Four Types of Chaotic Dynamics
in Cellular Flames
A principal concern of the considerable experimental work since Markstein's study has been the measurement of the size of the cells as a function of the parameters, such as pressure, fuel type, flow rate and equivalence ratio (Mitani and Williams, 1980; Vantelon, Pagni and Dunsky, 1986). Most studies were conducted at atmospheric pressure, and the parameters were chosen to produce large numbers of cells in order to minimize the effects of the boundaries. There is little mention of the dynamics, except to note the high degree of cell motion (Markstein, 1964) and the variation in cell size which necessitates averaging.
The incessant motion of cellular flames was first explained by Sivashinsky (1983) who performed numerical studies of the two-dimensional Kuramoto-Sivashinsky equation. He showed that the temporal evolution of a 5x5 square lattice of cells appeared quite similar to the dynamics observed in cellular flames. He used the term, "chaotic self-motion", to describe this nonperiodic motion which was intrinsic to the flame dynamics.
We have reported new experimental observations (Gorman, el-Hamdi and Robbins, 1993) in which cellular flames on circular porous plug burners form ordered states consisting of concentric rings of cells. In the parameter range in which these ordered states are found, four kinds of chaotic dynamics are observed: 1) ordered states in which ordered rings of cells subtly change their shape and size, but not their average position; 2) disordered states in which the ring structure is broken, and the cells move around in an irregular manner; 3) intermittently ordered states in which concentric rings of cells abruptly appear, mostly for short times, but occasionally for very long times; and 4) pulsating-cellular flames in which the pulsating radial state interacts with an ordered cellular state. Each of these examples of chaotic motion arises from dynamics intrinsic to cellular flames. In this paper we describe, compare and contrast the spatial and temporal characteristics of these four types of dynamics. Some relevant theoretical studies are also discussed.
Our experimental set-up has been described elsewhere (el-Hamdi, Gorman and Robbins, 1993). A 5.62 cm circular, water-cooled stainless steel porous plug burner, designed by Patrick Pagni (Vantelon, Pagni and Dunsky, 1986) and manufactured by McKenna Products of Pittsburg, CA, is placed in a low pressure combustion chamber made from process glass pipe. The flow rate of the premixed gases and the ambient pressure of 1/2 atm. in the chamber are controlled to 0.1% using MKS Instruments controllers. At this pressure a steady flame front is 0.5 mm in thickness and sits 5 mm above the porous plug. Ordered states with cell numbers from eight to thirty are easily stabilized using isobutane-air or propane-air mixtures.
A camera views the flame through a mirror at the top of the chamber. In the printing of the individual frames the regions between the cells appear black because of the limited dynamic range of videotape. These cusps and folds are not regions of extinciton; rather, they correspond to regions of lower optical intensity than the brighter cells.
The motion of the flame front is recorded on videotape. Individual frames are digitized, enhanced and printed to depict the motion. A small region of the flame front is imaged on a photodiode whose output is sent to a real-time spectrum analyzer for direct monitoring of the dynamics or to a microcomputer for later analysis. The power spectra presented in this paper are 16 averages of 4096-point spectra.
In section 2 the spatial and temporal characteristics of representative examples the four types of chaotic states are presented. Their stability boundaries in parameter space are described. In section 3 the theoretical studies relevant to each chaotic state are reviewed. In section 4 the implications of these results are discussed.
The five successive frames of videotape, which run down the page, indicate a steady pattern. However, the power spectrum of the light intensity shown in Figure 2a, has a broad background extending from 0 to 25 Hz, with a shoulder about 7 Hz. Such a power spectrum is characteristic of every ordered state.
The temporal evolution of this state is difficult to depict on paper. The motion of the individual cells does not change the overall appearance of the ring pattern so successive frames of videotape look identical. An image processing technique is used to demonstrate the motion of the cells. In Figure 3 two frames of the 12/6/1 state, separated by nine frames, are digitized and then subtracted. The absolute values of the differences are displayed as bright regions and the resulting image is multiplied by fifteen. The final differential image, displayed in Figure 3, shows that the intensity differences are strongest near the cusps and folds at the boundaries of the cells. All the observed ordered states exhibit these small amplitude chaotic oscillations in which individual cells change their size and shape in a subtle, irregular manner.
