Complex network analysis of spatiotemporal dynamics of premixed flame in a Hele–Shaw cell: A transition from chaos to stochastic state

Intrinsic flame front instabilities, which give rise to a rich variety of spatiotemporal patterns and the subsequent self-turbulization in a freely propagating premixed flame, have long been of great interest in nonlinear problems of chemically reacting fluid physics.⁴ They are predominately driven by the thermal-diffusive effect^5,6 and/or the hydrodynamic Darrieus–Landau-type effect.⁷ The spatiotemporal dynamics due to intrinsic instabilities can be represented by nonlinear evolution equations such as the Sivashinsky equation⁸ and the Michelson–Sivashinsky (MS) equation.⁹ Many theoretical and numerical studies have used nonlinear evolution equations to clarify the behaviors of wrinkled flames^10–12 and propagating flames in a stagnating point flow,¹³ a channel,¹⁴ and a spherical chamber.^15,16 An essential configuration for dealing with intrinsic instabilities is a Hele–Shaw cell with a narrow gap between two parallel plates. Almarcha et al.¹⁷ have recently reported that the pole trajectories and the cell size statistics of downwardly propagating premixed flames in a Hele–Shaw cell can be physically explained by the MS equation. We have more recently proposed a nonlinear evolution equation incorporating the buoyancy effect and numerically studied the spatiotemporal dynamics of a flame front in a Hele–Shaw cell from the viewpoint of complex networks.¹⁸ The key feature of our equation is that the dynamical state undergoes a significant transition from low-dimensional to high-dimensional deterministic chaos owing to the promotion of buoyancy. The irregular formation of large-scale wrinkles induced by the Rayleigh–Taylor hydrodynamic instability is strongly associated with the emergence of high-dimensional deterministic chaos.

Random noise, which is an inevitable attribute in nature, is well known to have an important role in the formation of complex spatiotemporal dynamics in a wide spectrum of biological, engineering, and physical environments. The interaction of nonlinearity and stochasticity results in the emergence of several intriguing phenomena involving noise-induced transitions,^19,20 stochastic bifurcations,²¹ and noise-induced chaos.²² An important issue in intrinsic flame front instabilities is the elucidation of the interaction between deterministic dynamics and random noise. The stochastic version of nonlinear evolution equations is a fundamental spatially extended system to deal with this issue. Olami et al.²³ were the first to discuss the effect of random noise on the growth of an unstable propagating flame front, showing phase diagrams as functions of the system size and the additive white noise level in the noisy MS equation. Radisson et al.²⁴ have recently clarified a random splitting mechanism along the flame front in a Hele–Shaw cell by estimating the probability density function of cell sizes and compared experimental results with numerical results obtained by solving a noisy equation similar to that used by Olami et al.²³ However, there are no detailed studies on how the presence of additive noise affects the flame front dynamics in a Hele–Shaw cell under various gravitational levels associated with the Rayleigh–Taylor hydrodynamic instability.

Complex-network-based time series analysis has increasingly become a versatile tool for revealing the dynamic behavior in nonlinear systems.²⁵ The applicability of the complex network approach to chemically reacting and thermal fluids has been shown in recent review articles.^26,27 The main purpose of this study is to elucidate the effect of additive noise on the spatiotemporal dynamics of the nonlinear evolution equation incorporating the buoyancy effect from the viewpoint of complex networks. On the basis of our recent study,¹⁸ we adopt two important networks in this study: the ordinal partition transition network^1,2 and the flame front network.³ The former network, which considers the symbolic dynamics approach based on permutational coarse-graining, is effective in dealing with the dynamical behavior in a time series. The latter network, which was proposed by Singh et al.,³ is useful for expressing the wrinkled flame front configuration.

This paper is organized as follows. Section II presents a brief description of the numerical computation and analytical methods in terms of complex networks. We present the results and discussion in Sec. III. Finally, a summary is given in Sec. IV.

We numerically solve the noisy nonlinear evolution equation under a periodic boundary condition as bellow. Note that the nonlinear evolution equation without an additive noise term was theoretically derived in our recent study.¹⁸ 

Here, $H$ represents the flame front fluctuations, $Ra$ (= $gγ/(6ε))$ is the normalized Rayleigh number,¹⁸ $g$ is the gravitational acceleration, $γ$ is the thermal-expansion parameter, $ε$ is the parameter on the Lewis number of the reactants, $ζ(x,τ)$ is a Gaussian stochastic process with properties $⟨ζ(x)⟩=0$⁠, $⟨ζ(x,t)ζ(x′,τ′)⟩=2δ(x−x′)δ(τ−τ′)$⁠, and $σ$ is the strength of the additive noise. The flame front propagates downward (upward) at $Ra>0$ (⁠$Ra<0$⁠), and the gravitational effect stabilizes (destabilizes) flame wrinkling in positive (negative) gravity with $Ra>0$ (⁠$Ra<0$⁠). In this study, the system size $L$ is set to 100. $L$ is discretized into $NL=500$ points.

