Wavelet Multiresolution Analysis of Stereoscopic Particle-Image

AIAA JOURNAL

Vol 40 No 6 June 2002

Wavelet Multiresolution Analysis of StereoscopicParticle-Image-Velocimetry Measurements in Lobed Jet

Hui Licurren

Kagoshima University Kagoshima 890-0065 JapanHui Hudagger

Michigan State University East Lansing Michigan 48824and

Toshio KobayashiDagger Tetsuo Sagasect and Nobuyuki Taniguchipara

University of Tokyo Tokyo 153-8505 Japan

A wavelet-based vector multiresolution technique was developed and applied to a high-resolution stereoscopicparticle image velocimetry system for studying the three-dimensional multiscale structure features of the lobed jetmixing ow The instantaneous three-dimensional velocity vectors were successfully decomposed into large- andsmall-scale velocity elds and existence of very strong multiscale turbulent structures were con rmed in the near eld of the lobed jet Within the central scale range of 16 mm the instantaneous pairs of large-scale streamwiseand horseshoe vortices can be clearly observed around the lobe peak regions and lobe troughs respectively andbegin to breakdown and spread outward at downstream The stronger small-scale streamwise vortices were foundin the lobe regions which rst spread outward along the lobes and then develop to the whole measured ow eldThese small-scale streamwise vortices also play an important role in the enhance mixing process The strongeralternate positive and negative peaks of small-scale axial velocity component which appear at the trailing edge ofthe lobed nozzle in the near eld indicate the existence of the KelvinndashHelmholtz or normal vortices

Introduction

T HE turbulentjet exhibitedcomplexstructureswith a wide rangeof coexisting scales and a variety of shapes in the dynamics

and its physics of mixing process is important in the engineering Itis a well-known fact that the streamwise vortices generated in a jet ow in addition to the azimuthal (or ring-type) vortices have beenfound to mix uid streams even more ef ciently The distortion ofazimuthal vortex structure can lead to streamwise vortices undercertain conditionsThe streamwise vortices in jet mixing ows canbe generated by many methods One of the methods is to use alobed nozzle to generate the large-scale streamwise vortices and toinduce rapid mixing which has been considered to be one promis-ing method for the enhanced jet mixing process More recentlyit has become obvious that lobed mixers provide the most mix-ing enhancement in the presence of a velocity differenceor normalvorticity component

The rst applicationof the lobedmixer was for jet noise reductionandnet thrust increaseof a turbofanjet engine1 becauseof themixingcharacteristicsof lobed mixers and their advantageover the convec-tional nozzle designs Now the lobed mixer has been applied to thecombustionchambers in enginethe spreadof pollutantsat industrialsites and so on and becomes an extraordinary uid mechanics de-vice for ef cientlymixing two different ow streams in determining

Presented as Paper 2001-0696 at the AIAA 39th Aerospace SciencesMeeting Reno NV 8ndash11 January 2001 received 5 March 2001 revision re-ceived 12 September 2001accepted for publication19 October 2001Copy-right cdeg 2002by theAmerican Instituteof Aeronautics andAstronautics IncAll rights reserved Copies of this paper may be made for personal or internaluse on conditionthat the copier pay the $1000per-copy fee to the CopyrightClearance Center Inc 222 Rosewood Drive Danvers MA 01923 includethe code 0001-145202 $1000 in correspondence with the CCC

currenAssistant Professor Department of Mechanical Engineering 1-21-40Korimoto limechkagoshima-uacjp Member AIAA

daggerResearch Associate TurbulentMixingandUnsteadyAerodynamicsLab-oratory

DaggerProfessor Institute of Industrial Science 4-6-1 Komaba MegurokusectAssistant Professor Institute of Industrial Science 4-6-1 Komaba