As the equivalence ratio is varied, the onset of disordered state is observed at different values of the flow rate and with different numbers of cells in the inner and outer rings. The transition line between the ordered and disordered states, the "melting curve", is difficult to determine because of the uncertainty in measuring the onset of the disordered states. Experience from our survey of parameter space suggests that at least two rings of cells are required before a transition to a disordered state is observed.
The measured intensity variation of the disordered states is approximately ten times larger than that of the ordered states. The power spectrum shown in Figure 2b has a high frequency fall-off similar to that of an ordered state. The cutoff frequency has increased to 50 Hz.
In order to demonstrate the characteristics of this dynamics more clearly, fifteen sequential frames of the intermittently ordered 9/3 state have been printed in Figure 4. An ordered 9/3 state is visible in frame 1. In frames 2 through 12 the ring structure is not identifiable, and it is only the outer boundary of the luminous area that suggests the circular geometry of the burner. In frames 13 and 14 the outer ring can be seen, but there is no hint of the presence of the three inner cells which abruptly reappear in frame 15.
The visual impressions of the disordered and intermittently ordered states are quite different. In the disordered state an approximately ordered state is seen all the time; but there is never a time at which the ordered state threatens to reappear. In the intermittently ordered state a rapidly changing, highly irregular cellular pattern, which does not even remotely suggest a ring structure, is present most of the time. An ordered state seems to suddenly materialize out of nowhere and remains for varying lengths of time. Most of the residence times of the ordered states are short, on the order of ten frames, but some are quite long, persisting for hundreds of frames. A histogram of the residence times has an exponential distribution, with a characteristic time of about 2 seconds (60 frames).
The disordered states are always observed at a flow rates larger than those of the ordered states. Once this critical value of the flow rate for the onset of disordered states has been passed, an ordered state never occurs again. The intermittently ordered states are observed at flow parameters that are interleaved between those corresponding to the ordered states. Their relative occurrences are shown in Figure 5 and Table 1.
Figure 5 is a composite picture depicting the occurrence of the intermittently ordered states with increasing flow rate in propane-air mixtures. These states are prevalent in propane-air cellular flames; they have not been observed in isobutane-air flames. In this figure a temporal sequence of frames, corresponding to the motion of each observed state, is displayed horizontally and the succession of states observed with increasing flow rate is displayed vertically. Ordered states with 1, 3, 4, and 5 inner cells can be seen respectively in rows a, c, e, and g. In alternate rows--b, d, f, h-- the corresponding intermittently ordered states are observed.
At flow rates above 7.35 lit/min, corresponding to the 9/3 ordered state, an intermittently ordered state is observed in which the 9/3 state abruptly appears, for varying lengths of time. Other ordered states (not shown) also appear. Ordered states in which the number of inner and/or outer cells changes by one cell, such as the 10/4, 9/4, or 10/3 states, are also observed. Different relative orientations of the inner and outer rings in the ordered states also occur, as can be seen by comparing frames 1 and 5 in row d of Figure 5. As the flow rate is increased, a state with four inner cells appears more often and for greater lengths of time, until at a flow rate of 8.39 lit/min the ordered 4-cell state becomes stable.
| Figure | State | Flow Rate | Flow Velocity | Eq. Ratio |
|---|---|---|---|---|
| 5a | 9/1 | 6.32 | 4.16 | 1.83 |
| 5b | 1HC | 6.49 | 4.27 | 1.85 |
| 5c | 9/3 | 7.35 | 4.84 | 1.85 |
| 5d | 3HC | 8.27 | 5.44 | 1.85 |
| 5e | 10/4 | 8.39 | 5.52 | 1.85 |
| 5f | 4HC | 8.62 | 5.67 | 1.85 |
| 5 | 11/5 | 8.80 | 5.79 | 1.85 |
| 5h | 5HC | 9.38 | 6.17 | 1.85 |
| 6a | Pulsate | 5.16 | 3.39 | 2.22 |
| 6b | Puls-Cel | 5.52 | 3.63 | 2.22 |
| 6c | Cellular | 6.10 | 4.01 | 2.21 |
The absence of an ordered state with two inner cells in this sequence of states in propane-air flames is related to our previously reported observation of the absence of an ordered state with two inner cells in isobutane-air flames. It would be expected that such an ordered state would be observed between those parameter values corresponding to ordered states with one and three inner cells. For a reason that has yet to be clearly identified, ordered states of cellular flames with two inner cells are not stable states of the system.
