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Velocity fluctuations in turbulent jets and flames
VELOCITY FLUCTUATIONS IN TURBULENT JETS AND FLAMES It. A. BECKEtl. AND A. P. G. BROWN
Department of Chemical Engineering, @~een's University, Kingston, O~dario, Canada The aim of this work was to develop simple and reliable techniques for turbulence detection in flames. Two approaches were followed, one based on properties of the equations of motion and the other based on properties of impact probes with differing geometry. The lateral equation of motion for a turbulent shear flow suggests that the static pressure field is directly related to the lateral normal Reynolds stress ~ (u~~) or ~ (u,fl }. Data on free and confined jets confirm it. Static pressure measurements in flames then indicate that the turbulence level there does not differ greatly from constant density jets. Velocity fluctuations are also sensed by a novel differential Pitot probe that exploits the difference in response between a square-nosed cylindrical probe and a sphere-nosed probe. The feasibility of a probe with linear response has been demonstrated. Measurements in flames again indicate tm'bulenee levels not radically different from those in constant density jets. cold jet study. In a symmetrical turbulent shear flow there is a static pressure defect that is directly related to the radial velocity fluctuations. We have studied the responses of these probes and have investigated the turbulence fields of a cold-air jet (the reference standard) and a propane-air diffusion flame. The probes are rugged, cheap, and easy to construct and use, and should provide research and industry with valuable new measurement tools.
Introduction
The characterization of turbulence in flames is made difficult by flame radiation, soot, free radicals, and large fluctuations in temperature, density, and composition. The limitations of the particle track technique and the turbulent diffusion experiment are well known. A water-cooled hot-film sensor has been developed ~ for hot-wire anemometry, but the problem remains of interpreting a response that consists of fluctuations in the heat toss of a constant-temperature cylinder. Theory Because of the lack of convenient experimental techniques, practical knowledge of turbulent The Static Probe flames has been largely confined to the fields of mean velocity, temperature, and composition. Under the usual "thin boundary layer" assumt)The understanding of flame-transport processes tions of a flow structure highly elongated ill the and structure can now hardly progress much mean-flow direction, the time-averaged radial further until some of the turbulence effects are equation of motion for a statistically steady, quantitatively known. We have therefore under- variable density, axisymmetrical turbulent shear taken in our laboratory a program in which flow with lateral buoyancy forces negligible, experimental methods are being developed and simplifies to used to study selected systems. Among the most powerful is the light-scatter technique for the detection of concentration fluctuations. ~,~ The present paper is, however, concerned with the exploitation of two simple devices for meas- where PB is the static pressure at the shear flow uring normal Reynolds stresses. The first is a edge r=B, and (~r2) is supposedly negligible at "differential Pitot probe" which utilizes the r=B. For a free iet discharging into a stitl difference in response of two rigidly coupled atmosphere, this is equivalent to Pitot tubes with differently shaped heads to detect transverse velocity fluctuations. The second is a flat-plate static pressure probe, first demonstrated by Miller and Comings 4 in a Equations (1) and (2) suggest that a true 1059
1060
TURBULENT FLAMES
static pressure probe can be used to measure to a good approximation the normal radial Reynolds stress (ou~2) or the quantity ~ (u~2}; since density fluctuations and u~2 should be negligibly correlated (zero correlated if density fluctuations and u~ are Gaussianly distributed), (pup }= ~ (u} }.
it is evident that a sl)here-nosed impact tube with a very small pressure tap should approximate A = 9/4. If a probe were constructed with A = 1/2, Eq. (6) would approximate to q = 89 ~, i.e., the mean x velocity could be measured.
The Differential Pitot The Impact Probe The difference between the pressure sensed by an impact probe (Pitot tube) at any point in a turbulent shear flow and the static pressure at the shear flow edge is
q = P-
PB "4-,
(3)
where U is the stream speed (U 2= U 2 4 - U , 2 4-U~2); C = C(0) is the impact coefficient for performance in a uniform stream, 0 being the yaw angle (the angle between the probe direction and the velocity vector), a may be called the turbulence scale factor; when the characteristic diameter of the probe tip D is very much smaller than the turbulence scale A, a = 1. Suppose the impact probe faces in the upstream (x) direction and suppose that in the involved range of 0, C = 1 effectively. Further, suppose = 1. Equation (3) yields, with small terms neglected, --
Suppose two impact probes are rigidly coupled with their tips far enough apart so that interaction is negligible, one with Eq. (5) response and the other with Eq. (6) response. Insert the combined probe into a tm'bulent shear flow so that the tips are at statistically equivalent positions--e.g., at equal radii in an axisymmetrical flow. The differential response is
~q = 89
~+ ~ 5 .
(s)
If, in practice, the condition (~ = 1 in Eq. (3) is not realized, the response of the differential Pitot will nonetheless be a measure of turbulence intensity. Since there is no evident reason why 9/r and u~ should be very differently responded to, we will assume that Eq. (7) remains valid in the modified form
Aq = ~flA~(u~ 2 + u,2) ,
(9)
where/3 is another turbulence scale factor. This
1 - % 2
(4) Now, Eq. (1) suggests t h a t / 5 _ P s and {~ (u~: + u~e) should very nearly cancel (and experience confirms it); thus, Eq. (4) effectively reduces to
q
=
T2
~o I--
A sine• = 1 -
A(U,2+Ur
~(v~>-
~Ap(~,, + W > .
(6)
From the inviscid flow pressure distribution around a sphere,
P -- Poe = ~pUo~2 (1 -- -~ sin2 0 ) ,
hole
; if_ p o h s h e d d i s k ,cooling w a t e r
(b) -~ll § 3.17m m
din.
2.
By the same arguments leading to Eq. (5) this reduces to
q
0.46ram din.
(5)
The case C = 1 is approximated by a thinwalled, squarely truncated hollow cylinder probetip geometry, and Eq. (5) is the predicted response of this probe in a large-scale turbulence when operated in the prescribed manner. Consider now the ease C = 1-
(a)
(7)
1.29m m
~
dia.
/ ~-OB3 m m din. 0.40 m m d/a. hole 4.75 m m din. s p h e r e ~J 12rnm
bore
b
FIG. 1. (a) the static pressure probe, and (b) the differential Pilot probe.