MegurokuparaAssociate Professor Institute of Industrial Science 4-6-1 Komaba

Meguroku

the length Because of these practical applications in engineeringthe investigationof thedetailedmixingmechanismsbecomessignif-icant Paterson2 provided the rst detailed experimental data in themixing region downstreamof axisymmetric lobed nozzles by usinglaser Doppler velocimetry (LDV) Originally it was believed thatthe mixing enhancement mechanism was solely caused by the in-creased interfacial area However his study found that the ow eldof downstream was dominated by strong secondary ow structuresalong with relatively large-scale streamwise vortices having scaleson the order of the lobe height which played a major role in theenhanced mixing process Also a horseshoe vortex on the orderof the lobe half-height was found to exist in the lobe troughs Thecontributionof these vortices to the overall mixing process was notclear but it was believed that these vortices were created by thelobe geometry and the trailing-edge Kutta condition Most of thelater studies concentratedon discovering the underlying physics ofthe two-dimensional lobed mixing process Werle et al3 found thatthe vortex formation process was an inviscid effect and the mixingprocess took place in three basic steps the vortices form intensifyand then rapidly break down into small-scale turbulence Eckerleet al4 studied mixing downstream of a lobed mixer at two velocityratios using a two-component LDV It was found that the break-down of the large-scale vortices and the accompanying increase inturbulentmixing are signi cant parts of the mixing processUkeileyetal56 appliedtheproperorthogonaldecomposition(POD) to a rakeof 15 single-component hot-wire data obtained in a lobed mixer ow eld They showed that the large-scale vortex as de ned bythe POD breaks down and the ow becomes more homogeneousMcCormick7 reported more detailed experimental investigation ofthe vortical and turbulent structureusing the pulsed-lasersheet owvisualization with smoke and three-dimensionalvelocity measure-ments taken with a hot wire Their study con rmed a new vortexstructure existing in addition to the well-known streamwise vortexarray and found that the streamwise vorticity deforms the normalvortex into a pinched-off structure that might also enhance small-scale turbulent mixing Recently the mixing process in a coaxial jetwhere the inner nozzle is a lobed mixer was experimentally stud-ied by Belovich and colleagues89 Detailed ow visualization andvelocity measurements with LDV were performed and differentmixing mechanisms for each velocity ratio were discussed

1037

1038 LI ET AL

During the past couple of years the development of particle im-age velocimetry (PIV) techniques has made it possible to providemore detailed information on ow structure such as the instanta-neous values of various ow quantities as well as their distributionand transientvariationHu et al1011 employedtwo-dimensionalandstereoscopicPIV systemsto measure the near ow eld of a lobed jetmixing ow The characteristics of the mixing process in a lobedjet mixing ow comparedwith a conventionalcircular jet ow werediscussedDespite the usefulnessof informationobtained by exam-ining the measured instantaneous ow elds and the time-mean tur-bulentquantitiesfurtherdetailof themixingprocessassociatedwiththe instantaneousmultiscale structures has not yet been clari ed

In the past decade there has been a growing interest in the use ofwavelet analysis for turbulent data This technique can track turbu-lent structures in terms of time and scale and extracts new informa-tion on turbulencestructuresThe continuouswavelet transformhasbeen proposed to analyze turbulent structures in terms of time andscale by Li andNozaki12 and Li13 The coef cientsof the continuouswavelet transformare known to extract the characterizationof localregularity continuously However the continuous inverse wavelettransform is unable to reconstruct the original function because themother wavelet function is a nonorthogonalfunction It is importantto reconstructthe originalsignalfromwavelet compositionin study-ing multiscale turbulent structures On the other hand the discretewavelet transform allows an orthogonal projection on a minimalnumber of independent modes and is invertible Such analysis canproduce a multiresolution representation and might be also usedto analyze turbulent ows Charles14 rst used the one-dimensionaldiscretewavelet transformto obtain local energyspectraand the uxof kinetic energy from experimental and direct numerical simula-tion data Staszewski et al15 identi ed the turbulentstructuresof theatmospheric boundary layer using the discrete wavelet transformLi et al1617 employed the discrete wavelet transform to evaluateeddy structuresof a jet in the dimensionof time and scale Li et al18

also applied the two-dimensional orthogonal wavelets to turbulentimages and extracted the multiresolution turbulent structures Toidentify the turbulent structures in the dimension of time and scalehowever there are no published studies that concern the extractionof multiscale turbulent structures from the PIV measurement vec-tor eld For the highly three-dimensional ow elds like lobed jetmixing ows the convectional analysis of three-dimensional PIVmeasurementresultscannot reveal the contributionof the multiscalestructures to the mixing mechanisms successfully

The aim of this paper is to apply a new signal processing tech-nique called vector wavelet multiresolutionanalysis to analyze thethree-dimensionalmeasurement results of a high-resolutionstereo-scopic PIV system in the near eld for providing a fundamentalunderstanding of the multiscale vortical structures and why vortic-ity dynamics greatly impact the mixing process of lobed jet

Two-Dimensional Discrete Vector Wavelet TransformHere we consider analyzing a two-dimensional vector eld

f x1 x2 The simplest way of constructing a two-dimensional or-thogonal wavelet basis is to take the simple production of two one-dimensional orthogonalwavelet bases

9mIn1 n2 x1 x2 D 2iexclm Atildeiexcl2iexclm x1 iexcl n1

centAtilde

iexcl2iexclm x2 iexcl n2

cent(1)

where m denotesthe dilationindexand n1 andn2 representthe trans-lation index The oldest example of a function Atildex for which theAtildemnx constitutes an orthogonal basis is the Haar function con-structedlongbeforethe termwaveletwas coinedIn thepast10 yearsvarious orthogonalwavelet bases have been constructed for exam-ple Meyer basis Daubechies basis Coifman basis BattlendashLemariebasisBaylkinbasis and splinebasisetc They provideexcellentlo-calizationproperties both in physical space and frequency space Inthis study we use the Daubechies basis with index N D 20 which isnot only orthonormalbut also has smoothnessand compact supportto analyze the vector eld

