oliver s. kerr and j.w. dold- flame propagation around stretched periodic vortices investigated...

8/3/2019 Oliver S. Kerr and J.W. Dold- Flame Propagation Around Stretched Periodic Vortices Investigated Using Ray-Tracing

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Combust. Sci. and Tech" 1996, Vol. 118, pp.101-125Reprints available directly from the publisherPhotocopying permitted by license only

1996 OPA (Overseas Publishers Association)Amsterdam B.Y. Published inThe Netherlands under

license by Gordon and Breach Science PublishersPrinted in India

Flame Propagation AroundStretched Periodic VorticesInvestigated Using Ray-TracingOLIVERS. KERR' and J. W. DOL[>21Department of Mathematics, City University, Northampton Square,London EC1V OHB, U.K.2Department of Mathematics, U.M. I.S. T., Sackville Street,Manchester M60 100, U.K.(Received 3 April 1996)We investigate the propagation of a thin flame that passes transversely through a per iodicarray of stretched vortices. The effect of different rates of stretching and different strengths ofvortices on flames with various flame speeds is calculated by the use of ray-tracing equations.The results show that the presence of stretched vortices enhances the propagation of flamesexcept in flows with very strong mixing where there can be an adverse effect on the mean speedof the flames through the vortices. Ultimately this can result in i t being impossible for flames topropogate at all past the vortices; instead they would re side so lely wi th in the cor es of thevortices.Keywords: Propagation; ray-tracing; vortices

NOMENCLATUREAkk(t), m(t)pSSTt

background strain rate of the flowdimensional wavenumber for the distance between vorticeswavenumber components for the raypressureflame speedmean speed of flame passing through vorticestime

101


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102 O. S. KERR AND J.W. DOLO

:T slow timeu velocity of the vorticesU background stagnation point flow (= (0, Ay , - Az))

Greek Symbolsc small expansion parameter (=S -.1)A nondimensional background strain ratev kinematic viscosityp densityr fast timei/t streamfunction for the vorticesi/tm", maximum value of i/tw vorticity

INTRODUCTIONThe propagation of premixed flames through a turbulent environment isvery complex. In order to obtain an insight into some of the interactionsthat occur between flames and these disordered motions it is instructive tofirst examine the propagation of a premixed flame through simpler flowfields that incorporate some aspects of fully turbulent flows. A brief reviewof some previous work in this area is contained in Dold, Kerr & Nikolova(1995). The flow that we consider here consists of a steady periodic array ofvortices which are maintained by stretching in the background flow. Thisenables us to look at the interaction of flames with both the vortices, suchas those found in turbulent flows, and the straining motions that are required to generate them.

In this paper we present results of some calculations of the propagationof a thin premixed flame through an array of periodic steady vortices. Thisflow consists of a background stre tched flow of the form U =(O,Ay, -Az),for A> 0, with a superimposed array of periodic vortices having their axis inthe z= 0 plane and parallel to the y-axis. This recently discovered family offlows (Kerr & Dold, 1994) which are solutions of the Navier-Strokes equation have two degrees of freedom: a measure of the rate of stretching, and ameasure of the strength of the inidividual vortices.

Although the flow is intrinsically three dimensional , it has the propertythat flames that initially have fronts with no y-dependency will continue to


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F LA ME P R OP A GA T IO N AROUND STRETCHED VORTICES 103

propagate with this structure. Thus we can look at the flame propagation inthis system of vortices as a two dimensional problem.We will be looking at thin flames, with a low heat release an d flame speedindependent of the flame curvature. It is possible to model the propagationof such flames by the use of an eikonal equation (Markstein, 1964).This wasdone for these flows by Dold, Kerr & Nikolova (1994). Their results showedthat flames could propagate along the array of vortices confined to a finiteregion around the plane z=O, the flames being unable to escape into theregion where the background flow speed towards the vortices is faster thanthe flame propagation velocity. They found that the effect of the vortices ont'he confined flame varies as the flame speed changes. Fo r a given strengthof vortices the flames with a speed much faster than a typical flow speed inthe vortices essentially passes through them with only a minor increase inthe mean propagation speed. As the flame speed reduces further the effectiveenhancement of the flame speed becomes much stronger until a peak valueis reached. Beyond this peak the average propagation speed declines until acri tical speed is reached beyond which the flames are unable to propagatepast the array of vortices.

