Astrophysical dynamics 1999/2000 : Proceedings of the...

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145.6

5 Results and Discussion

5.1 Utilizing different analyzing waveletsTo examine the effect of different analyzingwavelets, the x-component of the fluctuatingvelocity on the jet center line at 4/ =dx isanalyzed by the wavelet auto-correlationmethod with the help of the Morlet andMexican hat wavelets. Their wavelet auto-correlation coefficients ( )t,aRWR is presentedin Fig.1, and rainbow colors of Fig.1(a) and(b) are results of the Morlet and Mexican hat

wavelets, respectively. The traditional auto-correlation coefficient ( )tR is plotted at thebottom of the color figures (Fig.1(c)). Makingcomparison between Fig.1(a) and (b), the twomaps show very similar distribution, andnearly-periodic peaks appear at a=70-90ms.Because the Morlet wavelet has higherresolution than the Mexican hat wavelet in thefrequency space, two groups of periodicoscillations appear at msa 7050 −= using theMorlet wavelet, which cannot be found in themap of the Mexican hat wavelet. As tolocalization in time, from Fig.1 it is found thatthe positive and negative peaks of ( )t,aRWR

in the large period region correspond to largepeaks of ( )tR , and the peaks of ( )t,aRWR inthe small period region equivalent to smallpeaks of ( )tR , and both have almost samelocalization in time. From above results, it canbe concluded that the wavelet auto-correlationanalysis using the Morlet wavelet can obtainenough localization in both period and time. Inthe following section, we only use the Morletwavelet to analyzing the wavelet auto-correlation of a jet flow.

5.2 Comparison of MWC(a,ta,t )andRWR(a,ta,t )After choosing the analyzing wavelet, let us tocompare the modulus of the wavelet auto-correlation function ( )t,aMWC with its realpart ( )t,aRWR in Fig.2, which is obtained byanalyzing velocity fluctuation on the jetcenterline at x/d=8.5. Same as the waveletcoefficient mentioned above, ( )t,aRWR showsalmost same distribution as ( )t,aMWC . It isproper that ( )t,aRWR replaces ( )t,aMWC toexpress the strength of the wavelet auto-correlation. From Fig.2 ( )t,aRWR not onlygives exactly same the information as

( )t,aMWC , but also extracts local positive andnegative correlation within the period acentered at t . At the following section, weutilize ( )t,aRWR to describe the flow structureinstead of ( )t,aMWC . Comparing ( )t,aRWR

with ( )tR , it is found that the positive andnegative peaks of ( )tR correspond to thestrong positive and negative peaks of

( )t,aRWR in the range of a=90-110ms,

Figure 1: Wavelet auto-correlation coefficients offluctuating velocity on the centerline at x/d=4;

(a) using Morlet wavelet;(b) using Mexican hat wavelet;

(c) traditional auto-correlation coefficients.

��Ýià�Õ©ÏÐȸê6��a ��È�É�ÑÒÕ©ÏÐÈvÚNÈvÍ{Ø�Ô�ÍÎWÈvÍ{ئÈWÏÒ×ÛÝiÍ©È É/Ø � d \ � d ê�����ÚNÚó�É � 2ïÉGÌ�ÈW×iÈvØ�É�Õ]ئÔPã�ÎWÔ�ÏÐÏÐÈv×´É/ØÐÝÛÔ�ÍÎWÔEÈ96�ÎvÝÛÈWÍ{ØÐÑ ����� ��� m � Õ©ÑÐÝiÍ©àùئƩÈ��Ô�ÏÒ×ÛÈvØ�Ö}É�Ì�ÈW×iÈvØWó ä�� é\Ý�ØÐØÐÔ ä Õ]ØNÕ©ÑÐÝiÍ©à÷ØÐÆ©È��Ètø,ÝÛÎWÉ�ÍXƹÉPØÖ}ÉGÌ�ÈW×iÈvØ�ó�Î � ÅBϦÉ�Ë]ÝiئÝiÔ�͹É�×JÍ©Ô�ÏÒÚRÉ/×ÛÝ �vÈWË9É�Õ,ئÔ/ã?ÎWÔ�ÏÒÏÐÈW×ÛÉ/ئÝiÔ�Í�v �� � ó �¹ÏÐÔ�Ú �UÝ`��ê�ë�ëeg � ó

ê0g $��x����� � ����'b�G� �����W����'0�z� � � �s� ��� �� �Y�� �������� �� � �0���FLOW STRUCTURE IDENTIFICATION OF A TURBULENT SHEAR FLOW WITH USE OF WAVELETSTATISTICS