VELOCITY FLUCTUATIONS may be in error to the extent that any differential in the response to (u~2} has been neglected. Presumably the behavior of/5 in typical situations can be predicted from a set of well-chosen experiments. In any case, turbulence intensities in two different flows can presumably be compared if the turbulence scales are similar and the spectra are no~ radically different in the energycontaining range.
Apparatus The static pressure probe, Fig. 1 (a), was a flat-plate probe made from type 310 stainless steel with welded joints, similar in design to that of 5{iller and Comings.4 The flat face was highly polished and the edges were razor sharp. Two probes were used, one of 1.3-cm face diameter,
1061
the other 0.6 era. The 1.3-cm probe was large for the flows studied, hut comparison of the i)erfonnance of the two probes gave a correction for the effect of probe size. The differential Pitot probe, Fig. 1 (b), employed a squarely cut., type 310 stainless steel tube as one leg, and a water-cooled spherenosed probe as the other. The sphere was bearing ball. The jet rig consisted of a flow nozzle with its mouth in /,he plane of a 1.2 m diameter round table. The throat diameter was 0.635 cm. For a jet that entrains a constant, volume of air per unit of downstream distance, the effectively infinite plane boundary at, x = 0 promotes the well-defined boundary condition U~ = 0 outside the jet. The jet was traversed with a rig that allowed axial (x) and diametrical (y) positioning of the
.05
0
.05
0.2
0.11
~
i i
, i
4
.-4_
_.__O.J~---" 0
i
L 0.2
0.1
I
_P
_
sin2e Fzo. 2. The cfdibration curves of the differential Pitot probe. 9 the sphere-lJosed probe; ~ , the sqmm'nosed probe; and A, tile differentiM. Tile ordinates are: O I~, 1 - q/.'.,pU"; and A, Aq/'..,pU 2.
1062
TURBULENT
probes, and also a limited movement (z) normal to x and y so that the jet center could be accurately located at each axial position. The instrumentation included a micromanometer with a sensitivity of 0.01 mm water column, and a differential pressure transducer with a range of 3 em water and a sensitivity of 0.03 mm water.
Results The Probe Calibrations The probes were calibrated in a wind tunnel. The impact pressure coefficient of the static probe with the flat face parallel to the wind direction was zero. The calibration curves of the differential Pitot are shown in Fig. 2. The differential response is very nearly linear in sin2 0 up to sins 0 -- 0.18 or = 25 ~ a broad range. Up to 0 --- 10~ the slope is 1.04, while between 10~ and 25 ~ it is about 1.19. Since 0 = 10~ represents a considerable transverse velocity coml)onent, (U~ + U~) t/~ = 0.18U~, the value A = 1.04 is satisfactory for our purposes. Our working version of Eq. (8) is thus ~q = 0.52~(u~: + u J ) .
(10)
The calibrations of the individual impact probes are described by the equations C = 1 -- 2.04(sin~0)2, 5~
(11)
0 < 25 ~ for the square-nosed probe, and C = 1 - 2.047 (sin~ 0) 1'1~,
(12)
FLAMES
The Impact Pressure Field The impact pressure field was explored with the square-nosed probe. The radial profile in the air/air jet was self-preserving in the investigated region, 70 < x/ro < 270. The profile was represented in the usual form q/q~ = f(r/b2), where b~ is the half-value radius and smoothed values of the function are given in Table I. The data on the propane/air flame did not differ significantly in the mean from the curve of Table I, but a greater scatter was evident with excursions as large as • 0.05 at cross sections far from the nozzle plane. The center line axial profile for the air/air jet was described by
(q~/q,)ln = 0.087 ( x / r 0 - 10.3). The equation of the half-value radius was
b.2/ro = 0.0633 (x,/ro- 3).
The jets were operated at a nozzle momentmn flux of 0.475 kg m/see e. For the uniform density air/air jet this gave a nozzle Reynolds number of 45,000. In the absence of combustion the propane/air jet would, by virtue of the conservation of momentum, have had the same local Reynolds number far downstream. The temperature rise with combustion however decreases the local Reynolds number by a factor as large as 10. The visible length of the propane/air flame was 230 cm, or 360 nozzle diameters. The visible flame was lifted 18 cm, or 28 nozzle diameters, above the nozzle. No flame holder was used.
(14)
The propane/air flame data followed Eq. (14) closely, but followed Eq. (13) only up to x/ro = 120. There the slope shifted to a lower value, Fig. 3, giving (q~/q~)V2 -__ O.053x/ro + 3.1.
The Static Pressure Field Representative radial profiles of the static pressure are shown in Fig, 4. In the air/air jet, the profiles were essentially self-preserving. In the flame, they were narrower-skirted near the flow edge and became more so with increasing distance from the nozzle. The flame data also showed more random scatter at large distances from the nozzle (Fig. 4 shows data only for positions relatively near the nozzle). The centerline data, that indicate the ampli-
0 < 35~ for the sphere-nosed probe.
The Jets
(13)
TABLE I The radial profile of the impact pre.~sure in tile air/air free jet, measured with the square-nosed probe.
r/b.,,
q/q~
,'/b,.
q/qc
0 0.2 0.4 0.6 0.8 1.0 1.2
1. 000 0. 965 0. 885 0. 765 0. 630 0.500 0. 380
1.4 1.6 1.8 2.0 2.2 2.4 2.6
0. 275 0.188 O. 123 0. 076 0. 042 0.019 0. 000
VELOCITY F L U C T U A T I O N S
1063
20 T
Lo
T
!
CL _Q
'
'
i
.
I_
0 Oq
i
10
i
--
ET
,i
/ t
: ;
I
/'"
c "'//I ."
/
Tf i
....
~
i
:
i
.......
~.-
0 ET
i
0
.
/ - - / T~!
!
i
100
:
200
xlr" o Fie.. 3. Centerline i m p a c t pressure q~ and static pressure half-value radius bF. O - b p / r o . 0 ,IX, flame jet. - . . . . . , air/air jet.
,
(~t,/q~'J~j~-. A . . . ,
P -V
_~
A7
0
=
0
1
r/bp
2
FIG. 4. Radial profiles of the static pressure defect in free jets. B o t t o m set, the air/air jet; top set, the p r o p a n e / a i r flame jet. Distances from the nozzle x/ro are: O, 72; .~, 96; /L, 120; A, I44; d-, 16S; X, 192.
1064
TURBULENT FLAMES
'
t
:
,
I '
t
-
-7 f
[ ~
-
i-
i
0.3
i
(poo- p:) 1/2
v qc
!
