Does Your Flame Flicker?

Constant flickering and smoking occur when the teardrop shape of the flame is disturbed. This allows small amounts of unburned carbon particles (soot) to.
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Frustration makes it impossible that every pairs chosen from the three-coupled oscillators goes into anti-phase simultaneously because, when two pairs of the three synchronise in the anti-phase mode, respectively, the remaining third pair must synchronise in the in-phase mode. Nevertheless, frustration plays an important role in the dynamics of three-coupled oscillators 5 , because it may cause symmetry breaking that drives certain asymmetric synchronised modes 6. Earlier studies have demonstrated that several non-trivial mode patterns were observed in three-coupled oscillators of physical 7 , 8 , 9 , 10 , chemical 11 , and biological systems 12 , 13 , 14 as well as human group dynamics Since the symmetry-based understanding is model independent, it is expected to hold for synchronisation of three-coupled candle flames.

In the present work, we investigated the synchronised oscillations of three-coupled candle flames, wherein three oscillating flames are positioned at the vertices of an equilateral triangle. Through image analysis of the three-coupled flame dynamics, we examined the kind of synchronised modes observable in the system and their consistency with the symmetry-based understanding.

We also studied the relationship between the inter-flame distance i. We used candles made of paraffin, obtained from a grocery store in Japan, having a thin cylindrical shape with 9. To make an oscillator, we bundled three of the thin candles in parallel with tape; as a consequence, three wicks come close enough to each other so that the three individual flames merge at ignition.

Next we painted side surfaces of every bundle in black for convenience in our image analysis. Afterwards, these three bundles, each of which consists of three candles, are erected vertically on a table and equi-separated; the tips of the three bundles are positioned at the vertices of an equilateral triangle. Figure 1 shows a schematic representation of the experimental setup. A two-minute-long movie was recorded in grayscale per experimental trial, and then it was analysed to pursue the time variation in the oscillating flame sizes.

Given a grayscale image, in general, the brightness of each pixel is represented by an 8-bit integer ranging from 0 pure black to pure white.

The three-coupled candle flames turned out to exhibit four classes of synchronised oscillation modes. Essential features of the four modes are explained below in turn. The first synchronisation to be mentioned is the in-phase mode, in which all the three flames oscillate with an identical wave form. Figure 2 a demonstrates the flame dynamics of the in-phase mode in a one-second duration. The mean value deduced from many trials on the in-phase mode was Figure 2 b shows a snapshot of the oscillating flames in the in-phase mode.

During the oscillation, the visible parts of the flames are elongated and contracted periodically in the vertical direction. In addition, a transverse oscillation was observed; bowing to the center of the triangle left panel and arching backward right panel were repeated in the same frequency as that of the vertical oscillation. A footage can be found in the Supplementary Video S1. All the three oscillators obey an identical waveform. It should be noted that the in-phase mode observed in three-coupled candle flames is unstable.

The sustained time duration is only a few seconds at most, even when the system is free from external air flow. Moreover, the degree of synchronisation dissipates as time passes, followed by the occurrence of another synchronisation mode or the complete disappearance of synchronisation.


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This instability is in contrast with the stable oscillation in a single and paired candle flames 1. Another important feature of the time series variation shown in Fig. It is observed that all the three waveforms deviate significantly from a sinusoidal wave. That is, the waveform for a single period is slanted; the curve shows a gentle incline from a minimum e. The asymmetry in the waveform indicates the fact that a little larger time duration is required for the flames to complete a single period of elongation, compared with the time duration required for them to complete a single period of contraction.

The second synchronisation mode is the partial in-phase mode. The result indicates that the middle and right flames oscillate in nearly the same timing but the left flame oscillates with a delay of half a period. The snapshot presented in Fig. We have observed in experiments that, when the partial in-phase mode occurs, there seems to be no preference of a pair among the three oscillators to become in-phase coupled over the other two pairs.

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The pair was chosen stochastically and thus changed every experimental trial. Similarly to the case of the in-phase mode, the partial in-phase mode is also unstable and endowed with asymmetry in the waveform. The oscillation frequency is estimated to be of The alternately oscillating behaviour is visually confirmed by observing the footage in Supplementary Video S4. The rotative direction, clockwise or counterclockwise, was randomly chosen, as was confirmed by experiments.

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Again, the instability of synchronisation and asymmetry in the waveform were both observed in the rotation mode, too. The last but the most interesting synchronisation mode is the one demonstrated in Fig. In the death mode, no flame oscillates; all the three flames fall into stable combustion. The observation presented in Fig. Every flame stops oscillation and falls into self-sustained stable combustion, showing a markedly long and slender shape. The death mode involves two salient features: Firstly, the death mode is significantly robust and self-sustained, differing from the other three synchronisation modes.

Once the system falls into the death mode, it persists in a duration of several minutes or more. Even when external disturbance e. The panel shown in Fig. After the onset of the death mode left panel , we imposed crosswind to the flames several times during five minutes not shown. Nevertheless, the death mode turned out to regenerate cyclically with no sign of disappearance right panel and persists even after that. The second feature of the death mode is the slender shape of the combustion flame. In the death mode, the lateral width of the flame at the most bulge part i. This value is significantly smaller than the lateral width of the oscillating flames.