145.9

exist. In Fig.4 (b) the cross points in ( )tq ,aWC

increase compared with Fig.3 (b), this exhibitsthe increase of the periodic eddy motions with

various scales in quantity.Figure 5 shows the distributions of

waveletauto-correlation in the mixing layer at x/d=10and at a=15, 40, 75, 100 and 150ms in

( )t,aLRWR , y/d=2.5. The obvious periodicoscillations happen and the stronger periodicmotions appear obviously at a=75, 100 and150ms in ( )t,aRWR . This provides a map ofapparent multi-scale coherent structure in themixing layer. At t =170ms a structure of largebranching (from a positive peak) is clearlyobserved, and indicates that a complexperiodic large eddy structure, which containsperiodic small eddies, exists in the mixinglayer. Making a comparison between

( )t,aLRWR and ( )tR it is found that theinterference between the positive and negativemagnitudes at various periods around t mayproduce the zero value of ( )tR . However,there must be existence of correlation at someperiod, and this means that ( )tR hides somecoherent information. But the wavelet auto-correlation analysis can extract all informationexisting in the complex coherent flowstructure. From the distribution of the waveletphase difference ( )tq ,aWC between 0=t andt in Fig.5 (b), it is evident that the interval ofthe discontinue constant-phase lines increaseswith the period. At any given period or scale,we can obtain an angular frequency ( )tw ,a

(( )

ttq

∆∆= ,aWC

) by differentiating thecorresponding phase fields with respect to t .In fact, ( )tw ,a means the difference of theangular frequency between 0=t and t .Making a comparison between Fig.4 (b) andFig.5 (b), it is found that the average angle ofdiscontinue constant-phase lines are almostsame, and this shows the similarity of twoflow structure.

Comparing Fig.4 with Fig.5, it appearsthat in the mixing region the periodic motionextents to large scale and small-scale regions,and implies the process of the eddy growthand breakdown with downstream distance.

6 Conclusions

The following main results were summarized.

Figure 4: Wavelet auto-correlation analysis offluctuating velocity in the mixing layer at x/d=5 and

y/d=1;(a) wavelet auto-correlation coefficients;

(b) phase of wavelet auto-correlation function;(c) local wavelet auto-correlation coefficients;

(d) traditional auto-correlation coefficients.

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�j��'qs��� � �������b�� êGëHui LI

145.10

(1) The real part of wavelet auto-correlationfunction not only provide exactly same theinformation as their modulus, but also extractlocal information of positive and negative

strength within period centered at time delay.(2) The wavelet auto-correlation functionusingthe Morlet wavelet illustrates betterlocalization in frequency space than that of theMexican hat wavelet.(3) The wavelet auto-correlation analysis canextract the information of self-coherentstrength and phase in terms of period and timedelay.(4) The peaks of ( )t,aRWR or ( )t,aLRWR inthe large period region correspond to largepeaks of ( )tR , and the peaks of ( )t,aRWR or

( )t,aLRWR in the small period region areequivalent to small peaks of ( )tR . But from

( )tR the period of peaks can not be obtained.(5) The traditional auto-correlation can be seenas the results of interference among auto-correlation magnitudes of all periods at thetime delay, therefore, ( )tR had been hidingvery important features of coherent structure.(6) The periodic coherent structure of the largeperiod consists of that of small period, in otherwords, a periodic large eddy motion containsthe periodic smaller eddy motions.(7) The discontinuous constant ( )tq ,aWC linesand the range of zero ( )tq ,aWC correspond tothe zero ( )t,aRWR lines and the region of thepositive or negative peak of ( )t,aRWR ,respectively, and the cross-point of constant-phase lines equivalents to the zero point of

( )t,aMWC .(8) From the distributions of ( )t,aRWR on thejet centerline at x/d=4, it is found that theperiodic eddies at a=75ms for ms40≤t and ata=85ms for ms80≤t pass through the shearlayer.(9) In the shear layer of x/d=10, anyinformation of auto-correlation cannot beobtained from ( )tR , but the strong periodicmotions of eddy for a=75, 100 and 150ms areclearly observed by ( )t,aRWR .

References

1. Crow, S.C. and Champagne, F. H.; OrderlyStructure in Jet Turbulence, J. Fluid Mech., Vol.48,547-591, 1971.

Figure 5: Wavelet auto-correlation analysis offluctuating velocity in the mixing layer at x/d=10 and

y/d=2.5;(a) wavelet auto-correlation coefficients;

(b) phase of wavelet auto-correlation function;(c) local wavelet auto-correlation coefficients;(d) traditional auto-correlation coefficients.

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