'
t
i
0.2 '
-
0.1
!
T
---
1
0
100
200
x% FIG. 5. Static pressure defect and other quantities along the eenterline in free jets. 9 constant-density air/air jets; 9 propane/air flame jet. O 9 present data on static pressure defect; D, [(P~ -- P~)/plI/2/Uo, Miller and Comings4; -t-, (u~ 2 }1/2/U~, Corrsin and Uberoig; X, {Ur~}lJ2/U~, (]ibson. 1~ Miller and Comings' jet was planosymmetrical.
tude of the static pressure defect in the turbulent field, are shown in Fig. 5. There appears to be little difference between the air/air jet and the flame. The static pressure half-value radius bp in the air/air iet was simply related to the impact pressure half-value radius: be = 1.23b~- 0.8r0. The half-value radius for the profile of ql/2 was bl = 1.445b2, giving b2 = 0 . 8 5 b l - 0.Sro. In the flame, the behavior of be was more complex, like that of (qo/%) ~/2 (Fig. 3). One of us (HAB) had previously investigated the static pressure field of a confined axisymmetrical jet, 5 the same system described by Becker, Hottel, and Williamse~s in papers on the fields of velocity and nozzle fluid concentration. Because of space limitations, these results will be published elsewhere. However, it may be noted that, with due allowance for the differences between the free and confined jets, the findings are in excellent agreement with the present results.
The Differential Impact Pressure Field The differential impact pressures (measured with the differential Pitot probe) showed somewhat more scatter than the static pressure data. The data were therefore assembled in the simple form Aq/q~ = f(r/b~). Figure 6(a) shows that Aq/qc in the air/air jet was nearly uniform radially up to r = b2. The plateau value increases with downstream distance, from 0.041 at x/ro = 96 to about 0.060 at x/ro = 192. In the propane/air
flame, Fig. 6(b), the plateau is absent and the profile is much narrower skirted. The centerline maxima are however similar to those in the air/air jet.
Discussion Our object in this work has been to develop simple probes for turbulence detection, applicable in flames as well as in cold flows. In these terms, the static pressure probe may be judged a success. According to the radial equation of motion which reduces to Eq. (1), the static pressure defect in a free turbulent shear flow equals the radial normal Reynolds stress ~ ( u J ) . We therefore show in Fig. 5 the data of Corrsin and Uberoi 9 on
VELOCITY FLUCTUATIONS
i
i
l
' "
!
!
!
,
! L! I
L
i .
.
.
i
D.06
..... ! -
i, _ _ . ,
1065
i
.
i
D,04
/
i --
0.06 ~ - - ~ - ' - v
,
i
o.o4
:i
0.02
--0
'
+! "" -i
%/
,
0.02
.....
! i!
!
0
i
r'/b 2 FIG. 6. Radial profiles of the differential impact pressure in free jets. Bottom set, air/air jet; top set, propane/air flame jet. [~A~7 4-X, nozzle momentum flux 0.475 kg m/sec; O , A W , nozzle momentum flux 0.158 kg m/sec. Distances from the nozzle x/ro are: O, 72; F ] , , 96; A A , 120; V Y , 144; 4-, 168; X, 192
I>
I I:>
0.06
Poo-P 2qc
[> 0.04
o.,.
I! !
0.02
0
>
f"~o..
r'/bl
Fw,. 7. Radial profiles of static pressure defect and radial velocity fluctuations in constant-density air/air jets, 0 , (P~ -- Pc)/2qo present data, x = 96 r0; [:> <1, (u, ,2 )/U~ ~, Gibson, x/ro = 100. Gibson's jet was skewed; ~ denotes r.h.s, points and ~ l.h.s, points.
1066
TURBULENT FLAMES
accurate interpretation of the impact probe response. Thus, for constant density flows, the ordinate in Fig. 5 approximates [" (Po~ -- P)/p]I/2/U~. Gibson ~~ measured only one radial profile of (Ure), at x = 100r0. A further comparison, as direct as possible, between these data and our static pressure data for x - - 9 6 r 0 is shown in Fig. 7. Gibson's air iet was distorted for some reason so that his turbulence profiles do not fold neatly about the center of the mean velocity profile. Thus the 1.h.s. of his (u~2} profile follows our data exactly, while the r.h.s, data fall high. Nevertheless, his results and ours agree exactly on the jet centerline and the differences elsewhere are not great in terms of (u~e)ve. Curtet and Ricou n have measured velocity fluctuations in confined jets by hot-wire anemometry, and these results compared to the previously mentioned static pressure data 5 further support the validity of Eq. (1) and the flat-plate static pressure probe. To indicate the close similarity between the static pressure fields of axisymmetrical and pianosymmetrical jets, Fig. 4 includes Miller and Comings' data 4 on the static pressure defect along the centerplane of a plane constant density air/air jet. Figure 5 suggests an ultimate value of [ (Pr -- p~)/pjl/e as high as 0.39U~. It is evident in any case that self-preservation of velocity fluctuation amplitude develops very slowly, requiring perhaps 200 nozzle diameters of downstream translation. If the validity of the static pressure probe for measuring ~ (u~ } is now conceded, then it appears from the static pressure distribution in the propane/air fame that the normal Reynolds stresses there, at least in the radial direction, differ rather little from those in constant density jets. The only marked dissimilarity is in the half-value radius, Fig. 3. it seems, however, that the near invariance of be with x in the range 120 < x/ro 180 cannot continue to much higher values of x; the impact pressure half-value radius b~ grew linearly with x, be -- 0.063 ( x / r o - 3), and it is inconceivable that the balance between bp and be can shift very far in either direction. The results obtained with the differential Pitot probe are encouraging but suggest that the sphere probe used was too large relative to the turbulence scale. On a jet centerline, (u~e) = (u~2) exactly, and so Eqs. (2) and (10) suggest that Aqr162 = 2.08 (P~ -- Pr The data on the air/air jet, Fig. 6, however give Aq~/q~ = 0.67(Pr -- Pr Clearly the probe was reading only one-third the expected fluctuation energy. The profiles for the air/air jet are however of the expected shape and radial scale, and the ratio (P~,--P~)/Aqr is virtually a constant.
