POD, LSE and Wavelet decomposition: Literature Review

by Stanislav Gordeyev

Department of Aerospace and Mechanical Engineering

University of Notre DameNotre Dame, IN

1 Introduction

The problem of turbulent uid ow remains the outstand-ing unsolved problem of classical physics. Motivated bythe fact that turbulence is the rule rather than excep-tion in ows of technological importance, there has beenextensive theoretical and experimental work to explainthe underlying physical processes. Osborne Reynolds [1]was the �rst who in 1894 recognized the importance ofstudying turbulent ows and introduced the idea of de-composing the velocity into time mean and uctuatingcomponents. The so-called Reynolds-averaged Navier-Stokes equation has traditionally been the starting pointfor many investigations.

Contrasting the idea that the turbulence is completelyrandom, numerous experiments over the last decades haveshown the existence of large-scale quasi-organized vorti-cal structures in a variety of free and wall bounded turbu-lent ows. These so-called coherent structures (CS) ap-pear to be dynamically important and play a key role indetermining the macrocharacteristics of the ow such asmass, momentum energy and heat transports, entrainingcombustion, chemical reactions, drag and aerodynamicnoise generation. Theodorsen [2] and Townsend [3] dis-cussed an existence of organized motion (eddies) in tur-bulent shear layers. A variety of visualization techniqueswere applied in order to �nd and investigate a large-scalestructures in ows. One of the �rst visualizations weredone by Brown & Roshko [4]. They investigated a planeturbulent mixing layer using two gases with di�erent re-fraction properties. Shadowgraphs have reveled a well-'organized' almost two-dimensional vortex-like coherentstructures, convecting at nearly constant speed. Thosestructures were similar to the instability structures in alaminar shear layer, but have attributes typically associ-ated with turbulent ow: 'randomness' and broad energyspectrum. Large-scale structures were found not changedby changes in Re number, it a�ects only in producing�ne-scale turbulence, superimposed on the structures. Apairing process was suggested as a mechanism of creat-ing the CS. Similar observations were made by Winant& Drowand [5] using dyes in water tunnel in the initial

vortex layer at moderate Re numbers.

The importance of coherent structures in terms of theircharacteristics and dynamical roles was and still is un-der a serious discussion. Cantwell [18] in 1981 revieweda research of organized motion in di�erent types of tur-bulent ows, discussed possible ways like discrete vortexdynamics and large eddy simulation to investigate dy-namics of coherent structures, their applications in tran-sition and control of mixing. Hussain [35], [38], [34] haveintroduced a de�nition of coherent structure as a con-nected turbulent uid mass with instantaneously phase-correlated vorticity over its spatial extent and emphasizedvorticity as a characteristic measure of CS. He proposed atriple-decomposition of the ow into a mean ow, CS andincoherent turbulence, discussed a possible origins of CS,their time evolution as a strange attractor in phase space,called for importance of dynamics, profound practical sig-ni�cance, understanding of turbulent measurements andcontrol.

In order to qualitatively describe these structures, aseveral experimental techniques were developed to extractCS. Conditional sampling techniques (Kaplan [6], Anto-nia [10] ) use a conditional criterion in order to investigatestructures. A linear-stochastic estimation (LSE) ( Adrian[11] ) uses 2-point second-order correlation functions interms of conditional ow patterns, other techniques likeVITA technique [29] were used to investigate a spatialshape of CS.

The disadvantage of all conditional techniques is thedependence on a conditional criteria. The �rst rigoroustheoretical approach to investigating coherent structureswas developed by John Lumley [13],[14] and is termed theProper Orthogonal Decomposition (POD). This uncondi-tional technique uses a second-order statistics in orderto extract organized large-scale structures from turbulent ows. The mathematical basic of POD is a Karhunen-Lo�eve expansion ( Karhunen [16], Lo�eve [17] )

In 80's a new technique of extracting CS - wavelettransformation ([4]) was developed. This technique uti-lizes so-called localized waveforms, which possess localproperties in both Fourier and physical spaces and ap-pears as a generalization of Fourier transformation. It

1

provides a local information about spectrum, scales andpositions of events in ows. Because this technique doesnot require a priori any information about structures, nowit is widely used in analysis of intermittent events andstructures in turbulent ows.Below three major techniques, POD, LSE and Wavelet

Transformation will be discussed. Brief theories, as wellas application to aerodynamical problems are provided.Also VITA, WAG and Windowed Fourier transformationtechniques are brie y mentioned.

