A nonlinear dynamic model of a thermal pulse combustor consisting of four coupled ordinarydifferential equations has been used to investigate the effect of sinusoidal perturbations in massflow rate. The system of equations was solved using ODE45 function of MATLAB. For externaldisturbances with low amplitude, the system exhibits a period-doubling route to a possibly chaoticstate before extinction as the temperature of the combustor wall is reduced. However, at higheramplitudes, the transition to extinction is abrupt. The system characteristics are strongly affectedby the frequency of the external pulsation. At moderate to high amplitudes, no flame can besustained over extended ranges of intermediate frequencies. In the low frequency range, increase offrequency promotes transition of system characteristics towards extinction. However, atfrequencies that are close to integral multiples of the frequency of self-sustained oscillation, the timeseries for pressure exhibit near periodic behaviour. The minimum wall temperature at which flamecan be sustained is lowest at these frequencies and highest at frequencies that are approximatelymidway between these values. For the range of amplitudes of the external stimulus investigated, it isfound that increase in amplitude promotes early extinction.
NomenclatureA Combustor surface area (m2)Ae Combustor cross-sectional area (m2)B Pre-exponential factor (s-1)Cp Specific heat at constant pressure (J/kgK)DTP Diameter of tailpipe (m)f Friction factor (dimensionless)h Convective heat transfer coefficient (W/m2K)heff Effective heat transfer coefficient (W/m2K)Lc,1 First characteristic length (V/A) (m)Lc,2 Second characteristic length (V/Ae) (m)LTP Length of tailpipe (m)
im& Mass flow rate at combustor inlet (kg/s)
em& Mass flow rate at combustor exit (kg/s)
p Pressure (Pa)p0 Ambient pressure (Pa)
P p/p0 (dimensionless)
pe Pressure in tailpipe (Pa)
eP pe/p0 (dimensionless)
T Temperature (K)Ta Activation temperature (K)T0 Ambient temperature (K)
T T/T0 (dimensionless)
1 Research Scholar, Mechanical Engineering Department, [email protected]; Also at: Also at: TheUniversity Institute of Technology, The University of Burdwan, West Bengal, India.2 Professor, Mechanical Engineering Department, [email protected] Professor, Mechanical Engineering Department, [email protected].
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Te Temperature in tailpipe (K)
eT Te/T0 (dimensionless)
Tw Wall temperature (K)
wT Tw/T0 (dimensionless)
u Gas velocity in tailpipe (m/s)
u u/(Lc,2/τf) (dimensionless)
yf Fuel mass fraction (dimensionless)Zi /Vmi& (kg/m3s)
Ze /Vme& (kg/m3s)
ε Amplitude of mass flow rate oscillation(dimensionless)
γ Ratio of specific heats (dimensionless)κP Planck mean absorption coefficientρ0 Ambient density (kg/m3)σ Stefan-Boltzmann constant (W/m2K4)τc Characteristic chemical reaction time (s)τf Characteristic flow time (s)τh Characteristic heat transfer time (s)ω Angular frequency (rad/s)ωf Fuel consumption rate (kg/m3s)
I. Introductionulse combustors are a class of air-breathing engines that undergo self-sustained or forced oscillations, whichlead to low emission and higher heat transfer rates than steady combustors. The essential parts of a pulse
combustor consist of the combustion chamber and the tailpipe. The coupling between the dynamics of the twocomponents leads to the oscillatory nature of combustion. The operation and emission characteristics of thecombustor are extremely sensitive to the design details and operating conditions of the combustor. The pulsatingaction of these combustors makes these systems attractive for a variety of domestic, industrial and even propulsionapplications. The most widespread industrial use of pulse combustors has been in the area of drying1, 2, 3.Application of pulse combustors in solid and hazardous waste incineration has also been explored4. However,additional complexities in designing and commissioning of the combustors and lack of proper understanding ofthese combustors have severely limited their acceptability.
