[American Institute of Aeronautics and Astronautics 45th AIAA/ASME/SAE/ASEE Joint Propulsion...

American Institute of Aeronautics and Astronautics

1

Combustion Instability in a Turbulent Liquid-Fueled Swirl-Stabilized LDI Combustor

Tongxun Yi1 and Domenic A. Santavicca2 Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA

16802

Reported is an investigation of combustion instability in a turbulent, liquid-fueled, swirl-stabilized, LDI combustor. Phase-locked ICCD images of CH* chemiluminescence show little change in the flame structure across one pressure cycle, even when the pressure amplitude exceeds the mean pressure drop across the air swirler and the venturi. Depending on the working conditions, the one-wave mode, the half-wave mode, or both can be excited. The simultaneous excitation of the half-wave mode and the one-wave mode is inherently unsteady and unstable. Disturbances, even in small amplitude, can destroy the subtle balance between the two modes, causing one mode to grow and the other to decay. The time-varying amplitude of the two modes is rooted in the nonlinear response of the reacting swirling shear layer to inlet disturbances. An increase in the time scales for convection, evaporation, and chemical kinetics favors the excitation of the half-wave mode, and a decrease in these time scales favors the excitation of the one-wave mode. Unsteadiness is an intrinsic feature of thermoacoustic oscillations in turbulent combustors. An external disturbance pushes the state off the equilibrium trajectory, and it may take a number of revolutions for the state to return to a small neighborhood of the equilibrium trajectory. Because of the ubiquity of external and background disturbances, both the amplitude and the frequency of thermoacoustic oscillations are constantly time-varying. With decreasing pressure amplitude or the positive feedback between pressure and heat release, the limit-cycle thermoacoustic oscillations are increasingly vulnerable to external disturbances, as indicated by a larger neighborhood that the state trajectory roves over. The acoustic wave distribution along the combustion chamber deviates considerably from the natural one-wave mode. The natural acoustic modes refer to the acoustic eigenmodes in a combustion chamber which does not generate sound itself. Caution should be exercised when developing low-order models via Galerkin projection of the natural acoustic modes in gas turbine combustors.

Nomenclature A and B = complex amplitude of the downstream- and upstream-propagating acoustic wave, Pa; c = mean sound speed, m/s; f = frequency, Hz;

RF = the quarter-wave resonant frequency of the water-cooling tubing, Hz; G(…) = acoustic transfer function;

ck ω= = acoustic wave number;

cuM = = mean flow Mach number:

),(~ txP = complex dynamic pressure, Pa;

1 PostDoctoral Fellow, Department of Mechanical and Nuclear Engineering, AIAA member, [email protected]. 2 Professor, Department of Mechanical and Nuclear Engineering, AIAA member, [email protected].

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit2 - 5 August 2009, Denver, Colorado

AIAA 2009-5014

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


2

)(ˆ fP = spectra of the corrected pressure, Pa;

)(ˆ fPm = spectra of the measured pressure, Pa;

),(~1 txP and ),(~

2 txP = complex pressure at two upstream locations along the inlet pipe or the combustion chamber, respectively; u = the mean velocity, m/s; Xmax = the axial coordinate of maximum chemiluminescence, m; Rmax = the radial coordinate of maximum chemiluminescence, m; s = symbol of Laplace transform; ϕ = equivalence ratio; ρ = mean density, Kg/m3; ω = angular frequency, rad/s.

Acronyms DLE = dry low emission; FFT = fast Fourier transform; FN = the flow number of a liquid fuel nozzle; ICCD = intensified charge coupled device; ID = internal diameter, m; LBO = lean blowout LDI = lean direct fuel injection; LOS = line of sight; PVC = precessing vortex core; SMD = saunter mean diameter; PMT = photomultiplier tube; FWHM = full width half maximum;

I. Introduction Iquid-fueled LDI combustion, i.e. directly injecting liquid fuel into strong swirling air flows across the dome, is a promising DLE combustion technology for aircraft engines. LDI combustion is characterized by globally lean

but spatially non-uniform distribution of equivalence ratios. It represents an intermediate category between premixed and diffusion flames. LDI combustion generates more NOx than lean-premixed-prevaporized combustion, but it has advantages in terms of weight, size, and lean flammability. More importantly, LDI combustion is free from flashback. Combustion instability is a major technical challenge facing the development and operation of dry-low-emission (DLE) combustion systems [1][2]. The vulnerability of DLE combustion systems to self-excited pressure pulsations arises from several aspects. Firstly, a DLE combustor is particularly designed to operate at low equivalence ratios. The sensitivity of chemical kinetics to both external and internal disturbances exponentially increases with decreasing equivalence ratios towards lean blowout [3]; secondly, acoustic damping is low in DLE combustion systems, because of the reduced cooling air flowing across the liner. Acoustic dissipation via vortex generation across the orifices is an important damping mechanism, which is effective over a wide range of frequencies [4]; thirdly, the flame surface area of lean premixed and partially premixed combustion is more responsive to velocity perturbations than diffusion flames. In diffusion flames, heat release occurs along a surface of stoichiometry, and the chemical reaction rate is solely determined by the diffusion rate between fuel and air. But for lean premixed and partially premixed combustion, the flame surface area is subject to variations in both the velocity upstream of the flame front and the flame speed. In DLE combustion systems, the pressure drop across the dome is typically within 4% of the plenum pressure. Thus small-amplitude pressure oscillations, say within 1% or 2% of the combustor pressure, can cause large oscillations in the air flow rate. This results in large perturbations in the velocity

L


3

along the swirling shear layer, the equivalence ratio, the reaction heat, and consequently the instantaneous heat release rate.

Atomization and evaporation are unique processes of liquid-fueled combustion, which in a large extent determine the characteristics of spray combustion. Large droplets do not necessarily follow the ambient fluid motion [5]. In the case of a short convection time, droplets heating, evaporation, and combustion occur simultaneously [6]. It has been shown that, depending on the droplet number density and a dimensionless group combustion number, combustion may occur in different modes, namely isolated droplet combustion, external group combustion, and external sheath combustion [7]. However, the above theory does not address all the truth of swirl-stabilized, liquid-fueled, LDI combustion. A DLE combustor is specially designed to avoid or minimize the occurrence of “hot regions” and “hot spots”, i.e. stoichiometric combustion around droplets and droplets clusters. This is achieved by improving fuel atomization and enhancing droplets/vapor/air entrainment and mixing. When the droplet size decreases below certain thresholds, evaporation is no longer a rate-determining process, and fuel/air mixing and chemical kinetics start to dominate. A recent experiment on liquid-fueled combustion shows the invariance in heat release response to the fuel modulation amplitude [8], suggesting that neither droplet evaporation nor fuel/air mixing is the rate-determining process. This rig is featured with a radial air swirler, which has eight tangential-entry ports. Across the air swirler, there is significant generation of streamwise vortices, which enhances droplets/air entrainment and mixing.

