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Flame Propagation through Periodic Vortices
J . W. DOLD* an d 0. S. KERR
School of Mathematics, University of Bristol, Bristol BS8 1 m U.K.
I. P. NIKOLOVA
Inst itut e of Mechanics and Biomechanics, 4 Bonchev Str., Sofia, Bulgaria
The discovery of a new class of Navier-Stokes solutions representing steady periodic stretched vortices [l]
offers a useful test-bed for examining interactions between flames and complex flow-fields. After briefly
describing these vortex solutions an d their wide-ranging parameterisation in terms of wavelength and
amplitude, this article examines their effect on flames of constant normal prop agation speed as observed
through numerical solutions of an eikonal equation. Over certain ranges of vortex amplitude and flame-speed,
a corridor of enhanced flame passage is seen to be created as a leading flame-tip manages to leap-frog
between successive vortices. H owever, for large enough amplitudes of vorticity or small enough flame-speeds,
the flame fails to be able to benefit from the advection due to the vortices. It is shown that the leading tips of
such flames are effectively trapped by the stretched vortices.
INTRODUCTION
Am ongst other things, understanding the man-
ner in which flames interact with complex flow
fields is central to developing a well-found ed
theory of turbulent flame propagation. A nec-
essary component in developing any such un-
derstanding mus t lie with the nature of the
flow-field itself. In true turbulence, of course,
this is a very complicated and still poorly un-
derstood matter. However, a useful step for-
ward can be taken by studying selected flow-
fields, some more suitable than others, in order
to provide test cases for examining the re-
sponse of flames to known flows. The more
firmly such a test flow is rooted in the three
dimensional Navier-Stokes equations, the
more likely it is to have features in common
with real turbulence. In this spirit, Marble [2]
proposed a test problem consisting of a two-
dimensional decaying vortex in which initialflame and mixture p roperties could be speci-
fied as startup conditions (intended primarily
for non-premixed combustion), and this flow-
field has led to a number of studies of flame
interactions with diffusing vortices [3, 41.
*Corresponding author.
Presented at the Twenty-Fifth Symp osium (International)
on Com bustion, Irvine, California, 3 1 July-5 Augus t 199 4.
Ashu rst and Sivashinsky [5] employed an even
simpler test flow involving square cells of sinu-
soidal velocity variation, which do not satisfy
the Navier-Stokes equations.
The recent discovery of a new class of
Navier-Stokes solutions [l] provides another
potentially useful test-bed for the study of
flame and flow interactions, offering a number
of distinctive and useful features. These solu-
tions consist of steady periodic distributions of
vorticity, arising in a three-dimensional stagna-
tion-point flow field containing a unique direc-
tion of stretch and a unique direction of con-
verging flow from infinity. All of the vorticity
becomes aligned with the direction of stretch
and solutions are found in which the vorticity
becomes periodic in the third direction.
For a given stretch-rate, solutions with any
amplitude of vorticity distribution are found
for wavelengths of the distribution above a
critical wavelength. This results in a two-parameter family of solutions, depending on
both wavelength and amplitude. For increased
wavelengths the regions of strongest vorticity
become m ore and more separated; increased
amplitudes lead to more an d more rap id rota-
tion of fluid elements as they are eventually
sucked into a vortex and ejected along its axis.
The stretch plays the important three-dimen-
sional role of sustaining the vorticity against
diffusive decay.
COMBVSTIONAND FLAME 100: 359-366 (1995)
Copyright 0 1995 by The Combustion Institute
Published by Elsevier Science Inc.
OOlO-2180/95/$9.50
SSDI 0010-2180(94)00161-K
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36 0 J. W. DOLD ET AL.
The underlying uniformity of the vorticity
field in the direction of stretch also allows for
flame and flow interactions to be carried out
more simply as two-dimensional sub-problems,
depending only on the two directions of non-
uniformity in vorticity distribution . The roll-upthat can be produced by these vortices, their
repeated (periodic) occurrence along a plane
of converging flow, and their well-defined
structure both allow for a rich range of possi-
ble interactions and offer a useful te st bed for
investigating premixed and non-premixed com-
bustion in complex flows.
