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Effect of Near-Critical Swirl on Burke-Schumann Flame

by Kang-Ho Sohn+ and Zvi Rusak++ and Ashwani K. Kapila+++

Rensselaer Polytechnic Institute, Troy, NY 12180-3590

Abstract

The influence of a near-critical swirling flow on the shape of the Burke-Schumann flame

reaction sheet in a straight cylindrical chamber is investigated by asymptotic and numerical

analyses. A high Reynolds number laminar, isothermal, and incompressible flow is

assumed. An asymptotic analysis is developed to study the nonlinear interaction between

the swirl at near-critical levels and the fuel/oxidizer mixture mass fraction distribution. It is

first found that the leading-order changes of the velocity field from a columnar state can be

described by a nonlinear reduced-order model. Then, these flow changes are used to

calculate the corrections due to swirl to the classical flame sheet structure according to

Burke and Schumann solution. The resulting corrections apply to both lean and rich

conditions of combustion. The asymptotic results show a nice agreement with direct

numerical simulations, specifically as the flow Reynolds number is increased. This study

shows that as swirl is increased toward the critical level the flow decelerates near the

chamber centerline, accelerates near the chamber wall, and an outward radial speed is

developed. For a sufficiently high level of swirl a large separation (near stagnation) zone

appears around the centerline. As a result, for lean combustion, the flame becomes shorter

and more compact as swirl is increased. For rich combustion, the flame length

+ Graduate Student, Department of Mechanical, Aerospace, and Nuclear Engineering.

++ Professor, Department of Mechanical, Aerospace, and Nuclear Engineering.

+++ Professor, Department of Mathematical Sciences.

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decreases with the increase of swirl and then increases after vortex breakdown appears.

This work extends for the first time the theory of the Burke-Schumann flame sheet to

include the effect of swirl.

Introduction

Diffusive (non-premixed) combustion has attracted attention over many years since it is

considered to be safer and more economical than premixed combustion. The classical

solution of Burke and Schumann1 describes the steady-state flame sheet structure that

results from two co-axial jets of fuel and oxidizer that are injected into a straight cylindrical

chamber (see also Williams,2 pages 40-44). Their formulation studied the balance between

streamwise convection and transverse diffusion of the species according to only the species

conservation equation. Constant axial velocity and infinite rates of reaction were assumed

and effects of heat release due to chemical reaction, transverse convection resulting from

thermal expansion effects, axial diffusion of species, and viscous dissipation were

neglected. Despite these significant simplifications, in comparison with the physical nature

of flames, the Burke-Schumann approach describes the fundamental character of the flame

sheet and this model problem was extensively used to illustrate the characteristics of

diffusional combustion.

Williams2 (pages 73-76) used only the species and energy conservation equations and

extended the Burke-Schumann approach to compute the steady-state fields of the mixture

mass fraction and temperature in a diffusional flame for the case where the Lewis number

(Le) is unity. It should be noted, however, that the flow momentum equations were not

considered in the analysis; actually these equations cannot be satisfied with the solutions of

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the mass fraction and energy equations. The study of Roper3 modified the Burke-Schumann

theory to satisfy the continuity equation when the velocity and temperature fields are not

constant. Again, the flow momentum equations were not considered in the analysis. He

predicted a small effect of the modifications on the flame size in the case of a circular port

burner. The experiments of Roper, Smith and Cunningham3 showed agreement with the

predicted flame height according to the theory of Roper.4 Klajan and Oppenhiem5

developed an analytical approach to describe the effect of exothermicity on the shape of jet

diffusion flames and the structures of their temperature and flow fields. They constructed

self-similar solutions using the Dorodnitsyn-Howarth transformation of the compressible

flow equations and found agreement of the flame length with experimental data under zero

gravity conditions. Chung and Law6 revised the Burke-Schumann formulation to include

the effects of both streamwise and preferential diffusion of species and temperature for

flows with non-unity Lewis number. They used a perturbation analysis for flames at near

unity values ofLe and described the effect ofLe on the flame temperature and shape at

various Peclet number (Pe) flows. Li, Gordon, and Williams7 analyzed highly over-

ventilated laminar diffusion flames in an infinite atmosphere at Le=1. They used the

reaction sheet approximation and included the effects of buoyancy. A boundary-layer

approximation in the stream function coordinates was used to simplify the numerical

integration of the flow and species equations. The flow radial momentum equation was not

considered in the analysis. Computed results of the flame height as function of fuel flow

rate for various hydrocarbon fuels showed agreement with experimental data.