Based on the difference in categories for one-way valves mounted on a fuel or an air supply line, pulsecombustors are divided into two groups; one is mechanically valved and the other is aerodynamically valved(valveless). For valveless pulse combustors (also referred to as thermal pulse combustors), the pulsating action isself-sustaining and occurs due to the coupling between the combustor dynamics and flow in the tailpipe. This allowspulsating combustion even with steady inflow of reactants. However, fundamentals and applications of this thermalpulsed combustor remain till date largely unexplored.
Comprehensive models of valveless and valved pulse combustors have been developed by Richards et al.5 andMorel6 respectively. Several researchers have used the model of Richards et al.5 to explore the self-sustaineddynamics of thermal pulse combustors 7, 8, 9, 10. In addition to self-sustained oscillations, combustion dynamics maybe significantly affected by modulations of the reactant flow at the inlet. Such modulations may be both deliberate toalter the combustion characteristics and due to accidental fluctuations of the flow rate. Neuimeier et al.11 carried outa frequency domain analysis of a valved pulse combustor. They found that near resonance the oscillatory energyneeded to drive the combustor is only a small fraction of the energy generated by the oscillatory combustion.Marsano et al.12 investigated the behaviour of pulse combustors subjected to cyclic modulations of the mass flowrateat inlet. Their work focused on oscillations with large amplitudes and frequencies close to the natural frequency ofthe system. The results indicate that the phase between the temperature and pressure traces is strongly affected bythe oscillation characteristics. Kilicarslan13 evaluated the frequency of a valved pulse combustor and compared themodel predictions with experiments. Datta et al.14 investigated the effects of flow modulation but limited their studyto only a few low frequencies and amplitudes. The results were distinctly different from that of Marsano et al.12. Thepresent work investigates the effect of flow modulation over a wide range of frequencies and amplitudes. These
P
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results can be important for both assessing the effects of noise and other unintended fluctuations in the mass flowrate and developing strategies for controlling the dynamics of the combustor by modulating the flow rate.
II. Mathematical ModelThe major assumptions of the present model are similar to that of Richards et al.5. However, radiation is
considered in the pulse combustor model. The major assumptions of the model are: (1) Perfectly-Stirred Reactor(PSR) for the combustor; (2) slug flow in the tailpipe; (3) constant specific heat and ideal gas model for the reactantand product gases; and (4) single-step Arrhenius model for chemical kinetics and (5) convective and radiative heatloss to the wall. The model is described in terms of four ordinary differential equations. The derivation of theequations have been detailed in15 and is presented very briefly here. The first equation is derived from a lumpedmodel of conservation of energy within the reactor as
+−+−
++=
hw
e
f
i
chf
ii
T
ZZ
P
TTZ
P
T
dt
Td
τγ
ργ
ττττγ
0
2)1(
11(1)
Combining Eq. (1) with conservation of mass gives the equation for pressure as
+−
++=
hw
e
chf
ii
T
ZT
TZ
dt
Pd
τργ
τττγ 111
0
(2)
Conservation of fuel mass yields the equation for fuel mass fraction as
[ ]cc
p
fiff
if
P
T
h
TCyy
P
TZ
dt
dy
ττ11
.0
, ∆−−= (3)
Finally, momentum balance in the tailpipe gives the equation for gas velocity in the tailpipe as
( )f
c
TPe
e
e
TPc
f L
Du
ufP
P
T
LL
RT
dt
ud
ττ 2,
3
2,
0 1
21 −−= (4)
In the above equations, the dimensionless characteristic times are defined as
if Z
0ρτ = (5a)
weff
pc
h Th
TCL 001ρ
τ = (5b)
1
22/3
2
0
exp
−
−∆′
=T
Ty
T
P
TC
hB aF
p
ccτ (5c)
The chemical timescale, defined in Eq. (5c), is based on a single step global Arrhenius kinetics of the form
−ρ−=ω
T
TexpyyAT a
OF2
f2
1(6a)
For stoichiometric mixtures, using the relation that FOO yy ν= at all times, the above equation reduces to
−ρν−=ω
T
TexpyTA a2
F2
Of2
1(6b)
