Turbulence spectra from individual realization laser velocimetry data

Experiments in Fluids

Experiments in Fluids 3, 35-44 (1985)

© Springer-Vertag 1985

Turbulence spectra from individual realization laser velocimetry data D. V. Srikantaiah and H. W. Coleman

MHD Energy Center and Mechanical and Nuclear Engineering Department, Mississippi State University, Mississippi State, Miss. 39762, USA

Abstract. Two techniques - a modified correlation method and a

direct transform method - have been evaluated for use in making spectral estimates from randomly-sampled random data (such as turbulence data obtained with an individual realization laser Doppler velocimeter (LDV)). The effect of "bad" points (which usually appear randomly in actual LDV data sets) on the spectral estimates has been studied. Necessary modifications and extensions to the techniques have been determined based on studies using simulated data with known spectral characteristics. The direct transform method is found to have certain advantages over the correlation method. From the LDV measurements taken in a simulated coal-fired magnetohydrodynamic flow field, major differences in turbulence spectra obtained from the two methods are observed in the region immediately downstream of the combustor where the flow was evidently dominated by (uncorrelated) combustion instabilities. Spectra are reported for three axial positions, and comparisons with classic turbulent pipe flow data are presented.

1 Introduction

Spectral analysis is useful in determining the energy content and scales associated with a turbulent flow field. In a flow with combustion, spectral analysis can also be used to indicate flow regions dominated by combustion instabilities and flow regions in which the fluctuations are more characteristic of classic turbulence. The laser Doppler velocimeter (LDV) allows measurement of turbulent velocity fields in hostile environments in which previously no such measurements could be made. Laser beams are focussed to form a measurement volume, and particles in the flow traveling through this volume scatter light. The scattered light is collected and focussed onto a photodetector, which transmits a voltage modulated at the Doppler frequency to a signal processor. The Doppler frequency of the signal gives the velocity of the particle at the instant of light scattering. The particles arrive randomly at the measurement volume, and hence the output signal of an LDV used in a turbulent flow is a randomly-sampled random signal. If one can add enough particles to the flow so as to have an essentially continuous signal, the u s e of a fre-

quency tracker gives an analog signal output which can be analyzed using traditional equi-spaced sampling techniques for spectral analysis. In the research program at Mississippi State University (MSU), plans are made to use laser-based diagnostic instruments in large prototype and industrial size systems where it will be difficult or impossible to seed the flow, certainly to the extent where a continuous signal could be achieved. In the simulated coal-fired magnetohydrodynamic (MILD) flow in the MSU test stand, naturally-occurring particles in the flow were used as light scatterers for the LDV-system. The corresponding uncontrolled low data rate, combined with the use of a frequency counter which gave "individual realization" measurements, led to data sets which were composed of randomly-acquired samples of a random (turbulent) process. As an additional complication, such data sets typically contain randomly-distributed " b a d " points which are eliminated during analysis. (Definition of what is meant by "bad" points and the reasons for their presence are discussed later.) In such a situation, one is faced with spectral analysis of a randomly-sampled ranl dom signal where the data sets contain randomly-distributed "bad" points. The objectives of the present study were (1) to determine the most promising techniques for use in making spectral estimates from randomly-sampled random data, (2) to determine the modifications and extensions to these techniques necessary to allow their reliable application to actual L D V data, and (3) to determine the spectral characteristics of the flow field in the simulated coal-fired MHD flow in the MSU test stand.

1.1 Background Standard techniques of spectral analysis usually consist of determining the power spectra from the correlation functions of the signal through use of Fourier transforms. These techniques are well established for uniformly-sampled data (Blackman and Tukey 1958; Bendat and Piersol 1971). However, they are not directly applicable for the randomly-sampled case.

