Tulip Flame

On the “Tulip Flame” Phenomenon
CHRISTOPHE
CLANET and GEOFFREY
SEARBY’
Insitut de Recherche sur les Ph&omLnes Hors Equilibre, Laboratoire de Combustion et SystZmes Riactifs,
U.M.R 138 du CNRS - Vniversit& d’Aix-Marseille I et II, Service 252, Campus Universitaire de St. J@me,
13397 Marseille Cedex 20, France
We present an experimental study of the “tulip flame” phenomenon using high-speed photography. Contrary
to most previous studies, the work is in a simplified quasi-constant pressure configuration in a half-open tube.
It is shown that the salient features of the different stages of the flame propagation and shape can be
explained by a simple geometrical model of the interaction between the flame front and the gas dynamics. In
particular, the tulip flame results from an inversion of the flame front curvature caused by the deceleration
related to loss of flame surface area. Finally, the experimental results obtained by other authors in closed
vessels are in reasonable agreement with the analysis presented.
NOMENCLATURE
a(t)
“E
g(t)
k
L
P
R
r”
S
So
t0
t sphere
t wall
ttulip
t*
v,
U,ip
Uskirt
v,
v,
time-dependent
amplitude of front
wrinkling
sound speed
gas expansion ratio = pu/pb
time-dependent acceleration
wavenumber
length of tube or combustion chamber
instantaneous relative pressure at the
closed end of the tube
radius of tube
aspect ratio of combustion chamber =
R/L
flame surface area
cross section area of combustion chamber
instant of ignition
time at which flame propagation
changes from spherical to fingered
time at which flame touches side walls
of tube
time at which tulip flame inversion
occurs
= tU,/R reduced time
laminar burning velocity
velocity of flame tip
contact velocity of trailing edge of flame
skirt
volume of burned gases
volume of closed combustion chamber
Greek Symbols
flame front thickness
burned mass fraction
kinematic viscosity
density of unburned and burned gases
S/S,, ratio of flame area to burner
cross section
= &
: Characteristic
time in the
tulip flame experiment
time during which flame tip is decelerated
= ttulip - twall time delay after which
flame front curvature inverses or time
between shock and curvature inversion
in shock experiment
characteristic
frequency of Taylor’s
instability
INTRODUCTION
The propagation
of flames in tubes was first
observed by Mallard and Le Chatelier in 1883
[l]. They noticed that if a flame is ignited at
the closed end of a long half-open tube, the
* Corresponding author.
COMBUSTIONAND
axial coordinate of tip of flame
axial
coordinate of lower edge of skirt
zskirt
of flame
Z wall axial position of flame tip when the
flame skirt touches burner wall
axial
position at which tulip inversion
Gulip
occurs (flat front)
ztip
FLAME
105: 225-238
(1996)
Copyright 0 1996 by The Combustion Institute
Published by Elsevier Science Inc.
OOlO-2180/96/$15.00
SSDI OOlO-2180(95)00195-6
226
progression of the flame front is irregular with
inversions of the direction of progression of
the flame. The first photographs of the shape
of a flame propagating irregularly in tubes
were published by Ellis in 1928 [2]. He observed that, for closed tubes with a sufficiently
high aspect ratio (length/diameter
> 2 for
closed tubes), the shape of the flame changed
suddenly from a forward pointing finger to a
backward pointing cusp. After this inversion,
the cusped shape remained stable. In 1959
Salamandra et al. [3] coined the name “tulip
flame” to describe this curious phenomenon.
An example of the formation of a tulip flame is
shown in Fig. 1.
Various mechanisms have been proposed to
explain the formation of the tulip flame.
Dunn-Rankin et al. 141proposed that the flame
inversion is initiated by a radial gradient of
axial velocity just ahead of the curved flame
front. Similarly, Gonzales et al. [5] claim that
the “squish” flow generated between the curved
flame and the tube wall is crucial to the formation of the tulip inversion, Rotman and
Oppenheim [6] have claimed that vorticity generation in the unburned gases is responsible
for this phenomenon,
whereas Oppenheim
and Ghonheim
[7] invoke the role of
Tollmein-Schlichting
waves in the boundary
layer at the walls. Other authors [8] favor an
interpretation in terms of a manifestation of
the Darrieus-Landau
instability.
C. CLANET AND G. SEARBY
GuCnoche [9] observed that the tulip effect
appears both in closed tubes and also in halfopen tubes. Moreover, to understand the origin of the phenomenon. GuCnoche proposed
an analogy between the tulip flame inversion
and a shock wave experiment of Markstein
[lo]. In this experiment, Markstein investigated
the effect of a shock wave on a curved flame
front. His pictures show that the flame front
also undergoes an inversion of concavity, similar to the tulip phenomenon. The characteristic time for the inversion was shown to decrease as the shock intensity increases. In the
case of the tulip flame there is no shock, but
we will show that the change in flame surface
area after the flame reaches the tube walls
creates a deceleration that acts in a similar way
via the Taylor instability [ll]. Strehlow 1121and
Starke and Roth [13, 141 also believe that the
tulip inversion is a manifestation of the Taylor
instability. However, none of the above authors
has given quantitative evidence to confirm their
propositions. One of the main objects of this
paper will be to give such a quantitative proof.
We first describe the experimental apparatus
and qualitatively describe the different stages
of flame propagation preceding the tulip flame
formation. We then present the experimental
results and interpret them using a simple geometrical model. Finally, results are compared
with other published experiments.
