Dec 20, 1985 - Abstract. The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity ...
ExperimentsinFhids
Experiments in Fluids 5, 17-28 (1987)
© Springer-Verlag 1987
Power spectra of fluid velocities measured by laser Doppler velocimetry R. J. Adrian and C. S. Yao University of Illinois, Department of Theoretical and Applied Mechanics, Urbana, IL 61801, USA
Abstract. The power spectrum and the correlation of the laser Doppler velocimeter velocity signal obtained by sampling and holding the velocity at each new Doppler burst are studied, Theory valid for low fluctuation intensity flows shows that the measured spectrum is filtered at the mean sample rate and that it contains a filtered white noise spectrum caused by the steps in the sample and hold signal. In the limit of high data density, the step noise vanishes and the sample and hold signal is statistically unbiased for any turbulence intensity.
v ~1, ~2 ay a. r T8 ~o ~oC
kinematic viscosity arrival times in pdf, s root mean square of noise signal, m/s root mean square of u', m/s delay time = t2 - t~, s duration of a Doppler burst, s circular frequency, radians/s low pass frequency of signal spectrum radians/s
Other symbols List of symbols A A' B C
cross-section of the LDV measurement volume, m 2 empirical constant bandwidth of velocity spectrum, Hz concentration of particles that produce valid signals, number/m 3 diameter of LDV measurement volume, m am f ( ~ l , {21 u) probability density of ti and tj given u_(t) for all t, Hz 2 probability density for t j - ti, Hz f. n (t, c) number of valid bursts in (t, t') = N + n' U (t, t') mean number of valid bursts in (t, t') mean number of particles in LDV measurement Ne volume valid signal arrival rate, Hz mean valid signal arrival rate, Hz Ruu time delayed autocorrelation of velocity, m2/s2 S u • power spectrum of velocity, m2/s2/Hz times at which velocity is correlated, s tl , t2 arrival times of the bursts that immediately precede ti, tj t~ and t2, respectively, s tij tj-- ti, S averaging time for spectral estimator, s T integral time scale of u (t), s r. Taylor's microscale for u (t), s H velocity vector = U+ u', m/s Ht fluctuating component of velocity, m/s U mean velocity, m/s sampled and held signal, m/s b!m Greek symbols 7 (t) 7m (t) A co ~/ 2
noise signal, m/s sampled and held noise signal, m/s bandwidth of spectral estimator window, radians/s time between arrivals in pdf, s Taylor's microscale of length - UT~, m
( ) (I)
ensemble average conditional average estimate
1 Introduction Measurement of the power spectral energy density of fluid velocity by laser D o p p l e r velocimetry is complicated by the random, intermittent nature o f the L D V velocity signal caused by the r a n d o m arrival o f particles at the measurement volume. Special care must be used in the statistial analysis o f such data, and in the interpretation of results. Significant progress in understanding these problems has been made, b u t even the m e a s u r e m e n t of so simple a statistic as the mean velocity remains under current investigation. A comprehensive treatment for all types of statistical moments does not exist. The purpose o f this p a p e r is to investigate and evaluate procedures for the power spectral analysis o f an L D V velocity signal that is formed by sampling at the arrivals o f valid signal bursts, Fig. 1, and holding these values until another valid signal arrives. It is assumed that the average n u m b e r o f scattering particles in the measuring volume, N~, is significantly, less than one, so that the p r o b a b i l i t y of more than one particle' at a time in the measurement volume is small. In the terminology of Adrian (1983) Ne is the "burst density", and Ne ~ 1 is the low burst density limit. Sampled and held analog output signals o f the form described above are p r o v i d e d by most L D V signal processors.
18
Light,]flux ~~/
Experiments in Fluids 5 (1987) velocity u (t) when the samples occur at close intervals. The mean interval between successive random arrival times is inversely proportional to the mean data rate k?. An appropriate measure of the closeness of the data is the "data density", defined as the mean number of samples in one Taylor microscale of the velocity signal (Adrian 1983):
=bursttime I d.I Threshold
[ Br~ [
%j
.
.
.
.
data density = N T~.
(1)
.
The Taylor microscale is defined by Ui
((du/dt) 2) = a 2 / r 2 .