In this article, we begin by outlining briefly
the solution of the Navier-Stokes equations
that describes these vortices. We then presentthe results of a numerical study, using an
eikonal formulation [6], simulating the propa-
gation of thin premixed flames with constant
normal flame speed through periodic arrays of
stretched vortices. An increase in the overall
(or mean) speed of flame passage around the
vortices is found in some cases, as expected,
but it is also shown that flames w ith a normal
propagation speed that is too low are unable to
proceed from one vortex to the next. Dur ing a
part of their movem ent (from simp le flat initial
flame shapes) these flames resemble trapped
flames, which can also be calculated.
The modeling and calculations are carried
out for the case of sm all heat release, the so
called constant-density model, in which the
flow field is decoupled from the flame’s behav-
ior; this greatly simplifies the modeling. Al-
though it is clear that the behavior of a real
flame must be influenced by thermal expan-
sion, it is seen that the constant-density modelis sufficient to reveal a num ber of interesting
dynamical features of the movement of a flame
in this vortex field.
MODELS
Periodic Stretched Vortices
We can begin by examining a three-dimen-
sional flow field of the form
u= [ I]+ [:y], (1 )
so that u and u represent deviations in the x
and y components of velocity U(X , y, z), that
are independen t of a direction of uniform
stretch z, from a simple stagnation-point flow
field which involves no overall strain-compo-
nent in the x direction. Incompressibility, V *u= 0, requires that
u, + UY= 0, (2)
which can be satisfied by defining a d i s turbance
stream function I,!& y) such that
u = (c;, v= -I) X. (3)
Conservation of vorticity requires that
UOJX VW)’ hyw, + Aw + v(w,, + WYU )
(4)
where
Assum ing that a periodic distribution of vortic-
ity exists with fundamental wavenum ber k , w e
can use the length-scale for mom entum diffu-
sion, (v/h) ‘I2 to construct a dimensionless
strain-rate parameter
A’ = -$
which m ay also be thought of as the square of
a dimensionless wavelength of vorticity distri-
bution, based on the length scale for momen-
tum diffusion. It may also be thought of as a
Reynolds number based on the wavelength,
and the flow speed variation found over the
same length in the direction of stretch. A lsomaking lengths and velocities dimen sionless in
a straightforward way, appropriate dimension-
less equations become identical to Eqs. 2-5,
with the viscosity 1, set equal to unity (its
dimensionless value) and all primes, as in Eq.
6, dropped.
Full solutions can now be sought in the form
1c,= 2 [a,(y) + ib,(y)le’kx, a-k = akrk= --u:
b-, = -b,, a, = b, = b, = 0, (7)
where the real coefficients a k and b, are func-
tions of y only. The vorticity conservation
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FLAME PROPAGATION THROUGH PERIODIC VORTICES 36 1
equation requires that these functions satisfy
the sequence of ordinary differential equations
ar + hyu; + (A - 2k2)4 - k%yaZ
- k2Aa, + k4ak
+ i[b;111 hyb’; + (A - 2k’)b;l.
-k2Ayb; - k2A b, + k4b,]
= i C {(a;_, + ib;_,)I= --r
x [la; - /“a, + i(Zb;’ - 13b,)]
-(k - l)(a,_, + ib,_,)
x[a;’ - I’d, + i(b;’ - l*b;)]} (8)
for any k, together with a suitable set of
boundary conditions, namely
a;(O) = b,(O) = a,(4 = b,(a) = 0, (9)
which incorporates the symm etries that are
necessary for ensuring bo unded disturbances
at y = --. It is convenient also to define an
amplitude for any solution such that
A = +(max(t)) - min($>). (10)
In this form, if A > 1, solutions are found
for any value of A > 0 having a,(O) > 0, which
fixes the phase of the solution. In each case,
we use a combined shooting and continuation
method [l] to obtain the functions a,(y) and
b,(y), finding that these functions diminish
sufficiently rapidly with increasing k that typi-
cally very little contribution to the solution
arises after k = 7 or 8. An example of the fullflow-field associated with one solution is shown
in Fig. 1.
