Skip to main content

Full text of "NASA Technical Reports Server (NTRS) 19800004938: Amplification of Reynolds number dependent processes by wave distortion. [liquid fuel combustor stability"

See other formats

A/^3f\ -CK- 15^)73^ 



\. Report No. 

NASA CR-159732 

4. Title and Subtitle 

2. Government Acwviion No. 


3. Recipient's Catalog Np. 

5. Report Date 

November 197 9 

6. Performing Organization Code 

7. Author(s) 

M. Ventrice 

9. Performing Organization Name and Address 

Department of Mechanical Engineering 
Tennessee Technological University 
Cookeville, Tennessee 38501 

12. Sponsoring Agency Name and Address 

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

8. Performing Organization Report No. 

10, Work Unit No. 

11. Contract or Grant No. 
NCR 43-003-015 

13. Type of Report and Period Covered 
Final Report 

Jan. 1, 1972-Oct. 31, 1979 

14. Sponsoring Agency Code 

15. Supplementary Notes 


Richard J. Priem, Technical Monitor wes 

NASA Lewis Research Center 

Cleveland, Ohio date 


16. Abstract 

The object of the research v/as t ^ 

dependent process by wave distortion 

similar Reynolds number dependent pr 

with the operation of a constant-temp< 

interest was investigation of vaporizat 
associated with iiquid propellant rocks 

A series of experiments were car 
a Reynolds number dependent process 
process and the combustion process, 
both having the ultimate objective of 
to different chamber geometries and d 

The results indicate a high degr 

possibility of using the anemometer s^ 

nonlinear, aspects of a Reynolds numb 
mechanisms controlling instability. 



“rynolds number 
-Bsults to other 
as that associated 

plication of most 
" ustion typcially 

ive distortion on 
-I the anemometer 
-Jies were involved, 
izing behavior common 

jsses and the 
ility. The 
"he dominant 


NASA-C-168 (Rev. 10-75) 


The research summarized in this report was carried out by a number 
of individuals who functioned in various capacities. The initial idea 
for the research originated with Allan E. Hribar, who was also the first 
principal investigator. Later, Kenneth R. Purdy and finally Marie B. 

Ventrice served as principal Investigators. There have also been several 
faculty associates: John P. Wallace, Kenneth R. Purdy, Marie B. Ventrice, 

and John Peddieson, Jr.. The Ph.D. level graduate students who participated 
were Marie B. Ventrice, Jih-Chin Fang, and Kin-Wing Wong. The masters 
level students were John Bordenet, Jr. and Gary H. McDonald and A. Googerdy. 

The advice, encouragement, and support of Richard J. Priem, the technical 
monitor for the research, has been very important and is gratefully acknow- 
ledged and appreciated. 

Critical support was provided by Tennessee Technological University 
through the Department of Mechanical Engineering. In addition, the coope- 
ration of the Department of Engineering Science and Mechanics played an 
important role in the research. 


My 'V 



Chapter 1: Introduction and Background 1 

Chapter 2: Research 6 

Combustion Model 6 

The Analog. 7 

Initial Research Plan 10 

Analytical Open-Loop Studies 10 

Experimental Open-Loop Studies. .12 

Experimental Closed-Loop Studies 1A 

Pressure-Sensitive Closed-Loop Experiments . . 16 

Velocity-Sensitive Closed-Loop Experiments 17 

Theoretical Analysis of the Anemometer Output Signal . . 2.1 

Discussion of Closed-Loop Analog Investigations 2k 

Gasdynamical ly Induced Second Harmonic Sound Fields. 26 

Twin-Chambers Investigations, ... 29 

Mathematical Modeling 38 

Chapter 3: Conclusions 50 

References 53 


Introduction and Background 

The object of this report is to summarize the research and the signi- 
ficant results of the research which was carried out under NASA Grant 
NGR i«3"003"015 from January 1 , 1972, until the grant termination on 
October 31, 1979. 

The purpose of the research was to study high-frequency combustion 
instability in liquid-fuel rocket engines — instability in which the com- 
bustion process is coupled to acoustic waves. To do this, new experimental 
and analytical tools and techniques were developed. Basically, it was 
desired to develop a better understanding of and more insight into the 
fundamental phenomena associated with combustion instability, with the 
view that results of such studies would not be confirmed in application 
to liquid-fuel combustor instability, (a Reynolds Number dependent pro- 
cess), but would have general application to other Reynolds Number dependent 
processes. These other processes would include those associated with all 
types of combusters — solid- as well as liquid-fuel rocket engines, jet 
engines, and furnaces. 

Combustion instability has been a persistant problem in both liquid- 
and sol id- fuel rocket-engine development programs. In liquid-fuel rocket 
engines, the fuel and oxydizer are injected into the combustion chamber 
where they are vaporized and combustion takes place. The resulting hot 
gases continue through the chamber and are accelerated to supersonic 
velocities by means of a converging-diverging nozzle and ultimately leave 
the engine. This process is not smooth, but has associated with it back- 
ground osci 1 lations which are detected as fluctuations in gasdynamic 
variables such as pressure, density, temperature, and so forth. (These 
osci I lations can be a result of a number of phenomena such as turbulence 


In addition, some initial minimum amplitude of gasdynamic disturbance 
was necessary to excite instability. 

Analytically, some success had been achieved with modeling. It 
was generally accepted that combustion instability was primarily sensitive 
to fluctuations in the gas pressure and the velocity of the fuel drops 
relative to the gas. The problem of pressure-sensitive combustion in- 
stability had a long history of successful analytical treatment; but, 
because of Its greater mathematical complexity, much less work had been 
done with analysis of velocity-sensitive combustion instability. 

Acquiring the understanding of combustion instability needed to 
formulate rational design procedures depended critically on the develop- 
ment of realistic modeling techniques, and it was toward this end that 
this research was begun. Initially the emphasis was on development of an 
experimental analog of the combustion-acoustics interaction. Later, ana- 
lyti cal techniques were developed, partially in support of the experimental 
work and partially as a valuable study in its own right. 

Several specific observations led to the initiation of this research. 
First Priem [l] established a model in which combustion is viewed as. 
vaporization-limited (i.e., as a velocity-sensitive or Reynolds number 
dependent process). Later, Heidman [2, 3 ] showed that distorted acoustic 
oscillations (fundamental frequency oscillations plus high harmonic 
"distortion") affect the open-loop response of this vapor izati on- 1 imi ted 
combustion. The response is a measure of the degree of reinforcement of 
the acoustic oscillations and the combustion process. It is important to 
know how the instantaneous gasdynamic acoustic field combines with the 
acoustic fieid resulting from the combustion process. One of the better 
ways of reiating these two fields is to' find the correlation between the 
pressures of each. Such a correlation could then be used to determine the 


closed-loop response. But no explicit expression for the acoustic pres- 
sure which- resul ts from a particular combustion process was available; 
hence, Heidmann analytically comparand the existing pressure with the 
burning rate of the', combustion process, i.e., he formed an open-loop 


response. He used harmonic response factors to indicate the degree of 
reinforcement of the acoustic pressure by the combustion process and 
found that the response factors depended upon the amplitudes of the har- 
monic components and the phase between the acoustic pressure and burning 
rate harmonic components. Lastly, Hribar [A] noticed the similarity of 
the open-loop responses of a liquid-fuel droplet's vaporization rate and 
a constant-temperature hot-wire anemometer's energy transfer rate to the 
same gasdynamic environment. 

Combining the above observations, Hribar [i»] proposed the development 
of an analog technique for studying combustion instability, in which a 
constant- temperature hot-wire anemometer would serve as the analog to 
combustion. The advantages to such a system would be many. The analog 
would not involve an actual combustion process so that the many problems 
associated with making measurements in a hot, burning mixture would be 
avoided. The system could be more easily controlled; studies could be 
done with the analog which would not be possible using an actual combustion 
situation. (For example, parameters could be varied independently, a 
feature not possible in an actual combustion situation.) Also, the analog 
technique would be less expensive than actual experimental combustion studies 

Because of the above, the initial objective of the research was to 
develop an analog system for the study of combustion instabilities in liquid- 
propellant rocket engines, and then to use the analog to investigate the 
basic mechanisms governing these instabilities. Later, the research was 
expanded to include mathematical analysis techniques, initially to determine 


whether or not certain effects that were observed experimentally with the 
analog could be predicted mathematically. These analytical studies re- 
sulted in improved insight into analytical modeling techniques. 

