Flame propagation along a vortex axis

Transcription

Progress in Energy and Combustion Science 28 (2002) 477–542
www.elsevier.com/locate/pecs
Flame propagation along a vortex axis
Satoru Ishizuka*
Department of Mechanical Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739-8527, Japan
Received 12 August 2001; accepted 12 August 2002
Abstract
The propagating flame along a vortex axis is completely different from the ‘normal’ flames, which propagate in a tube or
expand spherically from an ignition point. Its propagating speed is nearly equal to the maximum tangential velocity in the
vortex. Thus, the flame propagation is governed not by a physico-chemical parameter, Su (the laminar burning velocity) but by
an aerodynamic parameter, Vu max (the maximum tangential velocity). Considerable efforts have been made to find the
characteristics of flame propagation; flame shape, speed, diameter, and steadiness of propagation, limits of propagation, the
Lewis number effects, and the aerodynamic structure, as well as whether pressure is raised behind the flame or not are all
important characteristics. In this article, the progress accomplished in the experimental, theoretical and numerical
investigations of the rapid flame propagation along a vortex axis is reviewed. Based on the knowledge of the flame
characteristics, modeling combustion in turbulence, which consists of fine-scale eddies, is discussed. q 2002 Elsevier Science
Ltd. All rights reserved.
Keywords: Combustion mechanism; Flame propagation; Flame speed; Vortex; Vortex breakdown; Back-pressure; Lewis number; Swirl
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Appearance and behavior of flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. Flame shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2. Steadiness of flame propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3. Flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Propagation limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. Concentration limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2. Aerodynamic limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Flame speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Steadiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2. Flame speeds as functions of the maximum tangential velocity. . . . . . . . . . . . . . . . . . . .
3.4. Pressure difference across the flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5. Flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Flame kernel deformation mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Vortex bursting mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. The original theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2. The angular momentum conservation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Tel.: þ81-824-24-7563; fax: þ 81-824-22-7193.
E-mail address: [email protected] (S. Ishizuka).
0360-1285/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.
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S. Ishizuka / Progress in Energy and Combustion Science 28 (2002) 477–542
4.2.3. A hypothesis based on the pressure difference measurement . . . . . . . . . . . . . . . . . . . . . .
4.2.4. A steady state, immiscible stagnant model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.5. The finite flame diameter approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.6. The back-pressure drive flame propagation mechanism . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.7. A steady-state back-pressure drive flame propagation model . . . . . . . . . . . . . . . . . . . . .
4.3. Baroclinic push mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Azimuthal vorticity evolution mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. Vortex breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Flame speeds: summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1. Flame speeds for typical flame diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2. Analogy between flows in vortices and gravitational flows . . . . . . . . . . . . . . . . . . . . . . .
6.2.3. Flame speeds for finite flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4. A note on Burgers vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.5. A general comparison between theories and experiments . . . . . . . . . . . . . . . . . . . . . . . .
6.2.6. An unresolved problem: finiteness of flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. Modeling turbulent combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
Flame propagation is one of the most basic problems in
combustion research. Related to explosion hazards in mines
[1,2], and also with the development of spark-ignited engine
and many of her premixed combustion devices, extensive
studies have been made of this subject. Early experiments
on the flame propagation in a tube have resulted in discovery
of the fundamental speed of flame, which is obtained by
dividing the flame speed by the ratio of the flame area to the
cross-sectional area of the tube [3]. This fundamental flame
speed is what we call today ‘the burning velocity’. For its
determination, two types of flames, stationary flames and
non-stationary (moving) flames, are used [4]. A spherically
expanding flame is one of the non-stationary flames. Under a
laminar flow condition and when free from any flame front
instability, the flame speed Vf is directly related to the
burning velocity Su ; which is a physico-chemical constant of
the mixture. In the case of a flame propagating in a tube, the
flame speed is given as
Vf ¼ Su
Af
;
Atube
ð1Þ
in which Af is the flame area and Atube is the cross-area of the
tube [3]. In a spherically expanding flame, the flame speed is
given as
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flame in a layered mixture, the flame speed is given as [5]
rffiffiffiffiffi
r
Vf ¼ Su u :
ð3Þ
rb
As clearly indicated above, these flame speeds are given as
functions of the burning velocity.
The propagating flame along a vortex axis has a
completely different aspect from these flames. A recent
study [6] predicts the flame speed in the form of
rffiffiffiffiffiffiffiffiffiffi
r
Vf ¼ Su þ Vu max 1 þ b :
ð4Þ
ru
Here, Vu max is the maximum tangential velocity in a vortex.
This is just an example of the results of recent studies; the
rigorous expression for the flame speed is under study.
The maximum tangential velocity is of the order of
10 m/s in a strong vortex, whereas the burning velocity is at
most 40 cm/s in most hydrocarbon fuels. Therefore, the
flame speed is controlled mainly by the aerodynamic
parameter Vu max ; not by the physico-chemical parameter
Su : There are some examples, whose flame speeds are
governed by other than physico-chemical factors. Turbulent
burning velocities ST ; are given as functions of the
turbulence intensity u0 ; and the simplest form is given as [7]
ST ¼ Su þ u0 :
ð5Þ
ð2Þ
An upward propagating flame near the flammability limits is
controlled by buoyancy and the flame speed is given as
pffiffiffiffi
Vf ¼ 0:328 gD;
ð6Þ
in which ru and rb are the densities of the unburned and
burned gases, respectively [4]. In the case of the propagating
in which g is the acceleration due to gravity and D is the
internal diameter of the tube [8]. Thus, the propagating
Vf ¼ Su
ru
;
rb
flame in a vortex has a different propagation mechanism
from other flames. Vortex bursting (breakdown) is the
possible mechanism; this was first pointed out by Professor
Chomiak in 1977 [9].
The phenomenon of vortex breakdown has been first
observed in the leading-edge vortices trailing from delta
wings [10], but it can also be observed in the swirling flows
in a duct, in rotating fluids in a container, and even in
combustion chambers [11]; there is some evidence that it
might occur in tornado funnels [12]. However, the vortex
breakdown in combustion is different from that in
conventional flows in that the density changes abruptly in
the breakdown region, due to combustion. Therefore, the
phenomenon of rapid flame propagation along a vortex axis
is also very important in the field of fluid dynamics in order
to understand correctly the vortex breakdown phenomenon
in constant-density flows.
To summarize the phenomenon of rapid flame propagation along a vortex axis seems to have the following
aspects of interest:
1. Flame propagation. The flame has a different propagation mechanism from the ordinary flames.
2. Flame – vortex interactions. Although the flame – vortex
interaction problem has been extensively studied [13],
little is known on the case when the vortex axis is
perpendicular to the flame surface.
3. Modeling of turbulent combustion. Chomiak [9], Tabaczynski et al. [14] and Klimov [15] have taken this
phenomenon into consideration in their turbulent
combustion models.
4. Flammability limits. Although the flammability limit is
defined as a propagation limit of a self-sustaining flame,
unexpected flame propagation can occur along a vortex
axis. The definition of flammability limits is obscured.
5. Combustion control. This phenomenon may provide a
new tool to control and enhance combustion in internal
combustion engines and industrial furnaces.
6. Fire safety. In a zero-g environment, where an artificially
created spinning gravitational field is formed, an unusual
flame spread may occur through this phenomenon.
7. Vortex breakdown. The rapid flame propagation
phenomenon itself can be regarded as an extension
of the vortex breakdown phenomenon in a constantdensity flow to that in a variable density flow.
This review is devoted first to a historical survey of the
studies of rapid flame propagation along the vortex axis in
Section 2, followed by the presentation of experimental
results to acquire basic, true knowledge on this phenomenon
in Section 3. Next, theories and numerical simulations are
presented in Sections 4 and 5, respectively. Relevant studies
of the vortex breakdown phenomenon in constant-density
flows are briefly reviewed in Section 6. A general discussion
is also made on the effects of vortex breakdown on modeling
479
turbulent combustion. Finally, conclusions and future
studies are presented in Section 7.
2. A brief history
To the author’s knowledge, Moore and Martin were the
first, to deal with flame propagation along a vortex axis.
Their report appeared in Letters to the Editor in the journal
Fuel in 1953 [16]. They used a glass tube 125-cm long and
47 mm diameter, one end of which was closed and fitted
with a 6-mm diameter entry nozzle tangential to the tube
circumference 1 cm from the closed end. They reported that
a tongue of flame, projected into the unburned gas within the
tube mouth, extended eventually to the closed end. They
emphasized that such flame flash-back occurred even when
the flow rate exceeded the critical value for blow off, if the
mixture was introduced not tangentially but straightforward.
Flame speeds were not measured. It was only mentioned that
‘the phenomenon was not stable; regular pulses of flame
passed down the tube with a velocity dependent on the
strength and rate of flow of the mixture’. This author was
unable to trace their following work, although it is written in
the last part of their report that ‘the investigation is
continuing’.
We had to wait nearly two decades to know the actual
flame speed along the vortex axis. In 1971, McCormack
measured the flame speed in the vortex rings of rich
propane/air mixtures [17]. In this experiment, the flame
speed was 300 cm/s. McCormack’s research was supported
by the Ohio State University, and followed by a work
performed with his co-workers, which appeared in Combustion and Flame in 1972 [18]. This time, they constructed a
bigger vortex ring generator and measured the flame speed
as a function of the vortex strength. The results are shown in
Fig. 1. It is seen that the flame speed increases almost
linearly with an increase in the vortex strength, and the
maximum flame speed reaches about 1400 cm/s. The flame
speed becomes much higher if pure oxygen is used as an
oxidizer. The mechanism for the high propagation speed,
however, was unknown. In his first paper [17], hydrodynamic instability, inherent to density gradient in a rotating
flow, was suspected. In their second paper [18], turbulence
was a candidate.
The results of McCormack et al. [18] have attracted keen
interest from other researchers. In 1974, Margolin and
Karpov made an experiment in an eddy combustion
chamber, which was a rotating vessel, 80-mm diameter
and 50-mm long [19]. They found that when a mixture was
ignited at periphery, the flame kernel first moved towards
the axis of rotation, and after reaching the axis of rotation,
the flame volume became cigar-shaped; that is, the
dimension of the volume along the axis increased much
faster than the volume along the radius. The radial flame
speed became lower than the flame speed in the quiescent
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as ‘vortex breakdown,’ termed ‘vortex bursting’. By noticing
pressure jump across the flame, and also considering
momentum flux conservation across the flame, he has
derived the following expression for the flame speed,
rffiffiffiffiffi
r
Vf ¼ Vu max u :
ð7Þ
rb
Fig. 1. Flame speed versus vortex strength [18].
mixture, whereas the axial velocity became much faster than
the flame in the quiescent mixture. As compared with the
combustion in the non-rotating case, the centripetal movement of the hot kernel to the axis of rotation resulted in a
rapid increase of flame area, and in a rapid pressure rise—
which is important from a practical viewpoint. Deformation
of the flame kernel was considered to enhance the axial
flame speed.
Also, Lovachev [20] considered that the deformation of
flame kernel in the centrifuge enhanced the flame speed in
vortices. In the course of his study on flammability limits, he
found that a flame spreads over a ceiling at a speed about
twice the flame speed in a quiescent mixture. He has pointed
out the analogy between the flame creeping over a ceiling
and the flame propagating along a vortex axis. Under the
ceiling, buoyancy, which acts on the hot burned gas, flattens
the shape of the hot kernel, resulting in the rapid flame
spread. In a vortex, the centrifugal force of rotation
suppresses expansion of the hot gas in a radial direction,
whereas it promotes elongation of the flame kernel along the
vortex axis [20,21].
However, a completely different mechanism was proposed in 1977 [9] when Chomiak proposed the concept of
vortex bursting for the rapid flame propagation along a
vortex axis, and developed a model for turbulent combustion
at high Reynolds number [9,22]. He considered that the rapid
flame propagation could be achieved by the same mechanism
Here, Vu max is the maximum tangential velocity in
Rankine’s combined vortex, and ru and rb are the density
of the unburned and burned gases, respectively.
Rapid flame propagation along a vortex axis has also
been taken into consideration in modeling turbulent
combustion by Tabaczynski et al. [14,23], Klimov [15],
Thomas [24] and Daneshyar and Hill [25]. Fig. 2 shows a
model by Tabaczynski et al. [14]. In this model, a flame was
assumed to propagate instantaneously along a vortex of
Kolmogorov scale, followed by combustion with a laminar
burning velocity. In the hydrodynamic model by Klimov
[15], the vortex scale was assumed to be much larger than
the Kolmogorov scale.
In 1987, Daneshyar and Hill [25] described the concept
of vortex bursting in more detail. By considering the angular
momentum conservation across the flame front, they have
obtained the pressure difference across the flame, DP, which
is equal to
"
2 #
0
0
r
2
DP ¼ ru u 1 2 b
ð8Þ
< ru u 2 ;
ru
in which u0 is rms of the velocity fluctuation and considered
to be equal to the maximum tangential velocity Vu max in
Rankine’s combined vortex. They considered further that
this pressure difference set-up a large axial velocity of
burned gas ua. By equating the pressure difference DP with
the kinetic energy of the burned gas rb u2a =2; they obtained an
Fig. 2. Model of burning, turbulent, small-scale structure [14].
expression for the axial velocity of the hot gas,
sffiffiffiffiffiffi
sffiffiffiffiffiffi
2ru
0 2ru
¼ Vu max
:
ua < u
rb
rb
ð9Þ
In modeling turbulent combustion, they introduced the
concept of average pressure difference, which was about
one-third of DP, resulting in
sffiffiffiffiffiffiffi
2 ru
0 2 ru
¼ Vu max
:
ð10Þ
ua < u
3 rb
3 rb
Finally, they gave the flame propagation speed in turbulent
combustion as
0 2 ru
;
ð11Þ
ut ¼ ul þ u
3 rb
in which ul is the adiabatic laminar burning velocity.
Since the works by Chomiak [9,22] and by Daneshyar
and Hill [25], related studies have been made. In 1983,
Zawadzki and Jarosinski investigated the effect of rotation
on the turbulent burning velocity [26]. They have found that
an intense rotating flow causes marked laminarization of
turbulent combustion, and as a result, the turbulent burning
velocity remains at the same level as the laminar burning
velocity [26]. In 1984, Hanson and Thomas investigated
flame development in rotating vessels to find that due to the
‘penciling’ effect, the flame is distorted from its otherwise
spherical shape; its surface area increases considerably, and
the combustion time is shortened in the rotating mixtures
[27]. It is interesting to note that in their paper, the pressure
difference between the burned gas and the unburned gas is
thought to be a driving force for the penciling effect. The
magnitude is the order of
DP ¼
1
2
W 2 rf2 ðru 2 rb Þ;
ð12Þ
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investigated the occurrence of the inward flow theoretically
[29]. Subsequent studies by Libby et al. in 1990 [30] and by
Kim et al. in 1992 [31] have revealed that three stagnation
points can appear in the case of strong rotation. This
complex structure of the flame in a rotating flow field may
provide a useful insight into the aerodynamic structure of
the flame, which propagates along a vortex axis through
vortex breakdown. In 1988, Sivashinsky et al. studied flame
propagation in a rotating tube to find that the flame speed
can be amplified in a rotating tube [32]. The effects of
rotation on a Bunsen flame were also investigated. In 1990,
Sheu et al. showed both experimentally and theoretically
that the cellular instability could be suppressed by rotation
[33]. Sohrab and co-workers have further investigated the
shapes of Bunsen flames under rotation [34,35]. Very
recently, Ueda et al. studied the Bunsen flame tip carefully,
to find that various tip behaviors, such as oscillation, tilting
and eccentric movement, are dominated by the Lewis
number of the deficient component in the mixture [36].
On the other hand, in 1984, a tubular flame was found to
exist in a stretched, rotating flow field [37]. Its characteristics have been studied both experimentally [37– 41] and
theoretically [42 – 46], and a survey on tubular flame
characteristics was published in this journal in 1993 [47].
This stationary flame study triggered a non-stationary flame
study in vortex flows. Using the same type of vortex flow as
in the work by Moore and Martin [16], a study on flame
propagation along a vortex axis was restarted [48], although
nearly four decades had already passed since their work.
In this study [48], the maximum tangential velocities, as
well as the flame speeds, were measured. Thus, this is the
first time that the relationship between the flame speed and
the maximum tangential velocity was obtained. Fig. 3 [48]
shows the results. It is seen that, as predicted by the theories
(Eqs. (7), (9) and (10)), the flame speed increases almost
in which W is the rotational speed and rf is the maximum
radius of the burned gas. Since the product of W and rf is
equal to the maximum tangential velocity Vu max ; the
pressure difference DP can be rewritten as
r
DP ¼ 12 ru Vu2 max 1 2 b :
ð120 Þ
ru
The coefficient 1/2 is obtained because in a rotating vessel a
forced vortex is formed. Thus, it is interesting to note that
both the flame deformation mechanism and the vortex
bursting mechanism give almost the same magnitude of
pressure difference for driving the flame.
In the mean time, the effects of rotation on various flame
characteristics have been studied from a fundamental
viewpoint. In 1987, Chen et al. made an experiment on a
binary flame in a stagnation point flow [28]. By rotating the
burner around a center axis perpendicular to the flame front,
they determined the extinction limit as a function of the
rotational speed, to discover a curious tendency in the limit.
This was caused by the occurrence of an inward, radial flow
on a stagnation plane. In 1987, Sivashinsky and Sohrab
Fig. 3. Relations between flame speed Vf and maximum tangential
velocity Wmax in various mixtures (the mean axial velocity Vm ¼
3:0 m=s; injector III) [48].
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linearly with an increase in the maximum tangential
velocity. The speeds, however, are much lower than the
predictions. In the predictions, the flame speed should be
several times as high as the maximum tangential velocity,
while the flame speeds measured are almost equal to or less
than the maximum tangential velocity. Another interesting
discovery in this study is that the propagating flame is
strongly influenced by the Lewis number. Fig. 4 [48] shows
the flame shapes of various mixtures. In rich methane (Fig.
4(b)) and lean propane mixtures (Fig. 4(c)), in which the
mass diffusivity of deficient component Di is smaller than
the thermal diffusivity k of the mixture, the head of the
flame is highly dispersed and weakened, whereas the head is
intensified in lean methane (Fig. 4(a)) and rich propane (Fig.
4(d)) mixtures of Di $ k: It is very interesting to note that in
a rich propane mixture, a flame of small diameter can
propagate in the vortex flow. A further study has been made
on the limit of propagation to find that flame propagation by
rotation is possible only if the modified Richardson number
exceeds the order of unity [49]. It has also been found from
static pressure measure measurements that unlike the usual
flames propagating in a quiescent mixture or in a onedimensional stream, pressure is raised behind the flame to an
extent of almost the same order of magnitude as predicted
by the vortex bursting theory [50].
About that time, Asato et al. restarted the vortex ring
experiments [51]. They used a vortex ring generator (which
diameter was almost the same as that used by McCormack
et al. [18]), to determine the flame speed, ring diameter, and
the translational speed of the vortex ring [51]. Since their
measured flame velocities were much lower than those
predicted by Chomiak [9] and Daneshyar and Hill [25], they
modified their theories by taking the finite flame radius into
consideration. It is regrettable, however, that they have not
Fig. 4. The shapes of propagating flames in (a) lean methane ðV ¼ 5:3%Þ; (b) rich methane ðV ¼ 11:9%Þ; (c) lean propane ðV ¼ 2:7%Þ; and (d)
rich propane ðV ¼ 7:7%Þ mixtures (the mean axial velocity Vm ¼ 3:0 m=s; injector III) [48].
measured the maximum tangential velocity. They estimated
the maximum tangential velocities simply by putting their
measured translational velocity into the Lamb’s equation
[52] and assuming that the core diameter was 10% of the
ring diameter [51]. This leads to a significant overestimation
(two or three times) for the actual maximum tangential
velocity. The Lewis number effects were also noted in their
following papers [53,54]. They further investigated the
propagating flame in stretched vortex rings, which were
obtained using vortex rings that impinged on a wall [55].
Later, they measured the flame diameter precisely and
constructed a theory which considered the effects of finite
diameter on the flame speed, although their maximum
tangential velocities were still obtained still using the
estimation [56,57].
Independently of the studies on the flame propagating
along a vortex axis, Ono and co-workers investigated the
flame propagation in a rotating disk [58,59]. Their
interesting finding was that at a very high rotational speed,
the flame ignited at the center, could not continue to
propagate outwards and the flame was extinguished at some
distance from the center. Flame extinction in a strong
centrifugal force field had been reported previously,
however, the extinction occurred when the flame propagated
not outward but inwards [60,61]. They also found that such
flame extinction occurred only for lean methane/air and rich
propane/air mixtures, whose Lewis number is less than unity
[58,59]. This result seems important in order to understand
the finite flame diameter, observed both in the vortex ring
combustion [17,18], and in the flame propagation in the
vortex flow [48].
In 1994, Atobiloye and Britter proposed a steady-state
model for the rapid flame propagation in a rotating tube [62].
They used Bernoulli’s equation to describe axial flows
accurately. Although their solutions were obtained numerically, their model gives much lower flame speeds than the
Chomiak theory [9] or the model by Daneshyar and Hill
[25]. It is interesting to note that in the limit of an infinitely
large diameter tube and if a free vortex is assumed, the ratio
of the flame velocity to the maximum tangential velocity Ui
pffiffiffiffiffiffiffiffiffiffiffiffi
is approximately given by Ui < 1 2 rb =ru ; i.e. in the limit
of an unconfined free vortex flow,
r
Vf < Vu max 1 2 b :
ð13Þ
ru
In 1995, Hasegawa et al. started a numerical simulation on
the flame propagation along a vortex axis [63]. They showed
that the flame propagation by the vortex bursting can occur
when the flame size becomes larger than the thickness of
laminar flame. Just after the work by Hasegawa et al.,
Ashurst proposed a different mechanism for the flame
propagation, termed as ‘baroclinic push’ [64]. He addressed
the baroclinic torque (a vector product of the density
gradient 7r and the pressure gradient 7P ), in his theory to
account for the rapid flame propagation along a vortex axis.
He asserts that this baroclinic torque evokes a vorticity v
around the flame through
dv
1
¼
7r £ 7P;
dt
r2
483
ð14Þ
where t is time, and the flame is accelerated by this vorticity.
His final expression for the flame speed UB is
pffiffiffiffiffiffiffiffi
t
1
2
pffiffiffiffiffiffiffi ðrM VM
UB ¼
Þ
ð15Þ
XF =rM :
SL ð1 þ tÞ
d 1þt
Here, t is the heat release parameter, which equals the
density ratio minus unity, d is the flame thickness, SL is the
laminar burning velocity, and XF is the length of the burned
gas in the axial direction. The vortex swirling motion, in
terms of the angular velocity, is given in a form
Vu
G
2
¼
½1 2 expð2r 2 =rM
Þ;
r
2pr 2
ð16Þ
where rM is the radius at which the vorticity is reduced by
e 21 the value at infinity, and VM is the approximate
maximum swirl velocity by setting the circulation as G ¼
2prM VM :
In their numerical simulation, Hasegawa and Nishikado
have shown that the baroclinic torque is produced around
the propagating flame [65]. Compared with the vortex
Fig. 5. The relation between flame speed Vf and maximum
tangential velocity Vu max in the vortex ring experiment [6].
