J.W. Dold, O.S. Kerr and I.P. Nikolova- Flame Propagation through Periodic Vortices

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Flame Propagation through Periodic Vortices

J . W. DOLD* an d 0. S. KERR

School of Mathematics, University of Bristol, Bristol BS8 1 m U.K.

I. P. NIKOLOVA

Inst itut e of Mechanics and Biomechanics, 4 Bonchev Str., Sofia, Bulgaria

The discovery of a new class of Navier-Stokes solutions representing steady periodic stretched vortices [l]

offers a useful test-bed for examining interactions between flames and complex flow-fields. After briefly

describing these vortex solutions an d their wide-ranging parameterisation in terms of wavelength and

amplitude, this article examines their effect on flames of constant normal prop agation speed as observed

through numerical solutions of an eikonal equation. Over certain ranges of vortex amplitude and flame-speed,

a corridor of enhanced flame passage is seen to be created as a leading flame-tip manages to leap-frog

between successive vortices. H owever, for large enough amplitudes of vorticity or small enough flame-speeds,

the flame fails to be able to benefit from the advection due to the vortices. It is shown that the leading tips of

such flames are effectively trapped by the stretched vortices.

INTRODUCTION

Am ongst other things, understanding the man-

ner in which flames interact with complex flow

fields is central to developing a well-found ed

theory of turbulent flame propagation. A nec-

essary component in developing any such un-

derstanding mus t lie with the nature of the

flow-field itself. In true turbulence, of course,

this is a very complicated and still poorly un-

derstood matter. However, a useful step for-

ward can be taken by studying selected flow-

fields, some more suitable than others, in order

to provide test cases for examining the re-

sponse of flames to known flows. The more

firmly such a test flow is rooted in the three

dimensional Navier-Stokes equations, the

more likely it is to have features in common

with real turbulence. In this spirit, Marble [2]

proposed a test problem consisting of a two-

dimensional decaying vortex in which initialflame and mixture p roperties could be speci-

fied as startup conditions (intended primarily

for non-premixed combustion), and this flow-

field has led to a number of studies of flame

interactions with diffusing vortices [3, 41.

*Corresponding author.

Presented at the Twenty-Fifth Symp osium (International)

on Com bustion, Irvine, California, 3 1 July-5 Augus t 199 4.

Ashu rst and Sivashinsky [5] employed an even

simpler test flow involving square cells of sinu-

soidal velocity variation, which do not satisfy

the Navier-Stokes equations.

The recent discovery of a new class of

Navier-Stokes solutions [l] provides another

potentially useful test-bed for the study of

flame and flow interactions, offering a number

of distinctive and useful features. These solu-

tions consist of steady periodic distributions of

vorticity, arising in a three-dimensional stagna-

tion-point flow field containing a unique direc-

tion of stretch and a unique direction of con-

verging flow from infinity. All of the vorticity

becomes aligned with the direction of stretch

and solutions are found in which the vorticity

becomes periodic in the third direction.

For a given stretch-rate, solutions with any

amplitude of vorticity distribution are found

for wavelengths of the distribution above a

critical wavelength. This results in a two-parameter family of solutions, depending on

both wavelength and amplitude. For increased

wavelengths the regions of strongest vorticity

become m ore and more separated; increased

amplitudes lead to more an d more rap id rota-

tion of fluid elements as they are eventually

sucked into a vortex and ejected along its axis.

The stretch plays the important three-dimen-

sional role of sustaining the vorticity against

diffusive decay.

COMBVSTIONAND FLAME 100: 359-366 (1995)

Copyright 0 1995 by The Combustion Institute

Published by Elsevier Science Inc.

OOlO-2180/95/$9.50

SSDI 0010-2180(94)00161-K



36 0 J. W. DOLD ET AL.

The underlying uniformity of the vorticity

field in the direction of stretch also allows for

flame and flow interactions to be carried out

more simply as two-dimensional sub-problems,

depending only on the two directions of non-

uniformity in vorticity distribution . The roll-upthat can be produced by these vortices, their

repeated (periodic) occurrence along a plane

of converging flow, and their well-defined

structure both allow for a rich range of possi-

ble interactions and offer a useful te st bed for

investigating premixed and non-premixed com-

bustion in complex flows.