The two-dimensionaldiscretevectorwavelet transformis de nedby

Wfm In1 n2 DX

i

X

j

fiexclx i

1 x j2

cent9mIn1n2

iexclx i

1 x j2

cent(2)

As in the Fourier series dilation by larger m compresses the vector eld on the x and y axes Altering n1 and n2 has the effect of slidingthe vector along the x and y axes respectively

The reconstructionof the original vector eld can be achieved by

fx1 x2 DX

m

X

n1

X

n2

Wf mIn1n29m In1 n2 x1 x2 (3)

Vector Wavelet Multiresolution TechniqueWavelet multiresolution analysis was formulated in the fall of

1986 and was applied to image processing in 1989 Since thenresearchers have been making widespread use of wavelet multires-olution analysis The goal of the wavelet multiresolution analysisis to get a representation of a signal or an image that is written ina parsimonious manner as a sum of its essential components Thatis a parsimonious representation of a signal or an image will pre-serve the interestingfeaturesof the originalsignalor image but willexpress the signal or image in terms of a relatively small set of co-ef cients It is a well-known fact that the vector eld often includestoo much information for multiscale vision processingA multires-olution algorithm can process fewer data by selecting the relevantdetails that are necessary to perform a particular recognition taskCoarse-to- ne searches process rst a low-resolution vector eldand then zoom selectively into ner scales information

In mathematics the multiresolutionanalysis consists of a nestedset of linear function spaces V j with the resolution of functions inincreasing with j More precisely the closed subspaces V j satisfy

cent cent cent V2 frac12 V1 frac12 V0 frac12 Viexcl1 frac12 Viexcl2 cent cent cent (4)

with[

j 2 Z

V j D L2lt2

j 2 Z

V j D f0g (5)

The basis functions for the subspaces V j are called scaling func-tions of the multiresolution analysis For every j 2 Z de ne thewavelet spaces W j to be the orthogonal complement of in V j iexcl 1 ofV j We have

V j iexcl 1 D V j copy W j (6)

where copy represents the orthogonal space sum and

W j W j 0 if j 6D j 0 (7)

that is any function in V j iexcl 1 can be written as the sum of a uniquefunctionin V j anda uniquefunctionin W j In L2lt2 the orthogonalbasis for W j is the family of wavelets that is de ned Thus L2lt2can be decomposed into mutually orthogonal subspaces and can bewritten as

L2lt2 D copyj 2 Z

W j (8)

In this study the procedure of the vector wavelet multiresolutionanalysis can be summarized in two steps

1) Compute the wavelet coef cients of vector data based on thediscrete wavelet transform of Eq (2)

2) The inverse wavelet transform of Eq (3) is applied to waveletcoef cients at each wavelet level and vector components are deter-mined at each level or scale

Of course a sum of these essential vector components can re-cover the original vector eld The vector wavelet multiresolutionanalysis can perform an extraction of the multiscale structures anddecomposethe vector eld in both Fourier and physicalspacesThistechnique is unique in terms of its capability to separate turbulencestructures of different scales

Experimental Setup and Stereoscopic PIV SystemA test lobednozzlewith six lobes as shownin Fig 1 is used in the

presentstudyThe width of each lobe is 6 mm and the heightof eachlobe is 15 mm (H D 15 mm) The inner and outer penetrationangles

LI ET AL 1039

of the lobed structures are microin D 22 deg and microout D 14 deg respec-tivelyThe equivalentnozzlediameter is designed to be D D 40 mmThe z axis is taken as the directionof the main stream the x ndash y planeis perpendicular to the z axis and is taken as the cross plane of thelobed jet The velocity components in x y and z directions are uv and w respectively

Figure 2 shows the air jet experimental setup used in the presentstudy A centrifugal compressor was used to supply airjet ows Acylindrical plenum chamber with honeycomb structures was usedto settle the air ow Through a convergent connection (convergentratio is about 501) the air ow is exhausted from the test nozzles

The velocity of the airjet exhausting from the test nozzle can beadjusted and the core jet velocity U0 was set at about 20 ms inthe present study The Reynolds number of the jet ow is about60000 based on the equivalent nozzle diameter D and the core jetvelocity A seeding generator which is composed by an air com-pressor and several Laskin nozzles was used to generate 2ndash3 sup1mDi-2-EthylHexyl-Sebact droplets as tracer particles in the jet owAccording to Raffel et al19 the time response of 1 raquo 5 sup1m oildroplets in an air ow is about 00004 s Therefore the expectedfrequency response of the tracer particles used in our experiment is2500 Hz