484
bursting theories, they concluded that as the baroclinic push
mechanism could better explain the inverse dependency of
the propagation velocity on the density ratio, as well as the
dependency on almost the second power of the circumferential velocity, and the diameter of the vortex tube [65].
Because there were many discrepancies in theories and
experiments, further studies were conducted. In 1996, Sakai
and Ishizuka experimented with a rotating tube [66]. In this
study, the maximum tangential velocity could be known
accurately by multiplying the rotational speed by the radius
of the tube. In 1998, Ishizuka and co-workers restarted the
vortex ring combustion [6]. In this study, the maximum
tangential velocity Vu max and the translational velocity U
were first determined by hot-wire anemometry for the cold
air vortex rings. Next, the relationship between the flame
speed Vf and the maximum tangential velocity Vu max ; in the
vortex ring combustion, was obtained with the aid of the
obtained U 2 Vu max relation, since the flame speed Vf and
the translational velocity U could be obtained at the same
time from a Schlieren photograph. Major findings from the
two studies [6,66] are as follows:
1. A steady state of flame propagation could not be
achieved in a rotating tube, but the flame speed is
almost constant in a vortex ring.
2. The slopes in the Vf 2 Vu max plane are much lower than
pffiffiffiffiffiffiffi
the value of ru =rb and nearly achieve unity for the
stoichiometric methane/air mixture (Fig. 5 [6]).
To account for the measured flame speeds, a theory,
termed as the back-pressure drive flame propagation theory,
has been proposed [6,66– 69]. This theory predicts the flame
speed as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ru
lVf l ¼
ðYSu Þ2 þ Vu2 max f ðkÞ
rb
ð17Þ
for the case when the burned gas is expanded only in the
axial direction, and as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
lVf l ¼ YSu þ Vu max 1 þ b f ðkÞ;
ð18Þ
ru
when the burned gas is expanded only in the radial direction.
Here, k is the ratio of the burning radius of the unburned gas
to the vortex core radius, Y is the ratio of the flame area to
the cross-sectional area of the unburned gas, and the
function f ðkÞ is given as
8 2
1
>
for k # 1;
< 2k
f ðkÞ ¼
:
ð19Þ
1
>
:1 2
for k $ 1
2
2k
The first terms in Eqs. (17) and (18) correspond to a velocity
induced by chemical reaction and the second terms
correspond to a velocity induced aerodynamically due to
rotation. It should be noted that the slope in the Vf 2 Vu max
plane almost achieves unity for the radial expansion case.
Quantitatively as well qualitatively, these theoretical
results are in good agreement with experimental results for
various mixtures of methane and propane fuels [70]. The
back-pressure drive flame propagation theory was originally
derived for non-steady flame propagation in a rotating tube
[66]. A recent observation of the vortex ring combustion,
however, indicates that a steady state of flame propagation
can be achieved for smaller Reynolds numbers [71]. Based
on this result, the back-pressure drive flame propagation
theory has been extended to a steady-state model to account
Fig. 6. Schematic of experimental set-up for vortex flow in a tube [48,50].
485
Fig. 7. Schematic of experimental apparatus for vortex ring combustion [6].
for the enhancement of the flame speeds in rich hydrogen/air
mixtures [67,72], in which the flame speed is greatly
increased, approaching the flame speed predicted by
Chomiak [9].
Very recently, Asato et al. measured the maximum
tangential velocity by hot-wire anemometry and they reexamined their results to find that the slope in the Vf 2
Vu max plane are near unity [73]. This strongly supports the
experimental results obtained by Ishizuka et al. [6,69,70]
Also, in a theoretical field, Umemura et al. proposed a new
mechanism, in which evolution of azimuthal vorticity is
responsible for the rapid flame propagation [74 – 77].
Hasegawa and co-workers have begun to study the flame
propagation in a straight vortex, both experimentally and
numerically [78,80]. Gorczakowski et al. have made an
experimental study of the flame propagation in a flow field
of rigid-body rotation to realize a new engine operated at
increased compression ratios, far from the knock limit [81].
Very recently, surveys have been made on theoretical
studies by Umemura [82] and also of experimental studies
and numerical simulations by Hasegawa [83].
3. Experiments
As described in Section 2, the phenomenon of flame
propagation along a vortex axis has not yet been understood
completely. In this chapter, we shall first look at
experimental results to obtain basic knowledge of this
phenomenon.
By now, many experiments have been made using
various types of vortex flows; they can be classified into
three groups [66]. The first type of experiments is the vortex
flow in a tube (Fig. 6) [48,50]; the second is flows in a vortex
ring (Fig. 7) [6] and in a straight vortex (Fig. 8) [80,83]; the
third experiment is the flow in a rotating tube (Fig. 9) [66]
and in a rotating vessel (Figs. 10 and 11) [27,58].
Fig. 8. Laser ignition at the core of one straight vortex [80,83].
486
Fig. 9. Schematic of experimental set-up for forced vortex in a rotating tube [66].
In case of the vortex flow, a combustible mixture is
introduced tangentially from one end of a tube, and the
mixture flows with the rotation in the tube. Finally, the
mixture exits from the other, open end. The vortex strength
is not constant in the tube; it decays almost inversely
proportional to the square root of the distance from the inlet
end [48]. The advantage of this method is that the vortex
flow is stationary and steady. This yields measurements of
Fig. 10. General arrangement for a rotating cylinder [27].
487
Fig. 11. Rotary combustion chamber arrangement [58].
static pressure as well as the maximum tangential velocity
by probing [50]. In case of vortex ring, or straight vortex, the
vortex strength is constant along the vortex axis; however,
the vortex moves forward at some velocity. To make matters
worse, mixing of the combustible gas with the ambient
mixture occurs at the same time. Very recently, experiments
were conducted in an atmosphere of the same mixture as the
combustible gas in the vortex ring [84,85]. In the third
case—of a rotating tube or a rotating vessel—a forced
vortex flow can be obtained. The vortex strength is uniform
along the vortex axis, and in addition, the flow is stationary.
However, the space is surrounded with solid walls. Hence,
complex phenomena such as tulip flames may occur. Thus,
any of the three methodologies is imperfect. In order to
understand the flame propagation phenomena correctly, we
need many types of vortex flows.
3.1. Appearance and behavior of flame
3.1.1. Flame shape
At first, we shall look at flame shape and flame behavior
in the vortex flows. As pointed out by Moore and Martin
[16], the flame is forced into a vortex center, and as a result,
the flame is convex towards the unburned mixture. Using a
similar apparatus to that of Moore and Martin [16],
observations have been made of the flame. Fig. 6 [48,50]
is the schematic of the apparatus. The diameter of the glass
tube is 31 mm and its length is 1 m. A combustible mixture
is tangentially introduced from a closed end and exits from
the other, open end. Fig. 12 [49] shows sequential
photographs of the propagating flame taken with a highspeed camera. The flame is convex toward the unburned
mixture, and it propagates into the tube, eventually to the
closed end. In the case of the Bunsen flame formed at an
open end of a rotating tube, buckling of the flame tip occurs
Fig. 12. High-speed photographs of the propagating flame in the
vortex flow (stoichiometric propane/air mixture, mean axial
velocity Vm ¼ 2 m=s; 120 frames/s) [49].
488
at some critical rotational speed, and the flame becomes
convex towards the unburned gas [33]. Above some
adequate rate of rotation, an axi-symmetric flame, which
is convex towards the unburned mixture, can propagate in a
rotating tube [33]. Therefore, the first point regarding to be
noted flame propagation is that the flame is convex towards
the unburned gas.
3.1.2. Steadiness of flame propagation
The second noteworthy point regarding flame propagation is that a steady state of propagation is usually not
achieved in the vortex flows. As seen in Fig. 12, the flame is
accelerated first from the open end where the mixture is
ignited. This is reasonable since the vortex becomes
stronger as the tangential inlet is approached. Fig. 13 [48]
shows the variation of the maximum tangential velocity
with the axial distance Z from the tangential inlet end. These
measurements were made by hot-wire anemometer under
cold flow conditions [48]. As seen in Fig. 13, the maximum
tangential velocity is decreased almost inversely proportional to the square root of the distance Z. This indicates
that the vortex decays mainly by viscous dissipation.
Halfway during flame propagation, however, the so-called
‘tulip flame phenomenon’ occurs (Fig. 14) [49]. The flame is
flattened and retarded. Thus, it should be noted that complex
phenomena might happen in a confined space such as a tube.
However, even if the rotation in a tube is constant and
uniform, such acceleration and deceleration occur. Fig. 15
[66] shows the variation of the flame speed Vf with the
distance from an ignition end X in a rotating tube [66]. The
flame is accelerated first, decelerated, and again, accelerated. On the other hand, in vortex ring combustion, a steady
state of flame propagation can be achieved, although under
limited conditions.
Fig. 16 [71] shows a Schlieren sequence of the vortex
ring combustion. Owing to the bulk of the vortex generator,
Fig. 13. Axial decay of maximum tangential velocity Wmax in the
vortex flow (the mean axial velocity Vm ¼ 3:0 m=s; injector III)
[48].
the light beam from the first mirror passed through the
vortex at an angle of about 308 to the direction normal to the
plane of the vortex ring. Hence, the vortex ring appears
elliptical. It is very difficult to conclude from this sequence
whether the flame propagates at a constant flame speed.
Fig. 17 [71] shows sequences of intensified image, taken
from the direction normal to the plane of the vortex ring.
The local flame speed can be obtained with reasonable
accuracy from this sequence. Their results indicate that the
flame speed is almost constant during propagation—within
30% if the Reynolds number, defined as Re ; UD=v (U is
the translational velocity of the vortex ring, D is the ring
diameter, and v is the kinematic viscosity of the mixture), is
less than the order of 104. Around Re < 104 a longitudinal
instability occurs, resulting in a periodic change in flame
speed. For Re $ Oð104 Þ; the flame speed is significantly
scattered because the turbulent vortex ring is established
[86]. It is important to note that precession around a vortex
axis may occur in any vortex whether it is the vortex ring or
the vortex flow in a tube.
Fig. 18 shows a sequence of the intensified images of the
propagating flame in the vortex flow of hydrogen/air
mixtures in a tube. The hydrogen concentration is 12.5%
and the mean axial velocity is 2 m/s. This picture was taken
using an ultra-violet lens and a high-speed video camera at
3200 frames/s. In Fig. 18, the picture starts from the top in
the left row, followed the arrow, and finally reaches the
bottom in the fifth row. It is seen that on the way of
propagation from the right to the left, the flame tip moves
upwards above the center axis in the first and second rows,
but it stops in the third row and then the tip moves
downwards in the fourth and fifth rows. It seems that
precession does occur and the rotational axis is constantly
oscillating, whether its extent is larger or smaller.
Very recently, experiments on line vortex have been
conducted. Fig. 19 [78,80] shows Schlieren sequences of
hydrogen/air mixtures. A tip vibrating behavior can be seen
in the straight vortex with the maximum tangential velocity
Vm ¼ 35:8 m=s: In this case, the flame speed can be
accurately determined from the Schlieren pictures, and in
addition, PIV has been applied to the vortex to determine the
velocity profiles. Fig. 20 [78,80] shows an example of the
outputs. The advantage of PIV is that the core diameter, as
well as the tangential velocity distribution, can be obtained
at the same time. It is also possible to obtain the pressure
distribution by integrating its profiles.
3.1.3. Flame diameter
The third point regarding the flame propagation is
that the head of the flame is intensified in the mixture
of Le , 1; whereas it is weakened in the mixture of
Le . 1: Fig. 4 [48] shows the flame shapes of various
mixtures. In most of the mixtures, the heads are blurred
and a distinct flame zone, such as a laminar flame zone,
cannot be identified. However, it can be seen that the
head region of the flame is intensified in a rich propane/
489
Fig. 14. Photographs showing flattening of the axisymmetric flame front, and appearance of the intense luminosity at the center (mean axial
velocity Vm ¼ 2 m=s; methane/air mixture, fuel concentration V ¼ 9:5%CH4, the two-inlet case, Fig. 14d, was taken during another run) [49].
air mixture (Fig. 4(d)), whereas the head is weakened
when the mixture is rich in methane (Fig. 4(b)) and lean
for propane (Fig. 4(c)) mixtures. The head is neutral for
a lean methane/air mixture (Fig. 4(a)). Near the
propagation limits, the flame diameter can become
small in the rich propane/air mixture, whereas it is
still larger in the latter mixtures, as mentioned.
In the case of a rotating vessel or disk, the dimension in
the radial direction is comparable to, or longer than in the
axial direction. Flame behavior in the radial direction can be
observed in detail. Fig. 21 [58] shows a Schlieren sequence
of lean methane/air mixture at 52 rad/s. After ignition at the
center, the flame propagates cylindrically first, but the flame
velocity is gradually reduced, and finally, it ceases to
propagate. Fig. 22 [58] shows the variations of the flame
radius with time for the lean mixture at different angular
speeds. When the disk is not rotating (the rotational speed
v ¼ 0), the flame can reach the wall. However, the flame
cannot reach the wall when the rotational speed is high. The
radial distance at which the flame can propagate becomes
smaller as the rotational speed is increased. The flame
extinction halfway at propagation could be seen only with
lean methane/air and rich propane/air mixtures. Hence,
the Lewis number of the deficient species seems to be
responsible for the flame extinction as it is responsible for
the finite flame diameter in the vortex flow (Fig. 4). Ono
et al. considered a shear flow, which is raised in the
unburned region by the expansion of the burned section, to
be a basic cause for the flame extinction at the finite flame
diameter. However, Gorczakowski and Jarosinski [87] have
490
Fig. 15. Variations of flame velocity Vf with distance from the
ignition end X in a rotating tube (methane/air mixtures; N ¼ 1210
rpm; Each symbol denotes one experimental run) [66].
pointed out from their Schlieren pictures that cooling at the
wall plays an important role in the flame extinction.
3.2. Propagation limits
3.2.1. Concentration limits
In the vortex flow in a tube, a flame can propagate in
most of the mixtures, which are within the flammability
limits determined by the standard method. Figs. 23 and 24
Fig. 17. Time sequence of intensified images of vortex ring
combustion (stoichiometric propane/air mixture, orifice diameter
Do ¼ 60 mm; driving pressure P ¼ 0:4 MPa) [71].
[48] show the flame propagation regions in the vortex flow
in a tube, using three injectors for methane and propane,
respectively. Here, Vm is the mean axial velocity obtained
by dividing the mixture flow rate by the cross-sectional area
of the tube. Note that in Injectors I– III, the total crosssectional area of four tangential slits are decreased, and
Fig. 16. Schlieren sequence of vortex ring combustion (stoichiometric propane/air mixture, orifice diameter Do ¼ 60 mm; driving
pressure P ¼ 0:4 MPa) [71].
Fig. 18. High-speed photographs of propagating flame in the vortex
flow (hydrogen/air mixture, fuel concentration V ¼ 12:5%; mean
axial velocity Vm ¼ 2 m=s; 3200 frames/s).
491
Fig. 19. Evolution of nitrogen-diluted hydrogen–oxygen flames (29.4% hydrogen) ignited by a pulsed laser, (a) in quiescent mixture, (b) in a
straight vortex with Vm ¼ 18:0 m=s and (c) in a straight vortex with Vm ¼ 35:8 m=s [78,80].
hence, the intensity of rotation at the closed end becomes
stronger for a fixed Vm : (Also note that with an increase in
Vm ; the intensity of rotation is increased.)
During weak rotation (Injector I), there are two regions
for flame propagation, the low-velocity region and the highvelocity region. The critical velocity, Vc ; in this figure is the
mean axial velocity of the mixture above which the flame
cannot propagate to reach the closed end, if the mixture is
introduced without rotation. In the low-velocity region, the
flame propagates through an aerothermochemical mechanism inherent to combustion—namely propagates when the
oncoming flow velocity is less than the velocity of flame
propagation related with the burning velocity, for example,
Eq. (1). On the other hand, in the high-velocity region,
the flame propagates through an aerodynamic mechanism
inherent to rotation. As seen in Figs. 23 and 24, with an
increase in the intensity of rotation, i.e. in the order of
Injectors I– III, the boundary velocity between the low- and
high-velocity regions becomes smaller and the area of the
high-velocity region expands. In the case of the present
horizontal tube (31 mm inner diameter and 1000 mm long),
the concentration limits for flame propagation determined in
a quiescent mixture are 5.5 and 13.1% for lean and rich
methane/air mixtures, respectively, and 2.3 and 8.0% for
lean and rich propane – air mixtures, respectively. Thus, a
flame can propagate slightly below the lean limit for lean
methane/air mixtures and slightly above the rich limit for
propane/air mixtures in the vortex flow [48].
In the case of a rotating tube (no axial velocity),
however, a remarkable result has been obtained for the
limits of propagation. Fig. 25 [66] shows the concentration
limit for flame propagation in a rotating tube which is
32-mm inner diameter and 2000 mm long. With an increase
in the rotational speed, the equivalence ratio f at the
propagation limit is increased for rich propane/air mixtures
and exceeds the rich flammability limit by the standard
method ðf ¼ 2:5Þ: It is unclear whether a self-sustained
flame is really established in this limit. However, it is
probable that this extension may be achieved by a
combination of two processes—the vortex bursting and
the flame intensification by the Lewis number effect. That is,
once a hot gas is introduced into the vortex core, the
preferential diffusion of the deficient species may occur
around the head of the involved gas. This sustains
combustion. A hot gas of low density is supplied, resulting
in higher pressure behind the flame. Thus, the flame
propagation is maintained.
Fig. 20. Velocity field around a vortex pair measured by PIV system
(moving velocity of the straight vortex U ¼ 20:8 m=s) [78,80].
492
Fig. 21. Photographs of flame propagation for M2 (0.52CH4 þ 2O2 þ 7.52N2) mixture in a rotary combustion chamber [58].
The Lewis number effect can also be seen in vortex ring
combustion. Fig. 26 [53] shows the propagation limits
determined for the vortex ring combustion by Asato et al.
With an increase in the maximum tangential velocity Vu max ;
the rich propagation limit for methane is steeply decreased;
whereas the lean propagation limit for methane and the rich
propagation limit for propane are not changed significantly.
These results are explained on the basis of the Lewis number
effect [53]. The Lewis number of rich methane/air mixtures
is larger than unity, hence, the flame tip is weakened in
burning, whereas the Lewis numbers of rich propane and
lean methane mixtures are less than unity, therefore, the
flame tip is intensified in burning. Based on the Lewis
number, the lean limit of propane ðLe . 1Þ is greater than
the lean limit of methane ðLe , 1Þ; and the rich limit of
methane ðLe . 1Þ is less than the rich limit of propane
ðLe , 1Þ:
However, there are some curious points in the results of
Fig. 26. Although the rich limit of methane is greatly
decreased, the lean limit of propane (whose Lewis number is
larger than unity as the rich methane mixture) is almost
constant, independent of the maximum tangential velocity.
The rich limit for propane (about 1.9) is much lower than the
standard flammability limit (2.5), although it is close to the
rich flammability limit in the vortex flow (Fig. 24). The rich
limit of methane at low maximum tangential velocities is
above 2.0, and hence, exceeds the rich flammability limit by
the standard method (1.67). It seems that an entrainment of
ambient air occurs since the experiments have been
conducted in air. An additional experiment should be
made with the same combustible mixture to positively
identify the concentration limits.
3.2.2. Aerodynamic limits
As seen in Figs. 23 and 24, there are two regions for
flame propagation. In the low velocity region, the flame
propagates at a speed corresponding to the reaction rate. In
the high velocity region, the flame propagates via an
aerodynamic mechanism inherent to rotation. A necessary
condition for the flame to propagate through rotation is
establishment of an axi-symmetric flame in the vortex flow.
To overcome some disturbing forces and accomplish the
formation of an axi-symmetric flame, the rotational speed
must exceed some critical value. A shear force, which is
directly proportional to the velocity gradient, may disturb
the formation of an axi-symmetric flame. The competition
between the driving, centrifugal force of rotation and the
disturbing, shear force can be characterized by the modified
Richardson number Rip ; which is defined as
Rip ;
1 ›r W 2
r ›r r
›U
›r
2
:
ð20Þ
Here, W is the tangential velocity, U is the axial velocity, r
and r are the density and the radial distance from the axis of
rotation, respectively [49,88]. At the lower boundary of the
high-velocity region, the modified Richardson number
seems to be the order of unity.
According to the results in the experiment [49], in which
the rotational strength is varied by changing the number of
Fig. 22. Process of flame growth for M4 (0.56CH4 þ 2O2 þ 7.52N2)
mixture in a rotating disk ignited at center [58].
Fig. 23. Flame propagation region of methane– air mixtures for
three injectors in the vortex flow [48].
slits through which a mixture is tangentially introduced, the
modified Richardson number Rip is 1.4 when there is one
inlet (the strongest case), 0.625 with two inlets, and 0.225
when there are four inlets. With eight inlets (the weakest
case), there is no high-velocity region, since the rotation is
too weak. Thus, the modified Richardson number at the
lower limit of the high-velocity region is not constant,
although the value is of the order of unity. The limit
aerodynamic condition for the occurrence of rapid flame
propagation along a vortex axis is still unclear. Competition
between the axial velocity and the turbulent flame speed
(because the turbulent intensity is strong near the axis of
rotation) is another possible mechanism.
Fig. 27 [89,90] shows the relationship between the flame
speed Vf and the maximum tangential velocity, Wmax ; in
the vortex flow [48] for various mean axial velocities Vm :
The flame speed is increased with an increase in Wmax for
any Vm : However, as the value of Vm is increased, the curve
shifts to the right side. The curve does not intersect the
lateral axis at the origin and it appears to intersect at some
finite value, which is slightly lower than its own Vm -value.
493
Fig. 24. Flame propagation region of propane–air mixtures for three
injectors in the vortex flow [48].
Fig. 25. Variation of equivalence ratio f at the propagation limit
with rotational speed N [66].
494
which magnitude is almost equal to the axial velocity;
otherwise the flame cannot overcome the oncoming axial
flow in the flame front to propagate through a mechanism
which is inherent to rotation.
3.3. Flame speeds
Fig. 26. Limits of flame propagation in vortex ring combustion [53].
For example, in the case of Vm ¼ 3 m=s; the flame speed Vf
is decreased with a decrease of Wmax ; and falls steeply, as if
it could intersect at a value slightly lower than Wmax ¼ 3:0
m=s: Note that the measured values of flame speed on the
curve for Vm ¼ 3 m=s are those at stations 3 – 9 in Fig. 6 from
the top. Thus, the flame speed decreases as the open end is
approached. At station 9, the rotation of the mixture is very
weak, and therefore the mixture flows mostly in the axial
direction, without rotation and its mean axial velocity is
Vm ¼ 3 m=s: Thus, it seems that flame propagation through
rotation requires some amount of rotational velocity in
Fig. 27. Relation between flame speed Vf and maximum tangential
velocity Wmax for various mean flow velocities, obtained with the
vortex flow experiment [89,90].
