Amplitude modulated instability in reactants plenum of a rotating detonation combustor

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Amplitude modulated instability in reactants plenum of a rotating detonation combustor Vijay Anand*, Andrew St. George, Ephraim Gutmark Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, OH 45221, USA

article info

abstract

Article history:

The pronounced interest in rotating detonation combustors (RDC) in recent years has

Received 19 December 2016

witnessed the investigation of multiple facets of the combustor, like reactants, injection

Received in revised form

schemes and combustor geometry. The issue of instabilities in RDCs is a nascent field, and

30 March 2017

requires both the identification, and the subsequent explanation of different instability

Accepted 31 March 2017

mechanisms. In particular, we are concerned with the low frequency instability exhibited

Available online xxx

by the detonation wave. This is attributed to two different types of low frequency instabilitiesdamplitude and frequency modulateddthat are discovered in the air plenum of an

Keywords:

RDC, and subsequently discussed. The occurrence of these instabilities is observed to

Low frequency combustion insta-

depend on the fuel injection scheme used and the air flow rates through the combustor.

bility

The amplitude modulated instability in the air inlet is spatially varying, and rotates in a

Rotating detonation engine

direction opposite to the direction of the detonation wave. At higher air flow rates, and

RDE

thus higher pressure ratios across the air injection, this instability disappears. A pre-

LFI

liminary hypothesis is proposed to explain this amplitude modulation.

Amplitude modulation

© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Coupling

Introduction Deflagration, the more commonly observed and studied combustion mechanism, is characterized by subsonic combustion waves that always have a miniscule pressure drop across them. Detonation, contrastingly, is distinguished by sonic to greater-than-sonic wave speeds, and is structured with a shock wave-chemical reaction zone coupling, with each entity sustaining the other through a feedback loop [1]. This usually produces a pressure gain of 13e55 in gaseous mixtures [2], a characteristic that has been seen as highly desirable in recent years due to a prospective increase in fuel efficiency since more work can be attained for a given supplied energy, in comparison to deflagration. Hence,

detonation combustors are being studied with increasing frequency and effort, to orient the future of propulsion and power towards the detonation mechanism. Rotating detonation combustors (RDCs), unlike pulsed detonation combustors (PDCs), are not valved, and courtesy of the rotating detonation spinning azimuthally in the kiloHertz regime, the exhaust flow field is quasi-steady. Hence, recent research efforts worldwide have almost exclusively concentrated on RDCs. Research into RDCs has intensified only in recent times, despite the combustor itself having been conceived and built in the 1960s [3]. Multiple factors like the reactants type, combustor geometry, thrust, oxidizer and fuel flow rates, and the rotating detonation wave speed at these conditions have been studied. The next obvious step is to identify and understand the various instabilities in RDCs. The

* Corresponding author. E-mail address: [email protected] (V. Anand). http://dx.doi.org/10.1016/j.ijhydene.2017.03.218 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Anand V, et al., Amplitude modulated instability in reactants plenum of a rotating detonation combustor, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.03.218

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Nomenclature _a m F t lfo/dfo wcb wap dcb lcb wfo Ws PR Dt DT

mass flow rate of air (kg/s) equivalence ratio time (s) length of the fuel injector holes/the diameter of the holes width of the combustor channel (mm) width of the air inlet slot (mm) radius of the combustor (mm) axial length of the combustor (mm) width of fuel injection slot with equivalent area (mm) average wave speed (km/s) ratio of time-averaged static pressure in air plenum and combustor during hot-fire detonation wave arrival time difference between two sectors (s) low frequency oscillation arrival time difference between two sectors (s)

field of combustion instabilities in RDCs is extremely nascent in comparison to the monumental work done in addressing the instabilities in gas-turbine combustors and rocket engines. Few studies exist at present that have addressed RDC instabilities to any appreciable degree [4e8]. The authors have addressed the four prominently occurring instabilities in an RDC of which low frequency instabilities (LFI) are a part of, and subsequently speculated on their mechanism in Ref. [4]. We use this abbreviation (of LFI) to be in continuation with the terminology used in gas-turbine combustors and rocket engines. Traditionally, instabilities having a frequency range of 1e500 Hz are termed LFI, while those between 500 and 1000 Hz are termed intermediate frequency instabilities (IFI, “buzzing” in rocket engines), and those greater than 1000 Hz are named high frequency instabilities (HFI) [9]. Though the frequency of the instabilities is more often than not a function of the geometry of the combustor, historically speaking, it has been beneficial to group the instabilities this way since the underlying mechanism is generally relatable for a given frequency range. For instance, resonant acoustic interaction with the supply feeds in both the gas-turbine combustors and rocket engines manifest themselves in the LFI range [9,10]. LFI in an RDC seems to be almost ubiquitous. A brief analysis of the pressure-time traces published by the different RDC facilities worldwide gives concrete evidence of the overarching existence of this instability [4,6,11e20]. However, most studies have not made an effort to address either the existence, or the mechanism behind LFI. Considering the crippling effects of LFI in rocket engines, supersonic inlets and hypersonic vehicles owing to their tendency to couple with the natural resonant frequency of the structure [21,22], it is imperative to acknowledge and treat LFI as we move forward with RDC research. Experimental and numerical studies have shown that the detonation wave tends to significantly alter the plenum dynamics owing to the high peak pressures [23e26]. A