u(t) l
ti
lj
t
Fig. 1. Burst processing in laser Doppler velocimetry
(2)
Hence, it characterizes the root mean square slope of u (t), and the data density represents the ratio of the mean time it takes u (t) to change by one standard deviation to the mean time between samples• It is easy to show that the root mean square error between u (t) and um (t), averaged over time, is of order a,/NT~ for large data densities. As the data density increases, Um(t) converges to u (t). The low burst density limit is of particular interest in this paper because the elimination of burst overlap avoids the "phase" or "ambiguity" noise that occurs in the high burst density limit (Adrian 1983; George & Lumley 1973; Adrian 1972). Within this range, high data density provides more information about the short time/high frequency behavior of the velocity than low data density. Thus, the combination of high data density and low burst density should yield optimum conditions for measurement of power spectra. High data density and low burst density can occur simultaneously because the LDV measurement volume can usually be made much smaller than the scales of motion of the fluid flow. As illustrated in Fig. 3, a turbulent eddy can be seeded with enough particles to resolve most of the scales of motion without introducing more than one particle into the measurement volume on aver-
Fig. 2. Sampled and held velocity for high data density •
1.1 Data density Holding the last known velocity is a simple form of interpolation between data, and it is a natural, appropriate procedure when the data are close enough, on average, to resolve the structure of the velocity fluctuations (Adrian 1983; Leneman & Lewis 1966). Figure2 illustrates the sampled and held signal Urn(t) derived from a random
°
•
•
-
°
,
o
u.,
Fig. 3. Eddy seeding in the high data density/low burst density limit
R. J. Adrian and C. S. Yao: Power spectra of fluid velocities measured by laser Doppler velocimetry age. This is possible whenever the measurement volume diameter dm is much smaller than the Taylor microscale of length, given approximately by 2-" U Tz according to Taylor's frozen field hypothesis. The fundamental impediment to the realization of low burst density/high data density operation is the difficulty in introducing high concentrations of scatterers into the fluid. High concentrations can often be achieved in water flows. For example, data rates of 4,000 samples/s at mean velocities of 0.5 m/s were reported in Buckles, Hanratty and Adrian (1984), corresponding to 0.125mm mean spacing between samples. However, seeding air flow with large concentrations of particles is much more difficult. Thus, our discussion of high data density signals will pertain primarily to liquid flows.
1.2 Spectral estimation from random data Previous research on LDV power spectral measurements has concentrated on the extreme limits of low data density or high burst density. George and Lumley (1973) developed a complete theory for the power spectrum of the instantaneous frequency of the high burst density signal. This theory assumed perfect frequency demodulation of the signal to obtain a continuous representation of velocity (plus phase noise) versus time. Buchhave, George and Lumley (1979) extended this analysis to include the effects of holding the signal during short drop-out periods that are correlated leith randomly occurring, low amplitude segments of the high burst density signal. Power spectral analysis methods for low data density signals are characterized by algorithms that estimate the spectrum in terms of individual velocity samples (Mayo et al. 1974; Scott 1979; Mayo 1979; George 1979; Gastor & Roberts 1975, 1977; Masry & Liu 1975). Mayo (1979) reviews this work. The arrival times of the velocity samples are assumed to be statistically independent of the fluctuating velocity, and described by Poisson statistics with a constant rate parameter. The set of arrival times {ti} generates a set of samples {ui}, and various algorithms have been developed which estimate the power spectrum by direct Fourier transformation of the samples, or by estimation of the autocorrelation followed by Fourier transformation. The best 'method is thought to be the "slotted time" or "discretized time" indirect computation studied by Gastor and Roberts (1975, 1977) and Mayo, Shay and Riter (1974), among others. In this algorithm data pairs are grouped according to the time window into which their time difference falls. The width or" the time window equals the time delay increment of the digital autocorrelation, so the correlation estimate is smoothed over this time, and the power spectrum computed by Fourier transforming the estimated correlation is a windowed estimate. A general conclusion from the foregoing work is that it is not necessary to satisfy Nyquist's criterion when
19
sampling randomly. With periodic sampling the sample rate must exceed twice the maximum frequency of the signal. With random sampling the mean sample rate may be less than this value; spectra can still be measured without aliasing because there is a non-zero probability that samples will occur at time intervals smaller than the mean interval, thereby providing information on the high frequency components of the signal. Removal of the Nyquist sampling requirement makes it possible, in principle, to measure high frequency spectra in experiments that produce low data rates, i.e. air flows with little or no particle seeding. In practice, it is difficult to make satisfactory measurements at frequencies significantly higher than the mean data rate, because the variance of the spectral estimator becomes large unless extremely long sampling times are used. The variance is approximated by (Gastor & Roberts 1975) var (Su (co))
S~,(co)
2~z { -
-
-
Su(0)
AcoT 1-~ ~
1} --=----
2NT.
2 ,
(3)
wherein the first term represents the uniform sampling variance, and the second term represents the additional variance created by random sampling. The additional variance is small if Su (0) ~ S. (co) 2N T., or if
N/B >>S~ (O)/S, (co),
(4)
where B is the bandwidth of S, defined by
2B = cr2/S~(0).
(5)
Since S. (co) is at least ~ times less than Su (0) in the high frequency range of most turbulent flow spectra, Eq. (3) implies that the added variance is large unless A?>>B. Consequently, as a practical matter, power spectra are not measured accurately at frequencies significantly above the mean sample rate, unless a very large number of samples is accumulated experimentally.