The above simplification enables us to eliminate the conservation equation for the oxidizer separately. The chemicaltimescale defined in Eq. (5c) assumes perfect mixing of the reactants. The parameter heff in Eq. (5b) reflects theeffective heat transfer coefficient that combines the convective and radiative heat losses. Radiation is modeled usingan optically thin gas approximation. With an optically thin model, the effective heat transfer coefficient can beexpressed as
( )( )TTTTLT4hh w22
wc30Peff 1
++σκ+= (7)
An expression for Ze needed to close the system is obtained from conservation of mass within the tailpipe as
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e
e
fe T
puZ
τ= (8)
Finally, flow in the nozzle connecting the combustor and the tailpipe is assumed isentropic, owing to short length ofthe section, although irreversibilities are present in both the combustor and the tailpipe. Thus the pressure and thetemperature in the tailpipe are related to the combustor variables through isentropic relations as
2f0p
2c
2
eTC2
LuTT 2
τ−= (9a)
and)1(
T
Tpp e
e
−γγ
= (9b)
The mass flowrate at the inlet is perturbed sinusoidally as
( )tSinZZ ii ωε+=′ 1 (10)
In the above equation, ε and ω denote the amplitude and angular frequency of perturbation respectively.The governing equations are solved as a system of coupled nonlinear ordinary differential equations using the
library function ODE45 of commercial package, MATLAB. This function uses a fourth order Runge Kutta method.The time-steps are selected adaptively by the integrator. In the present case, the upper bound of the time step was setat 10-5 seconds. However, the actual time step used is much finer (~2.5×10-6 seconds). This gives a data samplingrate of around 40000 Hz. Thus the generated dataset is sufficiently large for post-processing analysis.
III. Results and DiscussionsA model pulse combustor of specified geometry was investigated for dynamic response. The parameters used in
the computation are indicated in Table-I. Propane was used as the fuel and only stoichiometric mixtures of fuel andair were considered. The kinetic parameters were adopted from Ref. [5]. To initiate the reaction, the initialtemperature was raised to five times the ambient temperature. The choice of the initial time step affected only theinitial transience (t<0.1 seconds), which has not been considered for the analysis presented here (t>0.25 seconds).The pulsations in the mass flow rate are applied after 0.1 s.
The effects of modulations in the flow rate have been investigated by varying the frequency from 10 Hz to 250Hz for three amplitudes of 2%, 5% and 10% of the mean mass flow rate.
A. Effect of Wall TemperatureFigures 2 - 6 show the effects of modulating the mass flow rate at inlet at different wall temperatures. The results
for oscillations at two frequencies (10 Hz and 140 Hz) have been reported for the three amplitudes mentioned above.The frequencies chosen represent a low frequency perturbation and a frequency close to the resonant frequencyrespectively. Our earlier study 15 for self-sustained oscillations showed that the pulse combustor exhibits limit cyclebehaviour at a wall temperature of 1140 K for the conditions listed in Table I. As the temperature is lowered, thedynamics undergoes a series of period doubling bifurcations before the system becomes chaotic and ultimately leadsto extinction at a wall temperature of ~1040 K. To examine the effect of pulsations, the wall temperature is variedover the same range.
Figure 2 shows the time series data for pressure at different wall temperatures for a pulsation of 10 Hz frequencyand 2% amplitude. At 1140 K, the system dynamics is characterized by a superposition of the self-sustained limitcycle behaviour with the low frequency pulsation. At this low amplitude of the perturbation, the effect ofsuperposition is clearly visible. As the temperature is lowered to 1090 K, the combustion pattern starts becomingirregular. The alternate peaks attain lower values. However, due to the presence of the low frequency oscillations,the alternate high and low peaks are not as prominent as in unforced systems. The lower peaks become morepronounced at the troughs of the low frequency oscillations and less conspicuous at the crests. On further loweringthe wall temperature, the time series becomes irregular before extinction at 1060 K.