A pseudospectral method is adopted for the spatial derivative that uses the fast Fourier transform to transform the $H(x,τ)$ solution to Fourier space with wave number $k∈[−π/Δx,π/Δx]$⁠, where $Δx=L/NL$⁠. We evaluate the nonlinear term in real space and transform it back to Fourier space using the inverse fast Fourier transform. The numerical solution is then propagated in time using a modified fourth-order exponential time-differencing-time-stepping Runge–Kutta scheme²⁸ with time step $τ=0.1$ and sampling time interval $dτ=1$⁠. A random initial condition is imposed as $H(x,0)=ξ(x)$ with $⟨ξ(x)⟩=0$ and $⟨ξ(x)ξ(x′)⟩=2δ(x−x′)$⁠.

Ordinal patterns in a time series extracted by a symbolization technique have contributed to the sophistication of time series analysis. The central idea of ordinal (permutation) patterns was first introduced by Bandt and Pompe.²⁹ Inspired by the Bandt–Pompe methodology, Small and co-workers^1,2 proposed a transition network incorporating the concept of the Markov chain, referred to as ordinal partition transition networks. The importance of the ordinal partition transition networks has recently been shown by their group.^30–32 One of the authors has reported that the ordinal partition transition networks can capture the significant changes in combustion state.^33,34 On the basis of those studies^33,34 and similar to our recent study,¹⁸ we utilize the ordinal partition transition network in this study. For the construction of the ordinal partition transition network, the local temporal evolution of flame front fluctuations $H$ = $(H(τ),H(τ+1),…,H(τ+D−1))$ at $x$ = 0 is mapped into symbolic transition patterns, where $D$ corresponds to the rank order pattern length. The nodes in the network are represented by each permutation pattern $πi(i=1,2,…,D!)$⁠. The links between the nodes are expressed by the transfer probability $wij(=p(Γij))$ from the $i$th-order pattern to the $j$th-order pattern in a time series, where $Γij=πi→πj$⁠. By considering $wij$ between $πi$ and $πj$ as the definition of the information entropy, we estimate the transition network entropy $St(=−∑i=1D!∑j=1D!wijln⁡wij/(ln⁡D!2))$⁠.

Various entropies in the ordinal partition transition networks have recently been studied by McCullough et al.³⁰ and Sakellariou et al.³⁵ Unakarfov and Keller³⁶ have proposed the conditional entropy of the ordinal patterns in a time series, referred to as the conditional permutation entropy. The definition of $St$ in this study is basically identical to the conditional permutation entropy without the normalization by $ln⁡D!2$⁠. The relevance of the conditional permutation entropy to the ordinal partition transitional networks has been mentioned by McCullough et al.³⁰ In our preliminary test, forbidden patterns in the ordinal partition transition networks do not appear at $D=3$ for Brownian motion and white Gaussian noise with 10 000 data points, whereas they appear at $D=4$⁠. On this basis, we set $D=3$ for the estimation of $St$ in this study.

The natural visibility graph proposed by Lacasa et al.,³⁷ which directly transforms each point of a time series into nodes in an associated graph, has been adopted for various flame front and combustion instabilities.^38–41 Singh et al.³ proposed a flame front network constructed directly from the flame front shape using the visibility algorithm³⁷ and showed that the degree distribution in the network is useful for discussing the flame-turbulence interaction. In accordance with the work by Singh et al.,³ we construct the flame front network in this study. The node $vi$ in the network corresponds to the flame front position. Two nodes $vi$ and $vj$ are connected by links if the flame front does not exist on the two-dimensional line of sight from $vi$ to $vj$⁠. The adjacency matrix $Aij=1$ if $vi$ is visible to $vj$⁠; otherwise, $Aij=0$⁠. We estimate the degree $ki(=∑jAij)$ in the network.

