A study on the structure of turbulent diffusion flame: Properties of fluctuations of velocity, temperature, and ion concentration

COMBUSTION AND FLAME 4 1 : 2 6 1 - 2 7 1 (1981)

261

A Study on the Structure of Turbulent Diffusion Flame: Properties of Fluctuations of Velocity, Temperature, and Ion Concentration TOSHIMI TAKAGI,* HYUN DONG SHIN, and AKIRA I S H I O Faculty of Engineering, Osaka University, Suita, Osaka, 565 Japan

Simultaneous measurements of the fluctuation of flow velocity, temperature, and pogitive ion concentration are made in a turbulent round jet diffusion flame. Probability density function, correlation, spatial scale, and convective velocity are obtained with the object of getting a better understanding of the structure of the turbulent diffusion flame. Main results are: (1) The structure of the outer part of the flame is characterized by intermittent appearance of high temperature, high ion concentration, and high velocity gas in nonactive, low temperature and low velocity gas. However, on the contrary, in the central part of the flame the gas parcels of the fuel or air contain reacting gases or products mixed in the molecular scale, even though the mixing is incomplete due to the large-scale turbulence and the microstructural reaction zones that exist intermittently. (2) The shape of the high temperature gas or reaction zone is stretched especially in the axial direction. The difference of the profiles of scales obtained from the fluctuations of velocity, temperature, and ion concentration is noticeable. (3) Cross correlation and probability density function of temperature and velocity are intimately related to their time-averaged profiles, and their relations are qualitatively comprehensible if we consider that gas parcels exchange positions laterally by turbulent mixing while retaining their original temperature and velocity before the exchange as is presumed in the mixing length hypothesis. (4) Convective velocity of temperature fluctuation U-TCis nearly equal to the time-averaged flow velocity U, but UTCis a little less than U at the central part of the flame and a little larger than U at the outer part. The profile of the convective velocity of the ion concentration fluctuation Uic is more flattened as compared with/~.

INTRODUCTION In order to get the f u n d a m e n t a l aspects on the structure o f t u r b u l e n t diffusion flame, the properties o f the f l u c t u a t i o n o f flow velocity, temperature, a n d species c o n c e n t r a t i o n along with their time-averaged quantities are required to be investigated because the structure should be d o m i n a t e d b y the processes o f m i x i n g a n d reaction in t u r b u lent flow. F r o m this p o i n t o f view, the statistical properties o f the flow velocity f l u c t u a t i o n were investigated in a preceding s t u d y [ 1 ] . In the present s t u d y s i m u l t a n e o u s m e a s u r e m e n t s o f the fluct u a t i o n o f t e m p e r a t u r e and i o n c o n c e n t r a t i o n as well as o f flow velocity are m a d e a n d their interrelations are e x a m i n e d . * Correspondence is requested to be addressed to T. Takagi. Copyright © 1981 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017

Measurements o f velocity [ 1 - 4 ] , temperature [ 5 - 8 ] , a n d i o n c o n c e n t r a t i o n [7-9] fluctuation have b e e n made previously in flames b u t the interrelations b e t w e e n fluctuating quantities based on simultaneous detections are very limited. More such data is necessary to get a better u n d e r s t a n d i n g o f the structure o f t u r b u l e n t diffusion flames. In the present paper simultaneous measurem e n t s o f velocity, t e m p e r a t u r e , a n d i o n concentration f l u c t u a t i o n are made in the same flame in which time-averaged velocity, temperature, and species c o n c e n t r a t i o n s were previously determined. Statistical properties o f the f l u c t u a t i o n such as p r o b a b i l i t y density f u n c t i o n , correlation, spatial scale, and convective velocity are obtained in a t u r b u l e n t r o u n d jet diffusion flame. The microscopic structure o f the flame and the m i x i n g are discussed based o n the properties o f the fluctuations.