Figure 3 shows the time series data for pressure at different wall temperatures for 5% amplitude and 10 Hz. At1140 K, the time series pattern is similar to that for 2% amplitude. However, the effect of the low frequencymodulation is much more pronounced as the amplitude is larger. As the temperature is lowered, the temporalvariation remains similar up to 1110 K. However, as the temperature is lowered to 1105 K, combustion becomesextremely irregular and below that temperature, the flame extinguishes suddenly. A significant observation for this
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pulsation is the absence of the period doubling transitions towards extinction. For still higher amplitude of 10%, theextinction occurs at temperatures slightly above 1140 K.
The dynamics of the combustor at different wall temperatures subjected to a pulsation at 2% amplitude and 140Hz is presented in Fig. 4. At 1140 K, the system characteristics are close to limit cycle behaviour exhibited by theunforced system. As the wall temperature is reduced, the system undergoes a series of period-doubling bifurcationsas in the case of self-sustained oscillations. Superposed on this behaviour is a lower frequency perturbation. Suchlow frequency modulation has also been observed by Marsano et al.12 for sinusoidal perturbations with frequenciescomparable to the natural frequency.
Figure 5 shows the dynamic characteristics of pulse combustor pressure at different wall temperatures for inletflow modulation with 5% amplitude and 140 Hz frequency. At 1140 K, in this case also, the limit cycle behaviour isapparent. However, as the temperature is reduced, although the transition is qualitatively similar to that observed inFig. 4, the deviation from the self-sustained oscillation is much more pronounced. Below 1100 K, the time series ofpressure oscillations exhibit a period-2 behavior. The combustion pattern becomes increasingly irregular below 1075K till the flame extinguishes just above 1040 K.
Figure 6 shows the corresponding time series of pressure data for 10% amplitude and 140 Hz frequency. Over aconsiderable temperature range below 1140 K, the effect of forced oscillation is limited to early times andsubsequently the temporal variation of the naturally occurring dynamics is obtained. However, close to extinction,the effect of the forcing is felt throughout the cycle and tends to make the combustion more irregular.
The effect of wall temperature is summarised in Figs. 7 - 9 by means of bifurcation plot. In Fig. 7(a) the localpeak pressure values are plotted against the wall temperature in presence of a mass flow rate perturbation of 2% and10 Hz frequency. For temperatures above 1090 K, it is observed that a number of peaks appear that are close to eachother. The occurrence of multiple peaks is attributed to the existence of the low frequency modulation. Crests of thehigh frequency self-sustained oscillations, which constitute the local peaks, occur at different phases of the lowfrequency modulation giving rise to different peak values. However, below 1090 K, when self-sustained oscillationsshow period doubling 14, for the present pulsation also, two sets of pressure peaks are observed. However, thebifurcation is not very distinct. Further lowering of the wall temperature increases the scattering of the pressurepeaks till finally extinction is observed around 1060 K. This value is significantly higher than that of self-sustainedoscillations (~ 1040 K).
Figure 7 (b) presents the results for pulsations with 2% amplitude and 140 Hz frequency. A major distinctionfrom the results of Fig. 7(a) is a considerable reduction in the scatter of the peak pressures, particularly at hightemperatures. In fact, around 1140 K, the pressure peaks nearly coincide with each other, resulting in a limit cycle-like dynamics, as observed in Fig. 3 also. The bifurcation at 1090 K is also much more conspicuous than at 10 Hzfrequency. The extinction is also significantly delayed to a value close to that of self-excited systems.
Figure 8 shows the results for pulsations with 5% amplitude at a frequency of 10 Hz and 140 Hz. From Fig. 8(a), it is observed that at 10 Hz frequency, there is considerable scatter even at 1140 K and the system goes toextinction at ~ 1105K without any noticeable bifurcation. However, at 140 Hz, the extinction limit is lowered toabout 1045 K, which is slightly higher than that for self-sustained oscillation and flow modulation with 2%amplitude. However, at this amplitude, the bifurcations are not very clearly discernible.
