Flame propagation along a vortex axis - Hiroshima University · 2007-05-08 · Flame propagation along a vortex axis Satoru Ishizuka* Department of Mechanical Engineering, Hiroshima

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

2. A brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

3. Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

3.1. Appearance and behavior of flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

3.1.1. Flame shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

3.1.2. Steadiness of flame propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

3.1.3. Flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

3.2. Propagation limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

3.2.1. Concentration limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

3.2.2. Aerodynamic limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

3.3. Flame speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

3.3.1. Steadiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

3.3.2. Flame speeds as functions of the maximum tangential velocity. . . . . . . . . . . . . . . . . . . . 495

3.4. Pressure difference across the flame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

3.5. Flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

4. Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

4.1. Flame kernel deformation mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

4.2. Vortex bursting mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

4.2.1. The original theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

4.2.2. The angular momentum conservation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

0360-1285/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 36 0 -1 28 5 (0 2) 00 0 19 -9

Progress in Energy and Combustion Science 28 (2002) 477–542

www.elsevier.com/locate/pecs

* Tel.: þ81-824-24-7563; fax: þ81-824-22-7193.

E-mail address: [email protected] (S. Ishizuka).

http://www.elsevier.com/locate/pecs

4.2.3. A hypothesis based on the pressure difference measurement . . . . . . . . . . . . . . . . . . . . . . 508

4.2.4. A steady state, immiscible stagnant model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

4.2.5. The finite flame diameter approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

4.2.6. The back-pressure drive flame propagation mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 513

4.2.7. A steady-state back-pressure drive flame propagation model . . . . . . . . . . . . . . . . . . . . . 516

4.3. Baroclinic push mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

4.4. Azimuthal vorticity evolution mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

5. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

6.1. Vortex breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

6.2. Flame speeds: summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

6.2.1. Flame speeds for typical flame diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528

6.2.2. Analogy between flows in vortices and gravitational flows . . . . . . . . . . . . . . . . . . . . . . . 529

6.2.3. Flame speeds for finite flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530

6.2.4. A note on Burgers vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

6.2.5. A general comparison between theories and experiments . . . . . . . . . . . . . . . . . . . . . . . . 532

6.2.6. An unresolved problem: finiteness of flame diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

6.3. Modeling turbulent combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

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

Vf ¼ Su

ru

rb

; ð2Þ

in which ru and rb are the densities of the unburned and

burned gases, respectively [4]. In the case of the propagating

flame in a layered mixture, the flame speed is given as [5]

Vf ¼ Su

ffiffiffiffiffiru

rb

r: ð3Þ

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

Vf ¼ Su þ Vu max

ffiffiffiffiffiffiffiffiffiffi1 þ

rb

ru

r: ð4Þ

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

An upward propagating flame near the flammability limits is

controlled by buoyancy and the flame speed is given as

Vf ¼ 0:328ffiffiffiffigD

p; ð6Þ

in which g is the acceleration due to gravity and D is the

internal diameter of the tube [8]. Thus, the propagating

S. Ishizuka / Progress in Energy and Combustion Science 28 (2002) 477–542478

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 propa-

gation along a vortex axis seems to have the following

aspects of interest:

1. Flame propagation. The flame has a different propa-

gation 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], Tabac-

zynski 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 constant-

density 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

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 Combus-

tion 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], hydrodyn-

amic 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

S. Ishizuka / Progress in Energy and Combustion Science 28 (2002) 477–542 479

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 move-

ment 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 pro-

posed 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

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,

Vf ¼ Vu max

ffiffiffiffiffiru

rb

r: ð7Þ

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

DP ¼ ruu02 1 2

rb

ru

� �2" #

< ruu02; ð8Þ

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 rbu2a =2; they obtained an

Fig. 2. Model of burning, turbulent, small-scale structure [14].

Fig. 1. Flame speed versus vortex strength [18].


expression for the axial velocity of the hot gas,

ua < u0

ffiffiffiffiffiffi2ru

rb

s¼ Vu max


rb

s: ð9Þ

In modeling turbulent combustion, they introduced the

concept of average pressure difference, which was about

one-third of DP, resulting in

ua < u0

ffiffiffiffiffiffiffi2

3

ru

rb

s¼ Vu max


3

ru

rb

s: ð10Þ

Finally, they gave the flame propagation speed in turbulent

combustion as

ut ¼ ul þ u0


3

ru

rb

s; ð11Þ

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 12

W2r2f ðru 2 rbÞ; ð12Þ

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

DP ¼ 12ruV2

u max 1 2rb

ru

� �: ð120Þ

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

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 character-

istics 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

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].


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 one-

dimensional 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 numeri-

cally, 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

is approximately given by Ui <ffiffiffiffiffiffiffiffiffiffiffiffi1 2 rb=ru

p; i.e. in the limit

of an unconfined free vortex flow,

Vf < Vu max

ffiffiffiffiffiffiffiffiffiffi1 2

rb

ru

r: ð13Þ

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

dt¼

1

r2

� �7r £ 7P; ð14Þ

where t is time, and the flame is accelerated by this vorticity.

His final expression for the flame speed UB is

UB ¼t

dffiffiffiffiffiffiffi1 þ t

p

� �ðrMV2

MÞ1

SLð1 þ tÞ

� � ffiffiffiffiffiffiffiffiXF=rM

p: ð15Þ

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

r¼

G

2pr2½1 2 expð2r2

=r2MÞ�; 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

2prMVM:

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].