What follows in this report is a chronological description of the 
research and the significant results. Most of what was done has been 
documented in various papers, theses, and dissertations. Hence, specific 
details of the research will be kept to a minimum and the appropriate 
references will be mentioned in which more detailed descriptions can be 



AS was stated earlier, the Initial objective of the research was 
sa develop an aLlop systes. for th\ study of high frepuenc, combust, on 
.astabilities in lipuid-propellant rocket combustion chambers. Before 
considering the analog, an understanding of the phenomena beinp modeled 

is necessary# 

r„„h..srion Hodel . To characterize the combustion process, the 
conbustion model due to Priem [.]. in which the combustion is viewed as 

. tC OnllA 



rate of the liquid propellant droplets. 

and this vapo- 

rate W is given as 

W = Cj + C 2 Re 


Cl = R T ^ P-Pv 

u ^ 


0.6 p 

c - ^ — P In 

^ V V 


pD 1^ - 1/^1 


Re = 

where 0 is the inside diameter of the combustion chamber. is the mole- 
cular weipht of the propellant. P is the aolecular diffusion coefficient. 

C, is the concentration of propellant drops. is the universal gas con- 
stant. T is the gas temperature. P is the gas pressure. P„ is the vapor 
pressure of the fuel. is the Schmidt number, p is the gas density. 

H is the gas velocity. is the propellant droplet velocity and u is 
Sbe dynamic viscosity. Eguations (1) indicate that the vaporization rate 
is convection limited (Reynolds number dependent) with additional sens.t.v.ty 
to the gasdynamic and droplet variables. 

In Priem's model, the burning rate is sensitive to the magnitude of 
the surrounding gas velocity with respect to the drop velocity, i.e., it 
is sensitive to a "rectified" relative velocity. This causes higher 
harmonic frequency components of the burning rate to occur in response 
to fundamental frequency perturbations of the gasdynamic field. In addition 
to this method of generating higher harmonic components of the burning rate, 
and consequently of the gasdynamic variables, nonlinear gasdynamic effects 
will also generate higher harmonic components of the gasdynamic variables. 

As was mentioned earlier, during combustion in liquid-propellant rocket 
engines, the combustion process and the gasdynamic processes are coupled. 

Any oscillation in gasdynamic variables causes an osci 1 latory burning rate; 
and the oscillatory burning process serves as the energy source to sustain 
gas dynamic oscillations. Under certain conditions, when the energy supplied 
by an unsteady combustion process is sufficient to overcome the energy losses 
in the gasdynamic field, the coupling is self-sustained and the oscillations 
can build to large amplitudes, a condition referred to as unstable combustion. 

The Analog . The analog selected to model the vaporizati on- 1 imi ted 
combustion process was a hot-wire sensor and the associated series output 
of a constant temperature hot-wire anemometer, a phase change or time delay 
device, a power amplifier, and an acoustic driver. The heart of the analog 
is the anemometer which applies a voltage across the hot-wire, causing it 
to heat to some preset temperature (resistance) level. Any flow which passes 
over the wire tends to cool it, but the anemometer responds to counteract 
this by increasing the voltage across the wire. The bridge voltage of the 
anemometer E is related to the flow over the wire. This relationship can be 
expressed as 

E2 = 






rC4 = 

[T - T]A 
[T - T]B 

*- W 



where 0 is the diameter of the hot-wire, R is the operating resistance 

VI w 

of the hot-wi re, R^l^j is the resistance of the cable connecting the hot-wi re 

and probe to the anemometer, R is the probe resistance, R is the appropriate 

P s 

anemometer resistance, m. A, and B, are coefficients whose exact values are 
determined by calibration of the individual hot-wire (typical values are 
m = 0.^»A^»2, A = 0.5205 and B = 0.6729; details on hot-wire calibration can 
be found in [5], [6], [7], [8], is the length of the hot-wire, Is 
the gas (air) thermal conductivity evaluated at the film temperature = 

[T + T^]/2, T is the gas temperature, is the hot-wire operating tempera- 
ture, is the gas density evaluated at the film temperature, V is the 
component of the gas velocity normal to the stationary hot-wire and is 
the dynamic viscosity evaluated at the film temperature. (Discussions of 
the development of Equations (2) are given in several references by Purdy, 
Ventrice, and Fang [5, 6, 8, 9].) Equations (2) indicate that the square 
of the anemometer output voltage is analogous to the convection limited 
burning rate W given by Equations (l). To complete the analogy, a phase 
change or time delay device is used as the analog of the phase change or 
time delay associated with combustion, a power amplifier is used as the 
analogof the energy released during the burning of the propellant, and 
an acoustic driver is used to convert the energy associated with the 
anemometer and power amplifier to gasdynamic energy. 


In the analog model, the anemometer output is dependent upon the 
magnitude of the gas velocity normal to the hot-wire, i.e., it is sensi- 
tive to a "rectified" relative velocity. As in combustion, this causes 
higher harmonic frequency components of the anemometer output to occur 
in response to fundamental frequency oscillations in the gas velocity 
field. In addition to the nonlinearity of the analog process resulting 
in higher harmonic components of the anemometer output voltage, and con- 
sequently of the gasdynamic variables, nonlinear gasdynamic effects also 
generate higher harmonic perturbations of the gasdynamic variables. 

A series of events, analogous to those which occur in the combustion 
system, occur in the analog system. Acoustic perturbations in the gas- 
dynamic field are sensed by the hot-wire. The resulting anemometer output 
signal is amplified, after phase change or time delay, and fed to the 
acoustic driver, which generates gasdynamic perturbations proportional 
to the signal to it. These new acoustic perturbations combine with the 
previous ones to form a new gasdynamic field. As In combustion, in some 
situations the perturbations will reinforce each other and be sustained 
or increased in intensity; in other cases they will die out. 

A comment needs to be made concerning the anemometer output signal 
mentioned above. The anemometer actually has two output signals — E, the 
bridge voltage and E', the time-dependent or fluctuating* component of the 
bridge voltage, where E = E + E'. (The component E is the time-mean 
component.) It is E^ which is analogous to the burning rate W and E^ 
also has a time-mean and a fluctuating component, i.e., E^ = (£2) + (E^) ' . 
(The bar over a property is used to indicate the time-mean component and 
the prime designates the time-dependent component.) 

From the above it can be seen that E^ = E^ + 2EE' + (E')^ = (E^) + (E^) 
Only the fluctuating component of E would drive acoustic oscillations; 


hence, (E^) ' would be the required signal to be used in the analog system. 
However,, when E' is small relative to F (which is the case for the analog 
system). Fang [,8];has shown that ^'characterizes (E^)' to a satisfactory 
degree with respect to amplitudes, frequencies, and phases. Hence, E* 
was used as the feedback signal in the loop. 

Initial Research Plan . At the outset of the research, the above des- 
cription of the analog had not been fully conceptualized and the validity • 
of the description had not been establ ished. As the research progressed, 
the above description evolved. 

To initiate the analog studies, the following was proposed. 

1. To analytically perform an open-loop response factor analysis 
of the analog to see if the same results would be obtained as 
Heidmann obtained for vaporization-limited combustion. For 
these studies the pressure associated with an acoustic field 
would be compared with the resulting anemometer output (analogous 
to the burning rate in combustion) and a response factor. ob- 
tained. Various amounts and types of distortion would be 

added to determine stability predictions. This process would 
establish similarity between combustion and the analog. 

2. If the results of the above were in agreement with Heidmann, 
to experimentally perform an open-loop analysis of the ana- 
log to determine if the analog would perform as predicted 

by theory. 

3» To operate the analog system in the closed-loop mode to 
determine if the open- loop predictions of closed- loop be- 
havior were valid. The above three items, if successful, 
would confirm the validity of Heidmann's analytical technique 
and the prediction that distorted acoustic waves would .drive 
combustion, or the analog process, to instability. Once all 
of the above was accomplished, the analog system could then 
be used for further experiments designed to identify other 
factors important to unstable combustion. 

Analytical Open-Loop Studies . The nonlinear, in-phase response factor 
developed and used by Heidmann is defined as 

W' p' d (ut) 

— — Q 

(p')2 d (a,t) 

where W* is the time-dependent, dimensionless perturbation in the normalized 

burning rate (W* = WVW) and p* is the time-dependent, dimensionless 


perturbation In the pressure (p = p + p' = p (1 + p'/F) - P (1 + p') 

where p' = p' /p) . ' 

• \ 

The nonlinear, in-phase res 4 >onse factor used for the analog analysis 
is defined as • 

E' p' d (ut) 

^ s ^ 

^ (p')2 d (ut) 


where E' is the time-dependent, dimensionless perturbation in the anemo- 
meter bridge voltage, i.e., E' = E'/E. 

To obtain values for R^, the hot-wire's environment is specified. 

(As in Heidmann's work, the environment is considered to consist of a 
first tangential (IT) spinning resonant acoustic mode of vibration dis- 


torted by a second tangential (2T) spinning resonant acoustic mode of 
vibration. The relative magnitudes and phases can be varied.) This deter- 
mines p' . Knowing the hot-wire's environment, and the cal ibratlon equation 
for the particular hot-wire being used, enables E^' to be determined. These 
can then be integrated, as in equation 3, to obtain R^. 