‘trailing shock wave’ that is attached to the rotating detonation wave tends to travel into the reactants plenum, thereby significantly altering the dynamics [23e25]. Schwer and Kailasanath [23] note the existence of a subsequent reflected wave (from the back of the plenum) that is spawned when this trailing shock wave reaches the base of the reactants plenum. However, in their two-dimensional numerical analogue of RDC-plenum system with orificetype injectors, this reflected shock wave appears to have no effect on the detonation wave dynamics. Two different hypotheses were put forth in the past to explain LFI (previously characterized by periodic “waxing and waning” of detonation peak pressure strength) in an RDC. The authors [5] and Liu et al. [6] proposed that the LFI in the combustor could be due to periodic strengthening and weakening of the detonation wave, thereby periodically altering the reactants flow, and thus establishing feedback. However, both studies employed only one pressure sensor in the combustor and, as such, have drawbacks in accurately capturing the instability dynamics, thus corrupting the explanation. Additionally, the conjecture was not supplemented with any evidence. A sequel of the previously described study by the authors employed extensive pressure sensing in the combustor, air and fuel plenum. The sensors were distributed 120
from each other to obtain a comprehensive spatial view of the instability. A “locked-in” [27], azimuthally simultaneous, low frequency mode was observed at 235 Hz in the air inlet, at all air and fuel flow rates tested, which was then attributed to a probable Helmholtz resonation-type coupling between the air plenum and the combustor [15]. The locked-in oscillations in the air inlet manifest as a low frequency instability in the combustor, albeit at a broader frequency range [4]. Such a coupling has indeed been observed in gas-turbine combustors with a choked exit nozzle [28]. This study, however, deals with a notably different LFI in an RDC that is not locked-in across different flow rates, and is characterized by amplitude modulated oscillations in the air plenum. And, similar to the locked-in LFI (frequency modulated oscillations in the air inlet), the LFI treated in the current paper induces instabilities in the combustor as well. In subsequent sections, we present the experimental methodology and the concomitant results to support the claim that there are two distinct LFIs in RDCs. Then, we exclusively consider the amplitude modulated LFI, and discuss its characteristics.

Experimental methodology Hydrogen-air mixtures are used to operate the RDC at air flow _ a) of 0.2, 0.3 and 0.4 kg/s at different equivalence ratios rates (m (F). The supply temperature for the air is about 288 K. Rotating detonation wave onset and propagation in an RDC is not direct because it involves a complex combination of plenum coupling and deflagration-to-detonation transition (DDT) mechanism [29]. This produces a finite ‘transient/onset time’ which precedes steady-state RDC operation. In our previous study, we had established that this transient, onset time in RDCs is highly dependent on the flow rates and the presence of a choked exit [30]. For 0.4 kg/s (without nozzledas is the

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case here), the transient onset time was observed to be not more than 207 ms [30]. Bykovskii et al. [29] note onset times of 4e80 ms using varied ignition methods. Peng et al. [20] observed transient operation for roughly 7 ms after ignition. Hence, it is contended that the total testing time duration of 0.35 s used in this study is significantly longer than the detonation onset time, and hence can be concluded to be representative of steady-state RDC behavior. Note that the tested RDC attains thermal equilibrium far later (more than 3 s) after it attain operational steady-state [31]. The air and fuel flow rates are controlled by a closed-loop system of nitrogen-driven pilot regulators and a set of Flowmaxx sonic nozzles. The equivalence ratios discussed henceforth are global values estimated from the stagnation pressure and temperature upstream of the choked sonic nozzle. Three-dimensional numerical cold-flow simulations of the current type of mixing scheme (slot-orifice) have shown notable variations in the local equivalence ratio (varies from 0dairdto about 2.5) depending on the flow rates and geometry [32]. Despite the local equivalence ratio fluctuation, the overall macroscopic dynamics of RDC seems to follow clear trends, as evidenced by operating maps and detonation wave speeds [33e36]. Hence, for the purposes of the present study on RDC plenum coupling, we assume these local variations in reactivity inside the RDC annulus to have a negligible effect on LFI. Norgren VP50 proportional control valves (pilot) are linked to Norgren pilot-operated regulators to isolate electrical components from the primary fuel supply. GE Unik 5000 sensors are linked to the choked-flow nozzle assemblies to monitor air and fuel flow rates. Fuel flow is administered to the rig through a pneumatically-actuated BiTorq isolation ball valve located just upstream of the fuel plenum, which allows fuel flow rates to stabilize within 2 s of fuel introduction. Testing is done for a time period of about 0.35 s. The time-averaged stagnation pressure in the air plenum for these flow rates are about 2.4 bar, 3.3 bar and 3.8 bar respectively, during hot-fire operation. The static pressure evolution inside the air plenum, fuel plenum and combustor before and after ignition is shown in Fig. 1. Our prior experience with RDCs has shown that high speed flushmounted pressure sensors tend to permanently break if exposed to the RDC environment for more than a second [34]. This is usually circumvented by using stand-off tubes that attain the accurate frequency information, but this induces inaccuracies in pressure data [37]. Hence, longer operation times are avoided here since our goal in this study is to accurately capture the RDC dynamics. There is no zero/offset error in the sensors because this is removed through software before the hot-fire run. Thus, the systematic error in flow measurements is only from the scaling error of the instruments. The relative errors in the static pressure sensor and thermocouple are known from the associated instrument specifications. This is used to attain the uncertainty in the pressure and thermocouple sensors used in the reactants delivery, and is found to be ±0.069 bar and ±1 K (at the maximum output), respectively. This causes a ‘propagation of uncertainty’ from the sensors to the air and fuel flow rates, which in turn induces some error in the estimated equivalence ratios. The linearized error analysis of these uncertainties result in an estimate of 2.1% error in the air mass

