We experimentally study the dynamic behavior of intermittent combustion oscillations by time series analysis in terms of nonlinear forecasting, symbolic dynamics, and statistical complexity, including the detection of the change in dynamical state based on symbolic dynamics and graph networks. We observe sudden switching back and forth between irregular small-amplitude and regular large-amplitude pressure fluctuations. The nonlinear local prediction method, permutation spectrum test, and the Rényi complexity–entropy curve clearly identify the possible presence of chaotic dynamics in small-amplitude pressure fluctuations during intermittent combustion oscillations. The network entropy in ordinal partition transition networks allows us to capture a significant change in dynamical state switching between chaotic oscillations and noisy limit cycle oscillations.

A self-excited combustion instability, referred to as thermoacoustic combustion oscillations, arises as a mutual interaction between the heat release rate and longitudinal/transverse acoustic pressure fluctuations,1,2 giving rise to various nonlinear dynamic behaviors. This instability leads to the fracture damage of combustors and continues to pose significant challenges in engine development. The characterization and physical understanding of the transition route to thermoacoustic combustion oscillations are of much interest to the wider combustion science and physics community. Advanced analytical methods based on dynamical systems theory have provided us with an overarching understanding of the underlying nonlinear dynamics in thermoacoustic combustion oscillations.3–8 We have recently elucidated part of the dynamic behavior of pressure fluctuations during a transition and the subsequent well-developed thermoacoustic combustion oscillations in two types of turbulent combustor, a model staged aircraft engine combustor9 and a model rocket engine combustor,10 emphasizing the applicability of time series analyses in terms of symbolic dynamics and statistical complexity.

The transition from aperiodic to periodic oscillations and vice versa in nonlinear systems was found to mainly occur through three routes: period-doubling bifurcation, torus breakdown, and intermittency. Intermittency is a transitional state suddenly switching back and forth between bursts and limit cycle oscillations. A state of intermittency significantly emerges in turbulent combustors while transitioning from combustion noise to limit cycle oscillations and vice versa in terms of the Reynolds number,11–13 equivalence ratio,11–13 and flame density ratio.14 

The main purpose of this study is to obtain a better understanding of intermittent combustion dynamics in a model rocket engine combustor from the viewpoints of nonlinear forecasting, symbolic dynamics, and statistical complexity. As the model rocket engine combustor, we use a cylindrical combustor with an off-center installed coaxial injector, as in our previous study.10 We study the dynamical state during intermittent combustion oscillations by adopting three important analytical methods for the temporal evolution of pressure fluctuations inside the combustor: the nonlinear local prediction method,7 permutation spectrum test,15 and the Rényi complexity–entropy curve analysis.16 Graph networks consisting of nodes and links can express the relationship between various elements in real-world complex systems and have been adopted for a rich variety of nonlinear dynamical phenomena in many disciplines from physics to engineering.17,18 The importance of time series analysis in terms of graph networks to the treatment of thermoacoustic combustion oscillations has been identified for various types of turbulent combustor.19–24 These studies mainly consider three networks: visibility graphs,25,26 cycle networks,27 and recurrence networks.28,29

The ordinal partition transition network has been proposed by Small30 and McCullough et al.31 as a sophisticated transition network incorporating the concepts of the Markov chain and symbolic dynamics. On the basis of these studies,30,31 Kobayashi et al.32 have recently reported that the network entropy in ordinal partition transition networks is useful for determining the synchronized state between pressure fluctuations inside the combustor and those in the premixture region during degenerated combustion oscillations in a model gas-turbine combustor. However, the applicability of such transition networks from a practical point of view has not been studied for the model gas-turbine combustor or for a model rocket engine combustor. Gotoda et al. have conducted two important experimental studies on the early detection of blowout in a model laboratory-scale gas-turbine combustor using (I) the transition error based on orbital instability in phase space4 and (II) the mean degree in the natural visibility graph.22 Murayama et al.24 have shown the importance of the motif patterns in the horizontal visibility graph for the early detection of combustion oscillations in the same combustor. In this study, we examine the validity of the network entropy in ordinal transition partition networks as a potential diagnostic for detecting a significant dynamical change during intermittent combustion oscillations in a model rocket engine combustor. This has not yet been explored by the combustion community.

This paper is organized into four sections. In Sec. II, we briefly describe our experiments and analytical methods. The results and discussion are provided in Sec. III. We finally present a summary in Sec. IV.

