mi, deo and nathan 2005 - fast covergent itteratiuve scheme for filtering velcoity signals and...

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Fast-convergent iterative scheme for ltering velocity signals and nding Kolmogorov scales

J. Mi, R. C. Deo, and G. J. NathanSchool of Mechanical Engineering, University of Adelaide, South Australia 5005, Australia

Received 30 September 2004; published 13 June 2005

The present fundamental knowledge of uid turbulence has been established primarily from hot- and cold-

wire measurements. Unfortunately, however, these measurements necessarily suffer from contamination bynoise since no certain method has previously been available to optimally lter noise from the measured signals.This limitation has impeded our progress of understanding turbulence profoundly. We address this limitation bypresenting a simple, fast-convergent iterative scheme to digitally lter signals optimally and nd Kolmogorovscales denitely. The great efcacy of the scheme is demonstrated by its application to the instantaneousvelocity measured in a turbulent jet.

DOI: 10.1103/PhysRevE.71.066304 PACS number s : 47.27. i, 47.80. v

In his pioneering work 1 on turbulence, Kolmogorovderived, based on dimensional reasoning, the characteristiclength of the nest-scale turbulent motions to be

3

/ 1/4

, 1which is hence called the Kolmogorov length scale. In Eq.1 , is the kinematic viscosity and is the average dissipa-

tion rate of the turbulence kinetic energy given by e.g.,Hinze 2

= u i / x j + u j / xi u i / x j 2

with standard Cartesian tensor notation and summation onrepeated indices, where i or j =1, 2, and 3 represent thestreamwise, lateral, and spanwise directions, respectively.

The appropriate estimate of is of signicant importancefor improving our understanding of the ne-scale turbulence.However, great difculty occurs in directly measuring thischaracteristic scale. To obtain requires all the 12 terms of gradient correlations in Eq. 2 to be measured. This task cannot be realized by presently available experimental tech-niques. Accurate measurements of even one component of

u i / x j, as is well known, requires a multisensor probe withextremely high spatial and temporal resolution to incorporateeven the smallest scales of velocity uctuations. Moreover,several cross-correlation terms in Eq. 2 simply cannot bemeasured directly now 3 or in the foreseeable future, letalone direct measurements of and . In this context, itremains necessary to estimate from hot-wire measurementsof using the isotropic relation

= 15 u1 / x

1

2 3

and also Taylors hypothesis

u1 / x1 2 = U 12 u1 / t 2 , 4

where U 1 is the local streamwise mean velocity. Substitutionof Eq. 4 into Eq. 3 leads to

= 15 U 12 u1 / t

2 . 5

In addition, the Kolmogorov frequency is dened as

f K U 1 2 1 . 6

It is important to note that the nonltered or slightly l-tered velocity signals u im the subscript m means mea-sured is inevitably contaminated by high-frequency elec-tronic noise n , i.e.,

u im = u i + n . 7

This contamination causes both the spatial and temporal gra-dient variances to be overestimated, i.e.,

u im / x j 2 = u i / x j 2 + n / x j 2 , 8

u im / t 2 = u i / t 2 + n / t 2 .

The extra terms n / x j 2 and n / t 2 are the unwantednoise contributions. Therefore, to achieve accurate measure-ments of the velocity gradients, it is necessary that the rawvelocity signals u im be low-pass ltered at a specic cutoff

frequency f c to eliminate the effect of noise. The choice of f cis critical. Too high and it will not remove noise contribu-tions sufciently, while too low and it will wipe out somecontent of the signal. The right choice for f c is the Kolmog-orov frequency f K , i.e., the characteristic frequency of thesmallest structures. However, f K not only is a function of theow but also varies with spatial locations in the ow so thatit cannot be determined a priori . The method described in 4to determine f c in situ is complex, requiring two mechanicalanalog lters, a differentiator, a real-time spectrum analyzer,visual inspection, and optimization at each measurement lo-cation. This procedure is only realistic where the number of spatial locations or ow conditions is limited, and is prohibi-tive for experiments where these are large. In the absence of a simpler procedure, more arbitrary criteria are usuallyadopted, so that most previous measurements of must becontaminated by noise to some extent. The importance of this issue is also evident from 5 from which it is deducedthat even slightly over ltering u im at f c f K may cause sub-stantial underestimate of the velocity gradients.

From the above discussion it is obvious that substantialbenet would arise from a procedure that could correctlyobtain f K without prior knowledge of , or obtain both and f K solely from a nonltered signal of u1m. The present work aims to address this issue, i.e., to develop a simple and ef-

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fective scheme that can denitively obtain and f K fromu1m t , and thus a means by which to low-pass lter all thevelocity components u im t at f c = f K , thereby minimizing theeffect of noise on gradients u i / x j and other derived quan-tities such as structure functions.