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Fig. 1 e Static pressure variation before and after RDC ignition, obtained from capillary tube average pressure sensors.

flow rate and 2.8% in fuel mass flow rate, which in turn results in a maximum error of 3.4% (seen for the lowest flow rates) in equivalence ratio. The error in the discussed fundamental frequency is about 2.86 Hz (frequency resolution ¼ sampling rate/number of obtained samples). The standard deviation in the lap-to-lap detonation wave speed (Ws) acquired through a time of flight algorithm is not more than 5% of the average speed, for a given test point. A front and side-view schematic of the RDC and the associated instrumentation is shown in Fig. 2a. Air and fuel are supplied from two separate plenums (blue and green, respectively) to attain non-premixed mixing. Air is injected radially inwards through a circumferential slot and fuel is injected axially through a circumferentially distributed array of fuel orifices. This slot-orifice mixing scheme is visualized in Fig. 2b. The reactants mix through this orthogonal slot-orifice injection scheme and are ignited by a pre-detonator (grey tube). The pre-detonator tube is fed with opposing jets of ethylene and oxygen that are supplied at the headend through Parker solenoid valves. The stagnation pressure upstream of the valves is about 8 bar for both the reactants. Note that this stagnation pressure dictates the global equivalence ratio of the mixture inside the pre-detonator. However, our prior analysis of the initiation characteristics of the pre-detonator suggests that the energy input by this method of RDC ignition is about an order of magnitude smaller than the energy requirement to attain direct detonation initiation inside the RDC [38]. RDC ignition is thus possible across a wide range of pre-detonator equivalence ratios [38]. The only requirement is to have a detonation wave deposit into the RDC channel from the pre-detonator. This initial detonation wave is theorized to cause a complex deflagration-to-detonation transition (DDT) mechanism [38] that finally results in the deposition of a detonation wave into the RDC channel (red area). A spark plug that is located in between the two solenoid valves is actuated to ignite the pre-detonator mixture, which eventually evolves into a detonation wave inside the pre-detonator. The presence

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Fig. 2 e (a) RDC e front view, sectional cut and magnified RDC annulus (clockwise), (b) magnified view of the injection scheme, and (c) generic front view schematic of the fuel orifices layout in the two fuel schemes (Note: the pattern is repeated circumferentially).

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of a detonation wave inside the pre-detonator tube is verified using two ionization probes (not shown here) that accurately capture the speed of the supersonic wave. For a detailed description of the pre-detonator system, we direct the attention of the readers to our previous work [38]. The air injection slot area is maintained constant throughout the testing, whereas two different fuel injection schemes having the same total injection area are used. The fuel scheme is altered by using the required “fuel plates”. These are essentially stainless steel plates with the required fuel orifices pattern. The first fuel injection scheme (which we will call FP-I) has one row of circumferentially distributed orifices (diameter of orifice, dfo z 1 mm) whereas the second scheme (FP-II) has three rows of orifices (dfo z 0.75 mm). Thus, the total number of holes is higher for FP-II, but the area of the individual holes are smaller than that of FP-I. The length-todiameter ratio (lfo/dfo) of FP-I is 12.8 for each individual orifice, whereas the lfo/dfo of FP-II is about 17. A generic schematic of the two schemes is given in Fig. 2c. Both the schemes have orifices right at the outer wall of the RDC, i.e. by the opening of the air inlet. However, due to the three-rowed arrangement of FP-II, the fuel orifices extend to about 60% of the channel width, wcb. For FP-I, the orifices extends only to about 25% of the channel width. Other dimensions of interest of the RDC are given in Table 1. A detailed overview of the RDC geometry and the facility is presented in our previous publications [34,39]. The RDC under study has multiple instrumentation ports. There are three stations (I, II and III in Fig. 2a) that are sectorial and are distributed 120
from each other, with each station having four axially distributed rows of instrumentation. The remainder of the paper will resort to this nomenclature of color-coding (blue e station I, black e station II, red e station III) to represent the three sectors/stations in which the pressure sensors are placed. Note that a given station comprises of an axial array of four rows of instrumentation ports in the combustor and one port in the air inlet that is offset from the combustor ports by 20
. The small angular offset between the air inlet sensor and the combustor sensor is necessary to have pressure sensors in both the locations, which otherwise would not be possible since the finite lengths of the two pressure sensors would spatially block each other. This lets us monitor the RDC dynamics spatially by dividing the combustor into three sectors. Roman numerals are used to identify a given station and the usual numeration is used to identify a given axial port location. For instance, I-1 signifies an instrumentation port in the first row

Table 1 e RDC geometry. Part

Geometry measured

Dimension

Fuel scheme-I (FP-I) Fuel scheme-II (FP-II) Air injection

Length/diameter (lfo/dfo) Length/diameter (lfo/dfo) Slot width (wap) Total slot area Width (wcb) Inner diameter Outer diameter (dcb) Annulus area Length (lcb)