EXPERIMENTAL APPARATUS
Fig. 1. Superposition of three images of luminous emission
of flame front just before, during and just after formation of tulip flame. High-speed video recording at 4500
images/s. Time between selected images is 2.2 ms.
Propane-air flame @ = 0.70, in a tube 1.5 m x 0.05 m
radius.
Contrary to most previous studies, the work
was in a simplified configuration with a halfopen tube. This configuration has the advantage of being at constant pressure and eliminates complications of flame front motion
associated with compression of the unburned
mixture. The apparatus is shown in Fig. 2. The
walls of the burner were uncooled Pyrex glass
tubes. The tubes had radii of 2.5 and 5 cm and
lengths ranging from 0.6 m to 6 m.
The tubes were filled with a premixed
propane-air
mixture fed into the bottom of
the burner through an isolating valve. The
equivalence ratio was controlled using sonic
nozzles in the air and propane lines. Equivalence ratios were in the range 0.65 < @ < 1.3.
The laminar burning velocity ranged from 41
ON THE “TULIP
c---
Tulip Flame front
227
FLAME” PHENOMENON
I
2.R
Unburned
gas=
.
Pyrex tube
-
Pressure s
-
Spark electrodes
Fig. 2. Apparatus.
to 20 cm/s. The flame was spark ignited at the
closed end of the tube. A signal from the spark
generator synchronized the recording apparatus. A piezoelectric pressure sensor measured
the pressure at the closed end of the burner.
The signal was recorded on a numerical oscilloscope and post-processed on a microcomputer. Various high-speed imaging devices were
used to record the form and position of the
flame front as a function of time.
QUALITATIVE PRESENTATION
ORDERS OF MAGNITUDE
AND
As can be seen in Fig. 1, the flame front
assumes very different shapes between ignition
and tulip formation. We distinguish four stages
in the propagation of the flame. The first stage
occurs just after ignition, t, < t < tsphere. The
flame is small, unaffected by the presence of
the burner side walls and develops as a hemispherical flame expanding at the velocity of
burned gas production, typically tsphere= 10 to
20 m/s. The second stage of propagation occurs as the flame approaches the side walls,
t sphere < t < twall* The
flame shape changes
from hemispherical to finger shaped. The velocity of the tip of the flame increases exponentially and reaches a value (30-50 m/s),
nearly two orders of magnitude higher than
that of the spherically expanding flame. However, we will not be concerned here with the
details of the mechanism by which the front
propagation changes from spherical to fingered. The third stage occurs after the skirt of
the elongated flame touches the side walls of
the tube, twal, < t < ttulip. It is during this stage
of the propagation that the inversion of the
flame front is initiated. During this stage of
propagation, the flame surface area and the
propagation velocity decrease steadily. The
time at which the tulip inversion occurs is
found to be a function of the laminar burning
velocity and the radius of the tube. The time
scale for ttulip is typically = 40 ms for a stoichiometric flame of propane in a tube of radius
5 cm. The fourth stage of propagation concerns the dynamics of the front after the tulip
inversion, t > &,. It is during this last stage
that acoustic effects can dominate. The work
presented below will be mainly concerned with
the second and third stages of propagation, up
the formation of the tulip flame.
EXPERIMENTAL
DISCUSSION
RESULTS AND
The Role of Viscosity and Boundary Layers
The order of magnitude of the Reynolds number as defined by the tube radius and the
velocity of the flame tip, R, = 2.R.Qip/v, is
typically of order 10 5. Thus, except in the
boundary layer at the walls, viscous effects are
negligible compared to inertial effects. Numerical simulations of flame propagation in closed
vessels have already shown that nonviscous
codes can successfully simulate the tulip flame
phenomenon 14, 81 and that increasing the
numerical fluid viscosity attenuates the tulip
phenomenon 151.Moreover, Pocheau and Kwon
[15] have shown experimentally that an array
of spark igniters can produce an array of tulip
flames in which there is no physical boundary
between adjacent fingers. We have reproduced
their experiment and show an example of such
a double tulip flame in Fig. 3. This space
C. CLANET AND G. SEARBY
Fig. 3. Multiple exposure video image of twin tulip flame
formation. Propane-air
@ = 1.3 ignited by two sparks
symmetrically on left, outside field of view.
symmetry confirms that boundary layers are
not directly involved in the tulip phenomenon.
The Role of the Acoustic
Field
It has been suggested that the formation of the
tulip flame could be due to an interaction
between the flame front and acoustic waves in
the combustion chamber [16]. This appears to
be not so. Figure 4a shows the pressure
recorded at the bottom of 1.5 m tube, radius
5 cm, for two different flames. There is an
initial pressure rise which is approximately exponential. This pressure rise is associated with
the development of the finger flame, see below.
It is followed by a sharp pressure drop that
precedes the formation of the tulip flame,
marked by empty crosses. The pressure drop is
followed by strong pressure oscillations at the
resonant frequency of the tube and its harmonics. The period of the cold fundamental mode
is 0.017 s for this tube. Acoustic oscillations
are clearly present for both flames, even before the formation of the tulip flame. However
the sharp pressure drop and the tulip inversion
occur at a later time for the slower flame in
the same tube. Figure 4b shows the same data
in reduced coordinates. The time scale has
been reduced by R/U,, where R is the radius
of the rube and U, is the laminar burning
velocity. R/U, is the time scale for the flame
to reach the tube walls. The pressure has been
reduced by pUUE2,where p,, is the unburned
gas density. It can be seen that, in this representation, the transition from quasi-exponential rise to rapid drop occurs at the same
reduced time, as does the tulip formation.