3.3.1. Steadiness
Fig. 28 shows the variations of flame speed in various
mixtures in the vortex flow [48]. The flame speed is
increased as the tangential inlet is approached. This is
reasonable because the intensity of rotation is increased as
the tangential inlet is approached (Fig. 13). Thus, a steady
state of constant flame speed has not been achieved in this
vortex flow. Also, as pointed out in Fig. 15, a constant flame
speed is not achieved in a rotating tube, although the
rotational speed is constant along the axis of the tube. Thus,
a steady state of flame speed will not be achieved in a
confined space such as a small tube.
In an open space, however, a steady state can be
achieved. Fig. 29 [71] shows the variation of flame speed in
a vortex ring. The upper figure shows the variation of flame
diameter with time. In this case, the cylinder diameter of the
vortex ring generator is 160 mm and the orifice diameter is
90 mm. The driving pressure is 0.4 MPa. The solid and open
symbols correspond to the flames propagating on the right
and left halves of the vortex ring after ignition, respectively.
The flame diameters monotonically increase with time and
become constant. On the other hand, the flame speeds
decrease and increase similar to a sine wave. They are
almost constant, although they are scattered within a 30%
band. Further measurements of the flame speed have shown
that there are four types of flame propagation in the vortex
ring combustion: (1) steady flame propagation, (2) oscillatory flame propagation, (3) unsteady flame propagation
with acceleration and/or deceleration, and (4) random flame
propagation [71]. The steady flame propagation occurs for
Re ; UD=v # 104 : The oscillatory propagation originates
from a longitudinal instability of the vortex ring [91,92].
The random propagation occurs for Re $ 104 ; which is
caused mainly by the turbulent nature of the vortex ring
[86]. The unsteady propagation can be seen in the range
between the steady propagation and the random propagation
in the mapping of the mean flame speed Vf and the
maximum tangential velocity Vu max : According to very
recent research [93], the ratio of the square root of the
fluctuations in the flame speed to its mean speed is, at
smallest, 0.2 for propane/air mixtures in the steady flame
propagation regime. For the vortex rings of the stoichiometric hydrogen/air and methane/air mixtures, the ratios are
about 0.3 for a wide range of the Reynolds number. Thus,
the flame propagation in combustible vortex rings is not
steady but ‘quasi-steady’ in the strictest sense of the word.
This fluctuation seems to occur due to the precession of the
vortex core, as seen in Fig. 18.
495
Fig. 28. Spatial distributions of the flame speeds Vf (Z: the distance from the ignition end, the mean axial velocity: 3.0 m/s, injector III) [48].
3.3.2. Flame speeds as functions of the maximum tangential
velocity
The first measurement of the flame speed was made by
McCormack and co-workers [17,18]. The relationship
between the flame speed and the vortex strength is shown
in Fig. 1. From the theoretical viewpoint, however, the
relationship between the flame speed and the maximum
tangential velocity is more important. This was first
obtained in the vortex flow in a tube (Fig. 3) [48]. In the
case of the vortex ring, the relation was first obtained by
Asato et al. [51]. Fig. 30 [57] is one of their results. The
flame speeds were obtained by high speed Schlieren
photography, while the maximum tangential velocity,
Vu max ; is estimated by determining the translational velocity
of the vortex ring U, and by using Lamb’s relation with an
assumption of the tangential velocity distribution of
Rankine form,
G
8D
1
2
;
ð21aÞ
U¼
ln
d
4
2pD
Vu max ¼
Fig. 29. Variations of flame/core diameter ratio df =dc and local flame
speed Vf with time, showing an almost constant flame speed during
combustion (stoichiometric propane/air mixtures, orifice diameter:
90 mm, driving pressure: 0.4 MPa, broken line: mean flame speed)
[71].
G
:
pd
ð21bÞ
Here, G is circulation, D is the ring diameter, and d is the
core diameter.
It is seen in Fig. 30 that the flame speeds are much
lower than those predicted by Chomiak [9]. To obtain the
value Vu max ; however, it was assumed that the core
diameter was 10% of the ring diameter. This assumption
seems reasonable because the core diameters are 10.8% of
the maximum ring diameter in the Maxworthy experiment
[94] and 8.65% of the mean ring diameter in the Johnson
measurement [95]. In the Johnson’s experiment, the
cylinder diameter was 4 in. and the orifice diameter was
50 mm. In the experiment by Asato et al. [51], the
cylinder diameter was 220 mm and the orifice diameter
was 70 mm. Therefore, both of the generators are almost
the same in size.
Ishizuka and co-workers have attempted to measure
the maximum tangential velocity Vu max by hot wire
anemometry. The method is illustrated in Fig. 31 [6].
Although only one probe is shown in Fig. 31, two hot
wire probes are placed along a path where the ring
passes, and both U and Vu max are measured at the same
time for a traveling vortex ring. The value of U is
496
Fig. 30. Change in flame speed with maximum tangential velocity
for propane– air mixture [57].
obtained simply by dividing the distance between the
probes (2.5 cm), by the time required for the ring to pass.
The value of Vu max is obtained by setting the probe as
shown in Case I-a, and by subtracting the translational
velocity U from the maximum output value, which may
correspond to Vu max þ U:
Fig. 32 [6] shows the relationship between the
translational velocity U and the maximum tangential
velocity Vu max : In these measurements, the cylinder
diameter is 100 mm and the orifice diameters are 30, 40
and 60 mm. An almost linear relationship has been
obtained between U and Vu max for each orifice. If the
Lamb’s relation is applied to these results, a least square
fitting gives the core to diameter ratio d=D as 36% for
60 mm orifice, 39% for 40 mm orifice and 48% for 30 mm
orifice (only a line for D0 ¼ 60 mm is shown in Fig. 32).
Fig. 33 [69] shows the results for the 40-mm orifice, in
which the results regarding the core diameter, determined
by the two-peak method in case I-b, are also shown. It is
important to note that in case I-b, velocity changes rapidly
in magnitude and in direction, and the response of the hot
wire is poor; hence, the peak value becomes much lower
than predicted. Direct measurement of the value of d=D is
32.3%, which is smaller than the value of 39%, obtained by
fitting the Lamb’s relation to the results of U and Vu max :
Therefore, there is a discrepancy.
Fig. 31. Methods for measuring the maximum tangential velocity by
hot-wire anemometry [6].
Fig. 32. Relation between translational velocity U and maximum
tangential velocity Vu max of the vortex ring [6].
Fig. 33. The variations of maximum tangential velocity Vu max and
core/ring diameter ratio d=D with translational velocity U for the
orifice diameter Do ¼ 40 mm [69].
Fig. 34 shows pictures of the propagating flames in
vortex rings, taken very recently in the author’s laboratory.
The upper part of the vortex ring is illuminated by a laser
sheet, while the propagating flame, ignited at the bottom by
a corona discharge, is photographed by a high speed video
camera with an image intensifier. The cylinder diameter, the
orifice diameter, the piston stroke, and the driving pressure
are 160 mm, 70 mm, 15 mm, and 0.4 MPa, respectively. To
seed fine particles for laser tomography, the methane/air
mixtures are introduced into a heated pipe, on the wall
surface of which kerosene is dripped through a syringe, and
the mixture is supplied into the vortex ring generator. After
the mixture is ejected through an orifice by a piston,
kerosene is condensed, and small particles of kerosene can
be obtained. Due to a centrifugal force of rotation, the
number density of droplets is reduced in the core region.
Hence, the core region is represented as a dark zone in
Fig. 34. Johnson used a similar method to measure the core
497
Fig. 34. Photographs of the vortex ring combustion of propane/air
mixtures in air for equivalence ratio, (a) 0.6 and (b) 0.8. The upper
half of the vortex rings is illuminated with a laser sheet, and the
propagating flame, ignited at the bottom of the vortex ring, is
recorded with a high-speed video camera with an image intensifier.
The mixtures are doped with kerosene vapor, and the particles are
obtained by condensation of the kerosene vapor. The dark zone in
the upper half may correspond to the vortex core. A cylinder
160 mm in diameter and an orifice 60 mm in diameter are used to
generate vortex rings. The mean diameters of the vortex rings are
about 7 cm.
diameter of the vortex ring [95]. In Fig. 34 it is seen that
the flame propagates in the core region when the
equivalence ratio f ¼ 0:6; whereas the flame diameter
becomes larger and burning reaches the free vortex region
when f ¼ 0:8: Although the core size cannot be determined
accurately from these photographs, the ratio of the core
diameter to the ring diameter becomes 10 –20%. This value
is much smaller than the value estimated from the U 2
Vu max relation. Therefore, the assumption of 10% core
diameter (made in the paper by Asato et al. [51] and also by
McCormack et al. [18]), seems reasonable. When practically
determined, the maximum tangential velocities are much
498
lower than those estimated from the Lamb’s relation
(assuming 10% core diameter in the Rankine’s combined
vortex) because: (1) The tangential velocity distribution is
not the form of the Rankine’s combined vortex. Burgers
vortex may be preferable for approximating the profile. (2)
The Lamb’s relation is derived from an assumption that the
vortex is circular and the core diameter is much smaller than
the ring diameter. The vortex rings used are not circular nor
is the ratio less than 10%. (3) Although the Lamb’s relation
is derived under a laminar flow condition, most of the vortex
rings used are turbulent.
However, it should be noted that in the paper by Ishizuka
et al. [6] the relationship obtained between U and Vu max for
the cold air vortex ring are directly applied to derive the
relation between the flame speed Vf and the maximum
tangential velocity Vu max : Therefore, the Vf 2 Vu max
relation obtained does not depend on the ambiguity of the
core size. In addition, it is also important to note that the
translational velocity U can be obtained from a series of
Schlieren pictures by measuring a time during which the
ring passes two small rods, while the flame speed is also
obtained from the same series of Schlieren pictures. Thus,
the relationship between the flame speed Vf and the
maximum tangential velocity Vu max can be obtained
without ambiguity for each experimental run.
Figs. 35 – 38 show the results obtained by this method.
Figs. 35 and 36 [70] show the results for methane and for
propane, respectively. Figs. 37 and 38 [67,72] show the
results for hydrogen in air and in a nitrogen atmosphere,
respectively. In the case of hydrogen/air mixtures, the
densities in rich mixtures are significantly different from the
density of air. For example, the density of the rich mixture of
f ¼ 3:2 is one-fifth the density of air. The heat transfer
coefficients of rich hydrogen mixtures are also different
from that of air. Thus, the hot wire results for the cold air
cannot be applied to rich hydrogen mixtures. Then, LVD
measurements have been made for hydrogen mixtures to
obtain the U 2 Vu max relation absolutely [72]. Note that the
original results in Ref. [67] are corrected and presented in
Fig. 37.
It is seen in Figs. 35 and 36 that the flame speed is
increased almost linearly with an increase in Vu max ; and
the values of slope in the Vf 2 Vu max plane are almost
unity in various methane and propane mixtures. With
increasing Vu max ; the flame diameter is decreased monotonically. The solid lines are the predictions by Chomiak
[9] and by Daneshyar and Hill [25]. The broken lines and
the dotted curves are the predictions by the back-pressure
drive flame propagation mechanism, when the burned gas
is assumed to expand in the radial and axial directions,
respectively. The measured flame speeds are in good
quantitative agreement with those predicted by the backpressure drive flame propagation theory for the radial
expansion case, described later. In lean mixtures, however,
the flame diameter becomes very small with increasing
Vu max ; and the flame velocity falls below the broken line.
On the other hand, in rich mixtures, the flame diameter
does not become small and the flame velocity continues to
increase with increasing Vu max : Experiments in a nitrogen
atmosphere [96,97], and in an atmosphere of the same
mixture as the combustible mixture [84,85], show that for
larger values of Vu max ; the flame speed cannot become
larger in rich mixtures as well as in lean mixtures.
Therefore, diffusion burning of the excess fuel with the
ambient air may help the flame to propagate at larger
values of Vu max in rich mixtures (Figs. 35 and 36),
although the slopes remain at about unity.
In hydrogen mixtures, however, the situation is slightly
different from those in methane and propane. As seen in
Fig. 37 [67], the slopes in the Vf 2 Vu max plane are nearly
at unity in lean mixtures. In rich mixtures however, the
slope increases when the equivalence ratio increases. A
linear least squares fitting through the origin gives slope
values of 1.47, 1.70, 1.65, and 2.04 for F ¼ 1:0; 1.6, 2.4,
and 3.2, respectively [72]. The slight increase/decrease in
the slope around F ¼ 1:6 – 2:4 may be because the burning
velocity reaches its maximum around F ¼ 1:6 in hydrogen/air mixtures. The enhancement of the flame speed is
probably due to the secondary combustion of excess
hydrogen with the ambient air in a turbulent mode. In the
nitrogen atmosphere, however, the slope is decreased to
about unity, as seen in Fig. 38 (the solid circles and the
solid triangles are the flame speeds in the nitrogen
atmosphere).
Very recently, Asato and co-workers assumed Burgers
vortex to estimate the maximum tangential velocity in their
vortex ring experiment [73]. The relation they obtained
between the flame speed and the maximum tangential
velocity is shown in Fig. 39 [73]. It is seen that the slopes
in the Vf 2 Vu max plane are nearly at unity for various
methane/air mixtures. This strongly supports the results of
Figs. 5, 35 –38, obtained by Ishizuka et al. [6,67,69,70,72].
Finally, the results obtained in the straight vortex by
Hasegawa et al. are shown in Fig. 40 [80]. In his
measurements, PIV was used. This method is more
reliable than the hot wire method or the LDV method,
since the maximum tangential velocity and the flame
velocity can be determined in the burning vortex at the
same time. The flame speed is increased with an increase
in the maximum tangential velocity and the slope
gradually increases with increased density ratio, ru =rb ;
but remains almost at unity [80, Fig. 10]. This experiment
has been conducted in the atmosphere of the same mixture
as the combustible. Additional results are expected in the
near future.
3.4. Pressure difference across the flame
In the case of a one-dimensional premixed flame,
pressure behind the flame ðPb Þ is lower than
pressure ahead of the flame ðPu Þ; by an extent
ru S2u ðru =rb 2 1Þ; which can be easily derived from
the
the
of
the
499
Fig. 35. Variations of the flame speed Vf and the ratio of the flame diameter to the core diameter df =dc with maximum tangential velocity Vu max
in various propane/air mixtures ((K) Do ¼ 60 mm; (W) Do ¼ 40 mm; (A) Do ¼ 30 mm; solid symbols: (O, X, B) full mean flame speed). The
pffiffiffiffiffiffiffiffiffiffiffi
solid lines are the relations Vf ¼ Vu max 2kI ru =rb (Eqs. (26d) and (27d)); broken lines and dotted curves are the back-pressure drive flame
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
propagation theory for the lateral expansion case (Vf ¼ YSu þ Vu max 1 þ f ðkÞrb =ru ; see Eq. (31u)), and for the axial expansion case
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(Vf ¼ ðru =rb ÞðYSu Þ2 þ f ðkÞVu2 max ; see Eq. (31t)) [70].
500
Fig. 36. Variations of the flame speed Vf and the ratio of the flame diameter to the core diameter df =dc with maximum tangential velocity Vu max
in various propane/air mixtures ((K) Do ¼ 60 mm; (W) Do ¼ 40 mm; (A) Do ¼ 30 mm; solid symbols: (O, X, B) full mean flame speed). Solid
pffiffiffiffiffiffiffiffiffiffiffi
lines are the relations Vf ¼ Vu max 2kI ru =rb (Eqs. (26d) and (27d)); broken lines and dotted curves are the back-pressure drive flame
propagation theory for the lateral expansion case (Vf ¼ YSu þ Vu max 1 þ f ðkÞrb =ru ; see Eq. (31u)), and for the axial expansion case
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(Vf ¼ ðru =rb ÞðYSu Þ2 þ f ðkÞVu2 max ; see Eq. (31t)) [70].
501
Fig. 37. Variations of the flame speed Vf with maximum tangential velocity Vu max in various hydrogen/air mixtures (((K) Do ¼ 40 mm; (W)
Do ¼ 30 mm). Note that the values of Vu max in Ref. [67], measured by hot-wire anemometry, are corrected in these figures. Solid lines are the
relations Vf ¼ Vu max ru =rb (Eq. (26d)); broken lines are the back-pressure drive flame propagation theory for the lateral expansion case
(Vf ¼ Su þ Vu max 1 þ f ðkÞrb =ru ; k ¼ 2; Eq. (31u)).
equations for conservation of mass and momentum across
the flame as,
ru y u ¼ rb y b ;
Pu þ
ru y 2u
¼ Pb þ
ð22aÞ
rb y 2b ;
Pb 2 Pu ¼ 2ru S2u ðru =rb 2 1Þ:
ð22bÞ
ð22cÞ
Here, the coordinate is attached to the flame, subscripts u
and b denote the unburned and burned gases, respectively; P is the static pressure, y is the velocity, and
Su ð¼ y u Þ is the laminar burning velocity, in this
formulation. On the other hand, the vortex busting theory
predicts that the pressure behind the flame is higher than
the pressure ahead of the flame. To elucidate the validity
of the concept of vortex bursting, an attempt has been
made to measure the pressure variation across the flame
with the use of a micro-differential manometer and
conventional static probes [50].
The experimental set-up is shown in Fig. 6. The inner
diameter and length of the glass tube are 31 and 1000 mm,
respectively. Two stainless tubes, 1 cm apart, were inserted
into the tube on the axis of rotation, while a fine Pt/Pt-13Rh
thermocouple was immersed at their midpoint to detect the
flame arrival correctly. For this purpose, three holes of 3 mm
diameter were pierced at each station 1 – 9, placed at interval
of 100 mm, two holes at one side and one hole at its opposite
502
Fig. 38. Variations of the flame speed Vf with the maximum tangential velocity Vu max for rich hydrogen/air mixtures in the nitrogen atmosphere
((a) F ¼ 1:6; (b) F ¼ 2:4; and (c) F ¼ 3:2). Dotted lines are the relations Vf ¼ Vu max ru =rb (Eq. (26d)), broken lines are those by the backpffiffiffiffiffiffiffiffiffiffiffiffi
pressure drive flame propagation theory for the lateral expansion case (Vf ¼ Su þ Vu max 1 þ rb =ru ; i.e. Y ¼ 1 and k ! 1; see Eq. (31u)); solid
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lines and curves are those by the steady-state flame propagation model, Vf < Vu2 max þ ru S2u =rb and Vf ¼ Vu max 1 þ rb =ru (Eqs. (33e) and
(33f)). For comparison, the results in the air atmosphere are shown by open symbols [72].
side, as illustrated in the inset in Fig. 6. For convenience, the
pressure detected by the probe on the open end side is
denoted by P~ B (back side), and the pressure on the injector
side by P~ F (front side), respectively. In this measurement,
the gauze pressure PB ; and the pressure difference PB 2 PF ;
which is denoted by dP; have been measured.
Fig. 41 [50] shows the time histories of PB and dP
together with the temperature history at station 5 after a
quiescence combustible mixture of 9.5% methane, filled in
the tube, is ignited at the open end. After ignition, the
pressure PB first increases, and then becomes constant.
When the flame arrives at station 5, which can be detected
by a sharp increase of temperature, the pressure PB
abruptly decreases and takes an almost constant value.
The pressure PB decreases and then increases greatly when
the flame reaches the closed end. On the other hand, dP is
almost zero after ignition, but takes a negative value when
the flame passes the two probes, after that, dP becomes
zero again.
If we put the values of ru ¼ 1:122 kg=m3 ; ru =rb ¼ 7;
and the observed, mean flame speed Vu ¼ 82 cm=s
(which is in good agreement with the results by Coward
and Hartwell [3]), into Eq. (22c), we obtain that Pb 2
Pu is equal to 2 0.46 mmAq (2 4.53 Pa). This value is
in good agreement with the present results of
2 0.46 mmAq from the P-history. Also, the pressure
difference across the flame can be estimated from the
dP-history by integrating the dP signal with time. That
is, z being the axial distance, and ‘ being the distance
Fig. 39. Change in flame speed with the maximum rotational
velocity [73].
503
between two static probes,
Pb 2 Pu ¼
Fig. 40. Relation between axial propagation velocity and maximum
circumferential velocity for flames with different density ratios:
density ratio 5.3 (circle), 5.8 (triangle), 6.5 (square), and 7.2
(inverted triangle) [80].
ð dp
ð dP
V ð
ðVf dtÞ ¼ f ðdPÞdt:
dz ø
‘
‘
dz
ð23Þ
The obtained value was 2 0.80 mmAq, which was 70%
greater than the value of 2 0.46 mmAq. Nevertheless, it
was shown that the pressure behind the normal flame is
lower than the pressure ahead of the flame. This is
reasonable, since the gas stream is increased due to
expansion of the burned gas, and hence, the static
pressure is converted into the kinetic energy of the
burned gas, resulting in a decrease in the static pressure.
But in vortex flows the situation is changed. Fig. 42 [50]
shows the time histories of PB and dP in the vortex flow. The
mixture is a lean 6.85% methane/air mixture. After ignition,
the flame accelerates and propagates with a higher speed
than that in a quiescent mixture. Correspondingly, the
pressure PB monotonically increases, and its magnitude is
greater than that in a quiescent mixture. When the flame
arrives at station 5, the pressure PB further increases and
the pressure difference dP takes a positive value. Thus, it is
confirmed that the pressure behind the flame is higher than
the pressure ahead of the flame, and the aerodynamic
Fig. 41. The time-history of the gauge pressure PB and the pressure difference dP at station 5 for the flame propagation in a quiescent mixture of
9.5% methane/air mixture with 2 mm static tube [50].
504
Fig. 42. The time-history of the gauge pressure PB and the pressure difference dP at station 5 for the flame propagation in a rotating mixture of
6.85% methane/air mixture and the mean axial velocity ¼ 3 m/s with 2 mm static tube [50].
structure of the propagating flame in the vortex flow is
different from that in a quiescent mixture.
Fig. 43 [50] shows the variations of the pressure
differences Pb 2 Pu obtained directly from the PB -history
and estimated by integrating dP-history at each station for
the lean and stoichiometric mixtures. Due to the limitation
of the micro-differential manometer ð^10 mmAqÞ; only the
results from the dP-history were obtained for the stoichiometric mixture. The broken curves indicate the amount of
2
ru Wmax
; obtained from the measured values of Wmax (Fig.
13). These results give evidence for the vortex bursting
theory, which predicts that the pressure behind the flame is
higher than the pressure ahead of the flame by an amount of
ru Vu2 max {1 2 ðrb =ru Þ2 } (Eq. (8)).
3.5. Flame diameter
Finally, the results regarding the flame diameter are
briefly presented. As already shown in Figs. 35 and 36,
the flame diameter is decreased with an increase in the
maximum tangential velocity. However, due to diffusion
burning in air, the flame diameters of rich mixtures become
larger than those of lean mixtures. Very recently, a vortex
ring experiment was conducted in an atmosphere of the
same mixture as the combustible gas in the vortex ring
[84,85].