Figure 9 shows the corresponding results for amplitude of 10%. However, unlike the cases of 2% and 5%amplitudes, in this case, for a pulsation frequency of 10 Hz, the system undergoes extinction even at 1140 K. Hence,results are presented only for 140 Hz. As in the case of 2% and 5% amplitudes, in this case also, the extinctionoccurs at a wall temperature close to the extinction limit for self-excited systems. However, the bifurcations are notobvious.
B. Effect of Forcing FrequencyThe results in the previous section show that the dynamic characteristics are strongly dependent on the forcing
frequency. In this section, a more systematic investigation of the influence of the forcing frequency is carried out byexamining the system response to a wide range of frequencies from 10 Hz to 250 Hz at different amplitudes.
Figure 10 shows the time series data for the system pressure at a wall temperature of 1140 K for 2% amplitudeand different forcing frequencies. As the frequency increases from 10 Hz to 60 Hz, the time series for pressurebecomes increasingly irregular. With further increase in frequency to 70 Hz, the variation of system pressurebecomes more periodic in nature. However, the trend reverses at 90 Hz. As the frequency increases to 140 Hz, asobserved from Fig. 4, the temporal variations become nearly periodic. At frequencies higher than the naturalfrequency, the effect of forcing frequency is less pronounced till at 250 Hz, the system characteristics are similar tounforced system.
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Figures 11 and 12 show the corresponding results for 5% and 10% amplitudes respectively. At both theseamplitudes, no flame exists for extended ranges of intermediate frequencies unlike the case of 2% amplitude whereextinction is not observed at any frequency. This is discussed in greater details in Fig. 13. Figure 11 shows that at 10Hz frequency, the superposition of the low frequency pulsation and the high frequency self-sustained oscillations isclearly evident. At a frequency of 70 Hz, the combustion becomes very irregular. As the frequency increases to 120Hz, combustion becomes more regular and shows a near periodic behavior at 140 Hz. In contrast with the resultsshown in fig. 10, combustion becomes irregular again as the frequency increases from 140 Hz till it again becomesperiodic at 250 Hz.
Figure 12 shows that the system response at high amplitude is distinctly different. Although the behaviour at 10Hz and 70 Hz are qualitatively similar to that in Fig. 11, at 90 Hz, the amplitude of pressure fluctuation suddenlydecreases and the combustion becomes very irregular before becoming more regular at later times. At frequenciesclose to the natural frequency of the system, after an initial phase of unstable oscillation, characterized bycontinuous increase in amplitude, the combustion dynamics approaches a periodic character. At 170 Hz, the pressuredynamics shows “beat” like characteristics of alternate amplification and attenuation as seen in standing waves. At220 Hz, the pressure pulses show a period doubling bifurcation and finally at 250 Hz, in this case also combustionbecomes more regular but limit cycle behaviour is not approached.
A clearer picture of the effect of forcing frequency emerges from Fig. 13. Figure 13(a) shows that as thefrequency is increased at 2% amplitude, combustion becomes increasingly irregular as evidenced by the scatter ofthe pressure peaks up to 70 Hz. Beyond that the combustion becomes more regular and the pressure peaks clusteraround one another at 140 Hz, which is close to the natural frequency of the system. Beyond that, the pressure peaksget increasingly scattered till about 200 Hz. Beyond that, the combustion becomes increasingly regular. For 5%amplitude, no flame is observed between 10 Hz and 70 Hz and again between 80 Hz and 120 Hz. As in the previouscase, combustion is most regular at 140 Hz and 250 Hz. For 10% amplitude, even at a higher wall temperature of1150 K, the “no flame” ranges of forcing frequencies are wider and the pressure peaks are, in general, very scatteredwithout showing any definite pattern of variation.
