RATCHETING MOTION OF CONCENTRIC RINGS IN CELLULAR FLAMES
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
Physical Review Letters 76 (1996)pp. 228-231.
Abstract
Ordered states of cellular flames consist of concentric rings of luminous cells. These states are observed to bifurcate to ratcheting states in which one or more rings drift (~1 deg/sec) in a circular path, speeding up and slowing down in a characteristic manner.
Experimental results from three ratcheting states are presented:
one in which two rings of cells ratchet together; one in which an outer ring ratchets, locked with the inner ring through an angle, until the inner ring unlocks and snaps back to its original position; and one in which an outer ring ratchets and an inner ring remains fixed. These results are discussed in the context of bifurcations in symmetric systems.
Introduction
A substantial number of recent experimental studies have
documented the occurrence of ordered patterns and periodic
states in fluid flows and chemically reacting systems
(ref. 1).
We have previously described observations
(ref. 2) of ordered
states of cellular flames consisting of concentric rings of cells.
This paper reports the first observation of ratcheting motion
in which the slow (~1 degree/sec) drifting of a symmetric ring
in a circular path is punctuated by periods of abrupt,
faster angular displacement.
The appearance of traveling
wave states in translationally invariant systems is
well-understood
(ref. 3),
and such states have been observed in a
number of fluid systems
(ref. 4).
A stationary, symmetric ordered
pattern undergoes a symmetry-breaking bifurcation producing
a nonsymmetric pattern which uniformly propagates in either
characteristic direction. In circularly symmetric systems
such states appear as uniformly rotating states
(ref. 5).
We have observed rotating states
(ref. 6) in isobutane-air
cellular flames. One or more rings of cells rotate with
a constant angular velocity, and the cells assume an asymmetric
shape which depends on the direction of rotation.
These rotating states are found in widely separated regions
in parameter space from the ratcheting states. The largest
number of cells in an outer rotating ring is six.
The minimum value of the angular velocity for the rotating
states is 100 degree/sec. In contrast, the maximum value of
the average angular velocity for the
ratcheting states is 1 degree/sec.
The ratcheting states differ from uniformly rotating states
in at least four significant ways: the spatiotemporal symmetries
of these states are more complicated; their motions are not uniform;
the types of motion are more varied; and the average angular
velocity is two orders of magnitude smaller. We distinguish
different states of cellular flames by labeling them according
to the number of cells in each ring. A cellular pattern with
twelve outer cells surrounding six cells which encircle a single
inner cell is designated as the 12/6/1 state. Ratcheting states
are designated by attaching a W to the number of cells indicating
that a particular ring moves. Thus, 12W/6W/1 designates a state
in which the outer two rings ratchet.
In this paper the motions of three ratcheting states are described, and
measurements of angular displacement and angular velocity are presented. Therelevance of these results to dynamical models based on symmetry-breaking
arguments is discussed. The examples of ratcheting states are:
a state in which two rings ratchet together (12W/6W/1);
a state in which an inner and an outer ring both ratchet,
but the inner ring snaps back after a critical angle is reached
(13W/6W/1); and a state in which an outer ring ratchets around
a stationary inner ring (12W/5)
(ref. 7).
These three states are representative of the more than
fifteen ratcheting states we have observed to date.
Top
Experimental Observations
Cellular flames are formed in rich isobutane-air mixtures due
to the relative motion of the lighter oxidizer with respect to
the heavier fuel, creating hotter (brighter) cells of enhanced
chemical reaction due to the excess oxygen and cooler (darker)
cusps and folds of diminished chemical reaction due to reduced oxygen.
The dynamical variable is the local temperature, whose dynamics
can be measured using the intensity of the chemiluminescence at
a point in the flame front
(ref. 8).
In the experiments a 0.5 mm thick premixed flame is stabilized at a distance
of 5 mm above a 5.62 cm circular, water-cooled, porous plug burner which is
housed in a glass combustion chamber at a typical dynamic pressure of 1/2
atmosphere
(ref. 9).
The control parameters in the experiment are the total flow rate of the
premixed gases (isobutane and air) and the equivalence ratio of the
mixture. As these parameters are slowly changed,
the ordered states make transitions to other
ordered states with different numbers of inner or outer cells
or to dynamic states in which individual cells execute a hopping motion
(ref. 10) or rings of cells rotate
(ref. 6).
Ratcheting states are observed by abrupt (rather than slow)
changes in a control parameter, usually near the values at
which the corresponding ordered state is stable.