In this article, we begin by outlining briefly

the solution of the Navier-Stokes equations

that describes these vortices. We then presentthe results of a numerical study, using an

eikonal formulation [6], simulating the propa-

gation of thin premixed flames with constant

normal flame speed through periodic arrays of

stretched vortices. An increase in the overall

(or mean) speed of flame passage around the

vortices is found in some cases, as expected,

but it is also shown that flames w ith a normal

propagation speed that is too low are unable to

proceed from one vortex to the next. Dur ing a

part of their movem ent (from simp le flat initial

flame shapes) these flames resemble trapped

flames, which can also be calculated.

The modeling and calculations are carried

out for the case of sm all heat release, the so

called constant-density model, in which the

flow field is decoupled from the flame’s behav-

ior; this greatly simplifies the modeling. Al-

though it is clear that the behavior of a real

flame must be influenced by thermal expan-

sion, it is seen that the constant-density modelis sufficient to reveal a num ber of interesting

dynamical features of the movement of a flame

in this vortex field.

MODELS

Periodic Stretched Vortices

We can begin by examining a three-dimen-

sional flow field of the form

u= [ I]+ [:y], (1 )

so that u and u represent deviations in the x

and y components of velocity U(X , y, z), that

are independen t of a direction of uniform

stretch z, from a simple stagnation-point flow

field which involves no overall strain-compo-

nent in the x direction. Incompressibility, V *u= 0, requires that

u, + UY= 0, (2)

which can be satisfied by defining a d i s turbance

stream function I,!& y) such that

u = (c;, v= -I) X. (3)

Conservation of vorticity requires that

UOJX VW)’ hyw, + Aw + v(w,, + WYU )

(4)

where

Assum ing that a periodic distribution of vortic-

ity exists with fundamental wavenum ber k , w e

can use the length-scale for mom entum diffu-

sion, (v/h) ‘I2 to construct a dimensionless

strain-rate parameter

A’ = -$

which m ay also be thought of as the square of

a dimensionless wavelength of vorticity distri-

bution, based on the length scale for momen-

tum diffusion. It may also be thought of as a

Reynolds number based on the wavelength,

and the flow speed variation found over the

same length in the direction of stretch. A lsomaking lengths and velocities dimen sionless in

a straightforward way, appropriate dimension-

less equations become identical to Eqs. 2-5,

with the viscosity 1, set equal to unity (its

dimensionless value) and all primes, as in Eq.

6, dropped.

Full solutions can now be sought in the form

1c,= 2 [a,(y) + ib,(y)le’kx, a-k = akrk= --u:

b-, = -b,, a, = b, = b, = 0, (7)

where the real coefficients a k and b, are func-

tions of y only. The vorticity conservation



FLAME PROPAGATION THROUGH PERIODIC VORTICES 36 1

equation requires that these functions satisfy

the sequence of ordinary differential equations

ar + hyu; + (A - 2k2)4 - k%yaZ

- k2Aa, + k4ak

+ i[b;111 hyb’; + (A - 2k’)b;l.

-k2Ayb; - k2A b, + k4b,]

= i C {(a;_, + ib;_,)I= --r

x [la; - /“a, + i(Zb;’ - 13b,)]

-(k - l)(a,_, + ib,_,)

x[a;’ - I’d, + i(b;’ - l*b;)]} (8)

for any k, together with a suitable set of

boundary conditions, namely

a;(O) = b,(O) = a,(4 = b,(a) = 0, (9)

which incorporates the symm etries that are

necessary for ensuring bo unded disturbances

at y = --. It is convenient also to define an

amplitude for any solution such that

A = +(max(t)) - min($>). (10)

In this form, if A > 1, solutions are found

for any value of A > 0 having a,(O) > 0, which

fixes the phase of the solution. In each case,

we use a combined shooting and continuation

method [l] to obtain the functions a,(y) and

b,(y), finding that these functions diminish

sufficiently rapidly with increasing k that typi-

cally very little contribution to the solution

arises after k = 7 or 8. An example of the fullflow-field associated with one solution is shown

in Fig. 1.