The results obtained from the analytical open-loop studies of the 
analog process were qualitatively the same as those Heidmann obtained for 
vapor izati on- 1 imi ted combustion. There were small differences in the 
actual numbers involved, but this was to be expected since the equations 
for the two processes were slightly different. A scaling factor would have 
been necessary to make a direct numerical comparison between the two. This 
was not done since only qualitative agreement was necessary. 

The results showed that the addition of distortion to sinusoidal per- 
turbations of the environment of the hot-wire could increase the response 
significantly. Using the stability criterion of Heidmann — response factors 
greater than some value between 0.8 and 1.0 denoted unstable operation (the 
oscillations would grow with time) — the analysis showed that certain 


combinations of harmonic amplitude and phase angle would result in 
instability. The pure sinusoidal case was always predicted to be stable. 

More detailed discussions of the analytical' open- loop studies can be 
found in [5], [6], [ 9 ] and [II]. 

Experimental Open-Loop Studies . Since the analytical open- loop 
studies were successful, the experimental open-loop studies were carried 
out. The experimental technique for doing this was involved and is dis- 
cussed in detail in [ 5 ] and [6]. 

A simplified schematic diagram of the experimental system used is 
shown in Figure 1. The object was to correlate an electrical signal, 
which was proportional to the fluctuations in the pressure of the acoustic 
field in the vicinity of the hot-wire, with the fluctuations in the anemo- 
meter output voltage and, from this, to calculate the response factor for 
that particular acoustic field. A variety of acoustic fields were used 
starting with an undistorted field at frequency f^.^. and gradually adding 
to this distortion at so as to parallel the work done analytically. 

To obtain response factors the system illustrated in the data acqui- 
sition system of Figure 1 was used. No instrument was available to directly 
determine a response factor so a response factor was calculated from the 
correlation coefficient R. Again, details of this can be found in [5] and 
[ 6 ]. 

The results of this aspect of the research were important in estab- 
lishing the analog as a potential research tool. First of all, the experi- 
ments showed that it was possible to set up a physical system which could 
generate the same acoustic environment for a hot-wire probe that was in- 
vestigated analytically and which could be used to determine the response 
of the hot-wire voltage to that environment. Secondly, the experimental 
results were qualitatively the same as the analytical results--the response 


Function Generator with 

Generator | Phase-Adjust Control 

of the system to pure sinusoidal acoustic vibration was small, even when 
the magnitude of the acoustic pressure was large; the response could be 
increased by as much as an order of magnitude with respect to the sinu- 
soidal case by the addition of distortion; and the amplitude and phase 
of the distortion component, relative to the fundamental component, were 
the dominant factors in the increase in the response. 

Some problems did exist. The distorted acoustic fields could not 
be totally characterized by the equations resulting from the solution of 
the inviscid 1 inear wave equation. Viscous effects should probably have 
been included in the analysis to account for acoustic streaming and any 
other secondary phenomena which might have been occurring. Also, there 
might have been some problems associated with the assumption used to obtain 
calibration constants for the hot-wire. The assumption was that the steady- 
state, steady-flow calibration of a hot-wire is applicable even when the 
wire is used in an unsteady situation. 

Other discussions of the experimental open-loop investigations can be 
found in [5], [6], and [ll]. 

Experi mental Closed-Loop Studies . The closed-loop experimental studies 
posed many difficulties. Figure 2 is a very simplified schematic diagram 
of the experimental system. The test section's length-to-diameter ratio 
could be adjusted to two different values — one in which twice the frequency 
of the first tangential resonant mode of vibration was equal to the frequency 
of the second tangential -second longitudinal resonant mode of vibration, 

2^1 j = ^2T, 2L’ which twice the frequency of the first tangential 

was not equal to a resonant frequency of the chamber, 2f = f . Two chambers 

1 I X 

were used to establish the effect of chamber dimensions on instability. 

Referring to Figure 2, the gas-filled chamber could be excited by moving 
the switch into the upper position so that the function generator/amplifier 



Figure 2. Simplified Schematic Diagram of the Closed-Loop 
Experimental System 

system supplied the excitation signal to the driver (open-loop operation). 

By switching to its lower position, closed-loop operation could be obtained. 

One objective of the closed-loop investigations was to confirm the 
results of the open-loop work, i.e., if the system was operating with a 
certain pattern of distorted acoustic oscillations, would it become unstable? 
This was not the only objective, however. Another objective was to allow 
the system to function over a wide range of variables and determine under 
what conditions instability developed and to determine if these conditions 
were consistent with those associated with unstable combustion and, from 
this, to perhaps gain insight into why these particular conditions develop. 

To obtain a better understanding of closed-loop operation, the system 
was initially operated using a linear feedback device — the output of a 
microphone. This was a pressure-sensitive linear feedback process instead 
of the velocity-sensitive nonlinear feedback process of the anemometer 
analog system. Using the microphone, the characteristics of closed-loop 
system behavior at a single frequency could be studied and, ultimately, a 

comparison could be made of pressure and velocity sensitive feedback 
processes and the pressure and velocity sensitivity of the combustion 

A. Pressure-Sensitive Closed-Loop Experiments 

Referring to Figure 2, the general procedure followed was to excite 
the chamber in the open-loop mode with some predetermined signal by means 
of the function generator/amplifier system. The switch was then moved 
to its alternate (closed- loop) operating position and the system was 
observed to see what would evolve. Three basic responses could develop: 
the acoustic pressure in the chamber could die out; the acoustic pressure 
could build in intensity to such a level that the switch had to be moved 
back to its open-loop position to avoid burning out the fuse in the driver 
circuit; or, finally, some type of steady-state acoustic pressure field 
could be attained. It was the conditions under which the third response 
occurred that were of interest. This represented a neutral stability 
situation. During the course of the experiments, several parameters were 
altered systematically to determine their effect on the closed-loop operation 
of the system. These were the open-loop signal from the function generator/ 
amplifier system to the driver, the amount of time delay or amount of phase 
shift, the amount of amplification or gain, and the length-to-diameter ratio 
of the chamber. 

The overall results of the linear feedback study were as follows: 

1. The closed- loop sound field was independent of the character of 
the initial exciting sound field. The same closed-loop sound field resulted 
for a particular amount of gain and time delay or phase change in the loop, 
no matter v/hat the characteristics of the initial open-loop sound field. 

2. For a particular sound field to be self-sustained in closed-loop 
operation, the least ampl if icatlon was required when the existing and 


feedback sound fields were in phase with each other. If these two fields 
were not in phase with each other, more than this minimum feedback- loop 
amplification was required in or^er to have self-sustenance — the greater 
the phase difference between them, the greater the required feedback gain. 

3. In general, the higher the gain, the higher the level of the closed 
loop sound field.: If self-sustenance did not occur at a certain phase dif- . 
ference between the existing and feedback field, increasing the gain or 
adjusting the phase difference so that the overall phase change through the 
loop was closer to a net zero would make the system self-sustaining. 

4. Coexistence of two or more frequencies was possible whenever the 
gain was sufficient for each of them to be s'elf-sustained. They were not 
normally integral multiples of each other. 

5. None but resonant frequencies were self-sustaining. 

6. The iength-to-diameter ratio of the chamber did not effect the 
character of the results. 

B. Velocity-Sensitive Closed-Loop Experiments 

The general procedure followed was similar to that of the linear 
experiments--the chamber was excited in the open-loop mode with some pre- 
determined signal by means of the function generator/amplifier system (see 
Figure 2). The switch was then moved to its closed- loop position and the 
system was observed to determine what type sound field would evolve. 

The same three kinds of response could develop: the sound field 

could die out; the sound field could continuously build in intensity; or 
it could. reach some steady-state operating level. The same parameters 
could be altered--the open-loop excitation signal, the time delay or phase 
shift, the gain, and the length-to-diameter ratio of the chamber. 

It needs to be recalled here that the anemometer's response is non- 
lineai it is related to a "rectified" relative velocity. Because of this, 


Its output tends to be dominated by a component at 2f when exposed to a 
velocity field of frequency f, i.e.. If the chamber was initially excited 
with an IT sound field of frequency f^^, the anemometer would have as its 
output a signal with a strong 2f^^ component. 

It was found that three types of self-sustained operation could occur 
in the chamber for which 2f^.j. = f^^ depending on the time delay or 

phase change In the loop: (1) steady in waveform and level, having fre- 

quency components at f^^ and at Its higher harmonics; (2) unsteady in 
waveform and level, having frequency components at f^^ and Its higher 
harmonics; and .(3) transitional in waveform and level having frequency 
components at f f and higher harmonics of each. Figure 3 is a 
map of the lower limit of gain for self-sustenance at various values of 
time delay, with the type of sound field resulting Indicated by the 
symbols. These results can be partially explained using the results 
of the linear-feedback experiments and calculations of the net phase 
change through the loop for the fundamental and second harmonic frequencies, 
the results of which are listed in Table 1. 