Liquid-fueled combustion instability has been extensively studied for liquid rocket engines (LREs) [9-12]. Familiarity with combustion instability in LREs is obviously a good start, but whether the knowledge can be directly applied to LDI combustion requires further examination. Chemical kinetics is not a rate-determining process for LREs, because of the very high flame temperature, usually above 3000 K [11]. Thus, in LREs the heat release response is almost solely determined by atomization and evaporation. As early as in the 1950s, it was proposed that the unsteady evaporation process caused by droplets/acoustics interactions was a major combustion instability mechanism in LREs [9-12]. It was even believed that the unsteady mass addition alone was able to excited combustion instability [13]. Evaporation enhancement in oscillatory environment has been extensively reported in literature [14-16]. Undoubtedly, evaporation still remains as a key process for liquid-fueled LDI combustion, but depending on the details of atomization and fuel/air mixing, the combustion characteristics can be very different. In a well-designed LDI combustor, combustion can be very similar to lean premixed combustion [8][17]. However, for combustion near lean blowout, neither the evaporation process nor the chemical kinetics can be neglected. In the last two decades, a lot of attention has been given to combustion instability in lean premixed combustion systems [1][2]. However, systematic investigation of liquid-fueled combustion instability in DLE combustion systems is rather rare. Without sufficient understanding of the underlying mechanisms, the scientists and engineers will lack the tools for accurate prediction of the thermoacoustic behavior, which inevitably will limit our capabilities for combustor design and control design. Chishty et al. [18] performed a simulation of the droplets trajectory in non-reacting environment, and showed that the droplet size, the droplet numerical density, and the trajectory were modified during droplets/acoustics interactions. Note that in a DLE combustor, the velocity slip between droplets and the surroundings can be caused by both the unsteady hydrodynamic velocity along the reacting swirling shear layer and the acoustic velocity. In particular, if a pressure anti-node is located nearby the heat release zone, the velocity oscillations are solely associated with the unsteady hydrodynamic velocity. Eckstein et al. [19] found that, the unsteady droplet SMD of an airblast atomizer could be calculated using the steady-state atomization correlations, with the unsteady air velocity at the burner exit. In their experiments, air forcing was applied at 90 and 130 Hz at the combustor inlet. For a counter-swirl-stabilized combustor fueled with kerosence, Carcia et al. [20] observed variations in the excited acoustic mode with the air velocity, which was attributed to the differences in fuel/air mixing.

This paper is organized as follows. Firstly, a swirl-stabilized, liquid-fueled, LDI combustion rig and the instrumentation are described; secondly, ICCD images of CH* chemiluminescence under both stable and unstable conditions are presented and analyzed; thirdly, we discuss several features of combustion instability in liquid-fueled LDI combustion, including acoustic wave distribution along the combustion chamber, simultaneous excitation of the one-wave and the half-wave modes, and the effects of external and background perturbations on limit-cycle oscillations.

II. Experiment Setup Figure 1 shows the liquid-fueled, swirl-stabilized, LDI combustion rig and the instrumentation. This rig is

featured with a 60o axial swirler and a single-point pressure-swirl fuel injector (FN=0.7), both from the NASA Glenn Research Center. The swirl number at the swirler exit is 1.25, which is determined using the formula


4

suggested in Ref. 21. Preheated air, up to 773 K and 120 g/s, enters a quartz combustion chamber, 0.76 m in ID and 0.30 m in length, through a venturi. The 0.36-m-long inlet pipe has a uniform ID of 2.67 cm. The throat diameter of the venturi is 12 mm, and both the converging and diverging angles are 50o. The venturi creates a high-speed swirling jet across the throat, which prevents flame flashback. A stable and compact precessing vortex core (PVC) is formed immediately downstream of the dump plane. Jet-A is injected into the swirling air flow 8 mm upstream of the dump plane, i.e. 4.5 mm upstream of the venturi throat. An orifice plate with a blocking ratio of 95% is installed at 1.08 m downstream of the dump plane. Another orifice plate with a blocking ratio of 75% is installed 0.36 m upstream of the dump plane. The exit orifice plate has two functions. Firstly, it builds up combustor pressure, which can be up to 3 bar; secondly, it defines the acoustic boundary condition at the combustor exit. Pressure drop across the air swirler and the venturi varies with the air flow rate, the preheat temperature, and the equivalence ratio. For all results reported in this paper, the pressure drop is within 3~6% of the static pressure upstream of the air swirler.

The air flow rate is measured using a Vortex flowmeter from Omega, with a measurement uncertainty of 27 SLPM. The fuel flow rate is measured using a rotameter, based on a second-order curve-fitting polynomial. The measurement uncertainty is 5% of reading. An ICCD camera (576x384 pixels) from Princeton Instruments is used for flame imaging of CH* chemiluminescence. CH* chemiluminescence is also measured using a PMT from Hamamatsu Corp. (H7732-10, 185-900 nm). An interference filter centered at 430 nm with FWHM of 10 nm is placed in front of the PMT. The PMT is located 0.38 m downstream of the dump plane, and 0.60 m away from and perpendicular to the combustion chamber. Experiments are performed in a dark room. Static pressure is measured both upstream of the air swirler and downstream of the quartz combustion chamber, respectively, using two digital gauges (DPB1000B-100G) from Omega. The measurement uncertainty is 0.35 kPa. Dynamic pressure is measured at multiple locations along the inlet pipe and the combustion chamber, using PCB pressure sensors (PCB112A22). Self-designed water-cooling jackets are used for pressure sensor mounting. The quarter-wave resonant frequency of the water-cooling tubing is estimated around 1666 Hz, well above the unstable frequencies experienced in the rig. Pressure measurements are corrected as follows,

)2

cos()(ˆ)(ˆR

m FffPfP π

= (1)

Here ( )fP̂ and ( )fPmˆ refer to the spectra of the measured and corrected pressure, respectively. f denotes the

frequency. FR refers to the quarter-wave resonant frequency of the water-cooling tubing. The sampling frequency is 10 kHz, and the data length is 10 s. Without explicit explanation, FFT is performed for data acquired within each second, and the reported value is the average of the results for the ten data segments. Although not shown in Fig.1, dynamic pressure is measured at the dump plane using 8.9-cm-long water-cooling tubing. The quarter-wave resonant frequency is estimated around 1000 Hz.

7.6 7.6 8.9

34.3 25.4 80.0

10.2 10.2 10.2 39.4

Orifice Plate

Dynamic Pressure

Quartz Chamber

Dynamic Pressure

Air

Orifice Plate

PMT

12.7

ICCD Camera

125


5

Figure 1. The combustion rig and the instrumentation. The length unit is cm. Temperature is measured at 49, 68, and 85 cm downstream of the dump plane using type-K thermocouples.

