the measurement of turbolence with the lda

Ann. Rev. Fluid Mech. 1979. 11:443-503Copyright © 1979 by Annual Reviews Inc. All rights reserved

THE MEASUREMENT OF 8148TURBULENCE WITH THELASER-DOPPLER ANEMOMETER

Preben Buchhave1 and William K. George, Jr.

Department of Mechanical Engineering, State University of New Yorkat Buffalo, Buffalo, New York 14214

John L. LumleySibley School of Mechanical and Aerospace Engineering, Cornell University,Ithaca, New York 14853

1 INTRODUCTION

1.1 The Laser-Doppler Method

The laser-Doppler anemometer (LDA in the following) is an instrumentfor research on fluid flows that has the potential of allowing local time-resolved measurements of fluid velocity without disturbing the fluidphenomenon under investigation. However, as in the case of its maincompetitor in the field, the hot-wire anemometer, the application of theLDA to turbulence measurements is not trivial, and the problems (andchallenges) facing the experimenter grow with increasing levels of turbulentfluctuations. This article reviews the status of turbulence measurementswith the LDA and outlines the achievements of the past few years andsome of the remaining problems. The article deals with only thosemethods allowing time-resolved measurements of the velocity, and thuswe do not treat such signal-processing methods as spectrum analysis ofthe Doppler signal or time-averaged photon correlation.

The laser-Doppler method is based on the measurement of the Dopplershift of laser light scattered from small particles carried along with thefluid. Because of the motion of the particles the frequency of the scatteredlight will be shifted by an amount coo = K’u, where K = Ks- Ki is thedifference between the wave vector of the scattered and incident light

1 On leave from DISA Electronics, A/S, Copenhagen, Denmark.

4430066-4189/79/0115-0443501.00

www.annualreviews.org/aronlineAnnual Reviews

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http://www.annualreviews.org/aronline

444 BUCHHAVE, GEORGE & LUMLEY

respectively and u -- (u,v,w) is the velocity of the medium (Figure 1-1).The stability and spatial coherence of the gas laser output allow the lightto be focused to a small spot within the fluid. Excellent monographs nowexist on the basic theory of the LDA, and the reader is referred to thesefor further information (Durst, Melling & Whitelaw 1976, Durrani Greated 1977). For the detection of the frequency shift, the opticalheterodyning or light-beating effect of a square law photodetector isutilized. The scattered light is mixed on the surface of the photodetectorwith a reference beam derived directly from the laser or with light scatteredfrom another incident beam. Figure 1-2 shows the optical arrangement,that is almost exclusively used today, and also serves to define the basicoptical parameters.

The function of the optical system is partly to direct and focus theincident beams to a small volume within the flow field, and partly tocollect the scattered light from this volume while simultaneously dis-criminating against ambient light and light scattered from regions outsidethe volume. For clarity it is desirable to distinguish between the regionof space defined by the intersection of the two incident beams, which wedesignate the probe volume, and the region in space from which Dopplersignals are detected, which we designate the measuring volume. Themeasuring volume is more difficult to define than the probe volume, sinceit depends on the field of view of the detector, on the overall gain in thesystem, and even on the method of signal processing. The probe volumeis usually defined by the 1/e2-boundary of the modulation depth of theinterference pattern existing in the beam intersection region. WithGaussian laser beams the probe volume is an ellipsoid, and we have thefamiliar expression for the probe-volume dimensions in terms of the

Fi#ure 1-1 Wave vector diagram showing directions of incident and scattered light andthe velocity vector.


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LASER-DOPPLER ANEMOMETER 445

half-axes of the ellipsoid in the coordinate system defined in Figure 1-2(a,b, and c) or the standard deviation of the Gaussian intensity distribu-tion (o-x, % and az):

2a = 4crx - d~ (l.l,la)cos (0/2)’

2b = 4% = dl, (1.1.1b)

d~(1.1.1c)2c = 4az = sin (0/2)’

where dl, the beam waist of the focused laser beam in the measuringvolume, may be expressed as

4 )tdI - (1.1.2)

n A0

and where A0 is the convergence angle of the focused beam, and 2 thewavelength of the laser light.

The photodetector responds to the square of the total light fieldincident at each point of the detector surface, di slut Iz, where Et =E1 +E2, and E1 and E2 represent the scalar field strength at the detector

~ detector fringe

/~

mode

Brago cell ~ ~

~mode

Figure 1-2 Optical configuration of typical LDA.


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surface of the scattered light and the reference beam in the reference beammode or the two scattered beams in the dual-scattered beam mode. Theresulting photocurrent is the sum of contributions from each point ofthe detector surface:

i~cfa ,EtJZdA=fa ’Ell2dA+fa [E212dA

+ I. (E1E~+E~E2)dA, (1.1.3)d

where Ae is the total detector area.In case of a single particle at location xv(t), the resulting photocurrent

may be written as

isp(t) i~(t) + i2 (t)+ 2~i2 (i) COS [~o t + ~v (1.1.4)

where i~(t) and i2(t) are the currents which would be caused by eachincident beam acting alone and where the Doppler frequency ~o is given

by

~o = [4~ uv/2] sin (0/2), (1.1.5)

where uv is the x-component of the instant~eous particle velocity uvand ~v is a constant phase term depending on the initial location of theparticle. The ac part of the signal can be written

isp,ac(t) IW[Xv(t)] COS [~ O t + ~V]’ (1.1.6)

where w(x) = 2v/i~(x) i2(X) may be regarded as a weight function describ-ing the intensity of the modulated part of the detector signal receivedfrom space location x, and I includes the scattering characteristics of theparticle.

In general, when many particles are present simultaneously in themeasuring volume, the Doppler signal will consist of a slowly varyingpart (the dc part or pedestal) and a higher frequency term (the ac Doppler modulated term):

i(t) = i~ (t) i2 (t) + 2~) cos [co t +~b(t )], (1.1.7)

where now

09o -- [4~m0/2_] sin (0/2), (1.1.8)

and it(t), i2(t) and ~b(t) are now randomly fluctuating quantities resultingfrom the superposition at the detector surface of the scattered optical


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LASER-DOPPLER ANEMOMETER447

fields from all illuminated particles. Normally, there are many interferencefringes within the measuring volume, in which case il(t), i2(t), and q~(t) slowly varying relative to the cosine term: Uo --- (u0, Vo, wo) represents thevelocity of the fluid in the measuring volume. The exact definition of Uodepends on the circumstances of the measurement, and further discussionis deferred to the description of particular cases later in this article.

From Equations (1.1.5) and (1.1.8) we see that the LDA instrumentmeasures the x-component of the instantaneous velocity Uo ; the instru-ment is totally insensitive to the other velocity components vo and Wo.However, the particle transit time or residence time is determined by themagnitude of the total velocity vector. As we discuss in Sections 2 and 3,the properties of the Doppler signal and the noise level and accuracyof the system as a whole are determined by the magnitude and directionof the total velocity vector relative to the orientation of the optical beams.Furthermore, it should be noted that the relation between the Dopplerfrequency and the velocity uo is linear; thus, calibration is limited to thedetermination of the coefficient of proportionality given by the laserwavelength 2 and the beam intersection angle 0. Complete characteriza-tion of the turbulent velocity field requires simultaneous determinationof all three velocity components. For practical and economical reasons,however, many investigations are carried out with a one-componentsystem that in some cases creates special problems because of the unknownmagnitude and direction of Uo.

The properties of the Doppler signal and thus the proper method ofsignal processing depend strongly on the number of particles present inthe measuring volume and the mode of operation of the detector optics(Drain 1972, Lading 1973, Hanson 1974). If the probability that two more particles are present simultaneously in the measuring volume isnegligible, the best signal-to-noise ratio of the Doppler signal is obtainedin the so-called incoherent detection mode, in which a large detectoraperture is used to collect a maximum of scattered light from a singleparticle. The detector optics is fully able to resolve the interference fringepattern existing in the probe volume (this mode of operation is oftenreferred to as the fringe mode). If many particles simultaneously occupythe measuring volume, the photocurrent contributions from the individualparticles add incoherently at the detector surface, id(t)= Y4,(t), wherei,(t) is the detector current caused by each individual particle independentof the others. However, in the many-particle case a better signal-to-noiseratio can be obtained using the coherent detection mode. In this modethe receiver aperture is kept so small that the scattered light from allparticles add coherently on the detector surface: ia(t)~: [EE,(t)] 2. In thecoherent detection mode the signal strength can be enhanced by using a


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strong reference beam and the signal-to-noise ratio improved with theaddition of more particles, whereas the signal-to-noise ratio is independentof particle number in the incoherent detection mode.

The type of signal encountered in actual measurements depends on thefluid and the type and amount of seeding material added. In order tocontrol the characteristics of the particle motion it is desirable to workonly with monodisperse scattering particles uniformly distributed in thefluid. When this is not possible the experimenter can in most casesexert a certain amount of control over the size of particles contributingto the signal accepted by the processor by adjusting signal gain and insome cases by setting levels for the rejection of very low- or very high-amplitude signals. Continuous, many-particle signals are most oftenencountered in measurements in liquids, whereas Doppler signals frommeasurements in gases are often of the single-burst type. In this articlewe do not enter the subject of light scattering from small particles or thequestions concerning the proper seeding methods or materials, but againrefer the reader to the monographs on LDA measurements and thearticles by Farmer (1972), Durst & Eliasson (1976), and Adrian & Orloff(1977), where these problems are discussed and references to specializedpapers on the subject may be found.

1.2 Physical Parameters Affecting the Doppler Signal

As indicated above, the LDA responds to the component of fluid velocitynormal to the interference fringe planes in the probe volume. The actualphysical process underlying the laser-Doppler method can be describedas a measurement of the phase difference of two optical waves scatteredby small particles in the probe volume. The time rate of change of thisphase difference can be interpreted as a measure of the suitably definedfluid velocity relative to the interference fringe planes in the probe volume.Evidently this interpretation requires that the particles follow the fluidmotion with negligible slip due to fluid shear or acceleration. This aspectof the problem is taken up in Section 5; in the meantime it is assumedthat the particles follow the fluid motion exactly. The only other physicalparameter of the fluid directly affecting the phase of the scattered lightis the refractive index. Variations in the index of refraction may causesmall random shifts in the phase of the optical beams and thereby in thelocation of the interference fringe planes. Larger optical inhomogeneitiesmay cause the measuring volume to move about randomly or cause thebeams to miss each other, resulting in a loss of signal or a reduced signal-to-noise ratio. Optical index fluctuations are primarily a result of densityfluctuations (in turn caused by pressure or temperature fluctuations) or,in the case of turbulent mixing or combustion, a result of concentration


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fluctuations. The effect of optical index fluctuations on the spectral pro-perties of the Doppler signal is treated in Section 4. Meanwhile weconsider the fluid to be of constant density and refractive index.

In addition, the signal phase is influenced by electronic noise fromvarious sources, of which the most important are the detector shot noiseand the thermal noise in the input circuit to the signal processor. Usuallyit is possible to operate the system in such a way that the detector shotnoise is dominant (shot-noise-limited signal), and in this case the signal-to-noise ratio is directly proportional to the power of the incident laserbeams.

In the following discussion it is convenient to introduce the concept ofthe "ideal LDA," in which shot noise and thermal electronic noise maybe considered negligible, and where the optical system may be consideredperfect. A perfect optical system would be one in which no optical aberra-tions or alignment errors existed, so that the interference fringe systemwas undistorted and at a fixed location in space. For the purpose ofinvestigating the inherent properties of the LDA as an instrument forturbulence measurements, we assume an ideal LDA for the remainder ofthis paper; the influence of noise and optical distortions is not considered.

1.3 SignaI Processor Operation

In addition to the above considerations, two other main factors determinethe performance of the LDA in turbulence measurements: first, the typeor appearance of the Doppler signal, i.e. whether the signal may be con-sidered "quasi-continuous" or whether it consists of single bursts inter-spersed with periods of low-power shot noise; second; the type of signalprocessor employed, and even to a certain degree the mode of operationof the signal processor.

Details of the principles and modes of operation of the various types ofLDA signal processors are not described here; the reader is referred tothe ample literature on the subject. For our discussion of ,turbulencemeasurements it suffices for the moment to distinguish between twodistinct operating modes: one, the frequency demodulator type (of whichthe frequency tracker may be taken as the prime example) operating oncontinuous signals,~ and two, the burst processor, which essentiallymeasures the time for a particle traversal through the measuring volumeor through some finite length within the measuring volume. An exampleof the latter kind of processor is the burst counter or LDA counter,which measures the time for an integer number of zero crossings of ahigh-pass filtered Doppler burst. Other systems operating in the burstmode have been developed, e.g. the filter bank (Baker & Wigley 1967)and the burst correlator (Fog 1976).