12.8 17 1.02 mm 490 mm2 7.5 mm 139 mm 154 mm 760 mm2 125 mm

Combustor channel

5

of the first station. A total of nine PCB piezoelectric pressure sensors are useddthree in the air inlet (blue circles/tabs in Fig. 2a), three in the combustor (red circles/tabs in row 1 of all three stations in Fig. 2a) and three in the base of the fuel plenum (green circles/tabs in Fig. 2a). The fuel plenum sensors are on the same angular plane as the combustor sensors, but their radial location (measured from the RDC axis) is offset inwards from the RDC annulus by about 6 cm. It is to be noted that data from the fuel plenum sensors are not discussed in the current paper. There are no piezoelectric sensors in rows 2, 3 and 4 of any of the stations. The data from the PCB piezoelectric pressure sensors are acquired at 1 MHz. In addition to the high-speed pressure sensors, three lowspeed capillary tube averaged pressure sensors (CTAP) are used at a 1 kHz acquisition rate. The CTAP arrangement consists of a long plastic tubing (about 120 cm) with an inner diameter of approximately 3 mm, at the end of which a low speed static gauge pressure sensor is attached. While the common notion is that CTAP sensors give a time-averaged value of static pressure, and have been extensively used in RDCs, recent discussions have questioned this assumption [40,41]. Irrespective, we resort to this sensor setup to attain the nominal static pressure values while protecting the sensors from the hostile RDC environment. One CTAP sensor is placed in the air plenum, fuel plenum and the combustor, each (not shown in the figure), to ascertain the average nominal pressures in the air and fuel plenum, and the combustor, respectively. The three sensors in the combustor (row 1 e stations I, II and III) are located 1.9 cm away from the RDC headwall. The air inlet sensors are at a location that is 2.54 cm from the combustor outer wall. The three air inlet sensors (blue circles/tabs in Fig. 2a) obtain the plenum dynamics during rotating detonation in the combustor. Both numerical and experimental studies have shown that the rotating detonation wave causes considerable pressure feedback into the air inlet and a concomitant momentary localized occlusion due to its peak detonation pressure being higher than the plenum feed pressures [15,23,26,42,43]. Schlieren imaging of a two-dimensional analogue of the slotorifice RDC mixing scheme by Bedick et al. [25] revealed considerable shock wave leakage from the detonation wave (using a timed pre-detonator to simulate rotating detonations) into both the air and fuel plenum. However, they note that rather strong shock waves are inducted into the air inlet owing to its higher injection area which in turn heightens the exposure to the detonation wave. In contrast, only relatively weak “pressure waves” leaked into the fuel plenum owing to the probable viscous damping caused by the small orifice size. Fotia et al. also note “pressure waves” that propagate into the fuel plenum due to detonation wave propagation in their two-dimensional RDC analogue [24]. Hence, the common consensus seems to be that the slotted air inlet is prone to shock wave leakage from the detonation wave during each of its lap, whereas the multi-holed fuel injection is only prone to a weak pressure wave, which is presumably just an acoustic wave. We have shown that this localized feedback into the slotted air inlet, through shock wave leakage, can be used to effectively capture (using piezoelectric sensors) not just the air plenum dynamics but also the detonation wave speed [15].

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Results and discussion Two types of low frequency instabilities (LFI) Rotating detonation combustor operation in the current facility exhibits low frequency instabilities at most operating conditions, as described in Refs. [4,5]. We had statistically analyzed LFI (in terms of the number of detonation laps occurring per “packet” of waxing and waning) using only a single pressure sensor in the RDC annulus for two air flow rates spanning a range of equivalence ratios for three different fuel injection schemes, and found there was an absence of a clear trend [5]. This motivated another study which used pressure sensors in the air inlet and the combustor [15]. A singular mechanism of LFI was conjectured to produce the low frequency oscillations (150e500 Hz) seen in the combustor and the air plenum. However, upon a closer inspection, we observe two distinct physical mechanisms in the air inlet that seemingly produce unstable detonation propagation inside the combustor at the low frequency regime. Depending on the fuel injection scheme used and the air flow rates, the oscillation in the air inlet is either amplitude modulation (AM), or frequency modulated (FM), or both occurring in tandem. FM LFI is only observed when FP-II (lfo/ dfo ¼ 17) is used. AM LFI is observed only at the air flow rate of 0.4 kg/s and does not appear at 0.2 kg/s and 0.3 kg/s. To distinguish the two LFIs, it is imperative to study the pressure traces in both the combustor and the air inlet for the geometric case that sustains both (FP-II). Fig. 3a is an arbitrary pressure trace from the combustor that shows highly unstable detonation wave propagation that varies sinusoidally in strength as the detonation wave moves circumferentially through the three instrumented sectors of the RDC, implying a continual variation in the rotating detonation wave strength. Fig. 3b gives a pressure time series when there only FM oscillation in the air inlet. The carrier signal is the leaked shock wave from the detonation propagation through the three sectors. This carrier pressure signal, however, can be seen to have a sustained base pressure modulation (thereby, being frequency modulated) that is azimuthally simultaneous. That is, all three sectors are prone to sinusoidal increases and decreases in the base pressure. We will call this ‘spatial uniformity’ in the air inlet. This case can be contrasted with Fig. 3c, which shows pressure traces depicting very high spatial non-uniformity, despite also exhibiting FM oscillations (once again at 235 Hz). Here, ‘spatial non-uniformity’ is defined as the phenomenon where for a given arbitrary temporal window, a particular sector of the RDC exhibits notably higher peak pressures (of subsequent detonation laps) than the other sectors. The alternating sectoral strength is depicted in Fig. 3c, which shows stronger subsequent waves in station I (blue), followed by station III (red), and finally station II (black). This is an amplitude modulation (AM) of the carrier shock wave pressure signal. In our previous work [15], the base pressure oscillation was discovered to be locked-in at 235 Hz in the air inlet (the locked-in FM instability is an additive modulation, and thus, both the low and the high frequency signals can be easily observed in the FFT plot in Fig. 3d and the spectrogram in Fig. 3e).