If we assume that turbulence scale and energy spectrum had similar effects in the flame to those in the air/air jet, then the flame data in Fig. 6 taken in conjunction with the static pressure results indicate that ~ (uo2} was greatly diminished relative to the air/air jet. The results obtained at the standard nozzle momentum flux of 0.475 kg m/sec e were unfortunately fragmentary owing to a constructional difficulty with the probe which could not be remedied in time. Most of the results shown are thus from earlier tests in which the nozzle momentum flux was 0.158 kg m/sec e.
Conclusion The validity of the static pressure probe as an indicator of radial velocity fuctuations appears to be reasonably well established by the results reported herein. If so, the present data give quantitative indications of velocity fluctuation levels in flames. The application of the differential Pitot is in a more developmental stage. However, the validity of the device is unquestionable; it must give true results when the turbulence scale is large relative to the diameters of the probe noses. The important fact revealed by the present work is that probes can indeed be constructed for which the differential response is proportional to sin 2 ~ over a broad range. The effects of turbulence scale are now under investigation. We are mapping the fields of mean temperature and intermittency in the flame, and these results together with a detailed consideration of the impact pressure field will be presented elsewhere. The present work has covered only the half flame length nearest the nozzle, the region where meaningful readings could be obtained with the present instruments. It is hoped that refinements will permit study of the remaining half.
Notation Subscripts 0 in the nozzle stream at discharge c on the jet centerline far outside a free jet, r--~ ~r B at the iet edge, r = B
Variables x, r, c~ cylindrical polar coordinates, 0x in the nozzle plane, jet centerline at Or U (x, r, ~) velocity vector, with components U,, Ur, U~
VELOCITY
1067
FLUCTUATIONS
u~, u~, u, components of velocity fluctuation Uo =~ U~(x = O, r < r0) nozzle velocity U~(x) = U~(x, r = 0) eenterline mean velocity Us (x) = U,(x, r = B) mean velocity at jet edge P static _pressure Pr = P(x, r = 0) centerline mean static pressure PIJ (x) =-- D (x, r = B) mean static pressure at jet edge q mean response of impact pressure probe relative to the static pressure at r--+ *r r0 nozzle radius bx(x) half-value radius of ql/2 be (x) half-value radius of q be (x) half-value radius of/5 B(z) radius of outermost jet edge, approximately B = 4bl, where all mean jet variables become asymptotic in r, beyond which nozzle fluid never appears O fluid density Other variables are defined in context.
ACKNOWLEDGMENT The work was supported by grants from the National Research Council of Canada.
IIEFEIIENCES 1. Hot Film and Hot Wire Anemomelry. Bullelin TB5, Thermo-Syslems ha:., St. Paul, Minne.sota, 1967. 2. ROSENSWm(~, It. E., ItOTTm~, H. C., AID WILLIAMS, G. C.: Chem. Eng. Sei. 15, 111 (1961). 3. BECKER, H. i . , HOTTEL, I1. C., AND WILIAAMS, G. C.: J. Fluid 5.Iech. 30, 259 (1967). 4. ~'IILI~ER, D. ][2. AND COMINGS, E. W.: J. Fluid Mech. 3, 1 (1967). 5. BEeKER, H. A.: Concentration Fluctuaiions in Ducted Jet Mixing, Sc.l). thesis, M.I.T., 1961. 6. BECKER, H. A, lh)TTEL, H. C., AND WIIA,IAMS, G. C.: Ninth Symposium (International) on Combustion, p. 7, Academic Press, 1963. 7. BECKER, 1-[. A., HOTTEL, i[. C., AND WILI,IAMS, G. C.: Tenth S.qmposium (International) on Combustion, p. 1253, The Combustion Institute, 1965. 8. BECKER, tI. A., IIOTTEL, II. C., AND WILLIAMS, G. C.: Eleventh Symposium (International) o*l Combustion, p. 791, The Conibustion Institute, 1967. 9. CORRSIN, S. AND UBEROI, M. S.: Further Experiments on the Flow and Heat Transfer in a Heated Turbulent Jet, NACA Rept. 998, 1950. 10. GIBSON, M. M.: J. Fluid Mech. 15, 161 (1963). 11. CURTET, P,., AND Ihcou, F. P.: Trans. ASME, Ser. D, J. Basic Eng. 86, 765 (1964).
COMMENTS N. A. Chigier, Sheffield University. The authors have shown a reduction in turbulence intensity in a flame when compared to an isothermal jet. Can we conclude from these experiments that there is no evidence of flame generated turbulenee in unconfined diffusion flames? H. A. Becker. Yes.
H. Eickhoff, Karlsruhe University. We too have made measurements of turbulence intensity by means of its effect on the static pressure. The water-cooled probes were calibrated in isothermal jets of air by comparing the static pressure and the radial component of turbulent normal stress, as determined by a constant-temperature hotwire anemometer (see Ref. 1). As shown in Fig. 1, the turbulence is always lower than in the isothermal air jet. In the cold jet of town gas the turbulence intensity is even higher than in
the isothermal air jet, as evidenced by its more rapid spreading. Thus, when the jet is ignited, turbulence intensity decreases. Figure 1 shows also that the flame most influenced by buoyancy has the lowest turbulence intensity at z / d = 60. REFERENCE 1. EICKHOFF, II.: Statischer Druck und Turbulenz in drehsymmetrisehen Freistrahlen und Freistrahlflammen, Thesis, Universitat Karlsruhe (T.II.), 1968. H. A. Becker. The turbulence intensities found by Eickhoff for isothermal air jets are lower than any yet reported (see Figs. 5 and 7 of our paper). However, I see no reason to doubt his finding of relatively lower turbulence intensities in flames. I would only point out that: (1) his flame was of town gas, whereas ours was of propane; and (2) his flame data tend to co~> verge with his air data far fl'om the nozzle.
1068
TU[{BULENT FLAMES 0.2
L_
---
-t--
~
:-
I ~ ~
>IZ
"' kZ
0.1
r
O Z hi .J 3
I
~I 2S~<./ /~"
Isoihermol d,,~l Flames '.~-~a
7~
d= Nozzle Diameter
0.05
1
u Reo=7~14.104~ Flames Z~Reo=5,05,10~l Lifted ~Reo=5,05"104~ Flames o Reo=3,2d04 J Ad jocent
I,-
0
_
' ~
20
40
60
X d
80
I00
12.0
140
Figure I
Therefore, the differences between Eickhoff's data and ours may be quite real, and more work should be done to determine the effect of nozzlegas composition. Nevertheless, his data and ours
show the same tendency w:th distance from the nozzle. Finally, we agree that, in free turbulent diffusion fl~mes, there is no evidence of flamegenerated turbulence.