Fig. 44 [85] shows the variations of the flame/core
diameter ratio df =dc with the equivalence ratio F at a
condition of Vu max ø 11 m=s for methane and propane,
respectively. The flame diameter is determined with
intensified images. For comparison, results in air and
nitrogen are also presented, which were determined by
Schlieren photography. All measurements are made at the
quarter position of the vortex ring. Note that the core
diameters in air and nitrogen are about 1.5 times as large as
the vortex core in the same atmosphere, simply because the
vortex ring generator has been recently improved to obtain
more intense vortex rings. The core diameters obtained in
the same mixture experiments are about 12.5 mm. In
Fig. 44, the definition of the flame diameter is also shown
in the top illustrations. Usually, the diameter of the
luminous zone increases as the distance from the head is
increased. In lean and rich mixtures, however, this
diameter is saturated once. This saturated value is defined
as the flame diameter. In the near-stoichiometric mixtures,
however, the luminous zone diameter still increases.
505
In Fig. 44(a) and (b), enlarged images of the near-limit
flames are presented. In the near-limit lean methane and rich
propane mixtures, the flame diameters are very small and
burning is intensified at the head, whereas the flame
diameters are not as small and burning is weakened around
the head in the near-limit rich methane and lean propane
mixtures. These observations are in qualitative accordance
with the previous observations in the vortex flow (Fig. 4).
Due to strong flame curvature and flow non-uniformity
around the flame head region, mass and heat transfer across
a stream tube may occur. As a result, the flame suffers from
stretch, and the so-called ‘Lewis number effect’ may appear
in near-limit flame behavior.
4. Theories
McCormack first considered that the density jump across
the flame, and/or the shear waves in vortex rings, induce
flame front instability, resulting in a rapid flame propagation
along a vortex axis [17]. But, he rejected the instability
mechanism. Next, he and his co-workers suspected
turbulence [18]. But, they also discarded the turbulent
mechanism, since the speed had been unusually increased.
Until now, the major mechanisms postulated to explain the
increase in speed are as follows:
Fig. 43. The pressure difference across the flame obtained from the
PB-history and the dP history as a function of the distance from the
injection port Z [50].
Tentatively, the flame diameter is determined at a position
108 of angle behind the head of the flame. The true flame
diameter might be larger than this value.
Although the results are largely scattered, it is seen that
in the lean side, the flame diameters in the same mixture
atmosphere are largest. In the rich side, the flame diameters
in air are largest due to the secondary combustion between
the excess fuel and the ambient air, the diameters in nitrogen
are smallest, and those in the same mixture are midway
between.
In the same mixture, the flame diameter takes its
maximum around F ¼ 1:1; slightly in the rich side of the
stoichiometry, and the flame diameter decreases as the
mixture becomes leaner or richer. Outside the results
shown in Fig. 44(a) and (b), a flame cannot propagate. In
the case of methane, the flame diameter is very small
near the lean propagation limit, whereas it is larger near
the rich propagation limit. In propane, the flame diameter
is very small near the rich limit, whereas it is larger near
the lean limit.
1.
2.
3.
4.
Flame kernel deformation mechanism;
Vortex bursting mechanism;
Baroclinic push mechanism;
Azimuthal vorticity evolution mechanism.
Among these, the vortex bursting mechanism has
received considerable interest from many researchers. In
this section, various mechanisms are individually
described.
4.1. Flame kernel deformation mechanism
Margolin and Karpov [19] have focused attention on the
deformation of the flame kernel in the centrifugal field to
explain flame speed enhancement. When an ignition is made
at an off-center position in an eddy combustion chamber, the
flame first moves towards the axis of rotation. After reaching
the rotational axis, the flame becomes cigar-shaped and its
axial dimension increases much faster than the radial
dimension.
Assuming that the shape of the flame shape is a cylinder
with diameter D and length H, and assuming that the change
of the volume V is proportional to the flame surface S and
the visible burning velocity w, a simple relation can be
obtained:
dV
< Sw:
dt
ð24aÞ
506
Fig. 44. Variations of the flame/core diameter ratio df =dc with the equivalence ratio F of (a) methane/air and (b) propane/air mixtures
(Do ¼ 40 mm; P ¼ 0:4 MPa; Vu max ø 11 m=s. Top illustrations show the definition of the flame diameter) [85].
This results in
D2
dH
dD
þ 2DH
< 2ðD2 þ DHÞw
dt
dt
ð24bÞ
(note that in the paper by Margolin and Karpov [19], D and
H sometimes denote the radius and the half length of the
flame volume, respectively). In addition, they have
conducted an approximate analysis, leading to an expression
507
Fig. 45. Schemes of flame spreading (i) beneath a plane ceiling and (ii) along the rotating cylinder [20].
that
d D
w
ø
dt 2
2
d H
w
ø ðvtÞ
dt 2
2
H
ø vt:
D
ð24cÞ
Thus, the flame speed, dðH=2Þ=dt; increases with time
proportionally to the burning velocity w and also to the
rotational speed v. This result resembles the result by the
baroclinic torque mechanism proposed by Ashurst [64] in
that flame speed is accelerated as the flame propagates.
On the other hand, Lovachev [20,21] has pointed out the
analogy between flame spreading beneath the plain ceiling
and flame spreading along the rotating cylinder (Fig. 45)
[20]. He has considered that the flame kernel beneath the
ceiling is deformed by buoyancy and, as a result, the flame
propagates at a speed about twice (80 cm/s) the ordinary
speed (40 cm/s). But he did not give any theoretical
description of his concept.
Hanson and Thomas [27] have focused attention on the
‘penciling effect,’ which they called the flame speed
enhancement. They have mentioned in their paper that the
penciling effect is no more than the normal action of a
centrifuge flinging out components of higher density and
attracting the lighter components to the axis. If a spherical
bubble of radius rf and density rb is introduced on the axis
of a forced vortex of a fluid of higher density ru in a vessel,
unbalanced centrifugal forces are set up (Fig. 46) [27]. The
magnitude of the pressure difference DP between an
element of burned gas, and that of unburned gas at the
same radius rf ; with a rotational speed W, can be shown to be
of the order
DP ¼
1
2
W 2 rf2 ðru 2 rb Þ:
ð25aÞ
In their paper, it is written that if constant pressure at the
rotation axis is assumed, the higher pressure exists in the
denser gas, whereas if constant pressure at the wall is
assumed, the higher pressure exists in the less dense
medium.
Hanson and Thomas have pointed out that this pressure
difference drives the motion around the flame. They
considered that the pressure difference will be of the same
order as Eq. (25a) and the acceleration of the fluid will be
proportional to W 2 : The variation of the vertical diameter H
has been considered to follow the variation predicted by a
constant acceleration
H ¼ A þ Bt þ Ct2 :
ð25bÞ
It is interesting to note that the pressure difference across the
flame has been considered similarly by Chomiak in his
vortex bursting mechanism [9].
4.2. Vortex bursting mechanism
4.2.1. The original theory
Chomiak [9] is the first theorist who has pointed out the
nature of vortex bursting of flame propagation. He states [9]
“From the photographs given by McCormack, it follows that
the combustion causes a nearly discontinuous breakdown of
the vortex so we can assume after Benjamin that the process
is similar to a hydraulic jump. Then we can write for the
discontinuity surface a simple integral relation
ð
ð
ðp 2 p0 ÞdA ¼
rb y 2 dA;
ð26aÞ
A
A
which simply states that the pressure forces induced by the
rotation of the fluid are equal to the momentum flux due to
the ‘pulling’ of the flame inside the vortex. y is here the
bursting and, so the flame propagation velocity along the
vortex.”
His model is schematically shown in Fig. 47 [50].
Assuming the tangential velocity distribution of Rankine
508
4.2.2. The angular momentum conservation model
Daneshyar and Hill [25] have given a detailed explanation on the vortex bursting concept. They used the angular
momentum conservation equation in their derivation.
Assuming no axial motion and also assuming the tangential
velocity distribution of Rankine form, they obtained the
following relations using mass and angular momentum
conservation equations:
2
rffiffiffiffiffi
hb
ru
Vb
hu
r
;
¼
¼
¼ b:
ð27aÞ
hu
rb
Vu
hb
ru
Here, hu and hb are the diameters of the unburned gas and
the burned gas, respectively, and Vu and Vb are the angular
speeds of the unburned and burned gases, respectively.
By simply integrating the radial momentum equation
from r ¼ 0 (axis of rotation) to the infinity, the pressure
difference between the infinity and the axis of rotation in the
burned gas, P1 2 Pb ð0Þ; and that in the unburned gas, P1 2
Pu ð0Þ; are obtained. Thus, on the axis of rotation, the
pressure difference across the flame DP is obtained as
"
2 #
r
2
DP ; Pb ð0Þ 2 Pu ð0Þ ¼ ru Vu max 1 2 b
< ru Vu2 max :
ru
ð27bÞ
This pressure difference would set-up a large axial velocity
ua of the burned gas into the unburned region. Then, a
relation
DP < ru Vu2 max <
1
2
rb u2a
gives the magnitude of this velocity ua as
sffiffiffiffiffiffi
2ru
ua < Vu max
:
rb
Fig. 46. Flame ‘penciling’ effect in a forced vortex in a rotating
vessel [27].
form, and by ignoring the axial and radial velocities, the
pressure distribution in the vortex can be obtained from the
momentum equation
2
1 ›p
v
¼
:
ru ›r
r
ð26bÞ
Then, integration gives
p 2 p0 ¼ ru
ð1 v2
dr ¼ ru Vu2 max :
r
r
Consequently, y is given as
rffiffiffiffiffi
r
y ¼ Vu max u :
rb
ð26cÞ
ð26dÞ
ð27cÞ
ð27dÞ
To better explain the combustion in a small-scale vortex
tube, Daneshyar and Hill have proposed the concept of
average pressure and average axial speed in their paper [25].
If we assume that the average pressure difference (which is
integrated over the range from the center to twice the core
diameter), works on the bursting, the mean pressure is given
as
P1 2 P
3
ln 2
1
þ
< :
¼
16
4
3
ru Vu2 max
ð27eÞ
Here, the value of 1/6 in Ref. [25, Eq. (7.3)] is corrected to
3/16.
Thus, the average axial propagation velocity is given as
sffiffiffiffiffiffi
2ru
ua < Vu max
:
ð27fÞ
3rb
4.2.3. A hypothesis based on the pressure difference
measurement
To elucidate the validity of the theories by Chomiak [9]
and Daneshyar and Hill [25], Ishizuka and Hirano [50] have
attempted to measure the pressure difference across the
flame. Although their method was primitive, the results have
509
Fig. 47. Vortex bursting mechanism proposed by Chomiak [9,50].
elucidated that the pressure is raised behind the flame and
the extent of pressure rise is almost the same as that
predicted by Eq. (26c) or (27b). The flame velocities
measured, however, were much lower than those predicted
by Eqs. (26d) and (27d), and even by Eq. (27f) (Fig. 48).
Based on these results, it was pointed out that it could be
wrong to assume that the pressure difference was converted
into the momentum flux of the burned gas, or into the kinetic
energy of the burned gas. Then, a hypothesis was proposed
which states that the pressure difference must be used to
drive the unburned gas of high density, which is present
ahead of the hot burned gas of low density. That is,
1
2
ru Vf2 ø DPe < k2 ru Vu2 max :
ð28aÞ
Here, DPe is the effective pressure difference, which is
actually used to drive the flame, and k2 is a constant of the
order of unity. This yields an expression for the flame speed,
pffiffiffiffiffi
ð28bÞ
Vf < Vu max 2k2 :
To best fit the experimental results in the vortex flow, the
value of k2 is 1 for the stoichiometric and 1/3 for the near
lean limit methane/air mixtures, respectively (Fig. 48 [50] in
which the maximum tangential velocity Vu max is denoted as
Wmax ).
As seen in Fig. 4, the diameter of the lean flame is
smaller than that of the stoichiometric flame [48]; this may
result in less pressure difference across the flame. In fact, the
pressure difference measured for the lean flame was about
half the pressure difference for the stoichiometric flame
(compare the values Pb 2 Pu for (a) 6.85%CH4 and for (b)
9.54%CH4 at a condition of Vm ¼ 3 m=s in Fig. 43). As a
result, the value of k2 for the lean mixture becomes smaller
than that for the stoichiometric mixture.
4.2.4. A steady state, immiscible stagnant model
In 1994, Atobiloye and Britter [62] proposed a model in
which axial velocities are taken into consideration using the
Bernoulli equation. Their model is interesting in that the
predicted flame speed becomes much smaller than speeds
predicted by Chomiak [9] or Daneshyar and Hill [25], and
the slopes in the Vf 2 Vu max plane become less, or nearly
equal to unity. Fig. 49 [62] shows their model. They assume
that a heavy fluid such as the unburned gas, and a light fluid
like the burned gas, are separated by a thin diaphragm in a
tube, whose radius is R2 (Fig. 49(a)) [62]; and after rotating
the tube at a constant angular speed v, the interface takes the
form shown in Fig. 49(b) [62]. The denser fluid flows at
velocity u2 ; near the wall, while the lighter fluid flows at
velocity u1 ; in the center. A steady state of flame
propagation has been assumed, and two cases—a forced
vortex flow and a free vortex flow—are separately
discussed. Fig. 49(c) [62] shows the case of a forced vortex
flow of rigid-body rotation. The coordinate is attached to the
interface. Therefore, the denser fluid flows at velocity u1
from the left, and creeps over the wall at velocity c ¼
u1 þ u2 ; while the lighter fluid of radius R1 is at rest. Note
that in this model, the interface is treated as an immiscible
surface where there is no mass transfer, hence no
combustion. As a result, the continuity equation across the
flame (interface) is not considered and the flame speed
510
Fig. 49. A steady-state, immiscible, stagnant model by Atobiloye
and Britter [62]. (a) Sketch illustrating problem formulation I, (b)
sketch illustrating problem formulation II, and (c) configuration for
forced vortex structure.
4. Bernoulli’s theorem on stream lines, OE (axis of
rotation) and AC (wall):
Fig. 48. Comparison of the experiments with the theories by
Chomiak [9] and by Daneshyar and Hill [25] and the relations based
on a hypothesis [50].
predicted does not include any term related to the burning
velocity. This is in sharp contrast to the back-pressure drive
flame propagation mechanism, described later. Also, note
that the hot gas is treated as a stagnant body. For brevity
sake, only the case of solid-body rotation is described here.
The governing equations, which they have used, are as
follows.
1. The conservation of mass for the unburned mixture at
AA0 and CC0 :
ð
ru dA ¼ C1 :
ð29aÞ
P
1
þ u2 ¼ H:
2
r
5. Treatment of the lighter liquid (burned gas) as a stagnant
gas (stationary obstacle):
PO 0 ¼ PE :
r y ¼ const:
ð29bÞ
3. The radial momentum equation, assuming no radial and
axial velocities
2
›P
y
¼r :
›r
r
ð29cÞ
ð29eÞ
6. The conservation of momentum at AA0 and CC0 :
ð
ð
ru2 dA þ P dA ¼ C2 :
ð29fÞ
From Eq. (29a), the following relation can be obtained:
u1 ¼ cð1 2 x2 Þ;
ð29gÞ
in which
x ; R1 =R2 :
2. The angular momentum conservation on a streamline:
ð29dÞ
ð29hÞ
From the angular momentum conservation Eq. (29b), the
tangential velocity distribution at CC0 is given below. Note
that the tangential velocity at the interface differs in the two
fluids.
In the unburned gas:
!
v
R21
:
y¼
r2
r
1 2 x2
ð29iÞ
In the burned gas:
1
y ¼ 2 vr:
x
ð29jÞ
From Eq. (29c), the pressures at each point are given as
follows (PI is the pressure at the interface L0 ):
Between A and O:
PA ¼
1
2
ru v2 R22 þ P0 :
Between C and L0 :
PI ¼ PC 2 ru
v2
ð1 2 x2 Þ2
ð29kÞ
!
1 2
R
1 R41
R2 2 2R21 ln 2 2
:
2
2 R22
R1
ð29lÞ
Between L0 and O0 :
PI ¼
1
rb v2 R21 þ PO0 :
2x4
ð29mÞ
From Eq. (29d), the following relations are obtained:
Streamline OE:
PO þ
1
2
ru u2u ¼ PE :
ð29nÞ
Streamline AC:
PA þ
1
2
ru u21 ¼ PC þ 12 ru c2 :
ð29oÞ
By matching the pressures at the interface L0 obtained
through a route OACI and obtained through a route OEO0 L0 ;
an expression for c2 is obtained:
v2
1
2 4
2
2
2 2 rb
R
x
2
x
2
2x
ln
x
2
ð1
2
x
Þ
c2 ¼ 2
:
2
ru
2x2
ð1 2 x2 Þ2
ð29pÞ
511
This equation can be solved numerically and the value of
xð; R1 =R2 Þ is obtained as a function of rb =ru : Then, by
substituting this x-value into Eq. (29p) or (29q), the value of
cð; u1 þ u2 Þ can be obtained. Finally, the steady-state
propagation velocity u1 is obtained by putting c into Eq.
(29g).
Fig. 50 [62] shows the variations of x with the density
ratio rb =ru : As the density ratio approaches unity, the value
of x becomes larger. Thus, the flame (or hot gas) diameter of
lean mixture should be greater than that of the stoichiometric mixture. Fig. 51 [62] shows the non-dimensional
velocities of U1 ð; u1 =Vu max Þ and U2 ð; u2 =Vu max Þ: The
velocity of the hot gas U1 is lower than that of the unburned
cold gas U2 ; in addition, even for rb =ru ¼ 1=7 (which may
correspond to the stoichiometric mixture), the value of U1 is
about than 0.25. This is much lower than the actual flame
speed observed in a rotating tube [66].
Their solutions can also be obtained for the free vortex.
Fig. 52 [62] shows the relationship between the value of x
and the ratio of the core radius to the tube radius, að; h=R2 Þ:
Note that the value of x can be determined independent of
the density ratio for the free vortex; whereas the axial
velocities, U1 and U2 ; are dependent on the density ratio.
Fig. 53 [62] shows the variation of the non-dimensional
axial velocities, U1 and U2 ; with a for the density ratio, 1/7,
and Fig. 54 [62] shows the variation of U1 and U2 ; with the
density ratio in the case of a ¼ 0:05: In the case of free
vortex, the velocity of the hot gas becomes much faster, and
it increases with a decrease in the density ratio. That is, the
flame speed increases as the mixture approaches the
stoichiometry. Furthermore, the value of U1 ; which is
equal to u1 =Vu max ; is close to unity. In the limit of R2 ! 1
and a ! 0; the analysis gives
rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi
r
r
U1 ø ð1 2 x2 Þ 1 2 b < 1 2 b :
ru
ru
ð29sÞ
On the other hand, using a relation Eq. (29f), an expression
for c2 is obtained:
v2
1
2
2
ð1 2 x Þ ð1 2 x4 Þ
3
3
R22 x4 2 x2 2 2x2 ln x 2 x4 ln x
2
2
1 rb
2
þ
ð1 2 x2 Þ2 1 2 2 :
4 ru
x
c2 ¼ 2
ð29qÞ
By equalizing these two equations, Eqs. (29p) and (29q), for
c2 ; an equation for x can be obtained:
1
1
2 x2 þ x4 2 x6 þ x2 ln x þ 2x4 ln x
2
2
2
1 rb
ð1 2 x2 Þ2 ð1 2 2x2 Þ ¼ 0:
4x2 ru
ð29rÞ
Fig. 50. Variation of x with density ratio for the forced vortex [62].
512
Fig. 51. Variation of axial velocities with density ratio for the forced
vortex [62].
Thus, the slope in the Vf 2 Vu max plane becomes much
smaller than the value of ru =rb of the original theory [9],
and becomes almost equal to unity, in accordance with most
of the experimental results such as those in Figs. 35 and 36.
4.2.5. The finite flame diameter approximation
Experiments by Asato et al. [57] have found measured
flame speeds to be much lower than the predictions by
Chomiak [9] or by Daneshyar and Hill [25]. They have
pointed out that the flame is not planar, but convex towards
the unburned mixture, and the flame diameter is small.
Asato et al. have modified the theories by taking the flame
diameter into consideration, and by introducing the concept
of average pressure difference, originally introduced by
Daneshyar and Hill [25]. Their model is shown in Fig. 55
[57].
Asato et al. have assumed that the core radius a remains
unchanged in the unburned and burned gases. They start
Fig. 52. Variation of x with a for the free vortex [62].
Fig. 53. Variation of axial velocities with a for the free vortex
(density ratio ¼ 1/7) [62].
with the pressure distributions of the Rankine’s combined
vortex for the unburned and burned gases
8
2 2
2 2
1
>
< P1 2 ru v a þ 2 ru v r ðr # aÞ;
;
ð30aÞ
Pu ¼
2 4
>
: P1 2 1 ru v a ðr $ aÞ
2
2 r
8
2 2
2 2
1
>
< P1 2 rb v a þ 2 rb v r ðr # aÞ;
Pb ¼
:
ð30bÞ
2 4
>
: P1 2 1 rb v a ðr $ aÞ
2
2 r
They have concluded that only the pressure difference in the
flame tip area influences the flame propagation. If we denote
the radius of the flame tip as a, the mean pressures in the
unburned gas and the burned gas are given as follows:
ð2a
1
Pu 2pr dr ;
ð30cÞ
P u ¼
2
pð2aÞ
0
Fig. 54. Variation of axial velocities with density ratio for the free
vortex ða ¼ 0:05Þ [62].
Fig. 55. The finite flame diameter model by Asato et al. [57].
P b ¼
ða
ð2a
1
P
2pr
dr
þ
P
2pr
dr
:
b
u
pð2aÞ2 0
a
ð30dÞ
Here, the upper limit of the radius for integration is taken as
twice the radius of the vortex core. Similar to Chomiak’s
hypothesis, the momentum conversion is assumed to be
ð
ð
2
ðP b 2 P u ÞdA ¼
rb Vfth
dA:
ð30eÞ
A
A
Finally, the flame speeds are obtained as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
!ffi
u
>
u ru
>
a
a2
>
t
>
V
2 1 1 2 2 ða # aÞ;
>
< 2a u max
rb
4a
Vfth ¼
: ð30fÞ
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
>
1
r
3
a
>
u
>
þ ln
ða . aÞ
21
: Vu max
2
4
rb
a
513
Using the flame tip measurements, the values of Vfth are
obtained and shown in Fig. 56 [57], together with the
measured flame speeds. The values of Vfth become smaller
than the values by Chomiak, but they are still much larger
than the measured flame speeds. However, it should be
noted that in this figure, the values of the maximum
tangential velocity are estimated on the assumption that the
core diameter is 10% of the ring diameter. This overestimates the maximum tangential velocities and the actual
values are presumed to be about half the values indicated in
Fig. 56. In addition, there are two weak points in this mode.
First, it is assumed that the core radius remains unchanged
both in the unburned and burned gases. This means that the
burned gas must expand only in the axial direction. Second,
it is assumed that the burned gas pressure far downstream of
the flame acts on the unburned gas. This means that the
burned gas should be stagnant, as in the model by Atobiloye
and Britter ðPE ¼ PO0 Þ: The radius of flame tip a can be
considered as the diameter of the hot stagnant gas column.
Thus, this theory requires experimental evidence to validate
these assumptions.
4.2.6. The back-pressure drive flame propagation
mechanism
The back-pressure drive flame propagation model was
first applied to the unsteady flame propagation in a rotating
tube [66], and next to the vortex ring combustion [6]. Later,
this model was extended to include the effects of finite flame
diameter on the flame speed [68]. The validity of the theory
has been examined by comparison with experimental results
in vortex ring combustion [69,70], and it is shown that the
theory can describe the experimental results quantitatively
as well as qualitatively. Very recently, a steady-state model
has been developed to account for the enhancement of flame
speed in rich hydrogen/air mixtures [72].
