Coincidence of turbulent-nonturbulent interface and hot-cold interface in a plane turbulent wake

Coincidence of turbulent-nonturbulent interface and hot-cold interface in a plane turbulent wake

Pergamon Mechanics Research Communications. Vol, 23, No. 1, pp. 91-102 1996 Copyright @ 1996 Elsevier Science Ltd Printed in the USA. All rights resc...

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Pergamon

Mechanics Research Communications. Vol, 23, No. 1, pp. 91-102 1996 Copyright @ 1996 Elsevier Science Ltd Printed in the USA. All rights rescxved 0093-6413/96 $12.00 + .00

0093-6413(95)00082-8

COINCIDENCE OF TURBULENT-NONTURBULENT INTERFACE AND HOT-COLD INTERFACE IN A PLANE TURBULENT WAKE

S. Firasat Ali" and E. A. Ibrahim "Associate Professor, Aerospace Science Engineering Department, Tuskegee University, Tuskegee, AL 36088

(Received 13 June 1995; accepted for print 22 September 1995)

Introduction

Plane and axisymmetric jets, plane and axisymmetric wakes and mixing layer are the two dimensional free turbulent shear flows which may be labelled as a class of the simplest types of turbulent shear flow. In these flows the detection of turbulent-nonturbulent interface and measurement of conditional averages have been reviewed by Antonia [1]. For the study of plane turbulent wake, a circular cylinder has been traditionally used to form the wake. A plane wake can also be formed by placing a flat plate with its plane parallel to the mean flow-direction. A special advantage of using a fiat plate would be the absence of a wake producing blunt body with the inevitable local separation and large pressure gradients. The entire wake development out of the two back to back boundary layer would occur essentially without pressure gradient and would be controlled entirely by turbulent transport. Measurements of velocity fluctuation and Reynolds stress in the wake of flat plate are reported by Cbevray and Kovasznay [2]. Sekundov and Yakolevskii [3] have used different lengths of thin flat plates and measured velocity and temperature fluctuations in the near wake. For the study reported in this paper plane turbulent wake is obtained downstream of a heated flat plate. An objective of heating the plate has been examination of the coincidence or identity of turbulent-nonturbulent interface with hot-cold interface (heated flow-nonheated flow interface). Antonia [1] provides a review of experimental investigations on turbulent-nonturbulent interface detection. He acknowledges that velocity gradients, velocity components, and passive scalars

91

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S. FIRASAT ALI and E.A. IBRAHIM

(usually temperature) have been used to mark the turbulent-nonturbulent interface. Some of the studies of free turbulent shear flows, referred in Antonia [1], have verified the coincidence of the two interfaces that is turbulent-nonturbulent interface and hot-cold interface but their verification has been based on observation of fluctuating velocity (or fluctuating vorticity) and temperature signals on oscilloscope or oscillograph. The study reported here aims at developing statistical appreciation of the coincidence of the two interfaces. For this purpose signals precisely corresponding to the turbulent-nonturbulent interface and to the hot-cold interface are identified and their auto and cross correlations are studied. Measurements of mean velocity, mean temperature, and intermittency are also reported in this paper in order to identify the flow. On the setup described here measurements of conditional averages of velocity and temperature and the space-time correlations have also been carried out and reported in detail in Ali and Kovasznay [4]. An apparent disadvantage of heating the plate had been the resulting difficulty in obtaining velocity signal because of the effect of fluid temperature on the heat transfer characteristics of the hot-wire. This problem has been resolved to an acceptable degree in a separate study reported in Ali [5]. Equipment and Experimental Procedure A thin aluminum flat plate was mounted in a low speed wind tunnel such that the plane of the plate was horizontal and parallel to the mean flow direction. Each of the two boundary layers formed on the upper and the lower sides of the plate was tripped near the leading edge of the plate by a coil spring. This arrangement gave rise to a plane turbulent wake behind the trailing edge of the plate (Fig. I). The wind tunnel was an open return type provided with a centrifugal blower which was driven by a 37 kW induction motor and was located at the upstream end of the tunnel. The test section was 0.52 m wide x 3.60 m high x 9.70 m long. To allow for the boundary layer growth, the width of the test section was flared by about 20mm in the whole length. The free stream turbulence level u/U0 was 0.35 %, where u is the rms value of instantaneous velocity fluctuations in the strearnwise direction and U0 is the free stream velocity. This level was considered acceptable since the conditionally averaged values of u/U0 in the turbulent zone up to 1200 mm downstream of the plate were 2.4% or higher. More details about the wind tunnel can be seen in Blackwelder [6].