Department of Chemical Engineering, @~een's University, Kingston, O~dario, Canada The aim of this work was to develop simple and reliable techniques for turbulence detection in flames. Two approaches were followed, one based on properties of the equations of motion and the other based on properties of impact probes with differing geometry. The lateral equation of motion for a turbulent shear flow suggests that the static pressure field is directly related to the lateral normal Reynolds stress ~ (u~~) or ~ (u,fl }. Data on free and confined jets confirm it. Static pressure measurements in flames then indicate that the turbulence level there does not differ greatly from constant density jets. Velocity fluctuations are also sensed by a novel differential Pitot probe that exploits the difference in response between a square-nosed cylindrical probe and a sphere-nosed probe. The feasibility of a probe with linear response has been demonstrated. Measurements in flames again indicate tm'bulenee levels not radically different from those in constant density jets. cold jet study. In a symmetrical turbulent shear flow there is a static pressure defect that is directly related to the radial velocity fluctuations. We have studied the responses of these probes and have investigated the turbulence fields of a cold-air jet (the reference standard) and a propane-air diffusion flame. The probes are rugged, cheap, and easy to construct and use, and should provide research and industry with valuable new measurement tools.
Introduction
The characterization of turbulence in flames is made difficult by flame radiation, soot, free radicals, and large fluctuations in temperature, density, and composition. The limitations of the particle track technique and the turbulent diffusion experiment are well known. A water-cooled hot-film sensor has been developed ~ for hot-wire anemometry, but the problem remains of interpreting a response that consists of fluctuations in the heat toss of a constant-temperature cylinder. Theory Because of the lack of convenient experimental techniques, practical knowledge of turbulent The Static Probe flames has been largely confined to the fields of mean velocity, temperature, and composition. Under the usual "thin boundary layer" assumt)The understanding of flame-transport processes tions of a flow structure highly elongated ill the and structure can now hardly progress much mean-flow direction, the time-averaged radial further until some of the turbulence effects are equation of motion for a statistically steady, quantitatively known. We have therefore under- variable density, axisymmetrical turbulent shear taken in our laboratory a program in which flow with lateral buoyancy forces negligible, experimental methods are being developed and simplifies to used to study selected systems. Among the most powerful is the light-scatter technique for the detection of concentration fluctuations. ~,~ The present paper is, however, concerned with the exploitation of two simple devices for meas- where PB is the static pressure at the shear flow uring normal Reynolds stresses. The first is a edge r=B, and (~r2) is supposedly negligible at "differential Pitot probe" which utilizes the r=B. For a free iet discharging into a stitl difference in response of two rigidly coupled atmosphere, this is equivalent to Pitot tubes with differently shaped heads to detect transverse velocity fluctuations. The second is a flat-plate static pressure probe, first demonstrated by Miller and Comings 4 in a Equations (1) and (2) suggest that a true 1059
1060
TURBULENT FLAMES
static pressure probe can be used to measure to a good approximation the normal radial Reynolds stress (ou~2) or the quantity ~ (u~2}; since density fluctuations and u~2 should be negligibly correlated (zero correlated if density fluctuations and u~ are Gaussianly distributed), (pup }= ~ (u} }.
it is evident that a sl)here-nosed impact tube with a very small pressure tap should approximate A = 9/4. If a probe were constructed with A = 1/2, Eq. (6) would approximate to q = 89 ~, i.e., the mean x velocity could be measured.
The Differential Pitot The Impact Probe The difference between the pressure sensed by an impact probe (Pitot tube) at any point in a turbulent shear flow and the static pressure at the shear flow edge is
q = P-
PB "4-
(3)
where U is the stream speed (U 2= U 2 4 - U , 2 4-U~2); C = C(0) is the impact coefficient for performance in a uniform stream, 0 being the yaw angle (the angle between the probe direction and the velocity vector), a may be called the turbulence scale factor; when the characteristic diameter of the probe tip D is very much smaller than the turbulence scale A, a = 1. Suppose the impact probe faces in the upstream (x) direction and suppose that in the involved range of 0, C = 1 effectively. Further, suppose = 1. Equation (3) yields, with small terms neglected, --
Suppose two impact probes are rigidly coupled with their tips far enough apart so that interaction is negligible, one with Eq. (5) response and the other with Eq. (6) response. Insert the combined probe into a tm'bulent shear flow so that the tips are at statistically equivalent positions--e.g., at equal radii in an axisymmetrical flow. The differential response is
~q = 89
~+ ~ 5 .
(s)
If, in practice, the condition (~ = 1 in Eq. (3) is not realized, the response of the differential Pitot will nonetheless be a measure of turbulence intensity. Since there is no evident reason why 9/r and u~ should be very differently responded to, we will assume that Eq. (7) remains valid in the modified form
Aq = ~flA~(u~ 2 + u,2) ,
(9)
where/3 is another turbulence scale factor. This
1 - % 2
(4) Now, Eq. (1) suggests t h a t / 5 _ P s and {~ (u~: + u~e) should very nearly cancel (and experience confirms it); thus, Eq. (4) effectively reduces to
q
=
T2
~o
A sine• = 1 -
A(U,2+Ur
~(v~>-
~Ap(~,, + W > .
(6)
From the inviscid flow pressure distribution around a sphere,
P -- Poe = ~pUo~2 (1 -- -~ sin2 0 ) ,
hole
; if_ p o h s h e d d i s k ,cooling w a t e r
(b) -~ll § 3.17m m
din.
2.
By the same arguments leading to Eq. (5) this reduces to
q
0.46ram din.
(5)
The case C = 1 is approximated by a thinwalled, squarely truncated hollow cylinder probetip geometry, and Eq. (5) is the predicted response of this probe in a large-scale turbulence when operated in the prescribed manner. Consider now the ease C = 1-
(a)
(7)
1.29m m
~
dia.
/ ~-OB3 m m din. 0.40 m m d/a. hole 4.75 m m din. s p h e r e ~J 12rnm
bore
b
FIG. 1. (a) the static pressure probe, and (b) the differential Pilot probe.
VELOCITY FLUCTUATIONS may be in error to the extent that any differential in the response to (u~2} has been neglected. Presumably the behavior of/5 in typical situations can be predicted from a set of well-chosen experiments. In any case, turbulence intensities in two different flows can presumably be compared if the turbulence scales are similar and the spectra are no~ radically different in the energycontaining range.