Table 1. Net Phase Change Through the Loop at 
Various Time Delays for f^^ and 2f^^ 

Time Delay, 

Net Phase 

Change, degrees 






















T r 





□ □ o 




|_ O Steady iT Mode 
A Unsteady iT Mode 
O Transitional iT , iT~iL Mode 

J 1 1 > 

22.0 22.5 23.0 23-5 

Time Delay (msec) 

Figure 3. Self-Sustained Map of the Hot-Wire Closed-Loop 
System (2f = f^j 

First, Figure 3 follows a repetitive pattern. The basic pattern is 
that recorded between 22.69 snd 23-70 msec. This span of time delay, 

1.01 msec, is equal to one period of a signal at f^-j.. Starting at 22.69 
msec, the net phase changes for the f and 2f^j components are both 359 
degrees. Since the existing and feedback sound fields are almost in phase, 
a minimum amount of gain is needed in the loop. At 22, 80 msec, where the 
net phase changes are 39 and 78 degrees respectively, more gain is needed 
for self-sustenance. At 23-21 msec, the net phase changes are 183 and 7, 
respectively. Since the fundamental feedback component is almost 280 
degrees out of phase with the existing fundamental component, the gain 
required becomes a maximum. When the existing and feedback sound fields are 
l80 degrees out of phase, the level of the feedback sound field must be 
greater than that of the existing field if self-sustenance is to be attained 
the sound fields at f^.^. are constantly interfering with each other. The 


level of the self-sustained sound field is unsteady but the frequency 
components remain the same. This behavior resulted in the closed-loop 
sound field appearing "unsteady.” At a time delay of 23. 3 A msec— another 
relative minimum region— the net phase change for the fj.^. and 2f^^ com- 
ponents are 232 and 104 degrees, respectively. Both values of time delay 
are significantly different than 180 degrees so less gain is needed for 
self-sustenance, but not as little gain as at 23.70 msec where the phase 
changes have returned to 360 and 359 degrees, respectively. 

One time delay in Table 1 has not been mentioned— 23.00 msec, at which 

the net phase changes are 111 and 222 degrees, respectively. These values 

of net phase change differ from zero by more than they do at 22.80 msec 

and, hence, should require more gain instead of less for self-sustenance. 

However, another factor effects the results at 22.80 msec. As mentioned 

earlier, a iT, iL sound field and its higher harmonics sound fields existed 

along with the iT sound field and its higher harmonic sound fields in this 

transitional region. This second series of sound fields reduces the gain 

needed for sel f— sustenance. This new sound field appeared in some cases 

due to the following characteristics: (l) f was 1135 Hz while f ‘ 

II, lU 3T 

was 2262 Hz. Hence, fg^ was approximately twice f^^ ^ 1 ^. This is similar 

2fj^ = f^j It was concluded that whenever a resonant frequency was 

twice another resonant frequency, a selfrsustained closed-loop sound field 
would be possible if the gain was high enough and the time delay was close 
enough to an optimal value. For this reason, the appearance of f ^ and 
its higher harmonics, at certain time delays, was to be expected. (2) The 
IT, IL sound field requires more energy to drive (it is less responsive) 
than a IT sound field. Also, fg^ is not exactly twice f . Hence, 

for most time delays, the self-sustained sound field is naturally dominated 
6y f jy and its harmonic harmonics and f^^ 


does not develop. 

Another significant finding of the nonlinear experiments was that 
the resultant closed-loop sound fields do not depend on the form (fre- 
quencies and phases) of the exci^ting sound field. However, the gain 
required to initiate self-sustenance did vary with the waveform and 
level of the exciting sound field. When the open-loop sound-pressure 
levels for various resonant modes were the same, the minimum gain required 
to initiate self-sustenance occurred when the exciting sound field was 
the IT field. Also, for a given exciting sound field, the higher the 
sound-pressure level, the lower the gain required to initiate self-sustenance 

Other closed-loop nonlinear feedback experiments were also carried out 
with the chamber for which 2f^.j. = confirmed the results dis- 

cussed here. 

Finally, of great significance, it was not possible to achieve self- 
sustenance in the chamber for which 2f^.j. was not a resonant frequency. 

C. Theoretical Analysis of the Anemometer Output Signal . 

To fully understand the non 1 inear- feedback experimental results, it 
was necessary to analyze the anemometer output which resulted upon exposure 
of the hot-wire sensor to various sound fields. 

Considering only the IT mode, the bridge voltage of the anemometer 
was calculated. In Figure k the AC bridge voltage is plotted versus 
dimensionless time. (The curve is flat in the vicinity of a>t = 0, tt and, 

2n becasue of the assumption that whenever the calculated values of E, 
using Equations (l), are less than E^, the no-flow anemometer output, E, 
is set equal to E^. This means that during periods of low or zero air 
velocity, the bridge voltage is equal to Performing a Fourier series 

analysis, the harmonic components of the analytically predicted AC bridge 
voltage resulting from a .IT sound field can be calculated. They are shown 
in Figure 5« The AC signal is clearly dominated by the second harmonic. 


Figure Analytically Predicted AC Bridge Voltage 
of the Anemometer Responding to a IT 
Sound Field 


Figure 5. Harmonic Components n of the Analytically 

Predicted AC Bridge Voltage of the Anemometer 
in Response to a IT Sound Field 

which is typical of the rectification process. The presence of the first 


harnranic is primarily due to the temperature difference terms [T - Tl in 


Equations (1). 

Considering a "distorted" sound field produced by the summation of a 
iT and a 2j , 2l sound field, the bridge voltage for various values of P 21 
was calculated, where P 21 is the ratio of the amplitude of the sound'pres~ 
sure of the 2 T, 2 L sound field to that of the iT sound field and a Fourier 
series analysis was performed. The results are shown in Figure 6 and 7. 

It can be seen that distortion increases the component of E‘ at the funda- 
mental frequency and that it decreases the component of E' at the second 

Figure 6. Harmonic Components n of the Analytically 

Predicted AC Bridge Voltage of the Anemometer 
for P 21 = 0, 0.5» and 1.0 


From the experimental and analytical nonlinear feedback investigations 
discussed in the previous two sections, a general description of the tran- , 
sitlon process from an open-loop sound field to the resultant closed-loop 
sound field can be given phenomena log i ca 1 ly without describing the details 
for individual circumstances. 

Consider that an open-loop sound field at some resonant frequency f 
is triggered in the chamber. An AC anemometer voltage results which has 
the frequency characteristics shown in Figure 5* There are components at 
all frequencies, f, 2f, 3f, etc., but the components at the even frequencies 
dominate, with the 2f component being strongest. When this signal continues 
around the loop and replaces the Initial open-loop signal (i.e., when the 
loop is closed), the produced sound field (feedback sound field) is pre- 
dominately at 2f — twice that of the pre-existing sound field. 


The acoustics of the chamber are important here. If 2f is a resonant 
frequency of the chamber, a relatively strong 2f acoustic field will develop. 

If it is not, even if the driving at this frequency is strong, the acoustic 
field at 2f will be weak. Also, the higher harmonic components of the feed- 
back signal will not result in strong acoustic disturbances at these fre- 
quencies, first, because it is unlikely that they are also resonant fre- 
quencies, and second, because even if they were resonant, it is relatively 
more difficult to drive the higher resonant frequencies than the lower 
resonant frequencies. 

When the two sound fields, the initial at f and the feedback at 2f , 
combine, the resulting sound field contains both frequencies. The hot-wire 
senses this new sound field and responds with an AC anemometer voltage 
that has both fundamental and second harmonic components. Again, the 
acoustics of the chamber play an important role. The chamber selectively 
responds at f and 2f, when these are resonant frequencies, and does not re- 
spond well to the higher harmonic components of the input signal. The 
dependence of the components of the anemometer signal at f and 2f on the 
relative magnitude of the acoustic field in the chamber at f and 2f was 
shown in Figure 7. Figure 7 indicates that adding even a relatively low 
level acoustic field at 2f compared to the initial f acoustic field results 
in a significant component of the feedback signal being at the fundamental 
frequency f. 

Again, the new feedback sound field is superimposed upon the existing 
sound field which also consists of the fundamental and second harmonic com- 
ponents, and a new feedback signal is sent to the driver. Similar existing- 
feedback interaction (closed- loop) processes will continue as long as the 
sound field exists. 