36 For randomly-sampled data, the extent, in terms of maximum frequency range, to which proper spectral estimates can be made depends upon several different parameters, including the mean sampling rate and the sample size considered for analysis. Shapiro and Silverman (1960) showed, theoretically, that alias-free spectral estimates can be made if the sampling is Poisson distributed. However, they did not provide a n y practical tool or algorithm that would easily be applicable to real data, such as the data from an individual realization LDV system used in fluid flow measurements. Much work can be found in the literature aimed at finding a practical technique or algorithm to make reliable spectral estimates from randomly-sampled data. The techniques, generally, fall into two categories correlation methods and direct transform methods. The techniques described by Mayo etal. (1974) and Gaster and Roberts (1977) seemed the most promising to serve as a starting point in this study. The method of Mayo et al. is a correlation method in which the autocorrelation functions are computed first by a special "slotting" technique, and then the power spectral density functions are determined using Fourier transforms. The method of Gaster and Roberts is a direct transform method wherein the power spectral density is first computed directly and then, if needed, the autocorretation can be obtained through use of inverse Fourier transforms. Relatively few turbulence spectra from LDV measurements have been reported. Those which have been were obtained under laboratory conditions where the flow was seeded with additional particles and thus some control over the data rate was established. In some cases (Tsuji and Morikawa 1982, Takagi et al. 1981, Lading and Edwards 1975, Berman and Dunning 1973) a frequency tracker was used, allowing determination of spectra from the standard technique for equi-spaced samples. In others (Reis et al. 1982, Meyers and Clemmons 1979, Wang 1975, Smith and Meadows 1974) the modified correlation method of Mayo et al. (1974) was used for spectral analysis of the randomly-sampled data. This method has been found to give rise to ambiguity spectrum levels due to uncorrelated data. The ambiguity spectrum can be eliminated by omitting the "zero-lag" term in the autocorrelation estimates (Gaster and Roberts 1975). However, the modified correlation ("slotting") method of Mayo et al., without eliminating the "zero-lag" information, has been widely used, almost as a standard technique, to present turbulence spectra (see, for example, the references quoted above). As shown in this paper, the direct transform method of Gaster and Roberts eliminates the ambiguity spectrum and has certain advantages over the correlation method of Mayo et al. Few investigations have used the direct transform method to determine turbulence spectra from randomly-sampled LDV data. There has been one previous application of this method by Roberts et al. (1980) to

Experiments in Fluids 3 (1985) obtain turbulence spectra from LDV measurements in a seeded air jet. The previous investigations which have reported turbulence spectra from LDV data have been under tightly controlled laboratory conditions in which there was control over the data rate. The present paper examines how effectively the spectral techniques can be used to determine turbulence spectra from flow situations with an uncontrolled and relatively low data rate. In the following, the basic techniques for spectral analysis of randomly-sampled random signals are discussed, results of application of these techniques to simulated data sets and modifications to the techniques are presented, and the techniques are then applied to determine spectral estimates from LDV data obtained in a simulated coal-fired MHD flow field.

2 Spectral analysis of randomly-sampled random signals In this section, the spectral techniques of Mayo et al. (the correlation method) and Gaster and Roberts (the direct transform method) are briefly described. 2.1 Correlation method In this method an estimate of the autocorrelation, Rk (~), corresponding to lag time, r = k A ~, and for k # 0 is given by S U M ( k A ~) , k=l,2 ..... m (1) Rk(~)=R(kAz) H ( k A z) where 3~ is the slot width, S U M ( k A y ) is the sum of all the lag products of samples, say x (ti) x (tj), falling in the slot k, H ( k Az) is the number of such products contained in the slot k and m is some predetermined m a x i m u m number of slots. For k = 0, that is, at zero lag time, the autocorrelation is simply computed from

1 U R0 (z = 0) = R (0) = ~ - ~=~1x~

(2)

where N is the total number of samples. Note xi = x (t;) in the above equations denotes the i-th sample value at time t; with the sample mean previously subtracted from the data. The one-sided power spectral density function, S ( f ) , is then computed, in the same manner as in the case of the standard technique for equispaced samples (Bendat and Piersol 1971), from

=2At

R 0 + 2 ~ ' ~ Rrcos

+ ( - 1)kRm

r= 1

f= mfc '

k=0,1,2,

""'

m;

1 2Az

(3)

D. V. Srikantaiah and tL W. Coleman: Turbulene spectra from individual realization laser velocimetry data where f~ is the Nyquist cutoff frequency. Equation (3) gives raw estimates of the PSD functions. Smoothed PSD estimates can be obtained using a suitable smoothing window. A few remarks about the selection of the parameters A~, m, N and the mean sampling rate, v, in this slotting technique are worthwhile mentioning. The errors in the time domain in estimating t h e autocorrelation functions are of the order of + Ar/2. In order to keep these errors small, Az has to be much smaller than the mean time interval, Arm, between the data points. However, A z must also be large enough compared with Attain, the minimum time observed between data points in the data set, for a meaningful average to be obtained using Eq. (1). The selection of A r and the maximum lag number, m, also depends on the extent to which the signal is correlated, the resolution required and the m a x i m u m frequency of interest. It should be noted, however, that one loses the information at frequencies greater than 1/Ar in the slotting process. N is usually chosen to be of the order of 105 or more.