The periodic nature of this mode can be seen in the rhythmic nature of the expansions and contractions. The nonperiodic nature of the mode manifests itself by the varying position and the changing number of bright regions that appear on successive expansions. The power spectrum of this mode shows both sharp peaks due to the radial pulsation as well a broad background due to the chaotic motion of the cells .
At larger values of the flow rate a 9/1 ordered state of cellular flames is observed with no pulsations. At lower values of the flow rate, the cellular structure is not present, and the flat flame front pulsates in the radial mode in which it periodically and uniformly changes its radial extent. At still lower values the flame front extinguishes. The occurrence of these states with increasing flow rate is shown in Figure 6 a-c. A steady flame front is not observed at equivalence ratios exhibiting the pulsating-cellular interaction.
Bayliss and Matkowsky (1992) and Bayliss, Matkowsky and Riecke (1993) have published studies of numerical simulations using full equations that describe the thermodiffusive instability. In both studies they considered a one-dimensional circular (ribbon) flame concentric with a source of premixed gas directed outward along the radius. Steady ordered states with cell numbers from three to nine were found in the parameter range considered. Our preliminary experiments using a porous plug in the shape of a slot have also found that a one-dimensional ordered array of cells is steady and does not have chaotic motion, suggesting that the motion observed on the circular burner is a two-dimensional effect.
Nicolenko (1993) conducted numerical studies of the two-dimensional Kuramoto-Sivashinsky equation using a square geometry with periodic boundary conditions. His results capture four elements of the phenomenology of ordered states observed in our experiments: the metastability of a state with two inner cells, a maximum of six cells forming around a point, a transition from order to disorder, and chaotic dynamics of ordered states with motion concentrated around the cusps and folds.
The geometries of these studies are different: these experiments use a circular porous plug burner, Shtilman and Sivashinsky (1990) considered a square geometry with a large number of cells, Bayliss and Matkowsky (1992) considered a circular line propagating radially inward, and Nicolenko (1993) considered a square geometry with small number of cells. Each of these predicted disordered states is qualitatively similar to the experimental results shown in Figure lb.
The numerical simulations of Sivashinsky (1983) using the thermodiffusive model and of Michelson and Sivashinsky (1977, 1982) using the hydrodyanmical model both found a state in which the cells continually and chaotically recombine. It is not clear from the visual appearance of these states whether these numerical results correspond to a disordered state or to an intermittently ordered state.
In the language of dynamics the evolution of the system in time is described by the motion of a point in state space. The signature of heteroclinic connections is that the system visits the neighborhood of ordered states for varying lengths of time.
Each ordered state corresponds to a fixed point which becomes unstable as the total flow rate is increased. For isobutane-air flames there is always an ordered state (with different numbers of inner and outer cells, corresponding to a different point) that becomes stable and no intermittently ordered states are observed. For propane-air flames there are ranges of flow rates in which ALL the fixed points are unstable. The system is attracted to the neighborhoods of these fixed points where it is repelled along the unstable direction. An orbit which connects different fixed points is called a heteroclinic orbit, hence the name, heteroclinic connection.
In an experiment an intermittently ordered state is observed when the system passes near one of the unstable fixed points. The system can remain in the vicinity of the fixed point for various lengths of time corresponding to a range of residence times. When the system moves outside the neighborhood of the fixed point, it is moving along the unstable manifold connecting the fixed points and the ordered state is no longer discernible. Such motion is depicted in frames 2-13 of Figure 4. Notice the qualitatively different spatial character of this state compared with that of the disordered state.