Fig 1 Test lobed nozzle

Fig 2 Airjet experimental setup

Fig 3 Schematic of the stereoscopic PIV system

Figure 3 shows the schematic of the stereoscopic PIV systemused in the present study The objective jet mixing ows were il-luminated by a double-pulsed NdYAG laser set (New Wave 50-mJpulse) with the laser sheet thickness being about 2 mm Thedouble-pulsed NdYAG laser set can supply the pulsed laser at thefrequencyof 15 Hz The time intervalbetween the two pulsed illumi-nationswas settledas 30 sup1s Two high-resolution(1008pound 1016pix-els) cross-correlation charge-coupled device (CCD) cameras (TSIPIVCAM10-30) were used to do stereoscopicPIV image recordingThe two CCD cameras were arranged in an angular displacementcon gurationto get a bigoverlappedviewTo have the measurement eld focused on the image planes perfectly tilt-axis mounts wereinstalled between the camera bodies and lenses and the lenses andcamera bodies were adjusted to satisfy the Scheimp ug conditionIn thepresentstudy the distancebetween the illuminatinglaser sheetand image recording plane of the CCD camera is about 650 mmand the angle between the view axes of the two cameras is about 50deg For such an arrangement the size of the overlappedview of thetwo image recording cameras for stereoscopicPIV system is about80 pound 80 mm

The two-dimensional particle image displacements in every im-age planes were calculated separately by using Hierarchical Re-cursive PIV (HR-PIV) software20 The Hierarchical Recursive PIVsoftware is basedon a hierarchicalrecursiveprocessof conventionalspatial correlationoperation with offsetting of the displacement es-timatedby the former iteration step and hierarchicalreductionof theinterrogation window size and search distance in the next iterationstep Because 32 pound 32 pixel windows and 50 overlapping gridswere used for the nal step of the recursive correlation operationthe spatial resolution of the present stereoscopicPIV measurementis expected to be about 2 pound 2 pound 2 mm

According to the comparison of the simultaneous measurementresults of our stereoscopicPIV system and a LDV system the errorlevel of the instantaneousvelocitydata obtainedby the stereoscopicPIV system was less than 20

Results and DiscussionInstantaneous Three-Dimensional Multiscale Velocity Fields

To gain insight into the multiscale ow structuresvector waveletmultiresolution analysis is applied to the three-dimensional mea-surement results of PIV In the present study the measured three ve-locity components of 64 pound 64 are used The instantaneous velocityvector ux y is rst decomposed into three compositionsui x ywith different wavelet levels or scales where i represents the scaleThen three velocity vector compositions ui x y within differentscale ranges are produced based on vector wavelet multiresolutionanalysis To describe the frequency character of vector componentat each wavelet level the central scales of three velocity vectorcompositions were determined by Fourier transform The velocityvector composition of wavelet level 1 which corresponds to thecentral scale of 16 mm is employed to describe the large-scale ow

1040 LI ET AL

structureThe sum of velocityvectorcompositionsof wavelet levels2 and 3 which corresponds to the central scale of 6 mm constructsthe smaller-scale ow structure Of course the measured velocityvector ux y can be written as the sum of three velocity vectorcompositions ui x y that is

ux y D3X

i D 1

ui x y (9)

Figure 4a shows an instantaneousvelocity vectors of the stereo-scopicPIV measurementin the crossplane [(x y-planeview] over-lapping on the correspondingthe contour of the axial velocity com-ponent at a downstream location of z=D D 05 The monochromemappings have been assigned to the value of axial velocity com-ponent and the highest concentrationis displayed as white and thelowest as a black These are the original data before the waveletdecomposition The contour of the axial velocity component ex-hibits the geometry of the lobed nozzle approximatelySeven well-de ned peaks representingthe six lobes and the central core can beclearlyobservedThe vectorplot shows instantaneousirregular owstructures that imply the multiscale structures The larger velocitycomponent appears in the lobe regions The irregular streamwisevortices can be seen to be in the same con guration as the trailing-edge geometry of the lobed nozzle

The analysis results of the instantaneous velocity vectors of thestereoscopicPIV measurement(Fig 4a) basedon the wavelet vectormultiresolution technique are shown in Figs 4b and 4c in whichthe two different scale components of instantaneous velocity eldcan be seen It has been veri ed that the two decomposed instan-taneous vector components can be summed to obtain the originalvector eld of Fig 4a It provides a validation for the present dataanalysis technique Figure 4b displays the large-scale structures ofthe original velocity eld with a central scale of 16 mm The contourof the axial velocity component exhibits well-de ned geometry ofthe lobed nozzle The vector plot shows clearly six pairs of large-scale counter-rotatingstreamwise vortices around either side of thelobe peaks This location is the region of the formationand intensi -cation processesof the large-scalestreamwise vorticesgeneratedbythe lobed nozzle around the lobe peak regionsThese vorticescorre-spond quite well to the irregular vortices that appear in Fig 4a Thevector plot also clearly reveals that the cross-stream ow expandsindeed outward along the lobes and ambient ow ejects inwardin the lobe troughs which results in the generation of large-scalestreamwise vortices From the velocity vector plot the new vorticescan be clearlyobservednear the lobe troughswhich are also formedas a result of lobed nozzle around the lobe trough regionsThey alsoplay a important role in the enhance mixing process The insignif-icant differences between Figs 4a and 4b imply that large-scalestructures dominate the ow eld near the nozzle The ow struc-