Let us rst inspect the contributions of noise to the mea-sured kinetic energy u1

2 and dissipation . This is illustratedusing the one-dimensional spectral forms of u1

2 and forisotropy, which are see 6

u12 =

0 E 1 k 1 dk 1 9

and

= 15 0

k 12 E 1 k 1 dk 1 , 10

respectively, where E 1 is the one-dimensional spectrum func-tion and k 1 is the wave number in the streamwise x1 direc-tion. To demonstrate the inuence of noise on u1

2 and , weconsider the model spectrum function e.g., 7,8

E 1 k 1 = C K 2/3k 1

5/3 exp k 1 3/4 11

where C K is a constant determined empirically by experi-ments and = 32 C K . Note that Eq. 11 has been veried by alarge body of experimental high-Re data see 9 which ref-erences relevant experiments . Using Eq. 6 and k 1=2 f / U 1 Taylor hypothesis , Eq. 11 may be rewritten as

E 1 f = C K 2 / U 15/3 f 5/3 exp f / f K 3/4 . 12

Suppose the noise-contaminated velocity spectrum to be E 1m = E 1 + E n, where E n is the noise contribution. Then,from Eqs. 9 and 10 and k 1 =2 f / U 1, we can obtain

u1m2 = 2 U 110

E 1 + E n df 13

and

m = 120 3 U 13

0 f 2 E 1 + E n df . 14

Using Eq. 12 with C K =1.7 and then =2.55 e.g., 10 weillustrate in Fig. 1 the spectral density distributions of u1m

2

and m with E n =0 and E n =103 f / f K 2. This demonstratesthat, when f c f K , the contributions of the noise to u1m

2 andto m are very different. For example, if f c =10 f K , the ratiosu1m

2 u12 / u1

2 and m / are approximately 3.3% and2000%, respectively. This indicates that the high-frequencynoise contamination, if not properly ltered out, has an ex-tremely large impact on m while its inuence on u1m

2 , ormore generally on u im

2 , is very much less.Next we examine the effects of noise on the measured

Kolmogorov scales m and f Km. Let us express the measureddissipation m by

m = + n = 1 + n / = C 15

with C = 1+ n / 1, where n denotes the noise contribu-tion. Substituting Eq. 15 into Eq. 1 leads to

m

= 3 / C 1/4 = C 1/4 . 16

Then, from Eq. 6 , we obtain

f Km = C 1/4 f K . 17

Accordingly, the error in m, m, or f Km from high-frequencynoise can be measured by the departure of C p from unity. Asseen from Eqs. 15 17 , the exponent p varies from p =1for m to p =1/4 for m and f Km. This implies a much greatererror for m than for m and f Km. For example, the case m=5 results in an error of 400% overvalued , while the cor-responding error is only 33% underestimated for m and50% overestimated for f Km. It follows that, if we can rel-ter u im at f Km calculated from Eq. 6 and then recalculate m,

the noise part n or C 1 see Eq. 15 will reduce dra-matically. Subsequent recalculations of m, f Km, and then mwill further reduce their errors until m , m , and f Km f K . In principle, this analysis does not require any as-sumptions such as isotropy or Taylors hypothesis. That is,were the true or all terms of Eq. 2 measurable, it wouldresult in such an iterative scheme by which all noise con-taminations of u im i =1,2 ,3 can be optimally ltered andtherefore by which the true , , and f K can be found frompostprocessing of sampled signals of u im. However, since thedirect measurement of is impossible at present, and in theforeseeable future, this scheme currently can be implementedonly by invoking the isotropy assumption Eq. 3 and Tay-

lors hypothesis Eq. 4 . Here we propose it in Fig. 2 andprovide more details below.The original velocity signal U 1m

0 t is measured by a hot-wire probe and sampled at a very high frequency 20 kHz,say . Application of analog lters is not necessary, i.e., noltering is needed f c

0 = . Otherwise, use a high cutoff frequency f c

0 . With the signal collected, rst calculate m0

from Eq. 5 and then m0 from Eq. 1 . Next, substituting

m0 into Eq. 6 results in f Km

0 . If f c0 f Km

0 / f c0 , where

is a threshold of convergence and should be small, say,103 , lter u1m

0 at f c1 = f Km

0 digitally using, e.g., MATLAB .

FIG. 1. Color online Power spectra of u1m2 and m with and

without noise contamination.