0010-2180/81/060261+11502.50

262

EXPERIMENTAL PROCEDURE The turbulent diffusion flame is formed around an axially symmetrical turbulent fuel jet issuing vertically upward from a circular nozzle of 4.9and 6.0-mm inner and outer diameters [1]. The fuel of the nozzle fluid is a mixture of hydrogen (H2) and nitrigen (N2) whose volume ratio is 0.68:1. A small amount of methane is added to generate positive ions in the reaction zone. The average velocity of the fuel jet at the nozzle exit is 20.4 m/s, which is surrounded by a coaxial air stream of uniform velocity 5.1 m/s. A laser Doppler velocimeter (LDV) equipped with a frequency shift system by Bragg cell and a frequency tracker is applied to detect instantaneous flow velocity. The applicability of this system was examined in Ref. [1]. The instantaneous gas temperature is measured by a thermocouple (Pt - Pt.Rh.13%), 25.4 -gm in diameter, compensated electrically for the time lag of the response due to the heat capacity of the thermocouple element [10]. The compensator is adjusted using the local time-averaged time constant ~ of the thermocouple that is determined by the method of heating the thermocouple by alternating current [5, 7, 8]. The thermocouple with the compensator is supposed to follow the temperature fluctuation of 2-3 kHz. An electrostatic probe is used to detect the positive ions, which are formed near the reaction zone [7-9]. The tip of the probe, made from a Pt. Ph.13% wire,is 0.1 mm in diameter and 1.5 mm in length, and the other part of the wire is covered with quartz and copper tube whose base is water cooled. The outer diameter of the copper tube is selected to be as small as 1.6 mm to minimize the disturbance of the flow. Negative voltage of - 2 2 . 5 V is supplied to the tip of the probe and the ion current is measured. When the temperatuer or ion current is measured simultaneously with flow velocity, the scattering particles of talcum powder added to the flow for the use of the LDV are liable to stick to the thermocouple or ion probe. The data reduction is made within about 15 sec after starting the addition of the particles to minimize the effect of the deposition of the particles on the probe.

T. TAKAGI ET AL. The fluctuations of the velocity, temperature, and ion current are detected in the flame where the time-averaged flow velocity, turbulence intensity, temperature, and gas species concentration are predetermined. Gas species concentration is analyzed by a gas chromatograph after gas sampling [111.

EXPERIMENTAL RESULTS AND DISCUSSIONS The Profiles of Various Time-Averaged Quantities In Figs. 1(a) and (b) radial profiles of various timeaveraged quantities are shown at the typical cross sections where the distances from the fuel nozzle tip L are 90 and 160 mm. The abscissa r is the radial distance from the flame axis. U, T, I, u', T', and I ' are time-averaged flow velocity, temperature, ion current and the rms of their fluctuating components, respectively, o' is the rms of the fluctuating component of the flow velocity V in the radial direction. At L = 90 ram, H 2 and 02 species coexist apparently at r = 4 - 8 mm, and a temperature peak lies near stoichiometric condition at r = 6 mm. The fluctuating intensity of u' or T' becomes large where the radial gradient of the time-averaged velocity 0 or the temperature T is large, respectively. The peak of the ion current i prof'tle shifts a little to the fuel-rich region, which corresponds to the tendency previously found in a laminar diffusion flame [12]. At L = 160 mm, the coexistence region of H z and 0 2 species extends to the flame axis. The observed time-averaged coexistence does not mean the coexistence in molecular scale due to the large-scale mixing of turbulence. Temperature peak lies near r --- 6 mm but temperature is high in the central region of the flame. The time constant of the thermocouple Tis dependent on the location but it is within 4 - 9 ms in the present experiments. The fluctuations discussed next are at the typical sections of L = 90 and 160 nun of the same flame in Fig. 1.

STRUCTURE OF TURBULENT DIFFUSION FLAMES g o

,

~

Properties of Velocity, Temperature, and Ion Current Fluctuations and Their Correlations

3O 4 E ->

i

u'.lO

Typical examples of simultaneous recording of fluctuations of velocity, temperature, and ion current along with the real time t are shown in Fig. 2 at the central and outer parts of the flame. The detecting points of velocity, temperature, and ion current are located close to each other, but they are slightly shifted, about 1 mm in the axial direction. Similar simultaneous measurements of fluctuating properties are made at several points and their cross correlations Rxv ( 0 , O) are listed in

02

•

20 o

o

\

u

10 E

v,.o

ty)

E It,.,

0

1 0 ~

263

o.