Figure 4c shows the effect of changing the
acoustic time scale. The pressure is recorded in
tubes of different length for the same stoichiometric flame. It can be seen that the tulip
inversion occurs at a time (real or reduced)
which is independent of the tube length. For
tube lengths greater than 4 m, the tulip inversion occurs at a time shorter than that required for the pressure disturbances to propagate to the end of the tube and back to the
flame. Figure 4c also shows that the tulip formation time does not always coincide with the
first pressure minimum, as suggested by Figs.
4a and 4b where it is a coincidence that half
the acoustic time, 2.L/c = 8.7 ms, is equal to
the tulip inversion time which scales (as will be
shown) as R/U,, and is of the order of 8.5 ms
for a stoichiometric flame.
Figure 4d shows the results of an experiment
in which strong acoustic losses are introduced.
The burner consists of two aligned tubes separated by a small gap of width e, which was
varied from 0 to 10 mm. When e is zero, the
burner is the same as before. When e =
10 mm, the fundamental mode of the burner is
strongly damped. The tulip flame was formed
just after the gap in the burner. This figure
shows that the initial pressure rise and the
time of tulip formation are little affected even
by strong acoustic losses. However, the acoustic oscillations generated after the tulip formation are increasingly damped as the acoustic
losses are increased.
The above results show that combustion excited pressure waves are present, but they do
not provide the driving mechanism for the
tulip inversion. Moreover, the time scale for
the tulip phenomenon is closely related to the
time taken for the flame to reach the walls of
the burner and not to an acoustic time. However, acoustic waves play an important role
after the tulip formation, as already remarked
by Guenoche [9]. In particular, we have observed that the depth of the tulip cusp is enhanced or attenuated according to the direc-
ON THE “TULIP
229
FLAME” PHENOMENON
Pressure in tube tar two flame velocities
Dlfferent burning velocities, reduced coordinates
40000
zS
9
20000
P
!!
a
::
k
0
B
s -20000
-o-
U, = 0.42 m/s
%
-10000
I
0.00
0.02
Time
0.04
after
-40000
0.06
0.00
0.08
0.10
0.20
Incresslng scoustic losses
Different tube lengths, reduced coordinates
hllip
tlva//
,
--be=1
-e=2mm
8
-e-L=4Sm
-100000
-40000
0.4
0.6
Reduced time t&IL/R
0.50
(b)
(a)
0.2
0.40
0.30
Reduced time t+l,IR
ignition (see)
0.8
1.0
(c)
I--t
0
mm
I
0.1
I
0.2
0.3
0.4
Reduced time ML/R
(d)
Fig. 4. Evidence to show tulip inversion is not related to acoustic phenomena: (a) Pressure at closed end of tube for hvo
different burning velocities. Unfilled crosses show formation of tulip flame. Tube 1.5 m x 0.05 m radius. Period of
fundamental acoustic mode 17 ms. (b) Same data as a), but in reduced coordinates. (cl Pressure at closed end of tube for
stoichiometric propane-air flame in tubes of different lengths. Tube radius = 2.5 cm. Arrows mark time at which flame
skirt touches wall, I,.,,, and time at which tulip inversion occurs, ttulip. (d) Pressure at closed end of tube 1.5 m long X 5 cm
radius for different acoustic losses, introduced by a cylindrical gap in walls. Gap, of width e, is 0.6 m from the closed end.
tion of the acoustic acceleration at the flame
front during the tulip inversion. Moreover, in
half-open tubes or closed tubes with a high
aspect ratio (> 201, the acoustic velocity can
be sufficient to periodically reverse the apparent direction of flame propagation and the
cusped tulip flame is observed to break into
cells, oscillating at the acoustic frequency.
These oscillating cells are probably the parametrically excited structures studied experimentally by Searby 117, 181 and observed
numerically by Gonzales et al. [19], but are
beyond the scope of this present study.
The Dynamics of the Flame Tip
The global progression of the flame front was
filmed using a high speed Kodak video camera
at 4500 images per second. Two images are
shown in Fig. 5. The lower image shows the
finger-shaped flame just after the skirt of the
flame has touched the side wall of the tube. At
shorter times the luminous flame front extends
down to the base of the burner. The upper
image shows the flame at a later time, just
after the formation of the tulip. We have measured the position of the center of the flame
230
C. CLANET AND G. SEARBY
Fig. 5. Two images of luminous emission from finger flame.
Ignition on left. Lower image: flame just after skirt of
flame reaches tube walls. Upper image: flame just after
tulip inversion. Infrared emission from burned gas also
visible to left of flame surface. Propane-air mixture, @ =
0.90, tube radius = 2.5 cm.
and its area increases exponentially during the
development of the finger. The velocity of the
flame tip for this flame, of equivalence ratio =
0.7, is of the order of 23 m/s at t = twa,,. For
stoichiometric flames, the tip velocity is of the
order of 80 m/s. The velocity of the contact
point of the skirt of the flame is almost constant. The skirt rises up the tube faster than
the tip, = 24 m/s, indicating that the surface
area of the flame decreases during this stage.
The tulip formation occurs a considerable time
after the flame has touched the walls.