1.3 Velocity biasing Low burst density LDV data are biased toward high velocities because more high speed particles pass through the measuring volume per second than low speed particles. Consequently, as originally proposed by McLaughlin and Tiederman (1973), velocity samples must be weighted by appropriate statistical factors to arrive at averages that are not biased. It has been shown (George 1977; Hosel & Rodi 1977) that the random burst times of each Doppler burst, rB in Fig. 1, can be used as weighting factors for single-time statistics, since fast particles have short burst times, and vice versa. Alternatively, Dimotakis (1976) has suggested dividing each Sample by an analytically derived function that describes the probability of sampling the observed velocity vector. This approach may be used approximately when two-dimensional velocities are measured. Edwards (1981) and Edwards and Jensen (1983) have considered the effects on single-time statistics
20
Experiments in Fluids 5 (1987)
of dead times between samples (as a method of reducing velocity bias), and Stevenson, Thompson, Bremmer and Roesler (1980) and Dur~o, Laker and Whitelaw (1980)~ among others, have examined biases in the mean velocity experimentally. The only velocity biasing studies to treat the autocorrelation and power spectrum appear to have been those of Buchhave and George (1979) and Buchhave, George and Lumley (1979). That work deals with bias correction of low burst density data with high or low data density by burst time weighting. Provided that its underlying assumptions are satisfied by the experimental conditions, this method is correct, in principle. It has not, however, been tested extensively in experiments.
1.4 Sample and hold signals Two aspects of sampled and held LDV velocity signals have been examined previously: single-time statistics of low burst density signals (Edwards & Jensen 1983), and one-time and two-time statistics of high burst density signals with short drop-out periods (Buchhave et al. 1979). The latter case is complimentary to the low data density case in that data is present most of the time and missing for short intervals. While there is no well developed theory for the case of low burst density and high data density, it is obvious that the sampled and held signal is an accurate representation of the true velocity when the data density is sufficiently high. Hence, many investigators (e.g., Ferreira 1978; Pfister et al. 1984; Gollub et al. 1980) have obtained power spectra from such signals in practice by performing analog spectrum analysis, or by uniformly sampling the sampled and held signal and performing digital spectral analysis. The present work was motivated by observations of distortions in the high data density power spectra measured by Ferreira (1978). These distortions were obviously caused by the finite value of the data density in Ferreira's experiments, and they raised questions concerning the magnitude of the data density needed to justify this approach, the form of the distortion, and the parameters that affected it. Our analysis begins with a general treatment of the two-time statistics of the sampled and held signal. Two special cases are considered in detail: low fluctuation intensity flows with unrestricted data density, and un-restricted fluctuation intensity flow with high data density. We shall see that velocity biasing effects are negligible in each of these cases.
(6)
(7)
and the sample times t1 and t2 are separated by the time delay z = t 2 - tl, Fig. 4. The corresponding correlation and power spectrum of the sampled and held signal are 1
oo
S,m (co) = ~ _ ~ e
-j`°" R,,~,, (r) dr,
(8)
and R
....
(7g)
=
(9)
( H m (tl) b/m (t2)) .
In practice, one commonly uses time averages, and t~ is evaluated at a sequence of uniformly spaced times. However, we shall find it convenient to work in terms of ensemble averages; then tl and t2 can be any arbitrarily selected times. The relationships between the correlation and spectrum of the true velocity and the sampled and held velocity are derived by noting that bl m (t l) = bt (ti) ,
bl m (t2)
=
/g (lj)
(10)
where t i and tj are the random arrival times of:the bursts that immediately precede the sample times t 1 and t2. Then,
R.oum(r) = (u (ti) u (tj)).
(11)
The brackets in (1 i) average over the randomness in u (t), if the flow is turbulent, and the randomness in the arrival times. The latter, associated with the random locations of the particles in the fluid, occurs in all types of flows. The sources of randomness can be isolated in two ways. First, the average in (11) can be re-written as a conditional average of u(ti)u(ti) given that the vector field u(x, t) is known, followed by an average over all vector fields in the statistical ensemble describing the flow:
R . . . . (z)
=
( ( u (ti) u (tj) ],(x, t))).
(12)
The conditional average treats u(x, t) as a known vector field for all space and time, and averages over the random arival times, as determined by the rate at which u(x, t) sweeps randomly located particles into the LDV measurement volume. The statistics of the arrival times, and hence
,=~_..~x9 i I
tj l
I
I
j--
I
~
I /
x I
I i
~--
~1~1"d~1 tl
The power spectrum of a stationary random process u (t) is 1
R,u (r) = (u (tl) u (t2) )
i ill
2 Two-time statistics of the sampled and held signal
S~ (co) = ~ - 5o~ e-j o,, R.~ (z) dr,
where the autocorrelation is
----
"c
I--
u(t) urn(t}
I
~ i i i
I I
~2+dgz
t_
I
/
t2.
_1 --I
Tl
Fig. 4. Arrival times and sample time intervals, ti and tj are the arrivals the immediately precede t 1and t2
R. J. Adrian and C. S. Yao: Power spectra of fluid velocities measured by laser Doppler velocimetry the conditional average in (12), depend implicitly upon the flow field. The outer average averages over all random flows. If the flow is unsteady, but non-random, the average of the conditional average is just the inner, conditional average. In the second method Eq. (11) can be written as R ..... (r) = >.