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 circumfer-

ential 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

the value offfiffiffiffiffiffiffiru=rb

pand 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

lVf l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

ðYSuÞ2 þ V2

u maxf ðkÞ

rð17Þ

for the case when the burned gas is expanded only in the

axial direction, and as

lVf l ¼ YSu þ Vu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

rb

ru

f ðkÞ

r; ð18Þ

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

f ðkÞ ¼

12

k2 for k # 1;

1 21

2k2for k $ 1

8><>: : ð19Þ

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].


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 re-

examined 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. 7. Schematic of experimental apparatus for vortex ring combustion [6].

Fig. 8. Laser ignition at the core of one straight vortex [80,83].


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. 9. Schematic of experimental set-up for forced vortex in a rotating tube [66].

Fig. 10. General arrangement for a rotating cylinder [27].


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 high-

speed 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. 11. Rotary combustion chamber arrangement [58].

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].


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 propa-

gation 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 pro-

portional 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, acceler-

ated. 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,

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 $ Oð104Þ; 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/

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].


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

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].


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

[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 cross-

sectional area of four tangential slits are decreased, and

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].

Fig. 16. Schlieren sequence of vortex ring combustion (stoichio-

metric propane/air mixture, orifice diameter Do ¼ 60 mm; driving

pressure P ¼ 0:4 MPa) [71].

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].

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).


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 high-

velocity 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 mechan-

ism 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. 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].

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].


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

›r

›r

W2

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. 21. Photographs of flame propagation for M2 (0.52CH4 þ 2O2 þ 7.52N2) mixture in a rotary combustion chamber [58].

Fig. 22. Process of flame growth for M4 (0.56CH4 þ 2O2 þ 7.52N2)

mixture in a rotating disk ignited at center [58].


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.

Fig. 23. Flame propagation region of methane–air mixtures for

three injectors in the vortex flow [48].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].


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

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

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) oscil-

latory 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 stoichio-

metric 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.

Fig. 26. Limits of flame propagation in vortex ring combustion [53].

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].


3.3.2. 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,

U ¼G

2pDln

8D

d2

1

4

� �; ð21aÞ

Vu max ¼G

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

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].

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].


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. 30. Change in flame speed with maximum tangential velocity

for propane–air mixture [57].

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. 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

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

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. 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.


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 mono-

tonically. 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 back-

pressure 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 hydro-

gen/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, the

pressure behind the flame ðPbÞ is lower than the

pressure ahead of the flame ðPuÞ; by an extent of

ruS2uðru=rb 2 1Þ; which can be easily derived from the


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

solid lines are the relations Vf ¼ Vu max

ffiffiffiffiffiffiffiffiffiffiffi2kIru=rb

p(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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ f ðkÞrb=ru

p; see Eq. (31u)), and for the axial expansion case

(Vf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðru=rbÞðYSuÞ

2 þ f ðkÞV2u max

q; see Eq. (31t)) [70].


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

lines are the relations Vf ¼ Vu max

ffiffiffiffiffiffiffiffiffiffiffi2kIru=rb

p(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


p; see Eq. (31u)), and for the axial expansion case

(Vf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðru=rbÞðYSuÞ

2 þ f ðkÞV2u max

q; see Eq. (31t)) [70].


equations for conservation of mass and momentum across

the flame as,

ruy u ¼ rby b; ð22aÞ

Pu þ ruy2u ¼ Pb þ rby

2b; ð22bÞ

Pb 2 Pu ¼ 2ruS2uðru=rb 2 1Þ: ð22cÞ

Here, the coordinate is attached to the flame, subscripts u

and b denote the unburned and burned gases, respect-

ively; P is the static pressure, y is the velocity, and

Suð¼ y uÞ 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

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

ffiffiffiffiffiffiffiru=rb

p(Eq. (26d)); broken lines are the back-pressure drive flame propagation theory for the lateral expansion case

(Vf ¼ Su þ Vu max


p; k ¼ 2; Eq. (31u)).


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 ~PB (back side), and the pressure on the injector

side by ~PF (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 20.46 mmAq (24.53 Pa). This value is

in good agreement with the present results of

20.46 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. 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


p(Eq. (26d)), broken lines are those by the back-

pressure drive flame propagation theory for the lateral expansion case (Vf ¼ Su þ Vu max

ffiffiffiffiffiffiffiffiffiffiffiffi1 þ rb=ru

p; i.e. Y ¼ 1 and k !1; see Eq. (31u)); solid

lines and curves are those by the steady-state flame propagation model, Vf <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2u max þ ruS2

u=rb

qand Vf ¼ Vu max

ffiffiffiffiffiffiffiffiffiffiffiffi1 þ rb=ru

p(Eqs. (33e) and

(33f)). For comparison, the results in the air atmosphere are shown by open symbols [72].

Fig. 39. Change in flame speed with the maximum rotational

velocity [73].


between two static probes,

Pb 2 Pu ¼ð dp

dzdz ø

ð dP

‘ðVf dtÞ ¼

Vf

‘

ððdPÞdt: ð23Þ

The obtained value was 20.80 mmAq, which was 70%

greater than the value of 20.46 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. 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].

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].


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 stoichio-

metric mixture. The broken curves indicate the amount of

ruW2max; 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

ruV2u max{1 2 ðrb=ruÞ

2} (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.

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].


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.

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:

1. Flame kernel deformation mechanism;

2. Vortex bursting mechanism;

3. Baroclinic push mechanism;

4. 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

dt< Sw: ð24aÞ

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].


