Taylor-Couette burner for studies of hydrodynamically unstable weakly turbulent premixed flames

Inc. Comm. Heat Mass Transfes Vol. 29, No. 1. pp. 15-24, 2Oil2 Copyright 0 2002 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/02/$-see front matter

Pergamon

PII: SO7351933(01)00320-7

TAYLOR-COUETTE BURNER FOR STUDIES OF HYDRODYNAMICALLY UNSTABLE WEAKLY TURBULENT PREMIXED FLAMES

R. C. Aldredge Department of Mechanical and Aeronautical Engineering University of California, Davis, CA 95616-5294

(Communicated

by J.P. Hartnett

and W.J. Minkowycz)

ABSTRACT The design of a novel, new Taylor-Couette burner for studies of the propagation of hydrodynamically unstable flames in weakly turbulent flow is presented. The new burner allows for turbulence generation by cylinder rotation at constant cylinder radii ratio and for convective stabilization of stationary flames at a fixed location in the burner, for long-time flame diagnostics. The burner is designed for investigation of the development of the hydrodynamic flame instability with minimal influences of the Saffman-Taylor instability. 0 2002 Elsevier Science Ltd Introduction

Freely propagating planar flames are known to be unstable to large-scale flow-field fluctuations, such that small-amplitude

wrinkles that develop in the flame surface will grow and result in acceleration

of the flame [ 11. This is because the reactant flow speed decreases

(increases)

along streamlines

approaching the flame in regions where the flame surface is convex (concave) toward the reactants, and as the motion of the flame is governed primarily by the local reactant flow speed, the flame will tend to become more wrinkled, as both convex and concave bulges grow in amplitude and variations in the local reactant flow speed increase over time. increase in flame surface area.

Acceleration

This hydrodynamic

dynamics of the flame and of the incompressible

of the flame is a consequence instability associated

of the resulting

with coupling between the

reactant flow was discovered independently

by Darrieus

[2] and Landau [3]. A planar flame acceleration influences

will be unstable

if the Darrieus-Landau which act to suppress

to reactant-flow

hydrodynamic flame wrinkling.

fluctuations

and undergo

wrinkling

and

instability is stronger than any existing stabilizing Two stabilizing

mechanisms

are well known,

Vol. 29, No. 1

R.C. Aldredge

16

those associated with thermal-diffusive one-dimensional stabilized

acoustic waves has been recently established

by thermal-diffusive

wavelength

effects and buoyancy [ 11, and a third mechanism associated with

effects

wrinkles are stabilized

for sufficiently

[4, 51. Small-wavelength

large reactant

by buoyancy for downward

wrinkles are

Lewis numbers,

while large-

Acoustic waves

flame propagation.

propagating along the axis of flame propagation behave in a manner similar to that of buoyancy-induced gravity waves, inducing a local oscillating acceleration field that dampens flame wrinkles [6]. Unlike buoyancy, however, the influence of these one-dimensional even for flames that are not propagating

downward

acoustic waves is expected to be stabilizing

and, thus, is independent

direction of a gravitational field [6]. The strength of the hydrodynamic

of the presence of or

instability itself increases with the

strength of the reactant mixture, as represented by the amount of heat released per mass of reactants or the value of the planar, laminar flame speed.

Therefore,

whether flame propagation

in a given reactant

mixture will be stable or not depends on the strength of the mixture, as determined by the equivalence ratio and the extent of heat loss to the burner walls, as well as the size of the combustor and the presence of a gravitational or acoustic acceleration field. As a practical matter, it is of interest to determine the relationship between the burning rate of a premixed flame and characteristics

of the reactant flow.