Th e work presented here consists of a more extensive survey of theinteraction between the rate of stretching of the vortices, the strength of thevortices and the flame propagation speed. Ins tead of solving the eikonalequation as was done previously, the approach taken is to use ray-tracingtechniques to find the path of the fastest p ropagating flame through theperiodic array of vortices. This enables the average flame speed to be calculated much more readily and a more systematic investigation of theseinterrelations to be made. In the following sections we will describe brieflythese background flows, give the ray-tracing equations used an d describesome of the results obtained.

THE BACKGROUND FLOW

Th e flow consists of steady deviations from the stagnation point flow

U(x,y,z)=(O,Ay, -Az) (1)

with A> O. Th e fully nonlinear, time-independent velocity an d pressure perturbations, u an d p, to this flow satisfy


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104 O. S. KERR AND J. W. DOLDIu-Vu+UVu +UVU = - - Vp + vV2u with V'U = 0 (2)p

where p is the density of the fluid, and v the kinematic viscosity.We look for perturbations to the stagnation point flow that are indepen

dent of the y-coordinate, and have no perturbation velocity component inthis direction. The solutions we find are periodic in the x-direct ion, withperiod 27C/k where k is some prescribed wavenumber. We can nondimensionalise the above equations, using k- ' as a length scale and vk as avelocity scale, and express these equations in terms of a non-dimensionalstream function i/J, with

Then. the continuity equation will be satisfied, and (2) transforms to

(4)

with the non-dimensional vorticity, co, given by

(5)

Here A. = A/vk2 is a non-dimensional measure of the strength of the converging flowcompared to the rate of viscous dissipation. I f we insist that theperturbation stream function decays to 0 as Z-> ct:) then we can define as ameasure of the ampli tude of the vortices the maximum value of the streamfunction, i/Jmax' This is the Reynolds number of the vortices based on thelength scale of their periodicity. The choice of i/Jmax is entirely independentfrom that of A.. For any given strength of stagnation point flow vortices witharbitrary amplitude can be found.

Details of the method of obtaining solutions to these equations can befound in Kerr & Dold (1994). Typical solutions are shown in Figure 1.Herethe flows are projected onto the x-z plane. This shows three typicalexamples illustrating the variation in the flows as the strength of the stagnation point flow, A., and the amplitude of the vortices, i/Jmax' change. As theamplitude increases the vortices become stronger. As A. increases the vortices are confined to a narrower region near the plane z=O. If A. decreases


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FLAME PROPAGATION AROUND STRETCHED VORTICES 105-3 -2 -1 o 2 3

4

3

2

o-1

-2

-3

-4

4

3

2

o

-1

-2

-3

-4

1TO.51T-5+'--rL-f-J-.--'--.-'---,-l.---J,--L,-J--.-'--+-'r--'-,r-'---r'--+-...J.,-....L.,...L-r -5

(a) -1TFIGURE I Stream lines of the flow due to the vortices in a s tagnat ion point f low projectedonto the x-z plane. These examples have (a) A=4,I/Im.,= 10, (b) A=4,I/Im.,=40 and (c)A= 1 2 , I / I ~ , = 4 0 .


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106 O. S. KERR AND J. W. DOLO3 -2 -1 o 2 3

-4

-2

4

-1

3

o

2

7T.57TO.57T

3

4

2

o

-2

-1

-3

-4

.5 - + J - , . l - + . . . . . L , . . . . L , . . L - . , . L - + - 4 - y - - + - - - . J , . . . . . . L , . . L ~ + - . . l , - . L , - 4 - - 5(b) -7T

FIGURE I Continued

then the region occupied by the vortices increases, but solutions to theequations cease to exist that decay as z-> CIJ for A 1. A three dimensionalrepresentation of the flow is shown in Figure 2.