Apparatus The static pressure probe, Fig. 1 (a), was a flat-plate probe made from type 310 stainless steel with welded joints, similar in design to that of 5{iller and Comings.4 The flat face was highly polished and the edges were razor sharp. Two probes were used, one of 1.3-cm face diameter,
1061
the other 0.6 era. The 1.3-cm probe was large for the flows studied, hut comparison of the i)erfonnance of the two probes gave a correction for the effect of probe size. The differential Pitot probe, Fig. 1 (b), employed a squarely cut., type 310 stainless steel tube as one leg, and a water-cooled spherenosed probe as the other. The sphere was bearing ball. The jet rig consisted of a flow nozzle with its mouth in /,he plane of a 1.2 m diameter round table. The throat diameter was 0.635 cm. For a jet that entrains a constant, volume of air per unit of downstream distance, the effectively infinite plane boundary at, x = 0 promotes the well-defined boundary condition U~ = 0 outside the jet. The jet was traversed with a rig that allowed axial (x) and diametrical (y) positioning of the
.05
0
.05
0.2
0.11
~
i i
, i
4
.-4_
_.__O.J~---" 0
i
L 0.2
0.1
I
_P
_
sin2e Fzo. 2. The cfdibration curves of the differential Pitot probe. 9 the sphere-lJosed probe; ~ , the sqmm'nosed probe; and A, tile differentiM. Tile ordinates are: O I~, 1 - q/.'.,pU"; and A, Aq/'..,pU 2.
1062
TURBULENT
probes, and also a limited movement (z) normal to x and y so that the jet center could be accurately located at each axial position. The instrumentation included a micromanometer with a sensitivity of 0.01 mm water column, and a differential pressure transducer with a range of 3 em water and a sensitivity of 0.03 mm water.
Results The Probe Calibrations The probes were calibrated in a wind tunnel. The impact pressure coefficient of the static probe with the flat face parallel to the wind direction was zero. The calibration curves of the differential Pitot are shown in Fig. 2. The differential response is very nearly linear in sin2 0 up to sins 0 -- 0.18 or = 25 ~ a broad range. Up to 0 --- 10~ the slope is 1.04, while between 10~ and 25 ~ it is about 1.19. Since 0 = 10~ represents a considerable transverse velocity coml)onent, (U~ + U~) t/~ = 0.18U~, the value A = 1.04 is satisfactory for our purposes. Our working version of Eq. (8) is thus ~q = 0.52~(u~: + u J ) .
(10)
The calibrations of the individual impact probes are described by the equations C = 1 -- 2.04(sin~0)2, 5~
(11)
0 < 25 ~ for the square-nosed probe, and C = 1 - 2.047 (sin~ 0) 1'1~,
(12)
FLAMES
The Impact Pressure Field The impact pressure field was explored with the square-nosed probe. The radial profile in the air/air jet was self-preserving in the investigated region, 70 < x/ro < 270. The profile was represented in the usual form q/q~ = f(r/b2), where b~ is the half-value radius and smoothed values of the function are given in Table I. The data on the propane/air flame did not differ significantly in the mean from the curve of Table I, but a greater scatter was evident with excursions as large as • 0.05 at cross sections far from the nozzle plane. The center line axial profile for the air/air jet was described by
(q~/q,)ln = 0.087 ( x / r 0 - 10.3). The equation of the half-value radius was
b.2/ro = 0.0633 (x,/ro- 3).
The jets were operated at a nozzle momentmn flux of 0.475 kg m/see e. For the uniform density air/air jet this gave a nozzle Reynolds number of 45,000. In the absence of combustion the propane/air jet would, by virtue of the conservation of momentum, have had the same local Reynolds number far downstream. The temperature rise with combustion however decreases the local Reynolds number by a factor as large as 10. The visible length of the propane/air flame was 230 cm, or 360 nozzle diameters. The visible flame was lifted 18 cm, or 28 nozzle diameters, above the nozzle. No flame holder was used.
(14)
The propane/air flame data followed Eq. (14) closely, but followed Eq. (13) only up to x/ro = 120. There the slope shifted to a lower value, Fig. 3, giving (q~/q~)V2 -__ O.053x/ro + 3.1.
The Static Pressure Field Representative radial profiles of the static pressure are shown in Fig, 4. In the air/air jet, the profiles were essentially self-preserving. In the flame, they were narrower-skirted near the flow edge and became more so with increasing distance from the nozzle. The flame data also showed more random scatter at large distances from the nozzle (Fig. 4 shows data only for positions relatively near the nozzle). The centerline data, that indicate the ampli-
0 < 35~ for the sphere-nosed probe.
The Jets
(13)
TABLE I The radial profile of the impact pre.~sure in tile air/air free jet, measured with the square-nosed probe.
r/b.,,
q/q~
,'/b,.
q/qc
0 0.2 0.4 0.6 0.8 1.0 1.2
1. 000 0. 965 0. 885 0. 765 0. 630 0.500 0. 380
1.4 1.6 1.8 2.0 2.2 2.4 2.6
0. 275 0.188 O. 123 0. 076 0. 042 0.019 0. 000
VELOCITY F L U C T U A T I O N S
1063
20 T
Lo
T
!
CL _Q
'
'
i
.
I_
0 Oq
i
10
i
--
ET
,i
/ t
: ;
I
/'"
c "'//I ."
/
Tf i
....
~
i
:
i
.......
~.-
0 ET
i
0
.
/ - - / T~!
!
i
100
:
200
xlr" o Fie.. 3. Centerline i m p a c t pressure q~ and static pressure half-value radius bF. O - b p / r o . 0 ,IX, flame jet. - . . . . . , air/air jet.
,
(~t,/q~'J~j~-. A . . . ,
P -V
_~
A7
0
=
0
1
r/bp
2
FIG. 4. Radial profiles of the static pressure defect in free jets. B o t t o m set, the air/air jet; top set, the p r o p a n e / a i r flame jet. Distances from the nozzle x/ro are: O, 72; .~, 96; /L, 120; A, I44; d-, 16S; X, 192.
1064
TURBULENT FLAMES
'
t
:
,
I '
t
-
-7 f
[ ~
-
i-
i
0.3
i
(poo- p:) 1/2
v qc
!
'
t
i
0.2 '
-
0.1
!