E <

.i,'/-(

,.-2

~

i

3 m/sec O:14.Tm/sec

U

% 0

5

0

10 r

15

Ioo.c

T

T:1132"C

mrn

[io.,o-.A

(a) U

/ / ~ N

3O

u' i0

L

\

o,

o

E -> i

0.1

20 ,z~

t

sec

0.2

(a)

~o

-~

U

i 3 rn/sec

a;

0=6.8 m/see

0

m.. 10 "
~

!.10-z

\ ~2 ;.~o'\ ~

_,,.^-2 ,, ,u

i 300,C T=~35'C

? _~

¥0

0

5

t0

15 r mm

20

(b) Fig. I. Radial profiles o f various time-averaged quantities; (a) L = 90 m m and (b) L = 160 m m .

6

oi~

t

se¢

0'2

(b) Fig. 2. Examples of simultaneous recording of fluctuations of velocity, temperature, and ion current (L = 160 m m ) ; (a) r = 0 m m a n d ( b ) r = 12mm.

264

T. TAKAGI ET AL. TABLE 1 Simultaneous Cross Correlation Rxy(O, (a) L = 90 mm

r mm 0 2.5

5 8 10 r mm 0 12

O)

RuT(O, O)

RvT(O, O)

RTI(O,O)

-0.58 -0.63

0.01 -0.34

0.72 0.62

-0.54 -0.38 0.62 0.36 0.71 0.09 (b) L = 160 mm RuT(O, O) R Ul (0, O) -0.36 -0.24 0.65 0.47

0.25 0.63 0.66

RTI(O,O) 0.64 0.76

Table 1. The subscripts X and Y signify two quantities considered, including flow velocity in axial or radial direction U or 1I, temperature T, and ion current I. The correlation R x y(8, r) is del'med by

R x v ( 8 , r) = X(x, t) " Y(x + 8, t + r) / (X2(x, t)"

y2(x + 8, 0 ) 1/2

[1]

where x or t is the location and time of the detection. 8 is the interval between detecting points and r is the time lag. R x y(0, 0) means the cross correlation between fluctuating quantities of X and Y at the same point (8 = 0) and the same time (r = 0). The cross correlation in the present study, however, reaches a maximum at the small time lag corresponding to the small interval (about 1 mm) between detecting points. The maximum correlation is selected to be listed in Table 1. Examples of probability density function of fluctuations of temperature P(T) and ion current P(/') are shown in Fig. 3. The abscissa is the fluctuation components of temperature T - 1" or ion current I -- ] divided by their intensity T' or I', respectively. The Gaussian distribution is shown by the broken line for the sake of reference. The corresponding probability density fucntion of flow velocity P(U) is described in Ref [ 1 ]. The following items are noted from Figs. 1 to 3 and Table 1. 1. Fluctuations of flow velocity in axial direction U and temperature T correspond in the same

phase at the outer part of the flame (Fig. 2(b)), but in the reverse phase at the central part of the flame (in Fig. 2(a); the reverse phase is not always clear). Its tendency is qualitatively confirmed by the correlation RuT(0, 0) in Table 1 where RUT(0, 0) is positive in the outer part (r > 8 mm) and negative in the central part (r < 5 mm). It is noted that the positive sign of RuT(0, 0) corresponds to the same gradient of the radial profiles of time-averaged velocity U and temperature T (negative gradient of U and negative gradient of T for r => 8 mm in Fig. 1), and the negative sign of RuT(O, 0) corresponds to the reverse gradient of U and T negative gradient of 0 and positive gradient of T for r < 5 mm in Fig. 1). 2. The correlation R v r ( 0 , 0) between radial flow velocity Vand temperature Tis proportional to radial sensible heat transport by turbulence fluctuation. RvT(O, 0) in Table 1 tends to zero at the flame axis and outer part of the flame and the positive or negative sign of RvT(O, O) corresponds to the negative or positive radial gradient of time-averaged temperature T, respectively. 3. Fluctuations of temperature T and ion current I correspond almost always in the same phase (Figs. 2 (a) and (b)), which is confirmed by the positive and large values of the correlation RTt(0, 0) in Table 1. But it should be pointed out that a small value of the correlation RTI(0, 0) is obtained at r = 5 mm of L = 90 mm in Table 1. This can be understood, because the shapes of the probability density fucntion of T and I near the point at r = 5 mm o f L = 90 mm are significantly different and have opposite signs of skewness, which indicates that T and 1 have inherently different properties of fluctuation. 4. Probability density function of temperature P(T) in the outer part (r > 1 0 mm) in Fig. 3 has a nonsymmetrical shape having positive skewness and a sharp peak, which indicates the intermittent existence of high temperature gas in the relatively low temperature gas. On the contrary, at r = 6.0 mm where time-averaged temperature T has a peak as shown in Fig. 1, P(T) has negative skewness, which indicates

STRUCTURE OF TURBULENT DIFFUSION FLAMES

265

5

0.5

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0.51 05

P(T ) 0.5..

