Spiral Dynamics of Pulsating Methane-Oxygen Flames
on a Circular Burner
Kay Robbins
Department of Computer Science
University of Texas at San Antonio
San Antonio, TX 78249
And
Michael Gorman, Jill Bowers, and Robert Brockman
Department of Physics
University of Houston
Houston, TX 77204-5005
Abstract
A premixed flame stabilized on a circular porous plug burner produces
a uniform, steady luminous flame front. Throughout much of the parameter
range hydrocarbon-oxygen mixtures form spiral-shaped fronts. In methane-
oxygen flames at low pressure the flame undergoes a sequence of
bifurcations: 1) periodic rotation of a spiral front; 2) precession in a direction
opposite to its rotation, corresponding to doubly-periodic “petals-out”
meandering; and 3) intermittent jumps corresponding to linear excursions of
the tip, which occur after the spiral front has reached the boundary of the
circular burner. We use Karhunen-Loeve (KL) analysis to find the
coefficients of the dominant KL spatial eigenfunctions. Their phase space
portraits and power spectra provide a description of the dynamics as flow
rates are reduced and the system destabilizes. We discuss how these
experimental results relate to previous theoretical studies that assume
Euclidean symmetry for the experimental configuration.
1
1. Introduction
Spiral motion has been the subject of experimental1-3 and theoretical
studies4,5 in many types of media, especially excitable media, ranging from
closed chemical systems described by the Belousov-Zhabotinsky reaction to
cardiac arrhythmias. Many of these studies have observed periodic dynamics
of spiral arm rotation and doubly periodic dynamics of meandering.
This paper describes an experimental study of a combustion system,
which also exhibits both periodic and doubly-periodic spiral motions and, in
addition, non-periodic spiral motion. We report observations of spiral states
in pulsating premixed flames as the flow rate is adjusted towards the
extinction limit (the lean-burn limit). The distinguishing feature of these
experiments is the interaction of the spiral flame front with the circular
boundary of the burner. We report the observation of four states: a state with
approximately periodic front motion; a state with doubly periodic
meandering; a weakly nonperiodic state; and a chaotic state. The latter two
states are characterized by interactions of the spiral front with the boundary,
followed by excursions of the tip along a line away from the boundary.
A variety of techniques are used to analyze the spatiotemporal
dynamics. Each of these techniques makes an important contribution to the
understanding of the dynamics.
1.1 Pulsating and cellular flames on porous plug burners
Sivashinsky6-7 first showed that the Lewis number, which is defined as
the ratio of thermal diffusivity to mass diffusivitya controls combustion
dynamics for the thermodiffusive instability. This instability can give rise to
either cellular flames or pulsating flames in different parameter regimes.
Cellular flames8-11 are observed for Lewis numbers less than a critical value,
and pulsating flames12-13 are observed for Lewis numbers greater than a
second critical value.
Flames stabilized on porous plug burners have several advantages
over other experimental configurations for studying combustion dynamics.
Buckmaster14 has shown that heat loss to the burner moves the stability
boundaries of both pulsating and cellular flames closer to a Lewis number of
one, which is typical of most heavy hydrocarbon-air mixtures. The burner
stabilizes the flame front in the laboratory frame over a range of flow rates
and equivalence ratios. A flame front that propagates freely in a tube loses
heat both by radiation and also by heat conduction at the edges, which are in
aThe inverse of this Lewis number is the Lewis number used to parameterize fluid
convection.
2
contact with the walls of the tube. In contrast, a flame front attached to a
solid porous plug loses a much larger amount of heat throughout its entire
area by back conduction to the burner.
Pulsating flames are observed for configurations in which thermal
diffusion dominates mass diffusion. They typically occur in (lean) heavy
hydrocarbon-oxygen flames that tend to produce spiral-shaped fronts and in
(rich) light hydrocarbon-air flames that tend to produce standing waves. In
standing wave states, different spatial regions of the flame front have the
same phase with respect to one another. In contrast, spirals and other
traveling wave states are distinguished by a phase difference among different
regions of the flame front. Pulsating flames typically have characteristic
frequencies in the range 20100 Hz, which is above the 15 Hz Nyquist
frequency of standard video techniques.