III. Flame Structure

A. Stable Combustion No combustion instability is observed when three baffle plates are installed inside the combustion chamber.

Stable combustion is characterized by low pressure amplitude and random phase relationship between pressure and CH* chemiluminescence. ICCD images of CH* chemiluminescence are taken within a large range of working conditions, with the air flow rate from 22.2 to 38.9 g/s, the preheat temperature from 473 to 773 K, and the equivalence ratio from 0.50 to 0.20. Frame-to-frame variations in global CH* intensity are within 2%. Figure 2 shows the light-of-sight (LOS) and 2D ICCD images of CH* chemiluminescence, at the air flow rate of 22.2 g/s and the preheat temperature of 373 K, 473 K, and 773 K, respectively. The LOS images are shown at the top row, with the 2D images at the bottom row. The 2D images are converted from the LOS images using Abel Deconvolution [22]. Abel Deconvolution assumes perfect axisymmetry of the flame, which is only true for laminar flames. However, for the present combustor, the dominant feature of the reacting flow field, including the precessing vortex core, is still of 2D axisymmetric nature. Thus the processed images using Abel Deconvolution still provides useful information for the flame structure.

(a) Preheat temperature 373 K

Ф=0.50 0.46 0.42 0.340.38 0.30 0.26 0.22

Static Pressure

Thermocouple

Cooling Water

Axial Swirler

Single-Point Fuel Injector

Dump Plane

Quartz Upper Wall

Quartz Lower Wall

Dump Plane (x=0)


6

(b) Preheat temperature 473 K

(c) Preheat temperature 773 K

Fig.2 Line-of-sight and 2D ICCD images of CH* chemiluminescence. The air flow rate is 22.2 g/s. The intensity scale is the same for each row of images. The orientation of images is shown in Fig.1.

At the preheat temperature of 373 K and with equivalence ratios above 0.34, heat release is rather uniformly distributed within a large portion of the combustor. However, with the preheat temperature above 373 K, the heat release zone shrinks to a “doughnut” ring. The heat release zone is smaller at 773 K than at 473 K. The differences can be attributed to faster evaporation and chemical kinetics at higher preheat temperature. The diameter of the doughnut ring decreases with reductions in the equivalence ratio from 0.50 to 0.34. At the equivalence ratio of 0.34, the doughnut ring converges to the axis. With further decreases in the equivalence ratio, the heat release zone becomes shorter, elongated, and tilted off the axis. The remarkable variations in the flame structure can be attributed to the cone angle of the swirling shear layer. Gas expansion across the flame front promotes vortex breakdown and gas recirculation, which helps to open up the cone angle. A larger cone angle improves fuel/air entrainment and mixing. At equivalence ratios below 0.30, the heat release zone appears like a jet flame, because of the substantially reduced cone angle. The increasing intensity of CH* chemiluminescence with decreasing equivalence ratios can be attributed to the less efficient fuel/air mixing, because of a smaller cone angle. This also implies that combustion occurs around the droplets within the doughnut ring. However, the flame is purely blue near lean blowout. From Fig.2, one can see that in most working conditions, major heat release is accomplished within the swirling shear layer, before it reattaches to the wall. The ICCD images show that combustion characteristics of the present liquid-fueled, swirl-stabilized, LDI combustion rig is mainly determined by droplet evaporation, droplets/vapor/air entrainment, and mixing. This is true even when the preheat temperature reaches 773 K. Stable and continuous combustion is maintained at very low equivalence ratios down to 0.18.

To quantify the spatial distribution of heat release, Fig.3 shows the axial center of CH* chemiluminescence. With increasing preheat temperature, the heat release zone moves closer to the dump plane, which can be attributed to the increasing rate of evaporation and chemical kinetics. At the same air flow rate but at higher preheat temperature, the velocity slip between droplets and the surroundings are larger, which also favors finer atomization and faster evaporation. For pressure-swirl fuel injectors, the droplet size increases at the approach of lean blowout. Thus the

Ф=0.50 0.4 0.4 0.30.3 0.3 0.2 0.2

Ф=0.50 0.4 0.4 0.30.3 0.3 0.2 0.2


7

enhancement of evaporation and chemical kinetics at higher preheat temperature is more pronounced at low equivalence ratios. Figure 3 (b) shows that, the heat release zone moves closer to the dump plane at a higher air flow rate. This is because a higher air velocity improves atomization, evaporation, and consequently the flame speed. Note that, this observation is opposite to lean premixed combustion, where the flame length usually increases with the air velocity. Figure 3 also shows that the flame front consistently moves closer to dump plane with decreasing equivalence ratios, which is also in sharp contrast to lean premixed combustion. This can be attributed to the increasing chemical reaction rates at the approach of lean blowout. The burning velocity of spray flames is a function of the droplets SMD along with other parameters. Some researchers have observed that the flame speed of spray flames can be faster than fully premixed and prevaprized combustion, because of the “rugged and undulated” flame surface area [23].

9

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27

30

33

0.18 0.24 0.30 0.36 0.42 0.48 0.54Equivalence Ratio

Axi

al C

ente

r(mm

)

373 K 473 K 573 K

673 K 773 K

9

12

15

18

21

24

27


Axi

al C

ente

r (m

m)

573K&22.2g/s 573K&27.8g/s573K&33.4g/s 573K&38.9g/s773K&22.2g/s 773K&27.8g/s

(a) (b)

Fig.3 Axial center of CH* chemiluminescence. (a) Air flow rate 27.8 g/s; (b) The preheat temperature 573 K and 773 K.

Figure 4 shows the radial coordinate (Rmax) of maximum chemiluminescence. The radial distribution of heat release is determined by the flame speed, the shear layer velocity profile, and gas recirculation associated with the PVC. Figure 4 (a) shows Rmax at different preheat temperature. The air flow rate is 27.8 g/s. For equivalence ratios below 0.34, the radial distribution of heat release is only slightly affected by the preheat temperature, since the heat release zone has converged to the axis. Below the equivalence ratio of 0.34, the flame starts from the forward stagnant point of the PVC, and develops along the swirling shear layer. The heat release zone appears in a heart shape, indicating gas recirculation along the axis. Without gas recirculation, stable combustion could not have been maintained at equivalence ratios below 0.20. Figure 4 (b) shows that, for equivalence ratios above 0.36, the heat release zone moves further away from the axis with decreases in the air velocity. This can be explained by the balance of the flow velocity upstream of the flame front and the laminar flame speed. With equivalence ratios above 0.36, Rmax moves towards the axis with increasing preheat temperature, because of the increasing evaporation rate and flame speed. Figure 5 shows the axial and radial coordinates (Xmax and Rmax) of maximum chemiluminescence. Here the equivalence ratio is decreased from 0.50 to 0.20, and the preheat temperature is 473 K and 773 K, respectively. At 773 K, the trajectory of maximum chemiluminescence roughly follows the same cone angle. But at 474 K, there is a substantial increase in the cone angle across the equivalence ratio of 0.32. By referring to the 2D images in Fig.2, one can see that the doughnut ring converges to the axis at the equivalence ratio of 0.32.