Fig. 56. Relationship between flame speed and maximum tangential velocity in the finite flame diameter model [57].
514
It is generally accepted that the Bernoulli equation can
be applied to a streamline, even if the flow is rotational.
However, in Bernoulli’s equation,
1 2 ð dP
V þ
¼ const:
2
r
ð31aÞ
the second, integral term cannot be obtained in an explicit
form across the flame, because the density changes not only
with pressure P, but also with temperature and with the total
molar number. Note that in the theory of Atobiloye and
Britter [62], the Bernoulli equation is applied only to the
unburned mixture of constant density. The burned gas is
treated as stagnant. But they are unable to obtain explicit
solutions. Their results, in Figs. 50 – 54, were obtained
numerically. Instead the Bernoulli equation, the backpressure drive flame propagation theory uses momentum
flux conservation across the flame. The momentum flux
conservation equation has been used in the Chomiak theory,
but it is assumed that the pressure difference is converted
into the momentum flux of the burned gas (Eq. (26a)). The
back-pressure drive flame propagation theory instead, uses
the momentum balance equation as part of the form,
Pu ð0Þ þ ru Vu2 ¼ Pb ð0Þ þ rb Vb2 :
ð31bÞ
In the following, the back-pressure drive flame propagation
theory is briefly described.
Fig. 57 [68,69] schematically shows the present model.
We take the axis of rotation as the z-axis. In the model by
Daneshyar and Hill [25], the axial velocity is assumed to
be zero. However, we admit the existence of axial flow.
We assume that from left to right, the unburned gas of
radius Ru flows at the velocity Vu ; and only a part of
radius ru is burned in flame area A, to be a burned gas
of radius rb ; which flows at the velocity Vb : To avoid
confusion, we assume that the flame also moves from left
to right at the velocity Vf : The non-burning gas between
ru and Ru occupies a region between rb and R0u and flows
at velocity V 0u ; behind the flame. The pressures in the
unburned and burned gases are given as functions of the
radial distance r, Pu ðrÞ and Pb ðrÞ; respectively. In Ref.
[25], the unburned mixture is assumed to expand only in
the radial direction. However, we also admit axial
expansion. Axial expansion is expressed in the relative
velocity change from Vu 2 Vf to Vb 2 Vf ; whereas the
radial expansion is expressed by the burned/unburned gas
radius ratio
1r ; rb =ru :
ð31cÞ
As for rotation, we assume the tangential velocity
distribution of Rankine’s form. For the unburned gas
ahead of the flame, we denote the rotational speed and
radius of the forced vortex core as Vu and hu =2;
respectively. Behind the flame however, these values are
given in a different manner, depending on the burning
area. That is, when the burning is limited within the
forced vortex region, as shown in Fig. 57, we denote
the rotational speed of the burned gas as Vb ; and the
rotational speed and radius of the forced vortex core of
the non-burning gas as V0u and h0u =2; respectively. When
the burning reaches the free vortex region, we denote the
rotational speed and radius of the forced vortex core of
the burned gas as V0b and h0b =2; respectively, and the
circulation of the non-burning gas as G0u : Because of
limited space, only the former case ðru # hu =2Þ; is
described here. The other case can be solved in a similar
manner [68].
There are three regions: (I) the burning region 0 #
r # ru and 0 # r # rb ; (II) the non-burning region in a
forced vortex ru # r # hu and rb # r # h0u =2; and (III)
the non-burning region in a free vortex hu =2 # r # Ru and
h0u =2 # r # R0u : For each of the three regions, we consider
mass continuity and angular momentum conservation, i.e.
1. Mass continuity:
ð
ru ðVu 2 Vf Þ2pr dr ¼
ð
rb ðVb 2 Vf Þ2pr dr:
ð31dÞ
2. Angular momentum conservation:
ð
ru ðVu 2 Vf ÞVuu 2pr2 dr
¼
ð
rb ðVb 2 Vf ÞVub 2pr 2 dr
ð31eÞ
(in which the tangential velocity distributions are given
as follows):
Fig. 57. The back-pressure drive flame propagation mechanism.
Illustration shows the case when the burning within the forced
vortex core ðru # hu =2Þ [68,69].
(
Vuu ¼
Vu r
ðr # hu =2Þ;
Vu h2u =4r
ðr . hu =2Þ;
ð31fÞ
Vub
8
Vb r
>
>
<
0
¼ Vu r
>
>
: 0 02
Vu hu =4r
Here k is a measure of the burning region normalized by the
core radius,
ðr # rb Þ;
ðrb # r # h0u =2Þ;
ð31gÞ
0
k;
ðhu2 = , rÞ:
Next, we consider the momentum flux in the z direction.
Since the flame runs faster on the axis of rotation, we
concentrate our attention on this point of the flame. If we
assume that the momentum flux before combustion is
equal to the flux after combustion, the following relation
can be obtained:
Pu ð0Þ þ
ru Vu2
rb Vb2 :
¼ Pb ð0Þ þ
ð31bÞ
From the relations, Eqs. (31d) and (31e), the unknown
variables behind the flame are obtained implicitly with the
use of Vf ; which should be determined from
Vb 2 Vf ¼ ðru =rb ÞðVu 2
Vb ¼
h0u =2
V0u
¼
Vf Þ=12r ;
Vu =12r ;
¼ ðhu =2Þ1r ;
f ðkÞ ¼
1 2
2k
ðk # 1Þ;
12
1
ðk $ 1Þ:
2k2
From the mass continuity for Region I,
ð31mÞ
in which Y is a ratio of the flame area A to the crosssectional area of the unburned mixture,
ð31nÞ
By putting the above equation into Eq. (31h), we obtain
the relation
ð31oÞ
By substituting Eqs. (31m) and (31o) into Eq. (31b), a
quadratic equation is obtained for Vf : Its solution is
obtained as follows:
rSY
1
1
Vf ¼ 2 u u
12 2 2
ru 2 rb
r
2
rb
1r
u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
r
1
u
ru rb ðYSu Þ2 1 2
þðru 2 rb ÞDP:
ð31pÞ
rb 12r
He r e, DP i s th e p re s su r e ris e b e hi n d th e fla me
DP ; Pu ð0Þ 2 Pb ð0Þ; which is obtained by integrating the
radial momentum equation ›P=›r ¼ rVu2 =r; with the tangential velocity distributions of Eqs. (31f) and (31g). The
densities for r $ Ru and r $ R0u are assumed to be ru : The
final expression for DP, including the case when the burning
reaches the free vortex region ðru $ hu =2Þ; is given as
follows:
1
r
ð31qÞ
DP ¼ ru Vu2 max 1 2 2 1 þ b 2 1 f ðkÞ :
ru
1r
ð31sÞ
The solutions are simplified for two extreme cases, an axial
expansion
case ð1r ¼ 1Þ and a radial expansion case ð1r ¼
ru =rb Þ; given as follows:
lVf l ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ru
ðYSu Þ2 þ Vu2 max f ðkÞ
rb
r
lVf l ¼ YSu þ Vu max 1 þ b f ðkÞ
ru
ðradial expansion; 1r ¼ ru =rb Þ:
ð31jÞ
ð31lÞ
YSu ru
þ Vf :
rb 12r
and the function f ðkÞ is given as
ð31iÞ
R0u ¼ Ru 1r :
Vb ¼
ð31rÞ
ðaxial expansion; 1r ¼ 1Þ;
ð31kÞ
Y ; A=ðpru2 Þ:
ru
;
hu =2
ð31hÞ
V 0u 2 Vf ¼ ðVu 2 Vf Þ=12r ;
Vu ¼ YSu þ Vf ;
515
ð31tÞ
ð31uÞ
The first term is a component of flame velocity, which is
induced by chemical reaction, and the second term is a
component, which is induced aerodynamically by rotation.
In contrast with the model by Atobiloye and Britter, this
model also considers the burning rate Su : In a simple case,
when k ! 1 and Y ¼ 1 (plane), the flame speeds are given
as follows:
lVf l ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ru 2
S þ Vu2 max
rb u
ð31vÞ
ðaxial expansion; 1r ¼ 1Þ;
r
lVf l ¼ Su þ Vu max 1 þ b
ru
ðradial expansion; 1r ¼ ru =rb Þ:
ð31wÞ
Note that the slope in the Vf 2 Vu max plane is at about unity,
in accordance with the experimental results. Therefore, the
back-pressure drive flame propagation mechanism aptly
describes the experimental results quantitatively as well as
qualitatively.
The back-pressure drive flame propagation theory
however, has two weak points. The first point is the
tangential velocity distribution in the burned gas. The
tangential velocity distribution behind the flame may not
have the form of Rankine’s combined vortex. As in the
model by Atobiloye and Britter (Eqs. (29i) and (29j)), or as
in the model by Umemura shown later, we must consider
the angular momentum conservation r y ¼ const: on each
streamline. If this correction is made, the flame speed for the
516
radial expansion in the range 0 # k # 1 is given as:
lVf l ¼ YSu
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
r
r
1
þ kVu max 1 þ b þ u ln 1 2 b 1 2 2 :
2ru
rb
ru
k
ð31xÞ
Note that Eq. (31u) for radial expansion is still valid for
k $ 1; and Eq. (31t) for axial expansion is valid for any k.
Also note that the second term on the right side of Eq.
(31u), is almost equal to ð0:8 , 1:0Þ Vu max for 0:3 # k # 1
and ru =rb ¼ 7: For very small flame diameter (such a flame
may not exist as shown in Figs. 35 and 36), the backpressure drive flame propagation theory gives the flame
speed as
pffiffi
lVf l ø kVu max = 2
ðaxial expansionÞ;
ð31yÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffi
r
lVf l ø kVu max 22 u ln k
rb
ð31zÞ
ðradial expansion : k # 0:01Þ:
Note that in Eq. (31z), the flame speed is increased with an
increase in the unburned to burned gas density ratio ru =rb ;
while the flame speed for moderate flame diameters is
almost insensitive to this ratio, since the ratio works in a
reversed form rb =ru in Eq. (31w).
The second weak point of the back-pressure drive flame
propagation theory is that the momentum balance equation
is not consistent for the Galilean transformation. This has
been pointed out by Lipatnikov in a personal communication [98], which is briefly described in the following:
Lipatnikov note. Let us consider the same problem in
another coordinate system moving with a constant speed U,
with respect to the basic coordinate system. Then,
V~ u ¼ Vu 2 U and V~ b ¼ Vb 2 U:
ð32aÞ
We must obtain
V~ f ¼ Vf 2 U:
ð32bÞ
The insertion of Eqs. (32a) and (32b) into Eq. (31b) modifies
the latter expression since the following additional terms
ðrb 2 ru ÞU 2 þ 2Uðrb V~ b 2 ru V~0 u Þ
ð32cÞ
arise on the right side of Eq. (31b). Thus, V~ f ¼ Vf 2 U; is
not the solution to the problem in the moving coordinate
system, and therefore, the model under consideration is not
invariable with respect to the Galilean transformation.
The cause of this basic inconsistency is the use of steady
equations for modeling the unsteady case. The steady case
corresponds to Vf ¼ 0; and Vu;21 is associated with the
flame propagation speed. However, Eq. (31m) implies that
just ahead of the flame Vu;20 ¼ YSL ; in this steady case (Y is
assumed to be unity in the original Lipatnikov note). Thus,
the use of a constant Vu is incorrect.
4.2.7. A steady-state back-pressure drive flame propagation
model
As mentioned previously, a recent observation with an
image intensifier for a stoichiometric propane/air mixture
[71] indicates that an almost-steady flame propagation can
be achieved in the vortex ring combustion, if the Reynolds
number is less than the order of 104. A further observation
for a stoichiometric hydrogen/air mixture indicates that,
independently of the Reynolds number, the flame speed is
always varied and the ratio of the square root of the
fluctuations in the flame velocity to its mean flame speed is
about 0.3 [93]. Thus, whether the fluctuations are large or
small, a quasi-steady state can be achieved in the vortex ring
combustion. Fig. 58 [72] shows a steady-state model of the
back-pressure drive flame propagation mechanism.
In vortex ring combustion, the ignition and meeting
positions are at rest while the flame propagates at a constant
speed Vf. If a coordinate is attached to the flame, the
unburned gas approaches the flame at the velocity of Vf, and
the burned gas flows away at the velocity of Vf, while the
flame is at rest. The velocity of the unburned gas just ahead
of the flame front should be equal to the burning velocity Su,
and the velocity of the burned gas just behind the flame
should be equal to ru Su =rb : Thus, the area of the stream tube
varies from upstream through the flame position to
downstream.
Here, we look at the pressures on the axis of rotation
ðr ¼ 0Þ: The pressure upstream (z ¼ 2pD=4; D: the ring
diameter) Pu;2pD=4 ð0Þ; is very low due to a centrifugal force
of rotation, while the pressure just ahead of the flame Pu;02 ð0Þ
becomes higher than Pu;2pD=4 ð0Þ; because the axial velocity
is decreased from Vf to Su. In the burned gas side, the pressure
just behind the flame Pb;0þ ð0Þ; is higher or lower than the
pressure downstream Pb;þpD=4 ð0Þ; depending on whether
Vf $ ru Su =rb or Vf # ru Su =rb (although these pressures are
nearly equal to the pressure at infinity P(1) since the
centrifugal force of rotation is weak because of low
density).
The Bernoulli equation can be held even in a rotational
flow as long as the flow is steady and limited to a streamline.
However, we may adopt a less rigid relationship between the
momentum flux balance at the lowest and highest pressure
points, because the flow is disturbed at the larger Reynolds
number. This is not unusual; in the flow through a valve, a
pressure loss is always present. Then, the following relations
can be obtained:
ðiÞ if Vf $
ru
S ;
rb u
Pu;2pD=4 ð0Þ þ
ðiiÞ if Vf #
ru Vf2
¼ Pb;0þ ð0Þ þ rb
2
ru
Su ;
rb
ru
S ;
rb u
Pu;2pD=4 ð0Þ þ
ru Vf2
ð33aÞ
ð33bÞ
¼ Pb;þpD=4 ð0Þ þ
rb Vf2 :
517
Fig. 58. Steady-state model for the flame propagation in a vortex ring [72].
Assuming angular momentum conservation, the pressure
differences are given as follows:
Pb;0þ ð0Þ 2 Pu;2pD=4 ð0Þ
S
r
¼ ru Vu2 max 1 2 u 1 2 1 2 b f ðkÞ ;
Vf
ru
Pb;þpD=4 ð0Þ 2 Pu;2pD=4 ð0Þ
r
r
¼ ru Vu2 max 1 2 b 1 þ b f ðkÞ :
ru
ru
ð33cÞ
ð33dÞ
These
pffiffiffiffiffiffiffiequations are obtained by putting 1r ¼ Vf =Su and
ru =rb into Ref. [69, Eqs. (14) and (17)], respectively.
Substituting Eqs. (33c) and (33d) for Eqs. (33a) and (33b),
respectively, the flame velocities are obtained as follows:
ru 2
ðiÞ if Vf $ ru Su =rb ;
ð33eÞ
Vf <
S þ Vu2 max ;
rb u
ðiiÞ if Vf # ru Su =rb ;
r
Vf ¼ Vu max 1 þ b f ðkÞ:
ru
ð33fÞ
Vf is given as a solution of the third order algebraic equation
for Vf $ ru Su =rb : But it can be approximately given as a
solution of a quadric equation, since Su =Vf # rb =ru p 1: If
we solve the third order algebraic equation absolutely, the
solution is continuous at Vf ¼ ru Su =rb :
The solution, which consists of the solid line for Vf #
ru Su =rb ; and the solid curve for Vf $ ru Su =rb in Fig. 38, is
518
almost proportional to the maximum tangential velocity,
and hence similar to the solution obtained by the backpressure drive flame propagation theory (broken line),
except that it passes through the origin ðVf ¼ 0; Vu max ¼
0Þ: The steady-state model has been extended in order to
examine the enhancement of flame speed in vortex rings of
rich hydrogen/air mixtures in air [72].
It is reasonable to consider that the flame can propagate
radially; hence the pressure at the burned gas side is raised,
as in a spherically propagating flame. Fig. 59 [99] shows the
pressure distribution when a flame spherically propagates in
a combustible mixture in a soap bubble. The pressure
discontinuity pue 2 pae is due to surface tension of the soap
film. The pressure difference pup 2 pbp is due to the onedimensional nature of the flame. Note that the pressure
behind the flame pbp is less than the pressure ahead pbp
(Eq. (22c)).
According to the analysis by Takeno [99], the extent
of the pressure rise is given approximately as
ð1=2Þru S2u ðru =rb 2 1Þð3ru =rb 2 1Þ: In this case the burned
gas is completely at rest and the flame speed is increased to
ru Su =rb : In vortex ring combustion, the burned gas is not at
rest, neither is the flame propagation spherical. Thus, the
pressure rise may be smaller than that in the spherically
expanding flame. However, it is reasonable to expect that
the pressure is raised to some extent by this radial
~ DP~
combustion. If we denote this pressure rise by DP;
may be proportional to ru S2u : In rich hydrogen combustion,
the flame front is highly disturbed [72]. Thus, the laminar
burning velocity Su, should be replaced by the turbulent
burning velocity ST. Since the turbulent intensity is
considered to be the maximum tangential velocity in a
vortex [9,25,100], it is reasonable to expect that ST /
Vu max : Thus, the pressure rise may be given in a simple form
as DP~ < lru Vu2 max ; where l is an arbitrary constant. By
adding the pressure rise DP~ to the original pressure
difference across the flame, the final flame speed is obtained
from Eqs. (33a) and (33b) as
ðiÞ if Vf $ ru Su =rb ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ru 2
S þ ð1 þ lÞVu2 max ;
Vf ¼
rb u
ð34aÞ
ðiiÞ if Vf # ru Su =rb ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
ru
l:
Vf ¼ Vu max 1 þ b f ðkÞ þ
ru
ru 2 rb
ð34bÞ
This can explain the increase in slope of rich hydrogen/air
flame speed with increasing F (Figs. 37 and 38).
4.3. Baroclinic push mechanism
It is well known that the vector vorticity transport
equation can be written as [83,101]:
›v
1
¼ 2ðu·7Þv þ ðv·7Þu 2 vð7·uÞ 2 7
£ 7p
›t
r
1
1
ð35aÞ
þ7
£ 7·t þ
7 £ 7·t :
r
r
The first term on the right side of Eq. (35a) is the convection
of vorticity, and the second term is the vorticity production
due to stretch. The third term is vorticity decay due to
dilatation, and the fourth term is the production of vorticity
due to the baroclinic torque. The fifth and sixth terms are
the viscous diffusion and viscous dissipation, respectively
[83,101]. The viscous, fifth and sixth, terms are simply
written as vDv in Refs. [102,103].
In the past, the baroclinic torque has sometimes been
used to explain the flame front instability [104,105], which
has been called the Taylor instability [106,107], the
Rayleigh –Taylor instability [104], or Taylor – Markstein
instability [108,109]. Very recently, the baroclinic torque
has received considerable attention in scramjet engine
research [110,111], because it has the potential to achieve
rapid and efficient mixing of fuel and oxidizer in a
hypersonic flow. In a hypersonic flow, the instability of
the interface between two fluids of different densities is
called the ‘Richtmyer – Meshkov instability’ [112].
Ashurst [64] focuses attention on the baroclinic torque to
account for the rapid flame propagation along the vortex
axis; his model is shown in Fig. 60 [64]. By ignoring other
terms, he starts only with the fourth term,
dv
7r £ 7P
¼
:
dt
r2
ð35bÞ
Ashurst assumes that the tangential velocity distribution has
a form of
Fig. 59. Pressure distribution in the flame propagation in a soap
bubble [99].
Vu
G
2
½1 2 expð2r 2 =rM
Þ:
¼
r
2pr 2
ð35cÞ
519
increased with the flame propagation distance XF, whereas
the flame speed in the curved vortex is independent of the
distance of flame propagation.
An interesting effect of this theory is that the flame speed
is inversely proportional to the value of dSL. Thermal theory
predicts that the value of dSL is approximately equal to the
kinetic viscosity v. Since the kinetic viscosity is decreased
with an increase in pressure, the baroclinic push mechanism
predicts that flame speed increases with an increase in
pressure. It has been observed frequently by Kobayashi et al.
[113] that at elevated pressures, small-scale parts of the
flame front move quickly to the unburned side. This
observation may clarify the validity of the baroclinic torque
mechanism.
However, recent numerical simulations have shown that
the baroclinic torque works only at the early stage of
propagation [78,79]. This seems reasonable because the
baroclinic push theory ignores flame curvature and others
terms, except the baroclinic term. If the baroclinic push is
only effective in the early stage of flame propagation, it is
not valid for steady propagation and it is hard to demonstrate
experimentally the validity of this mechanism, because
ignition—whether by electric spark or laser beam—disturbs
the phenomenon.
Fig. 60. Baroclinic push mechanism by Ashurst [64].
4.4. Azimuthal vorticity evolution mechanism
He then roughly evaluates the pressure gradient and the
density gradient as follows:
Vu2
1
;
7P ¼
r
r
1
r 2r
t
7r ¼ upffiffiffiffiffiffib ¼ pffiffiffiffiffiffiffi ;
r
d ru rb
d tþ1
ð35dÞ
in which d is the flame thickness and t ; ru =rb 2 1: By
putting these relations into Eq. (35b), and by integrating the
equation over the cross-sectional area of the vortical core
from the vortex axis out to 3rM, Ashurst obtains,
dv
4:5t
2
, pffiffiffiffiffiffiffi rM VM
;
dt
d tþ1
ð35eÞ
in which VM is the tangential velocity at r ¼ rM : By
converting from a per-unit time basis to vorticity per-unit
length of the burned gas, whose length is denoted as XF, the
flame speed in the straight vortex is finally expressed as
pffiffiffiffiffiffiffiffi
t
1
2
XF =rM :
ð35fÞ
UB , pffiffiffiffiffiffiffi rM VM
SL ðt þ 1Þ
d tþ1
Umemura and co-workers have proposed a new
mechanism [74– 77,82]; they focus attention on a vortex
filament. Their model is illustrated in Fig. 61 [76]. This
filament is twisted by expansion of the burned gas in the
radial direction, while the angular velocity must be slowed
in order to conserve angular momentum. As a result, an
azimuthal vorticity is produced. This drives the flame,
resulting in rapid flame propagation. In their paper,
however, a quantitative description for the flame speed
was obtained in a similar manner as in the steady state,
immiscible stagnant model by Atobiloye and Britter [62],
except that the burned gas is not stagnant and there is a
convection through the flame. Their model is shown in
In the case of the curved vortex, such as the vortex ring, an
arc segment of about five core radii is considered to work on
the flame propagation to get
UB ,
t
9rM G
;
ðt þ 1Þ3=2 dSL Tp k2
ð35gÞ
in which G is the ring circulation, Tp is the time period to
create the flow at the orifice of the vortex ring generator, k is
the ratio of the orifice exit velocity to the maximum swirling
velocity. Note that the flame speed in the straight vortex is
Fig. 61. Azimuthal vorticity generation mechanism [76].