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It is however, important to note that many of these processors mayoperate in either the continuous, tracking mode or the burst modedepending on the choice of parameters such as signal level and bandwidth.As an example, the frequency tracker is normally applied to quasi-continuous signals. However, most trackers have a distinct minimumoperational input-signal level below which the tracking loop will bebroken and the instrument kept in a hold mode (dropoout protection). the input signal is lowered sufficiently, so that only the highest peaks ofthe fluctuating envelope are allowed to exceed the drop-out level, thetracker will operate in a single-burst detection mode. If the burst frequencyappears within the instrument’s "capture range," it will acquire, lock, andtrack each burst, but remain in drop-out condition in the time intervalsbetween bursts. Conversely, a typical burst-type instrument such as theLDA counter may operate in a continuous mode on a continuous signaland even provide an analog output through a digital-analog converter.

With these qualifications in mind we consider below in more detail theoperation of the tracker as a typical example of a continuous LDA signalprocessor providing essentially a continuous analog output and theoperation of the LDA counter as an example of the burst processorproviding one digital velocity sample per burst.

2 THE CONTINUOUS LDA

2.1 The Doppler Current

The statistical properties of continuous, many-particle Doppler signalscan now be regarded as well understood. The detector current wasinitially treated, as a superposition of random phase wave packets(Greated & Durrani 1971) and the methods of analysis of noise in demodulators applied to derive the statistical properties of the amplitudeand phase fluctuations of the envelope of signals from steady flows.Edwards et al (11971) applied the methods of light-beating spectroscopyto the optical spectrum and the resulting photocurrent and derivedexpressions for the spectral properties of the detector current in steady,nonuniform flow. They showed the spectrum from a Gaussian measuringvolume to be Gaussian with a half-width inversely proportional to thediameter of the measuring volume while the spectrum in nonuniform flowwas shown to be skewed and broadened as a result of gradients withinthe measuring volume.

George & Lumley (1973) utilized the fact that the turbulent velocityfluctuations and the particle distribution are statistically independent tocreate a framework for incorporating both the flow field statistics and thefluctuations caused by the random scattering center positions in a unified


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treatment. Since this is the only approach that has led to explicit resultsfor the instantaneous Doppler signal in turbulent flow, it is followedbelow.

It is convenient to represent the Doppler current by the followingexpression

i(t) = ~-~ i~p(a,t) g(a) (2.1.1)

where isp(a,t) is the time dependent current generated by the particle thatwas at a at t = 0, where g(a) is a function that accounts for the presence(or absence) of a particle at a, and where N(O is the instantaneous numberof particles in the volume. This expression is exactly equivalent to onein which the current is represented as the sum of the currents generatedby the particles in the volume.

The current generated by a single particle passing through the scatter-ing volume can be represented as

i~p(a,t) = Iw(x) cos K- (2.1.2)

where K is the previously defined scattering wavevector, x = x(a,t) is theposition of the particle that started at a, and Iw(x) is the envelope of thecurrent. The function w(x) is defined so that w(0) -- 1 and can be takento specify the spatial extent of the scattering volume by defining

= j" W(X) d3x (2.1.3)

A consequence of this is that the instantaneous number of particles in thevolume is

N(t) = w(x[a,t]) g( a) da a. (2.1.4)

If we operate only on the current generated by individual particles(Equation 2.1.2), we can in a straightforward manner obtain the velocityinformation from the time-dependent phase since

0(t)= K.x = K" u(a,t’)dt’ (2.1.5)

and

0(t).= K- u(a,t’). (2.1.6)

We explore this attractive possibility later when we consider burstcounters. Unfortunately, for the continuous (many-particle) case con-sidered here things are more complicated.


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2.2 The Velocity Measured by the Continuous LDA

Ideally one would like to begin from the phase of the Doppler currentand identify a portion of the phase as the velocity signal. Unfortunately,because of the random character of the fluctuations in phase and theirmany sources, there is no straightforward way to accomplish this. Thatthis is so is readily obvious by substituting Equations (2.1.2) and (2.1.5) Equation 2.1.1 and observing that since w(x) is, in effect, time dependentbecause of the particle’s motion, it is not possible to separate directlythe velocity-dependent part of the phase.

George & Lumley (1973) recognized the apparent impossibility of thisapproach and proposed instead that for the continuous LDA an effectivevelocity be defined as an average over the volume of the velocities of theparticles present. Formally this can be written as

Uo(t) = N(t)------ff u(a,t) w(x[a,t]) g(a) (2.2.1)

This approach was particularly useful for the cases where the flowcould be assumed incompressible and the particles were statisticallyuniformly distributed throughout space. With several additional localassumptions it was possible to show that the statistics of this effectivevelocity could be separated from the other sources of random phasefluctuations and was therefore an appropriate (although possibly non-unique) choice for the measured velocity.

The relation of Uo to the Eulerian velocity at the center of the volumecan be computed from the following relationships:

Uo =-~. u(x,t) w(x) d3x (2.2.2)

Uo)2_- __1 F w2(x) u’2(x) (u0NV

+ ~ W(X) W(X’) U’(X) U’(X’) ’. (2.2.3)

The overbar is used to represent the ensemble average, primes are takento represent fluctuating values, and capital letters are hereafter used todenote mean values unless otherwise noted. In most cases of interest forthe continuous LDA, the expected number of particles in the volume, N,is large, and the first term in Equation (2.2.3) can be neglected.

From Equation (2.2.2), the effect of the spatial averaging on measure-ments of the mean velocity can be shown directly to depend (to first order)


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on the curvature of the profile. This was first pointed out by Edwardset al (1971), who treated only laminar flow. The results are seldomimportant for turbulence measurement since mean profile curvatureeffects are relatively less important than for laminar flow.

George & Lumley (1973) calculated the effect of the spatial averagingin Equation (2.2.3) on the measured spectrum in a homogeneous, isotropicturbulent flow for an ideal Gaussian scattering volume for which

w(x) = exp - E2-~ + ~ + (2,2.4)

They obtained results like those shown in F~igures 2-1 and 2-2 by usingTaylor’s hypothesis to convert temporal variations to spatial variations(i.e. converting frequency to wave number by co = kU) and defining atransfer function as

T(k) = Measured spectrum = Fro(k) (2.2.5)True spectrum FI l(k)’

where F~(kl) is the one-dimensional spectrum; and by using Pao’sspectrum for the turbulence. As for the hotwire, it is clear that disturb-

Fiflure 2.1 Turbulent spectra normalized in Kolmogorov variables showing effect ofspatial averaging (George & Lumley 1973).


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~.o 0.4

0.2

’LOG ~i

Figure 2-2 Typical transfer function resulting from averaging of turbulent fluctuationsacross the volume, ~ ~ (largest gcattering volume dimension)- 1 (George & Lumley 1973).

ances smaller than the largest dimension of the volume cannot bemeasured.

There are a number of extensions and refinements that could be carriedout. In particular, alternate formulations for the effective measuredvelocity could be constructed and their consequences explored, and theeffect of alternate volume shapes [w(x) different from Gaussian] could calculated. It appears unlikely that any additional fundamental knowledgewould be gained.

2.3 The Problem of Random Phase Fluctuations (Doppler

Ambiguity)

George & Lumley (1973) were able to show that if an effective displace-ment field Xo is defined by

dx o(2.3.1)Uo = dt ’

then the expression for the Doppler current (Equation 2.1.1) can reduced to

i(t) = (F2 + G2)1/2 cos (K" x0- q~), (2.3.2)

where

Gtan ~b = ~. (2.3.3)


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Note that F, G, and ~b are all time-dependent and in general randomquantities.

Most important, by_ assuming that the average number of particles inthe volume is large (N > 10) and that relative displacements due to theturbulence are governed by Gaussian probability distributions, it ispossible to show that the fluctuations of Xo (and hence Uo) and those F, G, and ~b are statistically independent. This fact and the additionalfact that F and G could be shown to be Gaussian random variablesallowed immediate application of the large body of research on FMnoise that has developed over the past forty years.

It is not the purpose of this article to review all of the consequencesof this on the design of continuous LDA processors, but rather to concen-trate on those aspects that bear directly on the interpretation of measure-merits of turbulent velocities. For a more complete treatment the readeris referred to George & Lumley (1973) and the papers by Duranni Greated (1973) and George (1976).

The ideal signal processor would remove the amplitude informationfrom Equation (2.3.2) and yield an output proportional to the derivativeof the total phase. Thus the "velocity-like" output of the detector, sayud(t), would be given by

ua(t) = Uo(t)- K- l&(t). (2.3.4)

Since the fluctuations Uo and & have been shown to be statisticallyindependent, the statistical quantities of this output signal most oftenof interest have the following properties :

The autocorrelations of Uo and ~ are additive

ua(t) ua(t’) = Uo(t) Uo(t’) + K-z ~b(t) (2.3.5)

The spectra of Uo and ~ are additive

S..(f) =- S.o(f) + K- 2S4(f). (2.3.6)

The mean square values of uo and ~ are additive

Ud2 = U(~ + K-2(~2. (2.3.7)

The probability densities of u0 and ~ are convolved :

p,~(u) p.o(X- u) pd,/K(x) (2.3.8)

Clearly it is not enough to be able to relate the volume-averagedvelocity to the turbulence (Section 2.2), but we must also have complete


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knowledge of.~ if we are to interpret the measurements. An unfortunatefact is that the spectral bands of uo and ~ always overlap so that a simpleremoval of the phase fluctuations by filtering is not possible. In fact,there appears to be no way to avoid this contamination although thereare several ways in which its effects can be minimized. These are mentionedlater.

TURBULENCE

i ~ " ~ ii17. Zit...i...:.-: .....ii" "]

ZCC

DISA

Figure 2-3 Simulated turbulence and LDA outputs from thr~e types of signal proeessors~a p~as¢ locR loop, a zero-crossing detector, and a frequency lock loop.


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LASER-DOPPLER ANEMOMETER457

2.4 The Statistics of the Phase Fluctuations

The reality of the phase fluctuations and their effect on attempts tomeasure turbulence from continuous signal processors is readily apparentfrom Figure 2-3, taken from Lading & Edwards (1976). The top tracerepresents the turbulent input signal to an LDA signal simulator, whilethe bottom three traces represent the outputs of various continuous LDAprocessors. If there were no random phase fluctuations in the signal, allfour traces should be identical. Instead, the bottom traces are similar andare distinctly different from the turbulence signal .at the top. The high-frequency contributions of the random phase fluctuations are readilyapparent; more subtle, but still present, are the lower-frequency contri-butions that overlap the frequency band of the turbulent fluctuations.

The statistics of the random phase fluctuations can be shown to beentirely determined by one parameter, A~0, called the Doppler ambiguitybandwidth (or broadening). (A~o)-1 is, in fact, related to the time envelope of the Doppler current remains correlated with itself and canbe determined from the following

F(t) F(t’) = G(t) G(t’)exp { - (Aogz)2/2}, (2.4.1)

F2 G2

where z = t’- t. The rate at which the envelope becomes uncorrelated canbe influenced by numerous effects. Of primary importance for turbulencemeasurement are: the loss of correlation because of the change in popu-lation of the scatterers, the loss in correlation due to the fact that meangradients in the volume cause particles to beat against each other, andthe same effect due to fluctuating gradients in the volume. These arereferred to respectively as: transit time broadening, A~o~.; gradientbroadening, A¢oo ; and turbulence broadening, Acor. It can be shown that

(a~o)2 = (a~oL)z +(no, G)z +(A~or)z. (2.4.2)Each component can be .calculated separately and the appropriate equa-tions are shown in Table 1, For details the reader is referred to George &Lumley (1973) and Berman & Dunning (1973).

Table 1 Formulas for calculating ambiguity bandwidths [from Berman & Dunning (1973)and George & Lumley (1973)]

Transit time:

Turbulent:

Mean gradient :Total:


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Table 2 Statistical properties of phase fluctuations

Probability density:

Autocorrelation:

Spectrum:

An example of the relative importance of these three effects can befound in the measurements of Berman & Dunning (1973) in a turbulentpipe flow. Near the center of the pipe where the mean gradient was smallthe transit time and turbulence broadening terms were about equal anddominant. Near the wall, however, where both turbulence intensity andmean gradient were larger, the mean gradient and turbulence broadeningdominated while the transit time broadening was negligible.