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Rm

ax(m

m)

473 K 573 K

673 K 773 K

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Rm

ax(m

m)

573K&22.2g/s 573K&27.8g/s573K&33.4g/s 573K&38.9g/s773K&22.2g/s 773K&27.8g/s

(a) (b)

Fig.4 Radial coordinate (Rmax) of maximum chemiluminescence. (a) Air flow rate 27.8 g/s; (b) The preheat temperature at 573 K and 773 K.

Fig.5 The axial and radial coordinates of maximum chemiluminescence at the preheat temperature of 473 K and 773 K. The air flow rate is 27.8 g/s. The equivalence ratio is decreased from 0.50 to 0.20, with a stepwise decrement of 0.02.

Figure 6 shows the radial and axial distribution of CH* chemiluminescence at the air flow rate of 27.8 g/s and the preheat temperature of 773 K. Major heat release is accomplished within 20 mm downstream of the dump plane. With decreasing equivalence ratios toward lean blowout, the heat release zone moves closer to the dump plane, and the width of the major heat release zone becomes narrower. The peak intensity of CH* chemiluminescence increases with reductions in the equivalence ratio, indicating increasingly less efficient fuel/air mixing. This is true even when the preheat temperature reaches 773 K. From Fig.6 (b), one can see that, with decreasing equivalence ratios from 0.50 to 0.20, the major heat release zone moves towards the axis by more than 15 mm, and the width of the major heat release zone becomes narrower. The 2D images obtained using Abel Deconvolution are contaminated with noises near the centerline. The distribution of CH* chemiluminescence within 2.5 mm around the axis is determined by extrapolation of a third-order curve-fitting polynomial, which is developed based on the chemiluminescence intensity from 2.5 mm to 6 mm in the radial direction.

3

6

9

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7 8 9 10 11 12 13 14Xmax(mm)

Rm

ax(m

m)

473 K 773 K

Decreasing Ф

Decreasing Ф


9

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 10 20 30 40Axial Distance (mm)

CH

* Dis

tribu

tion

Phi=0.50Phi=0.44Phi=0.38Phi=0.32Phi=0.26Phi=0.20

0.00

0.01

0.01

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0 10 20 30 40Radial Distance (mm)

CH

* Dis

tribu

tion

Phi=0.50Phi=0.44Phi=0.38Phi=0.32Phi=0.26Phi=0.20

(a) (b)

Fig.6 Radial and axial distribution of CH* chemiluminescence. (a) Axial distribution; (b) Radial distribution. The air flow rate is 27.8 g/s, the preheat temperature is 773 K, and the combustor pressure is around 2.4 bar.

B. Unstable Combustion Without the baffle plates, self-excited combustion instability readily occurs. Both the half-wave and the one-

wave modes have been excited, sometimes even simultaneously. Figure 7 shows the phase-locked ICCD images at the air flow rate of 27.8 g/s, the preheat temperature of 423 K, and the equivalence ratio of 0.36. At this working condition, the one-wave mode of the combustion chamber is excited at 690 Hz. The phases shown in Fig.7 refer to the dynamic pressure at the dump plane. A trigger to the control unit is generated when the dynamic pressure goes below 3 kPa. The intensifier gate width is one-twelfth of the pressure cycle, i.e. 121 μs. Exposure starts after a specified delay, which spans over one pressure cycle with a stepwise increment of one-twelfth of the pressure cycle. At each phase, 100 images are taken. The phase delay shown in Fig.7 is the sum of the phase where the trigger is generated, the phase delay associated with the intensifier gate delay, and one half of the phase delay associated with the intensifier gate width. Figure 8 shows that the same phase relationship between chemiluminescence and pressure is obtained by both phase-locked ICCD imaging and PMT measurements. Thus the procedures of phase locking and analysis are correct. From Fig.7, one can see that there is no formation and shedding of large vortices during self-excited combustion instability. The pressure amplitude at the dump plane is about 85% of the mean pressure drop (10.3 kPa) across the air swirler. Phase-locked ICCD imaging is also done for combustion instability involving the excitation of the half-wave mode. Again across one pressure cycle, there are mainly variations in the chemiluminescence intensity rather than in the flame structure.

The absence of large vortex shedding is quite different from combustion instability in bluff-body-stabilized and dump combustors [12]. In these combustors, because of the strong backflow, the velocity profile exhibits a deflection point across the separating shear layer, thus the shear flow is particularly susceptible to the self-excited, absolutely unstable, hydrodynamic instability [24]. The fresh fuel/air mixture trapped within the large vortices suddenly burns up when they break up into small vortices around the wall reattachment region. Matveen and Culick [25] modeled vortex-induced heat release perturbations as pulses and developed a “kicked” oscillator model. Cohen et al. [26] observed that the self-excited, wake-mode, large vortex shedding alone, was capable of supporting large-amplitude pressure oscillations. In such a scenario, pressure oscillations were not associated with any acoustic mode. For bluff-body-stabilized combustors, it is found that large vortex shedding is most favored when air modulations are applied around the “preferred” frequency ranges. However, out of these “preferred” frequency ranges, even with large-amplitude air forcing, the heat release response can still be linear [27]. The swirling shear layer has faster growth and decay rates than the non-swirling counterparts, thus there are fewer opportunities for the initial disturbances to continuously grow into large vortices [21][28]. Several other researchers have made the same observations for swirl-stabilized combustion [19][29]. Daesik et al. [30] reported the formation of large vortex shedding under air modulations in a turbulent, swirl-stabilized combustor. However, flame stabilization in their combustor was mainly achieved by gas recirculation behind a recessed center-body instead of the aerodynamic PVC.


10

Fig.7 Phase-locked ICCD images for combustion instability involving the excitation of the one-wave mode of the combustion chamber.

-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

0 5 10 15 20 25

Snapshot Index

Chemiluminescence(a.u.) Pressure(x0.1, kPa)

-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Time(s)

Chemiluminescence(a.u.) Pressure x0.1, kPa)

(a) (b)

Fig.8 CH* chemiluminescence and dynamic pressure at the dump plane. Here the one-wave mode of the combustion chamber is excited. (a) ICCD measurements of CH* chemiluminescence; (b) PMT measurements of CH* chemiluminescence.

IV. Combustion Instability

A. Acoustic Wave Distribution along the Combustion Chamber Dynamic pressure is measured at three locations along the combustion chamber, i.e. at 35.6, 56.0, and 95.4 cm

downstream of the dump plane. To determine a globally representative sound speed, temperature is measured using type-K thermocouples at 48.9, 67.9, and 84.5 cm downstream of the dump plane. The radiative and convective heat losses are compensated for using the procedures described in Ref. 31. Temperature correction increases with the equivalence ratio from 124 K at ϕ=0.22 to 177 K at ϕ=0.36. The globally representative temperature is taken as the average of the corrected temperature measured at the three locations. For determination of the sound speed, the ratio of the specific heat is taken as 1.33 instead of 1.40, and the gas constant is 287 J/Kg/K.