Prob {1 arrival in
1 arrival in (42, g2 + dg2) and Oarrivalsin(¢2+d~2, t2) givenu},
=
The statistics of the arrival times can be modeled by a Poisson process with a random rate parameter, provided the location of each particle is random and completely independent of the location of any other , : (Erdmann & Gellert 1976; Edwards & Jensen 19~ ...... dmann & Tropea 1981; Adrian 1983). The last condition cannot be satisfied by large particles whose sizes preclude the presence of other particles over significant volumes of the flow. A general discussion of doubley stochastic Poisson processes in which the rate parameter depends upon another random process, such as the velocity, can be found in Snyder (1975). Given the velocity field, the conditionally averaged rate at which particles enter the LDV measurement volume can be expressed as (Buchhave et al. 1979)
Prob [1 arrival in (~1, ca +d¢1) and 0 arrivals in (¢1 + d¢l, t2) given u},
(14)
where A is the cross-sectional area through which particles with velocity u m u s t flow in order to produce valid LDV signals. Equation (14) assumes implicitly that the mean concentration of particles C and the local velocity vectoru are essentially constant over A. The probability of one arrival in ( t , t + dt) is h dt, and the probability of k arrivals in (t,t') is given by the Poisson distribution (Snyder 1975; Papoulis 1965) [n (t, t')] e e -"(t,t') Prob [k in (t, t')] = k! (15) Here the conditional mean number of arrivals is
(18)
= //;/(¢1) d~l e-n(¢a+d¢a'ta)fl(42) d¢2 e-n(¢2+¢2't~), ~1 4= ¢2, [fl(¢l) d¢l e-n(ga+d~l"t2) , ¢I = ¢2" Dividing by d¢ld~2 and taking the limit as d¢1 d¢2--+ 0 yields
f ( ¢ l , ¢2[ H) = f/ (~1) ?/(¢2) e-"(e"t*) e-"(¢2"2) q- /;/(~1)
e-n(gx't2)
3(¢2-- ~1).
(16)
(17a)
(19)
Using (12) and (19), the equation for the autocorrelation of the sampled and held signal is (20a)
R ..... ( t 2 - t l ) =
5 5U(~l) U(¢2)f(~l,¢2lu)d~id¢ ~--
tz
¢0 t 1
ta
= i d¢1 ~ d¢2_o,
(32)
R. J. Adrian and C. S. Yao: Power spectra of fluid velocities measured by laser Doppler velocimetry
23
provided that the process u (t) possesses a Taylor microscale. Then (35) oo l7 N e -~1 ~-~1R.. (i/) dr/+ e - ~ a2/N 2 T 2
ftij
.;il;i,, ......
m
--O0
S.,. (co) -
a
1
1 + co2/N2
.[S. (co) + 2 o-2/A~3 7 2] .
(36)
Equation (36) predicts that the spectral level of the step noise within the bandpass of the low filter vanishes as 3, whereas above ~ = 1/N it vanishes as A?-I. Hence increasing the data rate improves the spectral fidelity by increasing the bandwidth and reducing the noise contamination within the bandwidth.
• (ms)
3.1 Experimental and numerical results
',.'. X~(".;"" I,\': .:'t,;, 'i.~.:,,,., ~ ...,.. , ;',, ~.: ~ '~ v ' : ; ~ g ,
:,
't " : : "
•
5.0
'1~ (ms)
b
Fig. 6a and b. Conditional probability density of tij for constant arrival rate a N = 2 KHz; b N = 0,4 KJ-Iz
and Sum(co)
1
1 -it- c o 2 / j W 2
S~(co) - S R ~ ( t l ) e-Uqd~l 0
(33)
for the power spectrum of u. Thus, the effect of the sample and hold process is to low pass filter the true spectrum with low pass frequency A?/2 n. The additive term inside the brackets in (33) is a constant, independent of frequency, and it represents a white noise component. The low pass filtering is c~tused by the information loss that occurs during the hold periods, and the white noise component is created by the random steps that occur at new samples. We shall refer to the latter as "step noise". Note that the step noise is also low pass filtered. In the limit of high data density the step noise component can be approximated by observing that the exponentials in (32) and (33) will be small unless t/ is of order T~ or less. In this range we can write (Batchelor 1960). R~.(r/) = R . u ( 0 ) ( 1 - r / 2 / 2 T 2 ) ,
r/< Tz,
(34)
Measuring the frequency of an individual Doppler burst yields a sample of the velocity at the time of the burst, u (ti) that is almost always contaminated by a random noise component, 7(ti). The noise derives from several factors, including random frequency modulation of the Doppler burst caused by random noise in the signal from the photodetector; instrumental noise within the Doppler frequency signal processor; and velocity uncertainty associated with velocity gradients in the fluid and, perhaps, Brownian motion of the scattering particles (Adrian 1983; Buchhave et al. 1979). The input to the sample and hold process can be viewed as a continuous signal u (t)+ 7(t) that is sampled at the random arrival times tl. The effects of noise can be incorporated into Eqs. (32)-(36) by replacing R , , and S, by the autocorrelation and spectrum of the signal plus noise, respectively. The autocorrelation of the velocity plus the noise is equal to the sum of the autocorrelations of the velocity and the noise, and similarly for the spectrum, assuming that the noise and the velocity are statistically independent. According to Eq. (33) the spectrum of the sampled and held velocity plus noise is the low-pass filtered sum of the velocity spectrum plus the noise spectrum plus a white step-noise spectrum. The latter component depends upon the spectra of the velocity and the noise, but because of its wide bandwidth, the noise spectrum is expected to dominate. For this reason, we shall consider first the simple case in which the signal is pure noise• 3.2 Noise spectrum