¸

0.5:

' r=12

~' . r=lO mm mr=8 mm

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-3

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1

2

3

4

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,

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0

I

2

~

3

, , ,-, mm,O,mm .

i

, r=8mm

~ r=Omm

, r-2 mm

-

(b)

(l-i)/f Fig. 3. Probability density function of fluctuations of temperature and ion current (L = 90 ram), (a) P(T) and (b) P(/).

mm

266 the intermittent existence of low temperature gas in the relatively .high temperature gas. At r = 2 mm or 8 mm where radial gradient of timeaveraged temperature T exists, P(T) has a symmetrical shape, which seems to indicate that the gas parcels of lower and higher temperature come to the point with similar opportunity from the neighboring lower and higher temperature layers. A similar tendency is observed at the cross section ofL = 160 mm. Consequently, it is noted that the probability density function of the temperature P(T) is intimately connected with the profile of the time-averaged temperature T adjacent to the location considered. 5. The probability density function of ion current P(1) has a sharp peak, as compared with the Gaussian profile, and has a positive skewness in any part of the flame because the ion current signal always has the spike shape as in Fig. 2(b). 6. The microscopic structure of the flame is presumed to be as follows. In the outerpart of the flame, inactive, low temperature, and low velocity gas prevails with the intermittent appearance of the gas of high temperature, high ion concentration, and high velocity from the central part of the flame, because ion current is often zero and the spike of the intermittent appearance of high ion current corresponds to the peak of the temperature and flow velocity as shown in Fig. 2(b). In the central part of the flame the instantaneous minimum ion current and the temperature are quite high as shown in Fig. 2(a), which indicates that the gas parcels of fuel or air should contain reacting gases or products mixed in the molecular scale, even though the mixing is incomplete due to the large-scale turbulence, and that microstructural reaction zones should exist intermittently because the ion current has spikes. The microstructural reaction zone detected by the ion current does not necessarily mean the reaction zone with significant heat release because the ion current spikes are also observed on the centerline at L = 90 mm where O z concentration is zero. 7. As noted in (1), (2), and (4), the properties of fluctuation of temperature and velocity are significantly connected with their time-averaged

T. TAKAGI ET AL. profdes. The relations are qualitatively comprehensible i f we consider that the gas mixes by large-scale turbulence so that each pocket retains the properties of the adjacent pockets as is presumed in the mixing length hypothesis.

Time-Space Correlation, Scale, and Convection Velocity of the Temperature and Ion Current Fluctuation Examples of simultaneous recording of the temperature fluctuation from two recording thermocouples with interval 8L in the axial direction are shown in Fig. 4. The temperatures at the upstream and downstream point are recorded in the upper and lower traces, respectively. The fluctuations of the upstream and downstream are similar, which indicates the interference between the thermocouples is insignificant, even though the shape of the fluctuation is distorted as the interval 5 L increases. The time lag between the upstream and downstream fluctuation indicates the convection velocity. Similar recordings of temperature and ion current are made in the several locations at L = 90 mm and 160 mm. In order to obtain the spatial scale and convection velocity of the temperature and ion current fluctuation, one calculates time-space correlations, some examples are shown in Fig. 5. 8L, 8r, ~ are the intervals between two detecting points in the axial, radial, or circumferential directions, respectively. RTT(~L, r) or RU(~L, r) reaches a maximum at r corresponding to 8 L, indicating the fluctuation is convected downstream as well as indicating the convective velocity. RII(~L, T) is less than RTT(~L, T) for the same 8L, which indicates that the shape of the ion current fluctuation is liable to be distorted more than that of the temperature fluctuation. RTT(Sr, r) has a maximum at some positive value of r, which results from the fact that the convective velocity in the central part is larger than that in the outer part and consequently the eddy should be stretched. Examples of simultaneous (T = 0) space correlation of temperature or ion current fluctuation are shown in Fig. 6. The spatial scales are obtained