Fig 4a Instantaneous velocity eld of the stereoscopic PIV measure-ment results in the cross plane of zD = 05

Fig 4b Instantaneous velocity eld with the central scale of 16 mm inthe cross plane of zD = 05

Fig 4c Instantaneous velocity eld with the central scale of 6 mm inthe cross plane of zD = 05

tures with a central scale of 6 mm are shown in Fig 4c The velocityvector indicates the presence of the small-scale streamwise vorticesthat appeared around the six lobe positionsBy comparing Figs 4band 4c one nds that some of these small-scale vortices are con-tained in the streamwise vortices The larger velocity vectors canbe found in the lobe regions which enhance the small-scale mixingprocess The contour of the axial velocity component indicates thatthe alternatepositiveand negativepeaksalso appeararoundthe edgepositionof the lobed nozzle within the central scale of 6 mm It sug-gests that these peaks imply the existence of the KelvinndashHelmholtzor normal vorticeswhose axes are inclinedor aligned to the stream-wise direction because the normal vortex is formed as result of anaxial velocity difference and has a small scale at the cross-streamplane It can be seen that the shape of KelvinndashHelmholtz or normalvortex ring has almost the same geometry as the trailing edge ofthe lobed nozzle as it is expected Such structure features cannot beextracted by traditional techniques

Figure 5a shows the stereoscopic PIV measurement results ofthe cross plane at z=D D 1 From distributionsof the instantaneousvelocity vector eld and contour of axial velocity the geometry ofthe lobed nozzle can also be identi ed The large-scalestructuresofthe originalvelocity eld with a central scale of 16 mm are shown inFig 5b The vector plot clearly reveals that the streamwise vorticesor large-scale structures have spread outward along the lobes Theextent of the vortices is slightly beyond the area over which data

LI ET AL 1041

Fig 5a Instantaneous velocity eld of the stereoscopic PIV measure-ment results in the cross plane of zD = 1

Fig 5b Instantaneous velocity eld with the central scale of 16 mm inthe cross plane of zD = 1

Fig 5c Instantaneous velocity eld with the central scale of 6 mm inthe cross plane of zD = 1

Fig 5d Zoomed view of instantaneous velocity eld at two locationsof lobes with the central scale of 6 mm in the cross plane of zD = 1(Arrows show the positions of small-scale vortices)

were taken but still mostly visible From the vector plot three newcounter-rotatingvortex pairs can be clearly seen in the lobe troughregionsEach counter-rotatingpair representsthe legs fromadjacenthorseshoevortexstructuresThough the horseshoevortexstructureshave been visualized in previous studies710 this result providesunquestionableevidenceof their existencequantitativelySuch newcounter-rotating vortex pairs cannot be observed from the originalvelocity eld of Fig 5a The contour of velocity shows that themagnitudes of the axial velocity component decreased in the sixlobe regions compared to the preceding locationThe instantaneousvelocity eld with a central scale of 6 mm is plotted in Fig 5c Thezoomed view on instantaneous velocity vectors at two locations oflobesis showninFig 5dThe velocityvectorindicatesthatthe small-scale streamwise vortices such as indicated by arrows in Fig 5dhave strongly spread outward along the lobes By comparing thepreceding location the magnitudes of velocity vectors increase inthe lobe regions It is found that some of these small-scale vorticesare contained in the streamwise vortices It suggests that the small-scale vortices obtain the energy from the large-scale structures andenhancethe small-scalemixingprocessaroundthe loberegionsThecontour of the axial velocity component indicates that the alternatepositive and negative peaks also spread outward along the lobes Itimplies that the KelvinndashHelmholtz or normal vortex ring expendsradially outward as it is expected

The measured instantaneous velocity eld of the cross plane atz=D D 15 as shown in Fig 6a exhibits more complex structuresthan that in the upstream cross plane of z=D D 10 The instanta-neous large-scale velocity eld with a central scale of 16 mm ispresented in Fig 6b The contour of velocity indicates that the ax-ial velocity components have spread outward along the lobes Thevelocity vector plot shows that the large-scale streamwise vorticesbegin to exhibit signs of breakdownand spread outward The horse-shoes vortices disappear and develop to larger-scalevorticesThesestreamwise vortices can be clearly seen around the position of lobeand perform the signi cant large-scale turbulentmixing The largerambient owejects inwardcanbeobservedas a resultof thepresenceof these streamwise vortices Figure 6c shows the ow structureswith a central scale of 6 mm The small-scale streamwise vorticesalmost distribute in the whole measured ow eld They are muchmore activeat the positionof the trailing-edgegeometryof the lobednozzle The turbulenceappears to become more intense suggestingthat the vortex merging process is an unsteady interaction and thebreakdown of the large-scale streamwise vortices is accompaniedby a signi cant increase in turbulent intense and small-scale vor-tices The contourof the axial velocity component indicates that thealternatepositiveand negativepeaksdecreaseIt means the decreaseof small-scale spanwise vortices