It is appropriate to note that there seems to have often been confusionregarding the turbulent broadening part, AcoT. This broadening is quiteindependent of the effect of fluctuations in the effective velocity Uo, whichcontribute directly to a broadening of the Doppler current spectrum [forfurther information see George (1976)].

Once Aco is known, the above-mentioned statistical measures of ~ canreadily be calculated from the relationships summarized in Table 2 (from

0.5

0.4

0.:3

0.2

I I I I- 10 - 5 ,.~/~ 5 10

Figure 2-4 Probability density of random phase fluctuations.


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RN (~) = ~(t)~ (t

p (r) =

o I0 1 2

Figure 2-5 Autocorrelation of random phase fluctuations.

George 1976). These are plotted in Figures 2-4, 2-5 and 2-6. It is easy toshow that ~-~ -- ~. This of course assumes that our ideal detector has anunlimited frequency response.

2.5 Implications of Phase Fluctuations on TurbulenceMeasurementA number of implications of these results on attempts to measure turbul-ence are obvious. First, measurements of low intensity turbulence are notsomething the continuous LDA does well. Second, measurements of auto-correlations at small time delays and spectra at high frequencies aredifficult or impossible. Third, the accuracy of probability density deter-mination is entirely dependent on whether u’


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46O BUCHHAVE~ GEORGE & LUMLEY

0.~-0

0.35

0.30

0.25

0.20

0.15

ADAPTED FROM RICE (1948)

0.10 -

0.05 t

o I I I I I I0 1 2 3 4 5 6

Figure 2-6 Spectrum of random phase fluctuations.

Figure 2-7 shows an attempt to measure an autocorrelation in a gridturbulence. The finite value of the correlation at zero time delay reflectsthe finite bandwidth of the processor. Figure 2-8 shows a similar attemptto measure a velocity spectrum in a turbulent pipe flow. In this case, thehigh level of the turbulence spectrum relative to that of__the phase fluctua-tions indicates that a reasonably accurate estimate of ub2 could be madeif a filter were placed near the frequency at which the spectra are equal(shown by arrow). A similar spectrum measured in a grid turbulence shown in Figure 2-9. Filtering in this case would be less satisfactory.

It should be dear from the examples chosen that single-point statisticalmeasures can be taken with an LDA. It should also be clear that greatcare must be taken in interpreting the results. Moreover, significantbenefits are to be gained if the ambiguity noise can be minimized. Meansfor doing this are discussed in the next section.

Figure 2-7 Typical autocorrelations measured in turbulent flow using an LDA. Finitevalue for r = 0 reflects effect offinite bandwidth of detector.Figure 2-8 "Velocity" spectra measured in turbulent pipe flow showing effect of phasefluctuations at high frequencies (Berman & Dunning 1973).


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Figure 2-7

GRID TURBULENCE

RM ="’U°--~- ~ = 2500

X~ = 40 ....

Figure 2-8

10 2

,<

10-4

lO-S10°

’~~ PREDICTEDLEVELS

10’ 10~


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I-2 -1 log ~,

Figure 2-9 Turbulence spectrum in grid turbulence: x/M = 45; x representscircles represent hot-film anemometer (George & Lumley 1973).

LDA,

2.6 Minimizing and Eliminating the Effects of PhaseFluctuations on Turbulence MeasurementFor any given flow situation there exists an optimal size for the scatteringvolume that minimizes the value of A~0 and hence the spectral level ofthe phase fluctuations. This is readily evident from Equation (2.4.2) andTable 1. For more detail and examples see George (1972), Berman Dunning (1973) and George & Lumley (1973). George & Lumley considered the trade-off between spatial resolution and minimizing thespectral height of the phase fluctuations; in almost all cases the phasefluctuations are responsible for the limiting conditions on spectralmeasurements.

A more imaginative approach to minimizing the effects of phase fluc-tuations is simply to try and eliminate some of them. Wilmshurst & Rizzo(1974), noting that the maximum phase spikes correspond to the minimumof the Doppler envelope, used a tracker error signal weighted withenvelope amplitude, thereby eliminating a significant contribution to ~.In fact, any real tracker discriminates against these large phase fluctua-tions by tracking the Doppler signal only when the envelope is above agiven threshold. If the portion of the signal removed by this procedureis small compared with the length of the envelope (which is, in turn,usually much smaller than any flow time of interest), a significant reduc-


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tion in ~-~ appears to be possible. Unfortunately the implications of thison the phase fluctuation spectrum are not clear; that is, it is not clearwhether the level of the phase-fluctuation spectrum is reduced (thedesired result) or whether only the high-frequency part of the spectrumis removed.2 If only the latter, then the benefits are minimal.

Following a suggestion of Rowe, George (1974) showed that a sequenceof low-pass filters could be used to make accurate turbulence-intensitymeasurements of the spectrum of the zero crossings, even when the con-tribution of the phase fluctuations was large. This technique has beensuccessfully applied by numerous investigators to make reliable intensitymeasurements.

It is easy to show that the correlation of phase fluctuations betweennonoverlapping scattering volumes is zero (George & Lumley 1973,George 1976). This implies that the continuous LDA can be successfullyused for ambiguity-free, nonintrusive cross-correlation measurements.Examples of such use are Morton & Clark (1971) and Reed (1978). Maanen et al (1976), following a suggestion of George (1971; see alsoGeorge & Lumley 1973), built a special optical system for correlatingsignals from two volumes within the smallest turbulence scale, therebyobtaining the ambiguity-free, single-point turbulence spectrum.

In summary, it is possible to minimize the spectral level of the phasefluctuations in all situations by choosing the appropriate scatteringvolume. Moreover, even these phase fluctuations .can ¢ften be dealt withdirectly by filtering or subtraction, since their spectral level is knownexactly once A~0 is known. Finally the continuous LDA is at its verybest in measuring cross-correlations since it is both ambiguity free andnonintrusive (thus eliminating the familiar problem of the wake of theleading probe).

2.7 Dropout

No real tracker can track the fluctuating phase of the Doppler signalexactly. Momentarily or permanently the lock between tracker oscillatorand signal may be broken and the tracker encounters a so-called dropout.In modern trackers the tracking performance is monitored by a dropoutdetector, which indicates the presence of a dropout when the correlationbetween the signal and the tracker oscillator falls below a preset level.Under normal operating conditions dropout is associated with signalamplitude fluctuations caused by phase cancellations between manyparticles in the measuring volume rather than with the absence of

2 It appears to be straightforward to extend Rice’s calculations to include the spectrum

of the level crossings. To the best of our knowledge this has not been done.


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particles in the volume. Since these amplitude fluctuations occur with time scales of the order of the particle transit time through the measuring volume, dropout periods are normally short compared to the time scales of the flow. A typical dropout case is illustrated in Figure 2-10, which shows at the bottom the Doppler signal, next above the control voltage to the tracker oscillator, and at the top the tracker output including dropout periods, where the signal during dropout has been replaced by the last measured value. As discussed earlier, dropout can be beneficial if it discriminates against the largest phase fluctuations.

There are three ways of handling the signal during dropout : 1. setting the output equal to zero during dropout, 2. setting the output equal to the mean during dropout, and 3. holding the signal at its last value during dropout. In the following each of these alternatives is analyzed.

We first assume that the occurrence of dropout and reacquisition of signal are statistically independent of the velocity, and that the fractional dropout time is small and independent of velocity. This is a good approximation in the case of a many-particle continuous signal. The probability that the Doppler signal amplitude R is below a given value, R , say, is given by George (1976):

R2 Prob [ r < R , ] = 1 - exp [ - $1, (2.7.1)

where Pis the mean-square Doppler signal. If the dropout threshold is given, the fraction of time the tracker is in dropout condition can readily be computed.

Figure 2-10 Oscilloscope traces showing (from bottom): the Doppler current, LDA output (threshold at zero), and LDA output (threshold set to eliminate large phase fluctuations).


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Lumley, Buchhave & George (1978) suppose that the dropout timesare independently distributed with a probability of a dropout in anyinterval dt of #dr. They further suppose that the dropout process is aMarkov chain so that #dt is the transition probability from "in" to "out."The transitions to the tracking condition are also independently distri-buted with a transition probability from "out" to "in" in the interval dt ofvdt. Thus the expected "in-time" is 1/#, and the expected "out-time" is 1Iv.(The latter corresponds to the result computed from the amplitude prob-ability distribution above.) The probability of being "in" at any instant is(1 +#Iv)- 1, while the probability of being "out" is (1 + v/#)- 1. For allcases of interest v/# ~ 1, which is to say that the tracker is "in" most ofthe time.

To analyze the tracker output an indicator function I(t) was definedas + 1 when the tracker is "in" and 0 when the tracker is "out." Theexpected value of I is clearly f= (1 +~/v)-1. The tracker output signalcan thus be represented as

f(t) = f(t)" I(t), (2.7.2)

where f(t) is the output without dropout (the tracker response to theinstantaneous velocity).

Case I : Output equal to zero durin9 dropout We consider the statisticsoffgiven by Equation (2.7.2). It is straightforward to show that the meanis given by

while the mean-square fluctuation is given by

f,2 = f,2 12 + if2 + f,2] 1,2. (2.7.4)

I(t) can be shown to have a mean-square value given by

-- /~v /~(2.7.5)r2 = i0-i)

and an autocovariance function given by

~v e-(u+v)l’l ,~ kt e-~1~1.(2.7.6)Rx - (p + v)z - v

The analysis leads to the following expressions for the autocorrelationand spectrum off

R~ = ~2Ry + f2Rt -- RfRt (2.7.7)


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466 BUCHHAVE,GEORGE & LUMLEY

and

nvz 1 +(~/v)2 ’(2.7.8)

To get this particular form for the spectrum we have assumed that theperiods between dropouts 1/# are short compared to the integral scaleT. The effect of the dropout is to add a spike to the autocovariance anda low-passed white noise to the spectrum. The amount of noise is deter-mined by the mean square of the total signal (i.e. mean plus fluctuatingcomponent). If the periods between dropouts are long compared to theintegral scale, Rj- will dominate in the last term of Equation (2.7.7) andthe effect of the dropout will be merely a slightly reduced record length,but no significant addition of noise.

Case 2 : Output set equal to mean durin9 dropout We can represent thetracker output as follows:

~f(t) = f(t) I(t)+ [1 - I(t) ] (2.7.9)

It is easy to show that the mean is preserved in this case, i.e. ~ = f.The mean-square fluctuation, the autocorrelation, and the spectrum aregiven by Equations (2.7.4), (2.7.7), and (2.7.8) respectively with f set to zero. Thus the noise level introduced by the dropout on these measure-ments is considerably reduced since the spectral height of the noise, forexample, is now determined only by the dropout and the mean-squarefluctuating signal.

Case 3: Output holds last value This method is common practice incommercial trackers. The output signal can be repregented by:

~f(t) = f(t) I(t)+ [1 - I (t) (2.7.10)

where

f(t) = f(t,) for tk <= t < tk+t (2.7.11)

and tk (k = 1,2,...) are the dropout times.It is easy to show that in this case the mean and all higher moments of

f are identical to those of f. Thus, not only are the mean velocity andthe turbulence intensity accurately reproduced, but also the entire prob-ability density. Some noise is still present in the spectrum, but undernormal conditions the spectral noise power will be considerably lowerthan in the previous case. The autocovariance can be computed fromEquation (2.7.10)

R2 = ]2Rf + (Rs + R?- 2Rr?) R~ + 2i(1 - l) R~r?. (2.7.12)


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Assuming again T >> 1Iv and 1/V we see that the autocovariance isdominated by the first term, which is the desired autocovariance of thesignal, and the second term, which is a spike at the origin of duration -~1/# and a magnitude determined by the difference Ry + R?r - 2Rye, whichfor 1/kt ~ T is small compared to f,2. Thus the noise term in the spectrumwill be significantly reduced relative to the previous case:

(2.7.13)

where (A f) 2 represents the difference 2Considering the other extreme, 1Iv >> T, we must have RI2 -~ 0, since

f(t) will now most likely be uncorrelated__ with its value a_Lt the previousdropout. At the same time R~ ~ f,2 RI SO that R~RI -~ f,2 R~2 and alsoRIR~ ~- (1-12) R~. Thus the autocovariance reduces to

R7 ~- iR~+(1-i)f ’2 R2~ (2.7.14)

For long dropout times method 3 is more noisy than setting the outputto zero during dropout. However, 1Iv ~ T is not the usual mode ofoperation for the tracker.

