acoustic stability of large motors via small scale experiments. This expectation rests on a resolution of the factors influencing the stability such that each may be scaled individually. The stability of the motor itself may then be inferred by carefully considering the cooperative effect of these individually modeled factors. Of course, these ideas are not new. Many of the factors which will have to be known in order to determine motor stability were discussed at the Eighth Symposium on Combustion.1 In the present study, we wish to reexamine the status of this problem with particular emphasis on the problems of scaling. It will become apparent that much remains to be learned before the stability of full scale motors can be predicted confidently on the basis of small scale testing, although much valuable information is now becoming available. We hope that the discussion which follows will be of value to future research
994
MODELING
in providing guide markers to aid in the delineation of areas which demand exploration, as well as in noting other regions where research is well under way.
The Criterion for Acoustic S t a b i l i t y The Approach to the Problem For the discussion of the question of acoustic stability or instability in a solid propellant motor, there are certain thoughts which should be fresh in the mind. The basic question is how the acoustic losses compare with the acoustic gains. Thus, it will be essential to specify the various mechanisms whereby acoustic energy may be gained or lost in the rocket motor. What, then, are the sources of acoustic amplification and attenuation? Consider the schematic representation of such a motor according to Fig. 1. Here, we view the motor as an acoustic cavity containing as an acoustic medium the solid propellant and the gas, note the variety of physicochemical factors influencing the growth or decay of sound, and pause for a moment to consider the magnitude of our task. One way to proceed would be to a t t e m p t to write down all of the equations describing the acoustic field and then attempt to solve them. A perhaps less difficult path to follow, and one adequate for modeling purposes, might be to avoid solving the partial differential equations
GAINS; RESIDUAL CHEMICAL REACTIONS
BURNINJZONE (GAIN OR LOSS)
LOSSES" VISCOTHERMAL AND MOLECULAR RELAXATION
PRINCIPLES
by searching, instead, for similarity principles. I t would serve no discernible purpose for us to attempt to follow either of these two courses here. The thought of attempting realistically to write down the equations describing the acoustic response of the chemical reactions, as well as those describing, in general, the frequencydependent viscoelastic moduli of the solid propellant, is overwhelming. Thus, it is essential to forego the desire to model the motor as an entity. Instead, we shall assume that the relevant mechanisms may be studied experimentally, and then represented empirically in the analysis of stability. We must then evaluate the extent to which this approach will facilitate the assessment of acoustic stability via modeling.
The General Stability Criterion To make the problem at all tractable, we shall consider in detail only the question of instability in the presence of arbitrarily small disturbances, with only occasional references to some of the more difficult b u t sometimes important finite amplitude aspects. For present purposes, then, we may restrict our attention to the question of whether an acoustic field, as described by the linearized fluid dynamic and viscoelastic equations, would tend to grow or decay. Several vital points appear: (a) Throughout the chamber, the acoustic field will be described by the usual waveequation,
GAINS: CONVERSION OF FLOW ENERGY
~ TRANSMISSION LOSS
CHAMBER CASE LOSSES:VISCOTHERMAL AND TRANSMISSION GAINS: EXTERNAL INFLUENCES
Fro. 1. Schematic diagram of motor showing acoustic sources and sinks
SCALING PROBLEMS IN SOLID PROPELLANT ROCKETS
suitably modified by the steady flow of the burned propellant gas. In the typical case, acoustic losses and gains will be sufficiently small so that their effects can accurately be represented in terms of perturbations on the lossfree field. (b) The various boundaries, insofar as they affect acoustic stability, will be characterized by acoustic admittances. Since the burning surface is very thin compared with sound wavelengths of interest, it will be collapsed and regarded, for purposes of acoustic stability, as a surface. The condition for acoustic stability can be expressed symbolically in the following form~:
f
an boundarysurfaces
Each of these parts is susceptible to experimental determination under certain kinds of conditions, and in this way the stability of a motor may, in principle, be inferred from a study of the acoustic behavior of the propellant in a small scale test motor.
The Acoustic Field As long as the pressure oscillations are small compared to the mean pressure, the sound field is determined by a linear partial differential equation with apl)rol)riate boundary conditions. If the losses (and gains) are relatively small, it is
dS [Influx due to [mechanical work + convectional
 fgas and solid volumedV {volume dissipation (less gain)} < 0
The overbar denotes time average, dS is tlle surface element, and dV is the volume element. The surface integral extends not only over the exterior surface but also over the surface bounding the burning zone. Both the gas and the solid will be assumed homogeneous insofar as their elastoacoustic properties are concerned. It is usually convenient to represent the volume dissipation or gain of acoustic energy by means of generalized viscosities. If amplification should be produced, by residual chemical reactions for example, the gas viscosity would be negative. For the solid dissipation it will be necessary, in general, to specify both the dilatational and the shear viscosities. Equation (1) expresses the balance of acoustic gains and losses as a sum of contributions arising from each of the several surfaces, and from the volume of the gas and the solid. It is most important not to lose sight of the obvious but important fact that if the boundary conditions at the surfaces are specified in the usual way (in terms of admittance), and if the generalized viscoelastic constants of the media are known, then it becomes possible to assess the stability for any field by carrying out the indicated integrations. One should probably not be surprised that the acoustic field itself sometimes turns out to be readily sealable. Thus the problem before us divides itself naturally into three distinct but interrelated parts: (a) (b) nents (c)
995
Specification Specification of the surface Specification
of the acoustic field; of the loss (or gain) compoadmittances; and of the viscosities.