T
---
1
0
100
200
x% FIG. 5. Static pressure defect and other quantities along the eenterline in free jets. 9 constant-density air/air jets; 9 propane/air flame jet. O 9 present data on static pressure defect; D, [(P~ -- P~)/plI/2/Uo, Miller and Comings4; -t-, (u~ 2 }1/2/U~, Corrsin and Uberoig; X, {Ur~}lJ2/U~, (]ibson. 1~ Miller and Comings' jet was planosymmetrical.
tude of the static pressure defect in the turbulent field, are shown in Fig. 5. There appears to be little difference between the air/air jet and the flame. The static pressure half-value radius bp in the air/air iet was simply related to the impact pressure half-value radius: be = 1.23b~- 0.8r0. The half-value radius for the profile of ql/2 was bl = 1.445b2, giving b2 = 0 . 8 5 b l - 0.Sro. In the flame, the behavior of be was more complex, like that of (qo/%) ~/2 (Fig. 3). One of us (HAB) had previously investigated the static pressure field of a confined axisymmetrical jet, 5 the same system described by Becker, Hottel, and Williamse~s in papers on the fields of velocity and nozzle fluid concentration. Because of space limitations, these results will be published elsewhere. However, it may be noted that, with due allowance for the differences between the free and confined jets, the findings are in excellent agreement with the present results.
The Differential Impact Pressure Field The differential impact pressures (measured with the differential Pitot probe) showed somewhat more scatter than the static pressure data. The data were therefore assembled in the simple form Aq/q~ = f(r/b~). Figure 6(a) shows that Aq/qc in the air/air jet was nearly uniform radially up to r = b2. The plateau value increases with downstream distance, from 0.041 at x/ro = 96 to about 0.060 at x/ro = 192. In the propane/air
flame, Fig. 6(b), the plateau is absent and the profile is much narrower skirted. The centerline maxima are however similar to those in the air/air jet.
Discussion Our object in this work has been to develop simple probes for turbulence detection, applicable in flames as well as in cold flows. In these terms, the static pressure probe may be judged a success. According to the radial equation of motion which reduces to Eq. (1), the static pressure defect in a free turbulent shear flow equals the radial normal Reynolds stress ~ ( u J ) . We therefore show in Fig. 5 the data of Corrsin and Uberoi 9 on
VELOCITY FLUCTUATIONS
i
i
l
' "
!
!
!
,
! L! I
L
i .
.
.
i
D.06
..... ! -
i, _ _ . ,
1065
i
.
i
D,04
/
i --
0.06 ~ - - ~ - ' - v
,
i
o.o4
:i
0.02
--0
'
+! "" -i
%/
,
0.02
.....
! i!
!
0
i
r'/b 2 FIG. 6. Radial profiles of the differential impact pressure in free jets. Bottom set, air/air jet; top set, propane/air flame jet. [~A~7 4-X, nozzle momentum flux 0.475 kg m/sec; O , A W , nozzle momentum flux 0.158 kg m/sec. Distances from the nozzle x/ro are: O, 72; F ] , , 96; A A , 120; V Y , 144; 4-, 168; X, 192
I>
I I:>
0.06
Poo-P 2qc
[> 0.04
o.,.
I! !
0.02
0
>
f"~o..
r'/bl
Fw,. 7. Radial profiles of static pressure defect and radial velocity fluctuations in constant-density air/air jets, 0 , (P~ -- Pc)/2qo present data, x = 96 r0; [:> <1, (u, ,2 )/U~ ~, Gibson, x/ro = 100. Gibson's jet was skewed; ~ denotes r.h.s, points and ~ l.h.s, points.
1066
TURBULENT FLAMES
accurate interpretation of the impact probe response. Thus, for constant density flows, the ordinate in Fig. 5 approximates [" (Po~ -- P)/p]I/2/U~. Gibson ~~ measured only one radial profile of (Ure), at x = 100r0. A further comparison, as direct as possible, between these data and our static pressure data for x - - 9 6 r 0 is shown in Fig. 7. Gibson's air iet was distorted for some reason so that his turbulence profiles do not fold neatly about the center of the mean velocity profile. Thus the 1.h.s. of his (u~2} profile follows our data exactly, while the r.h.s, data fall high. Nevertheless, his results and ours agree exactly on the jet centerline and the differences elsewhere are not great in terms of (u~e)ve. Curtet and Ricou n have measured velocity fluctuations in confined jets by hot-wire anemometry, and these results compared to the previously mentioned static pressure data 5 further support the validity of Eq. (1) and the flat-plate static pressure probe. To indicate the close similarity between the static pressure fields of axisymmetrical and pianosymmetrical jets, Fig. 4 includes Miller and Comings' data 4 on the static pressure defect along the centerplane of a plane constant density air/air jet. Figure 5 suggests an ultimate value of [ (Pr -- p~)/pjl/e as high as 0.39U~. It is evident in any case that self-preservation of velocity fluctuation amplitude develops very slowly, requiring perhaps 200 nozzle diameters of downstream translation. If the validity of the static pressure probe for measuring ~ (u~ } is now conceded, then it appears from the static pressure distribution in the propane/air fame that the normal Reynolds stresses there, at least in the radial direction, differ rather little from those in constant density jets. The only marked dissimilarity is in the half-value radius, Fig. 3. it seems, however, that the near invariance of be with x in the range 120 < x/ro 180 cannot continue to much higher values of x; the impact pressure half-value radius b~ grew linearly with x, be -- 0.063 ( x / r o - 3), and it is inconceivable that the balance between bp and be can shift very far in either direction. The results obtained with the differential Pitot probe are encouraging but suggest that the sphere probe used was too large relative to the turbulence scale. On a jet centerline, (u~e) = (u~2) exactly, and so Eqs. (2) and (10) suggest that Aqr162 = 2.08 (P~ -- Pr The data on the air/air jet, Fig. 6, however give Aq~/q~ = 0.67(Pr -- Pr Clearly the probe was reading only one-third the expected fluctuation energy. The profiles for the air/air jet are however of the expected shape and radial scale, and the ratio (P~,--P~)/Aqr is virtually a constant.
If we assume that turbulence scale and energy spectrum had similar effects in the flame to those in the air/air jet, then the flame data in Fig. 6 taken in conjunction with the static pressure results indicate that ~ (uo2} was greatly diminished relative to the air/air jet. The results obtained at the standard nozzle momentum flux of 0.475 kg m/sec e were unfortunately fragmentary owing to a constructional difficulty with the probe which could not be remedied in time. Most of the results shown are thus from earlier tests in which the nozzle momentum flux was 0.158 kg m/sec e.