The flat plate was 2.40 m long and 0.50 m wide. It had constant thickness of 1.60 mm except the last 0.60 m of the length at the downstream end where it was tapered down symmetrically so that the trailing edge was 0.25 mm thick. For heating, both sides of the plate were covered by thin flexible heating panels (Silicone-rubber heaters, Electroflex Heat Inc., Bloomfield, Connecticut). Fach panel had the dimensions 0.28 m x 0.48 m x 1.2 mm thick. Seven panels were glued on each side of the plate so that 0. I0 m of the plate length at the upstream end and 0.34 m of the length at the downstream end were left uncovered. The total power dissipated by the heating panels was 3000 watts at a nominal current of 8 amperes. The same plate without the heating panels was used by Chevray and Kovasznay [2].

INTERFACES IN TURBULENTFLOWS

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The coordina~ system with a sketch of the model is shown in Fig. 1. Most of the measurements in the wake w e ~ obtained in the x-y plane at a distance of 230 mm from one of the vertical walls; z was taken equal to zero on that x-y plane; x=0 and y = 0 at the trailing edge of the plate. At x •0 the fully developed turbulent boundary layers were each 50 mm thick and the maximum temperature rise was 30" C. The wake development was studied from xffi0 to xffi3200 mm but more detailed measurements were made at xffil200 ram. High overheat (0.6) constant temperature hot-wire anemometers in association with very low overheat (0.003) constant current cold-wire were used for obtaining simultaneous signals for air temperature and two components of flow velocity at a given (x,y) point. The high overheat wiles formed a typical X-array to provide x and y components of velocity after their signals were corrected by cold-wire signal. The cold-wire was placed at 1 mm + 0.2 mm upstream of the X-army to sense the temperature of the flow. All probes were made of 3.8 micron diameter bare tungsten wire with a typical length of 2 mm spot welded to 0.4 mm diameter steel prongs. To obtain voltage signals proportional to velocity components, the analog circuit developed for this purpose and the calibration procedure have been described in Ali [5]. The v sensitivity of the probe was determined using the standard X-array calibration procedure of yawing the probe in the free stream flow (Kovasznay et al [7]), where v is the rms of the instantaneous velocity fluctuation in the y direction. For linear operations including differentiation and integration at various stages in the analog circuits ADI 118 A operational amplifiers were used with appropriate resistors and capacitors. Digitec DC voltmeter 6003 A, DISArms voltmeter 55 D 35, Honeywell galvanometer 104 WIG, Honeywell multichannel visicorder 1108 and Tektronics oscilloscope 503 were included in the equipment used for measurements and observations. In the present study it is assumed that appreciable value of Ov//)x implies appreciable value of a vorticity component (av/ax - &u/&y) and hence represents turbulent zone in the flow. Therefore ~v/ax is used as the criterion function to obtain intermittency function L,(t) which is defined as being unity in the turbulent zone and being zero in the nonturbulent zone. The intermittency detector circuit based on ~v/ax was the same as reported by Kibens et al. [8], except for a few changes as described by All and Kovasznay [4]. Another intermittency function, 6(0 is defined as unity in the hot zone and zero in the cold zone. The temperature signal is obtained from a "cold wire" located in the wake at a point where the intermittency function signal is desired. To take care of the drift in the free stream temperature, a temperature signal 0o(t) is obtained from another "cold wire" located in the free stream. To take care of possible heat conduction between the hot and cold zones and the electronic noise in the measuring instruments it is assumed that 0(0 - 0o(t)> ~rs represents hot zone and 0(t) - 00(t)< 0rs represents cold zone, where 00 is the free stream temperature. The temperature Ors is called the threshold temperature and it is well above the electronic noise level and well below the average hot zone temperature. The signal 0(t) - 00(t) is amplified by using PAR tm 113 preamplifier. Due to low frequency response of the temperature sensing probes (3 db point at 0.3 kHz) the preamplifiers is followed by a frequency compensation circuit to obtain a frequency response of up to 3kHz. Then the signal is fed to a comparator with a preadjusted threshold corresponding to 0rs so that the output of the comparator is a random square wave with positive and negative values corresponding to hot and cold zones, respectively. The final interraittency