(1)
fruitful to consider them as producing perturbations in tile field that would exist in their absence. The strengths of these sources and sinks of acoustic energy are then determined from this "zeroorder" field, and the balance of gains and losses in turn decides the stability of the sound field. Let us consider the "zeroorder" field equations.
The Acoustic Field in the Solid. To begin with, the field is complicated by the presence in the rocket chamber of two different mediathe solid fuel and the product gas. If we confine our attention to a single frequency component, the displacement S characterizing the viscoelastic motion of the solid obeys the usual differential equation s Cd2 grad div S  c~~ curl curl S + ~o~S = O, where Cd and c8 are dilatational and shear phase velocities determined by the elastic constants of the solid. (The time dependence exp (/~t) is assumed.) The familiar frequency vs length similarity principle obviously follows directly from this equation. This principle states that, for frequeneyinvariant boundary conditions and elastic constants, it would be possible to model the field by trading smaller dimensions for higher frequency. Unfortunately, in the rocket motor problem, both elastic constants and boundary conditions on the solid surfaces are frequencydependent, so that such modeling does not apply in general. This is not necessarily a serious setback, however, because one of the things theory has accomplished with some rigor is the stlecifieation of acoustic fields, at least for regular geom
996
MODELING
etries, whenever the elastic constants of the media are known. Of course, propellants may exist which are sufficiently viscous to absorb whatever small fraction of incident acoustic energy might succeed in penetrating the solidgas interface. In that event, the viscoelastic motion of the solid could legitimately be neglected in the determination of the zeroorder acoustic field. These considerations are intended mainly to emphasize the necessity for measurement of the viscoelastic moduli of solid propellants. Such data arc now becoming available to us, and considerably more will soon be appearing. 7 The Acoustic Field in the Gas. Although it is not possible entirely to separate consideration of the field in the gas from that in the solid, let us at least focus our attention for a moment on the gasfilled cavity. The immediate question is the determination of this field. Here, again, we would have a frequencydimension similarity rule if it were not for the fact that again the boundary conditions are usually frequencydependent and, in addition to this, the acoustic properties of the gasfilled medium are dimensionally dependent through the dependence of the mean flow field. The sonic nozzle continues to represent a fundamental difficulty, although some approximate theory presumably applicable to nontypical, very long nozzles has appeared recently, s I t does seem significant, however, that in transverse modes the nozzle admittance may have a negative real part for some frequencies. This presumably indicates conversion to sound of energy transported into the nozzle via convection. Perhaps the familiar onedimensional theory 9 will prove of value for the axial modes although it is based on the long nozzle approximation and mean flows which seem unrealistic for short nozzles. In any event, it is still impossible to discuss with confidence the effect of nozzle scaling. For this reason it appears feasible at the present time to determine the acoustic field in the gas only for primarily transverse modes of long rockets where the nozzle effects may be ignorable, and for side nozzle test motors, where the effect of subsonic orifices has been treated for axial modes. I~ These "nozzles" are generally characterized b y admittances having negative real parts. Apart from the nozzle and the problems which arise from the head and tail cavities, there are other difficulties which arise from the mean flow of the gas. Qualitatively, these difficulties arise partly because sound travels faster downwind than it does upwind and partly because oscillatory energy transport in a mean flow can occur not only via sound waves but also via entropy waves propagating with the flow speed. At sufficiently
PRINCIPLES
high frequency, the entropy wave dissipates within a thin boundary layer near the burning surface where it is generated, 11 and its effect on the zeroorder acoustic field can safely be ignored. At low frequencies, however, the entropy wave may persist throughout the chamber. In such a case, the fluid dynamic field is not properly represented in terms of sound waves alone. This introduces further complexities into the treatment of low frequency modes of small motors. In summary, it should be clear that several basic problems in the determination of the acoustic field in solid prol)ellant rocket motors remain inadequately treated from a theoretical point of view and inadequately explored from an experimental point of view. Idealized Example. Having noted that many of the problems connected with the determination of even the lossfreei.e., zeroorderacoustic field are still unsolved, let us attempt to gain some appreciation of what may be accomplished when the acoustic field is known by considering a highly idealized example adapted from ref. (3). Consider a long cylindrical motor without a head cavity, and restrict attention to the relatively high frequency, primarily transverse, modes. The solid propellant will be assumed to have a viscoelastic damping length that is short compared with the thickness of the grain. If the porttothroat ratio is rather large, but the port Mach number small, it will seem reasonable to approximate the acoustic field in the propellant channel by neglecting the mean flow and regarding the port plane as an acoustic velocity node.* Then the acoustic pressure and velocity (in cylindrical coordinates r, ~b, z) are approximated by the real parts of P = /sP0 exp (/~t) cos (m~b) cos ( ~ h z / L ) J m ( a r )
(2) and i u =   grad p, pr where a 2  (,.,Vc~)  (h~r/L) 2, and e2 = 3'Dip. (/hp0) is the acoustic pressure amplitude, o~ = 27r times frequency, p is the mean gas density,/5 is the mean pressure, m and h are the azimuthal and axial mode indices, and the frequency is determined by the condition that the propellantgas interface correspond to an acoustic velocity * For the conditions outlined here, the convective flow term in Eq. (1) can be neglected because the contributions from the propellant surface and the port plane tend to cancel.