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The newly ltered velocity signal u1m1

is thus generated.The above process may be repeated as many times asnecessary to generate u1m

2 , u1m3 , . . . , u1m

N , and then u1m

2 / t , u1m3 / t , . . . , u1m

N / t , until f c N f Km

N / f c N or

until f Km N , m

N , and m N have converged satisfactorily. From

the converged data, we nally obtain = m N , f K = f Km

N , and = m

N .Based on Eqs. 15 17 , both f Km

i and mi should con-

verge quickly to their respective asymptotic values. This istrue as demonstrated as an example in Table I for the case of f c

0 =10 f K , with reference to Figs. 1 and 2. Clearly, for thiscase, m , m , and f Km f K just in two iterations,with an accuracy of 99.8%.

To validate the present scheme for real measurements, theinstantaneous streamwise velocity =U 1 + u1 is obtained us-ing hot-wire anemometry along the centerline of a two-dimensional plane jet issuing from a rectangular w h=340 5.6 mm2 slot, with aspect ratio w / h =60. Here onlya brief description of the jet facility is given as details maybe found in 11 . To ensure statistical two-dimensionality,two parallel plates 2000 1800 mm 2 are attached to theshort sides of the slot so that the jet mixes with ambient uidonly in the direction normal to the long sides, following, e.g.,Gutmark and Wygnanski 12 . The jet exit velocity is U j

8 m/ s, which corresponds to a Reynolds number ReU jh / of approximately 3000.Velocity measurements are performed over the region

20 x1 / h 160 using a single hot-wire tungsten probe.

The hot-wire sensor is 5 m in diameter and approximately0.8 mm in length, aligned in the spanwise x3 direction. Ve-locity signals obtained at all the measured locations are low-pass ltered with a high and identical cutoff frequency of f c =9.2 kHz. Then they are digitized at f s =2 f c =18.4 kHz viaa 16-channel, 12-bit analog-to-digital converter on a personalcomputer. The sampling duration is approximately 22 s. Thewire calibration is conducted using a standard pitot tube inthe jet potential core near to the exit where u1

21/2 / U j 0.5%.The above measurements yield the original velocity sig-

nals U 1m0 t = U 1m

0 + u1m0 t , which are corrected for the hot-

wire length of 0.8 mm using Wyngaards approach 13 , and,consequently, the original time derivatives along the jet cen-

treline at x1 / h 20 are obtained as follows:

u1m0 / t u1m

0 / t = f s u1m0 t + f s

1 u1m0 t with t = f s

1 .

18

In this region, the original mean velocity U 1m0 not presented

is found to follow closely the relation U c / U j x1 / h 1/2

here U c denotes the centreline mean velocity , which is re-quired by self-preservation of the jet. This relation and alsothat for the half-velocity width, i.e., L1/2 / h x1 / h, are wellsatised by previous data e.g., 4 obtained in the far eld.

For high-Re ows, it is usually considered that the dissi-pation of turbulent kinetic energy out of the smallest-scalestructures is equal to the supply rate of the turbulence energyfrom the large-scale structures, which is of order U 0

3 / L0,where U 0 and L0 are the local characteristic velocity andlength scales see, e.g., 14 . Based on this argument, weobtain U c

3 / L1/2 , by taking U 0 = U c and L0 = L1/2 , for aplane jet. It follows that self-preservation of the ow furtherrequires

hU j3 = C x1 / h

5/2 19

and

FIG. 2. Iterative scheme of digital lter to obtain and f K from u1m.

TABLE I. Use of the iterative scheme for f c0 =10 f K .

Iterationsi f c

imi m

i f Kmi

0 10 f K 17.0 0.5 2.0 f K 1 2.0 f K 1.10 0.98 1.02 f K 2 1.02 f K 1.02 0.998 1.002 f K

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/ h = C x1 / h5/8 , 20

where C and C are constants determined by experiments.Indeed, both Eqs. 19 and 20 have been veried by Anto-nia et al. 4 who found that C 1.3 and C 0.94 Re j

3/4 .These relations provide a rigorous basis against which to

validate the proposed iterative ltering scheme using our jetcenterline velocity signal U 1m

0 t = U 1m0 + u1m

0 t for x1 / h 20.Figure 3 presents the results of m and m calculated from

the original signal and after three iterations of digital lter-ing. The iterative procedure is involved in obtaining m

i from5 via 18 and m

i and f Kmi from 1 and 6 . As with the

example presented earlier, it is found that only three itera-tions N =3 are required for m

i , mi , and f Km

i at all measured

locations to converge to their asymptotic values when setting =103 note that a smaller value of requires more itera-tions; e.g., N =5 if =104 . Also shown in Fig. 3 are Eqs.19 and 20 using C 1.3 and C 0.94 Re j

3/4 obtainedby Antonia et al. 4 .

Figure 3 demonstrates that the data of m3 and m

3 agreealmost identically with the curves of Eqs. 19 and 20throughout the measured region. This provides strong sup-port both for the validity of the iterative scheme and for thevalues C 1.3 and C 0.94 Re j

3/4 . It is also noted that,while m

0 lies well above Eq. 20 , m0 falls below Eq. 19 .