94

S. FIRASATALI and E.A. IBRAHIM

signal is obtained after this square wave is passed through a delay-time circuit. LaRue [9] and Antonia [1] have discussed the roles of threshold level and delay time when the intermittency function is constructed from a criterion function. To obtain I~ in the present study, the threshold 0r. = 20C and the delay time r . = one msec. The intermittency detector circuit using temperature as criterion function is considerably simpler than the one using velocity or vorticity fluctuations. The circuit diagram is shown in Fig. 2. In this circuit diagram, the delay time circuit is adopted from Kibens et al. [8]. For a typical hot-wire calibration [5], the linear relation between Nusselt number and the square root of Peclet number showed a standard deviation of 0.004 for twenty data points. For a typical cold-wire the temperature coefficient of resistance was found to be within -1- 3 % . Here Nusselt number is defined as Nu = (h-h,0d/k and Peclet number is Pe = pcpUod/k, where h is the heat transfer coefficient at free stream velocity and temperature, h. is the extrapolated value of h at U0 = 0, d is wire diameter, k is thermal conductivity, p is density and cp is specific heat at constant pressure. The uncertainties in the velocity and temperature measurements were +0.2 m/s and -I- 0.30C, respectively. The hot-wire probes were traversed in x, y, and z directions with an uncertainty of -I- 0.5 mm. For detecting the locations of hot-cold and turbulent nonturbulent interfaces an uncertainty of + 0.6 mm was present due to the spacing between the hot-wires and cold-wire. The selection of the threshold values for the detection of the interface locations were influenced by electronic noise and nonturbulent zone velocity fluctuations resulting in an additional uncertainty of + 0. i mm. The total uncertainty in locating the two types of interface was within + 1.2 mm. Upon normalizing it with ~m, the momentum thickness, we get + 0.2 on the dimensionless time scale, T = rU0/~m, where ~- is the delay time of the signal. ExPerimental Results and Discussion According to Townsend [10], a plane turbulent wake is self-preserving if the variation of any mean quantity over any plane, x = constant, is expressible non-dimensionally through suitable scales of length and velocity, b. and U,, as a universal function of y/b,. Here, b, is the half width of the wake based on velocity defect and U, is the velocity defect at the center plane. The scales of length and velocity must be functions of x only. Measurement of mean velocity and mean temperature have been carried out across the wake at several downstream stations. At a given x station the velocity scale is the maximum velocity defect U, (x) = Uo - U u ( x ) and the temperature scale is the maximum temperature rise 0,(x) = 0,~(x) - 00. For every station there are two points y = Yl and y = Y2 where the velocity defect is exactly one half of its maximum value, U (Yl) = U(Y2) = Uo - 1/2U,, then (y2-yl)/2 = bu and (y2+y0/2 = y° so that bu(x) is the half width, y = yo defines the centerplane of the wake. b, (x) is chosen as the length scale so that the non-dimensional normal coordinate is ~ =(y - yo)/b..