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FLAME PROPAGATION AROUND STRETCHED VORTICES 107

-3 -2 -1 o 2 3

(c) -1T -O.51T oFIGURE I Continued

O.51T 1T

The solutions shown here have a rotational symmetry about the origin,as do all the solutions in Kerr & Dold (1994). However this is not arequirement of governing equations, but a condition imposed when calcu-


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FLAME PROPAGAflBNAND THE WAY-PRAG1WG EQUATIOHS


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The first of these, Figure 3(a), shows a series of flame fronts when the flamehas a normal velocity S=8. In this case the flames easily pass from onevortex to another, gaining speed by passing on the side of each vortex withthe gas moving in the same direction as the flame. This leads to the flamedeveloping a flame "corridor" along the vortices. The flames here cannotescape from the region of vortices because the advection towards the planez=0 limits their propagation away from the vortices. The second example,Figure 3(b), shows an example with S=3.5. In this case the flame can onlyjust escape from one vortex to the next through a narrow region near as tagnat ion point in the flow. For flame speeds much less than this theflames are unable to propagate from one vortex to the next. In both casesthe nearly parallel flame fronts outside this corridor represent the residue ofthe initial configuration which consisted of flame fronts of infinite extentperpendicular to the x-axis. Fa r from the plane z=O these flames willcontinue to propagate with a speed S in the x-direction, unaffected by

-3 -2 -I o 2 3

z 0-1

-2

-3

-4

-5-3 -2 -1 0 2 3

(a) XFIGURE 3 Numerical solu tions of the eikonal equat ion showing flame fronts as a flamepropagates past a periodic array of vortices with .l= 6 and "'mn = 20 with the flames passingfrom right to left. In (a) the flame has speed S=8 and in (b) it has speed S=3.5. The dashedlines show the paths of the fastest propagating flame found with ray tracing.


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110 O. S. KERR AND 1. W. DOLO-3 -2 -I o 2

(b)-3 -2 -I o

x2 3

FIGURE 3 Continued

vortices. These flames can propagate from this region into the vorticesconfined near the plane z=O. Thus the mean speed of flames passing thevortices cannot be less than S in these calculations. Dold et al. looked at thedevelopment of solutions with time. The flames in the corridors settle downto a steady state. From the time taken for each flame to propagate acrossthe region it was possible to find the effective flame speed past the vortices.The results of such an investigation are shown in Figure 4. Each point onthis graph involved solving the eikonal equation, a time-dependent partialdifferential equation, over a two-dimensional region in space, and allowingit to evolve until the initial t ransients had decayed. Thus each point on thegraph involved a significant amount of computing.Our objective is to exploit further the analogy of flame propagation in a

premixed gas with sound propagation by looking at the path of the flamesby using the ray-tracing technique. This approach looks at any small part ofa wave or flame front and describes how it propagates, t racing out a "ray"as it progresses.


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FLAME PROPAGATION AROUND STRETCHED VORTICES II I

STS

1.6

1.2

4

ASe

8AS

FIGURE 4 Fractional increase in the effectivemean speed, ST' of flame passage through theperiodic vortices as a function of "'mallS found from numerical simulations using the eikonalequation. In the two cases the vortices have ,pmu=20 and "'mu=40. In both cases A=6.

The equations (see, for example, Lighthill , 1978) for such a ray are

dk au aw-=-k--mdt ax ax'dm au aw-= - k - - m - +m l ,dt az az

(6)

(7)

(8)

(9)

Where the coord inates of the ray are (x(t), z(t)), and the orientation of theray (or local normal to the flame front) is given by the wavenumber vector(k(t), m(t)). The non-dimensional flame speed is S. The first two equationsdescribe the propagation of the flame as the sum of the local fluid velocityplus the flame velocity in the direction of the normal to the flame. The


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\12 O. S. KERR AND J. W. DOLO

second pair of equations describe the effect of the fluid flow on the orientat i o ~ of the flame front. The propagation of the ray only depends on theorientation of the wavenumber vector, (k,m), and not on its magnitude.

In this investigation we look for the fastest flame path through the vortices. Because the overall passage of a flame around the vortices will be ledby the portion of flame that manages to move fastest through the vortices,we can argue that any flame approaches this shortest passage-time fromgeneral initial conditions. Because of the periodicity of the vortices thefastest flame will also be periodic. Hence we can restrict ourselves to looking for periodic "rays". We look for rays which for some initial point andorientation will, at some later time T, be in the equivalent position andorientation one spatial period further on, i.e. we want to find x(O), z(O),k(O)/m(O) and Tsuch tha t

x( 1)= x(O)+ 2n,z(1) = z(O),

k( 1)/m(1) = k(O)/m(O).