The above discussion explains why an acoustic field having components 

at both f and 2f results from the anemometer analog system. The two-frequency 


characteristic is a result of the nonlinear nature of the feedback mechanism 
and the acoustics of the chamber. In addition to the factors mentioned above, 
the amount of time delay or phase change in the loop is also of importance. 

The more out of phase the feedback signal is with respect to the initial 
perturbation in the chamber, the more amplification will be needed for self- 
sustenance as was illustrated by Figure 3* 

Because of the nonl inear character of the hot-wire response, all the . 
self-sustained closed-loop sound fields of the hot-wire closed-loop operation 
must contain higher harmonic components. If the chamber is not responsive 
at the 2f frequency, i.e., if 2f is not a resonant frequency (or close to a 
resonant frequency), the sound field will have difficulty being sustained; 
hence, the geometry of the chamber is or primary importance. 

More extensive discussions of the experimental closed-loop investigations 
can be found in [8], [12], [I3l, [1^1, and [15]. 

Gasdynami cal ly Induced Second Harmonic Sound Fields 

The closed-loop experimental studies indicated that the second harmonic 
sound field played an important role in instability. The second harmonic 
could originate from two sources. It could be directly driven by the pri- 
mary energy source (combustion in rocket engines; the acoustic driver of the 
analog process); or it could result from nonlinear gasdynamic effects which 
cause resonant acoustic fields at a given frequency to induce sound fields 
at higher harmonic frequencies. In the closed-loop work it was not entirely 
clear how much second harmonic was contributed by each source; hence, an 
Investigation was made of gasdynamical ly induced second harmonic sound fields. 

The object of these experiments was to determine the intensity and spatial 
distribution of the gasdynamical ly induced second harmonic sound fields. 

Three chamber configurations were used: one in which 2f^^.j. = f ^ 


in which 2f _ = f,- ,, and one in which the doubled frequency was not a 
resonant frequency, 2f = f . Three investigations were carried out. 

First, each chamber configuration was driven at f^.p at a particular sound 
level and the resultant second harmonic (2s) sound field was probed (2fj^y = 
fas), at five different sites, to determine the variation in amplitude and 
phase. Then, one location was chosen, and the amplitude of the iT sound field 
was varied and the level of the resulting 2s was measured. Lastly, the .. 
location of the driver was changed to determine the effect of driver location. 

Initially it was expected that, for those chambers for which f 2 s was a 
resonant frequency, the 2s would have the amplitude and phase characteristics 
of the resonant sound field. This was found to be only partially true. The 
results of the first experiments showed the gasdynamic (2s) sound fields were 
dependent on chamber geometry — changing the length of the chamber changed the 
spatial distribution of the second harmonic. Some effort was directed at 
characterizing the various 2s sound fields. The distributions appeared to 
be a combination of the resonant sound field closest in frequency to f 2 s, 
plus an additional component. The additional component appeared to be in 
part composed of a radial acoustic mode and in part some unknown component. 
Also, the results were compared to the analytically predicted gasdynamic 
sound fields obtained by Maslen and Moore [l6]. The comparison indicated 
the 2s was not entirely described by Maslen and Moore, but was closer to 
being a summation of the Maslen and Moore prediction, plus an additional 
component having the form of the resonant sound field closest in frequency to 
f 2 s* The work involved in trying to obtain analytical expressions which 
vx>uld ch.iractcr i ze the measured 2s sound fields was extensive when compared 
to the usefulness of the results obtained, so these efforts were terminated. 

The results of the second experiments showed that the level of the 2s 
was dependent on the level of the fundamental and that to obtain a significant 


amount of gasdynamically generated second harmonic, an intense fundamental 

sound field would be required. Typical results for the 2f^.j. = f 

iT, 3L 

chamber are shown in Figure 8. Thi's figure plots the ratio of the intensity 
of the 2s sound Held, at a particular location in the chamber, to the 
intensity of the IT sound field against the relative intensity of the IT 
sound field, when compared to ambient pressure. There were only relatively 
small variations in- the results for the other two chamber configurations. - 

2 s 


Figure 8. Relative Amplitude of the Gasdynamic Second Harmonic 
Sound Field as a Function of the Amplitude of the 
Fundamental Sound Field at Probe Site 2 at R = 1.0 [17] 



When the location of the driver was changed, i t was found that the 
overall amplitude of the second harmonic sound fields were of the same 
order of magnitude as had existed previously. A variation did occur in 
the spatial distribution of the second harmonic in some cases. The cause 
of this is unknown but is probably a combination of several factors. For 
instance, there were always slight irregularities in the driver sound 
field — the first tangential. These irregularities in the fundamental 
varied v/ith driver location and from experiment to experiment. Since the 
fundamental was the origin of the gasdynamic second harmonic, some changes 
in the second harmonic were to be expected. Also, the driver location 
used initially was a good location for actually driving a second harmonic . 
resonant sound field; the later driver location was not. 

Further details of the second harmonic studies can be found in [l 7 l, 
[18] , and [19] . 

The above studies of the gasdynamical ly induced second harmonic sound 
fields gave some insight into the nature of these sound fields, but did 
not evaluate the contribution of these sound fields to instability. A 
different type experiment was needed for that purpose. The twin-chambers 
investigations served that need. 

Twin-Chambers Investigations 

Simulation of actual combustion is attained in the experimental system 
by operating in the closed-loop mode — the sensor responds to oscillations 
in the chamber and an electrical output results, which is amplified, 
phase-shifted and/or time delayed, and the resulting signal sent to an 
acoustic driver mounted on the chamber, which in turn generates new oscil- 
lations, and so on. Altering the various parameters in the loop, such as 
amplification or phase, alters the operating characteristics; for example, 
the oscillations may die out or become very intense. This usually occurs 


so rapidly that it becomes difficult to make detailed studies of the 
phenomena and their relationships to one another. To overcome these 
difficulties, use of a twi n-chambeVs system was initiated. 

Referring tO' Figures 9 and 10, for the twin-chambers studies, the 
driver on the first chamber, chamber A, was activated by a signal origi- 
nating from some combination of signals from two function generators. 

This resulted in oscillations in chamber A, to which the sensor responded. 
The resulting signal was then amplified and sent to the driver on the 
second (or twin) chamber, chamber B, which resulted in oscillations in B. 
This would have been the feedback sound field in A if chamber A were 
operating closed-loop. The signals to the drivers and/or the sound fields 
in each chamber were then compared and these comparisons indicated whether 
or not a single chamber operating closed-loop under the same conditions 
would be stable or unstable. (If the fundamental and second harmonic 

Figure 9* Schematic Diagram of the System Used for Pressure 
Sensitive Open-Loop Simulation of Closed-Loop 
Operation [17). 




Figure 10. Schematic Diagram of the System Used for Velocity 
Sensitive Open-Loop Simulation of Closed-Loop 
Operation [I?]. 

frequency components of the feedback sound field were in phase with, and 
had the same or greater amplitudes than the original sound field, oscil- 
lations would be sustained or magnified if the system were operated 

The object of the studies was to examine the effect of second harmonic 
content on the operating characteristics of the two types of systems--the 
one having a linear feedback mechanism (a microphone which responds linearly 
to pressure fluctuations) and the one having a nonlinear feedback mechanism 
(the constant-temperature hot-wire anemometer which responds nonlinearly to 
velocity fluctuations)--, to examine the importance of second harmonic to 
self-sustained, closed-loop operation and to determine the intensity of the 
second harmonic for those situations for which self-sustenance was predicted. 

Three sets of twin-chambers were used — the ~ 2L’ ^^iT ~ ^iT 3L’ 

and the 2f^.j. = f^. 

unear avste.. referring ro figure the procedure «es as 
fouerrs. muuuv cwo funcUon generacors were set Co produce signals 

3, , . f and 2f1 These signals V-’f 

. „..d the resulting sound field and the output 

Chamber A. The Aerophone sensed the , „ The 

of the microphone was amplified and sent to the driver on cham er . 

,hase between the Ihitla. signals at f and 2 f were adjusted until the phase 

between the f and 2f components of the sound field In A was the same as 

the Phase between f and 2f In S. Keht . the amplifiers were 

the amplitude of the pressure at f in A and B was the same. The P ^ 

of the signal at 2f to the summer was then varied between zero and t 

A 1p\/p1s of the resulting sound 

„„it of operation of the system and the levels 

nelds in each chamber was noted. (After this adjustment there was s . ^ 
relative time delay between the sound fields In the two ebambers-the 
delay necessary In the loop for sustained closed loop operanon.) 