In Table 2, the fuel injection schemes used, the air flow rates and equivalence ratios tested for each of the scheme, and the corresponding presence/absence of the two LFI typesd(1) sectoral spatial non-uniformity of peak detonation feedback pressures in the air inlet (AM), and (2) locked-in base pressure oscillation in the air inlet at 235 Hz (FM)dis indicated through a binary “yes/no” (Y/N). This dual classification is arrived at qualitatively by visually studying the complete pressure trace for a given operating point. Points which exhibit spatial uniformity are marked “no”, whereas those operating points that do not exhibit any uniformity are marked “yes”. As seen in Fig. 3b and c, this qualitative analysis is a rather efficient process of delineation since AM LFI exhibits markedly different, visually distinct pressure traces characterized by notable variation in peak pressure strength among the three RDC sectors. We are left with two important details from this tabulation. First, the locked-in base pressure FM oscillation is non-existent when the fuel injection scheme is FP-I irrespective of the air flow rate or equivalence ratio. Second, the spatial non-uniformity instability exists for both FP-I and FP-II for 0.2 kg/s and 0.3 kg/s, but vanishes for 0.4 kg/s (higher pressure ratio of 3.2 across the air plenum and the combustor, as obtained from CTAP sensors). Thus, it is apparent that the presence of FM LFI in the air inlet is a function of the fuel injection scheme used, in some way, and the AM LFI is dependent on the pressure ratio across the air injection. The remainder of the analyses is on the test cases utilizing FP-I, so that only AM LFI is taken into account. FM LFI and its propensity to occur only for FP- II is to be dealt with in a separate paper. Fig. 4aec give the magnified pressure-time traces (it can be seen that the base pressure does not oscillate) from the three air inlet pressure sensors for an operating _ a ¼ 0.2 kg/s and F ¼ 1.0. Note that the three images point at m are continuous in time, i.e. Fig. 4a ends at t ¼ 0.302 s and Fig. 4b starts at t ¼ 0.302 s, and so on for Fig. 4c. The facet to observe here is the direction of rotation of the detonation wave. The rotating detonation wave (as observed by the shock wave leaked into air inlet) moves in the counter-clockwise direction (black / red / blue/station II / station III / station I) in the represented pressure traces. The spatially varying AM oscillation, however, exhibits the reverse order (blue / red / black/station II / station III / station I). To clarify, the sinusoidal amplitude modulated variation is exhibiting a clockwise rotating motion. This kind of behaviordthe spatially varying sinusoidal AM instability in the air inlet moving in the opposite direction to the rotating detonation wavedis witnessed whenever there is AM LFI. It is thus apparent that AM LFI is characterized by the spatially varying detonation strength is not caused due to a stationary phenomenon in the air inlet or the combustor, but rather, a lowspeed rotatory event in the air inlet. Since it is, in essence, a low frequency modulation of the amplitude of the high frequency carrier signal, it does not appear as a distinct frequency peak in FFT plots in the frequency ranges of interest (100e500 Hz). Instead, AM LFI appears as a frequency peak offset from the carrier signal peak frequency (red arrow in Fig. 4d). Such an offset frequency band is not observable in the spectrogram (red arrow in Fig. 4e) indicating the lack of a distinct preferred frequency for AM LFI, unlike FM LFI.

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Fig. 3 e (a) Pressure series from the combustor, (b) pressure series from the air inlet (0.4 kg/s) showing FM LFI, (c) pressure series from the air inlet (0.2 kg/s) showing FM and AM LFI, (d) FFT plot marked with FM LFI, and (e) spectrogram plot showing pronounced activity at about 235 Hz.

Analysis of AM instability This section deals with the particular characteristics of AM LFI with respect to its association with the RDC air flow rate, detonation wave speed and directionality. Three operating points are chosen that best show the intricacies of AM LFI and

perform case-studies on it. A peak-tracking algorithm was developed to capture the peak pressure (explained in detail in Ref. [4]) of the feedback from the rotating detonation wave on the air inlet from all the three air inlet sensors. This algorithm is further developed to have the capability to predict the detonation wave directionality (clockwise/counterclockwise)

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Table 2 e Test/injection scheme conditions and associated instabilities. Air _ a kg/s m 0.2 0.3 0.4