(10)(II)(12)

Then Twill be the time taken to pass through a full period of the vortices,and the non-dimensional average or effective flame speed will be ST= 2n/T.In the above formulation of the problem the flame travels from left to right.Because of the symmetry of the flow, the results obtained are equivalent tothose that would be obtained for a left travelling flame.Because of the additional symmetry about the origin that we have assumed for the vortices under consideration, we can make the further simplification which makes the periodic rays easier to calculate that after half aperiod, T/2, we will have

x(T/2) = x(O)+ rr,z(T/2) = - z(O),

k(T/2)/.m(T/2) = -k(O)/m(O).

(13)(14)(15)

This reduction results in it being easier to find solutions, especially for lowflame speeds. The solutions that satisfy these boundary equat ions can befound using standard techniques.


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F LA ME P RO PA GA TI ON A RO U ND STRETCHED VORTICES 113

RESULTS

The ray-tracing equations were solved numerically using a standard fourthorder Runge-Kutta technique. Some examples of the paths calculated areshown in Figure 5. These show a variety of the behaviours of paths found.Some of these have to be treated with caution. Although it would be meaningful for a sound wave to pass around a vortex and cross its original path,this is obviously not possible for a flame as once the flame has passed the fuelwould have been consumed and the gas could no longer support combustion.However, ray paths that continuously propagate into unburnt gases will notviolate such a restriction. This criterion will be satisfied by the ray corresponding to the fastest propagating flames under consideration here.

-3 -2 -1 o 2s . , . . L _ . . - L - - - L _ . . . L . _ . . . L _ - ' - - - J . , - S4

2

4

2

z 0 0-I -I

-2 -2

-3 -3

-4 -4

-5 -5-3 -2 -I 0 2

(a) XFIGURE 5 Paths of (a) the fastest flame travelling from right to left past vortices with 1 = 6and 1 j J ~ . = 20. The solid lines have flame speeds (from straightest to most curved) of S = 256,128,64,32, 16,8 and 4. The dashed line is for a flame speed of S = 3.5. The paths in (b) showhow rays with S = 4 start ing from x = 3,,/2 diverge from the fastest path if their initial condit ions are varied by about I%. As flame-tips, it is clear that some of these rays would becomeunphysical when crossed by another flame-tip propagating in another direction, thereby forming a cusp on the flame-surface. This does not happen to the leading flame-tip represented bythe perfectly periodic ray.


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114 O. S. KERR AND J. W. DOLO-3 -2 -I o 2 3

S+ - - -L - - - " - - - - - L - - - ' - - - - " - - - -TS

4

2

z 0 0-I -I

-2 -2

-3 -3

-4 -4

-s -s-3 -2 -1 0 2

(b) xFIGURE 5 Continued

The solutions of the ray-tracing equations can be found readily for anyflame speed S and a given background flow. In this way we can ca lculatehow the effective flame speed of a flame past an array of vortices varies as afunction of the flame speed. A typical example of a plot of the effective flamespeed, ST' as a function of the actua l flame speed, S, for a given s trength ofstagnation point flow and ampli tude of the vortices is shown in Figure 6.This shows not only how there is a maximum value of the flame speedenhancement just as was found by Dold et al., but also other features thatwould not have been practical to find by solving the e ikonal equa tion . Inthe limit of fast flames, i.e. St/!m" or t/!m"jS is very small, an initial glanceat Figure 6(a) may lead one to believe that (ST-S)jS is proportional to IjS.However a closer examination of this limit, which is shown in more detail inFigure 6(b), reveals that in this region (ST-S)jS is proportional to I jS2.The asymptotic behaviour of this weak enhancement in the overall speed offlame passage can be predicted by a multiple-scales analysis (see Appendix),the result of which is also shown in Figure 6(b). However, this quadratic