This results in

D2 dH

dtþ 2DH

dD

dt< 2ðD2 þ DHÞw 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

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].


that

d

dt

D

2

� �ø

w

2

d

dt

H

2

� �ø

w

2ðvtÞ

H

Døvt: 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 12

W2r2f ðru 2 rbÞ: 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 W2: 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 relationðAðp 2 p0ÞdA ¼

ðArby

2 dA; ð26aÞ

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

Fig. 45. Schemes of flame spreading (i) beneath a plane ceiling and (ii) along the rotating cylinder [20].


form, and by ignoring the axial and radial velocities, the

pressure distribution in the vortex can be obtained from the

momentum equation

1

ru

›p

›r¼

v2

r: ð26bÞ

Then, integration gives

p 2 p0 ¼ ru

ð1

r

v2

rdr ¼ ruV2

u max: ð26cÞ

Consequently, y is given as

y ¼ Vu max

ffiffiffiffiffiru

rb

r: ð26dÞ

4.2.2. The angular momentum conservation model

Daneshyar and Hill [25] have given a detailed expla-

nation 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:

hb

hu

¼

ffiffiffiffiffiru

rb

r;

Vb

Vu

¼hu

hb

� �2

¼rb

ru

: ð27aÞ

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 Pbð0Þ; and that in the unburned gas, P1 2

Puð0Þ; are obtained. Thus, on the axis of rotation, the

pressure difference across the flame DP is obtained as

DP ; Pbð0Þ2 Puð0Þ ¼ ruV2u max 1 2

rb

ru

� �2" #

< ruV2u max:

ð27bÞ

This pressure difference would set-up a large axial velocity

ua of the burned gas into the unburned region. Then, a

relation

DP < ruV2u max < 1

2rbu2

a ð27cÞ

gives the magnitude of this velocity ua as

ua < Vu max


rb

s: 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

ruV2u max

¼3

16þ

ln 2

4<

1

3: 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

ua < Vu max


3rb

s: ð27fÞ

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

Fig. 46. Flame ‘penciling’ effect in a forced vortex in a rotating

vessel [27].


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,

12ruV2

f ø DPe < k2ruV2u 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,

Vf < Vu max

ffiffiffiffiffi2k2

p: ð28bÞ

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

Fig. 47. Vortex bursting mechanism proposed by Chomiak [9,50].


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:ðru dA ¼ C1: ð29aÞ

2. The angular momentum conservation on a streamline:

ry ¼ const: ð29bÞ

3. The radial momentum equation, assuming no radial and

axial velocities

›P

›r¼ r

y 2

r: ð29cÞ

4. Bernoulli’s theorem on stream lines, OE (axis of

rotation) and AC (wall):

P

rþ

1

2u2 ¼ H: ð29dÞ

5. Treatment of the lighter liquid (burned gas) as a stagnant

gas (stationary obstacle):

PO 0 ¼ PE: ð29eÞ

6. The conservation of momentum at AA0 and CC0:ðru2 dA þ

ðP dA ¼ C2: ð29fÞ

From Eq. (29a), the following relation can be obtained:

u1 ¼ cð1 2 x2Þ; ð29gÞ

in which

x ; R1=R2: 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:

y ¼v

1 2 x2r 2

R21

r

!: ð29iÞ

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.

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].


In the burned gas:

y ¼1

x2vr: 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 12ruv

2R22 þ P0: ð29kÞ

Between C and L0:

PI ¼ PC 2 ru

v2

ð1 2 x2Þ21

2R2

2 2 2R21ln

R2

R1

21

2

R41

R22

!:

ð29lÞ

Between L0 and O0:

PI ¼1

2x4rbv

2R21 þ PO0 : ð29mÞ

From Eq. (29d), the following relations are obtained:

Streamline OE:

PO þ 12ruu2

u ¼ PE: ð29nÞ

Streamline AC:

PA þ 12ruu2

1 ¼ PC þ 12ruc2

: ð29oÞ

By matching the pressures at the interface L0 obtained

through a route OACI and obtained through a route OEO0L0;

an expression for c2 is obtained:

c2 ¼ 2v2

ð12 x2Þ2R2

2 x4 2 x2 2 2x2 ln x21

2x2ð12 x2Þ2

rb

ru

� �:

ð29pÞ

On the other hand, using a relation Eq. (29f), an expression

for c2 is obtained:

c2 ¼ 2v2

ð12 x2Þ21

ð12 x4Þ

R22

3

2x4 2

3

2x2 2 2x2 ln x2 x4 ln x

�

þ1

4

rb

ru

ð12 x2Þ2 122

x2

� ��: ð29qÞ

By equalizing these two equations, Eqs. (29p) and (29q), for

c2; an equation for x can be obtained:

1

22

1

2x2 þ x4 2 x6 þ x2 ln xþ 2x4 ln x

21

4x2

rb

ru

ð12 x2Þ2ð12 2x2Þ ¼ 0: ð29rÞ

This equation can be solved numerically and the value of

xð; R1=R2Þ 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 u2Þ 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 stoichio-

metric mixture. Fig. 51 [62] shows the non-dimensional

velocities of U1ð; u1=Vu maxÞ and U2ð; u2=Vu maxÞ: 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=R2Þ:

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

U1 ø ð1 2 x2Þ


rb

ru

r<


rb

ru

r: ð29sÞ

Fig. 50. Variation of x with density ratio for the forced vortex [62].


