Chaotic Dynamics Near the Extinction Limit
of a Premixed Flame on a Porous Plug Burner
M. Gorman (gorman@uh.edu),
M. el-Hamdi
(el-hamdi@uhphys.phys.uh.edu)
Department of Physics
University of Houston
Houston, TX 77204-5506
K. A. Robbins (krobbins@runner.utsa.edu)
Division of Computer Science
The University of Texas at San Antonio
San Antonio, Texas 78249
Combustion Science and Technology 98 (1994)pp. 47-56.
Our previous experiments on the dynamics of premixed flames on circular porous plug burners have demonstrated a variety of periodic pulsating flames with different spatial and temporal characteristics. In the radial mode the circular flame front expands and contracts, periodically changing its radial extent. As the system parameters are adjusted near the extinction boundary, a sequence of transitions is observed. First, the radial mode loses its circular symmetry but remains periodic. Next, it undergoes a transition to a chaotic state. Finally, it makes a transition to extinction. All periodic pulsating modes are separated from the extinction boundary by a region of chaotic dynamics. The characteristics of chaotic flame dynamics beyond the extinction limit are discussed. The spatial and temporal characteristics of this chaotic mode are disucssed and compared with the relevant theoretical studies.
Our experimental observations (el-Hamdi, Gorman, Mapp and Blackshear, 1987) in methane-air flames on a circular porous plug burner have demonstrated a variety of periodic modes with different spatial and temporal characteristics. In this paper the chaotic dynamics of the radial mode near and beyond the extinction limit will be described. The spatial and temporal motion of the flame front will be correlated with features in the power spectrum of the light intensity emitted at a point in the flame front. This particular example exhibits some of the characteristics of the period-doubling transition to chaos, one of the simplest and most studied routes to chaos. The implications of chaotic dynamics for the determination of the extinction boundary will be discussed, and the experimental observation of the phenomenon of transient chaos will be described. All pulsating modes that border the extinction boundary become chaotic before extinction.
The experimental set-up is described elsewhere (el-Hamdi, Gorman, and Robbins, 1993). These experiments are conducted on a water-cooled, 5.62 cm circular, porous plug burner housed in a low-pressure combustion chamber made of process glass pipe. The flame is viewed by a Silicon Intensified Target Camera through a mirror at the top of the chamber. All these experiments were conducted with a dynamic pressure of 1/2 atmosphere. The emitted optical intensity of the flame front is monitored by imaging part of the flame front on a photodiode whose output was sent to a real-time spectrum analyzer where the dynamics is monitored. In a typical experiment the type of fuel and pressure remain constant while the flow rate and equivalence ratio are varied. A co-flow of nitrogen gas is used to reduce the shear between the premixed gas and the vacuum in order to suppress aerodynamics instabilities. Our previous measurements (el-Hamdi, Gorman, Mapp and Blackshear, 1987) of the radial, axial, and drumhead pulsating modes have outlined the stability boundaries of the modes as a function of these parameters.
In the thermodiffusive model of premixed flames pulsating flames arise from a competition between thermal diffusivity and mass diffusivity. Bayliss, Leaf and Matkowsky (1992) have conducted a theoretical study of the dynamics of methane-air flames near the extinction limit in a range of parameters which are directly relevant to the experiment discussed in this paper. They provide an extensive review of the relevant literature.
Certain papers played a crucial role in motivating our studies of the dynamics of premixed flames. Prior to 1980 it was generally thought that pulsating instabilities were not accessible for laboratory flames because the predicted Lewis numbers for the instability were prohibitively large. Margolis (1980) did a realistic numerical calculation for rich hydrogen- oxygen flames with heat loss to the burner and found periodic and period- doubled solutions prior to extinction. This result motivated Stephenson (1980) to investigate ammonia-oxygen flames near the extinction limit where he found axial pulsations in which the entire flame front oscillated along the burner axis. Buckmaster (1982) (see also Joulin, 1981) then calculated the effect of heat loss on the stability boundaries of pulsating flames and showed that larger values of the heat loss moved the stability boundaries closer to realistic values of the Lewis number.
The thermodiffusive model does not take into account the flow field, the thermal expansion of the gases or the effect of the porous plug. Both McIntosh and Clarke (1984) and McIntosh (1985) have incorporated these effects into a considerably more realistic model of a premixed flame stabilized on porous plug for one and two dimensional flame fronts respectively. The flow is assumed to obey a Darcy-type law within the porous plug. Such a theory allows direct, quantitative comparison between theory and experiment. These studies have presented the variation of the stability boundaries of pulsating flames with the paramters of the experiment, but they have not yet described the characteristics of the pulsating solutions that replace the steady state.