As x1 increases, f K decreases and thus the ratio f c0 / f K in-

creases so that the relative contribution of electronic noise

grows rapidly. Consequently, m0

and m0

show increaseddepartures from their true values with increased x1. Thisproblem may be responsible for errors in some previouslyreported data, since the delicate setting for the low-pass lter

described in 4 may not work very well due to the fact that f K is unknown a priori . A typical example of data that isapparently contaminated by noise can be found in 15 seetheir Figs. 3 and 4 , where various velocity and scalar struc-ture functions are reported. That these measurements arequite recent highlights the need for a more rigorous approachto obtain such data for basic research of turbulence.

Figure 3 also illustrates the effect of overltering u1m t .

In this case, the measured values of m and m can neveragree with the relations 19 and 20 . The results presentedfor this case are obtained from ltering all u1m

0 t along thecenterline at f c =350 Hz, which is one-half of f K for x1 / h=160. Interestingly, these measurements of m agree quitewell with those from previous studies 12,16 , where theirvalues of mh / U j

3 were converted from their reported data.Gutmark and Wygnanski 12 and Heskestad 16 did notoffer any detailed information about their lter settings.However it is quite clear that their measurements do notfollow relations 19 and 20 and therefore that their signalsmight well be overltered. Hence, their data offered wronginformation that Eq. 19 does not hold in the far eld of

their jet where self-similarity of the mean ow has devel-oped. The above observation provides further support for ouriterative scheme.

In summary, based on the denitions of the Kolmogorovscales and f K , a fast-convergent iterative scheme has beendeveloped to both optimally lter the noise-contaminatedhot-wire velocity signals and simultaneously nd the valuesof and f K . The scheme, implemented via use of the isot-ropy assumption and Taylors hypothesis, has been validatedrmly by its application to the instantaneous velocities mea-sured in a turbulent plane jet presented and other ows notpresented here . The proposed scheme is believed to havewide signicance for basic research on ne-scale turbulencebecause nearly all experimental studies in this eld to datehave used hot-wire measurements 17 . It has implicationsfor past measurements, and the conclusions based on them,which may suffer to an unknown extent from the contamina-tion of noise resulting from underltered or overltered data.Application of our scheme will allow future measurements todetermine the correct cutoff frequency, as well as and f K ,easily and unambiguously, so generating reliable data forbetter understanding of small-scale turbulence. Our schemeis much simpler and more rigorous than the previous scheme4 and even obviates the need for analog lters, differentia-

tors, and the like.The present scheme also applies for the measurement of

temperature scalar using cold-wire anemometer for esti-

mates of the Batchelor, instead of the Kolmogorov, scale.The support of the Australian Research Council is grate-

fully acknowledged.

FIG. 3. Color online Centerline variations of m hU j3 and

mh1 for the plane jet.

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1 N. Kolmogorov, Dokl. Akad. Nauk SSSR 30 , 299 1941Proc. R. Soc. London, Ser. A 434 , 9 1991 .

2 J. O. Hinze, Turbulence , 2nd ed. McGraw-Hill, New York,1975 , p. 218.

3 L. W. Browne, R. A. Antonia, and D. A. Shah, J. Fluid Mech.179 , 307 1987 .

4 R. A. Antonia, B. R. Satyaprakash, and A. K. M. F. Hussain,Phys. Fluids 23 , 695 1980 .

5 J. Mi and G. J. Nathan, Exp. Fluids 34 , 687 2003 .6 F. H. Champagne, J. Fluid Mech. 86 , 67 1978 .7 S. Corrsin, Phys. Fluids 7, 1156 1964 .8 Y.-H. Pao, Phys. Fluids 8, 1063 1965 .9 S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech. 268 , 333

1994 .10 L. M. Smith and W. C. Reynolds, Phys. Fluids A 3, 992

1991 .11 R. Deo, J. Mi, and G. J. Nathan unpublished .12 E. Gutmark and I. Wygnanski, J. Fluid Mech. 73 , 465 1976 .13 J. C. Wyngaard, J. Sci. Instrum. 1, 1105 1968 .14 H. Tennekes and J. L. Lumley, A First Course in Turbulence

MIT Press, Cambridge, 1972 .

15 E. Lvque, G. Ruiz-Chavarria, C. Baudet, and S. Ciliberto,Phys. Fluids 11 , 1869 1999 .

16 G. Heskestad, J. Appl. Mech. 87 , 735 1965 .17 K. R. Sreenivasan and R. A. Antonia, Annu. Rev. Fluid Mech.

29 , 435 1997 .

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