Thus, the slope in the Vf 2 Vu max plane becomes much

smaller than the value offfiffiffiffiffiffiffiru=rb

pof 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

with the pressure distributions of the Rankine’s combined

vortex for the unburned and burned gases

Pu ¼

P1 2 ruv2a2 þ 1

2ruv

2r2 ðr # aÞ;

P1 21

2

ruv2a4

r2ðr $ aÞ

8><>: ; ð30aÞ

Pb ¼

P1 2 rbv2a2 þ 1

2rbv

2r2 ðr # aÞ;

P1 21

2

rbv2a4

r2ðr $ aÞ

8><>: : ð30bÞ

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:

�Pu ¼1

pð2aÞ2

ð2a

0Pu2pr dr

� �; ð30cÞ

Fig. 53. Variation of axial velocities with a for the free vortex

(density ratio ¼ 1/7) [62].Fig. 51. Variation of axial velocities with density ratio for the forced

vortex [62].

Fig. 52. Variation of x with a for the free vortex [62].

Fig. 54. Variation of axial velocities with density ratio for the free

vortex ða ¼ 0:05Þ [62].


�Pb ¼1

pð2aÞ2

ða

0Pb2pr dr þ

ð2a

aPu2pr dr

� �: 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 beðAð �Pb 2 �PuÞdA ¼

ðArbV2

fth dA: ð30eÞ

Finally, the flame speeds are obtained as follows:

Vfth ¼

a

2aVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

2 1

� �1 2

a2

4a2

!vuut ða # aÞ;

1

2Vu max


rb

2 1

� �3

4þ ln

a

a

� �sða . aÞ

8>>>>><>>>>>:

: ð30fÞ

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 over-

estimates 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].

Fig. 55. The finite flame diameter model by Asato et al. [57].


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

2V2 þ

ð dP

r¼ const: 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 back-

pressure 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,

Puð0Þ þ ruV2u ¼ Pbð0Þ þ rbV2

b : 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, PuðrÞ and PbðrÞ; 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 h0

u=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 h0

b=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 # R0

u: For each of the three regions, we consider

mass continuity and angular momentum conservation, i.e.

1. Mass continuity:

ðruðVu 2 VfÞ2pr dr ¼

ðrbðVb 2 VfÞ2pr dr: ð31dÞ

2. Angular momentum conservation:

ðruðVu 2 VfÞVuu2pr2 dr

¼ðrbðVb 2 VfÞVub2pr2 dr ð31eÞ

(in which the tangential velocity distributions are given

as follows):

Vuu ¼Vur ðr # hu=2Þ;

Vuh2u=4r ðr . hu=2Þ;

(ð31fÞ

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].


Vub ¼

Vbr ðr # rbÞ;

V0ur ðrb # r # h0

u=2Þ;

V0uh

02u =4r ðh

02u = , rÞ:

8>><>>: ð31gÞ

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:

Puð0Þ þ ruV2u ¼ Pbð0Þ þ rbV2

b : 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=rbÞðVu 2 VfÞ=12r ; ð31hÞ

Vb ¼ V0u ¼ Vu=1

2r ; ð31iÞ

h0u=2 ¼ ðhu=2Þ1r; ð31jÞ

V 0u 2 Vf ¼ ðVu 2 VfÞ=1

2r ; ð31kÞ

R0u ¼ Ru1r: ð31lÞ

From the mass continuity for Region I,

Vu ¼ YSu þ Vf ; ð31mÞ

in which Y is a ratio of the flame area A to the cross-

sectional area of the unburned mixture,

Y ; A=ðpr2uÞ: ð31nÞ

By putting the above equation into Eq. (31h), we obtain

the relation

Vb ¼YSuru

rb12r

þ Vf : ð31oÞ

By substituting Eqs. (31m) and (31o) into Eq. (31b), a

quadratic equation is obtained for Vf : Its solution is

obtained as follows:

Vf ¼ 2ruSuY

ru 2 rb

1 21

12r

� �2

1

ru 2 rb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirurbðYSuÞ

2 1 2ru

rb

1

12r

� �2

þðru 2 rbÞDP

s: ð31pÞ

Here, DP i s the pressure rise behind the flame

DP ; Puð0Þ2 Pbð0Þ; which is obtained by integrating the

radial momentum equation ›P=›r ¼ rV2u =r; with the tangen-

tial 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:

DP ¼ ruV2u max 1 2

1

12r

1 þrb

ru

2 1

� �f ðkÞ

� �� : ð31qÞ

Here k is a measure of the burning region normalized by the

core radius,

k ;ru

hu=2; ð31rÞ

and the function f ðkÞ is given as

f ðkÞ ¼ 12

k2 ðk # 1Þ; 1 21

2k2ðk $ 1Þ: ð31sÞ

The solutions are simplified for two extreme cases, an axial

expansion case ð1r ¼ 1Þ and a radial expansion case ð1r ¼ffiffiffiffiffiffiffiru=rb

pÞ; given as follows:

lVf l ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

ðYSuÞ2 þ V2

u maxf ðkÞ

r

ðaxial expansion; 1r ¼ 1Þ;

ð31tÞ

lVf l ¼ YSu þ Vu max


rb

ru

f ðkÞ

r

ðradial expansion; 1r ¼ffiffiffiffiffiffiffiru=rb

pÞ:

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 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ V2

u max

r

ðaxial expansion; 1r ¼ 1Þ;

ð31vÞ

lVf l ¼ Su þ Vu max


rb

ru

r

ðradial expansion; 1r ¼ffiffiffiffiffiffiffiru=rb

pÞ:

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 ry ¼ const: on each

streamline. If this correction is made, the flame speed for the


radial expansion in the range 0 # k # 1 is given as:

lVf l ¼ YSu

þ kVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

rb

2ru

þru

rb

ln 1 2rb

ru

1 21

k2

� �� s:

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 back-

pressure drive flame propagation theory gives the flame

speed as

lVf l ø kVu max=ffiffi2

pðaxial expansionÞ; ð31yÞ

lVf l ø kVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffi22

ru

rb

ln k

r

ðradial expansion : k # 0:01Þ:

ð31zÞ

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 communi-

cation [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,

~Vu ¼ Vu 2 U and ~Vb ¼ Vb 2 U: ð32aÞ

We must obtain

~Vf 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 ruÞU2 þ 2Uðrb

~Vb 2 ruV~0uÞ ð32cÞ

arise on the right side of Eq. (31b). Thus, ~Vf 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 ruSu=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=4ð0Þ; is very low due to a centrifugal force

of rotation, while the pressure just ahead of the flame Pu;02ð0Þ

becomes higher than Pu;2pD=4ð0Þ; because the axial velocity

is decreased from Vf to Su. In the burned gas side, the pressure

just behind the flame Pb;0þð0Þ; is higher or lower than the

pressure downstream Pb;þpD=4ð0Þ; depending on whether

Vf $ ruSu=rb or Vf # ruSu=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

rb

Su;

Pu;2pD=4ð0Þ þ ruV2f ¼ Pb;0þð0Þ þ rb

ru

rb

Su

� �2

;

ð33aÞ

ðiiÞ if Vf #ru

rb

Su;

Pu;2pD=4ð0Þ þ ruV2f ¼ Pb;þpD=4ð0Þ þ rbV2

f :

ð33bÞ


Assuming angular momentum conservation, the pressure

differences are given as follows:

Pb;0þð0Þ2 Pu;2pD=4ð0Þ

¼ ruV2u max 1 2

Su

Vf

1 2 1 2rb

ru

� �f ðkÞ

� �� ; ð33cÞ

Pb;þpD=4ð0Þ2 Pu;2pD=4ð0Þ

¼ ruV2u max 1 2

rb

ru

� �1 þ

rb

ru

f ðkÞ

� �: ð33dÞ

These equations are obtained by putting 1r ¼ffiffiffiffiffiffiffiVf =Su

pandffiffiffiffiffiffiffi

ru=rb

pinto 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:

ðiÞ if Vf $ ruSu=rb; Vf <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ V2

u max

r; ð33eÞ

ðiiÞ if Vf # ruSu=rb; Vf ¼ Vu max


rb

ru

f ðkÞ

r: ð33fÞ

Vf is given as a solution of the third order algebraic equation

for Vf $ ruSu=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 ruSu=rb:

The solution, which consists of the solid line for Vf #

ruSu=rb; and the solid curve for Vf $ ruSu=rb in Fig. 38, is

Fig. 58. Steady-state model for the flame propagation in a vortex ring [72].


almost proportional to the maximum tangential velocity,

and hence similar to the solution obtained by the back-

pressure 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 one-

dimensional 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=2ÞruS2uðru=rb 2 1Þð3ru=rb 2 1Þ: In this case the burned

gas is completely at rest and the flame speed is increased to

ruSu=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

combustion. If we denote this pressure rise by D ~P; D ~P

may be proportional to ruS2u: 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 D ~P < lruV2u max; where l is an arbitrary constant. By

adding the pressure rise D ~P to the original pressure

difference across the flame, the final flame speed is obtained

from Eqs. (33a) and (33b) as

ðiÞ if Vf $ ruSu=rb;

Vf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ ð1 þ lÞV2

u max

r;

ð34aÞ

ðiiÞ if Vf # ruSu=rb;

Vf ¼ Vu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ

rb

ru

f ðkÞ þru

ru 2 rb

l

r:

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

›t¼ 2ðu·7Þvþ ðv·7Þu 2 vð7·uÞ2 7

1

r

� �£ 7p

þ 71

r

� �£ 7·tþ

1

r

� �7 £ 7·t : ð35aÞ

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

dt¼

7r £ 7P

r2: ð35bÞ

Ashurst assumes that the tangential velocity distribution has

a form of

Vu

r¼

G

2pr2½1 2 expð2r2

=r2MÞ�: ð35cÞFig. 59. Pressure distribution in the flame propagation in a soap

bubble [99].


He then roughly evaluates the pressure gradient and the

density gradient as follows:

1

r7P ¼

V2u

r;

1

r7r ¼

ru 2 rb

dffiffiffiffiffiffirurb

p ¼t

dffiffiffiffiffiffiffitþ 1

p ; 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

dt,

4:5t


p rMV2M; 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

UB ,t


p rMV2M

1

SLðtþ 1Þ

ffiffiffiffiffiffiffiffiXF=rM

p: ð35fÞ

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

ðtþ 1Þ3=29rMG

dSLTpk2; 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

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.

4.4. Azimuthal vorticity evolution mechanism

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

Fig. 61. Azimuthal vorticity generation mechanism [76].

Fig. 60. Baroclinic push mechanism by Ashurst [64].