This relationship

is well understood for one-

dimensional flame geometries such as for planar or spherical flames but not otherwise, for example when the flame surface is wrinkled as a result of the development fluctuations. intensities

For flame propagation of velocity fluctuations

turbulence.

of instability or from imposed velocity

in turbulent flow the burning rate is expected and the length and time scales characterizing

to depend on the

the structure of the

Specifically, the burning rate of a premixed flame may be enhanced as a result of either flame

wrinkling by large-scale velocity fluctuations.

velocity fluctuations

or local combustion-zone

Various theoretical predictions and experimental

speed to turbulence intensities have been reported in the literature. work has focused on the regime of high-intensity predictions,

modification

correlations

by small-scale

relating the burning

Essentially all of the experimental

turbulence (for a comparison

of measurements

and

see Aldredge et al. [7]), while most theoretical predictions of the burning rate for the low-

intensity regime have neglected influences of buoyancy and hydrodynamic coupling.

An exception is the

theoretical work by Aldredge and Williams [8], where solutions to linearized conservation equations were obtained

and correlated

to determine

influences

of reactant turbulence

intensity,

mixture strength,

buoyancy and flame structure on the burning speed. Such analyses based on linearization as well as those restricted UrIcI,

to small-amplitude,

=I+const+ilU,)2;

periodically

varying

w h ere U, and U,

reactant

flow

have

invariably

are the planar and wrinkled-flame

predicted

that

burning speeds,

respectively, and u’ is the intensity of velocity fluctuations in the reactant flow, considered small relative to U,

A fundamentally

different relationship,

from nonlinear analysis, with hydrodynamic

U,/U,= 1+ const (u’l U, )4'?, was

predicted recently

coupling neglected and the flame considered

as a passive

interface in a randomly stirred fluid [9]. The TC burner design presented herein would allow studies of flame propagation in low-intensity turbulence

and examination

of the relationship

between

U,

and u’, as well as the influence

of

TAYLOR-COUETTE

Vol. 29, No. 1

hydrodynamic coupling on this relationship.

BURNER FOR STUDIES OF FLAMES

17

Results of such studies would be useful for comparison with

future analytical and computational predictions that account for nonlinear hydrodynamic

coupling.

TC-Burner Background

Taylor-Couette

(TC) flow is generated

in the annulus between

two independently

rotating

concentric cylinders, and different regimes of flow are achievable by adjustment of the relative cylinder rotation rates [lo-121.

In particular, if the outer cylinder is fixed, then steady laminar Couette flow is

observed at low rotation rates of the inner cylinder, while steady toroidal vortices (Taylor vortices) are observed at moderately high inner-cylinder

rotation speeds.

At very large inner-cylinder

rotation speeds,

or when the cylinders are rotated rapidly in opposite directions, a featureless, turbulent velocity field is observed throughout the annulus [ 131. A TC apparatus was proposed

by Ronney et al. [14] for studies of isothermal

propagation in aqueous turbulent flow and later [ 151 for studies of high-temperature-flame gaseous hydrocarbon air mixtures.

A TC combustor designed for gaseous turbulent-flame

151 has several key advantages over other burning configurations These include: (a) turbulence

characteristics

uniform over most of the flame-surface

conditions

propagation in propagation [7,

used for turbulent-combustion

that are statistically

steady upstream

studies.

of the flame and

area; (b) turbulence intensities that are independent of the mean

flow speed along the direction of flame propagation; velocities and extinction

reaction-front

(c) a mean strain rate, known to affect burning

of turbulent flames, [16] which is zero over most of the annulus

width; and (d) the capability of generating either large-scale, low-intensity

or small-scale high-intensity

turbulence, depending on the relative speeds of the inner and outer cylinders [7].

Design Overview

The Taylor-Couette

burner proposed for studies of weakly turbulent flames is shown schematically

in Fig. 1. It consists of two variable-diameter

concentric cylinders that can be rotated independently

in

either direction. Between the two cylinders is an annulus of width 1 cm (I .3 cm) in the lower (upper) constant-diameter

section of the burner. The annulus is filled with a combustible mixture introduced at the

bottom of the burner that is ignited at the open, top end of the burner. A flame established at the top upon ignition then propagates downward and becomes convectively

stabilized in the region of varying cylinder

diameter because of the variable annulus area in that region, where the opposing upward reactant flow increases along the direction of flame propagation. The total height of the TC burner is 60 cm, with lengths of the upper and lower constant-diameter

sections of 15 cm and 35 cm, respectively,

and a IO-cm

height for the region of variable annulus area. The length of the upper section was chosen to allow enough time for settling of unsteady ignition-generated

perturbations

before the flame enters the variable-area

section. The outer cylinder may be of Pyrex construction to allow optical diagnostics of flame and flow-