SCALING PROBLEMS
IN SOLID PROPELLANT
node. In order to evaluate the stability criterion, as expressed by Eq. (1), it is necessary only to carry out the integrations indicated. This will, of course, require specification of the attenuation or amplification at boundaries and in the body of the gas, which will require the real part of the admittances of these boundaries and the acoustic damping length (or the generalized viscosity) of the burned gases. Accordingly, we turn now toward the specification of the sources and sinks of acoustic energy.
Surface Gains and Losses We now wish to discuss the evaluation of the net effiux or influx of mechanical work at each boundary as indicated by the surface integral in Eq. (1). It is customary to specify the matching conditions at boundaries in terms of admittances, and our attention is therefore first directed toward the admittance of the burning surface, which is believed to be the most important source of amplification.
The Burning Zone. It is true, of course, that if the acoustic field were known, and if the acoustic admittance of the burning surface were also known, then the evaluation of the contribution of the burning surface to instability would be a mere formality. It is for this reason that much emphasis is placed on determination of the acoustic response of a burning propellant as measured by its admittance. At the time of the Eighth International Combustion Symposium, only crude theoretical estimates of the properties and the order of magnitude of this admittance were available. Now, the first tentative measurements are becoming available, confirming the implication of the theory in general, but more importantly providing essential information pertinent to particular propellants. 12,~ Hopefully, other research directed toward this goal will also soon yield results. Nevertheless, although the general stability question cannot be answered yet, considerable progress already can be made under some specific circumstances. In order to carry through the sample example begun in the previous section, we note that the mechanical work part of the surface integral extending over the propellant surface reduces (in terms of the admittance of the burning layer) to   /
(Re Yb) r p I 9 dS,
(3a)
where Yb is the normal specific admi.'ttance associated with the burning zone, and Ep is the mean acoustic power flowing into the cavity through the burning surface.
997
ROCKETS
A few comments seem in order, here. If the burning of the propellant were not influenced by the component of velocity parallel to the surface (erosion), the admittance would be a constant not dependent on the acoustic field itself, except through frequency, and thus could be removed from the integral sign. In general, however, this simplification must not be effeeted. For the illustrative example to be carried out here, however, we shall restrict our consideration to a configuration where the effect of erosion is believed to be negligible. For such a simple case, and for the field assumed in the previous section, Eq. (3a) reduces to3 E:=
f~" d4,foLdz [ Re p(a)
 Re ( Yb)a jo
]' (3b)
+ 8m,0)(1 + ~h,0)
= l(~aL)(1
X I p0 ]2/~J,~2(aa) Re (Yb).
(3c)
We note that, if Re Yb ~ 0, the burning propellant amplifies the sound field.