Our previous experiments (el-Hamdi, Gorman, Mapp and Balckshear, 1987) using methane-air flames found periodic modes in addition to the axial mode, such as the radial and drumhead modes, demonstrating that periodic pulsating modes occurred throughout parameter space near the extinction limit. In this paper we discuss the chaotic radial-extinction mode as a representative example of the chaotic dynamics which separates each periodic mode from the extinction boundary.
In these experiments the radial mode is observed in methane-air flames in a parameter range between total flow rates of 2.5 to 5.0 l/min and between equivalence ratios of 0.7 to 0.9. In the thermodiffusive model pulsating flames are found for Lewis numbers greater than one, corresponding to rich methane-air flames. Both Buckmaster (1982) and McIntosh (1985) showed that, for sufficiently strong heat loss to the burner, the stability boundaries of pulsating flames could be pushed to Lewis numbers less than one, as observed in our experiments.
At a pressure of 1/2 atm. a steady premixed flame is a luminous disk, 0.5 mm thick, which sits 5 mm above the porous plug. In the periodic radial mode the flame front changes its radial extent. Near the onset of the radial mode, the flame front uniformly expands and contracts with a small amplitude, on the order of 1/10 of the radius of the flame front. As the mode grows in amplitude, the dynamics becomes more nonlinear. The contraction of the flame front is uniform, but the expansion becomes more abrupt. A typical behavior is illustrated in Figure la which shows five successive frames of videotape that occur between two consecutive minima of the oscillation. In frame 3 the flame front extends over the entire burner surface. It smoothly changes its radial extent until it reaches a minimum in frame 5 with a radius half that of the maximum. The expansion part of the cycle is demonstrated in frames 1-3. In frame 1 the flame front is near a minimum, which increases slightly in frame 2 and then abruptly covers the entire burner in frame 3. The power spectrum (2.84 l/min, Air; 0.299 l/min, Methane) of the emitted light intensity, shown in Figure 2a, has a sharp peak at 6 Hz and its harmonics.
As the parameters are moved further toward the extinction boundary (2.6 l/min, Air; 0.28 l/min, Methane) the radial oscillation loses its cylindrical symmetry. The ten sequential frames of videotape shown in Figure lb demonstrate the observed motion. The flame front takes the form of a dogbone-like shape as shown in frame 1. This shape expands to a maximum in frame 3, which then contracts in frame 4 and reaches a second minimum in frame 5 with orientation (approximately) at right angles to that in frame 1. In the next half-cycle the flame front again expands and contracts, returning to the original orientation of frame 1.
In the language of nonlinear dynamics, the system has period-doubled. The power spectrum of this state has acquired another peak at half the fequency. The system takes twice as long, ten frames instead of five, to return to the original state. The time between minima of the contractions (frames 5 and 10) remains approximately constant, resulting in sharp peaks in the power spectrum at frequencies corresponding to the inverse time for the motion depicted in frames 1-5 (6 Hz), and 1-10 (3 Hz). Even though the motion has lost its cylindrical symmetry, it is still periodic.
As the parameters are varied still further (2.4 l/min, Air; 0.27, Methane), the motion of the flame front becomes chaotic. The contractions of the dogbone shape remain periodic, but the spatial orientation of the dogbone precesses in an irregular manner. The nonuniformity of the rotation is illustrated in Figure 3 which shows the dogbone shape at equal time intervals of 45 cycles which corresponds to 450 frames or 15 seconds. Figure 3 illustrates the nature of the chaotic rotation. There is slow rotation (from 3a to 3b and also from 3c to 3d), punctuated by abrupt changes (from 3b to 3c and also from 3e to 3f).
Figure 4 shows the angular orientation of sequential alternate minima of the dogbone shape (such as frames 1 and 10 in Figure lb over a long data run. These measurements were made from a video monitor using a superimposed angular grid. A visual estimate of the longitudinal principal axis was used to determine the orientation of the dogbone shape. The regions of abrupt change in the angular orientation are marked with arrows. The time between these regions and the magnitude of the change in angular orientation are not regular.
This chaotic rotation, and the accompanying variation of the measured intensity of the flame front, manifests itself as a broad peak in the power spectrum, shown in Figure 2b. Notice that the peaks corresponding to the radial expansion and contraction of the flame front are still present and remain sharp because the time between contractions remains constant. The exponential fall-off of the broad background is indicative of low-dimensional chaotic dynamics, and many of the properties of the chaotic (strange) attractor describing this chaotic state can be computed. The computed dimension of this chaotic attractor is 2.5 + 0.4, emphasizing the relative simplicity of this chaotic dynamics.
In the idealized mathematical description of period-doubling in a one-dimensional map, a periodic solution will undergo an infinite sequence of (forward) period-doublings as the stress-parameter is varied. This infinite sequence terminates in an accumulation point, beyond which another infinite sequence of reverse, or "noisy" period-doublings occurs in which the sharp peaks of the period-doubling are accompanied by broad peaks. The three observed states--periodic, period-doubled and period-doubled with chaos--are elements of this sequence.