Thin Flame Propagation

It is useful to employ an eikonal formulation

for describing the propagation of a flame

through the nonuniform flow field u [6]. We

will consider only a flame with constant normal

flame-speed S and no density jump, so that the

hydrodynamics can be decoupled from the

combustion. This approximation certainly sim-

plifies some of the physics, but also retains the

potential for isolating strong interactions be-

Fig. 1. The distortion of a sheet of fluid arriving from

y= +m as it enters a region with periodic, steady

counter-rotating vorticity in a stretched flow field, satisfy-

ing the Navier-Stokes equations. The dark lines follow the

movement of individual fluid elements, showing that al-

most all fluid is sucked towards the core of a vortex.

tween propagation and nonuniform advection

[5, 7-91.

For this purpose, taking a function f to be

such that a flame lies at

f<x, t> = c, (11)

the proper dynam ical evolution of the flame is

reproduced if f satisfies the eikonal field equa-

tion [6]

f, + u. Vf = SiVfl. (12)

Moreover, solving for f using this model for-

mulation can be made to simulate many dif-

ferent flames in the same calculation since any

isopleth of f, th at is f(x, t) = c for any ualue

of c, can represent a flame. So lutions of Eq. 12

are thus able to provide a great deal of infor-mation about the combined advection and

propagation of flames around vortices [5, 7-91.

We solve Eq. 12 using a straightforward

characteristic method, based on the character-

istic velocity u - SVf/]V f], with second-order

differencing taken over a quadrant of 8 points

in the general direction from which the charac-

teristic arrives at any one po int. T his is found

to give solutions that are free from oscillations

and in which cu sps in flame sh ape are captured

cleanly. Also, being quadratic in order (indeed

having some cubic terms because of the g-point

differencing in two-dimensions) the method is

relatively accurate and free of num erical dif-



36 2

fusion even when using modest numbers of

points.

Stationary Thin Flames

As will be seen, stationary flames can also arisewhen the velocity field exactly balances a

flame’s normal propagation speed at every

point. This happens where

u*ri = -s, (13)

in which h(x) is the unit normal vector to a

flame surface at the point X. In general, this

defines either two unit norm al directions (if

1~ 12 S) or none (if 1~1< S). Given one un itnormal that lies perpendicular to a stationary

flame surface, the corresponding tangent direc-

tion i((x), for which s^. it = 0, with s^ rotated

(say) anti-clockw ise from 5, provides a differ-

ential equation for the path of the flame sur-

face x(s) as a function of arclength s, since

x(s) must satisfy

x, = i(X), s^*ri = 0. (14)

Thus, through any point x there are either tw o

or no possible tangents to any possible station-

ary flame. Any stationary flame solution that

might be found must also depend on the over-

all shape of the flame, so that Eq. 14 needs to

be solved subject to reasonable boundary con-

ditions, such as that a flame should not spiral

around itself (requiring a flame to subsist on

burnt gases) or disappear suddenly at a point

where JuI = S, although a closed limit-cycle

would be acceptable as a solution representing

a flame trapped by the suction into a vortex.

Shooting techniques prove well-suited to solv-

ing Eqs. 1 3 and 14 in order to determine

stationary flame-profiles.

RESULTS

Advection and Propagation

Taking a sample flow-field with h = 6, A = 20,and S = 8, Fig. 2 show s a sequence of contour

diagrams of the field variable f calculated from

the simple initial condition f = x. These initial

data represent a family of flames (each mem -

J. W. DOLD ET AL.

Fig. 2. Profiles of flames traveling from right to left through

a layer of steady periodic vorticity distribution, having

A = 6 and A = 20, at a constant normal flame speed of

S = 8 (as calculated using the eik onal Eq. 12 which repre-

sents flames as isopleths of f). Contours of f at different

dimensionless times are shown when: r = $; (b), t = 4; cc),

t = 1; and (d), t = 3. The centres of regions of counter-

rotating vorticity are marked by small fans. The develop-ment of a corridor of enhanced flame passage is clearly

seen.

ber depending on a choice of c) that start out

being aligned parallel to the y axis.