To a good approximation 7 (t) can be modeled as a wide band random process whose spectrum is nearly white within the frequency range of interest. (The instrumental noise has wide bandwidth, and the particle-dependent uncertainties are statistically independent from one burst to the next, implying a white noise spectrum.) Consequently, we will examine the behavior of the sampled and held spectrum when the noise spectrum and its correlation
24
Experiments in Fluids 5 (1987)
are represented by
10-1
2 a2/o9c S ~ - 1 + (o)/o9c)2' R7 7 =
o-72 e-mclTI
(38)
where o9c is a low pass cut-off frequency that is large compared to the frequency range of the signal spectrum and/or the mean sample rate. The power spectrum of 7m is 1 [ 2@/coc 2~2/A? ] Sy,= I+(Og/N) 2 1+(o9/o9c) 2 + I+N/Og~] ' (39) In the high data density limit, N/ogc + oo, (39) reduces to the original spectrum, showing that the effects of sampling vanish with increasing data density. The sampled and held spectra for various finite mean sampling rates are shown in Fig. 7 over a range of frequencies for which the noise spectrum is essentially white. The theory predicts that the sample and hold process attenuates the spectrum with a 1/o92 roll-off that is characteristic of a first order low pass filter, and increases the energy content of the low frequencies by adding step-noise as the mean data rate decreases. Thus, distortion of the spectrum occurs over the entire spectral range. The theoretical predictions for white noise have been verified by numerical and experimental tests. Numerical simulations were performed by generating a random, wide-band times series 7(0 using a random number generator, and the time series was sampled and held at random times obeying Poisson statistics. Subsequently, the sampled and held signal was Fourier analyzed digitally to obtain its power spectrum. The theory agrees well with the numerically simulated results, Wide-band noise signals have been generated experimentally by scanning an LDV measurement volume at a constant velocity through a motionless body of glycerine that was seeded with polystyrene latex particles. By proper adjustment of the particle concentration, signals of the low burst density type were created. Figure 8 illustrates typical spectral behavior of the output of the frequency tracker that was used to measure the Doppler burst frequencies (TSI Model 1090). The spectra were characteristically flat out to a frequency of the order of the mean sample rate over 2~, and then rolled-off approximately as 1/o92, in accord with the present theory.
3.3 Velocity plus noise spectrum The spectrum of the sampled and held signal in this case is, in general, (40) S(u+Y)m-- 1+O92/82
S u + S~' -
T (Ruu+R~y)e-N'ldrl 1 0
.
Using Eq. (36) and the noise spectrum model in Eq. (39) the foregoing equation becomes (41) 1 S(u+y) - l_ko92//V2
(Hz)
(37)
[
2~2/o9~
2a 2
Su'+ 1+(o)/o)2) + ~ 4 ~ , ~
2@/N ] 1 +N/ogc]
o ,, [] v 0
10-z
-: d 10-3 E
230 690 2070 2990 co
D ,_ 10-4
10_5
10-6
0
,
,
i
,
10
,
i
,
, ~ I\
102 Frequency (Hz)
;
103
104
Fig. 7. Spectra of a Poisson sampled wide band noise signal. , equation (39); symbols denote spectral values calculated from numerical simulation, o)c >>Ar 10-z
10-3 -~ d E ~ 10-4 co
~,
0- lO-S
10 -e
i
~
i
10
10z Frequency (0.u.)
103
104
Fig. 8. LDV measurements of the noise spectrum measured in stationary flow
It should be noted that when the input noise spectrum is flat over a wide band (i.e., o9c is large compared to the largest frequency of S, or N), its form is indistinguishable from the step noise spectra, each being essentially low pass filtered white noise. Then, the brackets in Eq. (41) contain S, plus three white noise terms.
25
R. J. Adrian and C. S. Yao: Power spectra of fluid velocities measured by laser Doppler velocimetry '
I ' ' I '
I
'
I
I
'
I
,
I
'
I
I
'
I
,
I
10-I
10-2
d E
10-3 (.n
I0 -4
10-5 0
10-6
t
I
I0
,
,
I
102
,
,
I
103
I
10
~
,
,
I
,
i
I
102 103 Frequency (o.u.)