By assuming small-amplitude acoustic oscillations, the pressure wave along the inlet pipe is described by the following 1D linear equation,

)1

()1

(),(~ M

kxtiM

kxtiBeAetxP −

++

−+=

ωω (2)

The complex amplitude of downstream- and upstream-propagating waves can be determined from pressure measurements at two locations along the combustion chamber,

86o 116o 146o 176o 206o 236o 266o

296o 326o 356o 386o 416o 446o


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)1

21

(

1

)1

21

(

2

ˆˆ

ˆˆ

Mxik

Mxik

Mxik

Mxik

Mxik

Mxik

Mx

Mxik

Mx

Mxik

ee

ePePB

ee

ePePA

−−

++

−−

−+

+−+

+

−

−=

−

−=

(3)

1̂P and 2̂P are the complex pressure amplitude at upstream locations 1x and 2x , respectively. The complex pressure amplitude is computed using FFT.

At the air flow rate of 38.9 g/s and the preheat temperature of 373 K, the one-wave mode of the combustion chamber is excited. With decreasing equivalence ratios from 0.36 to 0.22, the dominant frequency decreases from 702 Hz to 634 Hz. Figure 9 (a) shows measurements Vs. prediction for dynamic pressure 95.4 cm downstream of the dump plane. The prediction error is within 3.0% for the pressure amplitude. However, the predicted phase lags measurements by about 8o to 10o. The phase error is caused by several factors, including the different response among pressure sensors, slight variations in the water-cooling tubing length, temperature gradient along the water-cooling tubing, and more importantly the temperature gradient along the combustion chamber. Temperature along the centerline of the combustion chamber can differ by more than 200 K. The almost constant phase error can be easily accounted for. Figure 9 (b) shows measurements Vs. prediction for dynamic pressure at the dump plane. The prediction errors are within 5.0% in amplitude. Again there is an almost constant error in phase prediction, about 10o.

0.0

0.5

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ssur

e A

mpl

itude

(kP

a)

MeasurementsPrediction

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Equivalence Ratio

Pha

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Fig.9 Measurements Vs. prediction for dynamic pressure at (a) 95.4 cm downstream of the dump plane and (b) the dump plane. The air flow rate is 38.9 g/s, and the preheat temperature is 373 K. The one-wave mode of the combustion chamber is excited.

Acoustic wave distribution along the combustion chamber is defined as,

)(ˆ),(ˆ

),(sP

sxPjsxGm

== ω (4)

),(ˆ sxP and )(ˆ sPm denote the Laplace transform of the predicted acoustic pressure along the combustion chamber and the pressure at the dump plane, respectively. The origin, i.e. x=0, is located at the dump plane. Figure 10 shows the acoustic wave distribution at the air flow rate of 38.9 g/s, the preheat temperature of 373 K, and the equivalence ratio of 0.36. At this working condition, the dominant frequency is 702 Hz, about 4.7% below the natural one-wave mode frequency. The natural acoustic modes refer to the acoustic modes within a combustion chamber which only allows sound to propagate within but do not generate sound itself. The natural eigenfrequency for the one-wave mode is determined as 735 Hz, based on the gas temperature of 1561 K and the combustion chamber length of 1.05 m. From Fig.10, one can see that, there is a phase change of 180o across two surfaces, at x=0.445 m and x=0.993 m, respectively. This means that, pressure oscillations within this region are out of phases with those at the other parts of the combustion chamber. Here we flapped the wave form between x=0.445 m and 0.993 m from the upper plane to the lower plane. Obviously, the gain cannot be negative. However, this allows one to better see the similarity and differences between this curve and the natural one-wave mode of the combustion chamber. During self-excited combustion oscillations, the system resonant frequency differs from the natural acoustic eigenfrequency. This is somewhat similar to a second-order, positively-damped, linear oscillator, whose resonant frequency is lower than the natural frequency. The acoustic wave distribution differs from the natural one-wave mode in several aspects. Firstly, the pressure anti-nodes are no longer located at both ends of the combustion chamber, but at 0.17 m downstream of the dump plane and 0.72 m upstream of the exit orifice plate, respectively; secondly, the acoustic wave distribution is less than one wave form, which is commensurate with the system resonant frequency, which is 33 Hz below the eigenfrequency of the natural one-wave mode. Note that, although the terms like the one-wave mode and the half-wave mode of the combustion chamber are not precise when referring to self-excited combustion oscillations, for the convenience of expression, we still use them in the rest of the paper.

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Fig.10 Acoustic mode shape along the combustion chamber. The air flow rate is 38.9 g/s, the preheat temperature is 373 K, and the equivalence ratio is 0.36.

B. Acoustic Wave Distribution along the Inlet Pipe The above procedures for determination of the acoustic wave distribution are applied to the inlet pipe. Figure 11

shows the measurements Vs. prediction for dynamic pressure 8.9 cm upstream of the dump plane. The prediction is based on two pressure measurements, at 16.5 and 24.1 cm upstream of the dump plane, respectively. The air flow rate is 38.9 g/s, and the preheat temperature is 473 K. For equivalence ratios above 0.26, the dominant acoustic oscillations along the inlet pipe are associated with the excitations of the one-wave mode of the combustion chamber. At ϕ=0.24 and 0.22, the dominant acoustic oscillations are associated with the excitation of the one-and-a-


13

half wave mode of the combustion chamber. Figure 11 (a) shows the dominant frequency for pressure at the dump plane. The prediction errors are within 1.5% for the pressure amplitude and within 2o in the phases. The good match between measurements and prediction implies that the fluid dynamics along the inlet pipe can be accurately described as linear acoustic wave.

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Fig.11 Measurements Vs. prediction for dynamic pressure 8.9 cm upstream of the dump plane. The air flow rate is 38.9 g/s, and the preheat temperature is 473 K.

Figure 12 (a) shows the acoustic wave distribution at ϕ=0.38. The air flow rate is 38.9 g/s, and the preheat temperature is 473 K. At this working condition, the one-wave mode of the combustion chamber is excited at 721 Hz. The pressure amplitude at the dump plane is 6.55 kPa. As explained before, the origin, x=0, is specified at the dump plane. x=-2.5 cm and x=-34.3 cm refer to the location immediately upstream of the air swirler and the location at the inlet orifice plate, respectively. Acoustic oscillations along the inlet pipe are induced by combustor pressure oscillations. Pressure oscillations at the dump plane propagate upstream across the venturi and the air swirler into the inlet pipe, get bounced back and forth between the inlet orifice plate and the dump plane, and finally reach a standing wave pattern along the inlet pipe. Note that the inlet pipe is not a passive element during self-excited combustion instability. Acoustic oscillations along the inlet pipe actively contribute to combustion instability via variations in the instantaneous air flow rate into the combustor. From Fig.12 (a), one can see that, strong acoustic oscillations occur at both ends of the inlet pipe, and decays towards the middle point. Acoustic oscillations nearby the air swirler have no phase differences from those at the dump plane. But the phase difference keeps increasing further upstream along the inlet pipe, and becomes 180o at the orifice plate.