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behaviour quickly gives way to a s teady growth in 51'/5with r/Jm,,/5, whichcontinues until a maximum is attained. Beyond this maximum 51'/5 dropssharply. As the flame speed approaches a critical point the time of transit ofthe flames through the vortices increases rapidly and 51'/5 drops towardszero. Beyond this cri tical value the flames can no longer pass through theperiodic vor tices. Instead the flame path found by ray tracing becomestrapped in the core of a vortex. However such paths would be unphysical asthe rays would be passing into regions of burnt gas. In this situation, whereflames can no longer propagate from one vortex to the next, the com bus-tion will take place near the core of the vortices. These trapped flames willbe steadly and can be found by other means in a straightforward way (seeDold et al.l.The maximum value of 51'/5 gives the largest possible enhancement of the

rate of flame passage through the given vortices. This value can be foundfor a range of values of both the strength of the converging flow, A., and the

100 . 5+ - - - - - . - - - - - - - - - - , , - - - - - , - - - - - - . - - - - - - ' - - ,

2.0

1.0

1.5

(a]FIGURE6 The average flame speed ST/S as a function of !fimu/Sis shown in (a) for a flowwith ,,=6 and !fi=.=40. Flames cannot propagate for flame speeds beyond the vertical dashedline. In (b) the region near !fimu/S=O is enlarged, showing the quadrat ic convergence toST/S = I along with the large S asymptotic behaviour (dashed line).


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116 O. S. KERR AND J. W. DOLO

1.075

1.05

(J ):r 1.025

1.0-- '----

,,,r,

rI

I,I

I,,I

I,,I

II

I,I

0.4.3.10.975+----,----,.----,---,

0.0(b)

FIGURE 6 Continued

amplitude of the vortices, i/Jm." The results of such a calcula tion are shownin Figure 7(a) which presents the maximum level of this enhancement. Thecorresponding values of the flame speed, S, at which these maxima occur areshown in Figure 7(b). These show that the maximum enhancement of a flamefor a given i/Jma< occurs for values of A. around 4, and that the magnitude of thisenhancement increases as i/Jmax' The flame speed is enhanced by being advectedaround the vortices. For small values of A. the vortices are elongated in thez-direction and so the distance that the flame would be advected is increased,which leads to a diminishing of the speed enhancement. For large values of A.the vortices are more localized, and the time of transit of the flame isdominated by the time taken for the flame to pass between the vortices. Thispart of the flame path is relatively unaffected by the presence of the vortices,and so limits the possible enhancement. The second figure shows how theoptimum enhancement for larger values of A. or i/Jmax correspond to faster flamespeeds. Flames with lower speeds find it more difficult to escape from onevortex and continue to the next. Similar calculations can show how the criticalvalue of S varies as a function of.any other background flow parameters.


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10 20 40 60 80 100 200

108

4

6

80 100000

8

4

2+----,----- ' --------,r--- ' ------,--L,---- ' r-.. . .L-L--- ' : . . . . . .:1--210 200

10

A 6

(a)

10 20 40 60 80 100 200

10

8A 6

4

10

86

4

20 40 60 80 100(b)FIGURE 7 Contour plots showing (a) the maximum ratio of the mean speed of a flame acrossthe vortices to the flame speed, ST/S, and (b) the co rresponding flame speed S at which themaximum occurs. Thesecalculations arefor). and "'max inthe ranges 2 :::;;;1:::;;; 16and 10,::; l/Imu ::::;:200.


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118 O. S. KERR AND J. W. DOLDIn any given calcula tion one is free to choose the parameters S, i/lmax and

A. Obviously this much freedom allows much scope for investigating para-meter space. The above results concentrate on investigating the results ofvarying S for a given flow field specified by a choice of i/lmxand A. This isjust one possibility. Another example would be to fix Sand ). and find howthe fastest flame responds to variations in the strength of the vortices, foundby varying i/lm ' This would give an indication of the effect of different levelsof mixing in a configuration where the flow and flame speed are fixed. Theresults of such an investigation are shown in Figure 8. This shows how theflame is relatively unaffected by the presence of the vortices until the vortexamplitude, i/lmax' which is also a typical velocity in the vortex, becomescomparable to the flame speed. There is then a region where the graph isapproximately linear, which with the scalings used on the axes would indicate that the increase in ST/S was proportional to the increase in 10gtO i/lmx.This enhancement abruptly stops when i/lmxis close to 275, and it s tart s todecay. The po in t at which this enhancement ceases is not currently wellunderstood.