Volume Losses in the Gas. It is unfortunate that volume losses do not, in general, scale with volume of the rocket motor, but it is clear that the various attenuation mechanisms tend to be highly frequencysensitive, and of course the mode frequencies are, in turn, functions of chamber volume. There' are several sources of acoustic attenuation in the body of the gas filling the propellant channel. Absorption occurs through ordinary gas viscosity and heat conduction, but more importantly from the relaxation of acoustic energy into internal energy of the molecular constituents. 3 The relaxation loss of sound energy in the burned gases/~g can be expressed in terms of the corresponding attenuation constant ao, by a Eg = (~/4)La2(1 { ~m,0)(1 } (~h.0) X ]p012J~(aa)
[
Fm(aa) ~ a~: J,,(aa) J "y ~
(4a) where
F,,(x)  1  ( m / x ) ~ + [Zm'(x)/J=(x)] ~. (45) It is unfortunate that no experimental information which bears directly on the relaxation loss in hot propellant gases is available. Here is another research area where the application of known techniques is required in the resolution of the rocket instability problem, but, so far as the authors are aware, no research directed explicitly toward this problem is now in progress. In order to assess the possible significance of such losses,
998
MODELING
we shall have to resort to an idealized simple case, and use a crude orderofmagnitude estimate based on considering only the nitrogen component of the product gases (which is expected to become a much more effective absorber at rocket motor temperatures than it is at ordinary temperature). Assuming a 10% abundance of N2, the attenuation constant has then been estimated from data presented in ref. (3) to be ~g ~ 7.3 X 105(ff//5)
for 2500~
(4c)
where f is the frequency. I t should perhaps be mentioned that relaxation loss can be extremely sensitive to chamber temperature, and this point should be kept in mind when we consider hotter propellants. Further, one should be cautioned that this type of relaxation can be very sensitive to the presence of small amounts of gases such as H~, H20, etc. which are effective in energy transfer. The propellant gas is, of course, abundantly supplied with a variety of such species. The significance of this orderofmagnitude estimate is merely that it indicates that gas phase relaxation losses may well be important, and that they merit investigation. At this point we should note that relaxations in chemical reactions could be included here by specifying an appropriate ~a for them. Relaxations in the shift of equilibrium would be expected to contribute a positive ~ (damping) while those related to incomplete reactions might result in either positive or negative values o~ ~a. There has been little research in this area pertinent to solid propellants. ~a A further source of attenuation is the common presence of solid particles in the gas. The attenuation constant for small spherical particles (radius R, number density N) suspended in a gas has been calculated in reference 15. Loss arises both from heat transfer and momentum transfer. The major contribution arises from viscous damping and is expressed in terms of an attenuation constant given by ~

3vR N ,7(1 + z~) c p
16z,4 X
]
16%4 ~ 72zpa~ + 81(2zpe + 2zv + 1)52
(5) where z, = R(wp/2~)i, ~ is gas viscosity, and = p/density of solid <<1. Replacing aa by ap in Eq. (4a) gives an expression for the power loss due to particle damping. (Note that for a distribution of particle sizes N ( R ) d R , av must be integrated over the distribution.)
PRINCIPLES
Acoustic Loss at the Exterior Boundaries. Here, it is especially important to recognize that we are concerned with the possible build up of incipient disturbances, and that we therefore limit our attention to the usual linearized acoustic theory. I t seems probable that the acoustic losses for finite amplitude will be very much greater than the small loss predicted by the linear theory, in view of the wellknown severe enhancement of heat transfer rates which is often observed under severe oscillatory conditions. These considerations, however, presuppose that the mean temperature of the gas is equal to that of the wall, and that there is no mean flow of gas past the wall. In the rocket motor, both of these conditions are usually violated. Since the wall temperature is generally cooler than the gas temperature, the possibility of conversion of thermal energy into acoustic energy at the chamber wall must arise. In other words, we are not absolutely certain that the wall of the rocket motor is really an absorber rather than an amplifier of acoustic energy. Here is another area where both experimental and theoretical work would be of considerable importance to our problem. For the simple illustrative example to be considered, however, we shall be primarily concerned with the head cavity in which the mean flow can be assumed to vanish. If the difference between wall temperature and gas temperature is neglected, the head walt loss may be estimated from the usual acoustic wall loss theory. One obtains 3 k =  (7r/4)a2(~l/2poJ)89 po ]2Jm2(aa) (Pw/3f)
)'( (ac/w)2(1 'J($~,o)[(1[ 5 '  ) 8951 a~c: ,w2~ X Fm(aa) +
2 J='(aa)_]
aa J,,,(o~a) J"
(6)
0rderofmagnitude calculations suggest that the losses at other exposed walls and through the outer wall of the rocket will ordinarily be negligible. There is, of course, the possibility of acoustic energy input at the outer walls at the aerodynamic screaming frequencies of the motor in flight, as has been noted by McClure. 16 The significance of this potential power source, which might go far toward explaining some differences in stability observed between missiles on the thrust stand and missiles in flight, has not yet been evaluated. The importance of the nozzle as a sink of acoustic power has already been referred to in the discussion of the lossfree acoustic field, and will not be commented upon further here, except to note that the order of magnitude of the loss (or gain) component of the nozzle admittance as
999
SCALING PROBLEMS I N SOLID P R O P E L L A N T ROCKETS
obtained from long nozzle theory can be of the same order of magnitude as that characterizing the gain at the burning surface. 8,9 This fact, and experimental evidence, confirm that the nozzle can be an important constituent of the stability problem.