A GEOMETRICAL MODEL FOR FLAME
PROPAGATION
The Finger Flame
tip and also the position of the trailing edge of
the flame skirt as a function of time. A typical
result is shown in Fig. 6. These two quantities
are plotted along with the pressure at the base
of the tube. The acoustic oscillations visible on
the pressure signal are not visible on the trajectory of the flame tip. This again confirms
that acoustic pressure waves do not control the
flame front dynamics at this stage. The maximum pressure coincides almost exactly with
the instant, twa,,, at which the flame touches
the walls, Before the flame touches the walls,
the trajectory of the flame tip is close to exponential. This means that the flame accelerates
~,h,T
3oooo,,,,,,,,,,,,,,,,,,,,,,,
“;s
,o
z
20000
I-
,,,,,,, 6
.-
14
A^^^_
1 “U””
E
7
g
-10000
p
-20000
F
J
-30000
0
t
-Skirt
Tip
5
6
e
a
L
8
Pressure
??
l2
10
position
position
,”
P
P
PC -40000
I
-
2
-50000
0
We introduce a simple geometrical model for
the propagation of the finger flame in a halfopen tube. This model neglects pressure variations and relates the flame propagation speed
directly to the production rate of burned gas.
During the development of the finger flame,
the skirt of the flame is close to the tube walls,
see Figs. 1 and 5. The axial velocity of the
flame tip (lo-80 m/s> is very much higher
then the radial velocity of the skirt, which
travels towards the tube wall at a speed close
to the laminar burning velocity, U, = 0.2-0.4
m/s. The proximity of the tube wall prevents
the flame from pushing the unburned gas radially outwards.
The model is presented
schematically in Fig. 7. It is supposed, for simplicity, that the flame surface can be represented as a cylindrical finger of radius r with a
hemispherical tip. The effect of curvature on
the local burning velocity is neglected.
Using these hypotheses we calculate the velocity of the flame tip. The mass consumption
0.1
0.2
Reduced
0.3
+’
0
0.4.
Time t.UL/R
Fig. 6. Normalized superposition plot of pressure at closed
end, positions of center of flame tip and of trailing edge of
flame skirt, as functions of time. Flame positions normalized by tube radius. Arrows mark time at which flame skirt
touches wall, t,,,,, and time at which tulip inversion occurs, ttullp. Continuous line is a best fit exponential.
Tube
1 Flame front
Fig. 7. Simplified geometrical
times between tSphereand t,,,,.
model of flame front at
ON THE “TULIP FLAME”
PHENOMENON
231
rate of fresh gas is given by p,, * U, . S, where
gas density, U, the laminar
burning velocity, and S the area of the flame
front. This is also equal to the production rate
of burned gas. The time variation of the volume of burned gases, V, is given by
Growth retee of the finger flame
p, is the unburned
-di/,
dt
=E*U,.S,
(1)
where E = pu/pb is the ratio of unburned to
burned gas density. The effect of the small
pressure changes on the laminar burning velocity is neglected. Supposing that the flame is
cylindrical with a hemispherical cap, the volume of the burned gas is given by
V, = m2(ztip - Y) + 2/3rr3,
S = 27rrz,i,.
(3)
dZtip
z = U, compared with -
and
putting r = R, Eq. 1 can be rewritten ,d
dZtip
dt
‘tip
with
=I
1
_=7
2EU,
R
.
(4)
Equation 4 is easily integrated to yield
t-t
ztip
-_=e
R
sphere
7
50
100
150
200
250
300
Predicted growth rate (1 I r = 2.E.UL/ R) (8-l)
Fig. 8. Measured growth rate of finger flame for various
equivalence ratios and tube radii, plotted as function of
predicted growth rate. Straight line has a slope of unity.
(2)
where Ztip is the axial coordinate of the tip and
r is the radius of the flame finger. The total
flame surface area is given by
Neglecting
0
ratios. The results are presented in Fig. 8 which
shows the experimentally measured growth rate
as a function of the predicted growth rate,
l/7 = 2EUJR. The gas expansion ratio, E,
was calculated using the burned gas temperatures given in [20]. The laminar burning velocities were taken from [21]. The straight line
indicates equality of both times. This equality
is verified to within the 10% experimental error
bars.
It is concluded that the trajectory of the tip
of the finger flame between tsphereand t,,,, is
exponential with a growth rate that is given, to
a good approximation, by a simple geometrical
model.
(5)
The constant of integration, tsphere, is a measure of the time at which the initial spherical
flame changes to a finger flame. This simple
first order model thus predicts that the locus of
the flame tip should follow an exponential trajectory with respect to time. The characteristic
time of growth is a simple function of the gas
expansion ratio, the laminar burning speed and
the radius of the burner. Figure 6 plots an
exponential curve over the tip trajectory. It can
be seen that the agreement is good until the
flame skirt touches the burner wall at t = twa,,.
To compare the experimental
data with
model predictions, we have measured the characteristic growth rate of flames in tubes of
radii 2.5 and 5 cm and for different equivalent
Measurement
of tspherefwa,,, and ttulip
The time at which the flame skirt touches the
burner wall, twa,,, can be estimated from the
high-speed films using a plot of the position of
the trailing edge of the flame such as in Fig. 6.
The extrapolation to position z = 0 gives twa,,.
The relative error in twa,, is of the order of 5%.
It is also possible to use the pressure signal,
since, as already mentioned, the first maximum
of the pressure coincides closely with twa,,. This
method is simpler than the previous one but
less precise, f 10%. The results obtained with
these hvo methods, for two tube radii and for
various equivalence ratios, are presented in
Fig. 9 as a function of R/U,. It is found that
t wall = 0.26(R/U,) f O.O2(R/U,). There is no
232
C. CLANET AND G. SEARBY
area after the flame skirt has touched the walls
must be known.