Figure 1 shows the spatiotemporal patterns and extracted spatial variation in $H$ for different $Ra$ and $σ$ values. When $Ra=1$ and $σ=0.05$⁠, the flame front wrinkles owing to the formation of well-organized intrinsic cells with small-scales, forming low-dimensional chaos with arrayed structures. The low-dimensional chaos is retained for sufficiently small additive noise strengths but disappears with increasing $σ$⁠, leading to a noisy pattern. When $Ra=−1$ and $σ=0.05$⁠, high-dimensional chaos with a large-scale wavelength is created by the destabilization of the arrayed structures owing to the continuous interactions of cells. The high-dimensional chaos, which is formed as a result of the promotion of the Rayleigh–Taylor instability, vanishes at $σ=0.5$⁠, similarly to the case of $Ra=1$⁠.

Figure 2 shows the local temporal evolution of $H$ at $x=0$ for different $Ra$ and $σ$ values. $H$ exhibits aperiodic fluctuations at $Ra=1$ and $σ=0.05$⁠. The irregularity notably increases at $σ=0.5$⁠. We clearly observe the significant increase in the irregularity of $H$ with increasing $σ$ at $Ra=−1$⁠. Figure 3 shows the variation in $St$ as a function of $σ$ at different $Ra$⁠, together with the surface plot of $St$ as functions of $σ$ and $Ra$⁠. When $Ra=1$⁠, $St$ takes low values until $σ$ exceeds approximately 0.01, indicating that for sufficiently small noise strengths, the dynamic behavior of local flame front fluctuations remains low-dimensional chaos. It begins to take high values at $σ>0.02$ and becomes nearly constant at $σ>0.1$⁠. The saturated $St$ is approximately 0.96, indicating that the dynamic behavior of local flame front fluctuations at $σ>0.1$ is fully stochastic. Note that $St$ estimated for Brownian motion is approximately 0.96. In contrast, when $Ra=−1$⁠, $St$ is almost constant and is approximately 0.7 at $σ≤0.1$⁠, indicating that high-dimensional chaos is retained at higher $σ$ than low-dimensional chaos at $Ra=1$⁠. $St$ significantly increases as $σ$ exceeds 0.2 and reaches approximately 0.96. These results show that low-dimensional chaos at $Ra$ = 1 is more sensitive to changes in the additive noise than high-dimensional chaos at $Ra$ = –1. As shown in Fig. 3(b), a phase diagram of these different dynamical states clearly suggests a continuous transition between chaotic and stochastic regions.

We here discuss the determinism in $H$ subjected to additive noise in more detail. Various information entropies^29,30,35,36 incorporating the permutation patterns in a time series enable us to quantify the randomness in complex dynamic behaviors. However, they are not always valid for distinguishing between aperiodic deterministic and stochastic behaviors. The ordinal pattern-based analysis,^42–44 which estimates the number of forbidden patterns (missing patterns) in ordinal sequences that are obtained by the symbolization of a time series, is robust for distinguishing between these behaviors. Some forbidden patterns appear if nonlinear determinism is present in the temporal evolution of $H$⁠. In this study, we estimate the number of forbidden patterns in permutation patterns^43,44 (not transition patterns in the ordinal partition transition network) for different $Ra$⁠. The surrogate data method,⁴⁵ which is a reliable nonlinearity test in conjunction with time series analysis, is useful for examining the existence of determinism. The important point in the surrogate data method is the postulation of a null hypothesis on dynamic behavior in a time series that one aims to reject. The rejection of the null hypothesis indicates the possible existence of determinism in the original time series data. In this study, we also estimate $St$ for the surrogate data obtained by the iterative amplitude adjusted Fourier transformation (IAAFT).⁴⁶ Note that the null hypothesis for the IAAFT surrogate data method is that the nonlinear transformation of linear noise dominates the aperiodic fluctuations in a time series. Figure 4 shows the variation in the number of forbidden patterns $Nf$ normalized by the total number of possible permutation patterns (= $D$!) as a function of $σ$ at different $Ra$⁠, together with $St$ estimated for original and IAAFT surrogate data. Here, the red solid circles denote $Nf=0$⁠, whereas the red solid triangles denote the dynamical state for which the null hypothesis cannot be rejected. The number of surrogate datasets is 100. Similarly to a numerical study by Gotoda et al.,⁴⁷ $D$ is set to 5 for the estimation of $Nf$⁠. When $Ra=1$⁠, $Nf$ is approximately 0.65 at $σ=0.001$ and decreases with increasing $σ$⁠. It reaches zero when $σ$ exceeds approximately 0.03, indicating the loss of determinism in the dynamic behavior. The value of $St$ for the original data at $Ra=1$ takes significantly lower values than those for the surrogate data at $σ≤0.02$⁠, showing the possible existence of the determinism in the dynamic behavior. When $σ$ exceeds 0.03, it corresponds to those for the surrogate data, and the null hypothesis cannot be rejected. The important point to emphasize here is that the critical value of $σ$ for the loss of determinism nearly corresponds to that of $Nf$⁠. A similar trend is clearly observed at $Ra=0$ and $−1$⁠. The results shown in Figs. 3 and 4 lead us to the conclusion that our nonlinear evolution equation exhibits dynamical states that can be classified into a deterministic chaotic region, a region with coexisting chaos and noise, and a fully stochastic region. Note that the three classified regions appear in a noisy generalized Kuramoto–Sivashinsky equation describing a falling thin-film flow.⁴⁷ 