Figure 1 shows our experimental apparatus. It mainly consists of a cylindrical combustor with an off-center installed coaxial injector. The sizes of the H2/O2 injectors and combustion chamber are almost the same as those described in Ref. 10, except for the setting of the face plate around the O2 injector. Both injectors are mounted 72 mm from the center of the combustion chamber. In this study, N2 is issued from the face plate. H2 (O2) is issued from the outer (inner) injector with a diameter of 8.6 mm (4.9 mm). The mass flow rates of H2 and N2 are set to 300 and 500 l/min, respectively. The mass flow rate of QO2 is varied from 50 to 100 l/min so as to trigger combustion oscillations while keeping the mass flow rates of H2 and N2 constant. We measure the pressure fluctuations p at a sampling frequency of 102.4 kHz inside the combustor using pressure transducers (Kulite Semiconductor Products, Model ETM-375-100SG). As shown in Fig. 1, five pressure transducers (PT1–PT5) are mounted on the wall of the combustion chamber. In this study, we mainly adopt nonlinear time series analysis for pressure fluctuations in the representative pressure transducer PT1 to study the dynamic behavior of intermittent combustion oscillations.

FIG. 1.

Experimental apparatus.

FIG. 1.

Experimental apparatus.

Close modal

The distinction between chaotic and stochastic dynamics is a major challenge that must be addressed in modern nonlinear time series analysis. The ability to make short- and long-term predictions of the system behavior (which also has a high sensitivity to the initial conditions) is very useful for determining whether the system is undergoing a chaotic or stochastic process. The importance of nonlinear forecasting has recently been shown in a spatially extended system,33 yielding a clear distinction between chaos and pure stochastic dynamics. In this study, we apply a nonlinear local prediction method4,7 with the aim of exploring the possible presence of chaotic dynamics during intermittent combustion oscillations. Note that the prediction method based on orbital instability in phase space is an extended version of the Sugihara–May algorithm.34 In this method, the temporal variations in Δp(=p(ti+1)p(ti)) for t[0;Tf] are first divided into two intervals: library data corresponding to t[0;tL] and reference data corresponding to t(tL;Tf]. We compare the predicted Δp with the corresponding reference data, where t>tL. The prediction of Δp is made by updating the library data, keeping the size of the updated library data constant. We here define xfx(tf) as the final point of a trajectory in D-dimensional phase space, where x(t)=(Δp(t),Δp(tτ),,Δp(t(D1)τ)). Note that D is set to 5 in this study. In accordance with the prescription of Fraser and Swinney,35 we set the delay time τ in the phase space that yields a local minimum of mutual information. We search for the nearby xk(k=1,2,,K) from all x in the phase space and predict Δp(tf+T) after T steps using the nonlinearly weighted sum of the library data Δp(tk+T), as given by

Δp^(tf+T)=k=1KexfxkΔp(tk+T)k=1Kexfxk,
(1)

where T=TsΔt, with Δt being the sampling time of Δp and Ts time step. The prediction accuracy is estimated using the correlation coefficient C between the predicted Δp^(tf+T) and reference Δp(tf+T) pressure values [C=E[ΔpΔp^]/σΔpσΔp^, where E[ΔpΔp^] is the covariance between the measured and predicted Δp and σΔp(σΔp^) is the standard deviation of Δp(Δp^)]. An important point of this method is that if the observed dynamics is chaotic, C is large for a short prediction time and decreases with increasing prediction time tp.

The permutation spectrum test is useful for identifying the presence of nonlinear determinism in complex dynamics.15 The permutation spectrum consists of the frequency distribution of the permutation patterns πi(i=1,2,,D!) for each window in a time series of p and their standard deviation between the windows. The central idea of the permutation spectrum test is that if a nonlinear deterministic process (stochastic process) strongly dominates pressure fluctuations, zero standard deviation with some forbidden patterns (nonzero standard deviation without forbidden patterns) appears in the permutation spectrum. We briefly mention the set value of D. The number of permutation patterns is 4! (= 24) at D = 4, which is insufficient to capture the chaotic dynamics of small-amplitude aperiodic fluctuations during intermittent combustion oscillations in a model rocket combustor under a high Reynolds number. When D is 7, forbidden patterns appear for stochastic dynamics such as Brownian motion with sufficient data points (= 100 000). This suggests that D should be smaller than 6. On this basis, we set D=5 as a suitable embedding dimension in this study.