Although heteroclinic connections are believed to occur widely in a variety of fluid flows (Moffatt, 1986), the spatial and temporal characteristics of a fluid exhibiting this dynamics has not been studied in detail. None of the numerical studies of cellular flames has identified similar dynamics although heteroclinic connections are well known in studies of the one-dimensional Kuramoto-Sivashinsky equation (Kevrekidis, Nicolenko, and Scovel, 1990). The studies using the hydrodynamical model of cellular flames have not specifically discussed heteroclinic connections.
Margolis (1980) used realistic kinetics to model a hydrogen-oxygen flame in the presence of a burner. His numerical results of an oscillating flame front suggested that pulsating flames were indeed accessible in laboratory experiments. Our previous experiments on methane-air flames showed that pulsating flames occur throughout parameter space and that a variety of spatial and temporal characteristics are possible.
Buckmaster (1982) included the effect of heat loss to the burner and computed the changes to the stability boundaries of both pulsating and cellular flames. He showed that the stability boundaries are pushed toward each other and, for sufficiently strong heat loss, can be made to overlap, indicating an interaction between the two types of modes. The water-cooled porous plug Pagni burner provides a sufficiently large heat loss to move the stability boundaries for pulsating flames to accessible values of parameters, including an overlap with those of cellular flames. Another consequence of this overlap is that a steady flame front is not observed as the flow rate is varied. McIntosh (1985) made similar findings using the hydrodynamic model as the source of the instability. He incorporated a realistic model of the flow created by the porous plug.
Pulsating-cellular states are found in a parameter range near the onset of cellular flames, for both isobutane and propane flames. The radial mode, which occurs for lean methane-air flames, is found near the extinction boundary in rich propane-air and isobutane-air flames, in agreement with the thermodiffusive model. The interaction between the radial mode and the ordered cellular state can be adjusted by varying either the flow rate or the equivalence ratio. The spatial character of the pulsating-cellular state changes with the strength of the interaction.
Experimental factors also contributed to the observed irregular motion. First, with such a large number of cells, there is a competition between the natural hexagonal structure associated with the unbounded system and the ring of cells at the circular boundary. Second, most other studies were conducted at atmospheric pressure in an open room. The presence of a diffusion flame and the convection of the room air strongly perturb the cellular pattern. Third, we cannot overstate the importance of the design and construction of the Pagni burner for the observation of the ordered states. In other experiments (by us) using a porous plug with a square geometry, the ordered states were never found. The cellular flames had a motion that was reminiscent of pictures of the irregular structures reported in most other experiments.
The disordered states, the intermittently ordered states and the pulsating cellular states described here all bifurcate from the ordered states. Their dynamics can be found in the solutions to the equations that describe the flame front. The nonperiodic dynamics observed in previous experiments occurs because of boundary conditions and external noise which cause the system to constantly make transitions among ordered states with slightly different (large) numbers of cells. This motion overwhelms the intrinsic chaotic motion and makes it unobservable.
The chaotic motion of the ordered states in which ordered cells subtly change their shape and size is the most unusual and unexpected aspect of this study. This motion causes subtle, but important effects in the other states of cellular flames. The experimental data conclusively establishes that low amplitude, nonperiodic oscillations are characteristic of ordered cellular states. The difficult question is whether these oscillations are chaotic and intrinsic to the structure of cellular flames, or whether they are extrinsic, due to the amplification of external noise (from nonaxial velocity components of the premixed gases) by the cellular structure. The answer to this question involves highly technical issues in dynamics. The preponderance of our experimental evidence to date suggests that the chaotic motion is intrinsic, and that it is associated with the unique topological nature of the cusps and folds of cellular flames.
The current tools of analysis of chaotic systems grew out of studies of the properties of strange attractors that describe the dynamics arising from ordinary differential equations. The pictures in Figures 1, 4, and 5 demonstrate the considerable differences among the spatial characteristics of these states; yet none of the current measures of chaotic dynamics distinguishes among these four states. Calculations of the dimension of the chaotic attractor yield inconclusive results in each case. The power spectra have similar high frequency fall-offs. These techniques work for "low-dimensional" dynamical systems, such as chaotic pulsating flames which are described by coupled sets of ordinary differential equations, but fail badly when applied to more complicated systems, such as cellular flames, which are described by partial dlifferential equations.
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