In summary, dropout can affect all statistical properties of the trackeroutput. The magnitude of this effect depends on the manner in which thetracker compensates for this dropout. In the most usual mode of operationof the tracker, where dropouts are short compared to the flow integralscale, dropout is best handled by maintaining the last measured outputuntil the tracker has again acquired lock. This method preserves allmoments of the velocity distribution, and adds the least amount of noiseto the spectrum. Higher order approximations during dropout can beenvisioned, e.g. linear extrapolation from the last measured value. Suchmethods might further reduce noise in tracker output, but these have,to the best of our knowledge, not been tried in practice. If dropout periodslong compared to the integral scale occur, it might be best to set theoutput equal to a running average of previously measured values.

3 BURST-SIGNAL PROCESSING

3.1. Function of the Signal Processor

The operation’of the burst-signal processor (single-particle measurementor individual realization) differs radically from that of the quasi-continuoussignal processor. Whereas the latter in its normal mode of operationpresents a continuous output, possibly interspersed with periods ofdropout, the former detects the velocity of individual particles and in


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most cases bases the velocity output on a measurement of the time offlight across the measuring volume (or across a fixed number of inter-ference fringes in. the measuring volume). If we know tlhe fringe spacing,the timing information is equivalent to a velocity sample. The velocitysample is computed and presented at the output after the particletraversal, usually as a digital word, and thus does not represent a real-time analog output. The function of the processor is inherently a digital,random sampling of the velocity at a time determined by the particlearrival. In some processors the real-time clock is also used to providethe time increment between samples allowing, as we discuss later in thissection, the formation of the velocity autocovariance and spectrumfrom the randomly sampled data. If the seeding concentration is suffi-ciently high, a continuous analog output can be obtained through adigital-to-analog converter, but in most cases of interest the data ratewould be below that required to resolve small-scale turbulent fluctua-tions in a real-time mode. Thus, the main problem in burst detectionLDA signal processing is the extraction of statistical quantities from therandomly sampled data accumulated by the processor.

The best optical configuration for detection of single particles is thedual-beam, fringe-mode system that optimizes the collection of lightfrom single scattering particles. In the absence of particles in the measur-ing volume, the detector produces shot noise when ambient light andlaser light reach the detector from optical surfaces and windows. Thephotocurrent is thus a band-limited shot-noise signal superimposed onrandomly arriving Doppler bursts. Mayo (1975) suggests a triplystochastic model for the burst signal:

i(t)= ~ eg,’h(t-t,), (3.1.1)

where e, 9,, and h(t) are, respectively, electronic charge, the random-charge gain of the detector preamplifier, and the impulse response of thedetector/preamp/filter system. The random time for the occurrence ofthe nth photoelectron is denoted by t,. This process is a modulatedPoisson process (known as a Mandel-process from photon countingtheory), where the modulating function is the statistical mean photo-electron rate it(t), which in turn is related to the classical optical power the detector p(t) by:

qp(t)2(t)= h~’ (3.1.2)

where q is detector efficiency and hv the photon energy. The opticalpower p(t) reflects the superposition of the background photoelectron


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arrival rate 2b and the randomly arriving bursts:

p(t)=~ 2b+ Ijf(t--zj, uj, rj (3.1.3)

where Ij is a random peak amplitude parameter determined by particlesize and scattering function, and f(t,u,r) is a normalized optical systemresponse function. The peak of a burst thus is marked by rj. As indicatedf, which specifies the burst shape, is in general a function of particlevelocity uj and particle trajectory parameter U.

The triply stochastic nature of the photodetector signal is evident fromthis model. The occurrence of the photoelectron at t, is a Poisson processmodulated by the instantaneousclassical optical power p(t), which is inturn a stochastic process determined by the particle arrival timeswhich are again distributed as a modulated Poisson process. The functionmodulating the burst arrival times is determined by the velocity timefunction u(t) and possibly other effects correlated with the velocity. Thevelocity may itself be a stochastic function, e.g. Gaussian random turbul-ence. Mayo (1977) provides a FORTRAN program for the simulation a burst signal, which allows evaluation of signal-processor performancefrom assumed models of the statistical processes entering the signalsynthesis.

With sufficiently high laser power, the background current may beneglected and the signal-to-noise ratio of the detector current (defined asthe signal variance relative to the square of the signal mean at the peakof the burst) high enough to allow the wave form of the burst to be setequal to the classical optical pulse. For the case of two Gaussian laserbeams of equal power intersecting in a common focal volume as illustratedin Figure 1-2, we retrieve the well-known Gaussian burst in the limit ofinfinite signal-to-noise ratio:

ij(t) = I~w[x~(t)] [1 + cos (mot)], (3.1.4)

where x~ is the particle trajectory, cod the Doppler frequency, and w(x)is given by Equation (2.2.4).

3.2 The Velocity Measured by Burst Processors

REPRESENTATION Unlike the case with the continuous LDA, where wewere able only to postulate the form of the measured velocity and thenshow that this form was reasonable, for the burst processor we can writethe velocity signal directly since there are assumed to be only singleparticles present. Thus the velocity measured by the ideal burst processoris exactly the velocity of the particle producing the burst. The only


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470 BUCHHAVE, GEORGE& LUMLEY

U .... INSTANTANEOUS VELOCITY

[] NO PARTICLE IN VOLUME

fl ~ PARTICLE~~RESEiT

/ \ []\

\

~ t

Typical velocity signal from individually arriving particles.

Figure 3-1 Typical uo(t) from Equation (3.2.1) illustrating how the velocity is sampled individually ariving particles.

question is whether the randomly arriving samples are capable of re-producing the statistics of the desired Eulerian velocity field. We showbelow that the answer to this question is in the affirmative /f the burstsignal is analyzed for the period of time the particle is in the volume.

To examine this further we write for the measured velocity

Uo(t) = w[x(a,t)] u( a,t) 9( d~a (3.2.1)

in the same form as that assumed for the continuous case.3 In effect, thisreproduces the instantaneous velocity of the scattering particle only whileit is in the scattering volume. A typical Uo(t) is shown in Figure 3-1.Assuming incompressibility and mapping from the Lagrangian to theEulerian frame this can be written

uo(t) = w(x) u[x(a,t)] ql (x,t) d3x, (3.2.2)

where

;ox = a + u(a, tl) dr1 (3.2.3)

and

~7(a) ~ #l(x,t). (3.2.4)

The statistics ofyl(X,t) for statistically uniform seeding to second ordercan be shown to be given by (George, Buchhave & Lumley 1978)4

3 The average over the number of particles in the volume is no longer necessary since,by hypothesis, there is never more than one.

’* George & Lumley (1973) and George (1976) igt~ored the time dependence of the function. The differences, with one exception, are inconsequential.


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LASER-DOPPLER ANEMOMETER 47 1

(3.2.5)

&,t) gl(x’ , t f ) = P P ( W I X ’ , t ’ ) f P L 2 ,

where p(x,r 1 x’,t’) is the probability that the particle at x at time t has moved to x’ at time t’ and p is the expected number of particles per unit volume. An excellent approximation to p(x,t 1 x’,t’) for all practical pur- poses is:

(3.2.6) p(x,t I x’,t’) -+ b(x’- x- Uz)+p2.

MEAN VALUES Once the above problem for the statistics of gl(x,t) is un- coupled from the statistics of the velocity field, then it is straightforward to show that the average of uo is given by

f

uo(t) = p J u(x,t) w(x) d3x . (3.2.7)

To understand thc implications of this consider the simple case where u(x,t) is independent of x,t and w(x,t) = 1 within the volume and zero outside. (Note that these correspond to a uniform mean velocity and an on-off scattering volume ; the on-off volume is an excellent approximation to the volume seen by a burst processor if we ignore the angle effects discussed later.) Assuming this, we have

-

__ - uo(t) = p v u(x,t). (3.2.8)

Thus the measured mean velocity is directly proportional to the desired Eulerian mean. I t should nut be inferred from this that one need merely average realizations to achieve reasonable averages. Equation (3.2.8) assumes that the velocity has been measured during all of the time the particle is in the volume and the factor pV accounts for the portion of time that it is not.

To see this, assume that we are determining the mean values by time- averaging,

(3.2.9)

where it is assumed that 7 is su6ciently long. Equation (3.2.8) implies that

(3.2.10) l T a N Io uo(t) dt = u(x7t) dt N pV u(x,t)

where ii(x,t) is the information we desire. Then it follows that


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~ ~- ~ Uo(t) dr. (3.2.11)

But I~VT is exactly the fraction of time that the signal Uo(t) is nonzero.Thus, the correct mean is given by averaging only during those periodswhere there is a signal.

Most real processors measure the average velocity during the burst;this implies that the velocity must be approximately constant during itstraversal of the volume. Moreover, since there is only a single realizationduring each particle passage, then the realization must be weighted by thetime the particle would contribute to the integral; that is, the residence(or transit) time. This is easily seen by approximating Uo(t) as constantwhile there is a signal and writing the integral as a sum:

_ 1 Inr ZiUo(ti)Ati

/aVT~ Uo(t) dt Y,iAt i ’(3.2.12)

where u0(t~) represents the ith realization and ~t~ the residen~ time that particle. Note that the direction of the particle and its velocity areirrelevant ; only its x-component and residence time matter.~

The idea of transit-time weighting of the realizations and the precedingproof were first presented by George (1976). HOsel & Rodi (1977) laterpresented a less rigorous derivation based on mean transit times andsuccessfully applied it to measurements of mean velocity in a jet. Somecommercial burst processors now provide the residence-time informationnecessary for using Equation (3.2.12) and those derived below.

TURBULENCE IN~NSI~ES Now we examine the second-order statisti~of the effective velocity. We square and average Equation (3.2.2) andapply the results of Equation (3.2.5) to obtain

d3x d~x’

~ The residen~ weighting may be viewed as a correction that creates an effective samplerate independent of velocity; referring to Figure 3-2, the actual sample rate is seen to beR = ~A ~ ~ ~. The effective samNe rate of residence-time weighted samples may be written:

where V is the volume of the measuring volume, At is the mean residence time for particleswith velocity u andTis the corresponding average trajectory length through the measuringvolume. The resultant rate is just the average number of particles in the volume, independent


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We first consider the case t = t’ for which p(x,t lx’,t’) = 6(x-x’). Thiscase is of particular interest since it corresponds to the mean-squarevelocity. From Equation (3.2.13) we obtain:

U~+u-7~= # [ f w2(x) U2(x)dax + f w3(x)u’2(x)d3x1

+ [terms of order (# V)2]. (3.2.14)

The second term is negligible compared with the first and is not of interesthere since we are interested only in individual realization anemometrywhere the particle concentration is low. (Note that the reverse was truefor the continuous LDA where the first term was neglected with respectto the second.)

It is clear that we can readily identify the mean-square fluctuation wedesire in Equation (3.2.14) if U(x) = constant. Again assuming w(x)= within the volume and zero outside, and assuming u’2 to be constant overthe volume, we obtain

ub2 = #V u’2. (3.2.15)

Analysis of this in terms of time integrals leads to the same conclusion asbefore: The residence-time weighted average provides the correct meansquare fluctuating velocity (George 1976). Thus

-- Xi[u(ti)-- U]2Atlu’z = , (3.2.16)

EiAti

where U is computed from Equations (3.2.12).It is tedious but straightforward to interpret Equations (3.2.7) and

(3.2.13) for nonideal scattering volumes and nonuniform velocities. Theonly. difference is that now volume-averaged information is obtained.

Fifure 3-2 Measuring volume cross section.


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The effect on the statistics of this volume averaging is identical to thatin the continuous case mentioned earlier. Note that the presence of amean gradient will contribute an apparent turbulence intensity evenwhen no turbulence is present because of the contribution of the firstterm in Equation (3.2.14). A particular problem associated with directionaleffects of real finite volume processors is discussed in Section 3.3.