The above results clearly reveal that the system behaviour differs significantly at low (2%) and high (10%)amplitudes. At low amplitudes, the system response approximates that of a linear system while at high amplitudes,the nonlinear features start dominating16. The results at 10% amplitude clearly indicate the nonlinear behaviour ofthe system.
The combined effects of wall temperature and frequency are summarized in Fig. 14. It is observed that ateach of the three amplitudes, the minimum wall temperature, which sustains combustion attains a minimum at 140Hz. Flame is sustained up to lower temperatures for frequencies that are close to integral multiples of the naturalfrequency while the extinction wall temperatures are highest at frequencies midway between these values.
IV. ConclusionsThe effect of sinusoidal flow pulsations on the dynamics of a thermal pulse combustor was investigated using a
nonlinear model for the dynamics of thermal pulse combustors consisting of four coupled nonlinear ordinarydifferential equations. The system characteristics were examined over a wide range of pulsation frequencies andwall temperatures for different amplitudes.
It was observed that at low amplitudes for low frequencies and frequencies comparable to the natural frequencyof the system, the system undergoes transition from periodic to chaotic state through a series of period-doublingbifurcations as the combustor wall temperature is reduced. However, at higher amplitudes, the system undergoesabrupt extinction at low frequencies while at frequencies, comparable to the natural frequency, extinction occurs atwall temperatures nearly equal to that observed in self-sustained systems.
At a given wall temperature, as the frequency is varied, the system characteristics undergo several transitions. Atlow and moderate amplitudes, nearly periodic behaviour is observed at frequencies close to that of self-sustainedoscillation. At moderate and high amplitudes, no flame can be sustained over extended ranges of intermediatefrequencies.
Flame is sustained up to lower temperatures for frequencies that are close to integral multiples of the naturalfrequency while the extinction wall temperatures are highest at frequencies midway between these values.
References
1Eibeck, R.A., Keller, J.O., Bramlette, T.T. and Sailor, D.J., Pulse Combustion: Impinging Jet Heat Transfer Enhancement,Combustion Science and Technology, 94, 147 – 165, 1993.
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2 Strumillo, C., Zbicinski, I., Smucerowicz, I. and Crowe, C., An analysis of a pulse combustion drying system, ChemicalEngineering and Processing, 38, 593–600, 1999.
3 Zbicinski, I., Strumillo, C., Kwapinska, M. and Smucerowicz, I., Calculations of the pulse combustion drying system, EnergyConversion and Management, 42, 1909 – 1918, 2001.
4 Stewart, C.R., Lemieux, P.M. and Zinn, B.T., Application of Pulse Combustion to Solid and Hazardous Waste Incineration,Combustion Science and Technology, 94, 427 – 446, 1993.
6 Morel, T., Comprehensive Model of a Pulse Combustor, Combustion Science and Technology, 94, 379 – 409, 1993.7 Rhode, M. A., Rollins, R. W., Markworth, A. J., Edwards, K. D., Nguyen, K., Daw, C. S. and Thomas, J. F., Controlling Chaos
in a Model of Thermal Pulse Combustion, J. App. Phys., 78, 2224 – 2232, 1995.8 Narayanaswami, L. and Richards, G.A., Pressure-Gain Combustion: Part I – Model Development, Trans. ASME J. Engg. Gas
Turbines Power, 118, 461 – 468, 1996.9 H.M. Heravi, J.R. Dawson, P.J. Bowen and N. Syred, Primary Pollutant Prediction from Integrated Thermofluid–Kinetic Pulse
Combustor Models, AIAA Journal of Propulsion and Power, 21, 1092 – 1097, 2005.10 Mukhopadhyay, A., Datta, S. and Sanyal, D., Effects of Tailpipe Friction on the Nonlinear Dynamics of a Thermal Pulse
Combustor, Trans. ASME J. Engg. Gas Turbines Power,.11 Neumeier, Y., Zinn, B.T. and Jagoda, J.I., Frequency Domain Analysis of the Performance of a Valved Helmholtz Pulse
Combustor, Combust. Sci. Technol., 94, 295 – 316, 1993.[12] Marsano, S., Bowen, P.J. and O’Doherty, T., Cyclic Modulation Characteristics of Pulse Combustors, Proc. Combust. Inst.,
27, 3155 – 3162, 1998.[13] Kilicarslan, A., Frequency Evaluation of a Gas-Fired Pulse Combustor, Int. J. Energy Research, 29, 439 – 454, 2005.14 Datta, S., Mukhopadhyay, A. and Sanyal, D., Modeling and Analysis of the Nonlinear Dynamics of a Thermal Pulse
Combustor, Paper # AIAA 2006-4396, 42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 9 - 12 July2006, Sacramento, California
15 Datta, S., Mondal, S., Mukhopadhyay, A., Sanyal, D. and Sen, S., An Investigation of Nonlinear Dynamics of a Thermal PulseCombustor, Combust. Theory Modell., in press, 2008.