In a physical system it is common to only observe parts of this sequence. Although period-doubling transitions have been observed in other fluid systems, the chaotic radial mode is one of the few examples in which correspondence between the spatial characteristics of the periodic and chaotic elements of the motion and the features of the power spectrum can be clearly demonstrated.
The characteristics of our experimental results exhibit the many of the qualitative features of the chaotic dynamics presented by Bayliss, Leaf and Matkowsky (1992). Their geometry is somewhat different from ours. They consider an annular geometry in which the flame resides in a region between concentric cylinders. Gas is fed in from the inner cylinder and extracted through the outer cylinder. As in our experiments, they examine the pulsating modes that arise for lean methane-air flames. They find a sequence of (three) period-doubling transitions and one chaotic solution as they changed the flow parameters.
The observed spatial characteristics of the flame front are in good agreement with these numerical results. Their periodic solution is sinusoidal near the transition point and becomes more nonlinear as the parameter is varied beyond the transition point. A similar behavior is observed in Figure la. In the first two frames the flame front is near its minimum. An abrupt expansion in which the flame front suddenly covers the entire surface is seen in frame 3, followed by a more gradual contraction in frames 4 and 5. This process then repeats.
In describing the nature of the first period-doubled solution, they find that the reaction zone compressed to a small spatial region in which rapid spatial variations in the temperature of the flame front occur. Such a description corresponds to frames 1, 5, and 10 of Figure lb. In the expansion phase of the cycle the reaction zone increases in size, producing a more gradual spatial variation in the temperature, corresponding to frames 4 and 8 of Figure lb. The essential feature of the period-doubled solution is that it is still periodic in spite of the loss of cylindrical symmetry.
The third dynamical regime is the chaotic oscillation. The spatial characteristics of the chaotic solution were not discussed, so it is not possible to make a comparison for this case.
Previously we have presented an extensive discussion of the use of power spectra in identifying deterministic chaos (el-Hamdi, Gorman and Robbins, 1993). Our measured power spectrum of the radial-extinction mode shown in Figure 2a has a clear exponential fall-off, indicative of low-dimensional deterministic chaos. This power spectrum was computed by averaging 16 sequential 4096-point spectra computed from a 70K data point time series corresponding to 1000 trips around the attractor. A long data run allows all regions of the attractor to be visited such that its average properties emerge. The time series of Bayliss et al. was not sufficiently long for their power spectra to exhibit such an exponential fall-off. In our experiments a real-time spectrum analyzer is used to directly monitor the dynamics. Our instantaneous experimental power spectrum of the radial- extinction mode resembles that of Bayliss et al..
Their reconstructed attractors, the one-dimensional nature of the Poincare section, and the computed Lyapanov exponents provide conclusive evidence that their computed solutions are chaotic. The standard techniques of chaotic dynamics can be applied to describe the chaotic motion of pulsating flames.
In the language of combustion science the flame front disappears at the extinction limit. In the language of nonlinear dynamics the chaotic radial mode, whose power spectrum is shown in Figure 2b and whose motion is depicted in Figure 4, is described by a chaotic attractor. The extinction boundary is defined as the set of points in parameter space at which this chaotic attractor loses stability to the fixed point attractor corresponding to extinction.
Just beyond the extinction boundary, the chaotic attractor is still present. Its existence can still be seen in the persistence of the chaotic motion of the flame front beyond the extinction limit. In dynamics terms all chaotic orbits now terminate on the fixed point. Some practical consequences of this situation have been observed in our experiments.
If the flame is in a chaotic state and the parameters are slowly changed beyond the extinction limit, the dynamics of the system will continue motion along the chaotic attractor (chaotic oscillations) for a time, known as the kick-out time, after which the oscillations decrease in amplitude as the solution spirals into the fixed point of extinction. Bayliss et al. demonstrate this phenomenon in Figure 10 of their paper.
Because the magnitude of the kick-out time is on the order of minutes, the extinction boundary is difficult to determine by a slow variation of the flow rates. Each change of the flow parameters can be made in seconds, so that a wait of ten minutes, or more, at each parameter value becomes quite cumbersome.
There is a subtle difference between slowly changing parameters and abruptly changing them. In a slow change the size of the attractor changes only slightly and the system almost always stays on its chaotic orbit. In an abrupt change of parameters the attractor suddenly changes its size and the system can suddenly find itself outside the chaotic attractor. The chaotic oscillations will then immediately cease, and the system will spiral into a fixed point.
This situation implies that the best way to precisely determine the extinction boundary is to make repeated abrupt transitions of significant magnitude to parameter values that are successively closer to the extinction boundary. These abrupt transitions have a higher probability of producing states that immediately decay to extinction.