Figure 12 (b) shows the acoustic wave distribution at ϕ=0.24. The air flow rate is 38.9 g/s, and the preheat temperature is 473 K. At this working condition, the one-and-a-half wave mode of the combustion chamber is excited at 968 Hz. The pressure amplitude at the dump plane is 1.72 kPa. The strongest acoustic oscillations are still observed nearby the inlet orifice plate, implying that the inlet orifice plate is almost “perfectly” rigid, i.e. the acoustic velocity nearby the inlet orifice plate is nearly zero. However, different from Fig.12 (a), the pressure amplitude immediately upstream of the air swirler is no longer around a maximum, but much nearer to a minimum.


14

Since the acoustic wave distribution along the inlet pipe can be accurately described as linear acoustic wave, the mode shape at a certain frequency is determined by the geometry, sound speed, and the acoustic boundary conditions. At 721 Hz the phase lag immediately upstream of the air swirler is zero, but at 968 Hz it becomes 78o. At 721 Hz the phase lag at the inlet orifice plate is 180o, but at 968 Hz it becomes 290o. The ratio of the flow acceleration force to the pressure drop across the air swirler and the venture determines the acoustic boundary condition upstream of the air swirler.

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Fig.12 Acoustic wave distribution along the inlet pipe. (a) Acoustic wave distribution at 721 Hz and ϕ=0.38. (b) Acoustic wave distribution at 968 Hz and ϕ=0.24. The air flow rate is 38.9 g/s, and the preheat temperature is 473 K.

C. Mode Selection Depending on the air flow rate, the preheat temperature, and the equivalence ratio, the one-wave mode and the

half-wave mode of the combustion chamber have been excited, sometimes simultaneously with comparable amplitude. Figure 13 shows the dominant frequency of acoustic oscillations. The frequencies above 600 Hz are related to the excitation of the one-wave mode, and those between 300 Hz and 400 Hz are related to the half-wave mode. Excitation of the half-wave mode is favored at lower preheat temperature, lower air flow rates, and surprisingly at higher equivalence ratios. The evaporation rate is slower at lower preheat temperature. The convection time is longer at lower air flow rates. The excitation of the half-wave mode at higher equivalence ratios is somewhat counterintuitive. However, from the ICCD images of CH* chemiluminescence, one can see that the chemical reaction rate in fact, is higher at lower equivalence ratios, as indicated by the stronger chemiluminescence intensity. It is also found that, at the same air flow rate and the same preheat temperature, with decreasing equivalence ratios from 0.50 to 0.20, the bulk velocity along the inlet pipe increases by about 10%. This is because, at low equivalence ratios, the air velocity across the exit orifice plate is lower. In general, one can see that, an


15

increase in the time scales of evaporation, convection, and chemical kinetics favors the excitation of the half-wave mode, and a decrease in these time scales favors the excitation of the one-wave mode.

The time scales for different processes overlap. Although the heat release zone is compact, it is inappropriate to assume that fuel is instantly consumed when it is transported to the flame front, because of the finite rate of evaporation and fuel/air mixing. There is no doubt that the fuel vapor surrounding the droplets or droplet clusters starts to be consumed when they reach the doughnut ring. However, because of the finite evaporation rate, the unevaporated droplets and unconsumed fuel vapor will be recirculated upstream from the rear end of the PVC. Probably the heat release dynamics within the doughnut ring should be better modeled as a local well-stirred reactor (WSR) instead of a flame sheet. The heat release response of a WSR is determined by the unsteady flow at the WSR inlet, the resident time, the evaporation process, chemical kinetics, heat loss, incomplete combustion, and so on [3]. Accurate determination of the WSR volume is kind of challenging, since it varies with the working condition and probably within a pressure cycle, too. In our experiments, the pressure amplitude at the dump plane often exceeds the mean pressure drop across the dome. When the combustor pressure reaches the peak, fuel is injected into a low speed or even stagnant air stream, resulting in an equivalence ratio significantly above the mean. When the combustor pressure reaches the valley, the air flow across the air swirler reaches the maximum, resulting in an equivalence ratio significantly below the mean. When the combustor pressure varies from the peak to the valley, the fuel-rich pockets are first being pushed to the flame front along the swirling shear layer. During this process, the spatial distribution of droplets is highly dispersed, i.e. the width of the denser spray is much narrower than that of the sparse spray. Note that the flame front is always located very near the dump plane, within 1.8 cm. Because of the unsteady velocity along the swirling shear layer, the convection time for the fuel to reach the flame front should not be computed using the mean velocity. Detailed analysis of the unsteady transportation process will be reported later [32].

(a1) 22.2 g/s and 373 K (b1) 22.2 g/s and 473 K

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Fig.13 The dominant acoustic frequency at different working conditions.

D. Simultaneous Excitation of Two Acoustic Modes At certain conditions, the half-wave mode and the one-wave mode of the combustion chamber are excited

simultaneously, sometimes with comparable amplitude. Figure 14 shows the spectra of combustor pressure and CH* chemiluminescence. Here the air flow rate is 22.2 g/s, the preheat temperature is 423 K, and the equivalence ratio is 0.50. The half-wave mode and the one-wave mode are excited at 371 and 699 Hz, respectively. One can also identify the first harmonics of the half-wave mode at 743 Hz, the first harmonics of the one-wave mode at 1400 Hz, the second harmonics of the half-wave mode at 1113 Hz, and the combinatory harmonics of the half-wave and the one-wave modes at 1070 (371+699) Hz and 1441 (371x2+699) Hz. The one-wave mode occurs at 44 Hz below the first harmonics of the half-wave mode. From the ICCD images of CH* chemiluminescence, one can see that major heat release occurs around the droplets within the doughnut ring. Velocity fluctuations along the swirling shear layer quasi-periodically alter the flow direction around the droplets and droplet clusters, resulting in unsteady evaporation and heat release at twice the frequency of acoustic oscillations.

Fig.14 Spectra of combustion pressure and CH* chemiluminescence. The air flow rate is 22.2 g/s, the preheat temperature is 423 K, and the equivalence ratio is 0.50. Both the one-wave mode and the half-wave mode of the combustion chamber are excited.