18

Products

Pro ucts

1.3cm annulus + width

+

15.4 cm diameter -_)

turbulent flame

1 cm 35 cm height

annulus wtdth

I

Rotating Inner Cylinder

Rotating Outer Cylmder

J-Reactant Inlet

Taylor-Couette

FIG. 1 burner for studies of weakly turbulent flames

Reynolds numbers based on the annulus width d and on the tangential velocity of the inner and outer cylinders may be defined.

Hence, Re, = ~,l;d

IV

and Re, = o,,r;,d

IV ,

where W, and W, are the

angular rotation rates, and c and 5 are the radii of the inner and outer cylinders, respectively; while the kinematic

viscosity

of the reactive mixture.

apparatus occurs when Re, > 50,000

The onset of homogeneous

turbulence

V

is

in the TC

if the outer cylinder is fixed [lo, 13, 171. However, if the outer

cylinder is rotated in the direction opposite to that of the inner cylinder a significantly Reynolds number Re__ = (Re, + Re,, )/ 2 of approximately

lower minimum

1000 defines the homogeneous-turbulence

regime. In this case, Re, and Re,, must be of the same order of magnitude, with the ratio Re, / Re,, being slightly less than unity. If this ratio is too large then steady Taylor vortices are present, while if it is too small then steady laminar Couette flow exists [13]. Although earlier TC combustion experiments focused primarily on flame propagation in high-intensity

have

turbulence, where Reov, is much higher than

1000, TC turbulence intensities as low as ten percent of typical hydrocarbon-air

laminar-flame

speeds

have also been generated for Reovr near the lower limit of the turbulence regime [7, 18, 191. The annulus width d of the TC burner must be carefully selected, as it must be not too small if heat loss to the cylinder walls is to have a negligible effect on the flame speed. The smallest annulus width of 1 cm, in the lower constant-area

section of the burner is over five times the quenching

distance for

Vol. 29, No. 1

TAYLOR-COUETTE

BURNER FOR STUDIES OF FLAMES

19

methane-air or propane-air flames propagating in tubes or between plates [20] and, therefore, heat loss to the cylinder walls should not be substantial

Geometry

of the Variable-Area

Section

The geometry of the variable area section of the TC combustor

is designed

so as to maintain a

constant cylinder radii ratio < lr, along the burner axis while providing for a decreasing annulus area along the direction (downward) of mean flame propagation, for convective flame stabilization. It is also desirable to minimize influences of the Saffman-Taylor

instability [21], which may cause flame wrinkling

and acceleration independent of influences of the Darrieus-Landau hydrodynamic

instability.

It is preferable to keep the radii ratio constant along the axis of the TC apparatus since this was an attribute

of

earlier

configurations. character

characterizing

In particular, experiments

for < /r,

sufficiently

studies

cold,

turbulent

fluid

flow

in

constant-annulus-area

have shown TC turbulence to have a nonstationary,

less than about 0.78 unless the TC Reynolds

numbers,

as defined

bursting

above, are

large [22]. With linear variations of the cylinder radii along the burner axis (slanted, but

straight cylinder walls), the following equations define the geometry of the variable-area section of the burner.