Fig. 9 a - c. Spectra of a simulated turbulence signal plus noise, a S, (co); h values of b) and frequency are dimensionless
The case of a turbulent velocity combined with wideband noise has been simulated numerically using the methods described above for the noise signal. The turbulent component was simulated by low-pass filtered white noise whose cut-off frequency lay significantly below coc and 3). The spectrum of the velocity plus the noise shown in Fig. 9a was measured by sampling at Poisson distributed random times with a large mean rate. The flat, high frequency section of the spectrum represents the noise level. As the mean sample rate decreases, the noise level is attenuated at high frequencies, and increases at frequencies below cut-off, N/2 n. For the spectra that are shown, the step-noise component depends primarily upon the noise spectrum. However, the velocity spectrum would contribute significantly if the mean data rate were low enough: Spectral measurements of the vertical velocity component in turbulent thermal convection in water between a hot lower plate and an insulated upper boundary have been made by Ferreira (1978). Measurements at various heights above the lower boundary are presented in dimensionless form in Fig. 10 using Deardorff's (1970) convection scaling. Below approximately 100 dimensionless frequency units the spectrum corresponds to the velocity spectrum, and above this frequency it corresponds to the combination of the noise spectrum and the step noise spectra. These spectra are flat over a short frequency range, then decrease as 1/co2. The frequency at which the roll-off occurs corresponds to the mean sample rate estimated for the experiments. It is clear that the velocity spectrum cannot
10
S,~ (co) obtained
,
,
10z
I
,
i
104
103
at N = 40; e S,, obtained at N = 20. The
10-1 Height (mm) o z, D
10-2
x +
6 33 105 115 130 1/,0
E cJ
&
c~
0
'0 O 0
10 -3
o "N 10-4
-5
-2
10-s
\ 10 -~
i
~
I
10 102 103 Dimensionless frequency Fig. 10. LDV power spectra of the vertical velocity measured at various heights in a 20 cm deep layer of water undergoing turbulent thermal convection (Ferreira 1978)
26
Experiments in Fluids 5 (1987)
be recovered accurately unless the data rate is large enough to isolate the shoulder by making it occur at high frequency. If the data rate is too low, the filtering action of the sample and hold attenuates the velocity signal as well as the noise. It is interesting that the measured power spectra are very similar in form to the spectra that are obtained from high burst density LDV signals, as described by George and Lumley (1973) and others. In that case, the white noise component is caused by random frequency modulation generated by the random overlap of LDV signal bursts ("ambiguity noise" or "phase noise"). In the present case, the low burst density assumption rules out ambiguity noise, and this has nothing to do with the observed spectral behavior. Frequency measurement errors are the source of Sy, but even when they vanish, the step noise associated with sampling u(t) alone will remain. Without careful interpretation, the spectra in Fig. 10 appear to represent turbulent flows with wide - 5/3 power law inertial subranges. When the measured spectra contain significant sampling uncertainty, it is not difficult to fit a spurious - 5/3 power law to them.
4 High data density correlation and spectrum
The preceeding section dealt with the case of low turbulence intensity, but arbitrary data density; and Eq. (36) shows how the spectrum of the sampled and held signal approaches the spectrum of u (t) in the limit of high data density. Consider Eq. (20b) in which n(41,h) represents the number of particle arrivals during the interval (41, h). This quantity depends on the volume of fluid that is swept through the measurement volume by the velocity field during the entire time interval, and it may fluctuate substantially in high turbulence intensity flow. However, if the data density is large, the mean time between arrivals is small, and the velocity field changes very little between arrivals. Then, n (41, t0 can be approximated by n(41,tl) ~ h ( 4 0 ( t l - 4 0 ,
(42)
and similarly for related quantities in Eq. (20b). With this approximation the equation for the correlation function becomes R ....
=(f~ d41H(41)/;/(41)C-h({a)(tl~-~l)
d41 u2(40h(~l) e -~(¢')(t2-¢1
r~ e -~(t-¢) ~ 6(t - 4)
,
(43)
wherein the averages are over the random elements u and n. The convolution integrals in Eq. (43) can be interpreted
(44)
for any non-zero velocity. Then each integral in (43) reduces to u (t), showing that Rum~m~ R~u
(45)
without velocity bias in the high data density limit. 5 Summary and conclusions