P,, ores Hand 12 are typical of the results Obtained for the unear 

system. The following observations can be made. Beferring to f ,gure 1 1 , 

.asdynamic second har.m,n1c content existed In each chamber. As second 

u hear A a lesser amount of second harmonic 
harmonic voltage was added to chamber A. a 

Tesulted in chamber B. The amount of second harmonic In chamber B was 
always less than that In A.' Adding second harmonic to A d,d not alter t e 

amplitude of be fundamental sound field in B. The only condl t Ions un er 

„H1ch the sound fields In each chamber matched occurred when A was not e, 

.Tiven at the second baram-nlc freguency-when primarily gasdynamic second 

harmonic existed, figure .2 shows the amount of second harmonic conten 

L v 4 :«-irm<i for the three chamber sizes, 

under these conditions Tor 


Figure 11. 

Levels of the Sound Fields at f and 2f in Chambers A and B 
as a Function of the Voltage at 2f to the Acoustic Driver 

on Chamber A; Linear System; 2f _ = f^_ [17] . 

Ratio of Fundamenta! to Second Harmonic Pressure 


A £i Ci££0.0 O Q\ 



A “a 




o a 


Non 1 I rear 

Conf iTjrat ion 


« «u"S 


i a" 



* ^2T. 2L 




' St. JL 


1 . • • 1 t « 1 




0. 1 

Pf/P . Ratio 

of Fundarienial 

to Aribicnt 




With a linear system, there was no natural mechanism which would 
enable any significant second harmonic osci 1 latiorv-other than the gas- 
dynamic one to be sustained during closed-loop operation. If a per- 
turbation occurred in the chamber at the second harmonic frequency, this 
perturbation would tend to die out. The second harmonic content would 
not be necessary for self-sustained closed-loop operation. These general 
results did not strongly depend on chamber geometry; similar results were 
obtained for all three chamber lengths. The results are consistent with 
the results of the closed-loop experiments. 

For the nonlinear system, referring to Figure 10, the procedure was 
as follows. Initially, the two function generators were set to produce 
signals at f “ fj^j and 2f. These signals were summed and sent to the 
driver on chamber A. The hot-wire sensed the sound field generated and 
the resulting output of the anemometer was amplified and sent to the driver 
on character B. The phase between the initial signals at f and 2f was 
adjusted until the phase between f and 2f of the sound field in A was the 
same as the phase between f and 2f in B. Next, the amplifiers were adjusted 
so that the amplitudes of the pressure at f and 2f in A matched the amplitudes 
of the pressure at f and 2f in B, respectively. The amplitude of the signal 
at 2f to the summer was then varied from zero to the upper limit of operation 
of the system and the variation in the results sound fields recorded. (Again, 
a tln»C"delay existed between the sound fields.) 

Figures 12 and 13 are typeial of the results obtained for the nonlinear 
system. The following results were observed. With no second harmonic voltage 
to the driver of chamber A, the sound field in B was dominated by second 
h.»rrx)nic content. As second harmonic voltage was added to A, the amount of 
second harmonic content in the sound field in B decreased in the chambers 
for which 2f^^ a at 2fj.j. = f^^ 2 ^^ and remained approximately the 




Figure 13. Levels of the Sound Fields at f and 2f in Chambers A and B 
as a Function of the Voltage at 2f to the Acoustic Driver 
on Chamber A; Nonlinear System; 2fj^.j. = f^^^ 3 ^ t^7]. 

same in the chamber for which 2f^^ = fx. There was an amount of second 
harmonic voltage which, when added to A, resulted in the same amplitude 
for the second harmonic sound fields in both chambers. Adding second 
harmonic content to A increases the amplitude of the fundamental sound 
field in B. It was necessary to have a strong second harmonic input to 
chamber A in order to obtain a strong fundamental sound field in B. There 
was a condition under which the sound fields in each chamber had the same 
amplitudes at both the fundamental and second harmonic frequencies and the 
same phases between fundamental and second harmonic. The amount of second 
harmonic existing at this state is shown in Figure 12 for each of the 
chambers. The results for the 2f = fx chamber tended to be somewhat 
errat ic. 


With the nonlinear system, a sound field at the fundamental frequency 

' y 

naturally resulted in a response primarily at the haH^tqonic frequency. The 
system would not be able to sustain itself under these conditions. It 
would be necessary to have a driver input at both the fundamental and 
second harmonic frequencies in order for the sound fields in chambers A 
and B to match. Second harmonic content would be necessary for self-sustained 
closed-loop operation. This was consistent with the results of the closed- 
loop investigations. 

What was not entirely consistent with the closed-loop results was the 
prediction that the chamber for which 2f^.j. was not a resonant frequency 
could be self-sustaining. With the twin-chambers experiments it is predicted 
that a 2fj^y = fx chamber could be self-sustaining if enough feedback ampli- 
fication were available in the loop. (A factor which is not noted in the 
figures is that it took approximately twice as much power to achieve the 
"matched” situation for the 2f^.j. = fx chamber as it did for either the 2f^^y — 

f or 2f = f chambers.) The twin-chambers results indicate 

iT, 3 T xT 2 T, 2L 

that, if the closed-loop experiments had had a more powerful amplifier and 
an acoustic driver able to withstand higher power input, perhaps closed- 
loop, self-sustained operation would have been achievable. 

Further discussion of the twin-chambers investigations can be found 
in I 17 ] and [I 8 ]. 


•• Mathematical Modeling. 

This research was begun because of a desire to determine whether or 
not certain effects that had been Observed experimentally could be pre- 
dicted mathematically. It was felt that this would be helpful in ex- 
plaining the experimental results to other engineers and scientists and 
in interpretation of the observed phenomena. In order that the pre- 
dictions of the analytical work could be easily understood, it was desired 
to avoid methods which required the simultaneous numerical solution of a 
system of partial differential equations. For this reason, the method 
of Powell [10], who analyzed pressure-sensitive combustion instability 
problems, was attractive. Powell [10] used certain approximations to 
deduce a single nonlinear wave equation governing a velocity potential 
and then solved this equation approximately using the Galerkin method. 

It was decided to use a similar methodology for the analysis of velocity- 

sensitive combustion instability problems. 

It was found that if a set of approximations siightiy different 
from those used in [lOl was employed, a wave equation could be derived 
which contained gasdynamic nonlinearities of all orders, rather than only 
those of order two as in [lOl. The assumptions involved in this analysis 
are discussed by Peddieson, Ventrice, and Purdy [18] and further details 
are given by Wong I20l. The appropriate wave equation is (in dimensionless 


a 0 + V<t • V (23 ♦ + VO > ^0/2) - V^O [1 - (y - 1) • 

tt ^ 

O^O . W2)] + W [1 - (y - =0 

where ^ is the gradient operator, 0 is the velocity potential, W is the 
burning rate, y is the ratio of specific heats for a perfect gas and t 


Is time. 


To be consistent with the work of previous 
simplify the computation effort, it was decided 
form of (l<) . This was done by assuming that 

♦ = e[J(z) + (|i(x, y, z, t)] 

W = w(z) + cw(x, y, 2 , t) 

investigators, and to 
to-deyelop a second-order 


where c is a measure of the initial disturbances, z is the axial coordinate 
(parallel to the main flow) in the combustor, and the quantities with the 
superposed bar are associated with the mean flow. Substituting (5) into 
(^«) , assuming w = 0(e) and w = 0(e) , neglecting al 1 terms of O(e^) and 
dividing by e, one obtains 

^tt*^ + W3^<|> + 2TTa^^ <j) - + e(27(|, . ‘ 

, ( 6 ) 

(y - <|. + w) = 0 


iT(z) = .r^wd5 (7) 

is the steady-state gas velocity. The perturbations of velocity, density, 

pressure, and temperature associated with are respectively 

->■ -»• 
u = 7(J) 

P = - [3^ 4) + u3^<(, + e(u^ - (2-y) (9^4.)2)/2] 

( 8 ) 

P=- y(9^'J' + "u 3j.<|) + e (u^ - (3^(J)).^)/2] 

T=- (y - 1)(3^(|. + 'u3^((> + eu^/2) 

It was first attempted to use the vaporization-limited combustion model 

of Priem [I] in conjunction with (6). In the present notation this can be 
written as 

w = (We^) [(1 - 3^.(f/2) (d - 3^({)/u, )2 + 

(u“ - - 1) 


where velocity of the liquid drops. When (9) is combined with 

(6), the result ;is a single equation for Because of the non-polynomial 
nonlinearities in (9), this equatio)i cannot be analyzed conveniently by 
the Galerkin method. For this reason, another version of the method of 
weighted residuals, the orthogonal collocation method, was used. The 
details of this method are discussed in [l8] and [20]. It was eventually 
concluded that this method was unreliable when applied to this type of 
problem and was abandoned. It is believed that this unreliability is 
related to the fact that the nonlinear terms in (6) have the correct form 
only for moderate values of e. This can be illustrated by the following 


Consider a nonlinear elastic system governed by the equation 

f + f + eN (f, 0 = 0 00) 

where f is the displacement, N is a nonlinear function, c « 1, and a 
superposed dot indicates differentiation with respect to time. Suppose 

N(f, f) = ff + 0(e) 

Then the second order form of (10) would be 

if + f + eN ff = 0 

This equation can be integrated once to yield 

log[(l + eN^f)/(l + eVo)] ' ‘ ^ ~ V 


( 12 ) 

(f2 - f^^)/2 = 0 (13) 

f(0) = f^. f(o) 


For eN « 1, (13) can be simplified to 

f2 + _ 2eN^fV3 + 0(e^N^O ^ 



This Is consistent with the original approximation leading to (12). For 

eN « 1 It Is expected that the solution of (12)"W+41 be a resonable 

approximation to the solution of (10). Equation (15) Indicates that the 
solutions are periodic as one would expect for free vibrations of an 
elastic system. 