0.08
z
JE 0.06
d
I1 0.04
c
r
0
Velocity of the Flame Skirt Contact Point
0.02
0
0
0.05
0.1
0.15
0.2
0.25
0.3
RlUL (s)
Fig. 9. Experimental values of time, tsphere, at which flame
propagation changes from spherical to finger shaped, t,,,,,
time at which flame skirt touches the burner wall and ttUliP,
time at which tulip shape is formed.
other significant dependence on gas expansion
ratio, tube radius, or laminar burning velocity.
The characteristic time at which the flame
propagation changes from spherical to finger
shaped, tsphere, cannot be measured directly but
can be evaluated using the model. Equation 5
yields:
t sphere
=
t wall
The position of the trailing edge of the skirt of
the flame will be denoted Zskirt and its propagation speed Qkirt. Figure 6 suggests that Uskirt
is constant. This was always the case in the
experiments. Uskirt is constant in one experiment but changes with equivalence ratio. In
the geometrical model this implies that the
skirt of the flame front is not exactly parallel to
the burner wall, as in Fig. 7, but is slightly
conical, making a small angle cy with the burner
walls. The apparent velocity of the contact
point between the front and the wall is then
qkirt = U,/sin(cr). From the data in Fig. 6,
Uskirt= 24 m/s, U, = 0.223 m/s, it is found
that (Y= 10m2 radians, so the angle between
the flame and the wall is very small and the
previous model remains valid. Suppose that the
angle (Y is inversely proportional to the aspect
ratio of the flame finger when the flame touches
the wall:
(6)
(7)
Z Wallis the position of the flame tip when the
skirt touches the wall at time twa,,. This position is directly measured from the films. In
Fig. 9 are also plotted tspherecalculated from
Eq. 6. It can be seen that tsphere= O.l(R/U’)
+ O.O2(R/U,), within experimental errors.
The inversion of front curvature leading to
the formation of the ‘tulip flame’ is a continuous process. The time of formation of the tulip
flame, ttulip, is the time at which the curvature
of the flame front changes sign, see Fig. 1. This
definition is somewhat arbitrary, but has the
advantage of being unambiguous and easy to
apply. Also plotted in Fig. 9 are values of ttulip.
The time at which the tulip inversion occurs is
also a linear function of R/U, and we find:
ttulip = 0.33(R/U,) * O.O2(R/UJ
Some authors consider the tulip formation to coincide
with the time that the flame first touches the
walls. It can be seen that this is not the case,
since ttulip = 1.29t,,,,.
To calculate the tulip inversion time from
the geometric model, the evolution of the flame
where S is a constant, expected to be of order
unity, since 2EU, is the propagation velocity
of a hemispherical flame cap. Figure 10 shows
a plot of the experimental values of the flame
contact velocity, Uskirt, as a function of Zwa,,/7.
Velocity
0
10
20
of Flame
30
40
Skirt
50
60
70
Fig. 10. Velocity at which contact point of flame skirt
moves down the tube as function of characteristic velocity,
Zwal,/~. Straight line slope 1.127.
ON THE “TULIP
The linearity of this plot shows that Eq. 7 is
functionally correct and the numerical value of
6 is 1.13 * 0.02.
Taylor and Richtmyer
Instabilities
That the physical origin of the tulip phenomenon is the flame deceleration is supported by an experiment due to Markstein [lo].
This showed that a flame shape inversion can
result from the interaction of a curved flame
front with a counter propagating planar shock
wave. The specific case of the instability of a
density interface subjected to a shock wave has
been treated, in a different context, by Richtmyer [22]. After recalling the main results concerning the Taylor and Richtmyer instabilities,
we construct a parallel between Markstein’s
experiment and the tulip instability and establish a comparison of the orders of magnitude
obtained in both cases.
In a Taylor instability, two fluids of different
density are submitted to an acceleration field
(along the z axis> perpendicular to their interface. The position of the interface is written:
a(x, 1) = a(t).cos(k.x), where a(t) is the timedependent amplitude and k the perturbation
wavenumber of (a Fourier component of) a
wrinkle on the interface. In the linear case,
u(t1.k e 1, neglecting viscosity and surface
tension effects, Taylor [ll] obtained the following oscillator equation for the amplitude:
d*a(t)
dt*
= k.g(t).
pp+; ;; .u(t),
+
(8)
where g(t) is the z-component of the acceleration and p+,_ are the fluid densities on the +z
and -z sides of the interface, respectively. For
constant acceleration, Eq. 8 leads to oscillation
or exponential growth according to the direction of the acceleration. The corresponding
frequency or growth rate is given by
~0’= k.g.
P+-
P-
p++
P-
233
FLAME” PHENOMENON
.
Richtmyer [22] has considered the case of an
impulsive acceleration, that is to say w,,r,,, CK
1, where r,,, is the impulse duration. In this
specific case Eq. 8 can be integrated
give
da
- = k.V.u,.
dt
p+-
P-
p++
P-
once to
(9)
’
where V = /;=c g(thdt. If the acceleration is
towards the lighter fluid (V.< p+ - p_ 1 < 01,
then the amplitude of the wrinkle will pass
through zero after a time Tin”:
Tinv
=
(
(p+-P-)
-k*V.(p++p_)
-l
(10)
I
If the shock intensity is reduced (i.e., I/ decreased), then the time, rinv, after which the
flame curvature inverses, is increased.