16 Huang, X., Development of Reduced-Order Flame Models for Prediction of Combustion Instability, Ph. D Thesis, VirginiaPolytechnic Institute and State University, Blacksburg, Virginia, USA, 2001.
InletTi,Yf,i Combustor Tailpipe
Fig. 1: Schematic of a Pulse Combustor
Te, Pe, Ue,Ae
T, P, Yf, A
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0.25 0.3 0.35 0.4 0.45 0.50.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(a) (b)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
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ensi
onle
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ress
ure
(c) (d)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 2: Effect of Wall Temperature on Forced Oscillations with 2% Amplitude and 10 Hz (a) 1140 K(b) 1090 K (c ) 1075 K (d) 1065 K
0.25 0.3 0.35 0.4 0.45 0.50.8
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1.2
1.4
1.6
1.8
2
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onle
ssP
ress
ure
0.25 0.3 0.35 0.4 0.45 0.50.8
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onle
ssP
ress
ure
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onle
ssP
ress
ure
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1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 3: Effect of Wall Temperature on Forced Oscillations with 5% Amplitude and 10 Hz (a) 1140 K(b) 1120 K (c ) 1110 K (d) 1105 K
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0.25 0.3 0.35 0.4 0.45 0.50.8
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1.4
1.6
1.8
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ensi
onle
ssP
ress
ure
0.25 0.3 0.35 0.4 0.45 0.50.8
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onle
ssP
ress
ure
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ress
ure
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0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 4: Effect of Wall Temperature on Forced Oscillations with 2% Amplitude and 140 Hz (a) 1140 K(b) 1090 K (c ) 1075 K (d) 1050 K
0.25 0.3 0.35 0.4 0.45 0.50.8
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1.8
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ensi
onle
ssP
ress
ure
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ssP
ress
ure
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ensi
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ssP
ress
ure
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1.4
1.6
1.8
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Dim
ensi
onle
ssP
ress
ure
Fig. 5: Effect of Wall Temperature on Forced Oscillations with 5% Amplitude and 140 Hz (a) 1140 K(b) 1090 K (c ) 1075 K (d) 1050 K
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0.25 0.3 0.35 0.4 0.45 0.50.8
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onle
ssP
ress
ure
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ress
ure
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ure
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ensi
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ress
ure
Fig. 6: Effect of Wall Temperature on Forced Oscillations with 10% Amplitude and 140 Hz (a) 1140K (b) 1090 K (c ) 1075 K (d) 1055 K
1060 1070 1080 1090 1100 1110 1120 1130 11401.2
1.3
1.4
1.5
1.6
1.7
1.8
Wall Temperature (K)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(a)Fig. 7: Bifurcation Diagrams for System Pressure at Different Wall
Temperatures for 2% Amplitude at (a) 10 Hz (b) 140 Hz
American Institute of Aeronautics and Astronautics092407
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1040 1060 1080 1100 1120 11401
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wall Temperature (K)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(b)Fig. 7(Contd.)