The mode of diffusional combustion was used in various systems. However, it may

suffer from flame instabilities, lift-off situations, and extinction as function of the

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Damkohler number (see, for example, Williams,2 page 80-84). In recent years it has been

found that exerting swirl on diffusion flames may help to eliminate blowout, reduce flame

lift-off distance, improve flame stability, and enhance combustion performance (see, for

example, the experimental results in Gupta, Lilley, and Syred,8 Marshall and Gupta,9

Lefebvre,10 Stephens, Acharya, Gutmark, and Allgoodn,11andCha, Lee, and Chung12). The

flow field of a combustion state with high levels of swirl has complex patterns including

large-scale internal separation (vortex breakdown) zones that create transverse convection

which significantly affect the flame length and shape. Therefore, the analysis of such states

requires modeling of the nonlinear flow and combustion interactions and is very difficult to

conduct. All of the current studies concentrate on extensive numerical simulations of the

flow and chemical reaction fields (see, for example, the works of Khalil and Spalding,13

Elgobashi,14 Habib and Whitelaw,15 and Shim, Sohn and Lee16). To the best of our

knowledge, there is no theoretical study of diffusive combustion with swirl.

As a first step toward developing a better understanding of the special effects of

swirling flows on diffusional flames, we focus in this paper on the influence of flows with

near-critical swirl levels on the classical flame structure of Burke and Schumann.1

Understanding this interaction can help in identifying the fundamental physical

mechanisms that develop in such a process of chemical reaction with complicated fluid

mechanics. We conduct an asymptotic study and numerical simulations and look for the

correlation between them. The analysis may also provide a reduced-order model to

describe the effect of swirl level, viscosity, fuel to air equivalence ratio, and chamber

geometry on the flame sheet shape and stability of the infinite-rate reaction process.

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We consider here the case where the swirl is near the critical level for vortex

breakdown, the flow is laminar and Reynolds number is high, and there is no heat release

as a result of the chemical reaction. In such a case, the solution of the flow field is

decoupled from that of the species. Also, the strong interaction between the swirl and small

viscous effects can be accurately described by the asymptotic approach of Wang and

Rusak17 which shows a nice agreement with the numerical simulations according to the

Navier-Stokes equations of Beran and Culick,18 Beran,19 and Lopez.20 In the present paper

we use this asymptotic solution to estimate the changes in the flame sheet structure from

the Burke and Schumann

1

solution due to swirl. These theoretical predictions have also

been numerically substantiated through extensive simulations of the flow field and flame

shape. The present work provides for the first time a theoretical model to study diffusional

combustion with swirl.

Mathematical Formulation

The mathematical formulation of the combustion problem with a swirling flow in a

straight pipe is based on several assumptions about the nature of the flow and the process

of reaction. We consider the limit of zero heat release and low Mach number. Then

energy balance implies that the flow is isothermal, i.e.,

T= T0 , while the gas law leads to

the conclusion that density is constant, i.e.,

r= r0. Following Burke-Schumann we also

consider infinitely fast chemistry so that the reaction is confined to a thin sheet.

The mixing rate is lower than the chemical reaction rate and the latter is often of negligible

importance for diffusion flame. The chemical reactions are typically completed in a very

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narrow region which suggests that the flame is identified as the surface on which the fuel

and oxygen rates of delivery are in stoichiometric proportions. In this study, we consider

reaction with an infinite rate but without any heat generation. Therefore, the temperature

field is constant and in the case of low Mach number flows the density is also nearly

constant. Then, the swirling flow may be assumed incompressible. It is also assumed that

the coefficient of inter-diffusion of the two gas (fuel/oxidizer) streams is constant. Then the

non-dimensional continuity, momentum, energy and species equations governing for a

steady, axi-symmetric, incompressible and viscous flow of a reactive non-premixed fluid

are:

( ) ,01 =+ xr wrur

(1)

uur+ wux - x02 v

2

r= -x0

2pr+x0

Re

1

rru( )

r

r

+uxx

x02

,

(2)

( ) ,12

0

0

+

=++

x

vrv

rRe

xwv

r

uvuv xx

r

rxr

(3)

uwr+ wwx = - px +x0

Re

1

rrwr( )r+

wxx

x02

,

(4)

( ) ,12

0

0

+=+

x

frf

rPe

xwfuf xx

rrxr

(5)

f=nYF - YO +YO,0

nYF,0 +YO,0, where n =

WO nO - nO( )

WF n F - n F( ).