In addition, the vertical length of the flame in the death mode is significantly larger than those of oscillating flames. This value is quite larger than the vertical length of the oscillating flames. In short, the death mode provides absolutely still flames with a vertically long and laterally slender shape. It should be emphasized that the death mode we have observed is essentially different from a mere temporal cease of the flame oscillation.


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    8. In general, combustion flame often shows a transient disappearance of fluctuation in the height of the flame. However, the mere transient disappearance of fluctuation is neither endowed with the self-sustaining property against external perturbation nor the slender shape with pronounced vertical height.

    In this sense, the death mode we have observed is a consequence of the coupling between candle flames, so-called a limiting case of the synchronised mode with no oscillation amplitude. The previous section focuses on the essential properties of four synchronised modes that the three-coupled candle flames have exhibited. It should be stressed that these four modes do not occur with equal probability; instead, which synchronised mode is preferred to occur is dependent on the inter-flame distance. Furthermore, a growth in will cause a decline in the coupling strength between the oscillating flames; therefore, the rate of occurrence of synchronisation is expected to decrease with increasing.

    To make clear the issue raised above, we have repeated many times the combustion experiment using different sets of three bundles, and examined the occurrence rate of synchronisation, P sync. This procedure allowed us to estimate P sync over the 48 samples or more of one-second movies for the given. Figure 6 presents the dependence of P sync on we have detected in experiments. The result clearly shows the decreasing trend of P sync with increasing as expected. This implies that the coupling strength between the oscillating flames is not a monotonic function of ; other physical or chemical factors may be responsible for the coupling strength, at least when three candle flames are more closely positioned.

    Figure 7 gives a chart that illustrates the probability with which each synchronised mode occurs for a given inter-flame distance.

    Flickering Flame

    The dominancy of the death mode in the chart is thought to stem from its self-sustained property and the instability inherent to the other three modes. This is attributed to the need of sufficiently strong coupling strength for the in-phase mode to take place, similarly to the case of paired oscillating flames 1 , 3.

    We have indeed checked that a much smaller tends to increase the rate of occurrence of the in-phase mode, although not shown in the chart because of difficulty in image processing; For the smaller , the bright areas of two adjacent flames looked overlapping in grayscale images, which made it difficult to distinguish between the two flame sizes.

    From a theoretical viewpoint, it would be interesting to capture the possible synchronised modes using the symmetric Hopf bifurcation theory 6. The theory provides an elegant mathematical approach for describing pattern formation in coupled oscillator systems. The distinct feature of the theory is the model independence.

    Synchronization in flickering of three-coupled candle flames

    In fact, in order to predict the possible synchronised mode, it suffices to know only geometrical symmetry of the system. This feature allowed to apply the theory to a wide variety of coupled oscillator systems and to obtain the list of possible synchronised patterns: It is natural to consider, therefore, whether or not our experimental results are consistent with those predicted from the symmetric Hopf bifurcation theory.

    A central hypothesis of the symmetric Hopf bifurcation theory is the presence of a network of n -coupled identical oscillators. The n -coupled oscillators are assumed to be in an equilateral n -gonal arrangement and to be endowed with identical couplings between the oscillators.

    Here, the dihedral group, D n , refers to the symmetries of n -gon, and S 1 denotes the circle group that represents the translational symmetry with respect to time. Specifically in the case of three-coupled oscillators, four synchronised patterns are predicted to take place: Our experimental results are in fair agreement with the theoretical prediction, in the sense that the former three synchronised patterns have been observed in the three-coupled candle flame oscillation.

    An exception is the last one of them, the partial anti-phase pattern. In our experiments, however, this pattern did not take place. Another noteworthy fact is the occurrence of the death mode in the three-coupled candle flames. In view of the symmetry-based understanding, the death mode may be recognized as an extreme case of the in-phase mode, since all the three oscillators exhibit identical time sequences. In summary, we have demonstrated that the symmetric Hopf bifurcation theory is effective to provide a list of possible synchronised patterns generated by three-coupled candle flames, while the partial anti-phase mode does not appear in experiments.

    As a final remark, we propose a conjecture on the mechanism of candle flame flickering. It should be stressed out that, despite the commonly-observed phenomena, the mechanism of candle flame flickering has not been entirely clarified. A hint for solving the problem can be taken from similar flickering phenomena observable in burner flames 19 , 20 , as previously discussed in ref. In the case of burner flames, flickering is attributed to the heated jet flow from the injection nozzle and the subsequent formation of large vortices nearby the flame surface Namely, spatial inhomogeneity in the flow velocity from the flame core to the ambient still air induces an instability in the flame surface, forming toroidal vortices immediately outside the luminous flame.

    The vortices stretch the flame vertically and create a detached puff Periodic detachment of the puff at the flame tip followed by regrowth of the main flame from the nozzle results in a flickering appearance of the burner flame. The film shows British society's shift to the right and documents the taking away of rights that had been bitterly fought for in the past. Above all, the film shows the dignity of the Liverpool dockers and of their wives. It's a deeply human film that shows how workers who had given their lives to their jobs were unfairly dismissed simply for sticking to their beliefs.

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