CORRELATIONS AND SPECTRA We now examine the possibility of measur-ing autocorrelations and spectra with burst processors. The correlationof the measured velocity is readily computed to be

Uo(t)Uo(t’)=;~u(x,t)u(x’,t’)w(x)w(x’)gl(x,t)gl(x’,t’)’d3xd3x’

(3.2.17)

Applying Equations (3.2.5) and (3.2.6) we

Uo(t)Uo(t+z)=#ffu(x,t)u(x’,t’)w(x)w(x’)~(x’-x-U~)d3xd3x’

I(z)

f f ’ (3.2.18)

II(z)

In the succeeding analysis we shall show that the first term contributesonly for very short time lags (z < d/U) whereas the second produces thevolume-averaged correlation (or spectrum) in the same manner as forthe continuous signal and is therefore the desired result.

We consider the first term only for now. Carrying out the integrationover the delta function we obtain

I(r) =/~ ; u(x,t) u(x+Uz, t + z) w(x) w(x (3.2.19)

If the turbulence is assumed frozen, the space-time correlation of thefluctuating component reduces to the turbulence intensity, i.e.

u’(x,t) u’(x+U~,t+~) = u’2(x),

and we are left with

I(z) = # ~{U(x) U[x+U(x)z] + u’2(x)} w(x) d3x.J

(3.2.20)


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Thus the contribution of the first term to the autocorrelation is simply acontribution determined by the volume and the mean transit time of theparticles.

For the case where the mean velocity is uniform and the turbulencehomogeneous this reduces to

I(z) laV[U2 -~ -U’2] pl(’~), (3.2.21)

where

Vp~(z) = w(x) w(x+Uz) d3 (3.2.22)

For the case where w(x)= 1 inside and zero outside the volume, this simply a spike at the origin of unit height whose width is proportional tothe mean transit time of the particles through the volume (,-~d/U). Notethat Equation (3.2.21) reduces to the result obtained in the previoussection for z = 0 as it should.

Since this first term I(z) contains no information of interest except z = 0, it can be considered noise that it is desirable to eliminate. Fortun-ately, this is easily done with the autocorrelation since time lags of theorder of a particle transit time are seldom of interest in turbulence. Thisis always the case if the streamwise extent of the scattering volume issmaller than the Kolomogorov microscale. A less restrictive conditionmight be d ~ 2u where 2u is the spatial Taylor microscale for the u-component of velocity and characterizes the small ~ behavior of theautocorrelation (since 2~ = ~.,,/U).

It is straightforward to show that this unwanted information contri-butes over the entire frequency range of the spectrum to o9 ~ zff ~ withheight St(co) ~/~V [U2+ >-~] z0 and rolls o ff a s co-2 thereafter where Zois the transit time. Note that the magnitude of this term can be consider-ably increased if significant velocity gradients are present across thevolume.

It will be seen below that in most burst-processing applications thiscompletely swamps the desired spectrum, which is proportional to ~ V)~.

(Recall that #V is expected number of particles in the volume, and isgenerally much less than unity.) Fortunately, since the correlation ofthis "transit time noise" is confined to the neighborhood of the origin, itcan be eliminated in most applications by removing it from the correla-tion before transforming to obtain the spectrum.

We consider now the second term and recognize it to be similar to theterm we retained in the analysis of continuous signal processors (Section2). Decomposing the velocity into mean and fluctuating parts yields


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II(~) = g~ U(×) g(~’) ~(~) w(~’) da~ ’

+#2ffu’(x,t) u’(x’,t’)w(x)w(x’)daxd3x’. (3.2.23)

The first term is precisely the square of the volume-averaged mean velocitygiven in Equation (3.2.7) and the second represents a volume-averagedspace-time correlation. If we assume a uniform mean velocity and afrozen turbulence we can write, using the volume averaged spectrum ofSection 2.2,

II(z) = U~ +(#V)~ By(Q,

where

By(z) = f eik. u~ F,~(k) d3k

(3.2.24)

= fu’(x,i) u’(x,t c) w(x) w(x’) d3x ’. (3.2.25)

Clearly Bv(z) is the desired information.Combining I(~) and II(r) we

[uo(t)- Uo] [Uo(t’)- Uo] = (/~ [U2+ u’ 2]p,(z)+ (/~ 2 By(z).(3.2.26)

Since /~V ~ 1, the second term is substantially smaller. However,since the first term is zero for all z > d/U, the second term can still beobtained for all but the smallest time lags. This is not the case with thespectrum, however, which will be completely dominated by the firstterm.

It is easy to improve the situation by first computing the autocorrela-tion from the randomly arriving samples, then keeping only the part forwhich z :> d/U. If the transit time is indeed less than any time lag ofinterest, the remainder of the autocorrelation can be transformed toyield the spectrum. Since the value of the autocorrelation at z = 0 hasbeen obtained by the earlier analysis of u’2 [Equations (3.2.15) and (3.2.16)],this value can be used to replace the missing information at the origin.Moreover, since we know that the autocorrelation is parabolic at theorigin, the gap can be substantial (relative to 2) before we lose anyinformation about the fluctuating velocity field, Thus the desiredinformation about the autocorrelation is given by the second term as


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u’(t) u’(t + z) = (# V)- -~ ub(t) u0(r’ +z) dr, (3.2.27)

where we have replaced ensemble averages by time averages.In practice the elimination of the spike at the origin can be accom-

plished by simply not considering products of realizations with themselves,but only those products of realizations with other particles. It is impor-tant to note that the autocorrelation (or any cross correlation) cannot becomputed simply by multiplying different realizations, grouping andaveraging, as this introduces a bias (see Sections 3.3, 3.5).

The analysis above shows that if we consider that the particle contri-butes to the signal while it is in the volume, all of the desired statisticalinformation can be obtained. As a consequence, the autocorrelation fora lag z can be computed only when there is a particle in the volume at tand one at t +z. This is illustrated in Figure 3-3. The interpretation ofthis analysis leads to the following approximation to Equation (3.2.27):

u’(t) u’(t + ~) -- ’’~ , (3.2.28)~,, Atij

where z = ti-t j, i < j, and Ati~ is the overlap time of the ith realizationand thejth realization displaced by time z. The denominator is essentially

Figure 3-3 Computation of the autocorrelation from burst processors using only theoverlap times for different particles. The middle trace is the upper trace displaced byamount z as shown.


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the total overlap time corresponding to realizations of u(t) u(t + since~u V)z T -~ ~i,~ Ati~. The overlap time is difficult to obtain in practice; anequivalent estimator for individual realizations is proposed in Section 3.5.

Similar considerations can be applied to single and joint probabilitydensities and to cross-correlations of different signals. Again the algorithmmust be determined by the time the one or more signals actually contri-bute to the appropriate time-integral expression. These are discussed indetail in Buchhave (1978) and George et al (1978).

3.3 Processing the Uncorrected Signal

Various methods of implementing residence-time weighting have beensuggested, but before discussing these and other correction techniqueswe briefly review the effects of processing the sampled data without apply-ing any corrections. The fact that the particles cross through the measuringvolume at a rate proportional to the instantaneous velocity (given uniformparticle distribution) gives rise to bias errors when statistical quantitiesare formed by direct ensemble averaging of the accumulated sample values.However, the bias effect associated with the fluctuating velocity is onlyone of a number of possible biasing effects occurring in burst-type LDAprocessing. The sampling rate may also be correlated with velocity direc-tion and with fluctuations in particle concentration resulting from densityfluctuations, mixing of different fluids, or chemical reactions.

If we assume an initially uniform concentration of scattering particlesand uniform density, we can determine that there are two main sources ofbias associated with the velocity of the fluid through the measuringvolume: correlation between sampling rate and velocity-vector magni-tude, and variation in the probability of a measurement with velocitydirection (variation of the cross section of the measuring volume). Froman examination of Figure 3-2, we can see that the rate at which particlescross the measuring volume is given by R = #Al ulwhere A is the cross-sectional area of the measuring volume normal to the streamlines. Weoperate with the usual assumptions in burst detection that the velocitydoes not change appreciably during the time it takes a particle to crossthe volume and that the probability of more than one particle in V atany time is negligible. Most signal processors, however, require a minimumnumber of zero crossings to execute a measurement. This fact has theeffect of reducing the effective cross-sectional area of the probe volumein a given direction to a value smaller than the geometrical cross sectionof the probe volume and results in "dead angles" in which no measure-ment is possible. Buchhave (1976) investigated the effect of the variationof cross section with flow direction on bias in LDA-counter measurementsand showed the effect to be significant and of the same order of magnitude


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as the velocity-magnitude effect. Expressing the cross section in termsof the velocity components u, v, and w, we may write

t)A(u) = abcp- 1--~ p2Q2 , (3.3.1)u

where

u 2 /9 2 w2

p2 = -~ + ~ + ~ (3.3.2)

and a, b, and c are given by Equation (i.I.1). Q =- Ne/Ns is the ratio of Ne,the minimum number of zero crossings or fringe crossings required for ameasurement, to N~-, the maximum number of fringe crossings available(corresponding to a particle trajectory along the x-axis). When Q ~ 0, approaches the geometrical cross section of the ellipsoidal probe volume,while increasing the value of Q reduces A, increases the directionaldependence and introduces dead angles. Figure 3-4 is a plot of A forparticle trajectories parallel to the x,y-plane (normalized with respect tothe geometrical cross section normal to the x-axis), versus ~b, the anglebetween u and the x-axis. These curves are valid for a monodisperseparticle distribution. Measured angular characteristics tend to show aless pronounced dead angle effect due to the presence of large particles.

The velocity bias effect on the formation of ensemble averages fromLDA burst signalprocessors was first studied by McLaughlin & Tiederman(1973) for the limited case of Q = 0 and a spherical measuring volume.Using Expressions (3.3.1) and (3.3.2) and assuming a data rate given R = t~ ]u]A, Buchhave (1976) later repeated the analysis including the

Figure 3-4(0 = 45°).

A

1.0

0.9

0.8

0.7

0.6

0,5

0.4

0.3

0.2

0.1

010 20 30 40 50 60 70 80

Measuring volume cross section as a function of angle in the x-y plane


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directional .dependence of the cross section for various velocity distribu-tions. Figure 3-5 shows the results for an assumed Gaussian velocitydistribution superimposed on a constant mean velocity. The McLaughlin-Tiederman analysis corresponds to the curves for Q -- 0. As is evidentfrom the figure, direct mean and rms values from the accumulated dataresult in gross errors for turbulence intensities above -,~ 10~, and theerrors increase as the angular dependence increases. The effect of bias isalso visible in velocity histograms formed from the measured data.Figure 3-6 (McLaughlin & Tiederman 1973) illustrates the resultingskewness in the measured velocity distribution. The correlation betweendata rate and velocity also results in bias in steady flow when gradients

Figure 3-5 Calculated bias error of mean and rms velocity measurements for three-dimensional Gaussian isotropic turbulence including effects of measuring volume crosssection.


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P(u)

oo

Figure 3-6 Velocity probability distribution for uncorrected two-dimensional velocity

fluctuations showing effects of bias and results of McLaughlin-Tiederman correction.

are present within the measuring volume. Kreid (1974) predicted theerrors introd~uced by steady gradients within the volume and found goodagreement between measured and calculated bias errors in laminar pipeflow. Figure 3-7 shows the expected gradient bias errors according tothese calculations.

Velocity error ~ --u

0.8 0.6 0,4 0.2Distance from center ~ = y/R

--9

7

6

4

3

2

0

Figure 3-7 Mean velocity bias in Poiseuille flow (after Kreid 1974).


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3.4 Bias Correction

In introducing this section it is important to recall that in Section 3.2 itwas shown that if time averages were formed by integrating only duringthe time measurement (i.e. only during particle residence times), no biaswould exist. For example, the mean velocity can be approximated by thesum of products of trajectory-averaged velocity values and residence timesin an unbiased manner by

~,iuiAtiU- EiAti’ (3.4.1)

where ui corresponds to the quantity measured by most burst-type LDAsignal processors. Thus complete velocity.bias correction is achieved (orbetter, bias is avoided altogether) if in addition to the velocity output measurement of the residence time is made available at the processoroutput to be used as a weighting factor in the computation of statisticalquantities.