At a farther downstream location of z=D D 4 the stereoscopicPIV measurement results are shown in Fig 7a The geometry of

1042 LI ET AL

Fig 6a Instantaneous velocity eld of the stereoscopic PIV measure-ment results in the cross plane of zD = 15

Fig 6b Instantaneous velocity eld with the central scale of 16 mm inthe cross plane of zD = 15

Fig 6c Instantaneous velocity eld with the central scale of 6 mm inthe cross plane of zD = 15

Fig 7a Instantaneous velocity eld of the stereoscopic PIV measure-ment results in the cross plane of zD = 4

Fig 7b Instantaneous velocity eld with the central scale of 16 mm inthe cross plane of zD = 4

Fig 7c Instantaneous velocity eld with the central scale of 6 mm inthe cross plane of zD = 4

LI ET AL 1043

the lobed nozzle almost cannot be identi ed from the instantaneousvelocity eld The higher magnitude of the axial velocity compo-nent is found in the center of the jet ow and decreases rapidly asa result of the intensive mixing of the core jet ow with ambient ow Figure 7b gives a clear picture of the large-scalestructurewitha central scale of 16 mm From the distribution of velocity vec-tors the large-scale vortices almost disappear and the streamwisevortices can be observed in the almost whole measured ow eldEspecially they are much more active in the jet core region Thevelocity eld with a central scale of 6 mm is given in Fig 7c Thevelocity vectors indicate that the active streamwise small-scale vor-tices are distributedin the whole measured ow eld The contourofthe axial velocitycomponentindicatesthat the alternatepositiveandnegative peaks rapidly decrease and become weak It implies thatthe spanwise vortex ring has broken down into many disconnectedvortical tubes

Instantaneous Multiscale Streamwise VorticityTo study the evolutionof multiscalestreamwisevorticesquantita-

tively the instantaneousstreamwise vorticity was calculated basedon the measured velocity data of the stereoscopic PIV and vec-tor wavelet multiresolutionanalysis The normalized instantaneouscomponent of streamwise vorticity Nzi at scale i can be de ned interms of the derivatives of the instantaneousvelocity components

Nzi DD

U0

sup3vi

xiexcl ui

y

acute(10)

where i stands for the scaleContours of the measured instantaneous streamwise vorticity at

z=D D 05 are shown in Fig 8a The monochrome mappings havebeen assigned to the vorticity values the highest concentration isdisplayed as white and the lowest as black and the positive andnegative vorticity are simultaneously denoted by solid and dashedlines respectively The alternative positive and negative peaks canbe clearly seen around the lobe edge positions which indicate thepairs of streamwise vortices However it is dif cult to identify thesmaller-scale vorticity using the measured instantaneous vorticityFigures 8b and 8c display the distribution of multiscale vorticity inthe lobed mixing turbulent jet The alternative positive and nega-tive vorticities aligned with the lobe which represent six counter-rotating large-scale streamwise vortex pairs located approximatelyon either side of the lobe peaks can be clearly observed in Fig 8bThese large-scale streamwise vortices as expected correspond tovortices appearing in Fig 8a Figure 8c shows the distributionof thesmall-scale vorticity with a central scale of 6 mm The alternativepositive and negative peaks can be clearly seen around the lobe po-sitions which indicate pairs of the small-scale streamwise vortices

At z=D D 1 the distributionof original instantaneousstreamwisevorticity (Fig 9a) indicates that the six large-scale streamwise vor-

Fig 8a Instantaneous streamwise vorticity distributions of the stereo-scopic PIV measurement results in the cross plane of zD = 05

Fig 8b Instantaneous streamwise vorticity distributions at the centralscale of 16 mm in the cross plane of zD = 05

Fig 8c Instantaneous streamwise vorticity distributions at the centralscale of 6 mm in the cross plane of zD = 05

tex pairsbreak into several smaller vorticesSeveral new streamwisevortex pairs can be found in the lobe trough regions which maybeimplies the existence of horseshoe vortical structures Contours ofthe large-scale vorticity with a central scale of 16 mm as shownin Fig 9b indicate that six counter-rotatinglarge-scale streamwisevortex pairs spread outward along the lobes The alternative posi-tive and negative vorticity which represent horseshoevortices canbe clearly observed in the lobe trough regions The distribution ofthe small-scale vorticity with a central scale of 6 mm is presentedin Fig 9c The stronger alternative positive and negative vortic-ity is distributed near the lobe positions and lobe trough regionswhich indicates more active small-scale streamwise vortex pairsThese small-scale streamwise vortices grow up and strongly spreadoutward along the lobes compared to the preceding location