2.2 Direct transform method In this method the smoothed one-sided power spectral density functions are directly computed from : S (f) = -~ - ~ x 2 (tj) D 2 (tj)],

i= ~

1

(4)

J J where v is the mean sampling rate, T is the length of the record and D (t) is the smoothing window. Other notation is as before, k may be noted that there is no limit on how many values o f f one may use evaluating S ( f ) . However, for N data values, there are at most N independent values of S ( f ) which may be computed (Mayo 1978). The spectral estimates can be computed at equi-spaced or logarithmically-spaced frequency values, f~, obtained from

f~=k/T,

k = 0 , 1,2 ....

(5)

or

f~=f6

2 (k/3)

, fo=l/T,

k=0,1,2 .....

(6)

If the final term in Eq. (4) is deleted, a false constant shift or bias is observed in the magnitude of the spectral estimates. This bias is proportional to aZ/v, where a is the standard deviation of x (t). The direct transform method can be applied to either one single block of length T or the entire record can be divided into n short blocks, each of length TB (approximated by T~ = NB A tin, where NB = number of points in a short block), and then the method can be individually applied to each short block, finally averaging the results

37

over n blocks. The averaging over the short blocks has the effect of smoothing and using D ( t ) = 1 in Eq. (4) may be justified in most cases (Gaster and Roberts 1977, and Roberts and Gaster 1978). Note that the short block concept can also be applied to the correlation method.

3 Studies on simulated data In order to test and evaluate the spectral analysis methods described previously, several examples whose spectral characteristics are known were considered. A randomlysampled sine wave signal, a randomly-sampled white noise signal and a randomly-sampled signal which would simulate a first-order spectrum were tested with the methods. The first-order spectrum was of particular interest as it has similarities to a typical one-dimensional turbulence spectrum. Only selected results for the firstorder spectrum will be discussed here. More details and the sine wave and white noise results are presented in Srikantaiah (1982). Comparisons of different spectral analysis techniques for randomly-sampled sine data, with regard to accuracy and computational speed, can be found in a recent paper by Bell (1983).

3.1 First-orderspectrum One example considered for testing and evaluating the spectral analysis methods was a first-order spectrum of the form,

S(f)-

2 1 + 10-4(2~f) 2"

(7)

A similar example was also considered by Gaster and Roberts (1977). The sample points with Poisson distributed time instants for this example were generated by an extension of a method described by Franklin (1965). Franklin described the simulation method for equi-spaced time intervals. The method can be suitably adapted for random time intervals (Srikantaiah 1982). The data were analyzed by using both the correlation method of Mayo et al. and the direct transform method. Various block sizes (short blocks), frequency ranges, mean sampling rates, sample size, etc. were considered. Typical resulting power spectra are shown in Fig. 1. The theoretical spectrum, Eq. (7), and the bias effect in the direct transform method are also shown in the figure. It can be seen that the estimated spectra from the correlation method and the direct transform true estimator agree reasonably well with the theoretical spectrum up to a maximum frequency equal to a value slightly over the mean sampling frequency. At higher frequencies the spectral estimates have large variabilities, producing negative values which are physically impossible. The spectral estimates were considered valid up to that frequency after which the first negative PSD point appeared. This is the

38

Experiments in Fluids 3 (1985) I

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ill

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....