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 : ruSL ¼ rbSb ¼ srbSL; ð36aÞ

the unburned gas : SLA ¼ pa2W ; ð36bÞ

the burned gas : sSLA ¼ pa02W : ð36cÞ

These equations lead to a relation:

a02 ¼ sa2

: ð36dÞ

2. Bernoulli equation for the unburned gas between points

A and O2:

PA þ 12ruW2 ¼ PO2

þ 12ruS2

L: ð36eÞ

3. Momentum flux conservation across the flame, i.e.

between Oþ (just ahead the flame) and O2 (just behind

the flame):

PO2þ ruS2

L ¼ POþþ rbðsSLÞ

2: ð36fÞ

4. Bernoulli’s equation for the burned gas between points

Oþ and D:

POþþ 1

2rbðsSLÞ

2 ¼ PD þ 12rbW2

: ð36gÞ

5. Equal pressure at points B and C:

PB ¼ PC : ð36hÞ

6. Tangential velocity distribution of Rankine’s form for

the unburned mixture:

y ¼

Vr for r # a;

Va2

rfor r $ a

8><>: : ð36iÞ

7. Angular momentum conservation on each streamline:

ry ¼ const: ð36jÞ

8. Momentum equation for radial direction:

›P

›r¼ r

y 2

r; ð36kÞ

which yields relations for the pressure difference in the

radial direction:

PB 2 PA ¼ ruV2a2

; ð36lÞ

PC 2 PD ¼1

2rbV

2a2 1 þ1

s

� �: ð36mÞ

Along the path ABCD, the following relation can be

obtained with the use of Eqs. (36l) and (36m):

PD 2 PA ¼1

2ruV

2a2 1 21

s

� �2 þ

1

s

� �: ð36nÞ

On the other hand, along the axis of rotation, AD, the

following relation can be obtained with the use of Eqs.

(36e)–(36g):

PD 2 PA ¼1

2ruW2 1 2

1

s

� �2

1

2ðs2 1ÞruS2

L: ð36oÞ

By equating Eqs. (36n) and (36o), an expression for the

flame speed is obtained as:

Vf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sþ 1

sðVaÞ2 þ sS2

L

r: ð36pÞ

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,

y ¼Vb

02

r

1

s2 1

� �þVr ¼ V r 2

ðs2 1Þf2a2

r

" #ð36qÞ

the final expression is given as follows:

Vf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sþ 1

sþ 2s ln

1 þ ðs2 1Þf2

sf2

" #f2ðVaÞ2 þ sS2

L

vuut :

ð36rÞ

These results resemble the results obtained by the back-

pressure drive flame propagation mechanism. That is,

Fig. 62. Model flow configuration in the azimuthal vorticity

generation mechanism [74,77].


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


p: For large values of Vu max; the flame speed

is almost proportional to the maximum tangential velocity,

although the slope in the Vf 2 Vu max plane is aboutffiffi2

p: This

is because the Bernoulli equation is used in the unburned gas

as well as the burned gas, although a momentum conserva-

tion 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 mech-

anism 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

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

VðrÞ ¼G0

2pr1 2 exp 2

r2

ðs0=2Þ2

! !; ð37aÞ

G0 ¼ ps0Vm0; 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 ; s0Vm0=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 circumfer-

ential 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

Fig. 63. Definition of effective vortex tube radius [77].


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


pwith 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. 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].


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].

6. Discussion

6.1. Vortex breakdown

As first pointed out by Chomiak [9], and very recently

emphasized by Umemura and Tomita [75], the phenom-

enon 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. 69. Relation between flame propagation velocity and the

maximum circumferential velocity [65].

Fig. 70. Contribution of each term for generation of the azimuthal

vorticity at the initial stage at 31 and 155 ms after ignition [83].

Fig. 67. Relation between maximum circumferential and propa-

gation 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].

Fig. 68. Relation between the Reynolds number of a vortex tube

RG ¼ Vms=v and proportionality factor of flame propagation

velocity in a vortex tube uV=Vm [63].


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 Lucca-

Negro 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 photo-

graph 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]. How-

ever, 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 combus-

tion. The burned gas expands through the density change,

causing an expansion of the vortex core. If the flow is a non-

rotating, one-dimensional flow, this expansion is restricted

in one direction; hence, it causes a pressure drop behind the

flame by an amount equal to 2ruS2uðru=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 constant-

density 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 Vxðr; xÞ; and the azimuthal velocity

Vuðr; xÞ; are measured with two-component optical fiber

laser Doppler anemometry in the backscatter mode. In this

experiment, the swirl parameters, defined as

S ;2VuðR=2; x0Þ

Vxð0; x0Þð38aÞ

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

VuðR=2; x0Þ is measured at half the radius of the nozzle exit

r ¼ R=2; and this azimuthal velocity VuðR=2; x0Þ; 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 ; 2R �Vxðx0Þ=v;where �Vxðx0Þ 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

Vxð0; xÞ øffiffi2

pVuðR=2; x0Þ: 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 been obtained for the onset of vortex

breakdown. That is, Sc øffiffi2

p:

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

Fig. 71. Sketch of experimental apparatus of a rotating system and a

nozzle for vortex breakdown [126].


defined as

Ro ;W

rpV: ð38cÞ

Here, W, rp and V represent a characteristic velocity,

length and rotation rate, respectively [129]. Usually, rp is

defined as the radial distance at which the swirl velocity is a

maximum, W is given as the axial velocity at rp; 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

Vxðrp; x0Þ at rp

Ro ;Vxðr

p; x0Þ

rpV: ð38dÞ

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.

If we assume that rpV is equal to Vu max and include the

relation W øffiffi2

pVu max in Eq. (38c) or (38d), the Rossby

number should be equal toffiffi2

p: Thus, the critical Rossby

numbers experimentally obtained are not consistent with the

critical swirl number of 1.4, and smaller than the

corresponding Rossby number offfiffi2

p: It should be noted

however, that the axial velocity is defined at rpð– 0Þ; not on

the axis of rotation.