1.2 Spirals in other combustion systems
Spirals have been observed in many other combustion systems.
Merzhanov and coworkers15-17 found helical propagation of combustion fronts
in solid-state combustion processes used to synthesize materials. For a
certain range of parameters, the combustion front is confined to hot spots
that propagate in a helical path along the surface of the sample. This process
is called “gasless combustion”, and the combustion front is referred to as a
“solid flame”.
Perlman and Ronney18 observed spiral flame fronts in an open tube
filled with lean, premixed butane-oxygen gas. Significant amounts of helium
gas were added to increase the Lewis number beyond the critical value
needed to observe pulsating spiral dynamics. The flat, uniform flame front in
these experiments propagates down the tube and intermittently assumes a
spiral shape.
Nayagam and Williams19 observed a spiral flame at the edge of a
rotating, downward-facing disk in near-limit combustion. The flame tip in
this system executes a meandering motion along a five-lobed petals-out
configuration. This experimental configuration produces a diffusion flame in
which the two reactants mix diffusively to create the combustion process. In
their experiments, the spirals arise from the fluid dynamics created by the
rotating disks.
Theoretical studies of combustion systems have also identified spiral
dynamics. Scott, Wang and Showalter20 used the Sal’nikov model of a
combustion process and found both target patterns and spiral waves. Their
work required a constant source of fuel to sustain the combustion reaction.
3
Panifilov, Bayliss and Matkowsky21 considered the same model with
parameters that are more representative of combustion processes. They found
both periodic and meandering spiral solutions that are stable solutions of the
equations.
Spirals have also been observed in cellular premixed flames
established on porous plug burners22. When the flame front makes a
transition to a state in which a single central cell rotates, this central cell
assumes an asymmetric spiral-like shape. This single spiral cell arises from a
parity-breaking bifurcation that produces the shape change, as described by
Coullet, Gunaratne and Goldstein23. In contrast, the rotating spiral front
discussed in this paper arises as a primary bifurcation from the uniform, flat,
circular state.
The remainder of the paper is organized as follows. Section 2 describes
the experimental configuration and introduces the analysis techniques used
to present the data. Section 3 reports the observations. Section 4 discusses
the relationship of pulsating spiral flames to theoretical descriptions of spiral
phenomena. Section 5 offers some concluding observations about burner-
stabilized flames as a vehicle for studying dynamics.
2. Experimental configuration, methods and procedures
2.1 Porous plug burner
Our experimental set-up has been described elsewhere24. A circular,
water-cooled, stainless steel porous plug burner is positioned at the center of
a glass combustion chamber. The large number of holes in the plug creates a
relatively uniform velocity profile of the upward propagating premixed gas as
it exits the burner. Combustion produces a flat flame front with a
propagation velocity equal and opposite to the flow velocity of the premixed
gas. A typical combustion chamber operating pressure of 250 mm Hg is
controlled to ±1 mm by an MKS Instruments pressure controller. The fuel
and oxidizer are mixed prior to introduction into the burner housing, and
their individual flow rates are controlled to ±0.001 lit/min by an MKS
Instruments flow controller.
In our experiments, a steady flame appears as a uniform, luminous
disk, 6 cm in diameter and 0.5 mm thick that sits 5 mm above the burner
surface. A 30 cm long glass tube with an inner diameter slightly greater than
the outer diameter of the porous plug is suspended over the burner. The
bottom of the tube sits 8 cm above the burner surface. This cylinder produces
a chimney effect that prevents the pump from disturbing the flow field by
4
pulling the burned gas to the side of the chamber after it exits the porous
plug. These modifications allow the flame front to exhibit its intrinsic
dynamics without significant perturbations by external factors. The observed
dynamics is reproducible to within the resolution of the experiment.
2.2 Video recording
A high-speed, high-sensitivity camera mounted on top of the
combustion chamber directly records the dynamics of the flame front. The
Redlake PCI-100 high speed (1000 frames/sec) CCD camera is equipped with
a Princeton Instruments Microchannel Plate Image Intensifier and an optical
coupler, giving the system a net optical gain of 60,000. Up to 2 GB of data can
be stored on the controlling computer.