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Fig.15 Time-varying spectra of combustor pressure and CH* chemiluminescence. The air flow rate is 22.2 g/s, the preheat temperature is 423 K, and the equivalence ratio is 0.50.

Simultaneous excitation of the half-wave mode and the one-wave mode is inherently unsteady and unstable. Figure 15 shows the spectra of combustion pressure and CH* chemiluminescence, which are time-varying. The amplitude of the two modes is constantly changing but with opposite trends. Figure 16 shows the time traces of acoustic pressure and CH* chemiluminescence. Time traces of the half-wave mode are obtained by filtering data using a 4th-order Butterworth bandpass filter within [330 410] Hz, and those of the one-wave mode are obtained using a 4th-order Butterworth bandpass filter within [630 710] Hz. When the energy of the half-wave mode starts to decay, the energy of the one-wave mode starts to grow, and vice versa. The transient period for the decay or growth of the half-wave mode is always commensurate with that of the one-wave mode. The sum of acoustic energy for the two modes is far away from being constant, thus the time-varying pressure amplitude is not caused by acoustic energy transfer. From Fig.16, one can see that, the growth or decay of acoustic oscillations always follows that of CH* chemiluminescence. In-phase heat release perturbations are the driver for thermoacoustic oscillations. The opposite trend in pressure amplitude among the two modes is rooted in the nonlinear response of the reacting swirling shear layer to inlet velocity perturbations. Firstly, the unsteady air flow along the reacting swirling shear layer modifies the flame kinematics, the flame surface area, and consequently the instantaneous heat release rate; secondly, the unsteady air flow introduces equivalence ratio perturbations, due to the differential acoustic impedance between the fuel and air lines, which modifies the laminar flame speed, the flame kinematics, and consequently the instantaneous heat release rate; thirdly, the unsteady air fluctuations enhance evaporation; fourthly, the acoustic pressure and temperature directly affect the chemical reaction rate via variations in the reactant density and flame temperature. When the pressure amplitude of one mode increases, say caused by an external or background disturbance, the unsteady air flow rate associated with this mode increases, resulting in larger heat release, and consequently stronger pressure pulsations. This process repeats. When the unsteady air flow and the reacting swirling shear layer are increasingly dominated by one mode, their response to the other mode decreases, resulting in increasingly weaker oscillations of the other mode. Because of the ubiquity of broadband background and


18

external disturbances, quasi-steady oscillations can only be maintained for short durations, say between 9.65s and 9.75 s and between 9.85 s and 9.95 s. The transient period is not constant. It is shorter if acoustic oscillations start to grow from relatively large amplitude, say from 9.80 s to 9.85 s. A much longer transient period is required if an acoustic mode gains energy from small amplitude, say from 10.05 s to 10.30 s. The interactions between the two modes very much depend on the working conditions. If the working condition only slightly favors the excitation of one mode, this mode will remain in small-amplitude oscillations, and can only occasionally diminish the oscillations of the other mode. Such an example is shown in Fig.17, where the equivalence ratio is 0.49.

Fig.16 Time traces of the half-wave mode (blue dashed curve) and the one-wave mode (green solid curve). The air flow rate is 22.2 g/s, the preheat temperature is 423 K, and the equivalence ratio is 0.50.

Fig.17 Pressure of the half-wave mode (blue dashed curve) and the one-wave mode (green solid curve). The air flow rate is 22.2 g/s, the preheat temperature is 423 K, and the equivalence ratio is 0.49.

E. Unsteadiness of Thermoacoustic Oscillations Unsteadiness is an intrinsic feature of thermoacoustic oscillations, which is true even if only one acoustic mode

is excited. Combustion instability is of nonlinear, limit-cycle, oscillatory behavior. The equilibrium trajectory is a 2D closed orbit in the phase plane. Whenever the state deviates from the equilibrium orbit, say caused by external


19

and background disturbances, the trajectory will no longer be closed. Because of the ubiquity of external and background disturbances, the limit-cycle oscillator is being excited all the time. It is worthwhile to point out that, the broadband background heat release perturbations are usually of low-dimension chaotic behavior, rather than being purely random. Figure 18 shows the phase portraits of pressure in a series of experiments. The air flow rate is 38.9 g/s, the preheat temperature is 373 K, and the equivalence ratio is decreased from 0.32 to 0.24. At these working conditions, the one-wave mode of the combustion chamber is excited. The amplitude of the half-wave-mode is less than 3% of the one-wave mode, thus its interference with the one-wave mode can be neglected. This allows one to better examine the effects of external and background perturbations on the system dynamics. To improve the signal-to-noise ratio, pressure is filtered using a 4th-order Butterworth bandpass filter centered at the resonant frequency with a passband of 60 Hz. Each point in Fig.18 represents a state, i.e. [p(n) p(n+4)]. p(n) refers to the nth sample of the filtered pressure. In the case of strong thermoacoustic oscillations, i.e. when heat release and pressure are strongly coupled, external and background disturbances have few opportunities to cause large deviations from the equilibrium trajectory. In such cases, the state is always confined within a small neighborhood around the equilibrium trajectory, as shown in Fig.18 (a). When the pressure amplitude or the positive feedback between pressure and heat release becomes weaker, the system is increasingly vulnerable to external and background disturbances. In such cases, the state roves over a larger neighborhood around the equilibrium trajectory, as shown in Figs.18 (b) and (c). When the pressure amplitude or the positive feedback between pressure and heat release decreases below certain thresholds, the equilibrium trajectory degenerates into a single point in the phase plane. For very small pressure amplitude, the system can be modeled as a closed-loop, slightly-damped, linear oscillator, which is still being excited by various disturbances. The damping ratio of this closed-loop linear oscillator has been used as an indicator for the proximity to the onset of combustion instability [33].

(a) ϕ=0.32 (b) ϕ=0.28

(c) ϕ=0.26 (d) ϕ=0.24

Fig.18 Phase portraits of pressure at different equivalence ratios. The air flow rate is 38.9 g/s, and the preheat temperature is 373 K. Shown here are 100,000 states.