520
6. Tangential velocity distribution of Rankine’s form for
the unburned mixture:
8
for r # a;
>
< Vr
:
ð36iÞ
y ¼ Va2
>
:
for r $ a
r
7. Angular momentum conservation on each streamline:
r y ¼ const:
ð36jÞ
8. Momentum equation for radial direction:
›P
y2
¼r ;
›r
r
ð36kÞ
which yields relations for the pressure difference in the
radial direction:
Fig. 62. Model flow configuration in the azimuthal vorticity
generation mechanism [74,77].
Fig. 62 [74,77]. In the first part of the papers by Umemura
et al. [74,75], only the mixture in a forced vortex is assumed
to burn. The governing equations they used are as follows:
1. Mass conservation:
across the flame : ru SL ¼ rb Sb ¼ srb SL ;
ð36bÞ
2
ð36cÞ
the unburned gas : SL A ¼ pa W;
0
ð36aÞ
2
the burned gas : sSL A ¼ pa W:
These equations lead to a relation:
0
a ¼ sa :
2
2
ð36dÞ
2. Bernoulli equation for the unburned gas between points
A and O2:
PA þ
1
2
ru W 2 ¼ PO2 þ 12 ru S2L :
ð36eÞ
3. Momentum flux conservation across the flame, i.e.
between Oþ (just ahead the flame) and O2 (just behind
the flame):
PO2 þ ru S2L ¼ POþ þ rb ðsSL Þ2 :
ð36fÞ
4. Bernoulli’s equation for the burned gas between points
Oþ and D:
POþ þ
1
2
rb ðsSL Þ2 ¼ PD þ 12 rb W 2 :
ð36gÞ
PB 2 PA ¼ ru V2 a2 ;
PC 2 PD ¼
1
1
r V2 a2 1 þ
:
2 b
s
ð36lÞ
ð36mÞ
Along the path ABCD, the following relation can be
obtained with the use of Eqs. (36l) and (36m):
1
1
1
PD 2 PA ¼ ru V2 a2 1 2
2þ
:
ð36nÞ
2
s
s
On the other hand, along the axis of rotation, AD, the
following relation can be obtained with the use of Eqs.
(36e)– (36g):
1
1
1
PD 2 PA ¼ ru W 2 1 2
ð36oÞ
2 ðs 2 1Þru S2L :
2
2
s
By equating Eqs. (36n) and (36o), an expression for the
flame speed is obtained as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2s þ 1
Vf ¼
ð36pÞ
ðVaÞ2 þ sS2L :
s
This analysis has been extended to a more general case,
in which only a part of the forced vortex region, r #
b ¼ fa; is burned. Fig. 63 [77] shows the model. By
considering the mass continuity for the burning and the
non-burning region in the forced vortex core, and also
by considering a tangential velocity distribution for the
non-burning region,
"
#
0 Vb 2 1
ðs 2 1Þf2 a2
ð36qÞ
y¼
2 1 þ Vr ¼ V r 2
r
s
r
the final expression is given as follows:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
#
u"
2
u 2s þ 1
1
þ
ð
s
2
1Þ
f
þ 2s ln
f2 ðVaÞ2 þ sS2L :
Vf ¼ t
s
sf2
ð36rÞ
5. Equal pressure at points B and C:
PB ¼ PC :
ð36hÞ
These results resemble the results obtained by the backpressure drive flame propagation mechanism. That is,
521
simulated using periodic boundaries. The initial vortex tube
was assumed to have a Gaussian vorticity distribution, thus
the circumferential velocity distribution was given by the
following equation
!!
G0
r2
VðrÞ ¼
;
ð37aÞ
1 2 exp 2
2pr
ðs0 =2Þ2
G0 ¼ ps0 Vm0 ;
Fig. 63. Definition of effective vortex tube radius [77].
the equation has two terms, the aerothermochemical term,
linked with the burning velocity SL, and the aerodynamic
term associated with the maximum tangential velocity Va ¼
Vu max : In the limit Vu max ! 0; the flame speed tends to
Vf ! Su ru =rb : For large values of Vu max ; the flame speed
is almost proportional to the maximum tangential velocity,
pffiffi
although the slope in the Vf 2 Vu max plane is about 2: This
is because the Bernoulli equation is used in the unburned gas
as well as the burned gas, although a momentum conservation equation has been used across the flame. On the other
hand, in the back-pressure drive flame propagation model, in
which the slope is at about unity, a more relaxed relation—
only the momentum flux conservation across the flame—has
been used, because the flow is turbulent and in addition, the
flame propagation is ‘quasi-steady’, as noted in Section
3.3.1.
Although the vorticity evolution mechanism sounds new,
its final expression for the flame speed and the procedure to
derive the equation are very similar to the model by
Atobiloye and Britter [62] or the back-pressure drive flame
propagation mechanism [6,66– 72]. Note that the vortex
breakdown can be explained on the basis of azimuthal
vorticity evolution as well as on the basis of pressure
difference. Hence, this azimuthal vorticity evolution mechanism can be categorized as a subspecies of the vortex
bursting mechanism, based on the pressure difference
presented in Section 4.2.
5. Numerical simulation
Only a few numerical simulations have been made on the
present flame propagation problem. Hasegawa et al. made
the first numerical study in 1995 [63]. Since they were
interested in modeling turbulent combustion, a small-scale
vortex tube was used in their study. A straight vortex tube
placed at the center of a cubic volume of 1 mm3 was
ð37bÞ
where the maximum velocity Vm0 was achieved at the point,
where r ¼ s0 =2 and s0 was regarded as the initial vortex
core diameter [63]. (Strictly speaking, the maximum
velocity is achieved at r ø 1:12ðs0 =2Þ; but the maximum
is almost the same as the value at r ¼ s0 =2 within 1%; see
Section 6.2.4). The subscript 0 denotes the initial condition.
A trapezoidal profile of temperature, with amplitude of
1960 K, was set initially at the center of the simulated
volume, causing two premixed flames to propagate
perpendicularly to the vortex tube in opposite directions.
The initial pressure was assumed to be constant. Note that
the mixture was 30%(2H2 þ O2) þ 70%N2. Thus, the
laminar burning velocity uL was 0.538 m/s, and the laminar
flame thickness d was 0.17 mm. The range in calculation
was Vm0 =uL ¼ 1:8 – 36:0 and s0 =d ¼ 0:18 – 1:71: Thus, the
Reynolds numbers, RG0 ; s0 Vm0 =v; for simulation were
small values, less than 89.6.
Fig. 64 [63] shows the temporal behavior of a premixed
flame propagating along a vortex tube, in which the ratio of
the initial maximum circumferential velocity to the laminar
burning velocity Vm0 =uL ¼ 36:0 and the ratio of the initial
core diameter to the laminar flame thickness s0 =d ¼ 0:94:
Note that in this simulation, the vortex itself is decaying,
hence the maximum circumferential velocity is decreased,
whereas the core diameter is increased with time. It is seen
in Fig. 64 that the flame propagates quickly along the vortex
axis and much slower outside. The vorticity in the burned
gas region decreases due to the expansion, and the vorticity
in the flame front is dissipated by the increased viscosity.
Fig. 65 [63] shows the temperature distributions at t ¼
7:0 for different initial circumferential velocities. It is seen
that the flame propagates faster as the circumferential
velocity is increased. Fig. 66 [63] shows the temperature
distributions for different initial core diameters. It is seen
that the flame propagates only slightly when the core
diameter is much smaller than the laminar flame thickness,
whereas the flame propagates faster as the core diameter is
increased.
Fig. 67 [63] shows the relationship between the flame
velocity and the maximum circumferential velocity of the
flame at the constant diameter of the vortex tube. The flame
speed increases almost linearly with maximum circumferential velocity, except at lower circumferential velocities
where no flame acceleration is observed.
Fig. 68 [63] shows the variations of the proportionality
factor, uV =Vm ; of the flame propagation velocity in a
vortex tube as functions of the Reynolds number. When
522
Fig. 64. Temporal behavior of iso-vorticity surface at lvl ¼ 0:3 and iso-temperature surface at T ¼ 0:9 in Case 8 ðVm =uL ¼ 36:0; s0 =d ¼ 0:94Þ:
Dimensionless unit time corresponds to 2.6 ms [63].
the Reynolds number is less than about 10, no flame
acceleration can be found. When the Reynolds number rises
above 10, the proportionality factor increases and reaches
unity at the Reynolds number of about 60.
To summarize, the following conclusions have been
obtained: (1) a premixed flame can be accelerated along a
vortex tube having a diameter comparable to the flame
thickness. (2) The flame propagation velocity is proportional
to the maximum circumferential velocity of the vortex tube.
(3) The influence of the vortex on the flame propagation can
be ignored when the Reynolds number is less than about 10.
When the Reynolds number of the vortex tube increases
above 10, the proportionality factor increases and reaches
unity at the Reynolds number of 60 [63].
Fig. 69 [65] shows variations of the flame speed with the
maximum circumferential velocity for different densities
and different core diameters. It is seen that flame speeds are
higher for ru =rb ¼ 2:63 than those for ru =rb ¼ 7:53: In
addition, it is seen that the increase in the flame speed is
proportional to the power of the maximum circumferential
velocity. This is contrary to the original vortex bursting
theory, which predicts a linear dependency of the flame
speed such as Vf ¼ Vu max ru =rb with respect to Vu max :
Thus, they have concluded that the baroclinic push
mechanism can better explain their numerical results.
Very recently, the transport equation of vorticity, Eq.
(35a), was analyzed to clarify the propagation mechanism
[83]. Fig. 70 [83] shows the vorticity obtained by integrating
Eq. (35a). The figures at the left are those at t ¼ 31 ms and
the figures on the right show those at t ¼ 155 ms: The white
color indicates that the vorticity provokes the flame ahead,
whereas the black indicates the vorticity pulling the flame
backwards. The top figures show the total vorticity, while
the second, third and fourth figures show the convective,
stretch, and baroclinic terms, respectively. It is seen that at
the early stage of flame propagation, t ¼ 31 ms; almost all
the vorticity which provokes the flame ahead is produced by
the baroclinic torque. Thus, the flame is forced to propagate
Fig. 65. Temperature distributions at t ¼ 7:0 for different initial
circumferential velocities. The initial core diameter is s0 =d ¼ 0:94
[63].
Fig. 66. Temperature distributions at t ¼ 7:0 for different initial core
diameters. The maximum circumferential velocity is Vm =uL ¼ 36:0
[63].
Fig. 67. Relation between maximum circumferential and propagation velocity of the flame at the constant core diameter of the
vortex tube. Propagation velocity uV shows only the effect of the
vortex tubes and excludes laminar burning velocity by subtraction
of uL [63].
by the baroclinic push. However, near the steady state of
flame propagation, t ¼ 155 ms; most of the total vorticity
around the flame tip is produced by convection and by
stretch. The baroclinic term does not contribute to the
vorticity around the tip region. The baroclinic torque is great
around the ignition point, where the vorticity by convection
and stretch take large negative values, which compensates
for the high positive value of vorticity by the baroclinic
torque. Thus, it is concluded that the azimuthal vorticity in
front of the flame, which is produced by convection and
stretch, provokes the flame propagation [79,83].
Fig. 68. Relation between the Reynolds number of a vortex tube
RG ¼ Vm s=v and proportionality factor of flame propagation
velocity in a vortex tube uV =Vm [63].
523
Fig. 69. Relation between flame propagation velocity and the
maximum circumferential velocity [65].
6. Discussion
6.1. Vortex breakdown
As first pointed out by Chomiak [9], and very recently
emphasized by Umemura and Tomita [75], the phenomenon of rapid flame propagation along a vortex axis can
be considered as a kind of vortex breakdown (bursting)
phenomenon. After the first observation of the vortex
Fig. 70. Contribution of each term for generation of the azimuthal
vorticity at the initial stage at 31 and 155 ms after ignition [83].
524
breakdown, in the leading edge vortices trailing from delta
wings [10], much experimental and theoretical research
has been conducted on this phenomenon. Reviews
[114 – 117] have also been published in the last three
decades and, very recently, a review article by LuccaNegro and Doherty [118] appeared in this journal.
Although many types of breakdown have been pointed
out [119], the two major varieties are a spiral- and a
bubble-type breakdown, which can be seen in a photograph taken by Lambourne and Bryer [120] (see Fig. 2 of
Ref. [121] or Fig. 1 of Ref. [115]). The explanation of
vortex breakdown has been disputed. The many proposals
are divided into three categories in which the basic ideas
are, respectively, (1) separation of a boundary layer, (2)
hydrodynamic instability, and (3) existence of critical state
[114]. Recently, an important role of evolution of negative
azimuthal vorticity has been addressed [122,123]. However, the simplest mechanism, which describes the onset
of vortex breakdown, is the appearance of a high-pressure
region. The pressure at a point on the axis of rotation is
abruptly raised to nearly ambient pressure, while the
pressure in the vortex core is kept low to balance a
centrifugal force of rotation. This pressure inequality
triggers vortex bursting. Adverse pressure gradient
induced in a divergent tube promotes this condition
[124,125]. The review by Delery [117] establishes that the
existence of a stagnation point forming on the centerline
of the vortical structure is a distinctive feature of
breakdown.
The difference between the vortex breakdown in
conventional flows and the rapid flame propagation along
a vortex axis is that, in the latter case, a density change is
always accompanied with the phenomenon, due to combustion. The burned gas expands through the density change,
causing an expansion of the vortex core. If the flow is a nonrotating, one-dimensional flow, this expansion is restricted
in one direction; hence, it causes a pressure drop behind the
flame by an amount equal to 2ru S2u ðru =rb 2 1Þ (Eq. (22c)).
On the other hand, if the flow is two- or three-dimensional
and rotating, the pressure behind the flame can be increased
as high as to the ambient pressure (or possibly more, due to
the secondary combustion [72]), because the pressure field
is governed mostly by the radial momentum equation, ›p=
›r ø 2ry 2 =r: As a result, a flame can propagate rapidly
along the vortex axis. Before discussing the way inequality
in pressure drives the flame, we will examine an interesting
result recently obtained for vortex breakdown in a constantdensity flow (water).
Fig. 71 [126] shows the experimental apparatus. A
honeycomb is rotated by a motor, and water flows in a tube,
with a rotational motion. The exit diameter of the
contraction zone D1 is 40 mm. A nozzle of exit diameter
D2 ¼ 25 mm is also used by being inserted into the former
nozzle. Axial velocity Vx ðr; xÞ; and the azimuthal velocity
Vu ðr; xÞ; are measured with two-component optical fiber
laser Doppler anemometry in the backscatter mode. In this
Fig. 71. Sketch of experimental apparatus of a rotating system and a
nozzle for vortex breakdown [126].
experiment, the swirl parameters, defined as
S;
2Vu ðR=2; x0 Þ
Vx ð0; x0 Þ
ð38aÞ
are used to analyze the vortex breakdown, where x0 is the
shortest axial distance measured from the nozzle exit plane,
at which the measurement of both components is possible
due to optical constraints: x0 ¼ 5 mm for the D2 nozzle and
x0 ¼ 24 mm for the D1 nozzle. The azimuthal velocity
Vu ðR=2; x0 Þ is measured at half the radius of the nozzle exit
r ¼ R=2; and this azimuthal velocity Vu ðR=2; x0 Þ; is nearly
equal to the maximum azimuthal velocity at a plane x ¼ x0 :
Fig. 72 [126] shows the critical values measured for
appearance Sca and disappearance Scd of breakdown in ðRe; SÞ
parameter space for each nozzle. Here, the Reynolds number
is defined as Re ; 2RV x ðx0 Þ=v; where V x ðx0 Þ is the mean axial
velocity in the jet and v is the kinematic viscosity of water. It
is seen that the critical values are about 1.4. This yields
pffiffi
Vx ð0; xÞ ø 2Vu ðR=2; x0 Þ:
ð38bÞ
Similar results can be found in the literature. The LDV
measurements in a swirling water flow in a slightly divergent
pipe indicate that, far upstream of the bubble nose, the axial
velocity is about 13 cm/s, while the maximum tangential
velocity is about 9 cm/s [127, Fig. 3]. Thus, the ratio of the
maximum tangential velocity to the axial velocity is about
1.4. In the LDV measurements on a swirling air flow in a pipe,
the upstream axial velocity is about 2.5 times, while the
maximum tangential velocity is about 1.75 times as large as
the mean axial velocity [128, Fig. 8]. This also gives the ratio
Vx =Vu max ø 1:4: Thus, in terms of the swirl number, a very
clear conclusion has beenpobtained
for the onset of vortex
ffiffi
breakdown. That is, Sc ø 2:
On the other hand, the criterion for the onset of vortex
breakdown has often been discussed on the basis of a
Rossby number (inverse swirl number), Ro, which is
525
The results obtained by Billant et al. [126] are somewhat
scattered; Roc ¼ 0:51 – 0:65 for the 40 mm nozzle and
Roc ¼ 0:80 – 0:84 for the 25 mm nozzle.
p
If we assume
pffiffithat r V is equal to Vu max and include the
relation W ø 2Vu max in Eq.
pffiffi(38c) or (38d), the Rossby
number should be equal to 2: Thus, the critical Rossby
numbers experimentally obtained are not consistent with the
critical swirl number of 1.4, and
pffiffi smaller than the
corresponding Rossby number of 2: It should be noted
however, that the axial velocity is defined at r p ð– 0Þ; not on
the axis of rotation.
If the Rossby number is defined using the axial
pffiffi velocity
on the centerline [117], the critical number of 2 has been
obtained. As addressed in the review by Delery [117], the
critical Rossby numbers are 1.4 in both the theories of
Squire [130] and Benjamin [131]. Although their theories
are based on the concept of critical state, and are somewhat
elaborate mathematically, the critical swirl value of 1.4 is
obtained simply in the paper by Billant et al. [126] as
follows.
Fig. 73 [126] shows the configuration of cone vortex
breakdown schematically. If the Bernoulli equation is
applied to a streamline of the vortex axis, the total head
H ¼ P=r þ ðVx2 þ Vr2 þ Vu2 Þ=2 leads
H¼
Fig. 72. Critical values for appearance Sca and disappearance Scd of
breakdown in (Re, S ) parameter space for each nozzle: (a) D1 ¼ 40
mm; (b) D2 ¼ 25 mm [126].
P0
V 2 ð0; x0 Þ
P
þ x
¼ 1;
r
r
2
ð38eÞ
where x0 is located well upstream of the stagnation point, P0
is the pressure on the vortex axis at the station x0 ; r is the
fluid density, Vx ð0; x0 Þ is the upstream axial velocity on the
vortex axis at x0 ; and P1 is the pressure at the stagnation
point. Far upstream of the stagnation point, the radial
pressure gradient is balanced by the centrifugal force;
defined as
Ro ;
W
:
rp V
ð38cÞ
Here, W, r p and V represent a characteristic velocity,
length and rotation rate, respectively [129]. Usually, r p is
defined as the radial distance at which the swirl velocity is a
maximum, W is given as the axial velocity at r p ; and V is
the core angular velocity and given as V ¼ limr!0 ðVu =rÞ
for the two-dimensional Burgers vortex.
A plot of the Rossby number Ro; versus Reynolds
number Reð; Wrp =vÞ; for a variety of numerical and
experimental studies of swirling flows, indicate that the
critical Rossby number is approximately 0.65 for the
Reynolds number greater than 100 for wing-tip vortices.
For leading-edge vortices, however, the critical Rossby
number becomes higher to be near unity [129, Figs. 1 and 2].
In the experiment by Billant et al. [126], the Rossby
number is also defined with the use of the axial velocity
Vx ðr p ; x0 Þ at r p
Ro ;
Vx ðr p ; x0 Þ
:
rp V
ð38dÞ
Fig. 73. Schematic configuration of cone vortex breakdown [126].
526
consequently,
P0 ¼ P1 2
ð1 V 2 ðr; x Þ
0
r u
dr;
r
0
ð38fÞ
where Vu ðr; x0 Þ is the azimuthal velocity and P1 is the
ambient pressure at infinity in the cross-stream plane x ¼ x0 :
If we assume that the pressure in the stagnation zone P1 is
equal to the pressure at infinity P1 ; the simple relation
ð1 V 2 ðr; x Þ
0
u
dr ¼
r
0
1
2
Vx2 ð0; x0 Þ
ð38gÞ
is obtained. In a particular case of Rankine’s combined
vortex, i.e. solid body rotation Vu ¼ Vr and Vx ¼ const: for
r # a and irrotational flow Vu ¼ V a2 =r and Vx ¼ 0 for r $
a; the left side of Eq. (38g) is equal to ðVaÞ2 : By noting that
Va ¼ Vu max ; this criterion reduces to
pffiffi
ð38hÞ
Vx ð0; x0 Þ ¼ 2Vu max ðx0 Þ:
In the case of a bubble state, the stagnant region is not
directly connected to the surrounding outer quiescent fluid.
Therefore, the relation P1 ¼ P1 must be replaced by the
weaker inequality P1 # P1 : In all other respects, the
previous reasoning holds and criterion (38g) becomes
ð1 V 2 ðr; x Þ
1
0
u
dr $ Vx2 ð0; x0 Þ:
r
2
0
ð38iÞ
For Rankine’s combined vortex, this inequality becomes
pffiffi
ð38jÞ
Vx ð0; x0 Þ # 2Vu max ðx0 Þ:
If the results of Eq. (38h) are applied to the case of the
propagating
flame in a vortex, the flame speed reaches
pffiffi
2Vu max ; if the burned gas expands infinitely in the radial
direction and the pressure behind the flame reaches the
ambient pressure. However, if the burned gas is confined
to a limited range in diameter, which is observed in the
experiments,
pffiffi the flame speed may become less than the
value of 2Vu max :
Concerning the finite diameter of the flame, a relevant
study of vortex breakdown exists. Fig. 74 [116] shows the
two-stage transition model proposed by Escudier and Keller
[116,132]. A bubble exists at the center, which is treated as a
stagnation zone. The pressure within the bubble is assumed
to be equal to the upstream stagnation pressure, and the
Bernoulli equation is applied to a center streamline. The first
stage, which establishes the breakdown criterion, incorporates the transition from the upstream flow to an intermediate
flow state, and the second transition, which has no bearing
on the breakdown criterion, is treated essentially as a
hydraulic jump. By considering the balance of the flow force
S (momentum flux), between the first flow state and the
second flow state, S1 ¼ S2 ; in which the flow force is defined
as
S;
ð
ðP þ rw2 ÞdF ¼ 2p
F
ðR
ðP þ rw2 Þr dr
ð39aÞ
0
a swirl parameter k, which is defined as
G1
2vd
¼
k;
pdW
W
ð39bÞ
has been obtained numerically for the occurrence of vortex
breakdown as a function of the core to radius ratio d=R: Here,
G1 is the constant circulation far upstream, d is the core
radius, W is the uniform axial velocity, and v is the angular
velocity of the vortex core, respectively. The numerical
results, which are shown in Ref. [116, Fig. 26], indicate that
the breakdown occurs at
8
pffiffi
<k ¼ 2
for d=R ! 0 ðfree vortexÞ;
:
ð39cÞ
: k ¼ 3:832 for d=R ! 1 ðforced vortexÞ
Fig. 74. Schematic diagram of proposed 2-stage transition [116,132].