It is very clear from this figure that flame

movement is enhanced where flow around the

vortex cores i s in the sam e direction as the

flame’s propagation. Where the flow inhibits

flame movement, directional effects quicklylead to the formation of cusp s where different

directions of propagation cross. In these calcu-

lations, flames assisted by advection in parts of

the flow also manag e to find their w ay to other

regions where advection assists flame-move-

ment in a way that enhances the overall flame

transit. A fter a long enough evolution it is

clear that a corridor of enhanced flame pas-

sage appears along the line of vortices. The

propagation through this corridor is led by anose of flame that appears to find an optimal

path through the flow-field so that it is able to

cross from one vortex to the next in the least

possible time.



FLAME PROPAGATION THROUGH PERIODIC VORTICES 363

Enhanced Flame Speed

By following the average rate of increase of f

within the corridor of enhanced flame-passage

(over a long enough time) the effective in-

crease in flame-speed can be estimated. Inparticular , with the initial condition f = x, the

effective flame speed is precisely

(151

the averaging being taken at any point or path

within the corridor of enhanced flame passage;

we, in fact, average firstly over the entire path

for which , at fixed y, f(x, y, t> has its highest

value, and secondly over time, usually over the

period from t = 1 to t = 2 or longer. Formula

15 is confirmed by the fact that I( = 1, as

ensured by the periodicity, for wh ich f<~ +

27r, y, t) = 27~ + f(x, y, t). Figure 3 presents

this overall effect of advection around the vor-

tices on the mean flame-speed S, for two

different fixed amplitudes A ranging over dif-

ferent values of S.

It can be seen that increasing the ratio A/S

serves to increase the effective flame speed

above the speed without vorticity provided thevalues of A/S do not grow too large; the

increase in effective flame speed is found to be

approxim ately linear when the vorticity level is

low (or flame-speed is high). At increased lev-

t

1 . 6

\

A = 4 0

Fig. 3. Fractional increases in effective mean-speed of

flame passage S, through the layer of vorticity distribu-

tion, as functions of A/S, from calculations of the eikonal

Eq. 12 using two fixed vorticity distributions, having A = 20

and A = 40, with h = 6. The variations for small A/S

(large 5’) are similar, and nearly linear, in both cases; at

smaller normal flame-speeds S the flames fail to take

advantage of the nonuniform flow-field.

els of vorticity, there is another effect which

manag es to reverse the gain in flame-speed

acquired at lower levels of vorticity. The

flame-speed decreases once A/S increases

above a threshold value, dropping very rapidly

in value. It is apparent from this that a point isreached where flames are unable to penetrate

through the arrays of vortices in any preferred

manner.

Flame Trapping

If a flame is unable to penetrate at some stage

from one vortex to the next, then the leading

parts of the flame m ight be expected to come

to a standstill. This leads one to search forstationary flame solutions. It turns out that

Eqs. 13 and 14 do yield stationary solutions for

trapped flames at sufficiently large values of

A/S.

Superimposed on a representation of the x

and y comp onents of the flow field for fixed

values for A and A, Fig. 4 shows a sequence of

flame-solutions trapped around the core of a

vortex. A s S increases, the flame is able to

advance further away from the vortex center,overcoming more of the flow of fluid that is

Fig. 4. Profiles of stationary flames trapped by the inflow

around a region of concentrated vorticity (having A = 6

and A = 10) with flow directions and magnitudes illus-

trated by the short lines. Trapped flames are shown for

normal propagation speeds of S = 1, 1.5, 2, 2.5. 3, and

3.42. Two regions for which (~1 < S (= 3.42), within which

a flame of speed S could not be held stationary by the

flow, are shown as shaded; and a flame of speed S = 3.42

is seen to just avoid touching these regions. This demon-

strates that trapped flames do not exist (for this flow-field)

at speeds of propagation above a critical value close to

S = 3.42.



364 J. W. DOLD ET AL.

sucked towards the vortex as an overall result

of the divergence of fluid along the line of any

vortex. This continues until a value S =

S&4 , A) is reached where a trapped solution

touches the region where IuJ I S. As pointed

out earlier, no stationary solution can existwithin these regions so that trapped solutions

cease to exist above this critical value of S.

That this value involves a close relationship

with the failure or success of flames in passing

between vortices is confirmed in Fig. 3 where

these critical levels (both for A = 20 and A =

40) are marked on the diagram and correspond

closely with the loss of enhanced propagation.