McLaughlin & Tiederman (1973) suggested a bias correction for onecomponent measurements, which has since been used by a number ofinvestigators. By weighting each sample by a weighting factor inverselyproportional to the measured velocity component of the ith sample, ui,a correction of the computed statistical quantities results, which givescorrect values in the case of one-dimensional flow fluctuations or steadyflow, but overcompensates when the velocity is fluctuating in two orthree dimensions. Another limitation is that the one-dimensional correc-tion does not take into account any directional dependence of themeasurement cross section. Using the weight l ui I-1 the expressions forthe corrected mean and mean-square fluctuating values become:

N

U~i=lN

andN

U~2 ~ i= 1

NN (3.4.2)

/~ (3.4.3)lu, 1-1

i=l

Figure 3-8 shows expected errors in mean and rms velocity measure-ments assuming a three-dimensional Gaussian velocity distribution and


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the application of the one-dimensional weighting factor. Again the curvesfor Q = 0 correspond to the analysis by McLaughlin & Tiederman, whileother values of Q correspond to cases where the finite-fringe-numberrequirement introduces additional angular dependence. The one-dimen-sional correction reduces the bias errors, but beyond approximately15~o fluctuations relative to the mean the errors increase quite steeply.However, for a value of Q about 0.2 (corresponding to, for example 8 of40 zero crossings) the directional dependence of the measurement crosssection is seen to balance to some degree the overcompensation intro-duced by the one-dimensional bias correction.

Bias effects and the various correction methods are presently beingstudied by a number of investigators. It still has not been firmly estab-

Figure 3-8 Calculated bias error of mean and rms velocity measurement in three-dimensional Gaussian isotropic turbulence including effects of measuring volume crosssection.


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lished under which conditions the assumptions leading to the expressionsfor the data rate in Equations (3.4.2) and (3.4.3) are valid. Some investi-gators have reported a weaker correlation between data rate and velocitythan expected from these expressions (e.g. Smith & Meadows 1974).Recent measurements by Karpuk & Tiederman (1976) and Quigley Tiederman (1977) in the viscous sublayer in pipe flow show good agreementbetween mean and rms values computed from LDA single-particlemeasurements corrected by one-dimensional weighting and hot-wiredata. The measurements were made with an optical system speciallydesigned to give a probe volume with small spatial dimensions in thedirection of the mean velocity gradient. The authors note that the datarate did not seem to be correlated with direction, but did not indicatethe ratio of the total number of fringes needed for the signal processingto the number available.

Recently Erdman & Gellert (1976) studied the correlation betweenvelocity and particle arrival rate and showed very good correlation in anair flow modulated by an acoustic horn up to frequencies where particlelag influences the particle motion.

Although the one-dimensional correction is of interest because it isrelatively simple to implement and gives good results in steady or one-dimensional flow, i.t seems more promising in turbulence measurements,especially of high intensity, to adhere to the method suggested by theexpressions developed in Section 3.2, i.e. the residence-time weighting.This method should give correct results for two- or three-dimensionalfluctuations of arbitrary high levels as long as the directional effectassociated with a minimum number of zero crossings is negligible. Thiscondition can be achieved either by adding periods to the burst by opticalfrequency shifting or by basing the signal processing of the burst on theavailable number of fringe crossings. The directional dependence is thenreduced to the geometrical cross section of the volume, and the residence-time weighting should give correct results independent of the particularshape of the volume even when the volume is truncated by the field of viewof the photodetector or by measurements close to a wall.

A number of methods of implementing the residence-time correctionhave been realized. H~sel & Rodi (1977) suggest an analog circuit thatmeasures the width of a burst at half height by rectification and low-passfiltering. The method of measuring the total number of fringes in a burst(adigitalmethod, which unlike analog methods~is not frequency dependent)has been adopted by some commercial producers of LDA equipment.Both the measured number of fringe crossings and the time between thefirst and last zero crossing (or the velocity based on the inverse of thattime) are made available simultaneously at the output. Experimental


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evaluation of the residence-time correction is still rather scant, but theresults of HiSsel & Rodi (1977) show good agreement between measuredand computed values for mean velocities.

Finally, concerning velocity bias it may be mentioned (as already pointedout by McLaughlin & Tiederman) that in simultaneous measurement ofall three velocity components (or in two-dimensional measurements inflows in which fluctuations in the third direction are negligible) it is ofcourse possible to compute the magnitude of the velocity vector ui, andassign a weighting factor proportional to luil-1 to each sample setvi, wi), and even to correct for directional dependence based on theknowledge of the direction of u. Such correction might be carried outon stored data points after the measurement, but it appears that thismethod has not yet been tried in actual measurements.

The question of whether other factors influence the data rate is stilllargely unresolved. Tiederman (1977) raised the question of whetherdifferences in signal-to-n0ise ratio of fast and slow bursts might influenc~the data rate. Other physical effects that cause correlation between datarate and flow velocity include density variations caused by pressure ortemperature fluctuations, mixing of fluids with different particle concen-tration, and chemical reactions. Asalor & Whitelaw (1976) derivedexpressions for the correlation between combustion-induced temperature,pressure, and concentration fluctuations and data rate based on assump-tions about the velocity-temperature and velocity-pressure correlationsin a diffusion flame. From the analysis and subsequent measurementsthe authors concluded that in this particular flow the bias effects due tovelocity fluctuations confirmed the hypothesis about the velocity~lata-rate correlation of McLaughlin & Tiederman. Velocity-pressure andvelocity-temperature correlation effects were found to be negligible.George (1976) discusses briefly the extension of the arguments of Section3.2 to flows with density fluctuations.

3.5 Correlation and Spectral Measurements

The measurement of covariance functions and spectra plays an importantrole in turbulence research. These functions reveal the inner structure ofthe turbulent motion and shed light on the dynamic processes of momen-tum and energy transfer between the spectral components of the fluctuatingvelocity. The experimental and computational methods for the determina-tion of covariance functions and spectra have undergone considerablechanges during the last decades in step with the technological developmentof measuring apparatus and computing facilities. These technical develop-ments include digital sampling and data storage techniques allowinglong-time averaging without drift problems, decreasing cost and increasing


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speed of digital computers, and the development and implementation ofmore sophisticated computational methods. The latest step in this on-going process is the development of integrated, hybrid analog/digitalcircuits allowing fast, inexpensive implementation of complex computa-tional methods arid transforms. Previous analog delaying and multiplyingoperations are replaced by digital manipulation of discrete-time samplesof the signal. Tradition and computational efficiency have in nearly alleases dictated periodic sampling, but mathematical examinations carriedout in the 1950s and 1960s on the effects of random sampling and therealization that certain forms of random sampling have advantages inspectral computation (e.g. alias-free spectra), plus the fact that the bursttype LDA signal processor provides data at random times correspondingto the random arrival of scattering particles in the measuring volume,have brought renewed interest and new developments in randomsampling spectral estimation.

Methods of spectrum computation of regularly sampled signals fromcontinuous LDA-signal processors are essentially identical to those usedin hot-wire data analysis, and the results are derived identically exceptfor the effects of dropout and ambiguity noise described in Section 2.Three basic methods of discrete estimation of the power spectrum exist:

1. the Blackman-Tukey method, based on the Fourier transform of theautocovariance function

2. the direct formation of the power spectrum from Fourier transformcoefficients of the sampled data

3. the digital filter or complex demodulation method

Many variations exist, and the optimum choice of time lag, sample size,and possible smoothing window depends on the type of spectrum expectedand the desired Characteristics of the resulting estimate.

Consider a record of length T consisting of N samples [u(ti)] (i = 2 ..... N), where ti is the time of measurement of the ith sample, and letus designate by C(r) and S(w) the time autocovariance function spectrum of the underlying continuous signal. The basic characteristicsdescribing the performance of a spectrum estimate are then the deviationof the expected value of the spectrum estimate ~(c,) from the true value,i.e. the bias given by

B = S(~o)-- E[~’(~o)], (3.5.1)

where E denotes mathematical expectation, and the variability given asan expression for the variance of a given estimate, e.g.

var [~(09)] = E[(~(og)-E[~(~)])2]. (3.5.2)


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These parameters are normally functions of the desired spectral resolutionor bandwidth B and the number of samples N.

The effect on spectral aliasing of various forms of random sampling wasstudied by Shapiro & Silverman (1960), who concluded that althoughnot all types of randomness in the sampling procedure lead to alias-freespectra, many sampling distributions exist, including the Poisson distri-bution, which will remove aliasing in computed spectra irrespective ofthe average data rate, albeit at the cost of increased variability. A numberof spectral estimates, equivalent to the Blackman-Tukey methods butbased on randomly sampled data, were presented at the 1974 PurdueWorkshop on laser anemometry (Thompson & Stevenson 1974). In thesemethods an estimated autocovariance function was formed from themeasured values of velocity u(tl) and the time increment since the lastvalidated sample, ti - t~_ 1. The time axis was divided into M slots of equalwidth Az, where Az is selected to give an effective spectral resolutionAo~ = 1/(2nAz) in the computed spectrum. An estimate of the autoco-variance function at nAz was formed by adding cross products betweensamples u(tl) and u(ti) separated by lag times ti-t~ falling in the rangenAz___(1/2)Az and counting the number of such products:

.E. ultC(nAz) = ,,s ; (n- 1/2)A~ < t~- t~ < (n + 1/2)A~. (3.5.3)

The denominator is simply the number of products in the window. Thespectrum was then obtained from the autocovariance estimate in theusual manner by Fourier transform after application of a suitable datawindow, D(nAz):

M

~1((D) = E C(/’//~) D(~/A’/7) ic°~nAz. (3.5.4)

The properties of this spectral estimator were analyzed by Gaster &Roberts (1975). First a more direct spectral estimator using all occurringlag times t~-ts was investigated:

1 N .

E IE< ,(t t cos

This estimator treats each cross product u(tg) u(ts) as a realization of theautocovariance function for a lag z = t~- ts. Note that zero-lag productsdo not occur. The spectral estimator $~(o2) was shown to be consistent


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and unbiased and to have a variance given by:

3~m [- a2 -]2var [-~2(co)] = 4T- ~S(co) + d , (3.5.6)

where a Poisson sampling process of mean rate v and a Gaussian randomprocess of mean-square velocity tr 2 were assumed. This expression showsthe increased variance introduced by the Poisson sampling; the penaltyfor the random sampling becomes appreciable for mean sampling rateslower than the corresponding Nyquist rate for regularly sampled data.

The spectral estimator (Equation 3.5.5) is impractical because of thedouble summation, but Gaster & Roberts did implement the time-slotapproximation (Equation 3.5.3) and showed, using simulated signals,that the variance of the former (Equation 3.5.6) is still a reasonableestimate for the variability of the latter. For constant slot width thespectrum is conveniently computed via the fast Fourier transform, butcomputation by direct implementation of Equation (3.5.3) for logarithmi-cally spaced slots is not unreasonable, since a high-frequency estimate atogr does not require summing over the full array of covariance values,but only for lags shorter than 1/co,.

The special features of these spectral estimators may be summarized :

1. no aliasing in the computed spectrum (a fact related to the LDA-method of sampling, not to the particular spectral estimator)

2. reasonably simple implementation, even for nonuniform lags3. insensitivity to added white noise due to the fact that only nonzero

lag products are used in the estimator4. an increased variance of the spectral estimate relative to regularly

sampled signals

In a more recent paper, Gaster & Roberts (1977) examine a directtransform of the sampled data:

~3(w) ----- u( ti) ei ~°t~ -- ~i u2( ti) D2( t:) " (3 .5.7)

This estimator is identical in form to the estimator used for regularlyspaced samples. However, in this case the time increments are randomand the terms in the sum are evaluated at the random sampling timesHowever, all samples are weighted equally and the factor t~-tl-originally multiplying each term is replaced by the average intervalbetween samples 1Iv and pulled outside the sum. It is shown that thevariability of this estimator, evaluated for Poisson sampling and assuminga Gaussian random process u(t), in general depends on the shape of the


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spectrum S(~o), the window D(t), and the length of the record length Tbut for large record length, approaches the value

var [~3(co)] = S(~o) + ~-~J . (3.5.8)

The first term represents the variability of the conventional periodogramwith periodic sampling, whereas the second represents the added vari-ability due to the random sampling. Data-smoothing windows or blockaveraging may be applied to form stable spectrum estimates. Gaster &Roberts suggest that in many cases it will be advantageous to use aspectral bandwidth that increases in proportion to the center frequencyof the estimate in order to obtain stable estimates at high frequencies.

So far only the slotted-time-lag autocovariance method has beenapplied to actual burst-LDA realizations (no residence-time weighting).Measurements were reported at the 1974 Purdue meeting by Smith &Meadows and by Scott. These measurements were in free jets in air.Other tests included measurements of acoustical vibrations with a loud-speaker. The basic feasibility of the burst-type LDA power-spectrummeasurement was proven. Later results have been reported by Mayo etal (1974) and by Bouis et al (1977). None of these measurements any attempt to correct for the bias effects discussed in Section 3.2.3 andbelow.