1100 1105 1110 1115 1120 1125 1130 1135 11401.3
1.4
1.5
1.6
1.7
1.8
Wall Temperature (K)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(a)Fig. 8: Bifurcation Diagrams for System Pressure at Different Wall
Temperatures for 5% Amplitude at (a) 10 Hz (b) 140 Hz
American Institute of Aeronautics and Astronautics092407
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1040 1060 1080 1100 1120 11401
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Frequency(Hz)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(b)Fig. 8: (Contd.)
1040 1060 1080 1100 1120 11401
1.2
1.4
1.6
1.8
2
Max
imum
Dim
ensi
onle
ssP
ress
ure
Wall Temperature (K)Fig. 9: Bifurcation Diagrams for System Pressure at Different Wall
Temperatures for 10% Amplitude at 140 Hz
American Institute of Aeronautics and Astronautics092407
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(a)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(b)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(c)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(d)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(e)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(f)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(g)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(h)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 10: Effect of Flow Pulsation Frequency on Oscillations with 2% Amplitude at Tw = 1140 K (a) 10Hz, (b) 60 Hz, (c) 70 Hz, (d) 90 Hz, (e) 140 Hz, (f) 160Hz, (g) 220 Hz and (h) 250 Hz
American Institute of Aeronautics and Astronautics092407
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(a)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Dim
ensi
onle
ssP
ress
ure
(b)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(c)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Dim
ensi
onle
ssPr
essu
re
(d)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(e)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(f)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(g)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(h)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 11: Effect of Flow Pulsation Frequency on Oscillations with 5% Amplitude at Tw = 1140 K (a) 10Hz, (b) 70 Hz, (c) 120 Hz, (d) 140 Hz, (e) 170 Hz, (f) 200Hz, (g) 220 Hz and (h) 250 Hz
American Institute of Aeronautics and Astronautics092407
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(a)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Dim
ensi
onle
ssP
ress
ure
(b)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(c)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(d)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
2
Time (s)
Dim
ensi
onle
ssP
ress
ure
(e)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(f)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(g)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
(h)0.25 0.3 0.35 0.4 0.45 0.5
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Dim
ensi
onle
ssP
ress
ure
Fig. 12: Effect of Flow Pulsation Frequency on Oscillations with 10% Amplitude at Tw = 1150 K (a)10 Hz, (b) 70 Hz, (c) 90 Hz, (d) 130 Hz, (e) 140 Hz, (f) 160Hz, (g) 220 Hz and (h) 250 Hz
American Institute of Aeronautics and Astronautics092407
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0 50 100 150 200 2501.3
1.4
1.5
1.6
1.7
1.8
Frequency(Hz)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(a)
0 50 100 150 200 2501
1.2
1.4
1.6
1.8
2
Frequency (Hz)
Max
imum
Dim
ensi
onle
sP
ress
ure
(b)Fig. 13: Bifurcation Diagrams for System Pressure at Different Forcing Frequencies at 1140 K and (a) 2%
Amplitude (b) 5% Amplitude and (c) 1150 K and 10% Amplitude
American Institute of Aeronautics and Astronautics092407
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0 50 100 150 200 2501
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Frequency (Hz)
Max
imum
Dim
ensi
onle
ssP
ress
ure
(c)Fig. 13: Bifurcation Diagrams for System Pressure at Different Forcing Frequencies at 1140 K and (a) 2%
Amplitude (b) 5% Amplitude and (c) 1150 K and 10% Amplitude
1020
1040
1060
1080
1100
1120
1140
1160
1180
0 50 100 150 200 250 300
Frequency (Hz)
Ext
inct
ion
Wal
lTem
pera
ture
(K)
2% amplitude
5% amplitude
10% amplitude
Fig. 14: Extinction Map on Wall Temperature-Frequency Plane
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