The Rényi complexity–entropy curve,16 which is a developed version of the complexity–entropy causality plane,36 is useful for discerning chaotic dynamics from stochastic dynamics. It is expressed as the variation in the derivative of CJS with respect to Sp(=dCJS/dSp) as a function of α, where CJS is the Jensen–Shannon statistical complexity (=Sp[P]QJS[P,Pe]), Sp is the Rényi entropy, QJS is the disequilibrium, and α is the entropic index. Similarly to Jauregui et al.,16 we compute the Rényi complexity–entropy curve for different embedding dimensions D,

Sp[P]=1lnD!{11αlni=1D!P(πi)α},
(2)
QJS[P,Pe]=DJS[P,Pe]Qmax,
(3)
DJS[P,Pe]=12(α1){lni=1D!P(πi)α(P(πi)α+1/D!2)1α+lni=1D!1D!α(P(πi)α+1/D!2)1α},
(4)
Qmax=12(α1)ln{(D!+14D!)1α(D!+1)1α+D!1D!}.
(5)

Here, P={P(πi)i=1,2,,D!}, with P(πi) being the existing probability of the permutation patterns in a time series of p, and Pe={1/D!,1/D!,,1/D!}. Note that when α=1, Sp represents the permutation entropy37 based on the definition of the information entropy.

The ordinal partition transition networks,30,31 which consist of the transition probability from one rank order pattern to another in a time series based on the Markov chain, are useful graph networks incorporating the concept of symbolic dynamics. In this study, we estimate the network entropy St in the ordinal partition transition networks constructed from pressure fluctuations. The transition probability matrix is used as the weighted adjacency matrix in the networks. We can obtain St by considering wij for the definition of the information entropy,

St=i=1D!j=1D!wijlnwijln(D!)2.
(6)

Here, wij=P(πiπj) is the existing transfer probability from the ith-order pattern to the jth-order pattern. Our preliminary test has shown that forbidden patterns in ordinal partition transition networks do not appear at D=3 for white Gaussian noise and Brownian motion with 10 000 data points, while they appear at D=4 as well as in the permutation spectrum. This suggests that D=3 should be considered for ordinal partition transition networks. In this study, we compute the network entropy St every 125 ms (the number of data points is 12 800). Therefore, D is set to 3 for the estimation of St in this study.

Figure 2 shows the temporal evolution of pressure fluctuations p for different oxygen volume flow rates QO2. The dynamic behavior of p at QO2=50l/min exhibits small-amplitude aperiodic oscillations. When QO2 increases to 64 L/min, large-amplitude periodic oscillations begin to occur intermittently. The formation of the periodic oscillations becomes prominent during intermittent combustion oscillations at QO2=100l/min. The temporal evolution of the power spectrum at QO2=100l/min is shown in Fig. 3. Two notable oscillations at frequencies of approximately 200 and 1000 Hz are formed during intermittent combustion oscillations switching irregularly back and forth between two states: one corresponds to the acoustic 0L mode in the longitudinal direction of the combustor and the other corresponds to the 1T mode. The latter mode becomes predominant during the formation of well-developed limit cycle oscillations. Pomeau and Manneville38 classified intermittent phenomena into three types: type-I, type-II, and type-III intermittencies. The Indian Institute of Technology Madras group14,39–42 has reported that type-II intermittency appears during intermittent pressure oscillations in various combustors. The presence of type-II intermittency is clearly identified using recurrence plots. On the basis of their studies,14,40–42 it is presumed that the type-II intermittency associated with Hopf bifurcation plays an important role in the intermittent dynamics of pressure fluctuations at QO2=100l/min.

FIG. 2.

Temporal evolution of pressure fluctuations p for different oxygen volume flow rates QO2. (a) QO2=50l/min, (b) QO2=64l/min, and (c) QO2=100l/min.

FIG. 2.

Temporal evolution of pressure fluctuations p for different oxygen volume flow rates QO2. (a) QO2=50l/min, (b) QO2=64l/min, and (c) QO2=100l/min.

Close modal
FIG. 3.

Temporal evolution of power spectrum during intermittent combustion oscillations at an oxygen volume flow rate of QO2=100l/min.

FIG. 3.

Temporal evolution of power spectrum during intermittent combustion oscillations at an oxygen volume flow rate of QO2=100l/min.