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Fig. 1. PSD f o r simulated f i r s t - o r d e r spectrum

practical limitation of the method although, in theory, an infinite number of alias-free spectral estimates can be made for uniform Poisson sampling in the direct transform method. Different sample sizes, 104 and above, were tried to see their effect on the computed spectral estimates. 105 seemed to be an adequate size. 5 x 105 points were also tried, but this became computationally expensive and the spectral estimates did not improve significantly as N increased from 105 to 5 x l0 s. 3.2 Spectral estimation from samples with arbitrarily discarded points

When applying spectral analysis techniques to actual data acquired with a typical individual realization LDV system, one encounters an additional complication. There are points included at random positions in the data set which must be considered "bad" points. These points arise from two causes. First, despite the sophisticated validation circuitry present in state-of-the-art LDV frequency counters, some points are validated which are not in fact "true" particle velocities but are rather a result of noise (optical, electronic, or both). This is a greater problem in a flow of the type in the MSU test stand than in tightly-controlled laboratory flows. These points are called "outliers" since they often lie many standard deviations (5 to 10) from the mean. Such points, though few in number, can have a

highly significant effect on the magnitude of the calculated standard deviation of the data set. It is common practice in LDV data analysis to calculate a mean, #, and standard deviation, G, for an original data set, eliminate all points lying outside _+ 3 G or +_4 a, and then consider the remaining points as the "true" data set for which a new mean and standard deviation are calculated and reported. The +_3 a criterion was used in this work. The second category of "bad" points are those for which the time interval between two successive measured points exceeds the maximum time interval which the LDV counter/interface/c0mputer system can count. (The maximum time interval for the MSU system was 0.131 second.) While these points are "good" (if they meet the _+3 a test) for calculating # and G~:they are "bad" from a spectral analysis viewpoint because they have unknown (erroneous) Ati' s. " '"~ It was necessary to investigate, after removing the randomly-occurring "bad" points from a data set, the effect on the calculated spectra for both the correlation and direct transform methods. A study was conducted on the simulated data of the sine signal and the first-order spectrum. Points lying beyond certain values of At and a were arbitrarily removed from the original data set to simulate both categories of "bad" points. Once the points are removed this way, the originality of the signal is lost. Suitable logical values (based on some specified criteria) can be introduced in the positions left vacant by the discarded points. Alternatively, the signal can be reconstructed as though there had never been any points in the missed positions. For example, consider that the time interval A t i , i+ 1 = t i + l - ti between sample values xi and xi+] was too large and overflowed. Then xi+] and Ati, i+l would be discarded and in their places would be xi+2 and Ati+u+2 (being taken as the time interval between xi and xi+2). After a preliminary study, the second method of reconstructing the signal was chosen for use in the present study. The results are shown in Fig. 2 for the first-order spectrum case. It is seen that as long as the total number of points removed arbitrarily from the original data set is within about one percent of the total sample size, the spectrum does not deviate very much from the original. Some noise is introduced because of the artificiality of reconstructing the signal, but that level is low and is generally smeared out in the averaging process. 3.3 White noise added to first-order spectrum

In an attempt to simulate the actual conditions that exist in an LDV data set, the case of white noise superposed over the first-order spectrum was considered. The amount of white noise added was such that the mean square value of the noise was roughly equal to that of the first-order spectrum signal. The data required for the white noise (Gaussian random numbers) were generated indepen-

D. V. Srikantaiah and H. W. Coleman: Turbulene spectra from individual realization laser velocimetry data

39

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Fig. 2. Effect of arbitrary removal of points from original data set on simulated first-order spectrum

dently of the first-order spectrum data. Both the correlation and direct transform methods were applied to obtain the spectral estimates. Further, in order to simulate an actual LDV data set, about one percent of the points, considered as "bad", was removed from the original data set. The spectral estimates from the reconstructed data set, as explained previously, were then obtained using t h e direct transform method. The resulting PSD estimates for this case are shown in Fig. 3. It is seen that the spectral estimates obtained from the "standard" correlation method of Mayo et al. are higher than those from the direct transform method. Also, this correlation method shows the ambiguity spectrum at the higher frequency end because of the presence of the noise in the signal. The spectral estimates computed for two different values of AT with the same frequency resolution are also shown to demonstrate the different ambiguity spectrum level in each case. These levels, of course, also depend on the amount of noise added. As mentioned previously, the ambiguity spectrum can be removed by omitting the "zero-lag" term in the autocorrelation (Gaster and Roberts 1975). The spectral estimates as a result of omitting the "zero-lag" information are also shown in Fig. 3. (The PSD estimates become negative beyond the last point shown. They also depend on m and AT). Although there is improvement in the PSD estimates, they are only good up to maximum frequencies less than half those obtained from the direct transform method. The direct transform method does not show an ambiguity noise spectrum at the higher frequency end. It is seen that the direct transform method filters out the ambiguity spectrum due to the uncorrelated noise. The spectrum from the reconstructed data set (one percent points removed from the original data set) does not deviate significantly from the spectrum obtained from the original data set.