The sensitivity of chaotic dynamics to initial conditions can also be demonstrated. If two "identical" (the initial and final flow rates have the same displayed values in both runs) experiments are performed, two significantly different results can be obtained. If the parameters are abruptly changed from values in the chaotic regime to values beyond the extinction limit, two different kick-out times, with substantially dlifferent magnitudes, seconds versus minutes, can be observed.
There are two reasons for this observed difference; both are related to the inability to conduct precisely identical experiments. The first reason is that the two points in parameter space are only specified to the accuracy of the experimental equipment. The flow rate is known to 0.1%. The second is that the exact time at which the abrupt change is made is not the same for the two experiments; so the parameters are never changed when the system is at the same point on the chaotic attractor.
The persistence of chaotic orbits into parameter regimes in which fixed points or limit cycles are the stable states of the system has been observed in other fluid systems (Gorman, Widmann, and Robbins, 1986). The resulting finite lifetime of the chaotic dynamics is called transient chaos.
All of these extinction-related phenomena have been demonstrated in our experiments. A quantitative, systematic study requires the re-ignition of the flame front for each of the many data runs.
The qualitative agreement between our experimental results and the theoretical description of Bayliss, Leaf and Matkowsky using the thermodiffusive model suggests that their implementation of this model provides a good description of the dynamics near the extinction limit. The theoretical study of McIntosh (1985) using a realistic model of a porous plug burner and the hydrodynamical model has established the parameter range in which chaotic pulsations should be observed. A nonlinear analysis using this model is necessary in order describe the characteristics of the nonsteady solutions.
Our measurements indicate that the entire boundary between the radial mode and the extinction limit is filled with variations of this general kind of dynamics. In other regions of this parameter space more elements of the period-doubling sequence, corresponding to more complicated spatial motions of the flame front, are observed. The flame front assumes ovoid shapes, qualitatively different from the dogbone shape presented here. As in this example, there is a range of parameters in which the flame front is temporally periodic but spatially doubly-periodic, and another range in which it is temporally periodic and spatially chaotic.
These experiments were part of a survey of parameter space. Other pulsating modes of premixed methane-air flames that border the extinction boundary have also been observed to exhibit chaos. These chaotic modes have considerably different spatial characteristics (for instance, they maintain a symmetry close to circular) from those of the radial-extinction mode.
Our results indicate that it is possible to undertake reproducible, quantitative measurements of the spatial and temporal characteristics of the chaotic dynamics of flames near and beyond the extinction limit. The ideas of nonlinear dynamics and deterministic chaos are essential for a proper description and understanding of these experiments.
This research was supported by a grant, N00014-K-0613, from the Office of Naval Research. We are especially grateful for numerous useful conversations with Bernard Matkowsky, Alvin Bayliss, Stephen Margolis and Gregory Sivashinsky and have benefited from discussions with John Buckmaster and Forman Williams. Manickam Neelakandan and Brian Pearson assisted in the measurements of the orientation of the chaotic radial mode and in preparation of the figures.
[1] Bayliss, A., Leaf, G. K., and Matkowsky, B. J. (1992). Pulsating and Chaotic Dynamics Near the Exinction Limit, Combust. Sci. and Tech. 84, 253
[2] Buckmaster, J. D. (1982). Stability of the Porous Plug Burner Flame, SIAM J. Appl. Math. 43, 1335.
[3] el-Hamdi, M., Gorman, M., Mapp, J. M. and Blackshear, J. I. (1987). Stability Boundaries of Periodic Modes of Propagation in Burner-Stabilized Methane-Air Flames, Combust. Sci. and Tech. 55, 330.
[4] el-Hamdi, M., Gorman, M., and Robbins, K. A., (1993). Deterministic Chaos in Laminar Premixed Flames: Experimental Classification of Chaotic Dynamics, Combust. Sci. and Tech., 94, 87.
[5] Gorman, M., Widmann, P. J., and Robbins, K. A. (1986). Nonlinear Dynamics of a Convection Loop: A Quantitative Comparison of Experiments with Theory, Physica l9D, 255.
[6] Joulin, G. (1981). Conductive Interactions of a Wrinkled Flame with a Cold. Flat Burner. Combust. Sci. and Tech. 27. 83.
[7] McIntosh, A. C. (1985). On the Cellular Instability of Flames near Porous Plug Burners, J. Fluid Mech. 161, 43.
[8] McIntosh, A. C. and Clarke, J. F. (1984). Second Order Theory of Unsteady Burner-Archored Flames with Arbitrary Lewis Numbers, Combust. Sci. and Tech. 38, 161.
[9] Margolis, S. B. (1980). Bifurcation Phenomena in Burner-Stabilized Premixed Flames. Combust. Sci. and Tech. 22, 143.
[10] Stephenson, D. A. (1980). Combustion Research Facility Newsletter 2, No. 4, Sandia National Laboratories.