If the Rossby number is defined using the axial velocity

on the centerline [117], the critical number offfiffi2

phas 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=rþ ðV2x þ V2

r þ V2u Þ=2 leads

H ¼P0

rþ

V2x ð0; x0Þ

2¼

P1

r; 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, Vxð0; x0Þ 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;

Fig. 73. Schematic configuration of cone vortex breakdown [126].

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].


consequently,

P0 ¼ P1 2ð1

0r

V2u ðr; x0Þ

rdr; ð38fÞ

where Vuðr; x0Þ 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 relationð1

0

V2u ðr; x0Þ

rdr ¼ 1

2V2

x ð0; x0Þ 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

Vxð0; x0Þ ¼ffiffi2

pVu maxðx0Þ: ð38hÞ

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) becomesð1

0

V2u ðr; x0Þ

rdr $

1

2V2

x ð0; x0Þ: ð38iÞ

For Rankine’s combined vortex, this inequality becomes

Vxð0; x0Þ #ffiffi2

pVu maxðx0Þ: ð38jÞ

If the results of Eq. (38h) are applied to the case of the

propagating flame in a vortex, the flame speed reachesffiffi2

pVu 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, the flame speed may become less than the

value offfiffi2

pVu 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, incorpor-

ates 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

FðP þ rw2ÞdF ¼ 2p

ðR

0ðP þ rw2Þr dr ð39aÞ

a swirl parameter k, which is defined as

k ;G1

pdW¼

2vd

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

k ¼ffiffi2

pfor d=R ! 0 ðfree vortexÞ;

k ¼ 3:832 for d=R ! 1 ðforced vortexÞ

8<: : ð39cÞ

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

ffiffi2

pVu max for a free vortex;

0:52Vu max for a forced vortex

8<: : ð39dÞ

Note that the flame speed reachesffiffi2

pVu 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

ffiffiffiffiffiffiffiffiffiffiffiffi1 2 rb=ru

pVu max for a free vortex

ðR2 !1; a! 0Þ;

0:25Vu max for a forced vortex

ðrb=ru ¼ 1=5–1=7Þ

8>>>>><>>>>>:

: ð40Þ

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 of free vortex, the value of slope

in the Vf 2 Vu max plane isffiffi2

pfor the vortex breakdown

(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 y0 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 recircula-

tion 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 non-

physical 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].


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 21.84 to 21.37,

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 high-

pressure 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 recent experimental result in which the axial

velocity becomesffiffi2

ptimes 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]:

Vf ¼ Vu max

ffiffiffiffiffiru

rb

r: ð26dÞ

2. The angular momentum conservation model by Dane-

shyar and Hill [25]:

Vf ¼ Vu max


rb

s: ð27dÞ

3. A hypothesis by Ishizuka and Hirano [50]:

Vf ¼ Vu max

ffiffiffiffiffi2k2

pðk2 # 1Þ: ð28bÞ

4. A steady-state immiscible stagnant model by Atobiloye

and Britter [62]:

Vf ¼ Vu max


rb

ru

rðx ; R1=R2 ! 0; a! 0Þ: ð29s0Þ

5. The finite flame diameter approximation by Asato et al.

[57]:

Vf ¼ Vu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

16

ru

rb

2 1

� �sða=a ¼ 1Þ; ð30g0Þ

Vf ø 1:07Vu max ða=a ¼ 1; ru=rb ¼ 7Þ: ð30g00Þ

6. The back-pressure drive flame propagation model by

Ishizuka et al. [69]:

axial expansion : Vf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ V2

u max

r

ðk !1; Y ¼ 1Þ;

ð31t0Þ

radial expansion : Vf ¼ Su þ Vu max


rb

ru

r

ðk !1; Y ¼ 1Þ:

ð31u0Þ

7. A steady-state back pressure drive flame propagation

mechanism [72]:

Vf øffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ V2

u max

rfor Vf $ ruSu=rb; ð33eÞ

Vf ¼ Vu max


rb

ru

rfor Vf # ruSu=rb: ð33fÞ

8. The baroclinic push mechanism by Ashrust [64]: Eq.

(35f) is rewritten in terms of ru, rb, Vu max and Su as

Vf < 1 2rb

ru

� � ffiffiffiffiffirb

ru

rrMV2

u max

dSu

ffiffiffiffiffiXF

rM

s

ðstraight vortexÞ:

ð35f 0Þ

9. The azimuthal vorticity evolution mechanism by Ume-

mura and Tomita [74,77]: Eq. (36p) is rewritten in terms

of ru, rb, Vu max and Su as

Vf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

S2u þ 2 þ

rb

ru

� �V2u max

s: ð36p0Þ

In accordance with the experimental result of vortex

breakdown in constant-density flow, the theory by Ume-

mura and Tomita [74,77] gives the proportionality offfiffi2

pin

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 12

ffiffiffiffigd

p; 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

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

uF ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbðr2 2 r1Þ

ðr2 þ r1Þ

s; ð42bÞ

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].