FP I PR

F

2 2.75 3.2

1.0, 1.2 1.0, 1.2 1.0, 1.2

AM Spatial non-uniformity Y Y N

Y Y N

FP II FM Lock-in N N N

by analyzing the time difference of the peak detonation pressures through three sectors (Dt) from the three pressure signals simultaneously, for subsequent laps. Thus, any switch in the detonation wave direction, which happens frequently in the RDC environment [13,38,44], is instantaneously captured. Note that such a direction shift capture necessitates that there be at least three circumferentially distributed sensors. Clockwise detonation wave rotation is associated with a positive detonation speed value, whereas counterclockwise rotation is represented with the negative of the detonation speed magnitude. Since the AM LFI is approximately sinusoidal among the different sectors (refer Fig. 4), it is possible to extract the envelope of the amplitude modulation in the three sectors by using the peak pressure obtained from the individual detonation wave passage. Now, this envelope of AM LFI in the three sectors is used to obtain the peak pressure of the envelope. Thus, in a manner similar to the computation of the detonation wave speed, the speed of the rotary AM LFI is estimated by getting the time index of the highest point of the sinusoids. The circumferential distance at the position of the air inlet sensors (which is 154 þ 25 ¼ 179 mm away from the central RDC axis) is then divided by these temporal differences of subsequent amplitude modulations (DT) to arrive at a value of AM rotary velocity (product of circumferential distance between each sensor in air inlet and time difference between each of the sensors) in the air inlet at the position of the pressure sensors. This algorithm is visualized in Fig. 5. It is emphasized that the absence of AM LFI (for 0.4 kg/s) does not produce this envelope of peak pressures, thereby enabling the algorithm to ascertain when there is no AM in the air inlet. In the current test matrix, the detonation wave flips direction often for all test cases run at 0.2 kg/s and 0.3 kg/s with the exception for the first case to be discussed below. Tests _ a ¼ 0.2 kg/s, F ¼ 1) multiple were repeated at this condition (m times, and we reached the conclusion that the rotating detonation wave is prone to random switching at this operating condition as well. This incurs an associated stochasticity in the direction of AM LFI. Thus, this AM instability is not rigidly repeatable for a given operating condition, as in the rotary speed of the AM LFI varies across different hot-fires since it is intrinsically linked to the detonation wave speed, which in turn is extremely random. But, the analysis of the pressure data from all the test points confirm that the AM LFI always propagates in a direction that is opposite to the detonation wave direction. In this sense, the first test case discussed below is a rather serendipitous event in that the detonation wave switched direction only three times throughout the whole test. The AM LFI seems to be “destroyed” every time the detonation wave changes direction since the detonation wave onset and propagation is fast and instantaneous, but the

N N N

F 0.87,1.0, 1.21 0.87,1.0, 1.22 0.92,1.0, 1.2

AM Spatial non-uniformity Y Y N

Y Y N

Y Y N

FM Lock-in at 235 Hz Y Y Y

Y Y Y

Y Y Y

mechanism behind the amplitude modulation is apparently not. Therefore, unlike the detonation wave speed direction, the AM LFI direction is not acquired since it exists for too short a duration to clearly define for a complete test. But, it is possible to qualify AM LFI direction at localized time intervals in the figures below. A magnified pressure-time trace of the three air inlet sensors, the complete pressure trace for the whole test duration of approximately 0.35 s, the normalized envelope of AM LFI, the AM LFI rotary speed, and the detonation wave speed with the associated directionality are given as subsections a, b, c, d, and e, respectively, in Figs. 6e8. The envelope of AM LFI is normalized with respect to the maximum value from the complete pressure signal, which happens to be around 3 bar for all the cases presented. This maximum value is from the initial detonation wave from the pre-detonator which is used to ignite the RDC. _ a ¼ 0.2 kg/s and F ¼ 1.0. From Fig. 6a it Fig. 6 deals with m can be seen that the rotating detonation wave is spinning counterclockwise and AM LFI (the sinusoidal overarching component) is clockwise. Fig. 6b shows the highly dynamic nature of this instability. Initially, there is incoherence and there is no marked sinusoidal spatially varying oscillation. However, from t ¼ 0.14 s, one could observe easily delineated sinusoidal variation existing until the end of the RDC hot-fire run. Additionally, it could also be seen that the sinusoidal variation gradually “thins” with time, occurring faster as time progresses. This kind of dynamic variation in the frequency of occurrence of a phenomenon is usually termed “bootstrapping”, a non-linear phenomenon that has been seen to occur in rocket engines [45], albeit for different reasons. The normalized pressure of the low frequency spatially instability is shown in Fig. 6c for all three sensors. The spatial variation and the faster occurrence of the instability throughout the test can be easily noticed. In fact, the plot of the AM LFI velocity vs. time (Fig. 6d) shows that initially the induced velocity is around 40 m/s, and continuously increases from t z 0.135 s to t z 0.275 s, after which time it plateaus to a speed of 100 m/s. The gradual increase in AM LFI velocity until a terminal condition can be better understood by analyzing the detonation wave speed and directionality plot shown in Fig. 6e. The rotating detonation wave establishes in the clockwise direction initially and rotates in this direction until t z 0.135 s. However, at t z 0.135 s there is a sudden stochastic flip in the direction to counterclockwise rotation, and from t z 0.135 s till t z 0.34 s, the rotating detonation wave exhibits stable directionality in the counterclockwise direction. It is thus apparent that the steady increase in AM LFI to a terminal value from t z 0.135 s is closely linked to the rotating detonation direction, i.e. as long as the detonation wave maintains its direction of rotation, the AM LFI in the air plenum maintains

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_ a ¼ 0.2 kg/s, F ¼ 1.0 between (a) t ¼ 0.295 s and t ¼ 0.302 s, (b) Fig. 4 e AM LFI pressure series from air inlet with FP e I at m t ¼ 0.302 s to t ¼ 0.31 s, (c) t ¼ 0.31 s to t ¼ 0.317 s, and (d) FFT plot and (e) spectrogram plot.

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Fig. 5 e Visualization of peak-tracking algorithm that computes detonation wave speed and AM LFI rotating speed through obtaining peak pressures and time indices.