The free stream velocity is 3.80 m/s. The free stream temperature is the same as room temperature which was maintained between 21°(2 and 240C. At the trailing edge of the plate, the conventional boundary layer thickness, # on either side of the plate is 50 mm, and the momentum thickness of the boundary layer, ~m = 6 mm. Chevray and Kovasznay [2] have

INTERFACES IN TURBULENTFLOWS

95

reported ~ = 55 mm and ~,, = 5.8 mm for the same plate without heating panels. The Reynolds number of the wake defined as Re = 2Uo6d~ is 3000. According to the principles of conservation of momentum and conservation of heat, the mean momentum defect (comequentiy the momentum thickness) and the mean heat flux are found to be constant along the wake (tee e.g. Tennekes and Lumley, [11]). The Taylor microscale or Taylor dissipation length, ), ffi 7.3 mm in the intermittent region at x/~= ffi 200. The maximum temperature rise 0, at the trailing edge of the plate is 30° C. Figure 3 indicates that at x/~,,--- 4 the flow is still boundary layer type. At the same x station, the mean velocity and mean temperature are not exactly symmetrical about ~ = 0. This asymmetry is attributed to the buoyancy effects as the boundary layer on the top side of the plate is unstably stratified. The initial asymmetry in the mean quantities does not seem to persist further downstream. Figures 3 and 4 show the mean velocity and mean temperature distributions at four streamwise stations. Self-preservation of mean quantifies is confirmed, although it is not surprising. If a half width be is defined for mean temperature distribution in the same way as for mean velocity distribution, then the squared ratio b2d b2, = 1.64. This ratio compares well with the value 1.70 obtained by Townsend [10] in the wake of a circular cylinder. In order to obtain heated plane turbulent wake, the heat is introduced at the wake producing fiat plate. Moreover PrandtiAEnumber for air (P,=0.7) is not much different from unity; hence it is expected that the turbulent-nonturbulent interface and the hot-cold interface would be coincidental in the wake. In heated jet, approximate coincidence of the two interfaces has been observed by Corrsin and Kistler [12] through oscillographic traces of the temperature and the velocity fluctuation signals. In heated boundary layer, Dumas et al. [13] have measured the distributions of intermittency factors 70 and 3', based on temperature and on velocity fluctuation respectively. They obtained identical profiles for T0 and ~,. Jenkins and Goldschmidt [14] have measured conditionally averaged velocity and temperature across the turbulent-nonturbulent interface in a heated plane jet. Both, the measurements of Dumas et al. [13] and those of Jenkins and Goldschmidt [14] support the Corrsin and Kisfler's [12] observation of the coincidence of the turbulentnonturbulent interface and the hot-cold interface. A crucial test to examine the coincidence of the two interfaces can be made if we simultaneously obtain two intermittency functions, L representing the turbulent and nonturbulent regions, and I0 representing the hot and cold regions. Figure 5 shows the traces of av/ax and 0 at a point in the intermittent zone in the wake. The signal av/ax is used to form Iv and the signal 0 is used to form I,. Based on temperature signal the intermittency function I~ and its derivative 8I/at or ~ are shown in Fig. 6. The spikes in ~ signal represent the interface locations; positive and negative spikes respectively correspond to the front and back of a "heated blob." By employing rectifiers in the electronic circuit (Fig. 2) ~ can be split into two signals, one consisting of positive spikes only and another consisting of negative spikes only; let us call them ~÷ and ~. respectively. Similarly |v signal represent turbulent-nonturbulent interface and it is split up into L+ and |v- signals.

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S. FIRASATALI and E.A. IBRAHIM

To examine the coincidence of the two front interfaces, we require the correlation coefficient