(6)

These dimensionless forms are based on the following reference properties. The axial

length scales with the pipe length 0l and the radial length with the pipe radius, 0r , with

000 rlx = being the length-to-radius ratio. We consider a sufficiently long pipe such that

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10 >> . The axial and circumferential speeds are scaled with the inlet axial speed 0U and

the radial speed with 00 xU . The speeds u, v and w are the non-dimensional radial,

circumferential, and axial components of the velocity, YF and YO denote the fuel and

oxygen mass fraction in reaction, and

f is the mixture mass fraction, derived for example

in [ ]. Also, WF and WO are the molecular weights of fuel and oxygen, respectively, i is the

stoichiometric coefficient for species i appearing as a reactant, i is the coefficient for

species i appearing as a product, is the stoichiometric oxygen-to-fuel mass ratio. The

dimensionless parameters appearing in these equations above are00

2

0

0

= , the

reference Mach number,

000 rURe = , the Reynolds number, and

D

UrPe

000= , the

Peclet number, where 0 is the flow constant density, is the constant viscosity, D is the

constant diffusion coefficient of fuel and oxygen, and

the ratio of specific heats. The

pressure

p appearing in the above equations is the pressure excess above the ambient value

p0 = r0RT0, in units of

M02 p0, i.e.,

p = p0(1+ gM02p) . We consider a sufficiently long

pipe and a largeRe such that the following asymptotic ordering applies:

x0 >>1, Re >>1, andx0

Re


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10 0


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Here, e.s.t. denotes an exponentially small term, in accordance with the Burke-Schumann

solution and the expected columnar state at the outlet for

x0 >>1. This is in accordance

with an expected columnar state at the outlet whenx0 is sufficiently large.

The pipe axis of symmetry is a streamline along which the radial and azimuthal velocity

components vanish, as do the radial gradients of the axial velocity and mixture fraction.

Consequently we set

u x,0( ) = vr x,0( ) = wr x,0( ) = fr x,0( ) = 0 for 0 x 1. (10)

Only the impermeability, but not no-slip, condition is enforced along the wall r=1. , in

accordance with the expectation that in a high Re flow the wall boundary layers are very

thin with respect to the pipe radius and a boundary-layer analysis can be conducted near the

wall to apply the no-slip condition. However, such a local analysis is beyond the scope of

this work. This condition ignores the influence of wall boundary layers, with the

expectation that this influence exerts at most a qualitative effect on the phenomenon under

study; see for example, the relevant discussion in [ Beran and Culick] and [Lopez ]. Also,

along the pipe wall, the radial gradient of the mixture mass fraction vanishes. Thus it is

assumed that

u x,1( ) = fr x,1( ) = 0 for 0 x 1. (11)

By virtue of the axisymmetry and the continuity equation (1), a stream function

( )yx, can be defined where the radial component of velocity is ru x= and the axial

component of velocity is rw r= . Let With 22ry = , yu x 2= , yw = , and the

azimuthal vorticity is r= where

= yy +y xx 2yx02

( ). Along the inlet we find that

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( )y0y= where ( )ywy 00 = and yw00 = . The circulation function K is defined as

rvK = and along the inlet we have and ( ) ( )rrvyKK 00 w== . Equations (15) can now

be expressed as The reactive flow equations which relate between the stream function

,

azimuthal vorticity, and the circulation functionKare derived from (1)-(5) and given by

,2

12

2

0

0

+=

(12)

( ),4

2

12

4 20

0

2

2

+

++=- y

xxyy

xyxxy

yxy

Re

x

y

Kc

cccyc (13)

( ).2

20

0

+=-x

fyf

Pe

xff xx

yyyxxy

y(14)

Note that the pressure is eliminated from the problem by the cross differentiation of (2) and

(4) and introduction of. The boundary conditions (7)-(11) become

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) .210fore.s.t,1,0,1,1,12where,1for0,0,0for1,0