The state of a cellular flame can be characterized either
by the positions of the luminous cells or by the locations
of the three-dimensional structures--cusps and
folds--which separate the cells. A line between
two cells appears to be a ridge or
fold in the flame surface, protruding upward away (~5 mm)
from the surface of the burner. Three or more folds intersect
to form a cusp. Figure 1a shows top and side
views of the 12W/6W/1 state. There is an optical
illusion associated with the side
view. Many observers see a pattern similar to waterdrops on the underside of aplate. The correct view is that the dark cusps and folds point away from the burner
surface. They are not visible in the top view of the flame because of the limited
dynamic range of 1/2\024 videotape (40 db).
Figure 1a-c
depicts the motion of the three states. These pictures were
obtained from the output of a Silicon Intensified Target camera which views theflame from above through a mirror mounted at the top of the combustion
chamber. The motion of the flame was recorded on videotape
along with a timecode for later
frame by frame analysis. Five sequential frames of digitized videotape taken at
equal intervals are shown for each state. Because the effects of
the motion are subtle, arrows have been drawn in order to
guide the eye, as if they were attached to
a particular cell. In Figure 1a
top and side views of the 12W/6W/1 ratcheting state
are shown. In this state both rings move together. The top view emphasizes the
motion of the luminous concentric rings of cells.
The corresponding frames of the
side view (from a low angle looking down onto the burner) show an ordered array
of the darker cusps and folds at various angles.
At most viewing angles, bright cells
in the background shine through darker cusps and folds in the
foreground along the
line of sight, chopping off the tops of the dark cusps.
The printed frames in Figure 1
suggest a rigid pattern; however, the ordered
patterns are not steady. The cells make small changes
in their size and shape; or, equivalently, the cusps and folds
execute small amplitude oscillations about their
equilibrium positions. This chaotic motion gives the drifting of the cells a
shimmering quality in direct visual observation.
The motion of the 13W/6W/1 ratcheting state of
Figure 1b most closely
resembles that of a mechanical ratcheting device.
In frames 1-4 both the outer ring and the inner ring rotate together,
until they reach an angle of approximately sixteen degrees.
In frame 5 they separate and the outer ring slides over the inner
ring as the inner ring (quickly) returns to its original position.
The locking and slipping then reoccurs with the next outer cell.
The motion of the 12W/5 ratcheting state is depicted in
Figure 1c.
While the inner ring remains fixed
(ref. 11), the outer ring
moves very slowly in frames 1 and 2 and significantly faster in frames 3-5.
The relative motion of the two rings in the 13W/6W/1 and the
12W/5 states produces a more complicated motion of the cusps and
folds than in the 12W/6W/1 state.
Quantitative measurements of angular cell positions were obtained from the
video recordings by evaluation of frames at one second intervals. An angular grid
with one degree of resolution was superimposed on each frame as it was displayed
on a large monitor. The leading edge of a single cell boundary in the direction
of
motion was taken as a measure of the cell position. The average angular
displacement was obtained by applying Simpson's rule to approximate the area
under the curve formed by the point and its two neighbors. This averaging
procedure reduces the effects of the faster (~5 Hz) chaotic motion which underlies
the slow ratcheting. The average velocity was computed using a centered
difference of the average displacement. Negative velocities are real. They arise
because the cells change their size and shape due to the chaotic motion of their
boundaries. At times the amplitude of this motion is larger than the angular motion
due to ratcheting, and the angular position of a cell decreases in sequential frames.
The angular displacement and angular velocity of a cell in outer ring of the
12W/6W/1 ratcheting state are shown in
Figure 2a-b
for a 710 second run. The
angular displacement plot shows intervals of abrupt motion superposed on an
average angular velocity of 0.6 degrees/second. The computed velocity shows sharp
maxima which occur in a definite sequence. These velocity maxima are doubly
periodic with measured values of the period (A--B) at 59.0 + 1.0 seconds and
the
period (A--A) at 100 + 1.5 seconds.
The angular velocity plot also suggests an
additional substructure of peaks which will
require an analysis at a higher temporal
resolution than the 1 Hz used in these plots.
The motion of the 13W/6W/1 ratcheting state involves the relative motion of
the inner and the outer rings. In order to demonstrate the variations of theangular velocity,
Figure 3
compares the angular displacement of a cell in the inner
ring (solid line) with the deviation in displacement
from uniform rotation (ref. 12)(of 1.1
degrees/second) of a cell in the outer ring (dotted line)
for 150 seconds of a 470 second run.
The motions of the two rings are highly correlated. Both rings move
together until the inner ring approaches its maximum angular displacement of
approximately sixteen degrees. The rapid snap-back of the inner ring is followed by
a significant deceleration of the outer cells as the two rings slide over each
other.
The angular displacement of the outer ring in the 12W/5 ratcheting state is
shown in Figure 4
for a 560 second run. The outer ring spends substantial time
with very low angular velocity, creating a
staircase-like structure in the angular
displacement plot. However, both the size of the
steps (net angular displacement)
and duration of the faster motion show substantial (~30%) variations indicatingthat the motion is not simply periodic. The angular velocity (not shown) has a
complicated structure of velocity maxima which implies that the space-time
symmetry of twelve cells moving around five cells may be more
intricate than the
periodic-looking plateaus in
Figure 4a would imply.