The above comparison of the results shows that the direct transform method gives better estimates of the PSD than does the correlation method (even after removal of the "zero-lag" term) and is thus found superior in the case investigated. Gross errors in calculated spectra can result if the correlation method is applied in the standard way which many researchers have been using, that is, without omitting the "zero-lag" information.

101~

.......

I

.......

N =100 0 0 0

10[ ~

I

Q

T + E S P E C T + M '

-

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~ 2 o

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,,,L

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10 -1 - -

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-u = 2 0 0 S a m p l e s / s

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~F

CORRELATION AT : 0/)005S m = lO0] (ZERO-LAG TERM REMOVED)

~

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40

Experiments in Fluids 3 (1985)

4 The experiment A schematic of the MHD test stand is shown in Fig. 4. The facility is a computer-controlled apparatus for the simulation of the gas stream conditions in the steamgeneration components downstream of the channel in a coal-fired M H D power plant. Air at 1.1 atm pressure and a flow rate of 228 kg/hr is preheated to 1000 °K by electrical resistance heaters and used to burn fuel oil in a combustor, obtaining a nominal temperature of 2500 °K. Fuel, ash and seed are injected into the combustion chamber through a hole at the center of an injector plate and air through four holes, placed symmetrically around the center hole, at an angular direction into the main stream. The fly ash (residues of coal combustion) and seed are injected into the hot combustion gases to simulate the chemical composition and high temperature of the ash/seed-laden gas stream of a baseline MHD power plant. To increase the flexibility and versatility of the test stand, the system downstream of the combustor is assembled from separate sections which can be used in various arrangements. The sections are all water-cooled and have 20.32 can inside diameter. Bare-wall and refractory-lined stainless steel pipe sections are available. The gas temperature decreases along the test stand as energy is transferred to the cooling water. Reynolds number based on diameter varies from about 6 x 103 at the combustor to about 104 at 7.5m downstream. Since the facility is used to develop and test optical diagnostic techniques for characterizing high temperature, particle-laden flows such as an MHD gas stream, optical instrumentation ports are located at various axial positions along the test stand. Ports located 1.5 m, 3.6 m and 7.5 m (or 7.4, 17.7 and 36.9 diameters) downstream of the combustor were used for the velocity measurements with the LDV system. The LDV system used was a commercial (TSI) singlechannel system using backscatter optics and a frequency counter. A Bragg cell was used for frequency shifting when required. The laser and the optics were mounted on a vibration-isolation table next to the test stand. The optics train was mounted on a stepping motor driven,

Exhaust A Fuel,seed&coalashin}ectors \ CombustorfNp× probes

Fig. 4. Schematic of the test stand

Dry fitter box(~ Cyc le ~scr, ~:rI[ I 1 \~. f~L~

single degree of freedom traverse. A microprocessor controlled the traverse and also acquired data from the frequency counter. After storing a block of data points, the microprocessor transferred the data through a minicomputer, which controls the test stand, to a magnetic tape unit where the data were stored for posttest analysis. The microprocessor was programmed so that on demand the mean, the standard deviation and the histogram of a 1000 data point block could be viewed for "real time" data anaylsis. More detailed descriptions of the test stand and LDV system are reported by Srikantaiah (1982) and Ali (1982). At each of the three axial locations where LDV measurements were made, two radial positions, y = 2.0 cm and 10.16 cm (centerline) measured from the pipe inside wall, were chosen for spectral analysis of the axial velocity measurements. Only axial velocity fluctuations were considered for the spectral analysis in this study. One hundred blocks of data (t02,400 data points) needed for the spectral analysis were collected for each given spatial location. The "bad" points, discussed earlier, were eliminated from the original data ensemble and a new mean, a new standard deviation and a new histogram were computed and were taken to be the "true" values. Although some distributions were slightly skewed, the distribution of velocity at each of the spatial locations was approximately Gaussian. Histograms of the sample interarrival times, At/s, in each ensemble of measurements were also obtained and they did show exponential behavior (Poisson distribution).