,210for,0,0,0,,0

,10for021,0,,2121,,00,

2

00

0

====

=


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as a result, so is the perturbation to the reaction-sheet location. However, as the swirl level

increase towards the critical, changes in the flow become more pronounced [give ref], and

correspondingly larger perturbations in the reaction-sheet location are anticipated. The

following asymptotic analysis estimates these changes. We begin by postulating the

asymptotic expansions

It is expected that the swirl and viscous stresses create transverse convective effects in

the reactive flow in the pipe that influence the shape of the flame sheet. To study the effect

of the near-critical swirl ( c where c is the critical swirl) and the slight viscosity

( )10


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From the boundary conditions (14), we find that the solution (15) has to satisfy

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) .210fore.s.t,1,0,1,1,1,210for0,0,0,0,0

,10for021,0,21,0,

1111

1111

1111

====

====

====

yyfyKyy

yyfyKyy

xxfxfxx

xxxx

xx

yy

c

y

y

(16)

Flow-Field Disturbance

We first solve the flow field perturbation. Following the analysis of Wang and Rusak17

we can show that in the leading-order terms

,1

0

01

= (17)

and

( ).2

Re

1

2

1

0

00

20

2

1

0

0

2

2

1

0

2

0

0

2 xK

yxKK

Ky

yy

yyy

y

yy

y

yey

y

ye

e

yy

y+

-= (18)

At order 1 we also find that

02

~~

2 10

0

20

2

00

20

1

1

=

+

+

(19)

( ) ( )( ) ( ) 210for,0,1,0

10for,021,0,

11

11

==

==

yyy

xxx

xy

y

(20)

Here we use 00~

KK = and let 2= . The boundary-value problem (19) and (20) has

nontrivial solutions only for specific eigenvalues values of . The first eigenvalue

2cc = of this problem was defined by Wang and Rusak17 as the critical level of swirl.

The critical eigenfunction is determined by

1 x,y( ) =y 1c x,y( ) = F y( )sinp

2x

, (17)

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where and c are found from the solution of

( ) ( ) .021,00with02

4

2

~~2

0

2

0

0

2

0

2

00=F=F=F

--W+F

y

x

w

w

wy

KK yyycyy

p(22)

The size of 1 is determined from the analysis of the second-order terms. We find that

02

~~2

2

~~

Re2

2

~~~

22

~~

1

2

~~

2

2

~~

2

120

2

00

1

0

00

20

000

2

1

0

0

0

0

2

0

0

02

0

020

200

0

21

2

0

0

20

2

00

20

222

1

0

0

20

2

00

20

111

=+

+++

+

+

+

+++

++

(23)

Here, we define ,1where


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Here

3232

211

4,

2,

3

4NM

NMNM

=== (26)

where

( ),

2

1~~ 321

0 0

0

023

0

00

230

1 dyy

w

w

wyw

KK

ywN

y

yy

y

yc

=

( ) ,2

~~2

21

020

2

00

2 dyywy

KKN y

y = (27)

( )dyyw

wyw

yw

KKN

yyyyyyy

c

+=

21

0 0

00

20

00

3

2

2

~~

2 .

Equation (25) has a real solution of 1 if and only if

Re3

16 0

2

31

3

x

N

NN

(28)

When ( ) Re316 02313 xNNN


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are special states which are limit and fold bifurcation points of solutions of the

axisymmetric Navier-Stokes equations.

From (15), (21) and (25), the asymptotic expansion of the stream function ( )yx, near

the critical swirl c is described by

( ) ( )

( )

( )

+

+=Re2

sin38

Re3

64

42, 0

1

31

0

3

2

2

2

2

0

xOxy

N

NNxNN

yyx

(31)

and the axial velocity is described by

( ) ( )

( )

( )

+= xyN

NNxNN

ywyxw y2

sin38

Re3

64

42,

1

31

0

3

22

22

0

. (32)

Flame Shape at Near-Critical Swirl

To study the flame shape in a near-critical swirling flow, we consider the expansion off

according to (15). At the leading order and when 10 >>x , the axial diffusion term in (13)

may be neglected and the function ( )yxfBS , is described by the same form of Burke-

Schumann1 equation

=

y

fy

yPe

x

x

f BSBS 20 .(33)

From (14), the function ( )yxfBS , must satisfy the conditions

( )

( )

( ) ( ) .10for021,0,

,21y0fore.s.t,1

,21for0

,0for1,0

==

=

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