Increasing the downstream distance to z=D D 15 as shown inFig 10a the distributionof the measured instantaneousstreamwisevorticityexhibitsmanyalternativepositiveandnegativepeaksTheyimply the multiscale of streamwise vortex pairs The geometry ofthe lobed nozzle almost cannot be identi ed However Fig 10bdisplays only the distribution of large-scale streamwise vorticitySeveral pairs of large-scale streamwise vortices can be clearly seenat the position of lobe The distribution of the small-scale vorticitywith a central scale of 6 mm is shown in Fig 10c As indicated in theprecedingvelocityvectorplotmanypositiveandnegativepeaksthatimply the small-scale vortices appear in the whole measured eld

1044 LI ET AL

Fig 9a Instantaneous streamwise vorticity distributions of the stereo-scopic PIV measurement results in the cross plane of zD = 1

Fig 9b Instantaneous streamwise vorticity distributionsat the centralscale of 16 mm in the cross plane of zD = 1

Fig 9c Instantaneous streamwise vorticity distributions at the centralscale of 6 mm in the cross plane of zD = 1

Fig10a Instantaneousstreamwisevorticity distributionsof the stereo-scopic PIV measurement results in the cross plane of zD = 15

Fig 10b Instantaneous streamwise vorticity distributions at thecentral scale of 16 mm in the cross plane of zD = 15

Fig10c Instantaneousstreamwise vorticity distributionsat the centralscale of 6 mm in the cross plane of zD = 15

LI ET AL 1045

Fig11a Instantaneousstreamwise vorticitydistributionsof thestereo-scopic PIV measurement results in the cross plane of zD = 4

Fig 11b Instantaneous streamwise vorticity distributions at thecentral scale of 16 mm in the cross plane of zD = 4

Fig11c Instantaneousstreamwise vorticitydistributionsat thecentralscale of 6 mm in the cross plane of zD = 4

Ata fartherdownstreamlocationof z=D D 4 fromthedistributionof the measuredinstantaneousstreamwisevorticityin Fig 11a manypositive and negativepeaks mainly distribute in the region of the jetcore The results of the wavelet multiresolutionanalysis are shownin Figs 11b and 11c Both the stronger large-scale and small-scalestreamwise vorticity are concentrated in the region of jet core Themaximum vorticity value of these streamwise vortices is found bedecreased when compared to that of z=D D 15

From the preceding discussion at different downstream crossplanes it can be seen that as the downstream distance is increasedthe size and strength of the large- and small-scale streamwise vor-tices generated by the lobed nozzle rst grow up and appear at thelobe regions Then they decay and beak up rapidly and can be onlyobserved in the region of jet core

ConclusionsTo reveal the three-dimensional multiscale structure features of

the lobed jet mixing ow vector wavelet multiresolutiontechniquewas developed to analyze the three-dimensional measurement re-sults of a high-resolutionstereoscopicPIV system in thispaperThistechnique is unique in its capability to decompose the vector datain both Fourier and physical spaces The following main results aresummarized

1) The instantaneous three-dimensional velocity vectors weresuccessfullydecomposed into large- and small-scale velocity eldsbased on the wavelet vector multiresolutionanalysis It suggests theexistence of very strong multiscale turbulent structures in the lobedjet mixing

2) The large-scale cross-stream ow expands outward along thelobes and the large-scale ambient ow ejects inward in the lobetroughs

3) Within the central scale range of 16 mm the instantaneouspairs of large-scale streamwise vortices and horseshoe vortices canbe clearly observed around the lobe peak regions and lobe troughsrespectively at z=d D 05 and 1 After z=d D 15 the large-scalestreamwise vortices begin to break down and spread outward

4) The stronger small-scale streamwise vortices have been con- rmed to exist in the lobe regions at z=d D 05 and 1 These vortices rst become intense and spread outward along the lobes and thendevelop to the whole measured ow eld after z=d D 15 It indicatesthat the small-scale streamwise vortices also play a important rolein the enhance mixing process

5) The stronger alternate positive and negative peaks of small-scale axial velocity component are found around the trailing edgeof the lobed nozzle at the location of z=D D 05 1 and 15 withinthe central scale of 6 mm These peaks indicate the existence of theKelvinndashHelmholtz or normal vortices

6) The stronger large- and small-scale streamwise vortices andaxial velocity component only appear in the center region of jet atz=D D 4