It seems that more experience with LDA spectral measurements isneeded before all aspects of error sources are clarified. A study (Wang1976, Asher, Scott & Wang 1974) of various sources of errors in relationto counter LDA data concludes that the quantizing of the output due tothe finite resolution of the counter itself is the greatest source of error inLDA single-burst spectral measurements. However, this analysis did notconsider the "apparent turbulence" caused by the finite dimensions ofthe measuring volume in the presence of gradients within the volume asdescribed in Section 3.2, nor did it consider the biasing effects introducedby using uncorrected data. It should be apparent that bias effects willsomehow modify the computed spectral estimator as well as mean andmean-square values, and that bias correction methods should be .appliedto spectral measurements as well.

In Section 3.2, it was shown that autocorrelation from burst processorscould be computed correctly without bias as

~ ui uj AtijR(z) ~ ,,s .; z = It,-- t~ [, (3.5.9)

~ ~tij


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where Ah~ is the amount of time a realization u~ with :residence time Ahat t, overlaps another realization u~ of residence time: Atj at tj = ti÷ z.Recall that self products (i =j) are not included in the sum since theyresult in the unwanted spike at the origin. This value (z~- 0) is computedseparately by Equation (3.2.16).

In practice, Equation (3.5.9) is not a useful algorithm since not onlymust the particle velocities and transit times be stored but also theirarrival times from which the overlap is computed. An estimator thatretains the essence of Equation (3.5.9) can be formed (Buchhave 1978)by simply multiplying each velocity realization taken at different times(or from different records) by its residence time and dividing by the sumof the residence times:

.~.ui uj Ah At~R’(T) ’’ J ; z = It,-tjl. (3.5.10)

~ At~ At~

This estimator is not biased by velocity-data-rate correlation if angleeffects have been minimized by frequency shift or similar techniques(cf the discussion about bias correction in Section 3.2). It is also easy see that this nonbiased algorithm can be fitted directly into the time-slotapproximation discussed earlier.

To this point it appears that the significance of the bias correction inthe formation of autocorrelation functions and spectra has not beenstudied. All attempts to date to reconstruct the autocorrelation fromburst LDAs have been biased since no attempt has been made to correctfor the residence time (or particle arrival rate). Some preliminary resultswere published at the 1978 Purdue meeting (Buchhave 1978a) and thequestion will be treated in more detail in a succeeding publication(Buchhave 1978b).

4 REFRACTIVE INDEX EFFECTS ON ILDAMEASUREMENT

4.1 Source and Magnitude of Refractive Index FluctuationsIf the laser beams of the LDA must propagate through a medium havingrandom index of refraction fluctuations, the ray paths will no longer bestraight, but will have random fluctuating cur’~ature, and the phase andamplitude of the arriving signal will fluctuate randomly. This is thephenomenon that causes jitter and scintillation of stellar images, andscintillation of extra-terrestrial radio sources, where the index of refrac-tion fluctuations are due primarily to temperature fluctuations in the


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atmosphere. There is an extensive literature on this subject [see, forexample, Monin & Yaglom (1975), Chapter 9].

The effect of index-of-refraction fluctuations on LDA measurementshas been considered by Lumley, George & Buchhave (1978), whom follow closely. These fluctuations arise primarily from fluctuations oftemperature or composition when the LDA is used in flows in whichheat transfer, chemical reactions, or combustion are taking place. Wefind in Section 4.3 that fluctuations of the order of 2 x 10- 6 are neces-sary for an appreciable effect. Let us consider what level of fluctuationsmight be met under various circumstances. In gasses, the index is usuallygiven for sodium light (2 = 0.5893/~m), and at 273 K and one standardatmosphere (101 kNm-2). The index under other conditions is approxi-mately proportional to density and is only a weak function of wavelengthin the visible range. Some representative values of (n-1)10’* understandard conditions are:

CH4:4.44 02 : 2.71

CO2 : 4.48 H20 : 2.49

Air: 2.93

(Weast 197%78, p. E-224). Taking air as an example, we can. write

P(n- 1)104 = 7.92 x -3 T---’ (4.1.1)

where P is absolute pressure in Nm-2 and T is absolute temperature inK. If we consider isentropic variations in the pressure in a compressibleflow and make use of a quasi-Gaugsian estimate relating pressure andvelocity fluctuations (Batchelor 1956, p. 182), we find that rms velocityfluctuations of the order of 70 msec-1 are necessary to produce index-of-refraction fluctuations of the necessary level. Since this corresponds toa 4~ turbulence level in a sea-level mean flow at Mach 5, we can usuallyneglect this source of refractive index fluctuations and take P = constant.For temperature fluctuations we can write

To,n’ = -(n- 1)o ~-~ (4.1.2)

where To and (n- 1)o are the values at 273 K, and 0’ is the temperaturefluctuation. Thus, for air at 273 K, dn/dO’ -- - 1.07 × 10- 6 K- 1, while at1300 K dn/dO’ --- -4.73 x 10-s K-1.

As far as composition fluctuations are concerned, to a first approxi-mation we may take the deviation proportional to the relative mass


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fraction X. Thus, for a two-component mixture, we obtain

70(4.1.3)dn/dX -- nl - n2 = (nl - n2)o ~-.

Considering, for example, a methane-oxygen mixture, we have dn/dX =1.73 x 10.4 at 273 K, and 3.63 x 10.5 at 1300 K. Since we may havemass fraction fluctuations of order unity, these are also representative ofthe values of n’.

In liquids, the index of refraction is considerably more sensitive totemperature. In water, for example, at 273 K, we have dn/dO’ = 9 x 10-5K- 1 for the absolute index of refraction for sodium light (Weast 197%78,p. E-223),

In addition, since the refractive index of various liquids differs usuallyin the first decimal, mixing of two liquids of differing indices can resultin values of dn/dX of order 10-1; glycerine-water results in 0.14, forexample. If we consider the case of sea water (Weast 1977-78, p. D-249)we find the index of refraction to be primarily a function of salinity(measured in parts per mille), with dn/dS = 1.8 × 10-4. If the LDA wereused in an experiment involving the mixing of sea water at 30%0 salinityand fresh water, we could have values of n’ of the order of 5.4 × 10- 3.

It is evident that there are a number of situations in which the LDAcould be used and in which substantial index of refraction fluctuationsmight be present; we must examine the sensitivity of the measurementsto these fluctuations.

4.2 Effect of Fluctuations on Ray Path and Phase in the

Geometrical-Optics Limit

We may assume that the wavelength of the radiation is short relative tothe size of the inhomogeneities and that the radiation is monochromatic.We then go to the geometrical-optics limit [see Tatarski (1961), Section6.1 ; Neubert (1970)]. If k is the wavenumber of the radiation, and n theindex of refraction, and if we write E = A exp[iS], where E is any com-ponent of the electric field, A is the amplitude, and S is the phase, weobtain the eikonal equation for the phase

S,iS,i -~ k2n2(x). (4.2.1)

From the canonical Hamiltonian solution to the eikonal equation(KravtsoV 1968), we obtain

S = So + k ~ n(s’) ds’ (4:2.2)


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where the integral is taken along the ray path Xi(s) given by the Fermatrelation

a ( axAn.i = ~ss tn --d~-s )" (4.2.3)

It is probably easiest to consider the fringe mode of operation, inwhich two beams (in the undisturbed state)

kt°: (k sin 0/2, k cos 0/2, O) = ki,

k~r): (-k sin 0/2, k cos 0/2, 0) = kr, (4.2.4)

form fringes which, if undisturbed, would lie stationary in planes perpen-dicular to the xl axis. Lines of constant phase difference of the two signalswill be lines of constant fringe phase. If these lines of constant fringephase move with a certain velocity, this will produce an error in theapparent particle velocity.

We may calculate

;o-- =dSk h(s’) ds’ + kn(s) (4.2.5)dt dr’

where ds/dt = (c~s/c3Xi)dXi/dt. To hold the phase difference constant, wemust allow xl to vary, corresponding to the fringe velocity. Hence,dXx/dt will consist of two parts, OX1/c3t+(c3X~/C3~l)d~/dt where ~ isthe coordinate of the ray on the initial wave front. We are not interestedin the motion of ¢ however, but in the motion that it produces at thescattering volume, so that we can write dX1/dt = c3X~/c~t+ ux, where u~is the velocity of the fringe pattern. Hence

(4.2.6)

where c3X~lc3t is half the sum of the ray terminal velocities and c3Xy/c3tis half the difference. We have selected rays that intersect instantaneously,so that the values of n at the end of each ray are the same. We may nowconsider n = h+n’, n’,~ ft. The first term in Equation (4.2.6) is of ordern’, and it is clear that c3Xj/c3t + ui is at most of order n’. Hence, to lowestorder we can neglect the departure of n and c3s~i)/c3Xj-c~s~’)/c3X~ fromtheir undisturbed values; we may also take the integrals in Equation(4.2.6) along the undisturbed rays. In addition, we take the termination


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of the rays on a plane perpendicular to the x2 axis, passing through thepoint of intersection of the undisturbed rays. Hence, X2 and x2 are equaland a fixed value, and the second term in square brackets in Equation(4.2.6~ vanishes. Since for the undisturbed rays we have ~s/OXj = kj/k,

- h(x. i) ds- h(x" r) ds , (4.2.7)fi 2 sin ~

where we have taken the undisturbed rays to be of equal length L.To evaluate ~gX~/~t, we must integrate Equation (4.2.3) twice, making

use of the assumption that n’ is small. To first order we obtain

Ot fi1- ~,~ ds, (4.2.8)

where we are neglecting the change in slope from the undisturbed valueas the ray pierces the plane, a term of the order of L(n’/~)sin (0/2),which is considerably smaller. Thus,

l (f0~ f0" )

(4.2.9)

S ---in~h(x" i) ds- h(x" r) ds

2 2

Physically, it can ~e seen what is happening: presuming that the phasesstay the same, if the beams swing in the same direction the fringes willmove with them. If the beams swing in opposite directions, however, thefringes will remain stationary. In addition, if the relative phases change,.the fringes will move. We may note that one of the standard techniquesfor examining flows without mean velocity is to shift the frequency of onebeam to produce a steady drift of the phase different, and hen~ of thefringes. To produ~ this effect, the parenthesis of the second term inEquation (4.2.9) is repla~d by A~/k.

The ratio of the first and second terms can be estimated by takingn,~ = n’/2,, h = n,~U = n’U/2,, where 2, is the Taylor microscale for theindex-ogrefraction fluctuations. The ratio is L sin (0/2)/2,. Since sin(0/2) ~ 0.1 in most situations, and L will usually be at least 3/, the integralscale, we have 0.3(Rt/15)~/~ for the ratio, where Rt = ul/v. If Rt = 167,the terms are comparable; if R~ = 1500, the terms differ by half an orderof magnitude, and their spectra by an order of ma~aitude. Laboratory

we can write


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values of R~ typically lie between these two values, so that although thefirst term is probably larger in most circumstances, it probably does notalways dominate. It is of course true also that path length and scatteringangle will differ in different experimental situations.

What is not apparent from Equation (4.2.9) however, is that the spectraof the two terms differ markedly. Lumley, George & Buchhave (1978)find that while the spectrum of the first term rises sharply as x~/3, thatof the second falls as x-2/3, so that the contaminating effect at the highend of the spectrum is felt from the first term long before that from thesecond term. This is because the first term depends on derivatives of theindex-of-refraction field one order higher than are present in the second.We limit our consideration to the first term, and refer the reader toLumley, George & Buchhave (1978)for a discussion of the second.

4.3 Beam-Swinfling Effects

It is relatively straightforward to show that the cross spectrum betweenthe two beam-swinging terms in Equation (4.2.9) is negligible for large since the region of correlation is of fixed extent as the beams diverge. Thus,the spectrum can be written as

+~o

l-L2u22rtLIc~4 h2 f I (~(r’i)+~(r’r))S,(r)d~c2d~¢3,(4.3.1)

where ~aU = ~ and we have used Taylor’s hypothesis; that is, we haveassumed that the principal source of variation contributing to the timederivatives appearing in Equation (4.2.9) is the convection of the frozenindex-of-refraction fluctuations across the ray path, and we have takenthe mean velocity in the 1-direction, Taking the spectrum of index-of-refraction fluctuations to be isotropic, so that S, = E,(x)/4~xz, whereE.(x) is the three-dimensional spectrum, and

En = 3~2/3 K-5/3 (4.3.2)qa ’

which is a reasonable approximation for any scalar (Monin & Yaglom1975, Section 23.5) we obtain for the spectrum

]5 fi2 12 " COS (Xll) ~/3. (4.3.3)

For this spectrum to rise to the same value as the ordinary one-dimensionalvelocity spectrum at the highest end, x~q = 0.55 (Tennekes & Lumley


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1972, p. 273) requires

h-~- = 3.52 Rt- 9[4 COS ~ (4.3.4)\Uj

If we take typical values of u/U ~ 10- 2, L/l ~ 3, Rt ~ 800, cos (0/2) ~ we have n’/fi = 1.96 × 10-6 required.