Thin Flame Propagation
It is useful to employ an eikonal formulation
for describing the propagation of a flame
through the nonuniform flow field u [6]. We
will consider only a flame with constant normal
flame-speed S and no density jump, so that the
hydrodynamics can be decoupled from the
combustion. This approximation certainly sim-
plifies some of the physics, but also retains the
potential for isolating strong interactions be-
Fig. 1. The distortion of a sheet of fluid arriving from
y= +m as it enters a region with periodic, steady
counter-rotating vorticity in a stretched flow field, satisfy-
ing the Navier-Stokes equations. The dark lines follow the
movement of individual fluid elements, showing that al-
most all fluid is sucked towards the core of a vortex.
tween propagation and nonuniform advection
[5, 7-91.
For this purpose, taking a function f to be
such that a flame lies at
f<x, t> = c, (11)
the proper dynam ical evolution of the flame is
reproduced if f satisfies the eikonal field equa-
tion [6]
f, + u. Vf = SiVfl. (12)
Moreover, solving for f using this model for-
mulation can be made to simulate many dif-
ferent flames in the same calculation since any
isopleth of f, th at is f(x, t) = c for any ualue
of c, can represent a flame. So lutions of Eq. 12
are thus able to provide a great deal of infor-mation about the combined advection and
propagation of flames around vortices [5, 7-91.
We solve Eq. 12 using a straightforward
characteristic method, based on the character-
istic velocity u - SVf/]V f], with second-order
differencing taken over a quadrant of 8 points
in the general direction from which the charac-
teristic arrives at any one po int. T his is found
to give solutions that are free from oscillations
and in which cu sps in flame sh ape are captured
cleanly. Also, being quadratic in order (indeed
having some cubic terms because of the g-point
differencing in two-dimensions) the method is
relatively accurate and free of num erical dif-
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36 2
fusion even when using modest numbers of
points.
Stationary Thin Flames
As will be seen, stationary flames can also arisewhen the velocity field exactly balances a
flame’s normal propagation speed at every
point. This happens where
u*ri = -s, (13)
in which h(x) is the unit normal vector to a
flame surface at the point X. In general, this
defines either two unit norm al directions (if
1~ 12 S) or none (if 1~1< S). Given one un itnormal that lies perpendicular to a stationary
flame surface, the corresponding tangent direc-
tion i((x), for which s^. it = 0, with s^ rotated
(say) anti-clockw ise from 5, provides a differ-
ential equation for the path of the flame sur-
face x(s) as a function of arclength s, since
x(s) must satisfy
x, = i(X), s^*ri = 0. (14)
Thus, through any point x there are either tw o
or no possible tangents to any possible station-
ary flame. Any stationary flame solution that
might be found must also depend on the over-
all shape of the flame, so that Eq. 14 needs to
be solved subject to reasonable boundary con-
ditions, such as that a flame should not spiral
around itself (requiring a flame to subsist on
burnt gases) or disappear suddenly at a point
where JuI = S, although a closed limit-cycle
would be acceptable as a solution representing
a flame trapped by the suction into a vortex.
Shooting techniques prove well-suited to solv-
ing Eqs. 1 3 and 14 in order to determine
stationary flame-profiles.
RESULTS
Advection and Propagation
Taking a sample flow-field with h = 6, A = 20,and S = 8, Fig. 2 show s a sequence of contour
diagrams of the field variable f calculated from
the simple initial condition f = x. These initial
data represent a family of flames (each mem -
J. W. DOLD ET AL.
Fig. 2. Profiles of flames traveling from right to left through
a layer of steady periodic vorticity distribution, having
A = 6 and A = 20, at a constant normal flame speed of
S = 8 (as calculated using the eik onal Eq. 12 which repre-
sents flames as isopleths of f). Contours of f at different
dimensionless times are shown when: r = $; (b), t = 4; cc),
t = 1; and (d), t = 3. The centres of regions of counter-
rotating vorticity are marked by small fans. The develop-ment of a corridor of enhanced flame passage is clearly
seen.
ber depending on a choice of c) that start out
being aligned parallel to the y axis.