References1Crouch R W Coughlin C L and Paynter G C ldquoNozzle Exit Flow

Pro le Shaping for Jet Noise ReductionrdquoJournal of Aircraft Vol 14 No 91977 pp 860ndash867

2Paterson R W ldquoTurbofan Mixer Nozzle Flow FieldmdashA BenchmarkExperimental StudyrdquoJournal of Engineering for Gas Turbines and PowerVol 106 No 2 1984 pp 692ndash698

3Werle M J Paterson R W and Presz W M Jr ldquoFlow Structure ina Periodic Axial Vortex Arrayrdquo AIAA Paper 87-0610 Jan 1987

4Eckerle W A Sheibani H and Awad J ldquoExperimental Measurementof the Vortex Development Downstream of a Lobed Forced Mixerrdquo Journalof Engineering for Gas Turbines and Power Vol 114 No 1 1992 pp 63ndash

715Ukeiley L Varghese M Glauser M and Valentine D ldquoMultifrac-

tal Analysis of a Lobed Mixer Flow eld Utilizing the Proper OrthogonalDecompositionrdquo AIAA Journal Vol 30 No 5 1992 pp 1260ndash1267

6Ukeiley L Glauser M and Wick D ldquoDownstream Evolution of PODEigenfunctions in a Lobed Mixerrdquo AIAA Journal Vol 31 No 8 1993pp 1392ndash1397

7McCormick D C and Bennett J C Jr ldquoVortical and TurbulentStruc-ture of a Lobed Mixer Free Shear Layerrdquo AIAA Journal Vol 32 No 91994 pp 1852ndash1859

8Belovich V M Samimy M and Reeder M F ldquoDual Stream Axisym-metric Mixing in the Presence of Axial Vorticityrdquo Journalof PropulsionandPower Vol 12 No 1 1996 pp 178ndash185

1046 LI ET AL

9Belovich V M and Samimy M ldquoMixing Process in a Coaxial Geom-etry with a Central Lobed Mixing Nozzlerdquo AIAA Journal Vol 35 No 51997 pp 838ndash84l

10Hu H Saga T and Kobayashi T ldquoResearch on the Vortical and Tur-bulentStructures in the Lobed Jet Flowby Using LIFand PIVrdquo MeasurementScience and Technology Vol 11 No 6 2000 pp 698ndash711

11Hu H Saga T Kobayashi T and Taniguchi N ldquoStereoscopic PIVMeasurement of a Lobed Jet Mixing Flowrdquo Laser Techniques for FluidMechanics edited by R J Adrian D F G Durao M V Heitor M MaedaC Tropea and J H Whitelaw Springer-Verlag Berlin 2002

12Li H and Nozaki T ldquoWavelet Analysis for the Plane Turbulent Jet(Analysis of Large Eddy Structure)rdquo JSME International Journal Fluidsand Thermal Engineering Vol 38 No 4 1995 pp 525ndash531

13Li H ldquoIdenti cation of Coherent Structure in Turbulent Shear FlowwithWavelet CorrelationAnalysisrdquo JournalofFluidsEngineering Vol 120No 4 1998 pp 778ndash785

14Charles M ldquoAnalysis of Turbulence in the Orthonormal Wavelet Rep-resentationrdquo Journal of Fluid Mechanics Vol 232 1991 pp 469ndash520

15Staszewski W J Worden K and Rees J M ldquoAnalysis of WindFluctuations Using the Orthogonal Wavelet Transformrdquo Applied Scienti cResearch Vol 59 No 23 1997 pp 205ndash218

16Li H Takei M Ochi M Saito Y and Horii K ldquoEduction of Un-steady Structure in a Turbulent Jet by Using of Continuous and DiscreteWavelet Transformsrdquo Transactions of the Japan Society for Aeronauticaland Space Sciences Vol 42 No 138 2000 pp 39ndash44

17Li H Takei M Ochi M Saito Y and Horii K ldquoWavelet Mul-tiresolution Analysis Applied to Coherent Structure Eduction of a TurbulentJetrdquo Transactionsof the Japan Society for Aeronauticaland Space SciencesVol 42 No 142 2001 pp 203ndash207

18Li H Takei M Ochi M Saito Y and Horii K ldquoApplication ofTwo-Dimensional Orthogonal Wavelets to Multiresolution Image Analysisof a Turbulent Jetrdquo Transactions of the Japan Society for Aeronautical andSpace Sciences Vol 42 No 137 1999 pp 120ndash127

19Raffel MWillertC andKompenhans J Particle ImageVelocimetrySpringer-Verlag Berlin 1998 p 249

20Hu H Saga T Kobayashi T Taniguchi N and Segawa S ldquoTheSpatial Resolution Improvement of PIV Result by Using HierarchicalRecursive Operationrdquo Proceedings of the 9th International Symposium onFlow Visualization EdinburghScotland UK 2000 Paper 137 pp 1ndash12

W R LempertGuest Associate Editor