1 .0- -1-==: : : : : : : : : :=- - - - - - - - , - - - - , - - - - - , - - - - , - - - - - - . - - - ,

1.5

S,IS 2.0

2.5

5 10 50 100 500 1000

FIG URE8 Graph of the enhancement of the name speed, ST/S, as a function of the ampli-tude of the vortices, rjlm"" for S = 8 and ), = 6. .


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Another possibil ity is to vary the strength of the background stagnationpoint flow, }., for a given level of non-dimensional mixing, ljJm,,' The resultsof such an investigation are shown in Figure 9. As a typical velocity invortices is approximately given by ljJm" the faster flames are relatively unaffected by the mixing of the vortices. As seen in Figure 1, reducing Alengthens the vort ices in the z-direction. As A is decreased towards 1, thevortices approach the exact solution, w Clcosx found by Craik and Criminale (1986) in which the flow has no velocity component in the x-direction;the effect on the passage of flames in this d irect ion is also therefore lost asA-d. As A increases vortices are changed from having this elongated structure towards becoming more concentrated around points in the z=O plane.The ini tial effect of introducing signif icant x-components in the velocityfield is to enhance ST' as is clearly seen in Figure 9. However, as the vorticesbecome more concentrated onto the z=0 plane and the incoming strainedflow from z= 00 increases through increasing }., the flames also becomeincreasingly const ra ined into a nar row band around z = O. As Figure 9makes clear , this limits the ability of flames to take advantage of the advection around the vortices. Because flames of higher propagation speed S areless const ra ined by the incoming flow from z= 00 , the values of ST forthese flames tend to fall away at higher values of A.

2.2

2.0

1.8

1.6

1.4

1.2

2 4 6 8 10A

20 40

FIGURE 9 Graph of the enhancement of the flame speed, ST/S, as a function of the s trengthofthe stagnation point flow, A, for I/tm.. = 80. The four curves arc for S = 10,20, 40 and 80.


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120

CONCLUSION

O. S. KERR AND J. W. DOLD

Ray tracing offers a very useful tool for calculating efficiently the passage ofany small section of flame through a complicated flow-field, provided the flamehas a normal speed which is independent of the curvature of the flame front.The use of this technique in conjuction with flows that are steady solutions ofthe Navier-Stokes equation reveals the most salient features of the way inwhich a real flow can enhance or hinder the passage of a premixed flame. Itshows how the flame with a propagation speed comparable to a typical vortexvelocity can utilise the flow to enhance its average speed past the array ofvortices. However, for larger vortex strengths this process is inhibited as theflames find it increasinglydificult to escape the tendency for the flow to entrainthe flame into the cores of the vortices. In addition the ray-tracing techniqueallows the quadratic enhancement of the flame speed for fast flames travellingpast vortices to be calculated in a straightforward manner.The value of Sr/S that has been focussed on here represents a proportion

ately increased passage time of flame-tips in a direction that is normal tothe axis of stretch of the vortices. It is not an overall reactant consumptionrate since this would depend on an entire flame-surface, although it muststrongly influence the reactant consumption rate since it would determinethe spread of combustion from one region to another. Its maximum, in thestudy carried out here, arises purely through kinematic effects and hasnothing to do with finite-rate chemical kinetics, extinction or flame curvature effects since only a constant normal flame-speed has been assumed. Anumerical study of some of these effects is being carried out.The technique of ray tracing has been used here to examine two-dimen

sional flame propagation in a steady flow. The technique is equally applicable to three-dimensional time-dependent flows.AcknowledgementsInitial work on this project began while a.s. kerr was supported by anEPSRC research grant and J.W. Dold was supported by an Advanced Fellowship from the EPSRC, held at the University or Bristol.References[1] Craik, A.D. D. and Criminale, W.O. (1986)"Evolution of wavelike disturbances in shearflows: a class of exact solutions of the Navier-Stokes equations," Proc. R. Soc. Land A,406, 13-26.[2] Dold, J. W., Kerr, O. S. and Nikolova, I. P. (1994)"Flame propagation through periodicvortices," Combust. Flame, 100, 359-366.