The Inversion
Experiment
Time in Markstein’s
Shock Wave
Markstein has observed the interaction of a
curved flame front and a shock wave [lo]. Full
experimental details are given in the original
paper, but the following analysis is new. A
stoichiometric butane/air mixture was ignited
at one end of a 7.6-cm-square closed tube. The
first stage of the propagation is the same as
described above. However before the growing
finger flame touched the burner walls, the flame
was subjected to a shock wave propagating
towards the burned gases. It can be seen from
Markstein’s pictures that, a short time after
the interaction, the flame front changes concavity in a way very similar to the tulip flame.
For the stoichiometric butane flame, the inversion of the front (flat flame) occurs 0.16 ms
after the passage of the shock wave. Equation
10 can be used to calculate the inversion time
in Markstein’s shock wave experiment. The
wavenumber, k, can be estimated by assuming
that the curved front in the burner tube of
width D, represents a period of a corrugation,
k = 277/D. Markstein gives the velocity jump,
I/, as -97.5 m/s and the ratio p-/p+ is the
gas expansion ratio, E = 7.4, for this flame.
These values give 7inv = 0.163 ms, which is
very close to the measured value. We conclude
that Ritchmyer’s model of the Taylor instability gives a good estimation of the flame front
inversion time in Markstein’s shock wave experiment. Moreover, there is an apparent simi-
234
C. CLANET AND G. SEARBY
larity of the shape and dynamics of the flame
front between this experiment and the tulip
inversion.
The Inversion Time in the Tulip Flame
Experiment
The main difference between the shock wave
and the tulip experiments is in the value of the
product oO~acc,where ~0’ is the time governing the evolution of the tip and race is the
duration of the acceleration. In the shock wave
experiment ~~~~ is very small compared to w; ’.
‘In the case of the tulip flame, the flame is
decelerated continuously by loss of flame surface area after the flame has touched the
burner walls, so w,,~..,, = 1 and the preceding
approximation cannot be used. This difference
does not refute that both phenomena may
have the same physical origin, but changes the
~order of magnitude of the inversion time.
To calculate the tulip inversion time, the
acceleration of the flame tip must be calculated. According to the geometric model, the
velocity of the flame front is governed by the
production of burned gases. After the skirt of
the flame has touched the burner walls, the
total area of the flame is given by:
Equation 11 can be used along with Eqs. 1, 2,
and the experimental relation 7 to calculate
the velocity of the flame tip after contact with
the burner wall:
dZtip
‘tip
= 7
7
-a-
OCCLUS, Tin” = ttulip -
‘k>
= $kr”g(t).dt
twall:
This average acceleration is substituted into
Eq. 8. Since the acceleration is negative, the
Tulip formation
Zwall
T2
SiOIl
(11)
- Z,kirt).
S = 2~r(Z*i,
This acceleration is time dependent and Eqs.
9 and 13 cannot be solved analytically. Equation 8 is integrated numerically using Eq. 13
for the acceleration and Zwa,, calculated from
Eq. 5, using the experimental values for twa,,
and &here- It is supposed that the curved front
in the tube represents a period of wrinkling,
and k is set equal to m/R. The time of inveris found by integrating Eq. 8
sion, ttulip - LII
until the amplitude of the wrinkle changes
sign. The results of this numerical calculation
for ttulip are plotted in Fig. 11 as a function of
the experimentally measured values of the tulip
inversion time. The model gives the correct
value of the phenomenon to within the 10%
error bars. The inversion time can also be
found more simply as follows: From Eq. 13 the
aueruge acceleration (g) seen by the flame
front during the interval, 7inv, is calculated
between the time at which the flame touches
the burner wall and the time the tulip inver-
time
(12)
t,
where T has the same meaning as before and
6 = 1.13 is the experimentally measured constant related to the contact velocity of the
flame skirt. This equation can be solved explicitly for Ztip and then differentiated to obtain
the acceleration of the flame tip, g(t):
g(t)
=
=
d2Ztip
7
-(a
0
0
-
I)+exp !I.?.!! .
i
7
1
-.‘-,.l.,.‘,..‘...i
0.02
0.04
t,u,,p
(13)
Fig. 11. Calculated
values
flame inversion occurs.
0.06
0.08
0.1
measured (s)
of time
ttulip at which
tulip
ON THE “TULIP
235
FLAME” PHENOMENON
solution of Eq. 8 is oscillatory and the tulip
inversion time is identified as one quarter of
the oscillation period. This leads to an implicit
equation for the inversion time:
.
(15)
The values of the inversion time obtained by
this approximate method are also plotted in
Fig. 11. The values are very close to those
obtained by direct numerical integration of
Eq. 8.
Thus, a simple geometrical model of flame
propagation in a half-open tube is sufficient to
explain the formation of the exponentially
growing finger flame and also the curvature
inversion leading to the formation of the tulip
flame. This model neglects all viscous and
compressible effects. Although acoustic waves
generated by the unsteady flame propagation
are visible in all but the longest tubes, this
model gives the correct order of magnitude for
the growth rate and inversion times, showing
that acoustic and viscous effects do not drive
the initial stages of the tulip flame formation.
However, the depth of the tulip cusp is strongly
decreased or enhanced according to the sign of
the acoustic acceleration at the time of inversion, which in turn depends on the aspect ratio
of the tube.
In a recent numerical study, Dold and Joulin
[23] have remarked that it is necessary and
sufficient to include inertial effects to observe
the tulip inversion. The model proposed here
(Eq. 8) is second order in time, meaning that
inertial effects are indeed included. However,
of all the mechanisms participating in the
Darrieus-Laundau
instability [24] we have
retained
only acceleration
effects
since
the Froude number is not small ((g ). S/
UL’= 1). This assumption reduces the Darrieus-Landau instability to the Taylor mechanism.