Close modal

A surrogate data method is a useful statistical test used in combination with nonlinear time series analysis to validate the determinism in irregular time series. We adopt the nonlinear forecasting method in combination with the small-shuffle surrogate data method43 for small-amplitude aperiodic pressure fluctuations during intermittent combustion oscillations. The null hypothesis of the surrogate data is that irregular fluctuations are independently distributed random variables. The surrogate data lose local structures in irregular fluctuations but preserve global structures.43Figure 4 shows the distribution of C in terms of tp for the original time series and small-shuffle surrogate time series at QO2=100l/min. Note that we generate 100 surrogate time series in this study. C at Ts=1 for the original time series, which corresponds to tp=9.8×106s, is approximately 0.90 with high predictability. It exponentially decreases with increasing tp. We observe short-term predictability and long-term unpredictability regions, showing the presence of chaotic dynamics based on orbital instability. C for the original time series does not correspond to its values for the surrogate time series. Therefore, the null hypothesis that irregular fluctuations are independently distributed random variables can be rejected. Figure 5 shows the permutation spectrum during small-amplitude aperiodic oscillations at QO2=100l/min. Zero standard deviation with some forbidden patterns appears in the permutation spectrum, identifying the presence of nonlinear determinism in the aperiodic oscillations. Note also that the null hypothesis of the small-shuffle surrogate data can be rejected in the permutation spectrum test because no forbidden patterns were observed for the surrogate time series. The variation in dCJS/dSp during small-amplitude aperiodic oscillations is shown in Fig. 6 as a function of α at QO2=100l/min. dCJS/dSp has positively sloped curves in the region 0 α0.25 at D=3,4, and 5, whereas it has a negatively sloped curve at D=6. Jauregui et al.16 reported that the appearance of the negative Rényi complexity–entropy curve at a high D is a signature of chaos. On the basis of the results obtained in Figs. 4–6, aperiodic oscillations in pressure fluctuations represent chaos.

FIG. 4.

Variation in correlation coefficient C as a function of prediction time tp for the original time series and the small-shuffle surrogate time series at an oxygen volume flow rate of QO2=100l/min.

FIG. 4.

Variation in correlation coefficient C as a function of prediction time tp for the original time series and the small-shuffle surrogate time series at an oxygen volume flow rate of QO2=100l/min.

Close modal
FIG. 5.

Permutation spectrum during small-amplitude aperiodic oscillations at an oxygen volume flow rate of QO2=100l/min.

FIG. 5.

Permutation spectrum during small-amplitude aperiodic oscillations at an oxygen volume flow rate of QO2=100l/min.

Close modal
FIG. 6.

Variation in dCJS/dSp as a function of α during small-amplitude aperiodic oscillations at an oxygen volume flow rate of QO2=100l/min.

FIG. 6.

Variation in dCJS/dSp as a function of α during small-amplitude aperiodic oscillations at an oxygen volume flow rate of QO2=100l/min.

Close modal

A multiscale complexity–entropy causality plane (CECP), which is a two-dimensional plane consisting of the permutation entropy and Jensen–Shannon statistical complexity, is useful for distinguishing complex dynamical states. We here adopt the multiscale CECP for large-amplitude periodic pressure fluctuations (see Ref. 40 for details of the CECP). Figure 7 shows the variations in the permutation entropy Sp and the Jensen–Shannon statistical complexity CJS as a function of the embedding delay time in phase space τ during large-amplitude periodic oscillations at QO2=100l/min, together with the CECP. Note that α is set to 1 for the estimation of Sp and CJS. Sp(CJS) significantly takes a value of almost unity (zero) every τ0.9ms, which correspond to the periods of the 1T mode. Small local values of Sp and CJS also appear every τ0.3 and 0.4 ms, which correspond to the period of the harmonics of the 1T mode. The dynamics of a stochastically driven van der Pol oscillator producing noisy periodic limit cycle oscillations shows a periodic local maximum (minimum) of Sp(CJS) in terms of τ.44 The shape of the trajectory on the plane corresponds reasonably well to that obtained by a stochastically driven van der Pol oscillator.44 This satisfactorily shows that the dynamical state during large-amplitude periodic pressure fluctuations represents noisy limit cycle oscillations. These results demonstrate that the dynamical state during intermittent combustion oscillations switches back and forth between chaotic oscillations and noisy limit cycle oscillations.

FIG. 7.

Variations in permutation entropy Sp and Jensen–Shannon statistical complexity CJS as a function of the embedding delay time in phase space τ during large-amplitude periodic oscillations at an oxygen volume flow rate of QO2=100l/min, together with the CECP.

FIG. 7.

Variations in permutation entropy Sp and Jensen–Shannon statistical complexity CJS as a function of the embedding delay time in phase space τ during large-amplitude periodic oscillations at an oxygen volume flow rate of QO2=100l/min, together with the CECP.