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[3] Kerr, O. S. and Dold, J. W. (1994)"Periodic Steady Vortices in a Stagnation Point Flow,"J. Fluid Mech., 276, 302-325.[4] Lighthill, J. (1978) Waves Ln Fluids, Cambridge University Press.[5] Markstein, H. H. (1964) Unsteady flame propagation, Pergamon Press.[6] Van Dyke, M. (1975) Perturbation Methods in Fluid Mechanics, Parabolic Press.

APPENDIX

Here we presenl the analysis of a flame propagating past an array of vortices in the limit of large flame speed, S, In this limit the flame passes pastthe vortices with only minor perturbation. In this motion there are two timescales, the time taken for the flame to propagate a period of the vortices,and a time scale associated with the cumulat ive effect the vortices on theflame. We will use a multiple-scales analysis (see, for example, Van Dyke,1954). This will enable us to find asymptotic solutions that are uniformlyvalid for all time. If we define

(16)to be our small parameter, then we have two time scales, a fast time,T = E- 1 t, and a slow time, f f = t and consider the variables to be functionsof these two time scales. The governing equations (6-9) then become

ax ax kaT + Eoff = W(x, z)+ (k2+m2) 1/2'

oz oz maT +6off = 6W(X,Z) - dz + (k ' +m2) 1/2'ok ok au ow-+6-= - 6 k - - 6m - ,aT off ax ax

am am au ow-+6-= - 6 k - - 6m - +6m.1..aT off oz OZWe also expand the variables in an asymptotic series in 6:

(17)

(18)

(19)

(20)

(21)


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122 O. S. KERR AND J.W. DOLO

These expansions are substituted into (17-20) and terms of like order arecollected together.

At leading order the equations are

These have solutions

ko( k ~ + m ~ ) 1 1 2 '

( k ~ + m ~ ) 1 1 2 '

ako-ar- = 0,amo=oaT .

(25)

(26)

(27)

(28)

(29)

(30)

(31)(32)

As we are looking for periodic flames confined to the region of the vorticesZo cannot grow linearly with T. Hence

(33)


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and hence(34)

for a flame travell ing in the direction of increasing x. Further, because of thesymmetry of the flow, the vertical position of the fastest ray would becentred on z=O, hence Bo=O.

The order E terms give the equat ions

ak l __ aco _ c au(xo,O)ar - a.r 0 ax '

(35)

(36)

(37)

(38)

At this point it is useful to introduce the expression for the stream functionfrom Kerr & Dold (1994). The stream function is given in terms of itsFourier components in x by

00ljJ(x,z)= L cn(z)cos(2n-l)x+dn(z)sin2nx.n=1

(39)

When this expansion is substituted into the order E equations the followingsolutions found:

aAo 00 c' (0) . d' (0)XI =AI( .r)-r 'dT - L -2 sm(2n-l)(r+A o)---'!..-2cos2n(r+Ao), (40)0;:1 n ~ n - I n

00+ L (cn-


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124 O. S. KERR AND J.W. DOLD

00 c"(O) d"(O)ml =D 1 +C o I Z_lsin(Zn-l)(r+A o) - .2-z cosZn(r+Ao) (43)n= 1 n nNote that the functions c.(z) and d.(z) are evaluated at z=O in these expressions as Zo =O. In the higher order expansions the relevant term fromthe Taylor expansion about z=O will be used. Again the requirement thatthe solutions remain periodic gives

D _ ac o _ O1- afT - . (44)At the next order there will also be the requirement that aAo/afT = O. Thisrequirement is introduced here to save some algebra.The order 0 2 equations are

(45)

(46)

(47)

(48)

Examining the equation for x2 we see that for this term to remain boundedall the constant non-oscillatory terms must cancel. These not only includethe terms such as the derivative of A I with respect to T, but also the termsobtained from the products of the Fourier expressions for the velocitycomponents and their derivatives. This gives the condition that


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(49)

or

(50)

Further, with the symmetries used for the vortices, all the c. (z) are evenfunctions and all the d. (z) are odd functions, so

b (0)= d. (0)= d (0)= 0This gives

!T d' (0)2 '0) " (0)

oliver s. kerr and j.w. dold- flame propagation around stretched periodic vortices investigated...

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