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

U ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gb

r2 2 r1

r1

r: ð42cÞ

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

uF ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbðr2 2 r1Þ

r2

s: ð42dÞ

If r1 ¼ 0 (cavity) is placed into Eq. (42b), the upper wedge

speed of the lighter fluid (cavity) becomes

uF ¼ffiffiffigb

p¼

1ffiffi2

pffiffiffiffiffiffiffigð2bÞ

p: ð42eÞ

This speed is different from the value of Eq. (42a) by a factor

offfiffi2

p; 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 density ratio r2=r1 is increased, whereas speed

approachesffiffiffigb

pin 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 ruV2u max{1 2 ðrb=ruÞ

2} (Eq. (27b)) or

approximately by ruV2u 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 rule) is used,

the flame speed is increased by a factor offfiffi2

p(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

Vf < Vu maxð1 2 x2Þ


rb

ru

rða! 0Þ: ð29s00Þ

2. The finite flame diameter approximation by Asato et al.

[57]:

Vfth ¼

a

2aVu max


rb

2 1

� �1 2

a2

4a2

!vuut ða # aÞ

1

2Vu max


rb

2 1

� �3

4þ ln

a

a

� �sða . aÞ

8>>>>><>>>>>:

:

ð30fÞ

3. The back-pressure drive flame propagation model by

Ishizuka et al. [69]:

axialexpansion : lVf l¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiru

rb

ðYSuÞ2þV2

u max f ðkÞ

r; ð31tÞ

radialexpansion : lVf l¼YSuþVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

rb

ru

f ðkÞ

r: ð31uÞ

Eq. (31u) is modified for angular momentum con-

servation on each streamline:

lVf l¼YSu

þkVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

rb

2ru

þru

rb

ln 12rb

ru

121

k2

� �� s

fork#1:

ð31xÞ

4. The steady-state model of the back pressure drive

flame propagation theory [72]

lVf l¼Vu max

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

rb

ru

f ðkÞ

rðVf #ruSu=rbÞ: ð33fÞ


Eq. (33f) is modified for angular momentum conserva-

tion on each streamline:

lVf l¼kVu max

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

rb

2ru

þru

rb

ln 12rb

ru

121

k2

� �� s: ð33gÞ

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

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ

rb

ru

þ2ru

rb

ln 12rb

ru

121

k2

� �� V2u max þ

ru

rb

S2u

s:

ð36r0Þ

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 in the axial expansion),

that the asymptotic value of YSu


pin 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

ffiffiffiffiffiffiffiffiffiffiffiffi1 2 rb=ru

p

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; 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

y ¼C

2prð1 2 e2Ar2=2vÞ: 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 ; limr!0

y

r¼

AC

4pv¼

AG1

4pv: ð43eÞ

Note that for a given G1ð– 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

Ar2

2v¼ j ø 1:2565; 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

rc ¼ffiffij

p ffiffiffiffiffi2v

A

rø 1:12

ffiffiffiffiffi2v

A

r¼ 1:12

ffiffiffiffiffiffiffiffiffiG1

2p

1

V

s; ð43gÞ

Vu max ¼G1

2p

ffiffiffiffiffiA

2v

r1 2 e2jffiffi

jp ø 0:638

G1

2p

ffiffiffiffiffiA

2v

r

¼ 0:638

ffiffiffiffiffiffiffiffiG1

2pV

s: ð43hÞ

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 1:70ruV2u max: ð43iÞ

In Rankine’s combined vortex, the pressure difference is

equal to ruV2u 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;


in the back-pressure drive flame propagation mechanism,

the slope may be increased from unity toffiffiffiffi1:7

pø 1:3; and in

the theory by Umemura and Tomita, the slope is also

increased fromffiffi2

pto

ffiffiffiffiffiffiffiffiffiffi2 £ 1:70

pø 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 stoichio-

metric 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

du


p: The parameter k in Eqs. (33g) and (36r0) is

estimated from a relation

k ; du=dc ¼ ðdf =dcÞffiffiffiffiffiffiffirb=ru

p: 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

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].


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

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 Tabac-

zynski 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


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.

In Region B (Fig. 80(c)), the axis of the coherent fine-

scale 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 fine-

scale 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 exper-

iments 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

Fig. 80. Distributions of the heat release rate and axes of the typical

coherent fine-scale eddy. (a) A region ðDHp ¼ 1:1Þ; (b) A region

(axial velocity), (c) B region ðDHp ¼ 1:1Þ; and (d) C region

ðDHp ¼ 1:0Þ [145].

Fig. 78. Contour surfaces of the second invariant and heat release

rate (green: Qþ ¼ 0:02; yellow: DHp ¼ 1:0) [145].

Fig. 79. Distribution of the axes of coherent fine-scale eddies in a

turbulent premixed flame (gray: axes, yellow: DHp ¼ 1:0) [145].


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 wake-

like 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

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 combus-

tion 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 Abdel-

Gayed 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.


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 under-

standing of this phenomenon. Additional experimental

works with PIV are also indispensable to obtain detailed

information on this phenomenon. From a practical view-

point, the rapid flame propagation phenomenon should

be applied to practical devises, to control or enhance

Fig. 81. Representation of measured swirl and axial velocity components [116,151].


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

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 flamm-

ability 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 aerodynami-

cally), 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 ‘quasi-

steady’ 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

Fig. 82. Flame configuration of lean methane/air mixture in a swirl-

type 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].


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,ffiffiffiffiffiffiffiru=rb

p; 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 measure-

ment. This is in sharp contrast to the ‘normal’ one-

dimensional 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 <ruV2

u 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 ruV2u 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 mech-

anism, based on the pressure difference across the flame,

can describe the flame speed even quantitatively. Among

the vortex bursting theories postulated so far, the back-

pressure 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 - Hiroshima University · 2007-05-08 · Flame propagation along a vortex axis Satoru Ishizuka* Department of Mechanical Engineering, Hiroshima