its direction. The amplitude modulation in the air inlet seems to exhibit a considerable transient operation followed by an apparent steady-state period where the rotary velocity of AM LFI is plateaued. This terminal rotary speed is approximately 100 m/s, for the present test condition. As will be shown below, this plateau in the rotary speed is not observed when the detonation wave flips direction frequently. _ a ¼ 0.3 kg/s and F ¼ 1.0) is in contrast to Fig. 6. The Fig. 7 (m magnified pressure series in Fig. 7a shows that while the spatial non-uniformity due to AM is still considerable, the oscillation time scale rapidly evolves as the instability rotates in the air plenum. A better view of the inherent stochasticity in the AM LFI magnitude and time scale can be observed in Fig. 7b, which presents the complete pressure-time trace for the test point. The normalized oscillation magnitude of AM LFI (Fig. 7c) also indicates the stochasticity for this test case of 0.3 kg/s. Since the equivalence ratio is the same as the prior case, one could contend that the variation seen here is produced due to physical, fluid dynamic effects as opposed to chemical. To elucidate, the results from our threedimensional simulation of cold-flow mixing characteristics of the current RDC geometry showed that the variation in local equivalence ratio is negligible across different air flow rates, provided the global equivalence ratio across the air flow rates is held constant [32]. Hence, there is no reason to assume that the detonation wave strength is going to vary at 0.3 kg/s, at F ¼ 1.0. The variation in AM LFI for 0.3 kg/s, must then, be an effect produced due to the increased pressure ratio across the air inlet (refer Table 2). Fig. 7c shows two other facets: a) the oscillation magnitude is considerably lower than the 0.2 kg/s case, and b) the direction of AM LFI is stochastic. The amplitude modulated sinusoids exhibit both blue / red / black pattern (clockwise) or the black / red / blue direction (counterclockwise), thereby indicating a lack of preferred direction. Analysis of AM LFI velocity shown in Fig. 7d also shows the effects of this stochasticity in magnitude. This is attributed to be a direct consequence of the rotating detonation wave directionality (which produces AM LFI in the air inlet, thereby creating a feedback loop). The current case has multiple random switches in the direction of detonation as seen in Fig. 7e. Throughout the test length, the detonation wave exhibits high propensity to randomly switch between clockwise and counterclockwise direction. As elucidated before, this arbitrary behavior of the detonation wave directionality is a well-observed phenomenon in RDCs. At present, the only proposed theory to explain this is from the numerical

simulation of Yao et al. [46] who found that improper mixing (which could be the result of AM in the reactants plenum) of the reactants facilitate the formation of random “hot spots” in the combustor, which cause explosion and subsequent generation of a detonation wave in the opposite direction. Note that detonation initiation through this kind of “explosion within an explosion” is an established phenomenon to explain detonation formation under certain conditions, as explained by Lee [1]. _ a ¼ 0.4 kg/s Advancing to Fig. 8, which is representative of m and F ¼ 1.0, the general behavior of AM LFI is seen to further evolve with additional characteristics not seen in the last two cases. There is an abrupt cessation of the instability at t z 0.108 s (Fig. 8a). This sudden termination of AM LFI can be better observed in Fig. 8b. From the moment of initiation until a time of t z 0.108 s, there is considerable amplitude modulation. But, after this time, there is no more AM LFI in the air inlet and the remainder of the test duration is characterized by the rotating detonation wave exhibiting the desirable behavior of repeatable peak-to-peak pressure magnitudes with a complete absence of spatial non-uniformity among the different sectors. The phenomenon is also seen in the normalized AM LFI envelope plot (Fig. 8c) which once again depicts the absence of AM LFI after an arbitrary time for this _ a ¼ 0.4 kg/s test. Additionally, the overall envelope of the m pressure feedback magnitude into the air inlet is notably lower than the 0.3 kg/s case, which in turn is lower than the case with an air flow rate of 0.2 kg/s. The AM LFI velocity is about 200 m/s on average initially (Fig. 8d). As noted in a prior section, the tested RDC did not acquire thermal equilibrium even after 3 s. Hence, this behavior is most probably due to the increase in pressure ratio across the air injection (Table 2) with increasing air flow rates. Assuming a rotating detonation wave of equal strength that is anchored to the RDC injection headwall for the three air flow rates (which is not the casedthe distance between the detonation wave and the injection plane increases with increasing air flow rate [47]), the higher air flow rate, and hence the higher pressure ratio, PR, across the injection, will naturally tend to inhibit the strength of the leaked shock into the air inlet [15]. For the lowest air flow rate of 0.2 kg/s, owing to its lowest PR among the three cases, the shock wave leakage into the plenum would be the highest. Since Cho et al. [47] have established that there is increasing ‘stand-off’ of the detonation wave from the injection plane as the air flow rate is increased, the penetrative effect of the trailing shock wave into the air plenum must be

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_ a ¼ 0.2 kg/s, F ¼ 1. Fig. 6 e Variables of interest acquired from air inlet sensors at m

lower at 0.4 kg/s in comparison to 0.2 kg/s. Finally, at t z 0.108 s, the rotating detonation wave attains an extremely steady periodicity and directionality that extends till the end of the test. This onset time of steady rotation can be observed

to be directly coincident with the abrupt absence of AM. It is imperative to note that the stable onset of the detonation wave entails with it a notable increase in the detonation propagation speed as seen in Fig. 8e.

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_ a ¼ 0.3 kg/s, F ¼ 1. Fig. 7 e Variables of interest acquired from air inlet sensors at m

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_ a ¼ 0.4 kg/s, F ¼ 1. Fig. 8 e Variables of interest acquired from air inlet sensors at m