Conclusion The validity of the static pressure probe as an indicator of radial velocity fuctuations appears to be reasonably well established by the results reported herein. If so, the present data give quantitative indications of velocity fluctuation levels in flames. The application of the differential Pitot is in a more developmental stage. However, the validity of the device is unquestionable; it must give true results when the turbulence scale is large relative to the diameters of the probe noses. The important fact revealed by the present work is that probes can indeed be constructed for which the differential response is proportional to sin 2 ~ over a broad range. The effects of turbulence scale are now under investigation. We are mapping the fields of mean temperature and intermittency in the flame, and these results together with a detailed consideration of the impact pressure field will be presented elsewhere. The present work has covered only the half flame length nearest the nozzle, the region where meaningful readings could be obtained with the present instruments. It is hoped that refinements will permit study of the remaining half.
Notation Subscripts 0 in the nozzle stream at discharge c on the jet centerline far outside a free jet, r--~ ~r B at the iet edge, r = B
Variables x, r, c~ cylindrical polar coordinates, 0x in the nozzle plane, jet centerline at Or U (x, r, ~) velocity vector, with components U,, Ur, U~
VELOCITY
1067
FLUCTUATIONS
u~, u~, u, components of velocity fluctuation Uo =~ U~(x = O, r < r0) nozzle velocity U~(x) = U~(x, r = 0) eenterline mean velocity Us (x) = U,(x, r = B) mean velocity at jet edge P static _pressure Pr = P(x, r = 0) centerline mean static pressure PIJ (x) =-- D (x, r = B) mean static pressure at jet edge q mean response of impact pressure probe relative to the static pressure at r--+ *r r0 nozzle radius bx(x) half-value radius of ql/2 be (x) half-value radius of q be (x) half-value radius of/5 B(z) radius of outermost jet edge, approximately B = 4bl, where all mean jet variables become asymptotic in r, beyond which nozzle fluid never appears O fluid density Other variables are defined in context.
ACKNOWLEDGMENT The work was supported by grants from the National Research Council of Canada.
IIEFEIIENCES 1. Hot Film and Hot Wire Anemomelry. Bullelin TB5, Thermo-Syslems ha:., St. Paul, Minne.sota, 1967. 2. ROSENSWm(~, It. E., ItOTTm~, H. C., AID WILLIAMS, G. C.: Chem. Eng. Sei. 15, 111 (1961). 3. BECKER, H. i . , HOTTEL, I1. C., AND WILIAAMS, G. C.: J. Fluid 5.Iech. 30, 259 (1967). 4. ~'IILI~ER, D. ][2. AND COMINGS, E. W.: J. Fluid Mech. 3, 1 (1967). 5. BEeKER, H. A.: Concentration Fluctuaiions in Ducted Jet Mixing, Sc.l). thesis, M.I.T., 1961. 6. BECKER, H. A, lh)TTEL, H. C., AND WIIA,IAMS, G. C.: Ninth Symposium (International) on Combustion, p. 7, Academic Press, 1963. 7. BECKER, 1-[. A., HOTTEL, i[. C., AND WILI,IAMS, G. C.: Tenth S.qmposium (International) on Combustion, p. 1253, The Combustion Institute, 1965. 8. BECKER, tI. A., IIOTTEL, II. C., AND WILLIAMS, G. C.: Eleventh Symposium (International) o*l Combustion, p. 791, The Conibustion Institute, 1967. 9. CORRSIN, S. AND UBEROI, M. S.: Further Experiments on the Flow and Heat Transfer in a Heated Turbulent Jet, NACA Rept. 998, 1950. 10. GIBSON, M. M.: J. Fluid Mech. 15, 161 (1963). 11. CURTET, P,., AND Ihcou, F. P.: Trans. ASME, Ser. D, J. Basic Eng. 86, 765 (1964).
COMMENTS N. A. Chigier, Sheffield University. The authors have shown a reduction in turbulence intensity in a flame when compared to an isothermal jet. Can we conclude from these experiments that there is no evidence of flame generated turbulenee in unconfined diffusion flames? H. A. Becker. Yes.
H. Eickhoff, Karlsruhe University. We too have made measurements of turbulence intensity by means of its effect on the static pressure. The water-cooled probes were calibrated in isothermal jets of air by comparing the static pressure and the radial component of turbulent normal stress, as determined by a constant-temperature hotwire anemometer (see Ref. 1). As shown in Fig. 1, the turbulence is always lower than in the isothermal air jet. In the cold jet of town gas the turbulence intensity is even higher than in
the isothermal air jet, as evidenced by its more rapid spreading. Thus, when the jet is ignited, turbulence intensity decreases. Figure 1 shows also that the flame most influenced by buoyancy has the lowest turbulence intensity at z / d = 60. REFERENCE 1. EICKHOFF, II.: Statischer Druck und Turbulenz in drehsymmetrisehen Freistrahlen und Freistrahlflammen, Thesis, Universitat Karlsruhe (T.II.), 1968. H. A. Becker. The turbulence intensities found by Eickhoff for isothermal air jets are lower than any yet reported (see Figs. 5 and 7 of our paper). However, I see no reason to doubt his finding of relatively lower turbulence intensities in flames. I would only point out that: (1) his flame was of town gas, whereas ours was of propane; and (2) his flame data tend to co~> verge with his air data far fl'om the nozzle.
1068
TU[{BULENT FLAMES 0.2
L_
---
-t--
~
:-
I ~ ~
>IZ
"' kZ
0.1
r
O Z hi .J 3
I
~I 2S~<./ /~"
Isoihermol d,,~l Flames '.~-~a
7~
d= Nozzle Diameter
0.05
1
u Reo=7~14.104~ Flames Z~Reo=5,05,10~l Lifted ~Reo=5,05"104~ Flames o Reo=3,2d04 J Ad jocent
I,-
0
_
' ~
20
40
60
X d
80
I00
12.0
140
Figure I
Therefore, the differences between Eickhoff's data and ours may be quite real, and more work should be done to determine the effect of nozzlegas composition. Nevertheless, his data and ours
show the same tendency w:th distance from the nozzle. Finally, we agree that, in free turbulent diffusion fl~mes, there is no evidence of flamegenerated turbulence.