The Cold Propellant Region. The loss of acoustic energy in the body of the solid propellant should be expected to have a significant effect on the stability of the system. A completely general treatment of a solid absorptive medium seems unlikely to meet with success. Perturbation methods, however, have been used to handle two extreme cases, namely, the cases where the damping length in the solid is either long or short compared to the web thickness. For the case of small viscous loss (long damping length) the solid must be treated as an acoustic medium bounded by gas medium. The motions of this twomedium system are calculated on a lossfree basis, and this result is then used to determine the losses which would ensue. Studies based on this standard perturbation treatment have been reported in previous papersY.28 Not unexpectedly, the effectof the solid turnsout to bedominant from time to time during the course of burning for geometries in which the solid participates heavily in the motion, and this effect is a source of intermittency in the stability. ~9 When such is the case, the power loss due to solid damping can be represented by adding to the burning surface admittance Yb [Eq. (3a)] the term Re (i~Sr/P,~)~_~, where St, P~ are the radial components of displacement and stress on the grain. This term is awkward to write explicitly, but its general nature can be discerned from the characteristic solutions of the vector wave equation. Thus for specified physical properties (Lamb moduli, viscosities, and density), we have
gas an admittance whose real part is (psCJ) 1 =
~pc/psCJ~(pc) 1
(psC/>>pC),
(7)
where the subscript s refers to the solid ( c / i s the sound velocity ia the solid appropriate to the modes under consideration). Thus the significance of the solid loss is determined by the relative magnitude of the real part of the admittance presented by the burning boundary, and that given by Eq. (7). As shown in reference 3, these two quantities will indeed tend to be of the same order of magnitude, and the amplifying ability can be expected to be somewhat reduced by losses in the solid. The frequency dependence of the solid propellant elastic constants is of considerable importance here, as is the fact that these constants tend to be very sensitive to temperature.
Sample Calculations As we have tried to emphasize, many sources of gains and losses have not been quantitatively studied in connection with the question of the assessment of linear acoustic stability of solid propellant rockets via small scale tests. But it is important to consider whether or not these mechanisms are really vital to the problem, or whether they can be disposed of because their effects must be trivial. This poses a question which should be resolvable by a quantitative study. It is in this light that the crude sample calculations presented herein are to be viewed, because they are intended primarily to suggest the importance of certain loss mechanisms. The attenuation or amplification arising from the various regions in Fig. 1 are conveniently expressed in terms of equivalent admittances at the burning surface by dividing the power by
[(~'/4)aL(1 +/fro,o) (1 [ ~h,o) X ]Po [2P~J~(aa)~.
Re (i~Sr/P,~)a = function of (m; hb, cob;a/b) for the mode with frequency w/27r, azimuthal node number m, and axial wave number h. Hence some scaling of calculated or experimental data is possible. Calculations for azimuthal ( h = 0) and for axial (m = 0) modes of a hypothetical propellant have been publishedY ,~8 If, on the other hand, the damping length in the solid is quite short, the impedance mismatch at the solidgas interface will be large. Consequently, nearly all of the acoustic energy incident on the solid surface will be reflected back into the gas, while the amount of energy which can be transmitted across the boundary will be dissipated. The solid then presents to the
These (real) admittances are summarized in Table 1, where explicit expressions are given that hold for the field under the assumption that the solid surface is a velocity node for the zeroorder acoustic field. We also indicate in the table the dependence of each gain or loss on the parameters describing the mode, the motor geometry, and the properties of the propellant and its product gas. Finally, we can restate in admittance form the
criterion for stability   R e Yb~ Re (Y~[ Yp~ Yo+ Y N + Yw) where the subscript s pertains to the solid, p to particles in the product gas, g to relaxation loss
1000
MODELING PRINCIPLES TABLE 1 Equivalent* Admittances for Sample Calculation
REGION
CONTRIBUTION TO GAIN () OR LOSS (+) IN ADMITTANCE FORM (c.g.s. units)
[ NONBURNING SURFACES
HEA~ WALL
NOZZLE PLANE
RELAXATION
Re YW=
' l+~h,o
PARAMETERS CHARACTERIZING THE SOURCE OR SINK
.