The autocorrelation and power spectrum of the signal derived from an LDV burst processor by holding between particle arrivals the last measured velocity has been analyzed for two cases: low fluctuation intensity and arbitrary data density; and high data. density and arbitrary fluctuation intensity. The limits on the fluctuation intensity are not crisply defined, but intensities below fifteen per cent are probably low enough to avoid velocity biasing errors and render the analysis valid. The spectrum of the sampled and held signal consists, at a minimum, of the velocity spectrum plus a white, "step noise" spectrum, each of which is low pass filtered by virtue of the holding action between samples. Holding results in missing high frequency information. The low pass filter frequency is equal to the mean data rate divided by 2 zc, and the attenuation is approximately 10% at a frequency equal to one-twentieth of the data rate. Thus, as a guideline, the mean data rate should be about twenty times the largest frequency at which undistorted measurements are desired. This criterion is quite stringent by LDV standards. It could be reduced by using a more sophisticated interpolation between data points, such linear interpolation, in the next simplest instance, or even spline fitting. Higher order interpolation of this sort would also reduce the magnitude of the step noise. Step noise arises from the jumps that occur at new, random sample times in the sampled and held signal. It obscures the low spectral energy density regions of the velocity spectrum, i.e., the high frequency end of a turbulent power spectrum. The ratio of the step noise power spectral density to the power spectrum of the velocity at zero frequency is a useful measure of the magnitude of this effect. It is easily shown that the ratio is given by 2 ~ / N S T2 S, (0)
ll
+
as first order low pass filters with time constants 1//z. The time constants fluctuate randomly with the velocity, but when the data density is high, the integrands approach Dirac delta functions
2 Ns T2 Tu
(46)
where Tu is the integral time scale of u(t), equal to S,(O)/o 2. Using well-known relations for turbulent flow (Tennekes & Lumley 1972), the ratio becomes 2 a2/3) 3 T~ S~(0)
30
- --
A'
(NTx) -3 R e ~
1,
(47)
R. J. Adrian and C. S. Yao: Power spectra of fluid velocities measured by laser Doppler velocimetry where Re~ is the microscale Reynolds number, and A' is an empirical constant of order one. Thus, if Re~ = 100, and A?T~ = 10, there will exist approximately three decades of spectral energy level between the spectrum at zero frequency and the step noise level associated with turbulent flow. The step noise level is lifted substantially by noise in the velocity measurements. The noise adds to the signal its own spectrum plus its associated step noise spectrum. According to Eq. (41), the step noise spectrum would be equal to the spectral level of the noise if the noise were low pass filtered at the mean data rate, coo= ~r. Consequently, the ratio of the velocity spectrum to the baseline noise spectrum (input noise plus step noise) will be less than the value indicated by Eq. (47) by an amount that depends upon the quality of the LDV signal. The ratio achieved in Fig. 10 is about two and one-half decades for most of the data, but only one and one-half decades close to the lower boundary where the mean square of the velocity fluctuation was relatively small. It is interesting, but not very surprising, to note that the mean data rate imposes a frequency limit, regardless of the method of data analysis. In principle, burst data analysis using slotted time techniques, and burst time weighting (Buchhave etal. 1979) can evaluate, without bias, the spectrum at frequencies greater than the mean data rate. However, the sampling error variance becomes large above the mean data rate, unless very long averaging times can be tolerated. For practical averaging time, the results of Saxena (1985) indicate that the variance of the unbiased slotted time estimator is comparable to the bias of the spectrum of the sampled and hold signal. Hence, high data rates are desirable. If high data rates can be achieved, then simple interpolation between data is as viable as the more complicated burst time weighted, slotted time approach.
Acknowledgement This work is based on research supported by National Science Foundation grant ATM 82-03521.
References Adrian, R. J. 1972: Statistics of laser Doppler velocimeter signals: Frequency measurement, J. Phys. E.: Sci. Instrum. 5, 91-95 Adrian, R. J. 1983: Laser velocimetry. In: Goldstein, R. J. (ed.): Fluid mechanics measurements, chapter 5. pp 155-244. Washington, D.C.: Hemisphere Batchelor, G. tC 1960: The theory of homogeneous turbulence. Cambridge: Cambridge University Press Buchhave, P.; George, W. K. Jr. 1979: Bias corrections in turbulence measurements by the laser Doppler anemometer. In: Thompson, H. D.; Stevenson, W. H. (eds.): Laser velocimetry and particle sizing, pp. 110-122. Washington, D.C.: Hemisphere
27