It is very clear from this figure that flame
movement is enhanced where flow around the
vortex cores i s in the sam e direction as the
flame’s propagation. Where the flow inhibits
flame movement, directional effects quicklylead to the formation of cusp s where different
directions of propagation cross. In these calcu-
lations, flames assisted by advection in parts of
the flow also manag e to find their w ay to other
regions where advection assists flame-move-
ment in a way that enhances the overall flame
transit. A fter a long enough evolution it is
clear that a corridor of enhanced flame pas-
sage appears along the line of vortices. The
propagation through this corridor is led by anose of flame that appears to find an optimal
path through the flow-field so that it is able to
cross from one vortex to the next in the least
possible time.
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FLAME PROPAGATION THROUGH PERIODIC VORTICES 363
Enhanced Flame Speed
By following the average rate of increase of f
within the corridor of enhanced flame-passage
(over a long enough time) the effective in-
crease in flame-speed can be estimated. Inparticular , with the initial condition f = x, the
effective flame speed is precisely
(151
the averaging being taken at any point or path
within the corridor of enhanced flame passage;
we, in fact, average firstly over the entire path
for which , at fixed y, f(x, y, t> has its highest
value, and secondly over time, usually over the
period from t = 1 to t = 2 or longer. Formula
15 is confirmed by the fact that I( = 1, as
ensured by the periodicity, for wh ich f<~ +
27r, y, t) = 27~ + f(x, y, t). Figure 3 presents
this overall effect of advection around the vor-
tices on the mean flame-speed S, for two
different fixed amplitudes A ranging over dif-
ferent values of S.
It can be seen that increasing the ratio A/S
serves to increase the effective flame speed
above the speed without vorticity provided thevalues of A/S do not grow too large; the
increase in effective flame speed is found to be
approxim ately linear when the vorticity level is
low (or flame-speed is high). At increased lev-
t
1 . 6
\
A = 4 0
Fig. 3. Fractional increases in effective mean-speed of
flame passage S, through the layer of vorticity distribu-
tion, as functions of A/S, from calculations of the eikonal
Eq. 12 using two fixed vorticity distributions, having A = 20
and A = 40, with h = 6. The variations for small A/S
(large 5’) are similar, and nearly linear, in both cases; at
smaller normal flame-speeds S the flames fail to take
advantage of the nonuniform flow-field.
els of vorticity, there is another effect which
manag es to reverse the gain in flame-speed
acquired at lower levels of vorticity. The
flame-speed decreases once A/S increases
above a threshold value, dropping very rapidly
in value. It is apparent from this that a point isreached where flames are unable to penetrate
through the arrays of vortices in any preferred
manner.
Flame Trapping
If a flame is unable to penetrate at some stage
from one vortex to the next, then the leading
parts of the flame m ight be expected to come
to a standstill. This leads one to search forstationary flame solutions. It turns out that
Eqs. 13 and 14 do yield stationary solutions for
trapped flames at sufficiently large values of
A/S.
Superimposed on a representation of the x
and y comp onents of the flow field for fixed
values for A and A, Fig. 4 shows a sequence of
flame-solutions trapped around the core of a
vortex. A s S increases, the flame is able to
advance further away from the vortex center,overcoming more of the flow of fluid that is
Fig. 4. Profiles of stationary flames trapped by the inflow
around a region of concentrated vorticity (having A = 6
and A = 10) with flow directions and magnitudes illus-
trated by the short lines. Trapped flames are shown for
normal propagation speeds of S = 1, 1.5, 2, 2.5. 3, and
3.42. Two regions for which (~1 < S (= 3.42), within which
a flame of speed S could not be held stationary by the
flow, are shown as shaded; and a flame of speed S = 3.42
is seen to just avoid touching these regions. This demon-
strates that trapped flames do not exist (for this flow-field)
at speeds of propagation above a critical value close to
S = 3.42.
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364 J. W. DOLD ET AL.
sucked towards the vortex as an overall result
of the divergence of fluid along the line of any
vortex. This continues until a value S =
S&4 , A) is reached where a trapped solution
touches the region where IuJ I S. As pointed
out earlier, no stationary solution can existwithin these regions so that trapped solutions
cease to exist above this critical value of S.
That this value involves a close relationship
with the failure or success of flames in passing
between vortices is confirmed in Fig. 3 where
these critical levels (both for A = 20 and A =
40) are marked on the diagram and correspond
closely with the loss of enhanced propagation.