This value of n’/h corresponds to temperature fluctuations of 1.83 Kin air at 273 K, or to fluctuations of 41 K in air at 1300 K. We mayconsequently expect this to be a significant effect in combustion. Methane-oxygen mass-fraction fluctuations of 5.39 × 10-2 at 1300 K are alsosufficient to produce these levels of n’/h. Note that the cut-off point,where Equation (4.3.3) crosses the velocity spectrum, goes as (n’/~l)2/3;

hence, nearly a full decade will be lost when the mass-fraction fluctuationsof CH4-O2 are O(1) at 1300 K. In addition, it must be noted that at thecross-over point the combined spectrum is twice the value of either, sothat the spectrum is contaminated by 10~o or more down to a wavenumberabout half the cross-over value. In water, of course, the situation is muchmore serious: 2.17 × 10-2 K will satisfy Equation (4.3.4), and 1 K willcause loss of a decade.

Note that the spectral level goes up quite rapidly as L increases; everyeffort should be made to keep L as small as possible, though it willprobably not be possible to reduce it much below a value of three.

Using the spectrum Equation (4.3.3), we can integrate to find the rmsvalue of the spurious velocity. This is useful because in many experimentalsituations a continuous signal is not obtained, so that the spectral informa-tion is not available. Hence, only the direct influence on the variance canbe determined. Integrating Equation (4.3.3) we obtain

\ ~t/ = 8.90 x 10- ~ 0-2/~?/~ cosIf we adopt as a criterion that the rms value of the spurious velocityshould be 10~o that of the real velocity, then this requires that

n,2 7/,, (cos ~) 8/3 (4.3.6)

Again using the values cited below Equation (4.3.4), we find a level n’/h = 1.86 × 10-6 required, or virtually the same as that arising fromthe criterion Equation (4.3.4). Hence, in this parameter range, a signal/noise ratio of unity at the high end of the spectrum, or 10~ contamina-tion of the rms value of the signal, are equivalent.


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We should also estimate the rms motion of the ray terminus, since thisrepresents an uncertainty in the particle location ; in addition, if the motionof the ray terminus becomes too large, it is possible for the beams to failto overlap, and thus produce no fringes. In this case, the signal is lost.It is simplest to write Equation (4.2.3) in coordinates along and perpen-dicular to the undisturbed ray path. Then, making the same assumptions,we obtain

-- rc(~)l/2n’2L3(~Ri1/*-l) (4.3.7,

~Taking the same values quoted below Equation (4.3.4) we obtain(X])1/z ,~ 0.5 #m, or 5 #m at ten times the level of n’/h. These small valuesare quite consistent with the fact that the velocities of the fringes occurprimarily at the highest frequencies. If sin (0/2) ~ 0.1, a typical value, thefringe spacing is rough!y 2.5 pm, if light of wavelength 0.5 pm is used.Thus, we are envisioning rms motion of between 20~ and 200~ of thefringe spacing. The beam diameter at the scattering volume is usually oforder 10 fringes, or 25_/~m. It is clear that only under extraordinarycircumstances would (X2~)1/2 be large enough for the beams to fail tointersect. However, if we consider the case of mixing of sea water at30°/°° salinity with fresh water, we obtain (X~)~/~ ~ 1.24 mm, which ismore than sufficient.

4.4 The Possibility of Correction

Correction of the measurements for these effects is possible in severalways. Obviously, if the spectrum can be measured sufficiently beyond the~oss-over point to reach a region of approximate ~/s dependence, thiscan be subtracted. In the absen~ of this, however, there are other pos-sibilities. If one beam can be passed through the same part of the flowand heterodyned with an unscattered beam, and the result processed bya detector, then from Equation (4.2.5) we get a direct measure

kU n,~ ds (4.3.8)0

(assuming Taylor’s hypothesis). This will have a spectrum given

( 0V/ 4~’2~n LUZ~eos~) (~O-~/~ (4.3.9)

and a variance given by

6"ll k2n’~L u2 (0’~/~/2~ z/ ,, )R ,4_ . (4.3A0


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Depending on the experimental situation, the value of L may not be thesame~. However, :if the various parameters can be estimated, an estimatefor n’2 can be obtained from either Equation (4.3.9) or Equation (4.3.12)and this used to correct the measurements.

5 PARTICLE PROBLEMS

5.1 Criteria for Particle Selection

We cannot consider here the various types of particles commerciallyavailable for flow seeding. We do consider, however, the criteria thatshould be used for particle selection.

The basic requirement is that the particles follow the fluid motiondown to the smallest scale of interest. We will presume here that thesmallest scale of interest is l lr/, where t/is the Kolmogorov microscale;this corresponds to ~r/= 0.55, where ~c is wavenumber. This is essentiallythe viscous cut-off (see Tennekes & Lumley 1972, p. 270). By followingthe fluid motion, we mean that the value of the spectrum of particle/fluidrelative velocity should be small compared withthe value of the spectrumof fluid velocity at the wavenumber in question. Following Lumley (1976),and taking the fluid velocity seen by the particle to be essentially theLagrangian velocity, we have

a~to2

l+a~o92 ~ I(5.1.1)

for to = 0.74 o/rh where al is the particle time constant, and 0 = (ve)the Kolmogorov velocity. If we take as a specific criterion that the velocityshould be in error by no more than 1~, we may write

alco<10-2 or ax <-- (5.1.2)= = 74"

Using e ~- u3/1 (Tennekes & Lumley 1972, p. 68), we may write Equation(5.1.2)

u ol/2 1al 7 *-z=< --74’ (5.1.3)

In a turbulent pipe flow of water at 1 msec- 1, diameter 6 cm, this givesroughlya~ -_< 2 × 10-4sec;,for the same flow in air wehave al =< 9 × 10-*sec. We may use the Stoke~ form for al (Lumley 1957):

d2 { Pva, = ~2 ~+ 1), (5.1.4)


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where pp is the particle density and p~ that of the fluid, and d is theparticle diameter. For use in air it is difficult to find particles of densitymuch less than 1 gc-a. For such particles, the value ofsec corresponds to d ,-- 17/~m. For the case of water, if we presume thatpp is still 1 gc- 3, we have d ~ 49 #m.

Since scattering particles typically in use have diameters of the orderof a few microns, we may be sure that in these typical situations there isno need for concern that the particles are not following the flow. It isnot hard to think, of situations with higher Reynolds numbers and shortercharacteristic times, however, where this might be a problem (e.g. rocketexhausts).

5.2 Influence of Nonuniformities of Particle Concentration

Two common seeding methods are likely to produce nonuniformities ofparticle concentration that can influence velocity statistics. In one, apoint source of particles is introduced upstream of the measurementlocation; in the other, only the turbulent part is seeded, in a turbulentflow having a turbulent/nonturbulent interface.

Let us consider first the case of a point source. For simplicity, we willtake homogeneous turbulence with a uniform mean velocity U. To sim-plify matters further, we can suppose that the particles are released alonga line perpendicular to the flow, parallel to the x3 axis, a distance L up-stream. In this way we need consider only wandering of the particles inthe x2 direction. The generalization from a line source to a point sourceis straightforward. Now, if the particles are released along the x3 axisbetween -dx2/2 < x2 < dx:/2 at xl = 0, and measurements are carriedout along the line xl = L, xa = 0, we will have a signal if X2(L,x2,0,t I t--L/U)e [-dx2/2, +dx2/2], where X2(x~tlt’ ) is the 2-position at t’ of thematerial point which was (or will be) at x at t (we are ignoring dispersionin the direction of the mean flow). If, for simplicity, we suppose that theturbulence is Gaussian, then X2 and u2(x,t) will be jointly Gaussian, withstandard deviations respectively 02 and ~r~. We have

It-L/U

X2(L, x2,0,t[t- L/U) = 2 q- I.l 2 [X(x,t[t’),t’] dt’. (5.2.1)dt

The mean value is ~2 -- x2, and the variance is

(X2 - x2)2 -- 2 uz2.Y-LL/U = 022, (5.2.2)

say, where ~-L is the Lagrangian integral time scale, and we have presumedthat L/U is at least several" times ~’L, a restriction probably not met inpractice. For the cross correlation, we may write


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500 BUCHI--IAVE, GEORGE & LUMLEY

-- It-L /U --/./2(X, t) [X2(x, tit" L/U)-- xz] = u22 p1.(t’-- t) dr’ ~- - u2~-L

dt

(5.2.3)

where Pt, is the Lagrangian autocorrelation coefficient. Hence, the auto-correlation coefficient is given by

_ ~U. (5.2.4)P= ~/ 2L

The probability density for -dx2/2 < X2 ~ +dx2/2, and the jointdensities for u2 to have an arbitrary value while - dx2/2 < X2 ~ + dx2/2,may be written down immediately, from which we obtain the conditionalprobability for u2, given that X2(L, x2,0,t ] t- L/U)e [-dx2/2, + dx2/2]

1~ tr l ~ e-t ~/1~1 - p~)~tu~/,l + o~/~.

(5.2.5)

If the option is taken, that the pro~ssor holds the last value duringdropouts (see Section 2.7), then the probability density for the totalsignal is the same as that for the signal during the on-times. Thus,Equation (5.2.5) will be the density for the total signal. We see that, the measurement point is not directly downstream of the release point,a spurious mean velocity will be introduced, given by

U2 = --p ~ X2 = +x2U/2L, (5.2.6)if2

while the variance is reduced:

u~(1 -~U/2L) (5.2.7)

even directly downstream of the release point. One can conceivably getsufficiently far downstream so that ~-~U/2L ~ 1, so that the effect isnegligible, but this is often beyond the range of the experiment. Roughlyspeaking, ~z = l/3u (Tennekes & Lumley 1972, p. 278), so that

’~-~ U 1U l-- -~ -- - (5.2.8)2L - 6 u L"

Since U/u is probably between 10 and !02, for an error of no more than10~ in the velocity variance, we require L/l >= 17 or 170 respectively ; thatis, of the order of 10~-102 boundary-layer thicknesses, jet diameters, etc.

The second method that can cause problems is uniform seeding of theturbulem part of a flow. For example, if a jet is uniformly seeded at the


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nozzle exit, then downstream the seeded material will correspond exactly with the turbulent fluid, and there will be no seeding in the nonturbulent part exterior to the turbulent/nonturbulent interface. If the processor holds the last value of the signal each time the signal drops out, the probability density of the velocity will be exactly that measured in the turbulent part of the fluid only, which is observed to differ considerably from that measured in both turbulent and nonturbulent parts (Hedley & Keffer 1974). Since in most flows, intermittancy extends deep into the flow, reaching the centerline in a wake (Townsend 1956), this is not a useful way to measure overall properties. It is, however, an ideal way to measure properties in the turbulent fluid only, a matter of considerable current interest. Such an application would eliminate the necessity of establishing threshold criteria for the velocity field, always more or less unsatisfactory. The results obtained should be essentially comparable with those obtainable with the hot-wire by conditioned sampling on a temperature signal, if the turbulent fluid is heated.

ACKNOWLEDGMENTS

William K. George and Preben Buchhave acknowledge the support of the US National Science Foundation, Meteorology Program under Grant ATM 76-02157 and the Fluid Mechanics Program under Grant ENG 76-17466. Preben Buchhave acknowledges the support of the Danish Natural Science Research Council and the Danish Council for Scientific and Industrial Research. John L. Lumley is pleased to acknowledge several helpful discussions with S . Liebovich and the support of the US National Science Foundation, Meteorology Program under Grant ATM 77-22903, and the US Office of Naval Research, Fluid Dynamics Branch. The authors are grateful to Eileen Graber for typing the manuscript and to Bonita Webb for assisting with the literature search.

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the measurement of turbolence with the lda

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