and a plot Ri0+~,÷ vs r, where the overbar denotes mean values. This is done with the help of the correlation function computer. In case of the coincidence of the two front interfaces R~0÷iv+ should be maximum and close to unity at T = 0. Similarly the coincidence of the two back interfaces can be examined by determining Ri0.iv.. Figure 7 shows Rie÷iv+ and Rie.kv.as functions of T. On the same graph the auto-correlation curve for ~,+ is also shown. The auto-correlation curve corresponding to iv. and ~. are identical to that of iv÷ and Ri0.|v. and has its peak value at T = 1.64 (T=rUohS.O. This indicates that on the average the back interface of the hot bulge leads the back interface of the turbulent bulge by T = 1.64. R~÷iv÷ has its peak value at T = -0.2. This indicates that on the average the front interface of the hot bulge lags behind the front interface of the turbulent bulge by T = 0.2. This immediately gives an impression that the hot-cold interface and the turbulent-nonturbulent interface are not identical. But it has been found that the lack of coincidence suggested by Fig. 7 is attributed to the imperfections in the electronic circuits and not to the actual physical phenomenon. The differentiators used in the detector circuit have to be followed with low pass filters in order to reduce the high frequency noise. But this is done at the cost of introducing a lag in the intermittency signal on the back interface side. A phenomenon of similar nature was observed by Corrsin and Kistler [12] as the lag introduced in the smoothing process. As v signal is differentiated two times and no differentiator is used for temperature signal, the time lag in Iv signal caused Ri0-i~-vs T curve to be shifted towards the positive side by T = 1.64. The temperature compensation circuit also caused a small shift, T = 0.2, in the intermittency function but this has been on the front interface side. Therefore the Ri0+iv+ vs T curve is shifted towards the negative side by T = 0.2. To estimate the time shifts due to differentiation and temperature compensation circuits, a square wave and an intermittent sine wave were used to represent the temperature and the velocity signals respectively. The square wave was alternating between the values 0 and 1 at a frequency nearly equal to the crossing frequency, so that the change from 0 to 1 would represent the front interface and 1 to 0 would represent the back interface. The intermittent sine wave was obtained from a high frequency (nearly same as the characteristic frequency of turbulence) regular sine wave. This was done by modulating it by the above described square wave. The interfaces represented by the intermittent sine wave and the square wave were identical. The waves were passed through respective intermittency detector circuits and correlation curves were obtained for positive and negative pulses which represented the front and back interfaces respectively. The peaks of cross correlation curves in relation with the auto correlation curves were found to be shifted by the same amount as for the real 0 and v signals. Therefore, we infer that the shifts shown in Fig. 7 are due to imperfect instrumentation and the corrected correlation curves would be shown as in Fig. 8. Figure 8 shows the data obtained at x/~, = 200, y / ~ = 9.5, where 3, = 0.5, 3' being the intermittency factor which is the mean value of I(t). Similar characteristics are observed at 3' = 0.25, 3' = 0.75. On the normalized time scale, T = 1.0 corresponds to the time interval analogous to the Taylor microscale. It is inferred from Fig. 8 that the hot-cold interface and the turbulent-nonturbulent interface are identical, at least they are well within the

INTERFACF~IN TURBULENTFLOWS

97

Taylor microscale. Since the peak values of Rt0+~+ and R~.t~.are rather small it is worthwhile to examine the cross correlation coefficient of the intermittency signals before differentiation. Fig. 9 shows the cross correlation coefficient R~ ~ and the auto-correlation coefficients of I0 and I~ vs T. The cross con'elation is defined as {le(t-z)

- Yo} {Iv(t)

- Yv}

The difference R~ ~ - R, ~ is also plotted in the same figure and it shows appreciable value, only for -4 < T < 4. This indicates that mostly the turbulent and hot blobs of small duration are not correlated whereas the larger blobs show a perfect con'elation. This may be due to the problem that the detection of the smaller blobs is highly sensitive to the choice of the intermitteney detection parameters, the threshold and the hold time. The non-correlation of the smaller blobs may also be a reason for low peak values of Ri0+i~+ and Ri0.t~-. Figures 10 and 11 show the distribution of the intermittency factor V and the normalized value of crossing frequency f~bJUo across the wake at three streamwise stations, where £, is the crossing frequency. Identical profiles for the two downstream stations indicate that the average width of the turbulent bulges and the average lateral position of the interface show invariance when they are scaled with the local momentum width, which was suggested by Corrsin and Kisfler [12] for self-preserving shear flows. The error function curve for 7 and the Gaussian distribution curves for f~bJUo have been drawn in Figs. 10 and 11 following the method shown by LaRue [9]. The ratio of the standard deviation • to ~ (where 7 -- 0.5 at ~/ -- ~) for 7 is 0.245 for positive ~/and it is 0.175 for negative 17. The ratio a to ~ for f~bJUoT is 0.245 for positive 7/ and 0.167 for negative ,7. The invariance of 7 and f-,bJU, was also observed by LaRue [9] in the self-preserving region of the wake of a circular cylinder. A detailed comparison of o to ~ ratios measured by different investigators is reported by LaRue [9]. In the present case the unsymmetric distributions of intermittency and crossing frequency about the center plane of the wake are attributed to the buoyancy effect due to unstable density stratification in the upper region of the wake (Ri ffi -.038) and stable density stratification in the lower region of the wake (Ri = +0.038). Here Ri = -g0,/[0oU,(0UJ~y)~J is the Richardson number, where g is the gravitational acceleration. Conclusion The coincidence of the turbulent-nonturbulent interface with the hot-cold interface has been established in the plane turbulent wake well within the Taylor microscale of turbulence. This is reasonable in view of the fact that the source of heat addition and momentum defect is the same, and the Prandtl number for air (Pr=0.7) is not much different from unity. At the turbulent-nonturbulent interface, the viscous superlayer is believed to be of the order of Kolmogrov microscale (Corrsin and Kistler [12]). The question of coincidence of the two interfaces within the Kolmogrov microscale could not be resolved due to the limitations of the equipment used. A point examined in the flow was essentially a sphere of 1.2 mm diameter because of the physical dimensions and spacing of the velocity and temperature sensors. Also,