Figure 5 shows the degree distribution $p(k)$ in the flame front network for different $Ra$ and $σ$ values. When $Ra=1$ and $σ=0.05$⁠, the degree $kp$ at which $p(k)$ takes a local maximum value is approximately 35 and remains nearly unchanged at $σ=0.5$⁠. $k$ at $σ=0.5$ takes a higher maximum degree (⁠$km∼380$⁠) than that (⁠$km∼210$⁠) at $σ=0.05$⁠. For $k≥200$⁠, $p(k)$ at $σ=0.5$ is notably higher than that at $σ=0.05$⁠. When $Ra=−1$ and $σ=0.05$⁠, $kp$ is approximately 55 and remains nearly unchanged at $σ=0.5$⁠. $p(k)$ for $k>200$ at $σ=0.5$ is notably higher than that at $σ=0.05$⁠, similarly to the case of $Ra=1$⁠. $kp$ at $Ra>0$ (⁠$Ra<0$⁠) decreases (increases) with increasing $g$⁠. $kp$ remains nearly unchanged regardless of the increase in $σ$ for all $Ra$ values until $σ$ exceeds approximately 2. An important point to note here is that $kp$ at $Ra=−1$ is larger than that at $Ra=1$⁠. As shown in Fig. 1, high-dimensional chaos is formed when $Ra=−1$⁠. Our recent study¹⁸ has clearly shown that the formation of the small-scale wrinkles of the flame front is more marked for high-dimensional chaos than for low-dimensional chaos. On these bases, $kp$ is an important indicator for identifying the small-scale wrinkles of the flame front. These results show that for all $Ra$ values, the additive noise at $σ<2$ does not change the small-scale wrinkles of the flame front, whereas it causes the emergence of the large-scale wrinkles owing to the increase in $km$ with $σ$⁠. A numerical study using the MS equation²³ found that the additive noise produces giant cups in the flame front. This supports the emergence of large-scale wrinkles with increasing $σ$⁠. The promotion of the Rayleigh–Taylor instability with increasing $g$ has a significant impact on both the small-scale and large-scale wrinkles of the flame front. Fern$a´$ndez–Galisteo et al.⁴⁸ have recently conducted a numerical simulation on the dynamics of an upwardly propagating flame in a Hele–Shaw cell under various gravitational strengths. They observed that the flame front deformation with convex curvature toward the burnt gas becomes elongated with increasing gravitational strength. Their important finding⁴⁸ supports the significant increase in $kp$ with increasing $g$ at $Ra<0$ and $σ<2$⁠.

Similarly to our recent study,¹⁸ we here study the local maxima in $H$ in terms of $x$ to obtain a better understanding of the effect of the additive noise on the flame front dynamics. Note that the local maxima in $H$ correspond to cusps in a study carried out by Almarcha et al.¹⁷ Figure 6 shows the temporal evolution of the local maxima in $H$ in terms of $x$ for different $σ$ values at $Ra=−1$⁠. We clearly observe that at $σ=0.01$⁠, the appearance and disappearance of flame wrinkles occur by the merging and dividing of contiguous wrinkles. The flame wrinkles divide more rapidly with increasing $σ$ and interact randomly with each other. A large amount of additive noise can interfere with the mutual interactions of flame wrinkles, leading to the loss of local stripe-like structures. The variation in the spatially and temporally averaged distance $dmean$ between local maxima is shown in Fig. 7 for different $Ra$ and $σ$ values. Here, $dmean=(1/(τt−500))∑j=501τt⟨d⟩j$⁠, $⟨d⟩j$ is the spatially averaged distance at time $τj$⁠, and $τt=1500$⁠. $dmean$ increases with $g$ and $σ$⁠. Note that the amplitude of $H$ significantly increases with increasing $σ$ (see Fig. 1), which is associated with the increase in $dmean$⁠. This indicates that the formation of large-scale wrinkles in negative gravity, which is induced by the promotion of the Rayleigh–Taylor instability and the additive noise, plays an important role in the increase in $dmean$⁠.