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Proposed mechanism From the above sections, we have enough information to hypothesize a mechanism behind the amplitude modulated instability in an RDC. AM LFI, as shown above, occurs as a rotary event in the air plenum. This rotation, with sinusoidal variation in peak pressures leaked into the air inlet, occurs in a direction opposite to the direction of the rotating detonation at all the points tested. Additionally, the manifestation of AM LFI seems to be inherently linked to the pressure ratios across the air inlet, since at 0.4 kg/s (highest pressure ratio in the current study), the AM LFI subsides. Let us consider the former observationdof a reverse rotary AM eventdfirst. Fotia et al. [24], in their two-dimensional experimental RDC study, have noted that the trailing shock wave (attached to the bottom of the detonation wave, as established by Schwer and Kailasanath [23]) moves at about 60% of the detonation wave speed. Owing to this relative velocity between the detonation wave in the combustor and the trailing shock wave in the reactants plenum, they postulated a “pressure beating” event when the next detonation wave lap interacts with the trailing shock wave in the plenum from the prior lap. However, their study was a ‘one’ detonation wave event, and hence the “pressure beating”, or more accurately, constructive/destructive interference was not observed. On the other hand, Schwer and Kailasanth [23], in their numerical simulation, have noted that this trailing shock wave (incident wave) travels to the base of the reactants plenum and gets reflected as another relatively strong wave. They tested two different reactant plenum depths (to vary the strength of the reflected wave) and concluded that this reflected wave did not impact the reactants conditions upstream of the next lap of the detonation wave. In Fig. 9, we have presented a schematic of the detonation wave, the trailing shock wave that is the incident wave on the base of the air plenum and the concomitant reflected wave. This model of the wave system is adapted from the twodimensional simulations of Schwer and Kailasanath [23]. Note that the whole system of waves moves with the detonation wave in the laboratory frame of reference. However, in a detonation-fixed reference frame both the incident wave and reflected wave have a relative velocity (broken arrows) that moves away from the detonation wave. The reflected shock wave moves in the opposite direction when the incident shock wave interacts with a concave surface. Additionally, multiple reflected waves are produced at discrete times and locations along the concave surface, depending on curvature and other

effects [48]. Evidence for this in an RDC can be seen in the Schlieren images of an air plenum (with curved base) exposed to a detonation wave, obtained by Bedick et al. [25]. Hence, one should expect a similar phenomenon of production of multiple reflected waves moving in the opposite direction of the trailing shock wave when it is incident on the concave curved surface of the base of the air plenum of the RDC. This is represented as a highly simplified multiple reflected waves in the schematic shown in Fig. 9. Thus, the trailing shock wave of the next lap of detonation wave should interact with the multiple reflected waves that are moving towards it, thereby causing constructive/destructive interference that moves in a direction opposite to the direction of the detonation wave. This effect could be understood by noting the air inlet sensors in the three sectors of the RDC (colorcoded as before). The trailing shock wave from the first lap of the detonation wave is recorded by the blue sensor. However, the constructive interference between the reflected wave from the base of the plenum and the trailing shock wave of the second lap should be captured as an amplitude modulation occurring in the opposite direction (hence captured by the red sensor), as is the case here. It is possible that this effect was not observed in the simulations [23] because it used an orifice injection system (instead of slots, which have considerably higher area and hence lower viscous damping) and a straight plenum base (since it was a two-dimensional study). We hypothesize here that AM LFI in the air inlet is probably due to this interaction of incident and reflected waves. This is an extension of the acoustic interference hypothesis that Fotia et al. [24] that is expanded to include the effects of the reflected wave as well, in an actual RDC. Naturally, one should expect this complex system of waves to develop much slower in comparison to the detonation wave. This could explain the lower speed of the rotation and the stochasticity in AM LFI when the detonation wave randomly switches direction. The absence of this effect for the flow rate of 0.4 kg/s could be due to the increased impedance produced by the higher pressure ratio, which might hinder the production of a strong trailing shock wave into the air plenum. Cho et al. [47] found that the ‘stand-off distance’ increases almost linearly with increasing air flow rates. This fact that the detonation wave significantly “lift-offs” from the RDC injection plane at higher flow rates might also produce weaker shock waves that might not significantly penetrate the air plenum. Visualization of the reactants plenum is required to further test the proposed hypothesis.

Fig. 9 e Schematic of the incident and reflected wave in the reactant plenum produced due to the detonation wave in the combustor. Adapted from Schwer and Kailasanath [23]. Please cite this article in press as: Anand V, et al., Amplitude modulated instability in reactants plenum of a rotating detonation combustor, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.03.218

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Conclusions [9]

Low frequency instabilities observed in rotating detonation combustors are linked to two distinctly different types of instabilities in the air plenum of RDCs. The first is recognized by frequency modulation in the air inlet, whereas the second is distinguished by amplitude modulation on top of the carrier signal that is the detonation wave. The occurrence of these instabilities seems to depend on the fuel injection scheme used and the air flow rates through the RDC. These instabilities occur either in tandem or exclusively depending on the operating conditions. The amplitude modulation instability was considered specifically for further analysis. Amplitude modulation occurs as a rotary event that is tracked through the different sectors of the RDC. This type of phaselagged low frequency rotating instability has been observed in rocket engines, as well (Fig. 9.7.1i in Ref. [9]). There, it is called a “precessing tangential mode” and is noted to have anywhere between five to hundred tangential wave laps per low frequency cycle. Here, it moves in a direction opposite to the direction of the detonation wave. Higher air flow rates, and thus higher pressure ratio across the air inlet, tend to remove this amplitude modulation. A hypothesis is proposed that attributes this instability to a probable constructive/ destructive interference that could occur between the trailing shock wave from the detonation wave to the reverse moving system of incident wave and reflected wave (from the base of the plenum) from the previous lap of the detonation wave. Further experimentation is required to accurately capture and analyze this phenomenon. Regardless, the preliminary findings suggest that the nature of RDC-reactants plenum coupling is a complex process that requires considerable attention.

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Please cite this article in press as: Anand V, et al., Amplitude modulated instability in reactants plenum of a rotating detonation combustor, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/j.ijhydene.2017.03.218