The last part of this paper will show that the
results obtained in the half-open tube can be
transposed to experiments performed in closed
vessels.
COMPARISON WITH EXPERIMENTS
CLOSED VESSELS
IN
This section compares the present results with
those of Starke and Roth [13] obtained in
the classical closed burner configuration. In
their experiment, Starke & Roth used an
acetylene/air
mixture. The laminar burning
velocities were taken from 1251.
In these experiments the pressure and density change considerably. It is convenient to
express the progression of the flame front in
terms of the burned mass fraction, 7. Equation
1 can then be rewritten:
(16)
where V, is the volume of the closed combustion chamber. The unburned gas density is now
a function of the burned mass fraction. In the
particular case n K 1, Eq. 16 reduces to Eq. 1.
Difficulties arise from the dependence of the
thermodynamic properties on n. In order to
make the calculation tractable some simplifying assumptions are made: the molecular
weight of the products are equal to that of the
reactants and the heat capacities cg, cu, do not
depend on temperature and pressure during an
adiabatic compression. Equation 16 can then
be written in a nondimensional form:
d?,
- = [l + y.(E - l).n]“‘.r*.z,
dt*
(17)
where
t* = t.U,/R,
Z = S/S,,
y = cp/cu,
r* = R/L. S, and L are the cross-section area
and the length of the combustion chamber,
respectively.
The surface area of the flame front is evaluated using the same model as for the half-open
tube, see Fig. 7. Using the same physical assumptions concerning the shape of the finger
flame and neglecting the radial propagation
velocity compared to the axial tip velocity,
Eq. 17 becomes a first-order homogeneous
equation for the burned mass fraction, 7:
dv
1+ f
i
.[l + y.(E - l).?$‘y
- (1 - 77)
i
= 2.dt*
(18)
236
C. CLANET AND G. SEARBY
Burned mass fraction
/
1.0,
I
!
I
d
are physically correct. In the initial stage of
propagation (77 < 11, the growth rate of the
finger flame in a closed vessel is related to that
of the open tube by a simple multiplicative
constant.
Comparison
0.0
0.2
0.4
0.6
0.8
Relative position of flame tip, i&I
1.0
L
Fig. 12. Burned mass fraction as function of relative position of flame tip in closed vessel experiment.
The solution of Eq. 18 gives the evolution of n
with time.
This model leads to a relation between the
flame tip position and the burned gas fraction:
=tip
-=
L
,,;
i
1
(1 - 77)
-
[l + y.(E - 1).7#y
*
(19)
This relation is plotted in Fig. 12 for E = 7
and y = 1.4. The essential result is that the
burned mass fraction remains small, n I 0.2,
while the flame tip position is in the first 60%
of the combustion vessel, qip/L < 0.6. This
allows Eq. 18 to be simplified in the limit
77-=K1,
=tip
r* + E.77
-=_
L
L
r* (E - 1)
dv
-v+a
and
= 2.E* i l+
where a =
3
E
dt*,
r*
Starke and Roth used cylindrical burners, 5 cm
in radius and with two different lengths, 38 and
76 cm. We calculate the magnitude of the
correction term for the growth rate of the
finger flame. In their experiment, E = 8 and
0.066 < r* < 0.132, so the correction given by
Eq. 20 is between 2% and 4%. In the experiments of Starke and Roth, the tulip always
appeared in the first 2/3 of the tube and we
may compare their data to ours.
Their gas velocity measurements are used to
obtain the time of arrival of the flame tip at
different positions in the vessel and thus deduce the growth rate of the finger flame. It is
supposed that the time of arrival of the flame
skirt at the burner wall, twa,,, coincides with
the time of maximum gas velocity ahead of
the flame. These values are summarized in
Table 1.
Figure 13 plots the growth rate of their
finger flame as a function of the value calculated from the present model, Eqs. 4 and 20.
The dependence is linear, but the experimental
values are a little higher then the calculated
values. The numerical simulation of Gonzales
et al. [5] shows a growth rate of 1250 s-’ (their
Fig. lb), whereas the growth rate calculated by
our model is in their case 1095 SK’. Figure 14
plots the value of twa,,, deduced from their
data, as function of R/U,. Here also, there is
a linear dependence, with a slope of 0.22, not
far from the value of 0.27 we found in the
These relations
3E + r*.(E - 1).
give the tra_jectory of the flame tip and yield
the first order correction to the growth rate of
the flame finger:
These two growth rates are equal in the case of
an infinite length tube (r* = 0) or for a non
expanding gas mixture (E = 1). These limits
with the Data of Starke and Roth
TABLE
1
Experimental Data from Starke and Roth
1.0
0.67
0.63
0.50
8.5
7.2
6.9
5.9
130
70.6
61.3
33.6
5
5
5
5
76
38
38
38
590
260
203
89
8.0
18
18
33
ON THE “TULIP
FLAME” PHENOMENON
Growth ratea of
finger
flame
700
600
600
400
300
200
100
0
.
0
100
200
Calculated
300
growth
400
600
rate (a-c)
Fig. 13. Experimental growth rate of finger flame deduced
from Starke and Roth’s data as function of the growth rate
given by the geometrical model. Straight line slope 1.26.
Arrival
0.04
I
3
E
0.02
of flame
,
I
- ..
.
.
1
0
0.05
/
0.1
anisotropic finger shape and accelerates. After
the skirt of the finger flame touches the side
walls, it takes a finite time for the tulip flame
to appear. During this period the flame is
continuously decelerated.