MODE + GEOMETRY (rn, tua, ha,__a. L' u,h)
L y~
GAS PHYSICALS (y, c, to,T/)
(,Re YN
MODE + GEOMETRY, FREQUENCY, BURNING RATE, SOUND VELOCITY, MEAN FLOW DISTRIBUTION
NO THEORY EXCEPT FORLONG NOZZLES
Re Yg =
i(+)
I ~ ( cL Eq. 141for ag )
Y
P RODUCT GAS
a
ag
DITTO WITH % ~ap
2
PARTICLES
MODE + GEOMETRY (m, r ha, a) GAS PHYSICALS (y, c,p, ag (involves ~) )
(% Involves R, N(R), p, p', c, ~7,~)
Re Yp = (of. Eq. (S) for ap)
ACOUSTIC LAYER Re Yb = COMBUSTION ZONE COMBUSTION LAYER
/REDUCED ~ ReSPECIFIC J ~ADMITTANCE"
MODE FREQUENCY STEADY STATE ~jS RESPONSE FUNCTION (GAS AND SOLID PHYSICOCHEMICALS) MODE + GEOMETRY
= Re ~I;rYrr la(Iowloss) COLD
)PELLANT
(m,~b, hb,~)
PROPELLANT PHYSICALS
Re Ys
(~,~,h',~,p s) 1
p, c;
(high.loss)
* (EACH ADMITTANCE IS NORMALIZED TO THE AREA OF THE BURNING SURFACE)
in the gas, N to the nozzle plane, W to the head wall, and b to the burning zone. The dependence on scale is only partly in evidence in Table 1 because the various elastic constants and damping lengths are, in general, functions of frequency which is a function of size and shape. Table 1 illustrates particularly how stability can be assessed when these parameters are known. In order to display the possible significance of the various mechanisms for which we have made the orderofmagnitude estimates given in the preceding we shall consider each of them individuaUy, pretending that each one, in turn, was dominant. Assuming a representative value of ~(O/P) for the net real part of the admittanee of the burning zone (9 is the mean velocity
of the hot gas emerging from the burning zone), we find that the contours of neutral stability for this mode are as shown in Fig. 2, which is borrowed from reference 3, where its derivation is discussed in more detail. The relevant feature for the present discussion is that each of the gas phase damping mechanisms could very well influence stability. The relatively small exposed head wall area accounts for the fact that these losses are very small for the motor under discussion, so that they are not shown in the figure. For the particular motor and mode under consideration, the curves of Fig. 2 constitute a set of similarity relations as summarized in Table 2 Fwhich is taken from reference 3] where it is noted that the theoretical straightline relation
SCALING PROBLEMS
IN SOLID PROPELLANT
ROCKETS
1001
SO0 w
.•
400
UUz
3O0 ARROWS SHOW
UNSTABLE REGION ,n N U.M O0 O'Y
2
3 PORT DIAMETEI~ Dp (In.)
5
6
FIG. 2. Stability map showing stable and unstable operating regions characteristic of various kinds of gas phase losses. (The response of the burning surface has been assumed frequencyindependent.) ship for small particle damping is in excellent agreement with experimental data of Brownlee and Marble. ~~ In order to illustrate the varied stability behavior which occurs when motors are scaled, we have prepared Fig. 3. For this illustration, we have considered two sources of attenuation in the body of the gas, viz. molecular relaxation and particle damping, and adopted the numerical values used in reference 3. We will recall that, during burning, a given motor with a fixed nozzle area is represented by a diagonal line TABLE 2 Scaling Rules*
LOSS MECHANISM
PORTION OF THE FORM OF THE Kn  D p PLANE STABILITY LINE THAT IS UNSTABLE
SMALL PARTICLES IN THE GAS
Kn oc Dp
PARTICLES OF SIZE A FEW MICROHS
Kn ~c Dp  I
PARTICLES OF SIZE A FEW TENS OF MICROHS
Kn ~
Dp  2 / 7
WALL DAMPIHG
Kn ~
Dp  2
MOLECULAR RELAXATION GAS DAMPIHG
g n o( Dp  2
LOWERRIGHT
LOWERLEFT
UPPER  RIGHT
* These mlal pertain to the propellant of Fig. 2 having a burning rata praosura. axpcmente n m 1/3, Kn Is the ratle of burning propellant to hassle thmet area, and Dp Is th;" pert diameters
segment as indicated on the figure (an internal burning cylindrical charge is being considered). Since the burning area to throat area ratio (K,) is dimensionless, the firing of a scale model of the motor is represented by shifting the line segment as shown in the figure (note the log scales). For the situation depicted, we can see that while the full scale motor should actually operate stably, tests on a scale model would indicate instability or stability depending on the degree of scaling. For example, the onesixth scale motor should be unstable over almost its entire burning period, whereas the onetwelfth scale motor would be stable. This illustrates theoretically the wellknown experimental fact that one cannot in general naively predict the stability or instability of a motor directly from the stability or instability of a scale model. Nevertheless, a careful analysis of scale model firings can give important information on the separate contributions of various mechanisms to the stability of the full scale engine.
Concluding Remarks We have been considering the problem of determining the stability of a rocket motor against small pressure perturbations, by small scale, rather than full scale testing. It seems to be clear that such a determination should be possible, at least in large measure. What is required,
1002
MODELING
PRINCIPLES
200
Ui.,~ u
#'7"
,

.c.,'/ z
~'I
'~ .~
z, 100 " ~ ,I
\
z
UNSTABLE I~EG/ON ~ . .
50
L
",~
\
5 10 PORT DIAMETER,Dp (in.)