Because it would correspond to zero speed of

propagation, A/S, must overestimate the pointof loss of enhanced propagation speed, which

cannot fall below the value of unity since flames

propagating at this speed are still to be found

outside the layer of nonun iform vorticity.

It is interesting to compare this result with

an unsteady solution of the eikonal Eq. 12 for

a value of S that is not high enough for the

flame to pass from one vortex to the next. Such

a calculation, at one time, is shown in Fig. 5. In

this diagram, certain contours show that an

initial flame distortion by the flow field at the

initial time h as come to a standstill as a bulb of

flame is unable to escape from the suck ing

Fig. 5. Profiles of flames travelling from right to left, under

the sam e conditions as in Fig. 2, but for a constant normal

flame speed of S = 3;. Isopleths of f are plotted at the

dimensionless time t = 1. The bottlenecking in between

the vortices, of contours at the nose of the flame, demon-

strates the difficulty that a flame of this propagation speed

experiences in escaping from the sucking effect of each

vortex, resulting in a much reduced mean speed of flame

passage and flame profiles that resemble, for a while, the

shapes of trapped flames.

effect of a vortex. This bu lb h as a very similar

form to the steady trapped flame shapes in

Fig. 4.

CONCLUSIONS

Some examples of a new family of periodic

stretched vortex solutions of the Navier-S tokes

equations have been used in this article to

illustrate part of the manner in which flames

can interact with complex flow fields. In partic-

ular, as solutions of an eikonal equation for

constant normal flame-speed show, the flames

are able to take advantage of nonuniform ad-

vection to travel more qu ickly. Arou nd one

side of any region of increased vorticity theadvection helps the flame’s movement. After

having been helped in this way, the flame then

has to be able to escape from the overall

suction of the vortex in order to be able to

move on to take advantage of the advection

around another vortex. The suction associated

with the stretching of vortices can therefore

hinder a flame’s movement while the advection

around vortices aids its overall movement.

Thus, the flow has within it the essential

ingredients both for enhancing the passage of

a flame through an array of vortices and for

holding back the flame, depending on the in-

tensity of the vortices and on the norma l flame

speed. In some cases, it has been shown that

flames could even become completely trapped

within regions of enhanced vorticity. It seem s

likely that these processes should reflect at

least part of the experiences a premixed flame

would undergo in a turbulent flow field.

One mechanism for enhanced flame move-ment in turbulence is not immediately avail-

able for study with the particular steady-flow

solutions described in this article. The propa-

gation of a flame along a vortex should also

provide a significant setup for non-uniform

flame advancement in which density changes

and curvature effects shou ld be very impor-

tant; this mechanism is likely to be especially

important if and when flames become trapped

within vortices, unless three-dimensional cur-vature effects then restrict flame speeds exces-

sively. The unsteady movem ent of vortices

themselves is also likely to be a major factor in

real turbulent flame propagation.



FLAME PROPAGATION THROUGH PERIODIC VORTICES 36 5

In spite of these m ore general aspects, this

article show s that the new Navier-Stokes solu-

tions described here do make available some

insights into the interplay between propagation

and advection. As a test-bed for combustion

problems, it complements significantly thetwo-dimensional initial-value vortex proposed

by Marble [2] and the simple doubly periodic

flow-field of Ashurst and Sivashinsky [51 for

some flame problems, by being steady

Navier-Stokes solutions and therefore, of ne-

cessity, stretched. These properties, and the

mixing-layer context within which the vortices

appear, make the solutions useful for examin-

ing both premixed and non-premixed burning

in the presence of non-un iform vorticity.

J. W . D o l d w a s s u p p o rt e d i n t h i s w o r k b y a n

A dv anced Fel low ship f rom t he S .E.R .C.; 0 . S .

Kerr w as suppor ted by a Research Grant from t h e

S .E.R .C. and I . P . N iko lov a w as suppor ted by an

E.C. V i s i t i ng Fe l low sh ip Gran t (No .

ER B351OP L920821) enabl ing her to spend th e

m on ths o f A pr i l t o June 1993 a t Br i s to l Un iv er-

s i t y .

REFERENCES

1.

2.

3.

4.

5.

Kerr, 0. S. and Do ld, J. W., J. Fluid Mech.,

276307-325 (1994).