Recording rates of 60 Hz, 125 Hz, and 250 Hz sample the entire CCD
array and store images at a resolution of 480 420 pixels. A 500 Hz recording
rate uses only the upper left corner of the CCD array, and the resolution
drops to 320 240 pixels. A 1000 Hz recording rate uses only a rectangular
strip of 320 160 pixels. The latter two recording rates require that the
camera be repositioned during the experiment in order to keep the flame
front centered in the frame, which makes quantitative comparison of data
with the other frame rates virtually impossible. All of the data presented in
this paper is sampled at 250 Hz. A data run of two to five seconds (500-1250
frames) is recorded and stored in an AVI format file using the Indeo codec.
Because of the high characteristic frequencies of pulsating flames,
most of their dynamics is not visible by direct viewing. Consequently,
important characteristics of the dynamics of pulsating flames are not known
until after the experiment is completed and the data is viewed and processed
off-line. In contrast, most of the characteristics of the dynamics of cellular
flames can be immediately identified by the operator.
2.3 Measurement of the tip motion
The tip motion of the spiral flame front is measured manually from
individual video frames. The operator locates the center of the spiral arm at
its narrowest point, places the cursor over this visually determined position,
and clicks the mouse to store the (x, y) location of the cursor in a file.
Software provided with the camera records this position relative to a
coordinate system with the origin in the upper left corner of the frame. The
principal difficulty in determining the tip position is that the luminosity of
the spiral arm decreases near the tip.
5
2.4 Preparation of the data for analysis
Each AVI movie is converted to individual frames, which are stored as
JPEG or TIFF images in separate files. The frames are numbered, and the
frame number is incorporated into the file name. Reflections from the surface
of the inner glass chimney appear as an outer ring of light in the images. In
order to remove this extraneous feature, as well as time codes and other
embedded video information, we apply a circular mask to the image, crop the
masked image, and convert the result to a vector by arranging its pixels in
row major form.
The camera remained in the same position and at the same resolution for
all of the experiments reported in this paper, allowing the same masking and
cropping operations to be applied to all of the images. The original images
are 480 420 pixels. A circular mask centered at pixel (233, 212) with a
radius of 185 is applied, and all values outside the circle are set to zero. The
images are then cropped to a size of 386 386 pixels using a box with upper
left corner at pixel (40, 19).
2.5 Analysis using Karhunen-Loeve (KL) decomposition
Karhunen-Loeve (KL) decomposition is a well-established technique
for finding an optimal basis or orthogonal coordinate system for a collection of
vectors. KL decomposition is sometimes called singular value decomposition
(SVD), proper orthogonal decomposition (POD) or principal component
analysis (PCA)25. We have previously used KL analysis in studies of the
dynamics of cellular flames26-27.
After cropping and masking, a data set has N time snapshots, each
consisting of L spatial values or pixels. The data set can be arranged in an L
N matrix U. Each column of U represents a time snapshot of the data set,
and each row of U represents a time series for a particular spatial sampling
point. We compute the L 1 data set average by averaging each row of U. We
subtract the row average from each element of that row prior to performing
KL decomposition.
KL decomposition finds a set of L 1 orthonormal spatial basis vectors
or modes, k, satisfying:
r
U k k ( k ) T (1)
k 1
6
Here r is the rank of U. The k are sometimes referred to as empirical
eigenfunctions. These functions are ordered by their energy contribution to
the data set, with 1 contributing the most energy.
Because KL is a proper orthogonal decomposition, the N 1 vectors k
are also orthonormal and represent temporal modes. Each k is a scalar.
Because of orthogonality, ak = kk is the projection of k on the data set (e.g.,
ak = UTk) and kk is the projection of k on the data set (e.g., k k = Uk).
The k can be computed as the eigenvectors of the L L matrix R =
UUT. If L > N, it is more efficient to find k as the eigenvectors of the N N
matrix C = UT U and then to project k on the data set to find k. The latter
approach is called the method of snapshots28. In the current situation L > N,
but N is still quite large. Solving the full eigenvalue problem for this system
using either approach (U is typically 90,000 5000) is prohibitively
expensive. We use a sampled KL technique combined with the method of
snapshots.