In Eq. (1) the subscript 1on the radii variables refers to the lower constant-diameter the parameters respectively,

section of the burner,

s, and s, are the slopes (relative to the burner axis) of the inner and outer cylinder walls, and z is the axial coordinate, measured from the bottom of the variable-area section of the

burner. Choosing

s, = t-s,,, where

r = < / 5, then gives a constant radii ratio along the burner axis,

through the variable-area section. With the TC burner dimensions as shown in Fig. 1, one has r = 0.86, s, = 0.17 and s,, = 0.2. The maximum cylinder radii, in the upper constant-area section, for a given radii ratio is chosen so as to ensure that the upward mean speed of the reactants in the upper section is about two-thirds of its value in the lower section of the combustor. In this way, a downward-propagating

flame

located between the two sections will experience increasing (decreasing) opposing reactant flow speeds if it propagates forward (backward) toward the smaller-area (larger-area) section, and stabilization of the flame at a fixed location in the variable-area section will be possible. The slopes of the cylinder walls in the variable-area section are defined by the height of the variable-area section once the maximum cylinder radii have been specified

and are made to be reasonably small, so that the direction of mean flame

propagation is essentially directly downward. It is important that the reactant-flow with the inverse of the characteristic

strain generated in the variable-area section be small compared

laminar-flame

reaction time. The former quantity can be estimated

20

Vol. 29, No. 1

R.C. Aldredge

as U, /h , where h is the height of the variable-area section, while the latter quantity is of the order of U, / 6 , where 6 is the flame thickness.

Therefore, the effect of the strain rate imposed by the reactant

flow on the structure of the flame in the variable-area section of the burner is of the order of 6 /h , less than 1O-3for h equal to 10 cm as shown in Fig. 1 and a representative ratio U, /U,

Suppression of

is desirable propagation in

and

Saffman-Taylor Instability

isolate influences TC burner.

the dynamics mixtures on

(DLI)

wavenumber,

as

on either

of the

the Darrieus-Landau

requires suppression

interfaces separating side of

estimate the instability

value of 6 equal to 0.05 mm; the

is considered to be of order unity for weakly turbulent flow.

propagating flame.

relative significance

such as

their

to the

In this

reactant

Joulin and

instability (STI)

from piecewise

on [21], which

having different

of the comparing

instability

the Saffman-Taylor

[23], one

to the

rate of

perturbation

solution of

of

Euler equations

the wavenumber-dependent

J, is

to represent

the STI,

fP,

J, =(j-1) PJJ,VJ,

-

f!J,

-U,W

(2) Jo_

g’P”-P,’

f&J,

-

fJJL I

In Eq. (2) k is the wavenumber of the flame-surface the products of combustion, the gravitational acceleration friction factors representing

constant (negative for downward flame propagation) the proportionalities

weak for downward

( j < 0 ), J, < 0 and flame-surface for sufficiently

U, is the local flame speed relative to

p, and P,, are the densities of the reactants and products, respectively,

strong downward

and f,

flame propagation

as defined in [23]. When the gravitational field is (0 < j < 1) or for upwardly

propagating

fields ( j > 1) only perturbations

When

flames

However,

having wavenumbers

larger than a critical value, defined by 0 < J, < 1, are unstable, while smaller-wavenumber corresponding

and f, are

perturbations are found to be unstable at all wavenumbers. gravitational

g is

of the mean pressure gradient with U, and U, in the

reactant and product regions of flow, respectively, sufficiently

perturbation,

perturbations

to 1, > 1 are stable.

j = 0 the magnitude

of J,

is a measure of the growth rate of unstable

flame-surface

perturbations due to the ST1 relative to that due to the DLI, otherwise it measures the combined influence of the ST1 and effects of gravity relative to that of the DLI. For sufficiently small wavelengths of flamesurface perturbation the ST1 would be negligible in importance relative to that of the DLI. It should be

TAYLOR-COUETTE

Vol. 29, No. 1

noted, however, that flame-surface thermal-diffusive

perturbations

21

BURNER FOR STUDIES OF FLAMES

of sufficiently

small wavelength

may be stabilized by

influences [1], and thus the strengths of both the DLI and the ST1 may be significantly

decreased at large wavenumber, depending on the thermodynamic It can be shown that the criterion for negligible

significance

properties of the combustible mixture. of the ST1 relative to that of the DLI

(I.!, ) <<1 with j = 0 ) may be expressed as

where

o,(P”~PlJ3’2-l

PUIPb--1

I

(4)