Marble, F. E., in Recent Adrtances in the Aerospace

Sciences, Plenum, New York, 1985, pp. 3955413.

Laverdant, A. M. and Can del, S. M.. Com hust. Sci.

Technol. 60:7Y (1988).

Ashurst, W. T., Lect. Not es Phys. 352:3-21 (1989).

Ashurst, W. T. and Sivashinsky, G . I., Cornhusf. Sri.

Tech&. 80:159-16 4 (1991).

Markstein, G. H., Nonsteady Flame Propagation, Perg-

amon, 1964, p. 8.

Kerstein, A. R., Ashurst, W. T. and Williams,

F. A., Phy s. Reel. A 37:2728 (1988).

Ashurst. W. T., Sivashinsky, G. 1. and Y akhot, V.,

Combust . Sci. Technol . h2:273-284 (1988).

Ashurst, W. T. and W illiams, F. A.. Twenty-Third

International .$mposium on Combustion, The Com-

bustion Institute, Pittsburgh, 199 0, pp. 543-550.

Receked 30 Nol’emher 1993; reked 27 Ma)’ 1994

Comments

S . I brah im, Br i t i sh Gas , UK. Your test cases

are at small values of ST/S (turbulent to lami-

nar burning velocity) which are of order of

l-l.6 and low turbulence intensity (u’/S). Haveyou done any tests at high turbulence level, in

order to confirm the effective flame speed trend

that you have obtained?

A u t h o r ’ s R e p l y . In this preliminary work we

have been working with relatively low magni-

tude vortices in order to establis h the initial

trends. Calculations involving larger ampli-

tudes are planned.

J. Chom iak , Cha lm ers Un iv ers i t y o f Techno logy ,

S w e d en . In turbulen t flows the vorticity is con-

centrated in vortex tubes randomly distributed

in space. Thus the situation of parallel vortices

you consider is very unlikely to occur.

A u t h o r ’ s R e p l y . The exact configuration of vor-tices that we consider is not intended to mimic

turbulence precisely, although the observations

from D.N .S. that you describe provide a strong

motivation for its study. The vortices that we

present here provide a useful test-bed for in-

vestigating complex flame an d flow interac-

tions, in which the fact that they are exact

steady Navier-Stokes solutions, sustained by

stretching, imparts m ore features in common

with real turbulent vortices than is found for

any other well-understood flows that have beenstudied in the same way to date, w ith the

possib le exception of Burge r’s vortex. It is also

worth mentioning that these particular



36 6 J. W. DOLD ET AL.

stretched vortices only fall into one range of

solutions within a wider class of steady solu-

tions that is still under investigation.

R . K l ei n , R W T H A a c h e n , G e rm a n y . 1. Is yourexact solution of the Navier-Stokes equations

extendable to a stack of parallel arrays of

vortices, so that you could describe interac-

tions of those “ flame tunn els”? This configu-

ration would be very close to actual turbulent

flame propagation. 2. Did you assess whether

inclusion of at least weak heat release in your

model is feasible?

A u t h o r ’ s R e p l y . 1. There is a large variety ofpossible steady solutions to the Navier-Stokes

equation in this stagnation point flow, more

than we have described here. W ithin these

possible solutions are some that have two par-

allel arrays of vortices stacked on each other,

with adjacent vortices having opposite senses

of rotation. It is not, however, possib le to have

more than two such arrays stacked on each

other. 2. We have considered the possibility of

thermal expansion in the vortices. T he govern-

ing equations do allow for steady solutions

where the chemical constituents, the tempera-ture, and hence the density vary only with x

and y, and not z. In principle we could find

solutions to these equations, and hence find

steady flame solutions with expansion effects

included , but this wou ld be relatively difficult

and we have not yet done so. Non-steady flames

with expansion would destroy the underlying

steady vorticity field as they propagate, requir-

ing a different means of study from that em -

ployed here. In such a study it is not clear thata small heat-release approximation would be

much easier to solve or interpret than order

one heat releases. However, it is a very impo r-

tant currently unsolved problem to determine

the relationship between highly non-uniform

flow fields and expansion in propagating flames.

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J.W. Dold, O.S. Kerr and I.P. Nikolova- Flame Propagation through Periodic Vortices