Suppose that N Is sufficiently large so that the inequality eN^ « 1 
o ^ 

Is not satisfied., Then (12) Is not a rational approximation to (10) and 
the solution of (12) may not have the characteristics of an elastic system. 
As an example, consider eN^ » 1. Then (13) can be simplified to 

f = f exp [e^ N ^ (f^ - f ^)/2] (16) 

o o o 

For f =0 this becomes 

f = f exp(e^ N ^ f^/2) (17) 

o o 

In this case the solution Is not periodic and grows without bound. The 
application of the collocation method to (6) produces a set of nonlinear 
ordinary differential equations governing the values of (J) at the collocation 
points. These equations contain nonlinear terms similar to the last term 
In (12). It appears that most choices of collocation points cause the co- 
efficients of some of these terms (equivalent to cN^) to become sufficiently 
large so that the equations predict behavior similar to that indicated by 
( 17 ) • This happens even in the absence of combustion terms. It may be 
that more success could be achieved by applying the collocation method 
directly to (1*), but this matter was not pursued. 

Since the collocation method could not be made to produce satisfactory 
results, It was decided to employ the Galerkin method. As mentioned pre- 
viously, this method cannot be used In conjunction with (9). Thus, (9) 
was replaced for purely velocity sensitive combustion with 



w = w n 

where n Is called the interaction i^ndex. This is analogous to the 
customary treatment;' of pressure sensitivity (see, for instance, Powell [10]). 

As in the case of pressure sensitivity, (l8) can be generalized to account 
for history effects as 

w = w n (u^ - u^*) 

where u^(x, y, z, t) = u(x, y, z, t - x) and x Is called the time lag. 

Equation (18) can be related to burning-rate laws meant to apply to special 
types of combustion processes such as (9). For an example of this, see 
Wong [20, 21]. With the use of (19), it was possible to achieve significant 


Peddieson, Wong, and Ventrice [19l discussed the application of (6) 
and (19) to transverse wave motion in a cylindrical combustion chamber. 

An approximate solution, based on the IT, IR, and 2T acoustic modes of 
the chamber, was used and the Galerkin method was employed to carry out 
a modal analysis. This resulted in a system of ordinary differential equa- 
tions that could be solved to determine the temporal behavior of the modal 
amplitudes. This was done numerically using a fourth-order Runga-Kutta 
method. (For further details, see [19] and [20, 21].) Stability boundaries 
were determined and the behavior of the pressure at the chamber wall was 
determined. Some typical stability boundaries are shown in Figures lA-17- 
It can be seen that all boundaries resemble rectangular hyperbolas in the 
n-e plane. A purely velocity-sensitive system will always be stable to 
Infinitesimal disturbances. Thus, the instabilities associated with Figures 
14-17 are triggered instabilities. 

It was felt that the numerical solutions discussed above could be usefully 
supplemented by using the method of multiple scales (see, for instance. 


Nayfeh [22]) to obtain approximate analytical solutions to the governing 
equations for-the modal amplitudes. To investigate the feasibility of 
doing this, Googerdy [23, 2l»] considered purely pressure-sensitive com- 
bustion and replaced (19) by 

w = - wn(9^(|> - 9^ 4>^)/e (20) 

Rather than deal with a cylindrical chamber, he considered the mathematically 
simpler problems of transverse motion in a narrow rectangular chamber and 
transverse motion in an annular chamber with a narrow gap width. The first 
problem is analogous to that of radial oscillations in a cylinder, while 
the second is analogous to the problem of combined radial and transverse 
motion in a cylinder. Both of these cases were treated numerically by 
Powell [10]. Using the method of multiple scales it was possible to produce 
most of Powell's results in closed form and to provide physical explanations 
for them. This gave confidence to the use of the perturbation method. 

McDonald [25, 26] then applied the method of multiple scales to the 
problem of purely veloci tysens i tive combustion instability in an annulus 
of narrow gap width. Transverse motions were treated. For standing waves, 
the simplest solution able to show the effect of quadratic nonlinearities 
has the form 

<J)(0, t) = fi (t) cos 0 + f 2 (t) cos 20 (21) 

where 0 is a polar coordinate which loactes radial cross sections in the 
annulus. Substituting' (21) and r=1 into (6) and (19), using the Galerkin 
method to carry out the modal analysis, and assuming that the gasdynamic 
non 1 inear i ties are negligible compared to the combustion non 1 inear i ties, 
results in the equations 


fj + fx + cx(ofi + 2wn fi f 2 ) “ 0 
fa + ^fa + c(of 2 - wn fi^/2) =0 

( 22 ) 


where o = w/e. Solving these equations by the methpd of multipie scales, 
subject to the initial conditions ~ 

fi(0) = f2(0) = 0. fi(0) = 1. f2(0) = 0 (23) 

leads to the uni formly-val id first approximations 

3 / 2 . 

fi = exp( - w t/2) sec[en(l - exp( - w t/2))/2 ] sin t + 0(e) 

fz = ■ (1/2 ^^) exp ( - w t/2) tan [en(l - exp( - wt/2))/2^^^] • 

( 2 ^ 4 ) 

sin 2t + 0(e) 


These simple solutions illustrate the basic features of the system. Both 
the secant and the tangent become infinite when their arguments' take on 
the value tt/ 2. Thus, i f en < ir2 , the arguments of the secant and tangent 
in (24) are less than tt/2 for all t, and fj and f 2 approach zero as .t ap- 
proaches infinity. I f en > tt 2 , on the other hand, these arguments reach 
ir/2 in a finite time, causing fi and fa to become infinite in a finite time. 
The former situation is stable, while the latter is unstable. The equation 

of the stability boundary in the n-e plane is 


n = 2 ir/e 

For this situation it can be shown that 

1 im f X = (2/tt) sin t 

t -V oo 

lim f2 = " (1/2 ^ tt) sin 2t 



Thus, a neutral oscillation is approached when mass addition due to unsteady 
combustion is exactly balanced by mass loss due to steady-state flow. 
McDonald [25, 26] has found similar solutions for traveling waves and show 
that the conclusions discussed above are qualitatively unchanged by the 
inclusion of gasdynamic non 1 inear i t ies. He has further shown that the 
qualitative aspects of the perturbation soiutions are in agreement with 
those of direct numerical calculations based on (22). This work illustrates 


vividly the usefulness of the combustion law (18), Closed“form solutions 
similar to (2fi) could never be obtained using a combustion law of the 


The research has resulted in useful experimental analog and analytical 
techniques for the study of Reynolds number dependent processes. The 
techniques are general and could be applied to any Reynolds number dependent 
process. During the course of the research, application to combustion in- 
stability in liquid propellant rocket engines was emphasized. 

Experimentally, the initial object of the work was to study the use- 
fulness of a constant-temperature hot-wire anemometer as an analog tool 
for the investigation of combustion instability. To accomplish this, 
analytical and experimental open-loop studies of the anemometer were carried 
out and compared to analytical open-loop studies by Heidmann [2, 3] of 
vaporization limited combustion. (Open- loop studies compare existing pressure 
perturbations and the resulting perturbations in the burning rate in the 
case of combustion, or in the anemometer output in the case of the analog, 
in order to predict actual closed-loop operation.) The comparisons indicated 
qualitative agreement among the three studies-the second harmonic frequency 
distortial of the fundamental frequency perturbation was the important 

factor in the development of instability. 

In closed-loop studies, the burning rate in combustion, or anemometer 

output in the analog process, feeds back upon itself, resulting in new 
pressure perturbations, which combine with the previous perturbations 
to form a new acoustic field, and so on. The analog was refined to run 
closed-loop using either the original nonlinear analog device-the anemometer 
or a linear analog dev ice-a microphone. Experiments were then run to 
investigate the validity of the open-loop predictions of closed-loop behavior 
These studies confirmed the importance of second harmonic content to insta- 
bility in the nonlinear case and gave insight into the reasons for this 


Knporunce-second harnonic content i? e natural^ result of the nonlinearity 
of the process and is necessary to sustaining pertorbayons at both the 
fundamental and second harmonic frequencies. 