20

FFT usually provides good information regarding the amplitude and frequencies of thermoacoustic oscillations. However, FFT is not suitable for analyzing fast transient events. Since combustion instability in turbulent environment is inherently unsteady, statistical analyses, including the probability density function (pdf), can provide better characterization of the system dynamics than FFT. Figure 19 shows the histograms of the pressure amplitude and the time taps within one pressure cycle. The sampling frequency is 10 kHz, thus each time tap corresponds to 0.1 ms. The number of taps within one pressure cycle is a good measure of the oscillation period or the frequency. The amplitude refers to the peak pressure within one oscillation cycle. Firstly, we identify three consecutive samples with the largest positive value and develop a quadratic curve-fitting polynomial, and then we identify three other consecutive samples with the largest negative value and develop another quadratic curve-fitting polynomial; secondly, we figure out the maximum and the minimum pressure based on the two quadratic polynomials. The peak pressure within a pressure cycle is determined as the average of the magnitude of the two extrema; thirdly, from the quadratic polynomials, we determine the instants at which pressure reaches the peak and the valley. The oscillation period is the interval between two consecutive pressure peaks or valleys. The usage of a quadratic polynomial to fit the samples around an extremum, is based on the Taylor expansion of the cosine function, at a small phaseα ,

)(211cos 42 ααα O+−≈ (5)

The probability density function shown in Fig.19 is based on pressure acquired within 20 s, which contains 12,945 pressure cycles at ϕ=0.22 and 13,732 pressure cycles at ϕ=0.32. With decreasing equivalence ratios from 0.32 to 0.26, combustion oscillations become weaker, and the amplitude varies from 4 to 10 kPa at ϕ=0.28, and from 1 to 8 kPa at ϕ=0.26. However, for the equivalence ratio at 0.32, 0.26, and 0.24, the period varies only within 0.2 time taps, i.e. less than 1.4% of the mean period. The pdf for the amplitude has a roughly flat hat, and is skewed except at ϕ=0.32. However, the pdf for the period is always symmetric and can be well approximated as a Gaussian distribution. At ϕ=0.24, the pdf for the amplitude is narrower than that at ϕ=0.28 and 0.26, but the pdf for the period is very much broadened. There seems to be no fixed relationship between the pressure amplitude and the period. In fact, the period has been observed to increase with the amplitude, and decreases with the amplitude, too. It is known that, for a limit-cycle oscillator, the period for one revolution along the equilibrium trajectory is a function of the oscillation amplitude [34]. But for combustion oscillations in turbulent environment, the state almost never follows the equilibrium trajectory. The thermoacousic oscillator is always being accelerated or decelerated by external and background excitations. The fact that the unsteadiness in the period remains small, despite large variations in the pressure amplitude, suggests the presence of strong forces which quickly pull the state into a small neighborhood around the equilibrium trajectory. At ϕ=0.24, the pressure period varies within a much broader range than that at ϕ=0.26, 0.28, and 0.32. This is because, for oscillations around an equilibrium point (instead of a 2D equilibrium trajectory), the system dynamics can be modeled as linear in the presence of small deviations from the equilibrium point, but nonlinear in the presence of large deviations from the equilibrium points. The oscillation period is a function of the nonlinearity and the pressure amplitude [34]. Note that since the pressure amplitude and the period is constantly time-varying, a meaningful phase can only be determined for a short period. Lieuwen reported that the phase drifted in an irregular manner during combustion instability in a lean premixed combustor [35]. The phase is a key variable for a class of empirical phase-shift controllers [36][37][38], which are designed to attenuate combustion instability. Yi et al. [36] developed an observer for online estimation of the dominant frequencies, which usually can be figured within one pressure cycle and a half in the case of a single unstable mode. The identification of the frequency using a short data set is meaningful, since the variations in the period remain small, within 1.4% around the mean in the case of strong thermoacoustic oscillations. In Ref.36, a special bandpass filter is used for delay-free reconstruction of the dominant mode, which has unit gain within a certain frequency range around the unstable frequency. This allows the reconstructed signal to have the same amplitude even if the frequency fluctuates around the mean.


21

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I. Conclusion • The present liquid-fueled, swirl-stabilized, LDI combustor is featured with a venturi downstream of a 60o axial

swirler. The flame structure is very sensitive to the equivalence ratio. Heat release occurs within a “doughnut” ring, which is located outside the precessing vortex core. With reductions in the equivalence ratio, the ring diameter decreases, and converges to the axis at the equivalence ratio of 0.34. With further decreases in the equivalence ratio, the heat release zone becomes shorter, elongated, and tilted off the axis. The remarkable variations in the flame structure can be attributed to the change in the cone angle of the swirling shear layer. Stronger gas expansion promotes vortex breakdown and gas recirculation, which helps to open up the cone angle and improves fuel/air entrainment and mixing.

• The major heat release zone is always located within 18 mm downstream of the dump plane, which can be attributed to the very stable and compact precessing vortex core. With decreasing equivalence ratios towards lean blowout, the heat release zone moves closer to the dump plane, and the chemiluminescence intensity becomes stronger, but the flame is purely blue.

• During self-excited combustion instability, there is no formation and shedding of large vortices, even when the pressure amplitude exceeds the mean pressure drop across the dome. This can be attributed to the fast growth and decay rates of the swirling shear dynamics.

• Acoustic wave distribution along the combustion chamber and the inlet pipe is determined from pressure measurements at two locations. A globally representative temperature and sound speed are assumed along the combustion chamber. The acoustic wave distribution along the combustion chamber differs from the natural one-wave mode, in both the wave length and the locations of the pressure anti-nodes. Caution should be exercised when developing low-order modes based on Galerkin projection of the natural acoustic modes. The natural acoustic modes refer to the acoustic modes in a combustion chamber which only allows sound to propagate within but does not generate sound.

• Depending on the working conditions, the half-wave mode and the one-wave mode of the combustion chamber can be excited, sometimes simultaneously. This can be attributed to the simultaneous occurrence of several physicochemical processes and the overlap in the time scales. The simultaneous excitation of the half-wave mode and the one-wave mode is inherently unsteady and unstable. Small external and background disturbances can destroy the subtle balance between the two modes, causing one mode to grow and the other to decay. The time-varying amplitude of the two modes is not associated with acoustic energy transfer, but is caused by the nonlinear response of the reacting swirling shear layer to inlet disturbances.

• Unsteadiness is an intrinsic feature of thermoacoustic oscillations in turbulent environment. A limit-cycle thermoacoustic oscillator is constantly excited by external and background disturbances. In the case of a large perturbation, it may take a number of revolutions for the state to return to a small neighborhood around the equilibrium trajectory, thus resulting in time-varying amplitude and frequencies. The limit-cycle thermoacoustic oscillator becomes increasingly vulnerable to external and background disturbances, when the deterministic, acoustics-induced heat release oscillations become weaker.

• The probability density function for the pressure amplitude has a top flat hat, and is skewed except at very strong thermoacoustic oscillations. But the probability density function for the oscillation period can be approximated as Gaussian. The small variations in the period in the presence of large variations in the pressure amplitude,


22

suggest the presence of large forces which quickly draw the state towards the equilibrium trajectory. The variations in the period are much larger for linear oscillations than for limit-cycle oscillations.

Acknowledgments Support from NASA John H. Glenn Research Center at Lewis Field under grant NNX07C98A, “Active

Combustion Control for Low-Emission Combustors,” is gratefully acknowledged. Clarence Chang is the project manager. This project is also supported by the U.S. Air Force Office of Scientific Research under grant FA9550-07-1-0451, “Advanced Thermally Stable Coal-Based Jet Fuels.” The authors would like to thank Brian Jones for assistance with Abel Deconvolution.

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