Top
Symmetry-breaking by Spatial Inhomogeneity
In the asymmetric periodic states which arise as a result of a secondary
bifurcations, the vector field moving along the group orbit
in phase space is forced
to remain constant by the symmetry of the system, producing uniform rotation ofthe pattern in physical space. Knobloch, Hettel and Dangelmayr
(ref. 13) have presented
one model which demonstrates how dynamics more complicated than uniform
rotation can arise in a translationally invariant system with periodic boundaryconditions. They break that symmetry by invoking the effects of
"inhomogeneities or spatial perturbations".
Their inclusion into the normal form of the bifurcation
equation leads to a "rich assortment of bifurcations".
In their model secondary bifurcations from the primary state of
an ordered pattern produce rotating waves
(RW) in which the phase of the pattern increases monotonically but nonuniformlywith time. Some RW states describe patterns which propogate
in either direction. As the driving parameter is changed,
more complicated states are found: RW
states "which travel first to the left then to the right"
(motion similar to that of the
inner ring of the 13W/6W/1 state) and RW states "which propagate nonuniformly
though always in the same direction undergoing regular
intervals of stasis followed
by fast propagation" (motion similar to that of the outer ring
in the 12W/5 state).
Their model captures some of the elements of observed ratcheting motions,
demonstrating that relatively simple considerations based on symmetry can lead
to
complex motions similar to those in our experiments. However, other mechanisms
may produce similar results. The application of any model to explain our data
requires explicit consideration of the circular symmetry of the experiment. The
dynamics of most ratcheting states are further complicated by interactions between
noncommensurate rings of cells, a situation not present in the transitionally
invariant model system.
Top
References
[1]M. C. Cross and P. C. Hohenberg,
Rev. Mod. Phys. 65, 851 (1993).
[2]M. Gorman, M. el-Hamdi and K. A. Robbins,
Combust. Sci. Technol. 98, 37 (1994).
[3]For relevant theoretical studies see,
P. Coullet, R. E. Goldstein and G. H. Gunaratne,
Phys. Rev. Lett. 63, 1954 (1989);
R. E. Goldstein, G. H. Gunaratne, L. Gil and P.
Coullet, Phys. Rev. A 42, 4338 (1990).
[4] For experimental studies see,
A. J. Simon, J. Bechhoefer and A. Libchaber,
Phys. Rev. Lett. 61, 2574 (1988);
G. Fairve, S. deCheveigne, C. Guthman and P. Kurowski,
Europhys. Lett. 9, 779 (1989);
M. Ribaud, S. Michalland and Y. Couder,
Phys. Rev. Lett. 64, 184 (1990).
[5] M. Golubitsky, I. Stewart and D. G. Schaeffer,
Singularities and Groups in
Bifurcation Theory
(Springer-Verlag, New York, 1988);
M. Krupa et al., SIAM J. Math. Anal.
21, 1253 (1990).
[6]M. Gorman, C. F. Hamill, M. el-Hamdi and K. A. Robbins,
Combust. Sci. Technol. 98, 25 (1994).
[7]The parameter values of isobutane-air mixtures
(total flow rate (lit/min), equivalence
ratio) at which these states are observed:
12W/6W/1 (7.61, 1.76); 13W/6W/1 (7.71,
1.73); and 12W/5 (7.49, 1.71).
[8] For a review of the theoretical studies
of the dynamics of cellular flames see
Combustion Theory,
F. A. Williams (Benjamin Cummings, Menlo Park, 1985) p. 269-372;
A. Bayliss, B. J. Matkowsky and H. Riecke, in
Numerical Methods for PDE's with
Critical Parameters,
edited by H. Kaper and M. Garbey, Kluwer Academic Publishers,
1994;
S. B. Margolis and G. I. Sivashinksy,
SIAM J. Appl. Math. 44, 344 (1990);
A. Bayliss and B. J. Matkowsky,
SIAM J. Appl. Math. 52, 396 (1992).
[9]
For a description of the experimental setup see
M. el-Hamdi, M. Gorman and K. A.
Robbins, Combust. Sci. Technol. 94, 83 (1993).
[10]
M. Gorman, M. el-Hamdi and K. A. Robbins,
Combust. Sci. Technol. 98, 77 (1994).
[11] The deviation was computed by subtracting the
measured value of the displacement from the best linear fit.
[12] In the 560 second data run the inner ring moves
once, rotating by 70 degrees during the time from 440 to 520 seconds.
[13] E. Knobloch, J. Hensel and G. Danglmayr,
Phys. Rev. Lett. 74, 4839 (1995).
Abstract