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the frequency range of the measurements was kept between 0 and 3 kHz. An increase in the frequency range would give rise to an undesirable level of electronic noise.The coincidence of the two interfaces within the Kolmogrov microscale may be examined with more advanced measurement techniques such as laser velocimetry. Another limitation of the equipment was that a clear identification of the turbulent-nonturbulent interface would not be obtained if the turbulent intensity u/U0 in the turbulent zone was lower than one percent. This was not a problem in the present work since the research was mainly concerned with the high intensity turbulence in the wake of the plate. It is proposed that more revealing information may be obtained by conducting a set of similar experiments with water and other fluids having Prandtl number appreciably different from unity. Acknowledgements S. F. Ali is grateful for the guidance and support provided to him by Dr. L. S. G. Kovasznay, Dr. Fazle Hussain, Dr. Hajime Fujita, and Dr. Ho Chih Ming. References 1. R. A. Antonia, Ann. Rev. Fluid Mech. 13, 131 (1981) 2. R. Chevray and L. S. G. Kovasznay, AIAA Journal 7, 1641 (1969) 3. A. N. Sekundov and Yakovlevskii, Fluid Dynamics (translated from Russian) 5, 1026 (1970) 4. S. F. Ali and L. S. G. Kovasznay, Structure of the Turbulence in the Wake Behind a Heated Flat Plate, Tech. Rep. 75-2, the Johns Hopkins University, Baltimore (1975) 5. S. F. Ali, Rev. Sci. Instr. 46, 185 (1975) 6. R. F. Blackwelder, large Scale Motion of a Turbulent Boundary Layer with a Zero and Favorable Pressure Gradient, Ph.D. thesis, the Johns Hopkins University, Baltimore (1970) 7. L. S. G. Kovasznay, V. Kibens, and R. F. Blackwelder, J. Fluid Mech. 41,283 (1970) 8. V. Kibens, L. S. G. Kovasznay, and L. J. Oswald, Rev. Sci. Instr. 45, 1138 (1974) 9. J. C. l.aRue, Phys. Fluids 17, 1513 (1974) 10. A. A. Townsend, The Structure of Turbulent Shear Flow, 2nd ed., Cambridge University Press, London (1976) 11. H. Tennekes and J. L. Lumley, A first course in turbulence, The M.I.T. Press, Cambridge (1972) 12. S. Corrsin and A. Kistler, NACA Report 1244 (1955) 13. R. Dumas, L. Fulachier and E. Arzoumanian, Comptes Rendus, Acad. Sc. Pads 274 A, 267 (1972) 14. P. E. Jenkins and V. W. Goldschmidt, A Study of the Intermittent Region of a Heated TwoDimensional Plane Jet, Rep. HL 74-45, Purdue University, Lafayette (1974)

INTERFACES IN TURBULENT FLOWS

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FIG. 1 The turbulent wake and the hot-wire configuration.

100

S. FIRASAT ALI and E.A. IBRAHIM

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R

0.6

O.B

I.O