Buchhave, P.; George, W. K. Jr.; Lumley, J. L. 1979: The measurement of turbulence with the laser-Doppler anemometer. Ann. Rev. Fluid Mech. 11,443-504 Buckles, J. J.; Hanratty, T. J.; Adrian, R. J. 1984: Separated turbulent flow over a small amplitude wave. Proceedings of the Second Int. Symp. on Appl. of Laser Anemometry to Fluid Mech. Lisbon, July 1984, LADOAN-Instituto Superior Technico Lisbon, Portugal pp. 15.4.1- 15.4.7 Deardorff, J. W. 1970: Convective velocity and temperature scales for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci. 27, 1211-1213 Dimotakis, P. E. 1976: Single scattering particle laser Doppler measurements of turbulence. In: Appl. of non-intrusive instrumentation in flow research, AGARD CP- 193, pp. 10.1- 10.14 Durgo, D. F. G.; Laker, J.; Whitelaw, J. H. 1980: Bias effects in laser-Doppler anemometry. J. Phys. E. 13, 443-445 Edwards, R. V. 1981: A new look at particles statistics in laser anemometry measurement. J. Fluid Mech. 105, 317-25 Edwards, R. V.; Jensen, A. S. 1983: Particle-sampling statistics in laser anemometers: sample-and-hold systems and saturable systems. J. Fluid Mech. 133, 397-411 Erdmann, J. C.; Gellert, R. I. 1976: Particle arrival statistics in laser anemometry of turbulent flow. Appl. Phys. Lett. 29, 408-411 Erdmann, J. C.; Tropea, C. D. 1981: Statistical bias in laser anemometry. Report SFB80/ET/198, Sonderforschungsbereich 80, Universit~it Karlsruhe Ferreira, R. T. D. S. 1978: Steady turbulent thermal convection. Ph.D. Thesis, University of Illinois, Urbana Gastor, M.; Roberts, J. B. 1975: Spectral analysis of randomly sampled signals. J. Inst. Math. Its Appl. 15, 195-216 Gastor, M.; Roberts, J. B. 1977: The spectral analysis of randomly sampled records by direct transform. Proc. R. Soc. Lond. Ser. A 354, 27- 58 George, W. IC Jr.; Lumley, J. L. 1973: The laser-Doppler-velocimeter and its application to the measurement of turbulence. J. Fluid Mech. 60, 321-362 George, W. K. Jr. 1977: Limitations to measuring accuracy inherent in the laser-Doppler signal. The Accuracy of Flow Measurements by Laser Doppler Methods, Proc. of LDASymposium, 1975. pp. 20"63. Washington, D.C.: Hemisphere George, W. K. Jr. 1979: Processing of random signals. Proc. of the Dynamic Flow Conference 1978, Skovlunde, Denmark, pp. 757-793 Gollub, J. P.; Benson, S. V. 1980: Many routes to turbulent convection. J. Fluid Mech. 100, 449-470 H6sel, W.; Rodi, W. 1977: New biasing elimination method for laser-Doppler velocimeter counter processing. Rev. Sci. Instrum. 48, 910-19 Leneman, O. A. Z.; Lewis,' J. B. 1966: Random sampling of random processes: mean square comparison of various interpolators. IEEE Trans. Autom. Control 11,396-403 Masry, E.; Lui, M. C. 1975: A consistent estimate of the spectrum by random sampling of the time series. Soc. Ind. Appl. Math. 28, 793-810 Mayo, W. T. Jr. 1979: Spectrum measurements with laser velocimeters. Proc. of the Dynamic Flow Conference 1978, Skovlunde, Denmark, pp. 851-868 Mayo, W. T. Jr.; Shay, M. T.; Riter, S. 1974: Digital estimation of turbulence power spectra from burst counter LDV data. Proc. of the Second Int. Workshop on Laser Velocimetry, Purdue University, West Lafayette, Indiana, March 1974, pp. 16-26 McLaughlin, D. K.; Tiederman, W. G. 1973: Bias correcting for individual realization of laser anemometer measurements in turbulent flows. Phys. Fluids 16, 2082- 2088 PaP0ulis, A., 1965: Probability, random variables and stochastic processes. New York: McGraw-Hill
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Experiments in Fluids 5 (1987)
Pfister, G.; Schgttzel, K.; Gerdt, U. 1984: Velocity-correlation measurements of oscillating flow and turbulence in rotational Couette flow. In: Adrian, R. J. et al. (eds.): Laser anemometry in fluid mechanics. Selected papers from the First Int. Symp. on Appl. of Laser-Doppler Anemometry to Fluid Mech., Lisbon, July 1982. LADOAN-Instituto Superior Tecnico, Lisbon, Portugal. pp. 37-43 Saxena, V. 1985: Power spectrum estimation from randomly sampled velocity data. In: Dybbs, A.; Pfund, P. (eds.): Int. Syrup. on Laser Anemometry, ASME, Miami Beach, Florida, Nov. 1985, FED-Vol. 33, pp. 209-219 Scott, P. F. 1979: Theory and implementation of laser velocimeter turbulence spectra measurements. Proc. of the Dynamic Flow Conference 1978, Skovlunde, Denmark, pp. 47-67
Snyder, D. L. 1975: Random point processes, New York: John Wiley & Sons Stevenson, W. H.; Thompson, H. D.; Bremmer, R.; Roesler, T. 1980: Laser velocimeter measurements in turbulent and mixing flows - Part 2. Air Force Tech. Rep. AFAPL-TR-792009, Part 2 Tennekes, H.; Lumley, J. L. 1972: A first course in turbulence. Cambridge MA: MIT
Received December 20, 1985
Announcements 2nd I n t e r n a t i o n a l c o n f e r e n c e on laser a n e m o m e t r y - advances and a p p l i c a t i o n s University o f S t r a t h e l y d e , G l a s g o w , S c o t l a n d , U K , S e p t e m b e r 2 1 - 23, 1987
The organising committee invite authors to submit abstracts on topics including: - General applications, - Flows with heat transfer, - Single-phase fluid mechanics, - Two-phase flow studies, - New systems and fibre optics, - Data processing, - Comparisons - computer solution, - Measurement and method.
Draft papers will be selected on the basis of 300 word abstracts. All papers will be referred before acceptance. Deadlines
December 1, 1986 February 1, 1987 June 1, 1987
Abstracts (300 words) Draft papers for refereeing Final paper
Abstracts should be submitted to: LDA Conference, Dr. S. M.
Fraser, Mechanical and Offshore Engineering, University of Strathclyde, Glasgow, G1 1XJ, U.K.