By noting that vd and W correspond, respectively, to Vu max
and Vf ; these yield
Vf ¼
8 pffiffi
< 2Vu max
for a free vortex;
: 0:52V
for a forced vortex
u max
:
ð39dÞ
pffiffi
Note that the flame speed reaches 2Vu max for the free
vortex. Thus, in regard to Eq. (38h), the pressure in the
stagnation zone seems to reach the pressure of the
surrounding outer quiescent fluid.
This two-stage breakdown model is similar to the model
by Atobiloye and Britter [62] in that a stagnation zone of
finite diameter is taken into consideration. In the model by
Atobiloye and Britter, the Bernoulli equation is applied to
the streamline on the axis of rotation and, it is also applied to
the streamline at the outer boundary of the vortex; while the
momentum flux conservation is applied to the high-density
flow. The difference between the two models is that the
velocity components are continuous across the interface in
the two-stage breakdown models [116,132], whereas the
tangential velocity at the interface differs in the two fluids in
the model of Atobiloye and Britter [62]. As the result, the
latter model predicts smaller flame velocities than those
given by Eq. (39d), so that
Vf ø
8 pffiffiffiffiffiffiffiffiffiffiffiffi
>
1 2 rb =ru Vu max
>
>
>
>
<
for a free vortex
>
>
0:25Vu max
>
>
>
:
for a forced vortex
ðR2 ! 1; a ! 0Þ;
:
ð40Þ
ðrb =ru ¼ 1=5 – 1=7Þ
From discussions above, it is found that the vortex
breakdown in swirling flows is qualitatively quite similar
to the rapid flame propagation along the vortex axis.
Quantitatively, however, there is a difference between the
two phenomena. In the case ofpfree
ffiffi vortex, the value of slope
in the Vf 2 Vu max plane is 2 for the vortex breakdown
527
(Eq. (39d)), whereas it is nearly at unity for the rapid flame
propagation in a tube (Eq. (40)).
Such a quantitative difference can also be found in the
generation of azimuthal vorticity. As clearly pointed out by
Umemura et al. [74 – 77,82], azimuthal vorticity is generated
at the onset of rapid flame propagation. Likewise, it is well
known that the azimuthal vorticity is generated in the vortex
breakdown. Fig. 75 [123] shows the calculated contours of
(i) stream function c, (ii) azimuthal vorticity h, and (iii)
tangential velocity y in a swirling flow by Brown and Lopez.
They consider that in the absence of viscous or turbulent
diffusion, a necessary condition for breakdown to occur
downstream of z0 is one in which a helix angle a0 of the
velocity exceeds a helix angle b0 of vorticity on some
stream surfaces. That is,
a0 $ b0 :
ð41Þ
Here, a0 ; y 0 =w0 ; in which y 0 and w0 are the azimuthal and
axial components of the velocity, respectively, and b0 ;
h0 =z0 ; in which h0 and z0 are the azimuthal and axial
components of the vorticity, respectively, and subscript 0
denotes some upstream station.
The value of a0 =b0 is 1.91 for the condition in Fig. 75.
Fig. 75(a) shows the contours at t ¼ 227; and Fig. 75(b)
shows those at t ¼ 250: Due to a slight divergence of
streamlines, the azimuthal velocity and the azimuthal
vorticity are reduced with distance downstream, and a
further divergence of these stream surfaces generates a
negative azimuthal component, leading to a small recirculation zone on the axis, rapid changes in azimuthal vorticity
ahead of this region where the streamlines diverge, and the
evident propagation upstream of the region of negative
azimuthal vorticity due to its own induced velocity.
Fig. 75(c) shows a corresponding development in a nonphysical case in which, for the above flow, at t ¼ 227; the
viscosity is suddenly doubled (the Reynolds number is
halved). Fig. 75(d) is a case in which the viscosity is
suddenly halved at t ¼ 227: A comparison between
Fig. 75. Calculated contours of (i) stream function c, (ii) azimuthal vorticity h and (iii) azimuthal velocity y for a flow with Vc ¼ 1:75; Wc ¼ 1:6
and Re initially 300. (a) t ¼ 227 and Re ¼ 300; (b) t ¼ 250 and Re ¼ 300; (c) t ¼ 250 following a reduction at t ¼ 227 in Re from 300 to 150;
(d) t ¼ 250 following an increase at t ¼ 227 in Re from 300 to 600 [123].
528
Fig. 75(b) and (c) shows that the subsequent effect of a
sudden increase in the viscosity is to diffuse the axial
vorticity and increase the initial divergence of streamlines,
but to reduce the magnitude of the maximum negative
component of the azimuthal vorticity from 2 1.84 to 2 1.37,
to reduce the size of the recirculation bubble. The reverse is
true for the sudden decrease in viscosity (Fig. 75(d)).
When we apply these numerical results to the actual
flame, two points of difference can be considered. The
first difference is in gas expansion. This results in an
abrupt increase in size and bulges of stream surfaces in
the vortex core, resulting in the appearance of a highpressure region, by which the flame is forcibly driven
ahead. Therefore, independent of the criterion for the
occurrence of vortex breakdown, a0 $ b0 ; the rapid flame
propagation can be achieved in a vortex once the mixture
is ignited and the combustion proceeds. The second
difference is the increase in viscosity with temperature.
Due to increased viscosity with temperature, the vorticity
is damped immediately and the vortical structure may
disappear. Thus, it is true that, similar to the vortex
breakdown in swirling flows, the evolution of vorticity
induces rapid flame propagation along a vortex axis. The
constant viscosity model, however, is inadequate to
describe the flame propagation quantitatively.
6.2. Flame speeds: summary
6.2.1. Flame speeds for typical flame diameters
In Section 6.1, studies on the vortex breakdown
phenomena in constant-density flows have been reviewed.
Of interest is a recentpexperimental
result in which the axial
ffiffi
velocity becomes 2 times the maximum tangential
velocity at the onset of breakdown. Based on this result,
relevant theories on the rapid flame propagation along a
vortex axis have been reviewed. Here, we summarize the
formulations for the flame speed in Section 4 and discuss
their validity.
In typical cases, when the radius of the burning region is
infinitely large, or when the radius is equal to that of the
forced vortex core, flame speeds are given as follows:
1. The original theory by Chomiak [9]:
rffiffiffiffiffi
r
Vf ¼ Vu max u :
rb
ð26dÞ
2. The angular momentum conservation model by Daneshyar and Hill [25]:
sffiffiffiffiffiffi
2ru
Vf ¼ Vu max
:
ð27dÞ
rb
3. A hypothesis by Ishizuka and Hirano [50]:
pffiffiffiffiffi
Vf ¼ Vu max 2k2 ðk2 # 1Þ:
ð28bÞ
4. A steady-state immiscible stagnant model by Atobiloye
and Britter [62]:
r
Vf ¼ Vu max 1 2 b ðx ; R1 =R2 ! 0; a ! 0Þ: ð29s0 Þ
ru
5. The finite flame diameter approximation by Asato et al.
[57]:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3 ru
21
ða=a ¼ 1Þ;
ð30g0 Þ
Vf ¼ Vu max
16 rb
Vf ø 1:07Vu max
ða=a ¼ 1; ru =rb ¼ 7Þ:
ð30g00 Þ
6. The back-pressure drive flame propagation model by
Ishizuka et al. [69]:
ru 2
axial expansion : Vf ¼
S þ Vu2 max
rb u
ð31t0 Þ
ðk ! 1; Y ¼ 1Þ;
r
radial expansion : Vf ¼ Su þ Vu max 1 þ b
ru
ð31u0 Þ
ðk ! 1; Y ¼ 1Þ:
7. A steady-state back pressure drive flame propagation
mechanism [72]:
ru 2
for Vf $ ru Su =rb ;
Su þ Vu2 max
ð33eÞ
Vf ø
rb r
ffiffiffiffiffiffiffiffiffiffi
r
Vf ¼ Vu max 1 þ b
for Vf # ru Su =rb :
ð33fÞ
ru
8. The baroclinic push mechanism by Ashrust [64]: Eq.
(35f) is rewritten in terms of ru, rb, Vu max and Su as
sffiffiffiffiffi
rffiffiffiffiffi
r
rb rM Vu2 max XF
Vf < 1 2 b
ru
ru
dSu
rM
ð35f 0 Þ
ðstraight vortexÞ:
9. The azimuthal vorticity evolution mechanism by Umemura and Tomita [74,77]: Eq. (36p) is rewritten in terms
of ru, rb, Vu max and Su as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
ru 2
r
ð36p0 Þ
Vf ¼
Su þ 2 þ b Vu2 max :
rb
ru
In accordance with the experimental result of vortex
breakdown in constant-density flow, the theory by Umepffiffi
mura and Tomita [74,77] gives the proportionality of 2 in
the Vf 2 Vu max plane for large values of Vu max : The theories
by Atobiloye and Britter [62], Asato et al. [57], and Ishizuka
et al. [69,72], however, give unity slope. In Eqs. (26d),
(27d), and (30g0 ), the density ratio appears in the form of
ru =rb ; whereas in Eqs. (29s0 ), (31u0 ), (33f), (35f0 ) and (36p0 )
it appears in the reverse form, rb =ru ; with respect to Vu max.
Thus, these equations, theoretically derived, contradict each
other. Similar controversy can also be found in the
formulations for gravitational flows.
6.2.2. Analogy between flows in vortices and gravitational
flows
In the case of gravitational flows, the driving force is the
difference in gravitational force working on two fluids of
different density. In the case of a wedge of fluid displacing a
heavier fluid from the under side of a horizontal plane
(Fig. 76(a)) [133], the speed of the cavity is given as
c1 ¼
1
2
pffiffiffiffi
gd ;
ð42aÞ
in which d is the depth of the flume, and g is the acceleration
due to gravity. This speed is obtained by applying the
Bernoulli theorem along the surface and using the balance of
529
flow force (i.e. momentum flux plus pressure force) between
the approaching and receding parts of the stream.
In the case of mutual intrusion of two fluids in a flume
(Fig. 76(b)) [134], the speed of the intrusion front is
obtained as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gbðr2 2 r1 Þ
uF ¼
;
ð42bÞ
ðr2 þ r1 Þ
in which r2 and r1 are the densities of heavier and lighter
fluids, respectively, and b is the depth of the fluid. Here, it is
assumed that the flow is symmetric and energy is conserved,
i.e. by allowing the kinetic energy gained by both fluids to
be the equal of the net potential energy gained by the lighter
fluid and lost by the heavier fluid.
In a model by von Karman, shown in Fig. 76(c) [134,
135], the speed of gravity current of density r2 advancing in
Fig. 76. Models for (a) steady flow past a cavity [133], (b) mutual intrusion [134], and (c) gravity current advancing in an ambient fluid [134,
135].
530
an ambient, lighter fluid of density r1, is obtained by
applying the Bernoulli equation to points A and B on the
boundary current to be
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r 2 r1
U ¼ 2gb 2
:
ð42cÞ
r1
In a model by Fannelop and Jacobsen [136], the motion of a
heavy fluid is considered on the basis of shallow-layer
theory, and the wave speed for this layer is derived as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gbðr2 2 r1 Þ
:
ð42dÞ
uF ¼
r2
If r1 ¼ 0 (cavity) is placed into Eq. (42b), the upper wedge
speed of the lighter fluid (cavity) becomes
pffiffiffi
1 pffiffiffiffiffiffiffi
uF ¼ gb ¼ pffiffi gð2bÞ:
2
ð42eÞ
This
pffiffispeed is different from the value of Eq. (42a) by a factor
of 2; since 2b corresponds to the flume depth d. Thus, Eq.
(42a) is contradictory to Eq. (42b) for the upper cavity
speed.
A contradiction also occurs in the speed of the lower
fluid front. In Eq. (42c), the speed of the layer is increased as
the densitypffiffiffi
ratio r2 =r1 is increased, whereas speed
approaches gb in the limit r2 =r1 ! 1 in Eq. (42b). Eq.
(42d) is in sharp contrast to Eq. (42c) in that the
denominator is the density of a heavier fluid, not the density
of a lighter fluid. Thus, the speeds of the intrusion layer of a
heavier fluid predicted by the three models (Eqs. (42b) –
(42d)), contradict each other.
In the problem of rapid flame propagation along a vortex
axis, the driving force is the difference in the centrifugal
forces of rotation working on the unburned gas of high
density and on the burned gas of low density. This difference
in force is given by ru Vu2 max {1 2 ðrb =ru Þ2 } (Eq. (27b)) or
approximately by ru Vu2 max (Eq. (26c)). If this difference in
force is considered to balance with the momentum flux of
the burned gas, the denominator in the equation for the flame
speed becomes the burned gas density rb (Eq. (26d)). If the
Bernoulli equation (energy conservation prule)
is used,
ffiffi
the flame speed is increased by a factor of 2 (Eq. (27d)).
If the inertia of the heavy, unburned gas is taken into
consideration, the density ratio ru =rb disappears (Eq.(28b)).
If the balance of flow force is considered, the proportionality
factor becomes about unity, and ru appears in the
denominator (Eqs. (31u0 ), (33f)).
Recently, detailed research [137] has been performed
on the lock-exchange problem, using various fluids of
different density as well as numerical calculations in the
experiments. It was concluded that the light-fluid front
along the underside is elongated, smooth, and generally
loss-free, and hence, the front velocity is in agreement
with Benjamin’s ideal theory [133] (Eq. (42a)). On the
other hand, the heavy-fluid front is blunt and gives more
evidence of mixing and other loss processes, and therefore,
its speed is close to the speed based on the flow force
balance. Thus, precise observations on the front shape are
indispensable in order to conclude which model is
appropriate for predicting the front velocity.
6.2.3. Flame speeds for finite flame diameter
Similarly, precise observations on the flame shape are
indispensable in order to predict flame speed accurately.
Our major concerns are the shape of the flame shape,
whether the flame area is constant or if it increases in the
propagation, whether the flow is in a laminar or turbulent
condition, and eventually, whether the flame propagation
is steady or unsteady, etc. As for the flame shape, some
theories have taken it into consideration: They are
summarized as follows:
1. In the steady state, immiscible stagnant model by
Atobiloye and Britter [62], the solutions are obtained
numerically. In the case of a free vortex in a rotating
tube, however, the flame speed is analytically
expressed as
r
Vf < Vu max ð1 2 x2 Þ 1 2 b
ða ! 0Þ:
ð29s00 Þ
ru
2. The finite flame diameter approximation by Asato et al.
[57]:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8
!ffi
u
2
>
u
>
a
r
a
>
t u 21 12
>
ða # aÞ
V
>
< 2a u max
rb
4a2
Vfth ¼
:
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
>
>
ru
3
a
>
> 1 Vu max
ða . aÞ
þ ln
21
:
2
4
rb
a
ð30fÞ
3. The back-pressure drive flame propagation model by
Ishizuka et al. [69]:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r
ð31tÞ
axial expansion : lVf l ¼ u ðYSu Þ2 þVu2 max f ðkÞ;
rb
r
radial expansion : lVf l¼ YSu þVu max 1þ b f ðkÞ: ð31uÞ
ru
Eq. (31u) is modified for angular momentum conservation on each streamline:
lVf l ¼ YSu
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
rb ru
rb
1
þkVu max 1þ
þ ln 12
12 2
2ru rb
ru
k
for k # 1:
ð31xÞ
4. The steady-state model of the back pressure drive
flame propagation theory [72]
r
ð33fÞ
lVf l ¼ Vu max 1þ b f ðkÞ ðVf # ru Su =rb Þ:
ru
Eq. (33f) is modified for angular momentum conservation on each streamline:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
rb ru
rb
1
lVf l ¼ kVu max 1þ
þ ln 12
12 2 : ð33gÞ
2ru rb
ru
k
5. The azimuthal vorticity evolution mechanism by
Umemura and Tomita [76,77]: Eq. (36r) is rewritten
in terms of ru, rb, and k as
Vf
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
rb
ru
rb
1
ru 2
2
¼ 2 þ þ 2 ln 1 2
12 2
Vu max þ Su :
ru
rb
ru
rb
k
ð36r0 Þ
In these formulas, two parameters are taken into
consideration. One is a parameter related with the flame
area, Y, the ratio of the flame area to the cross-sectional
area of the flow concerned. This parameter, however, is
concerned only with the aerothermochemical term in the
back-pressure drive flame propagation theory; it has been
shown in the experiment in a rotating tube [66] (hence,
the burned gas is expanded mostly inpthe
axial
ffiffiffiffiffiffi
ffi expansion),
that the asymptotic value of YSu ru =rb in the limit
Vu max ! 0 is in good agreement with the measured flame
speed.
Another parameter is concerned with the radius of the
burned gas. In the theory by Atobiloye and Britter [62], the
ratio of the burned gas to the tube radius x ; R2 =R1 is
introduced and solutions are obtained numerically. Eq.
(29s00 ) is the result in the case of a free vortex. It is seen that
with a decrease of x ; R1 =R2 (R1 is the flame radius),
the
flame speed is increased and approaches Vu max 1 2 rb =ru
in the limit of x ! 0: That is, the flame speed is increased if
the flame becomes more sharp-pointed.
In the other three theories, the flame shape is taken into
consideration through a ratio of the unburned gas radius to
the radius of the vortex core, i.e. a=a or k ; ru =ðhu =2Þ: With
a decrease in a=a or k, the flame speed is decreased in these
three models. It should be noted, however, that these
parameters are based on the concept of Rankine’s combined
vortex. The tangential velocity distribution of Rankine form
is assumed to obtain the above equations. The actual
tangential velocity distribution, however, is not of Rankine
form but of Burgers form. Their theoretical results should be
modified accordingly.
6.2.4. A note on Burgers vortex
Burgers vortex is a solution for the tangential velocity
component of the Navier – Stokes equation under the
conditions that the flow is incompressible, axi-symmetric,
and stretched. The radial and axial velocity components, u
and w, are given in such a way that [138]
uðradial velocityÞ ¼ 2Ar;
ð43aÞ
wðaxial velocityÞ ¼ 2Az;
531
ð43bÞ
in which A is the velocity gradient, r is the radial distance,
and z is the axial distance. This set of velocity components
satisfies the continuity equation, and the tangential velocity
distribution is obtained as
y¼
2
C
ð1 2 e2Ar =2v Þ:
2pr
ð43cÞ
Here, v is the kinetic viscosity and C is a constant, which can
be determined for considering the circulation at r ¼ 1; G1,
as
C ¼ G1 :
ð43dÞ
The velocity profile of Eq. (43c) tends to that of the free
vortex far from the center ðy / 1=rÞ; and tends to that of the
forced vortex of a rigid body rotation near the axis of
rotation ðy / rÞ: The rotational speed V is given as
V ; lim
r!0
y
AC
AG1
¼
¼
:
4pv
r
4pv
ð43eÞ
Note that for a given G1 ð– 0Þ; the rotational speed V
increases linearly with increasing A (the velocity gradient).
Also note that the rotational speed is inversely proportional
to the kinetic viscosity v.
The tangential velocity profile has a maximum. By
differentiating Eq. (43c) with respect to r, its condition is
obtained as
Ar 2
¼ j ø 1:2565;
2v
ð43fÞ
where j is a solution for an algebraic equation ex ¼ 2x þ 1:
Thus, the core radius rc, defined as the maximum tangential
velocity position, and the maximum tangential velocity
Vu max, are obtained in terms of (A/v, G1/2p), or in terms of
(V,G1/2p) as
sffiffiffiffiffiffiffiffiffi
rffiffiffiffiffi
rffiffiffiffiffi
pffiffi 2v
2v
G1 1
;
ð43gÞ
rc ¼ j
ø 1:12
¼ 1:12
2p V
A
A
rffiffiffiffiffi
rffiffiffiffiffi
G
A 1 2 e2j
G
A
pffiffi
ø 0:638 1
Vu max ¼ 1
2p 2v
2p 2v
j
sffiffiffiffiffiffiffiffi
G1
ð43hÞ
V:
¼ 0:638
2p
Note that with an increase of the velocity gradient A, the
core radius decreases, whereas the maximum tangential
velocity increases. By integrating the radial momentum
equation numerically, the pressure difference DP between
the center and the infinity is obtained as
DP ø 1:70ru Vu2 max :
ð43iÞ
In Rankine’s combined vortex, the pressure difference is
equal to ru Vu2 max : Thus, the driving force in Burgers vortex
becomes bigger in terms of Vu max. This results in an
increase in the proportionality factor in the Vf – Vu max plane;
532
Fig. 77. Comparison between measured flame speeds and theoretical predictions; (a) lean methane/air mixture (F ¼ 0:6; Su ¼ 0:097 m=s;
ru =rb ¼ 5:6), (b) stoichiometric methane and propane mixtures (F ¼ 1:0; Su ¼ 0:4 m=s; ru =rb ¼ 7:85 (mean)), and (c) rich propane mixtures
ðF ¼ 2:0; ru =rb ¼ 7:7Þ [85].
in the back-pressure drive flame propagation
pffiffiffiffi mechanism,
the slope may be increased from unity to 1:7 ø 1:3; and in
the theory by p
Umemura
and Tomita, the slope is also
ffiffi pffiffiffiffiffiffiffiffiffiffi
increased from 2 to 2 £ 1:70 ø 1:8:
6.2.5. A general comparison between theories and
experiments
Based on the above discussions, comparisons between
theories and experiments are briefly made in the following.
In the case of flame propagation in a rotating tube, a
steady state is not achieved. Since gas expansion in the
radial direction is restricted by the existence of the solid
glass wall, the axial expansion model of the back-pressure
drive flame propagation theory can accurately describe the
experimental results. A good agreement has been obtained
between the theory and the experiment [66, Fig. 6].
In vortex ring combustion, a steady state of flame
propagation can be achieved, although the flame speed is
usually fluctuated, and its magnitude sometimes reaches
nearly 30% of the mean flame speed for large Reynolds
number. Very recently, an experiment [85] has been
conducted in a ‘pure’ atmosphere of the same mixture as
the combustible gas of the vortex ring, in the sense that the
flame propagation is not influenced by dilution with the
ambient mixture, or the secondary combustion between
excess fuel and ambient air. Flame speeds have been
determined for lean and rich mixtures as well as stoichiometric methane/air and propane/air mixtures, and compared
with some theories, which have taken the finite diameter
into consideration. They are as follows:
1. Umemura and Tomita; steady-state, Bernoulli’s
equation: Eq. (36r0 );
2. Ishizuka et al.; steady-state momentum flux balance: Eq.
(33g);
3. Asato et al.; hot stagnant, effective pressure: Eq. (30f);
4. Ishizuka et al.; back-pressure drive, axial expansion: Eq.
(31t0 ).
Fig. 77 [85] shows the results for (a) lean methane, (b)
stoichiometric methane and propane, and (c) rich propane
mixtures, in which the variations of flame diameter with the
maximum tangential velocity are also presented. In the
steady state models by Umemura and Tomita and by
Ishizuka et al., the burned gas is assumed to expand in the
radial
direction;
i.e. the flame diameter df becomes
pffiffiffiffiffiffi
ffi
du ru =rb : The parameter k in Eqs. (33g) and (36r0 ) is
estimated from a relation
k ; du =dc ¼ ðdf =dc Þ rb =ru :
ð44Þ
Representative values of k, obtained from this relation, are
shown in the upper figures of Fig. 77. To avoid complexity,
the relations of Eqs. (36r0 ) and (33g) are shown with k ¼
0:08 and 0.12 in Fig. 77(a), with k ¼ 0:2 and 0.3 in Fig.