Neither viscosity, nor acoustic effects are
dominant in these stages of the development
of the tulip phenomenon. It thus follows that,
as long as the flame front thickness is small
compare to the tube dimensions, the only relevant parameters are the laminar burning velocity U,, the tube radius R and the non dimensional gas expansion coefficient E.
A geometrical model predicts the trajectory
of the flame tip to be exponential in time
during the anisotropic fin er ppFtiThe
charB
acteristic growth rate is - = L. * L ExperiR
skirt at wall
J
0
237
J
0.16
RW Cd
Fig. 14. Time of arrival of flame skirt at burner wall, t,,,,,
deduced from Starke and Roth’s data as a function of
R/l&. Straight line through data slope 0.22.
present experiments in half-open tubes. It was
not possible to estimate the tulip inversion
time from their published data. However these
comparisons tend to prove that the observed
phenomena are the same in the half-open tube
and in the closed vessel.
x CONCLUSION
Tulip flame phenomenon in a half-open tube
has been studied with a high speed camera.
There are four stages of flame propagation:
From ‘0 to ‘sphere the flame develops at a
constant speed with a hemispherical shape.
the flame has a very
From ‘sphere
to
&vall
mental measurements of &here and twa,, show
that these times are constant when expressed
in units of R/U,.
The formation of the tulip flame is a manifestation of the Taylor instability driven by the
deceleration of the flame tip. The inversion
time for flame curvature, predicted by this
mechanism, is in good agreement with the experimental results.
The model has been tested and results compared with those of Starke and Roth obtained
in the classical closed vessel configuration.
There is reasonable agreement.
Further work concerns the development of a
theoretical model able to predict the experimental values for the times, &here and twa,,, at
which the flame changes between regimes.
The authors wish to thank Alain Pocheau for
helpful discussions. This work was cam’ed out
with the help of financial support from the
D.R.E. T. under contract number 920074.
REFERENCES
Mallard and Le Chatelier, Ann. Mine, Paris Series
81274-377 (1883).
Ellis, 0. C. d. C., J. Fuel Sci. VII:502-508 (1928).
Salamandra, G. D., Bazhenova, T. V., and Naboko,
I. M., Seventh Symposium (International) on Combustion, Butterworths, London, 1959, pp. 851-855.
Dunn-Rankin, D., Barr, P. K., and Sawyer, R. F.,
Twenty-First
Symposium
(Intemarional)
on Combus-
238
C. CLANET AND G. SEARBY
fion, The Combustion Institute, Pittsburgh, 1986, pp.
1291-1301.
5. Gonzales, M., Borghi, R., and Saouab, A., Cornbust.
Flame 88: 201-220
6.
7.
8.
9.
10.
11.
12.
13.
(1992).
Rotman, D. A., and Oppenheim, A. K., Twenty-First
Symposium (International) on Combustion, The Combustion Institute, 1986, Pittsburgh, pp. 1303-1312.
Oppenheim, A. K., and Ghonheim, A. F., in 21st
Aerospace Sciences Meeting, AIAA-83-0470,
Reno,
Nevada, 1983.
N’Konga, B., Fernandez, G., Guillard, H., and Larrouturou, B., Combust. Sci. Technol. 87~69-89 (1992).
GuCnoche, H., in Non-Steady Flame Propagation
(Markstein, G. H. Ed.), Pergamon, London, 1964.
Markstein, G. H., Sixth International Symposium on
Combustion, Reinhold, New York, 1956, pp. 387-398.
Taylor, G. I., Proc. R. Sot. Land. A CCI:192-196
(1950).
Strehlow, R. A., Combustion Fundamentals, McGrawHill, New York, 1985.
Starke, R., and Roth, P., Combust. Flame 66~249-259
(1986).
14. Starke, R., and Roth, P., Combust. Flame 75:111-121
(1989).
15. Pocheau, A., and Kwon, C. W., in Colloque de l’A.RC.
“ModZsation de la combustion dans les moteurs ci
piston” (C.N.R.S.-P.I.R.S.E.M.Ed.1, Paris, 1989, p. 62.
16. Leyer, J.-C., and Manson, N., Thirteenth Symposium
(International) on Combustion, The Combustion Institute, Pittsburgh, 1971, pp. 551-557.
17. Searby, G., Combust. Sci. Technol. 81:221-231 (1992).
18. Searby, G., and Rochwerger, D., J. Fluid Mech.
231:529-543
(1991).
19. Gonzales, M., in 14th International CON. on the dynamics of explosive and reactive systems, Coimbra Portugal, 1993.
20. Strehlow, R. A., Fundamentals of Combustion,
Kreiger, New York, 1979.
21. Yamaoka, I., and Tsuji, H., Twentieth Symposium (International) on Combustion, The Combustion Institute, 1984, Pittsburgh, pp. 1883-1892.
R. D., Comm. Pure Appl. Math.
22. Richtmyer,
13:297-319
23. Dold,
(1960).
J. W., and
100:450-456
Joulin,
G., Cornbust.
Flame
(1994).
Zeldovich, Y. B., Barenblatt, G. I., Librovich, V. B.,
and Makhviladze, G. M., The Mathematical Theory of
Combustion and Explosions, Plenum, New York, 1985.
25. Gdttgens, J., Mauss, F., and Peters, N., Twenty-Fourth
Symposium (International) on Combustion, The Combustion Institute, 1992, Pittsburgh, pp. 129-135.
24.
Received 3 January 1995; revised 6 August I995