The sampled snapshot method uses a random sample of the time
snapshots to estimate k. After the row average based on the full data set is
removed from U, M time snapshots are selected at random from the N
available ones. Here M 2, the system exhibits meandering
characterized by a petals inward configuration. When 1 1, implies a petals inward configuration, which we do
not observe. In our experiments, the ratio 1 /2 is slowly increasing towards
1, resulting in a monotonic increase in center radius from state 550 to state
500. However, state 500, which has 1 /2 0.86 is still relatively far from
resonance. The evidence suggests that the observed excursions are a result of
impact of the enlarged pattern of circular meanderings with the boundary.
Still unexplained is the apparent pinning of the secondary frequency at 5.4
Hz in all three meandering states.31 Also, the power spectrum of state 500 in
Fig. 5 has an apparent resonance because 33.4 Hz is close to a harmonic of
5.4 Hz.
LeBlanc and Wulff32 have studied the effects of breaking translational
symmetry, while retaining rotational symmetry. Their models explain
boundary drifting, which has been observed in large aspect ratio experiments
with the BZ reaction33. They are also able to explain spiral anchoring, which
has been observed in experiments on cardiac tissue.34 With boundary drifting,
a spiral executes a meandering motion as it moves in a (mostly) concentric
path with the outer circular boundary. With spiral anchoring, the spiral
moves towards an inhomogeneity and becomes pinned at that point. Neither
of these phenomena has been observed in the flame experiment.
The overall picture of the evolution of the spiral dynamics is: 1) a
rotating flame front undergoes a supercritical Hopf bifurcation to a rotating
spiral arm, whose tip traces out a circle; 2) the front undergoes a secondary
Hopf bifurcation, which leads to a petals-out configuration of meandering
spirals; 3) as the fuel flow rate is further decreased, the radius of the circle
formed by the centers of the petals increases, which results in contact
between the spiral arm and the circular boundary; 4) further decreases in the
flow rate give rise to stronger front intensity variations, which result in a
longer spiral arm and even stronger interactions with the boundary. The
interactions produce more frequent and longer excursions of the spiral tip
before the front continues meandering. Although the system appears to be
19
slowly approaching resonance as the fuel flow rate is decreased, our
experiments suggest that the flame front destabilizes into more complex
behavior well before the resonance point is reached.
5.0 Summary and conclusions
Burner-stabilized premixed flames exhibit a wealth of interesting
dynamics accessible in the laboratory frame over a range of flow rates and
equivalence ratios, allowing controlled experimental exploration. An
important dynamical variable is the emitted chemiluminescence, which
allows the measurement of the spatial characteristics of the dynamics.
Burner geometry and type of fuel are also important control parameters. The
four spiral states reported in this paper share characteristics common to
previous theoretical and experimental studies of spiral dynamics. However,
the pulsating flame spirals also have features that are unique to this
experiment.
The experimental results presented here demonstrate the
complementary interplay among different analysis techniques used to study
systems that exhibit periodic and chaotic spatiotemporal dynamics. The video
frames and the KL eigenfunctions capture the shape of the spiral arm and
the motion of the flame front. The measurement of the position of the spiral
tip demonstrates its petals-out meandering motion. The tip excursions, the
phase space trajectories of the KL coefficients, and the power spectra of the
KL coefficients show the onset of first weak, then strong chaotic dynamics.
The most surprising aspect of this work is that even in the face of large
dramatic boundary collisions and a scale that fills the entire region, the front
maintains many characteristics of translational symmetry. Further
theoretical work will be necessary in order to support this experimental study
and analysis.
Acknowledgements
The authors would like to thank Martin Golubitsky, Dwight Barkley
and Gemunu Gunaratne for conversations about the nonlinear dynamics of
spirals. We appreciate the advice of Bernie Matkowsky and Alvin Bayliss
regarding the motions of spirals in combustion systems. Paul Ronney
provided the program used to calculate the Lewis numbers. This research
was supported by ONR grant N00014-97-1-0029 and by NSF ACI-9721348.
20
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