5 Discussion of results The power spectral density values were computed using both the "standard" correlation method and the direct transform method. Short blocks were used in both the methods. The power spectra computed at each position in the pipe are shown in Figs. 5-10. The autocorrelation estimates can be found in Srikantaiah (1982). The spectra from the original data sets were computed without dropping any of the "bad" point's. The time overflow points were set equal to the m a x i m u m time interval which the LDV counter/interface/c0mputer system could count between two successive measured points. As previously discussed, once the "bad" points are removed from the original data set the original signal is distorted. Reasonably good estimates of the spectra can be recovered by reconstructing the signal as though there had never been any points in the discarded positions. In the following, the spectra obtained from such an analysis will be called the reconstructed spectra. Spectra in each case were also computed from "true blocks" (called true block spectra below). True blocks are those which did not contain may "bad" data points. The spectra from the original data set, the reconstructed spectra and the true

D. V. Srikantaiah and H. W. Coleman: T u r b u l e n e spectra f r o m i n d i v i d u a l r e a l i z a t i o n laser v e l o c i m e t r y data 100

block spectra were all determined using the direct transform method and are shown in the figures discussed below. The elimination of points did not appreciably affect the spectra from the correlation method as long as 27max <

41

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It can be seen from the turbulence spectra measured at X = 1.5 m (7.4 diameters downstream of the combustor) that the velocities are uncorrelated and exhibit a noise type spectra (Figs. 5, 6). The flow in this region is evidently dominated by combustion instabilities and cannot be considered a turbulent flow in the traditional sense based on the evidence shown in the spectra. It was determined from studies on simulated data that the direct transform method filters out contributions from uncorrelated data, such as white noise, present in the signal, and that is the situation observed in these measurements. The correlation method does provide an ambiguous white noise type spectrum (which could have been eliminated by omitting the "zero-lag" information). At X = 3.6 m (17.7 diameters downstream) the intensity of turbulence is more at the 2.0 cm position than that at the centerline of the pipe (Figs. 7, 8). It can be seen, in the case of spectral estimation from the direct transform

N

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Fig. 6. Turbulence spectra at X = 1.5 m, y = 2.0 cm. M e a n data rate 513/sec

100

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(Z~T = O.O001s , m =100)

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Fig. 5. Turbulence spectra at X = rate 7962 sec

I,,,,I

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42

Experiments in Fluids 3 (1985) 10C

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,

[

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y = 2.0 cm. M e a n

data

method at the 2.0cm position, that the reconstructed spectrum, the true block spectrum and the spectrum from the original data set show greater differences than in any other case observed. An examination of the histograms of the instantaneous Velocities Uj and the interarrival times Ati revealed that there was a higher percentage of " b a d " points (3.66%) rejected than in any other case. Time-overflow points themselves were about 2.8 percent. Once these "bad" points were removed, the average data rate was larger than for the original data set. In this particular case, the true block spectrum can be taken to be the best estimate, b u t there were not enough points (only 9900) from the true blocks to get good averaged estimates. The reconstructed spectrum, however, agrees fairly well with the true block spectrum (even after 3.66% rejection of points). It should also be noticed that in this case reasonable spectral estimates are obtained for frequencies past the mean sampling frequency, v, the reason being that the histogram of Ati showed a higher percentage of data that were sampled in the smaller time intervals. This was the only such peculiar case observed out of all the data sets taken. In most of the measurements, the rejections of points were less than about one percent. The power spectral estimates at the two radial positions for the X = 7.5 m location (36.9 diameters downstream) are shown in Figs. 9 and 10. The shapes of the spectra are different from those measured at the same radial positions

Fig. 9. T u r b u l e n c e spectra at X = 7.5 m , y = 10.16 cm. M e a n d a t a rate 606/sec

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SPECTRUMFROM ORIGINAL DATA RECONSTPHPT~F~ ........ SPECTRUM

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CORRELATION METHOD(At = 0.0005 s , m = 1OO)

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Fig. 10. Turbulence spectra at X = 7.5 m, y = 2.0 cm. M e a n d a t a rate 107/sec

D. V. Srikantaiah and H. W. Coleman: Turbulene spectra from individual realization laser velocimetry data

10C

at the 3.6 m X-location. The spectra are sloping more toward the higher frequencies. A more fully developed and more organized steady turbulent pipe flow was expected in this region.

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