~\ 15
20 25 30
Fro. 3. The effect of scaling on the stability of the first tangential mode is shown for the values of gas phase damping parameters indicated. (The g a s  i.e., relaxationdamping length is given in the text.) Quite different effects can be obtained in other modes and with other values of the admittance of the burning surface. TABLE3 I m p o r t a n t P a r a m e t e r s R e q u i r e d f o r S t a b i l i t y Determination
MECHANISM
FUNCTION OF
ADMITTANCE OF BURNING LAYER
AMPLIFICATION OR ATTENUATION AT THE BURNING SURFACE
FREQUENCY, PRESSURE, 'l PROPELLANT COMPOSI. PROPELLANT TEMPERA i TION, CURING TIME, TURE, EROSIVE VELOCITY I METHODOF CURE, ETC.
GAS PHASE DAMPING LENGTH OR ATTENUATION COEFFICIENT
AMPLIFICATION OR ATTENUATION IN THE GAS PHASE
FREQUENCY, PRESSURE, PROPELLAHT TEMPERATURE
DITTO THE ABOVE
SOLID PHASE VISCO ELASTIC CONSTANTS
CONTRIBUTES TO DETERMINING MODE FREQUENCIES AND TO ATTENUATION IN THE SOLID PHASE
DITTO THE ABOVE
DITTO THE ABOVE
NOZZLE ADMITTANCE
CONTRIBUTES TO FREQUENCY, MODE, MEAN FLOW DISTRIBUTION, DETERMINING MODE NOZZLE SIZE AND SHAPE, SOUNDVELOCITY FREQUENCIES AND IN GAS, AND THE SOUNDFIELD, ITSELF. TO GAIN OR LOSSAT THE NOZZLE PLANE
QUANTITY
I
OTHER PARAMETERSMAY OCCASIONALLY BE IMPORTANT ALSO, SUCH AS THOSE DESCRIBING WALL LOSSES, RESONANTROD LOSSES, INPUTS DUE TO AERODYNAMICSCREAMING, ETC.
SCALING PROBLEMS
IN SOLID PROPELLANT
ROCKETS
1003
however, is measurement of the parameters heat transfer rates which may also occur during which are essential to characterizing the acoustic severe oscillation. Thus, it would seem entirely gains and losses. These will include at least those premature to attempt to catalogue the status of summarized in Table 3. We have attempted to the finite amplitude stability question. indicate in the text the extent to which these parameters still constitute essentially virgin Nomenclature ground insofar as research in solid propellants is concerned. The extent of the untilled areas is Inside radius of tubular grain (  1Dp) quite impressive, at least to us. a Outside radius of tubular grain There are other problems which we have b Sound velocity in the chamber gas avoided, such as those presented by irregular c Sound velocity in the grain ( = Cd for geometries, and by occasionally encountered c/ dilatational wave, = c, for shear wave) oscillatory instability at very low frequencies not Port diameter ( = 2a) corresponding to the normal modes of the motor. Dp Circular frequency ( = oJ/21r) These are among the incidental problems which f Axial index of acoustic mode [Eq. (2)] should be susceptible, at least in principle, to the h Bessel function of first kind of order m standard methods of attack, but which involve J~ Ratio of burning surface area to nozzle substantial new difficulties which have not been K~ throat area well explored. It is only natural that clarification Length of tubular grain in these areas should be delayed while the main L Azimuthal index of acoustic mode [Eq. interest lies in resolving the more commonly en m (2)3 countered problems. A further complication which was discussed N(R) Number density of particles of radius R in the gas briefly ensues from the fact that at low freAcoustic pressure quencies the acoustic boundary layer may be so P Acoustic pressure amplitude thick as to include within itself all or a sub pop Mean chamber pressure stantial part of the motor cavity, n In that event P Radial component of stress in grain the oscillatory field should not be represented by P,, Radial position in cylindrical coordinates only the acoustic wave, but must include also r Radius of solid particles suspended in the entropic wave, which is very sensitive to the R chamber gas mean flow distribution. It is not yet clear just Radial component of displacement in how important these considerations will turn Sr grain out to be in affecting the assessment of stability. Acoustic particle velocity The effect of erosive velocity on the ability of u Mean velocity of gas leaving burning the propellant to amplify or attenuate is another zone aspect of the stability problem which deserves Normal specific admittance both theoretical and experimental attention, as Y Axial position in cylindrical coordinates indicated both by crude theoretical assessment z and by experiment. 4'12 We would be in error, however, if we were to a Radial index of acoustic mode [Eq. (2)] leave the implication that the stability deterag Attenuation constant due to gas mination will be settled once the question of relaxation linear stability has been answered. First of all Attenuation constant due to particles in there is the question of stability against finite ap the gas amplitude disturbances. It has now been definitely established that solid propellant (like ~, Specific heat ratio of chamber gas liquid propellant) motors will occasionally be en ~h.0 Kronecker delta symbol countered which remain stable only so long as ~m,o Kronecker delta symbol they are not too seriously perturbed, a This probShear viscosity lem is one of considerably greater difficulty from 7/ k Lam6 modulus both the theoretical and experimental point of view, although it may be that measurement of h' Dilatational viscosity the finite amplitude parameters indicated in tt Lam6 modulus Table 3 will go a long way toward resolving the p Density of chamber gas finite amplitude question. There are further ~ Density of solid propellant difficulties which appear, such as the fact that p' Density of a solid particle in the chamber the solid propellant damping seems to be signifigas cantly altered after exposure to an oscillatory environment, and the question of the anomalous r Angular frequency (=27rf)
1004
MODELING PRINCIPLES ACKNOWLEDGMENT
The authors are especially grateful to Dr. F. T. McClure for many helpful discussions in connection with the preparation of this manuscript. This work was supported by the Bureau of Naval Weapons, Department of the Navy, under NOw 620604c.