Two important similarities in the characteristics of combustion in- 
stability and the closed-loop analog process were Identified. First, in 
combustion instability, a minimum perturbation amplitude is necessary to 
excite instability. In the analog this was also true. Second, in com- 
bustion instability, an initial sinusoidal perturbation becomes harmonically 
distorted once instability develops. Again, in the analog this was also 
found to be true. These studies also Indicated that chamber resonance might 
Playa more significant role in causing instability than had been anticipated 

Second harmonic content was not found to be important in the linear feedback 

The closed-loop analog studies did not fully indicate the difference 
in character between linear (pressure sensitive) and nonlinear (velocity 
or Reynolds number sensitive) processes, so a twin-chambers technique was 
developed for this purpose. In the twin-chambers work, instead of having 
the analog device feeding back on itself, the feedback signal was sent to 
an identical twin chamber so that the originating and feedback perturbations 
could be studied separately. These studies indicated, even more clearly 
than the closed-loop studies, that second harmonic was an integral part of 
a nonlinear process and necessary to the sustenance of the fundamental 
frequency perturbations. They also indicated that this was not true for 
linear case the linear case was primarily a single frequency process. 

All of the experimental results indicated that the nonlinear aspects 

of the combustion process were the dominant mechanisms control ing combustion 
instabi J I ty. 


Both the closed-loop and twin-chambers analog technique of studying 
Instability proved to be valuable niethods for gaining insight into the 
phenomena being studied and both haie potentiai for application to other 

similar problems, i 

The analyticalwork established a method for computing stability 
boundaries for velocUy-sensitive combustion. It was established that the 
method used moderate amounts of computer time and predicted results that 
were in qualitative agreement with the known characteristics of velocity 
sensitive combustion instability. The results of the numerical work were 
substantiated by comparison of approximate solutions obtained by perturbation 
procedures. In some cases, the perturbation solutions were obtained in 
closed form. It is felt that these explicit solutions are very helpful 
in understanding the physical nature of velocity-sensitive combustion 





1. Priem, R. J- and Heidmann, M. F. "Propellant Vaporization as a 
Design Criterion for Rocket-Engine Combustion Chambers, "TR R-67 « 

NASA, I960. 

2. Heidmann, M. F. "Amplification by Wave Distortion in Unstable 

Combustors." AIAA Journal , Vol. 9, February 1971, PP- 336-339. 

3. Heidmann, M. F. "Amplification by Wave Distortion of the Dynamic 

Response of Vaporization Limited Combustion," TN D-6287, NASA, 

May 1971. ’ 

k. Hribar, A. E. "An Investigation of the Open-Loop Amplification of 
Reynolds Number Dependent Processes by Wave Distortion," an un- 
solicited Proposal for Research submitted to NASA, June 1971., 

5. Ventrice, M. B. "An Investigation of the Open-Loop Amplification 
of a Reynolds Number Dependent Process by Wave Distortion," Ph.D. 
dissertation, Tennessee Technological University, June 197^. 

6. Ventrice, M. B. and Purdy, K. R. "An Investigation of the Open-Loop 
Amplification of a Reynolds Number Dependent Process by Wave Distortion," 
CR- 134620, NASA, May 1974. 

7. Fang, J. C. and Purdy, K. R. "Calibration of a Constant-Temperature 
Hot-Wire Anemometer for Very Low Reynolds Numbers," Mechanical 
Engineering Report, ME-44, Tennessee Technological University, 

December 1973. 

8. Fang, J. C. "On the Convection Limited Self-Sustained Acoustic 
Vibrations in a Closed-Closed Cylindrical Chamber," Ph.D, dissertation, 
Tennessee Technological University, August 1975. 

9. Purdy, K. R. , Ventrice, M. B. , and Fang, J. C,. "An Analytical 
Investigation of the Open-Loop Amplification of Reynolds Number 
Dependent Processes by Wave Distortion," Proceedings of the Tenth 
Southeastern Seminar on Thermal Sciences , April 1974, Pp. 1 76-204. 

10. Powell, E. A. "Nonlinear Combustion Instability in Liquid Propellant 
Rocket Engines," Ph.D. dissertation, Georgia Institute of Technology, 

11. Ventrice, M. B. , Purdy, K. R. , and Fang, J. C. "Vaporization Limited 
Combustion Stability Study Using an Analog Process," 11th JANNAF 
Combustion Meeting Bulletin , (Chemical Propulsion Information Agency 
Publication 26l), Vol. 2, December 1974, pp. 121-128. 

12. Ventrice, M. B. , Fang, J. C., and Purdy, K. R. , "Amplification of 
Reynolds Number Dependent Processes by Wave Distortion," CR-134917> 

NASA, June 1975. 


* 13. 

* li». 

* 15. 

* 17. 

* 18. 

* 19. 

* 20 . 

* 21 . 
22 . 

* 23. 

* 2k, 

* 25. 

Purdy, K. R. , Ventrice, M. B. , and Fang, J. C. "The Important 
Ingredients for Self-Sustained Unsteady Combustion, JZt Ji 


rnmh..^ tion Meeting Bulletin , (Chemical Propul s ion J nformat ion Agency 
Publ ication 273) , Vol. 2, December 1975, PP- 493-507. 

Ventrice, M. bI, Fang, J. C., and Purdy, K. R. 

Liquid Propellant Rocket Engine Combustion I nstab i 1 i t les , Al AA 
17 th Aerospace Sciences Meeting, Paper 79"0156, January 1979. 

Ventrice. M. B.,. Fang, J. C., and Purdy, K. R., 

Liquid Propellant Rocket Engine Combustion Instabilities, Ajjw 
Journal, to be published about December 1979. 

Maslen, S. H. and Moore, F. K. "On Strong Transverse Waves Without 
Shocks in a Circular Cylinder," Journal of Aeronautical Sciences, 

Vol. 23, No. 6, 1956, pp. 525 - 538 . 

Ventrice, M. B. and Purdy, K. R. "Pressure and Velocity-Sensitive 
Feedback Mechanisms and Their Importance to Self-SustainedUnsteady 
Liquid Propellant Combustion," 13th JANNAF Combustion Meeting Bulletin 
(Chemical Propulsion Information Agency Publication 2o1), Vol. 3, 
December 1976, pp. 131-144. 

Peddieson, Jr., J. , Ventrice, M. B. , and Purdy, K. R. "Effects of 
Lnlinear Combustion on the Stability of a Liquid Propel ant System, 
lljth JANNAF Combustion Meeting Bulletin , (Chemical Informa i 

Agency Publication 292), Vol. 1, December 1977, PP- 525-53B. 

Peddieson, Jr., J., Wong, K. W. , and Ventrice, M. B., "Stability 
Boundaries for Velocity-Sensitive Unsteady Combustion Using the 
Galerkin and Collocation Methods," 15th JANNAF C ombustion Meeting 
Bulletin, (Chemical Propulsion Information Agency Publication 297), 

Vol. 2, December 1978, pp. 461-472. 

Wong K. W. "Analysis of Combustion Instability in Liquid Fuel Rocket 
Motors," Ph.D. dissertation, Tennessee Technological University, 

August 1979 . 

Wong, K., Peddieson, Jr., J., and Ventrice 
Instability in Liquid Fuel Rocket Motors," 

M. B. "Analysis of Combustion 
CR 159733 , NASA, November 1979. 

Nayfeh, A. H. Perturbation Methods , Pure and Applied Mathematics, A 
Wi ley- Interscience Series of Texts, Monographs, and Tracts, 1972. 

Googerdy, A.. "Perturbation Solutions of Combustion Instability Problems 
M.S. Thesis, Tennessee Technological University, June 1979. 

Googerdy, A., Peddieson, Jr., J., and Ventrice 
Solutions of Combustion Instability Problems," 
September 1979. 

M. B. "Perturbation 
CR 159642, NASA, 

Chamber ," 

G. H. , "Stability Analysis of a Liquid Fuel Annular Combustion 
M.S. Thesis, Tennessee Technological University, August 1979. 


HcDonald. G. H., PeddUson, Jr.. J., f m'cr "wU' 

Analysis of a Liquid Fuel Annular Combust io«T Chamber, CR , 

•'Stabi 1 1 ty 

NASA, November 1979. 

All references marked with * are papers or reports which 
resulted from this research. In addition to the reports listed, 
semiannual reports were submitted every six months. Two of these 
semiannual reports, July through December 1973 and January through 
July 1975. were of special significance and were, therefore, as- 
signed CR numbers. They are listed in the references as numbers 
6 and 12. Also, informal reports were submitted on a somewhat 
monthly basis to the project's Technical Monitor, Dr. Richard . 
Priem. These reports have some details in them that do not appear 
In the published work.