Almarcha et al.¹⁷ have reported that the probability density function of the cell size, which was obtained by (i) an experimental study on a propagating propane/air premixed flame in a Hele–Shaw cell and (ii) a numerical simulation using the MS equation, can be expressed by the Gamma function. On this basis, we here discuss the probability density function $P(ξ)$ of the normalized distance between local maxima at $Ra=−1$ for different $σ$ values, where $ξ=d/⟨d⟩$ and $⟨d⟩$ is the ensemble-averaged distance. The obtained results are shown in Fig. 8. The distribution of $P(ξ)$ remains nearly unchanged regardless of the additive noise at $σ≤0.5$ and fits reasonably well with the following Gamma function, where $β=12$⁠:

In contrast, $P(ξ)$ at $σ=1$⁠, corresponding to a fully stochastic state, does not follow the Gamma function. As shown in Fig. 3, $St$ for $Ra=−1$ significantly increases beyond $σ∼0.1$ and takes approximately 0.87 at $σ=0.5$⁠. The distribution of $P(ξ)$ at $σ≤0.5$⁠, corresponding to the region with coexisting deterministic and stochastic states, can be expressed by the Gamma function. Radisson et al.²⁴ have shown in their model without thermodiffusive instability that even if the noise intensity significantly affects the mean cell size, the probability density function follows the Gamma function. Although the value of $β$ is larger than that in the previous studies,^17,24 these results suggest that the probability density function of the normalized distance between local maxima of the flame front obtained by our nonlinear evolution equation incorporating thermal-diffusive and gravitational effects can also be expressed by a Gamma function. Thus far, the effect of additive noise on low- and high-dimensional chaos in flame front dynamics has not been elucidated in many studies on nonlinear evolution equations.^{8–10,12,16,18,23,24} Our results will lead to a comprehensive understanding of the interaction between deterministic dynamics and additive noise during flame front instability in a Hele–Shaw cell under enhanced gravity.

Finally, in relation to noise-induced phenomena, there is an important point on dynamical transition of our nonlinear evolution equation. The Lorenz system is a well-recognized nonlinear dynamical system describing buoyancy-driven Rayleigh–Benard convection and can produce low-dimensional deterministic chaos. A numerical study⁴⁹ using a noisy Lorenz equation found that additive noise can give rise to low-dimensional chaos in a periodic oscillatory region. Following that study, we investigated here (not shown) the possible presence of noise-induced chaos in our system by the nonlinear forecasting methodology⁴⁷ based on orbital instability in phase space. We note that noise-induced chaos is not observed in the wide range of conditions of thermal-diffusive and gravitational effects.

We have conducted a systematic numerical study on the dynamical state of a noisy nonlinear evolution equation describing flame front dynamics in a Hele–Shaw cell by using analytical methods based on complex networks. The information entropy in the ordinal transition network and the degree distribution in the flame front network have been considered in this study. The high-dimensional chaos of flame front fluctuations at a negative Rayleigh number retains the deterministic nature for a sufficiently small additive noise. As the strength of the additive noise increases, the flame front fluctuations begin to coexist with stochastic effects, leading to a fully stochastic state. In contrast, a small amount of additive noise allows the low-dimensional chaos of flame front dynamics at a positive Rayleigh number to enter a fully stochastic state. The additive noise significantly promotes the irregular appearance of the merge and divide of small-scale wrinkles of the flame front at a negative Rayleigh number, resulting in the transition of high-dimensional chaos to a fully stochastic state.

This study was supported by the Tokyo University of Science Grant for International Joint Research and was also performed under the “Cooperative Research Project (No. R02/A31) of the Research Institute of Electrical Communication, Tohoku University.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The transition from chaos to stochastic dynamics occurs at lower noise levels for $Ra$ = 1 than for $Ra$ = $−$1 (see Fig. 3). This is clearly shown for an absolute scale of noise, but it would be important to examine whether this finding holds for relative noise levels. We here show the variation in $St$ in terms of a relative noise level defined as the ratio of the standard deviation of additive noise to that for the noise-free dynamic behavior of $H$ for $Ra=−1$ and 1 (see Fig. 9). Similarly to the distributions of $St$ obtained in Fig. 3, the transition from chaos to stochastic dynamics occurs at lower noise levels for $Ra=1$ than for $Ra=−1$⁠. In accordance with a recent study by Radisson et al.,²⁴ we display the noise level as an absolute scale in this study.