77(b), and with k ¼ 0:1 and 0.2 in Fig. 77(c). In the model
by Asato et al., and in the case of axial expansion, the value
of a=a or k is equal to df/dc. Thus, the relations of Eqs. (30f)
and (31t0 ) are shown with k ¼ 0:2 and 0.3 in Fig. 77(a), with
k ¼ 0:5 and 0.75 in Fig. 77(b), and with k ¼ 0:25 and 0.5 in
Fig. 77(c). The value of Y is assumed to be unity in Eq. (31t0 ).
It is seen that Eq. (31t0 ) underestimates, while Eq.
(30f) completely covers almost all the results except for
small values of Vu max. On the other hand, the model by
Umemura and Tomita, Eq. (36r0 ), overestimates the lean
methane flame speeds, however, it predicts well those of
the stoichiometric and rich propane mixtures in a range
of Vu max , 10 m/s. A steady-state momentum flux
conservation model, Eq. (33g), somewhat overestimates
the lean methane flame speeds, however, it predicts well
the results of the stoichiometric and rich propane
mixtures in a range of Vu max . 10 m/s. As a whole
Eq. (36r0 ) is better in low velocity region and Eq. (33g)
is better in the high velocity region for describing the
measured flame speeds. This is because for Vu max . 10
m/s, the Reynolds number, which is defined as Re ;
UD=v (U and D are the translational velocity and
diameter of the vortex ring, respectively, and v is the
kinematic viscosity), becomes the order of 104 and the
so-called turbulent vortex rings are formed [86]. Thus,
pressure loss occurs and the Bernoulli equation may not
be valid, resulting in a poor description of Eq. (36r0 ) in
the high velocity region. The angular momentum
conservation on each streamline sounds rigorous, but a
vortex motion decays rapidly behind the flame due to
high viscosity. This partly explains why the hot, stagnant
gas model by Asato et al. [57] predicts the results so
well.
However, the above discussion is based on Rankine’s
combined vortex. As noted in Section 6.2.4, the value of
slope of Eq. (36r0 ) will be raised to 1.8 for Burgers vortex,
and as a result, Eq. (36r0 ) may overestimate the experimental
results for almost all values of Vu max. Instead, Eq. (33g)
survives because the slope is raised to 1.3 for Burgers
vortex. Although Eq. (31t0 ) for axial expansion always
underestimates the results as long as Rankine’s combined
vortex is assumed, the validity of Eq. (31t0 ) for Burgers
vortex must be considered. At present, it is very difficult to
conclude which theory describes best the measured flame
speeds.
6.2.6. An unresolved problem: finiteness of flame diameter
As seen in Fig. 77, flame diameter decreases with an
increase in the vortex strength. In air and in a nitrogen
atmosphere, dilution of the combustible mixture by
entrainment of the ambient gas causes flame extinction,
resulting in a finite flame diameter. In a pure atmosphere,
however, dilution may not occur. Thus, the observed
decrease in flame diameter with increasing vortex strength
results from a pure interaction between the flame and the
flow. Although some theories have taken flame diameter
into consideration, they cannot describe the decrease of
flame diameter with vortex strength. In this sense, these
theories are not complete, merely semi-empirical.
The finiteness of the flame diameter in the vortex ring
combustion at the stage of the rapid flame propagation may
be closely related with the finiteness of the flame diameter
in a rotating vessel. Ono and co-workers [58,59] have
considered that the shear flow induced by gas expansion
533
extinguishes the flame, while Gorczakowski and Jarosinski
[87] considered that a heat loss to the wall extinguishes the
flame. The heat loss to the wall can be considered in the case
of vortex ring combustion as a heat loss to the ambient
mixture, due to flow non-uniformity. That is, because of
non-uniform flow, both heat and mass transfer can occur
through a stream tube [47,139– 144]. This results in a heat
loss around the head of the propagating flame in vortices.
Thus, flame stretch plays an important role on the finiteness
of flame diameter. The flow diverges toward a convex flame.
The flame suffers from stretch through non-uniformity of the
flow and through curvature of the flame. This may result in
flame extinction at some distance behind the head of the
flame. This flame stretch mechanism can explain the
observed Lewis number effect on the flame diameters
[85]. Apparently, the flame diameter characteristics of the
propagating flame are very similar to those of the tubular
flame, established in a rotating, stretched flowfiled [37 – 47].
For further discussion, detailed measurements on the flow
field by PIV—not only in the plane perpendicular to the axis
of rotation (Fig. 20) but also in the plane parallel to the axis
of rotation (Fig. 34)—are indispensable.
6.3. Modeling turbulent combustion
The phenomenon of rapid flame propagation along
vortices has received considerable attention in modeling
turbulent combustion [9,14,15]. In the model by Tabaczynski et al. [14], it is assumed that fast flame propagation in
a vortex of Kolmogorov scale occurs, followed by a laminar
combustion of the mixture outside the vortices. In the
Klimov model [15], rapid flame propagation occurs in a
vortex whose diameter is larger than the Kolmogorov scale;
this is followed by combustion in tubular flame geometry. A
recent direct numerical simulation of turbulent combustion,
however, does not yield evidence for their models. Fig. 78
[145] shows contour surfaces of the second invariant of the
velocity gradient tensor (green) and those of the heat release
rate (yellow). It is seen that there are many tube-like eddies
in the unburned gas. The turbulence in the unburned gas
consists of coherent fine-scale eddies. The mean diameter of
these eddies is about 10 times the Kolmogorov microscale
h, and the maximum azimuthal velocity is about half the
root mean square velocity fluctuation u0 [145,146]. In the
burned gas side, however, these eddies are dissipated
because of increased viscosity with temperature. Although
penetration of the hot burned gas into the unburned gas by
vortex bursting is expected, such penetration cannot be
recognized in Fig. 78.
Fig. 79 [145] shows the distributions of the axes of the
coherent fine-scale eddies, in which the visual diameters of
the axes are selected to be proportional to the square root of
the second invariant of the velocity gradient tensor. It is seen
that near the flame front, the magnitude of the solid body
rotation of the eddy decreases, while strong coherent
fine-scale eddies survive behind the flame front and are
534
Fig. 78. Contour surfaces of the second invariant and heat release
rate (green: Qþ ¼ 0:02; yellow: DH p ¼ 1:0) [145].
elongated in the streamwise direction due to the acceleration
caused by gas expansion [145]. This situation is reasonable
upon consideration of Eq. (27a), which indicates that the
angular speed is decreased by a factor of ru =rb through the
flame when the burned gas expands radially, or in Eq. (31i),
which indicates that the angular speed is decreased by a
factor of 12r ; where 1r is the expansion ratio rb =ru (Eq. (31c)).
However, even upon close observation, rapid flame
propagation along the vortex axis (which has been observed
in vortex ring combustion) may not occur in the coherent
fine-scale eddies.
Fig. 80 [145] shows typical interactions between the
coherent fine-scale eddy and the premixed flame that are
denoted by the Regions A, B, and C in Fig. 78. In Region A,
a coherent fine-scale eddy is impinging on the flame front
with an axial velocity towards the direction of the burned
gas (Fig. 80(a) and (b)), and the unburned mixture is
provided to the flame front by the axial velocity. As a result,
the reactions are enhanced.
Fig. 79. Distribution of the axes of coherent fine-scale eddies in a
turbulent premixed flame (gray: axes, yellow: DH p ¼ 1:0) [145].
Fig. 80. Distributions of the heat release rate and axes of the typical
coherent fine-scale eddy. (a) A region ðDH p ¼ 1:1Þ; (b) A region
(axial velocity), (c) B region ðDH p ¼ 1:1Þ; and (d) C region
ðDH p ¼ 1:0Þ [145].
In Region B (Fig. 80(c)), the axis of the coherent finescale eddy is parallel to the flame front, and the tube-like
structure of high heat release rate is observed along the axis.
However, as stated in the paper by Tanahashi et al. [145],
‘because the coherent fine-scale eddies have large azimuthal
velocity of the order of u0 , the eddy parallel to the flame
front can transport the unburned species into the flame front,
which results in the tube-like structure of high heat release
rate along the axis’. Therefore, it should be noted that this
tube-like zone of high heat release rate is not established by
the rapid flame propagation mechanism postulated by
Chomiak [9], by Tabaczynski et al. [14] or by Klimov [15].
In Region C (Fig. 80(d)), the axis of the coherent finescale eddy is perpendicular to the flame front [147]. The
axial velocity is towards the unburned gas near the flame
front, and the flame front is convex towards the unburned
gas [147]. The heat release rate becomes relatively low
compared with the intense combustion in Region A or in
Region B [145]. This heat release rate reduction appears to
be a feature of the flame, which propagates along a vortex
axis. As seen in Fig. 4 and in Fig. 44, the head of the flame is
often dispersed and weak in luminosity. Thus, the heat
release rate must be low at this head region. The vortex
scales, however, are largely different between the experiments and the DNS. It is clearly stated in the Comments of
the paper by Tanahashi et al. [145] that ‘we cannot observe
the flame penetration into the coherent fine-scale eddies in
turbulence’. This is probably because the diameters of the
vortices are too small.
According to the numerical simulation by Hasegawa
et al. [63], a flame cannot propagate along a vortex axis
when the diameter is smaller than the order of the laminar
flame thickness. The same section also states [145] that ‘the
ratio of diameter of coherent fine-scale eddies in the
unburned side is about 0.46 times that of laminar flame
thickness. Therefore, the flame penetration into the coherent
fine-eddy seems to be very difficult’. The DNS, however, is
limited to a stoichiometric hydrogen/air mixture at 0.1 MPa
and 700 K. In addition, it is noted in the Comments that ‘this
result does not deny the possibility of flame penetration into
the coherent fine-scale eddies. Now we are planning to
conduct DNS that can verify the possibility of flame
penetration in the eddy’ [145].
It seems that there exists still another barrier to the flame
penetration, even if the eddy diameter becomes bigger than
the laminar flame thickness. A large axial velocity in the
vortex may prevent the flame from propagating upstream.
As confirmed in the numerical simulations [145,146,148],
the mean tangential velocity profile is similar to that of
Burgers vortex. This means that there is also an axial
velocity, as indicated by Eq. (43b). The results of the
numerical simulations [144,145,147] show that there are
radial flows towards the center from the outside, and also
strong axial flow from the center towards the outside, not
only on the axis but also in the regimes far from the central
axis (Ref. [148, Fig. 8]). The large axial flow is the order of
u0 . These velocities are, of course, largely fluctuated. Thus,
the axial flow sometime resists the flame penetration and
sometime helps the flame to penetrate into the vortices.
Although flame behavior in unsteady flows is quite different
from behavior in steady flows [149,150], it is reasonable to
expect that the axial flow, which is positive in the average,
may prevent the flame from penetrating into the vortices.
Fig. 81 [116,151] shows two examples of measured swirl
and axial velocity profiles in the exit of the slit-tube vortex
generator. The jet-like profile in Fig. 81(a), is typical of the
supercritical flow upstream of breakdown, while the wakelike profile in Fig. 81(b), is typical of the subcritical flow
downstream, in which small disturbances coming from
downstream propagate in the upstream direction and
ultimately provoke the breakdown. If the velocity profiles
in coherent fine-scale eddies in the unburned gas are
supercritical, the flame penetration will not occur. For these
two reasons, i.e. smallness of vortex diameter and largeness
of axial velocity, penetration of the hot burned gas into
either the fine-scale eddies or the somewhat larger vortices
by vortex bursting seems not to occur.
However, it should be noted that breakdown could occur
in the turbulence even if penetration is impossible. As found
experimentally, an axial flow prevents the flame from
propagating upstream in the vortex flow (Fig. 27). Under
such a flow condition, the flame may not be able to penetrate
deeply in the vortex. There are, however, many vortices in
the turbulence. As a result of interaction between these
vortices, subcritical flow conditions may be achieved at
many locations in the turbulence. As a result, vortex
breakdown can happen locally at many locations although
535
vortex breakdown at each place may not be extended to deep
penetration of the flame into the vortex.
Let us reconsider Region A. In Region A, the vortex axis
is perpendicular to the flame front, and the unburned
mixture is provided to the flame front by the axial velocity,
resulting in enhancement of the heat release rate. This
situation is very similar to the combustion at the exit of the
swirl type tubular flame burner. Fig. 82 [37] shows a
photograph of the combustion with a tubular flame burner.
At the exits, intense combustion proceeds due to sudden
expansion of the flow and a subsequent formation of a hot
recirculation zone (also see Ref. [37, Figs. 3c and 6b], and
Ref. [39, Fig. 9]. Note that at the exit of the slit tube vortex
generator [152], vortex breakdown occurs without combustion due to sudden expansion.) This phenomenon is a vortex
breakdown, common in industrial furnaces, which use a
swirl combustor to stabilize and enhance combustion [116,
153]. The hot gas is impinged from the burned gas side due
to breakdown, while the unburned gas is impinged from the
unburned gas side with a large axial velocity. As a result,
intense combustion proceeds on the interface of the gases to
form an intense reaction zone.
Thus, if we examine this turbulent combustion, which
has turbulent intensity u0 , hot gas propagates upstream along
any vortex axis by vortex bursting with a flame speed Vf if
possible. This flame speed can be considered as the turbulent
burning velocity ST for this combustion. Namely
ST < Vf :
ð43aÞ
As indicated in Figs. 35 and 36, the flame speed Vf is almost
equal to the maximum tangential velocity Vu max in the
moderate range less than 5 –10 m/s, while it is saturated or
slowed down in ranges higher than 5 – 10 m/s. The boundary
velocity depends on the mixture stoichiometry. On the other
hand, the maximum tangential velocity is given to be nearly
equal to u0 ; i.e.
ST < Vf ¼ f ðVu max ; FÞ;
ð43bÞ
Vu max < u0 :
ð43cÞ
This means that the turbulent burning velocity ST, is
increased almost linearly with the turbulent intensity u0 , but
it is saturated for larger values of u0 .
The analogy between the Vf 2 Vu max relation and the
ST 2 u0 relation is unclear, however. Daneshyar and Hill
[25] have attempted to explain the ST 2 u0 relation by
introducing the mechanism of vortex bursting and the
concept of average pressure. Their result, Eq. (11), can
explain the experimental results summarized by AbdelGayed and Bradley [154]. However, the ST 2 u0 relation can
also be explained by different models [155 –157]. Fig. 83
shows an example of how to explain the ST 2 u0 relation
based on flamelet modeling [157]. Although the features of
bending and quenching in the curves are very similar to
those observed in Vf -Vu max plane, they are explained in a
different manner in their paper.
536
Fig. 81. Representation of measured swirl and axial velocity components [116,151].
From the discussion above, understanding the Vf 2
Vu max relation seems to be very important from a
fundamental viewpoint. However, there are still many
problems such as why the flame speed slows in higher
maximum tangential velocity; why the flame diameter
decreases with an increase in maximum tangential velocity;
why the flame is extinguished at a finite flame diameter, etc.
As compared with theories which have been developed for
vortex breakdown in the field of fluid dynamics, the theories
for the flame propagation are undeveloped. A rather
rigorous theoretical study by Umemura et al. [74 – 77,82]
has started very recently. Numerical simulations on this
rapid flame propagation are very few except in the works by
Hasegawa et al. [63,65,78– 80,83]. Further theoretical and
numerical studies should be done to gain complete understanding of this phenomenon. Additional experimental
works with PIV are also indispensable to obtain detailed
information on this phenomenon. From a practical viewpoint, the rapid flame propagation phenomenon should
be applied to practical devises, to control or enhance
Fig. 82. Flame configuration of lean methane/air mixture in a swirltype tubular flame burner (glass tube: 13.4 mm in inner diameter,
120 mm in length, fuel concentration: 5.4%CH4, the mean
tangential velocity at the slit: 3.0 m/s) [37].
Fig. 83. Comparison between experimental data of Abdel-Gayed
and Bradley and KPP results. Equivalence ratio ¼ 0.9 [157].
combustion, or to develop a new, high-compression engine
without knocks. In this regards, recent attempts by
Gorczakowski et al. [81] and Dwyer and Hasegawa [158,
159], both using a rotating vessel or tube with a closed end,
are specially noted.
7. Conclusions
The present article reviews past and recent studies on
rapid flame propagation along a vortex axis. First, a brief
historical survey has been made of related studies in this
subject, followed by reviewing experimental, theoretical,
and numerical studies. Relevant studies on the vortex
breakdown phenomena in swirling flows of constant density
have also been reviewed to discuss the mechanisms of rapid
537
flame propagation along a vortex axis. Basic features of the
phenomenon are summarized below:
(1) Flame shape. The flame is convex towards the
unburned gas. In most flames, the heads are blurred and a
distinct flame zone such as a laminar flame zone is difficult
to identify. However, the head is intensified in burning when
the mass diffusivity of a deficient—hence limiting—
component, is larger than the thermal diffusivity of the
mixture; whereas the head is weakened in burning when the
mass diffusivity is less than the thermal diffusivity.
(2) Limits. Rapid axial flame propagation can occur
when the rotation is adequately strong. Although it has not
yet been made clear, it is probable that the modified
Richardson number needs to be larger than the order of unity
for the occurrence of rapid flame propagation. If this
condition is achieved, a flame can propagate rapidly on a
vortex axis. The concentration limits for the rapid flame
propagation are close to the standard flammability limits of
mixture, or somewhat beyond the flammability limits due to
the Lewis number effect. It should be noted, however, that a
flame was observed to propagate far outside the flammability limit in rich propane/air mixtures in a rotating tube.
This suggests that a flame (hot gas), can propagate along a
vortex axis without any concentration limit, once a flame is
established, and if the aerodynamic condition is satisfied.
Note that the phenomenon of rapid flame propagation along
a vortex axis consists of two processes, the hot gas
movement at a high speed (which is induced aerodynamically), and the combustion which enables the transition from
the unburned gas of high density to the burned gas of low
density. It should also be noted that a numerical simulation
has shown that such rapid flame propagation may not occur
if the vortex diameter is smaller than the order of the laminar
flame thickness.
(3) Steadiness. Flame seldom propagates with a constant
speed in the vortex flow within a tube or in a rotating tube.
Flame acceleration and deceleration frequently occur. A
spiral mode of flame propagation, which presumably
corresponds to the precession of the vortex core, is also
observed. In vortex ring combustion however, a ‘quasisteady’ condition is achieved in the flame propagation. The
flame speed is always varied and the ratio of the square root
of the fluctuations in the flame speed to its mean value
attains about 0.3 in most of vortex rings of various mixtures.
A steady state of flame propagation is limited to vortex rings
of propane/air mixtures in which the Reynolds number is
less than the order of 104; the ratio of fluctuation is then
decreased to 0.2.
(4) Flame speed. The (mean) flame speed is closely
related first to the maximum tangential velocity in the
vortex, and secondly to its flame diameter. With an increase
in the maximum tangential velocity, the mean flame speed is
increased almost linearly while the flame diameter is
decreased monotonically. For higher maximum tangential
velocities, however, the flame diameter becomes smaller
and the flame speed is lowered from the otherwise straight
538
line. If the maximum tangential velocity is further increased,
the flame is quenched on the way to propagation, or the
mixture cannot be ignited at all. In most of the combustible
mixtures, the flame speed slope in the Vf 2 Vu max plane is at
about unity, independent of the equivalence ratio of the
combustible mixture. An exception is the vortex ring
combustion of rich hydrogen mixture in air, in which,
with an increase of the equivalence ratio, the slope is
increased up to the value, the square root of the unburned
to burned gas density ratio, ru =rb ; predicted by Chomiak.
(5) Flame diameter. First, rapid flame propagation along
a vortex axis occurs, followed by slow burning in the radial
direction. Thus, the burning area at the first stage of rapid
flame propagation is limited and the flame diameter is finite.
This flame diameter is decreased with an increase in the
maximum tangential velocity. For the mixture in which
the mass diffusivity of a limiting component is larger than
the thermal diffusivity of the mixture, the flame diameter
can become small, whereas for the mixture in which the
mass diffusivity of a limiting component is smaller than the
thermal diffusivity of the mixture, the flame diameter cannot
become small.
(6) Aerodynamic structure. The propagating flame along
a vortex axis has a peculiar aerodynamic structure; the
pressure is increased behind the flame, which has been
located experimentally by a simple static probe measurement. This is in sharp contrast to the ‘normal’ onedimensional flames, such as a flame propagating in a
quiescent mixture, or flame propagating in a non-rotating
stream, in which the static pressure is decreased behind
the flame. In accordance with the prediction, DP <
ru Vu2 max ; the pressure difference across the flame is
increased in the vortex flow with increasing Vu max ; and its
magnitude is of the order of ru Vu2 max :
(7) Flame propagation mechanism. At this point, four
different mechanisms have been proposed for rapid flame
propagation. They are (i) flame kernel deformation due to
centrifugal effects, (ii) vortex bursting due to pressure
differences across the flame, (iii) baroclinic torque, and
(iv) azimuthal vorticity evolution. The azimuthal vorticity
evolution mechanism can explain the flame driving
process qualitatively, whereas the vortex bursting mechanism, based on the pressure difference across the flame,
can describe the flame speed even quantitatively. Among
the vortex bursting theories postulated so far, the backpressure drive flame propagation theory, which assumes
the momentum flux conservation across the flame, and
does not use the Bernoulli equation on a streamline of
the axis of rotation, can fit the experimental data well.
On the other hand, it has been shown numerically that
the baroclinic torque mechanism plays an important role
only at the initial stage of propagation.
(8) Modeling of turbulent combustion. Recent direct
numerical simulations indicate that such rapid flame
propagation along a vortex axis may not occur in
turbulent combustion. This is because the diameter of
the coherent-fine eddies is much smaller than the
thickness of the laminar flame. A large axial flow also
may work as an inhibiting force to flame propagation.
As noted in the last part of Section 6, further studies are
necessary for complete understanding of the rapid flame
propagation phenomenon along a vortex axis. It has not yet
been made clear why flame ceases to propagate in the radial
direction after the first step of axial propagation. We
continue our work in this subject attempting to build an
integral theory to simultaneously describe the decrease in
flame diameter and the decrease in flame speed. For this
reason, PIV measurements on the velocity components of
radial and axial directions, as well as the tangential velocity
component, will be used for experimental verification.
Studies of the axial flame propagation in swirling flow will
be used to control and/or enhance combustion in practical
devices.
Acknowledgements
I am indebted to Professor Jerzy Chomiak, whose help is
greatly acknowledged, during many helpful discussions and
comments. I also appreciate stimulating discussions with
Professor Tatsuya Hasegawa, Professor Katsuo Asato,
Professor Shiro Taki, Professor Yukio Sakai, Professor
Toshio Miyauchi, Professor Mamoru Tanahashi, and Dr
Andrei N. Lipatnikov. Sincere thanks are offered to
Professor Takashi Niioka for allowing me the opportunity
to write this review, and for his continuing encouragement.
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Flame propagation along a vortex axis

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