STRUCTURE OF TURBULENT DIFFUSION FLAMES

T

L -- 90 mm

267

r -- 5

mm

SL = 1-9 mrn

1360"C

~ :

_L

s5o'c

T = 1360"c

[

T

L : 9 0 rnm

r= 5 turn

~L: 5-0 mm _L 550"C T = 1360 °C

- 1360 "C

T

L = 9 0 mrn

r=5rnm

SL =10.0 mrn

~

_L 550°C ? = 1360 °C

~ 0

1

I

I

25

50

550"C =1360.c

mse¢

Fig. 4. Examples of simultaneous recording of temperature fluctuation from two recording thermocouples with interval 6 L in the axial direction (L = 90 mm, r = 5 mm).

268

T. TAKAGI ET AL. 1.0

The following are noted from Fig. 6 and Table .

~0.5

0 1.0 .7 m m

t~ 0.5 H.1

0

I

1.0

cs

"•.7 •

mm

9 . 4 ~ 14-I I

.... I01KI 50 "-'_-

r

o

i

4 ~"

msec

Fig. 5. Examples o f time-space correlation of temperature and ion current fluctuation (L = 160 m m , r = 9.1

mm).

from Fig. 6 or the correlations at other detecting locations where the spatial scale l is defined as the interval 6 for which the simultaneous space correlation is 1/e (= 0.368). The spatial scales in the axial, radial, or circumferential direction are listed in Table 2 where the subscript T or I of l means the scale obtained from the temperature or ion current fluctuation and another subscript, L, r, or ~ means the scale in axial, radial, or circumferential direction, respectively.

1. The scale in the axial direction is always the largest and the scale in the circumferential direction is larger than that in the radial direction when they are compared at the same location. This means the shape of the high temperature gas eddy or reaction zone is stretched especially in the axial direction. The stretching is more noticeable in the outer part of the flame than in the central part. 2. The spatial scale obtained from ion current fluctuation is less than that from temperature fluctuation. Radial profiles of the spatial scales in the axial direction are shown in Fig. 7. luz means the scale obtained from the fluctuation of the velocity U. The hollow symbols are the spatial scales obtained from the time scale multiplied by the time-averaged velocity, where the time scale is defined by the time lag for which the autocorrelation of the fluctuation at the single detecting location is 1/e. The spatial scales obtained from the autocorrelation agree well with those obtained from the time-space correlation, which is indicated by the solid points. Radial profiles of IUL are almost parallel to those of the time-averaged temperature T; luz has a peak at near r = 6 mm of L = 90 mm, whose proffie is significantly different from that in the nonburning fuel jet as pointed out in Ref. [1]. The radial profde OflTL has a peak at larger r than the maximum temperature, probably because the burnt (not burning) gas prevails more at the outer part. 11L is almost always less than luL or lrL and the radial profile of lxz is relatively fiat. The smaller length scales obtained from the ion current fluctuation would result from the significant recombination of ions after combustion reactions. Convective velocities of the fluctuation of temperature UTC and ion current Ozc, along with the time-averaged flow velocity 0, are shown in Fig. 8. The convective velocity is defined by 6L divided by Tmax, where ~'max is the time lag for which the time-space correlation Rrr(6L, r) or RxI(6L, r) has the peak corresponding to the interval 8 L. Radial profiles of Urc are similar to those of Cr but Urc is a little less than Uat the central part of the flame and a little larger than U at the circum-

STRUCTURE OF TURBULENT DIFFUSION FLAMES

o

269

1.0 ~

1.0 tz) rv-

0

\\

0.5

L,o

""

o

q

05

v

@ 0

O

O0

10

20 30 SL,f;r, ~, mm

q .J

t~o

v

aT

01 lb

20

Fig. 6. Examples of simultaneous space correlation of temperature and ion current fluctuation (L = 160 m m , r = 9.1 mm).

ferential part. This tendency is similar to that of the convective velocity of velocity fluctuation in an isothermal jet [13]. The profile of U1c is more flattened than that of 0, and Urc is much less than 0 at the central part of the flame and much larger than 0 at the outer part. One probable reason for the much larger convective velocity of ion current O~c than U at the outer part of the flame may be the selective detection of high ion concentration accompanying higher axial velocity as shown in Fig. 2(b).