Because it would correspond to zero speed of
propagation, A/S, must overestimate the pointof loss of enhanced propagation speed, which
cannot fall below the value of unity since flames
propagating at this speed are still to be found
outside the layer of nonun iform vorticity.
It is interesting to compare this result with
an unsteady solution of the eikonal Eq. 12 for
a value of S that is not high enough for the
flame to pass from one vortex to the next. Such
a calculation, at one time, is shown in Fig. 5. In
this diagram, certain contours show that an
initial flame distortion by the flow field at the
initial time h as come to a standstill as a bulb of
flame is unable to escape from the suck ing
Fig. 5. Profiles of flames travelling from right to left, under
the sam e conditions as in Fig. 2, but for a constant normal
flame speed of S = 3;. Isopleths of f are plotted at the
dimensionless time t = 1. The bottlenecking in between
the vortices, of contours at the nose of the flame, demon-
strates the difficulty that a flame of this propagation speed
experiences in escaping from the sucking effect of each
vortex, resulting in a much reduced mean speed of flame
passage and flame profiles that resemble, for a while, the
shapes of trapped flames.
effect of a vortex. This bu lb h as a very similar
form to the steady trapped flame shapes in
Fig. 4.
CONCLUSIONS
Some examples of a new family of periodic
stretched vortex solutions of the Navier-S tokes
equations have been used in this article to
illustrate part of the manner in which flames
can interact with complex flow fields. In partic-
ular, as solutions of an eikonal equation for
constant normal flame-speed show, the flames
are able to take advantage of nonuniform ad-
vection to travel more qu ickly. Arou nd one
side of any region of increased vorticity theadvection helps the flame’s movement. After
having been helped in this way, the flame then
has to be able to escape from the overall
suction of the vortex in order to be able to
move on to take advantage of the advection
around another vortex. The suction associated
with the stretching of vortices can therefore
hinder a flame’s movement while the advection
around vortices aids its overall movement.
Thus, the flow has within it the essential
ingredients both for enhancing the passage of
a flame through an array of vortices and for
holding back the flame, depending on the in-
tensity of the vortices and on the norma l flame
speed. In some cases, it has been shown that
flames could even become completely trapped
within regions of enhanced vorticity. It seem s
likely that these processes should reflect at
least part of the experiences a premixed flame
would undergo in a turbulent flow field.
One mechanism for enhanced flame move-ment in turbulence is not immediately avail-
able for study with the particular steady-flow
solutions described in this article. The propa-
gation of a flame along a vortex should also
provide a significant setup for non-uniform
flame advancement in which density changes
and curvature effects shou ld be very impor-
tant; this mechanism is likely to be especially
important if and when flames become trapped
within vortices, unless three-dimensional cur-vature effects then restrict flame speeds exces-
sively. The unsteady movem ent of vortices
themselves is also likely to be a major factor in
real turbulent flame propagation.
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FLAME PROPAGATION THROUGH PERIODIC VORTICES 36 5
In spite of these m ore general aspects, this
article show s that the new Navier-Stokes solu-
tions described here do make available some
insights into the interplay between propagation
and advection. As a test-bed for combustion
problems, it complements significantly thetwo-dimensional initial-value vortex proposed
by Marble [2] and the simple doubly periodic
flow-field of Ashurst and Sivashinsky [51 for
some flame problems, by being steady
Navier-Stokes solutions and therefore, of ne-
cessity, stretched. These properties, and the
mixing-layer context within which the vortices
appear, make the solutions useful for examin-
ing both premixed and non-premixed burning
in the presence of non-un iform vorticity.
J. W . D o l d w a s s u p p o rt e d i n t h i s w o r k b y a n
A dv anced Fel low ship f rom t he S .E.R .C.; 0 . S .
Kerr w as suppor ted by a Research Grant from t h e
S .E.R .C. and I . P . N iko lov a w as suppor ted by an
E.C. V i s i t i ng Fe l low sh ip Gran t (No .
ER B351OP L920821) enabl ing her to spend th e
m on ths o f A pr i l t o June 1993 a t Br i s to l Un iv er-
s i t y .