Close modal

The degeneration process of combustion oscillations is of importance in the formation and sustainment of intermittency. We here discuss the degenerated behavior of combustion oscillations. It has been recently reported by Murugesan and Sujith45 that the significant decrease in heat release rate during the degeneration process in a bluff-body combustor reduces the driving of acoustics, resulting in the degeneration from high- to low-amplitude oscillations. In our preliminary test, we observe a slight decrease in OH chemiluminescence intensity in the degeneration process, but the degree of this decrease is not apparent compared with that in their experiment.45 In this study, we treat a turbulent H2/O2 inverse diffusion flame without a flame holder such as a bluff body. The periodic detachment/attachment of the flame base to the oxidizer–injector rim has a significant impact on the formation of combustion oscillations.10 The phenomenological mechanism of combustion oscillations in our study will differ from that in their study45 in the sense that (i) combustion oscillations in their study45 are sustained by the large-scale vortical structure generated by the wake behind the bluff body and (ii) their intermittent combustion oscillations are formed near lean blowout.45 The decrease in heat release rate might be related to the physical mechanism of the degeneration from high-amplitude oscillations to low-amplitude oscillations in our study, but we must carefully clarify this point in the future.

Figure 8 shows the time variations in p, Sp, and St, together with the extracted network structures. Note that α is set to 1 for the estimation of Sp as in the multiscale CECP. St takes large values during small-amplitude aperiodic oscillations, showing the high degree of randomness in p. As combustion oscillations are formed, two specific self-loop transition patterns, π1π1 and π6π6, become predominant, resulting in the significant decrease in St. These patterns correspond to the monotonically increasing and decreasing processes, respectively. The network entropy responds to the significant change in dynamical state from chaotic oscillations to noisy limit cycle oscillations and vice versa. Matsuyama et al.46 have recently obtained the spatiotemporal dynamics of well-developed combustion oscillations in a model rocket engine combustor by large-eddy simulation. Hashimoto et al.47 have revealed the driving region of combustion oscillations in the same model combustor using the transfer entropy and the order parameter based on synchronization theory. St for the numerical simulation data is approximately 0.4, with the formation of two dominant self-loop transition patterns (π1π1 and π6π6). This corresponds reasonably well to the value obtained in our experimental work. An interesting point to emphasize here is that some transition patterns (π3π3, π4π3, and π5π2) do not appear at t=0.625s. This means that forbidden transition patterns exist during small-amplitude aperiodic oscillations. Within the framework of the permutation spectrum test,15 small-amplitude aperiodic oscillations represent chaos. The ordinal partition transition networks are also useful for distinguishing between chaos and stochastic state. As shown in Fig. 8, the time variation in St corresponds to that in Sp during intermittent combustion oscillations. Thus far, Gotoda and coworkers have attempted to detect a precursor of combustion oscillations using two measures: the permutation entropy9 and the motif patterns in the horizontal visibility graph24 related to the permutation patterns. The applicability of the permutation entropy has also been discussed for different turbulent combustors by the Indian Institute of Technology Madras group48,49 and the Hong Kong University of Science and Technology group.50 In addition to the permutation entropy, the network entropy in ordinal partition transition networks has potential use for detecting the precursor of thermoacoustic combustion oscillations.

FIG. 8.

Time variations in the pressure fluctuations p, the permutation entropy Sp, and the network entropy in the ordinal partition transition networks St at an oxygen volume flow rate of QO2=100l/min, together with the extracted network structures. (a) t=0.625s and (b) t=1.25s.

FIG. 8.

Time variations in the pressure fluctuations p, the permutation entropy Sp, and the network entropy in the ordinal partition transition networks St at an oxygen volume flow rate of QO2=100l/min, together with the extracted network structures. (a) t=0.625s and (b) t=1.25s.

Close modal

We have experimentally studied the dynamic behavior of intermittent combustion oscillations in a model rocket engine combustor from the viewpoints of nonlinear forecasting, symbolic dynamics, and statistical complexity, including the detection of the change in dynamical state based on symbolic dynamics and graph networks. Aperiodic small-amplitude pressure fluctuations suddenly appear in the regular large-amplitude fluctuations generated by acoustic 0L and 1T modes. Intermittent combustion oscillations accompany chaotic dynamics in small-amplitude aperiodic oscillations, which are clearly identified by the nonlinear local prediction method, permutation spectrum test, and Rényi complexity–entropy curve analysis. The network entropy in ordinal partition transition networks is a useful measure for detecting a significant change in dynamical state switching between chaotic oscillations and noisy limit cycle oscillations during intermittent combustion oscillations. It is expected to become a much more sophisticated detector of combustion oscillations than the permutation entropy9 and the motif patterns in the horizontal visibility graph24 in the sense that temporal variations in the permutation patterns are considered.

We would like to express our thanks to Dr. Hiroyasu Saito and Dr. Naoki Hosoya (Shibaura Institute of Technology) for their invaluable support in our experiments, to Dr. Shingo Matsuyama (Japan Aerospace Exploration Agency) for providing the numerical data obtained by large-eddy simulation, and to Mr. Shuya Kandani (Tokyo University of Science) for his support in the computation of the network entropy in ordinal partition transition networks.

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