REFERENCES 1. Panel Discussion, Eighth Symposium (International) on Combustion, p. 904. Williams and Wilkins, 1962. 2. McCLURE, F. T., HART, R. W., and BIRD, J. F.: Solid Propellant Rocket Research, p. 295. Academic Press, 1960.
3. BIRD, J. F., McCLuRE, F. T., and HART, R. W.: Proceedings of the Twelfth Annual International Astronautical Federation Congress. To be published. Also available as The Johns Hopkins Univ. Applied Physics Laboratory Report TG 3358, June 1961.
4. McCLuRE, F. T., BIRD, J. F., and HART, R. W.: ARE Journal 32, 374 (1962). 5. MORSE, P. M. and FESHBACH, H.: Methods of Theoretical Physics, p. 151. McGrawHill, 1953. 6. Ibid., p. 142.
7. Proceedings of the Second Meeting of the Technical Panel on Solid Propellant Combustion Instability, The Johns Hopkins Univ. Applied Physics Laboratory Report TG 3714a, May 1961. 8. CULICK,F. E. C. : Ph.D. Thesis. Massachusetts
Institute of Technology, Technical Report No. 480 (1961). 9. CRocco, L. and CHENG, S.: Theory of Com
bustion Instability in Liquid Propellant Rocket Motors. Butterworths, 1956. 10. McCLURE,F. T., HART,R. W., and CANTRELL, R. H. : To be published. 11. HART, R. W. and CANTRELL,R. H.: On Amplification and Attenuation of Sound by Burning Propellants. A.R.S. Journal (to be published). Also available as The Johns Hopkins Univ. Applied Physics Laboratory Report TG 33511. 12. HORTON, M. D. and PRICE, E. W.: This volume, p. 303. 13. STRITWMATER,R., WATERMEIER,L., and PFAFF, S. : This volume, p. 311. 14. WmHT, H. M.: AFOSR Report No. TR6060; also Aeronutronics Technical Report No. U858, March, 1960. 15. EPSTEIN, P. and CARHART, R. : J. Acoust. Soc. Am. 25, 553 (1953). 16. McCLuRE, F. T.: Private communication, May, 1962. 17. McCLuRE, F. T., HART, R. W., and BIRD, J. F.: J. Appl. Phys. 31, 884 (1960); BIRD, J. F., HART, R. W., and McCLURE, F. T. : J. Acoust. Soc. Am. 32, 1404 (1960). 18. DETERS, O. J.: ARE Journal 32, 378 (1962). 19. RrAN, N. W., COATES,R. L., and BAER, A. D.: This volume, p. 328. 20. BROWNLEE, W. G. and MARBLE, F. E.: Solid Propellant Rocket Research, p. 455. Academic Press, 1960.
Discussion MR. S. L. BRAGG (RollsRoyce) : Dr. Lawhcad has shown that the modeling parameters can be kept constant, and the relative stability ratings of injector systems determined by tests on strip, annular, or segmental section models of liquid propellant rockets. Could the same technique be applied to solid propellant motors? The frequency would be aPproximately correct if strip or annular chambers were used, and the extra wall damping might not be significant if the major part of the damping is in fact caused by the particles in the stream. DR. R. W. HART (APL/The Johns Hopkins University): It is clear that the acoustic gainless balance is a function of quite a number of parameters and that modeling experiments in which one or perhaps a very few of these parameters were varied could be quite helpful. Of course, this would be particularly true if only one very significant
quantity were varied. With respect to the experiments to which Dr. Bragg refers the steady state flow might be well reproduced in the model. The oscillation frequency which is known often to be an important quantity could also be held substantially fixed. The acoustic field, however, would then not be correctly modeled in all dimensions. I do not believe that it is yet known just how important this lack of fidelity in the model might be. There would be other varied quantities, however, such as the exposed motor wall area, and probably the acoustic loss at the orifice. These variable factors would be expected to influence the validity of the modeling to an extent which would have to be determined. Two important questions would appear to remain unilluminated, however. If the model proved to be stable, no information would seem forthcoming as to the margin of safety, and it would be difficult to infer finite amplitude properties from the model because the acoustic field would, in general, not be faithfully modeled.