REFERENCES
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2.
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4.
5.
Kerr, 0. S. and Do ld, J. W., J. Fluid Mech.,
276307-325 (1994).
Marble, F. E., in Recent Adrtances in the Aerospace
Sciences, Plenum, New York, 1985, pp. 3955413.
Laverdant, A. M. and Can del, S. M.. Com hust. Sci.
Technol. 60:7Y (1988).
Ashurst, W. T., Lect. Not es Phys. 352:3-21 (1989).
Ashurst, W. T. and Sivashinsky, G . I., Cornhusf. Sri.
Tech&. 80:159-16 4 (1991).
Markstein, G. H., Nonsteady Flame Propagation, Perg-
amon, 1964, p. 8.
Kerstein, A. R., Ashurst, W. T. and Williams,
F. A., Phy s. Reel. A 37:2728 (1988).
Ashurst. W. T., Sivashinsky, G. 1. and Y akhot, V.,
Combust . Sci. Technol . h2:273-284 (1988).
Ashurst, W. T. and W illiams, F. A.. Twenty-Third
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Receked 30 Nol’emher 1993; reked 27 Ma)’ 1994
Comments
S . I brah im, Br i t i sh Gas , UK. Your test cases
are at small values of ST/S (turbulent to lami-
nar burning velocity) which are of order of
l-l.6 and low turbulence intensity (u’/S). Haveyou done any tests at high turbulence level, in
order to confirm the effective flame speed trend
that you have obtained?
A u t h o r ’ s R e p l y . In this preliminary work we
have been working with relatively low magni-
tude vortices in order to establis h the initial
trends. Calculations involving larger ampli-
tudes are planned.
J. Chom iak , Cha lm ers Un iv ers i t y o f Techno logy ,
S w e d en . In turbulen t flows the vorticity is con-
centrated in vortex tubes randomly distributed
in space. Thus the situation of parallel vortices
you consider is very unlikely to occur.
A u t h o r ’ s R e p l y . The exact configuration of vor-tices that we consider is not intended to mimic
turbulence precisely, although the observations
from D.N .S. that you describe provide a strong
motivation for its study. The vortices that we
present here provide a useful test-bed for in-
vestigating complex flame an d flow interac-
tions, in which the fact that they are exact
steady Navier-Stokes solutions, sustained by
stretching, imparts m ore features in common
with real turbulent vortices than is found for
any other well-understood flows that have beenstudied in the same way to date, w ith the
possib le exception of Burge r’s vortex. It is also
worth mentioning that these particular
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36 6 J. W. DOLD ET AL.
stretched vortices only fall into one range of
solutions within a wider class of steady solu-
tions that is still under investigation.
R . K l ei n , R W T H A a c h e n , G e rm a n y . 1. Is yourexact solution of the Navier-Stokes equations
extendable to a stack of parallel arrays of
vortices, so that you could describe interac-
tions of those “ flame tunn els”? This configu-
ration would be very close to actual turbulent
flame propagation. 2. Did you assess whether
inclusion of at least weak heat release in your
model is feasible?
A u t h o r ’ s R e p l y . 1. There is a large variety ofpossible steady solutions to the Navier-Stokes
equation in this stagnation point flow, more
than we have described here. W ithin these
possible solutions are some that have two par-
allel arrays of vortices stacked on each other,
with adjacent vortices having opposite senses
of rotation. It is not, however, possib le to have
more than two such arrays stacked on each
other. 2. We have considered the possibility of
thermal expansion in the vortices. T he govern-
ing equations do allow for steady solutions
where the chemical constituents, the tempera-ture, and hence the density vary only with x
and y, and not z. In principle we could find
solutions to these equations, and hence find
steady flame solutions with expansion effects
included , but this wou ld be relatively difficult
and we have not yet done so. Non-steady flames
with expansion would destroy the underlying
steady vorticity field as they propagate, requir-
ing a different means of study from that em -
ployed here. In such a study it is not clear thata small heat-release approximation would be
much easier to solve or interpret than order
one heat releases. However, it is a very impo r-
tant currently unsolved problem to determine
the relationship between highly non-uniform
flow fields and expansion in propagating flames.