CONCLUSIONS 1. Regarding the microscopic structure of the flame, inactive, low temperature, and low velocity gas is prevailing in the outer part of the flame with intermittent appearance of the gas TABLE2 Spatial Scales in Axial, Radial, and Circumferential Directions L mm

r mm

ITL

lTr

90

0.0 5.0 0.0 9.1

3.0 4.5 5.0 35.0

2.2 1.6 3.6 4.3

160

IT~ 2.0 7.7

llL

lit

2.1 2.4 2.8 11.0

1.5 0.8 1.0 2.3

lid 1.9 3.9

of high temperature, high ion concentration, and high velocity. On the contrary, in the central part of the flame the gas parcel of fuel or air contains reacting gases or products mixed in the molecular scale, even though the mixing is incomplete due to the large-scale turbulences and the microstructural reaction zones exist intermittently. . Correlation and probability density function of the fluctuations of temperature and velocity are intimately related to their time-averaged profiles as noted in (1), (2), and (4) of the previous subsection. The relations are qualitatively comprehensible if we consider that the gas parcels exchange positions laterally by turbulent mixing while retaining the original temperature and velocity before the exchange, as is presumed in the mixing length hypothesis. . The shape of the high temperature gas eddy or reaction zone is stretched especially in the axial direction, and this stretching is more noticeable in the outer part of the flame. The spatial scale from the temperature fluctuation has a maximum at larger r than the time-averaged maximum temperature while the scale from the velocity fluctuation has a peak near the location of the maximum temperature. The spatial scale from the ion current fluctuation is usually less than that from temperature or velocity fluctuation.

T. TAKAGI ET AL.

270

i2o

/ ::

I~L

L:9Omm

~. 15I a I ~ j L L ~

_.. 30

g

o

°"

L,,

~A

[,IULL

I

/~=160

m

/

lb r

mrn

r

rnm

Fig. 7. Radial profiles of the spatial scales in the axial direction.

4. The radial profdes of convective velocity of the temperature fluctuation UTc is nearly equal to the time-averaged flow velocity 0 but Urc is a little less than 0 at the central part of the flame and a little larger than 0 at the circumferential

v

20

I

~ ~ ~ =

90mm REFERENCES

10 F-

•

~rc

"~ U l ~

'

,

5

10

15

r

v

The authors thank Mr. Y. Takabatake for assistance in conducting experiments.

I

E

¢p oi

part. The profde of the convective velocity of the ion current fluctuation U1c is more flattened as compared with U.

mm

20

L=160mm z~

L,)

1o

o~ •

o"

0

Urc ~IC

,I

,1

5

10

15

r

mm

Fig. 8. Radial profiles of convective velocity of the fluctuation of temperature and ion current.

1. Takagi, T., Shin, H. D., and Ishio, A., Combust. Flame (in press). 2. BaUantyne, A., and Bray, K. N. C., Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, p. 777. 3. Glass, M., and Bilger, R. W., Combust. Sci. Technol. 18:165 (1978). 4. Chigier, N. A., and Yule, A. J., 17th Aerospace Sciences Meeting, AIAA, New Orleans, La., January 1979. 5. Kunugi, M., and Jinno, H., Seventh Symposium (International] on Combustion, Butterworths, London, 1959, p. 942. 6. Tanaka, Y., and Shimamoto, Y., Bull. JSME 22 (165):390 (1979). 7. Lockwood, F. C., and Odidi, A. O. O., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1975, 561. 8. Onuma, Y., and Shibata, M., Fifteenth Combustion Symposium (in Japan), The Combustion Institute of Japan, 1977, p. 133. 9. Suzuki, T., Hirono, T., and Tsuji, H., Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, p. 289. I0. Shepard, C. E., and Warshawsky, I., NACA Technical Note 2703, 1952.

STRUCTURE O F TURBULENT D I F F U S I O N FLAMES 11. Takagi, T., Ogasawara, M., Fujii, K., and Daizo, M.,

13. Wills, J. A. B.,J. FluidMech. 20(3):417 (1964).

Fifteenth Symposium (International] on Combustion, The Combustion Institute, Pittsburgh, 1975,

p. 1051. 12. Takagi, T., Shin, H. D., and Ishio, A., Sixteenth Combustion Symposium (in Japan), The Combustion Institute of Japan, 1978, p. 220.

Received 19 March 1980; revised 2 June 1980

271