6=alU, Prsvla

and /z and a are the flame-surface mixture, respectively.

perturbation wavelength and the thermal diffusivity of the reactive

In deriving Eq. (3) it is assumed, as in [23], that the friction factors f, and f, are

proportional to the absolute viscosity of the respective mixture and to the inverse of the square of the annulus width. Since the circumference wavelength, the criterion for insignificance

of the burner annulus is the largest possible

perturbation

of the ST1 according to Eq. (3) is in fact a constraint on the

cylinder radii for a given annulus width. It is also evident from Eq. (3) that more energetic flames (larger cr) require smaller-diameter when the expression circumference

for 6

burners for isolation of the DLI. Eq. (3) reduces to 2 < (2&j, given in Eq. (4) is used. Upon replacement

of a

/30v)d2

with the annulus

n(q + ru) a constraint on the outer cylinder radius for a given radii ratio and annulus

width is obtained, namely,

(5)

With the representative

parameter values CJL= 40 cm/s, p, / ph = 8,

r = 0.86, one obtains the criterion

v =

0.15 cm’/s and the radii ratio

7, ==K 31cm (r, <<52cm) for the lower (upper) constant-diameter

section of the burner where d = 1 cm (d = 1.3 cm). Since the actual outer-cylinder

radii are 7 cm and 7.7

cm in the regions below and above the variable-area section of the burner, respectively, one would expect the maximum strength of the ST1 to be about 22% (i.e., 7/31) and 15% (i.e., 7.7/52) of that of the DLI in the lower and upper constant-diameter

sections, respectively,

considering Eq. (2) and the expression for

the perturbation growth rate given in [23]. However, the relative strength of the ST1 is maximum only for

R.C. Aldredge

22

Vol. 29, No. 1

the largest possible perturbation wavelength (the annulus circumference) the wavelength. wavelengths

and decreases in proportion to

It may be more relevant to estimate the strength of the ST1 at smaller perturbation

where the influences

of the DLI are strongest since isolation of DLI effects is of key

concern. Theoretical and experimental

studies have found the wavelength of perturbations most unstable

to influences of the DLI to be on the order of 2 cm 14, 24-261. At this wavelength the strength of the ST1 is found to be only 1.3% and 1.8% of that of the DLI in the lower and upper constant-diameter respectively.

Notwithstanding

value of the most-unstable

these small relative contributions, perturbation

wavelength

the ST1 may significantly

sections, affect the

when the criterion of Eq. (5) is not adequately

satisfied.

Summary

The design of a novel, new Taylor-Couette burner for studies of the propagation of hydrodynamically unstable flames in weakly turbulent flow has been presented. generation stationary

by cylinder flames

rotation at constant

at a fixed location

importance of the Saffman-Taylor circumference

cylinder

The new burner allows for turbulence

radii ratio and for convective

in the burner, for long-time

flame diagnostics.

stabilization

of

The relative

instability has been minimized by appropriate selection of the annulus

and width for a given radii ratio.

Aeknowledement

The author would like to thank the NASA Glenn Research Center for supporting the work presented herein. Nomenclature

speed of laminar flame relative to reactants and to products, respectively turbulent flame speed turbulence intensity densities of reactant and product mixtures friction factors for reactant and product mixtures Reynolds number for inner and outer cylinders, respectively average of inner- and outer-cylinder Reynolds numbers rotation rates of inner and outer cylinders, respectively radii of inner and outer cylinders, respectively slopes of inner and outer cylinder walls in variable-area section laminar-flame thickness annulus width

Vol. 29, No. 1

TAYLOR-COUETTE

BURNER FOR STUDIES OF FLAMES

h

height of variable-area section of burner

V

kinematic viscosity of reactant mixture

;t,k

wavelength and wavenumber of flame-surface perturbation, respectively

cr

function of volumetric expansion across a laminar flame

j

ratio of importance of gravity influence to that of Saffman-Taylor

Jk

ratio of importance of gravity and Saffman-Taylor

23

instability

influence to that of the Darrieus

Landau instability

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Received November

2, 2001