2

2 Proper Orthogonal Decomposi-tion (POD)

2.1 Theory and Properties

The body of POD technique is Karhunen-Lo�eve (KL) de-composition. The idea is to describe a given statisticalensemble with the minimum number of modes. Let u(t)be a random generalized process with t as a parameter(spatial or temporal). We would like to �nd a determin-istic function �(t) with a structure typical of the membersof the ensemble in some sense. In other words, a func-tional in the following form needs to be maximized :

(R(t; t0); �(t)��(t0))=(�(t); �(t)) = � � 0 (1)

where R(t; t0) = Efu(t)�u(t0)g is the autocorrelation func-tion of u(t) and u(t0), Eff(u)g is the average or expectedvalue of f(u), (�; �) is a scalar product and the asteriskdenotes a complex conjugate. The classical methods ofthe calculus of variations gives the �nal result for �, ifR(t; t0) is an integrable functionZ

R(t; t0)��(t0)dt0 = ��(t) (2)

The solution of (2) forms a complete set of a square-integrable orthogonal functions �n(t) with associatedeigenvalues �n. It was shown that any ensemble of ran-dom generalized functions can be represented by a seriesof orthonormal functions with random coe�cients, thecoe�cients being uncorrelated with one another:

u =

1Xn=1

an�n ; Efanamg = �nm�m (3)

These functions are the eigenfunctions of the autocorre-lation with positive eigenvalues. The eigenvalues are theenergy of the various eigenfunctions (modes). Moreover,since the modes were determined by maximizing � (theenergy of a mode), the series (3) converges as rapidly aspossible. This means that it gives rise to an optimal setof basis functions from all possible sets.If the averaging is performed in time domain, then

u(t;x) (here t is a time parameter) can be representedas follows:

u(t;x) =

1Xn=1

an(t)�n(x); (4)

where a's are temporal coe�cients and �'s are the spatialeigenfunctions or modes.The transformation (2)-(3) is POD transformation.Because of discretization of experimental data, a vec-

tor form of POD is widely used. In this case u becomesan ensemble of �nite-dimensional vectors, the correlation

function R is a correlation matrix and the eigenfunctionsare called eigenvectors.

On practical grounds, (3) (or(4)) usually is representedonly in terms of a �nite set of functions,

u � uL =

LXl=1

al�l (5)

Brie y, some important properties of POD:

1. The generalized coordinate system de�ned by theeigenfunctions of the correlation matrix is optimal inthe sense that the mean-square error resulting froma �nite representation of the process is minimized.That is for any �xed L:

RL =

Z T

0

[u(t)�LX

n=1

an�n(t)]2dt! min (6)

i� �n(t) are KL eigenfunctions of (2).

2. The random variables appearing in an expansion ofthe kind given by the equation (3) are orthonormalif and only if the orthonormal functions and the con-stants are respectively the eigenfunctions and theeigenvalues of the correlation matrix.

3. In addition to the mean-square error minimizingproperty, the POD has some additional desirableproperties. Of these, the minimum representationentropy property and satisfaction of the continuityequations are worth mentioning.

4. Algazi and Sakrison [43] showed that Karhunen-Lo�eve expansion is optimal not only in terms of mini-mizing mean-square error between the signal and it'struncated representation (Property 1), but also min-imizes a number of modes to describe the signal fora given error.

5. For homogeneous directions POD modes are Fouriermodes [14].

In the case of large number of elements in second-order correlation tensor the direct method of �nding theeigenfunctions becomes practically impossible. Sirovich[36] pointed out that the temporal correlation matrixwill yield the same dominant spatial modes, while of-ten giving rise to a much smaller and computation-ally more tractable eigenproblem - the method of snap-shots. Mathematically, for a process u(t; x) instead of�nding a spatial two-point correlation matrix Rij =

1=MPM

m=1[u(xi; tm)u(xj ; tm)], where N is a number ofspatial points and solving (2) (N � N -matrix), one can

3

compute a temporal correlationM�M -matrix Amn overspatial averaging,

Amn =1

M

ZV

u(x; tm)u(x; tn)dx; (7)

where M is number of temporal snapshots and calculate�i(x) from um(x) = u(tm; x) series as

�i(x) =

MXn=1

bm;ium(x) (8)

where bm;i's are the solutions of the equation Ab = �b.UsuallyM � N and the computational cost of �nding �'scan be reduced dramatically. The method of snapshotsalso overcomes the di�culties associated with the largedata sets that accompany more than one dimension.The optimality of POD reduces the amount of informa-

tion required to represent statistically dependent data toa minimum. This crucial feature explains the wide usageof POD, also known as Karhunen-Lo�eve expansion in aprocess of analyzing data.In [50] the limitations of POD with temporal averaging

were discussed. It was shown that in this case the analy-sis uses only information that is close to a particular �nalstate of the system and thus cannot be used for the sys-tem which has a several �nal states. Also it was pointedout that the analysis de-emphasizes infrequent events, al-though they could be dynamically very important (burst-like events in a turbulent boundary layer). Alternative av-eraging techniques were proposed and shown to be moreinformative in terms of investigating the system dynam-ics. Delville [64] pointed out that POD technique can betreated as generalization of Fourier transform in inhomo-geneous direction.The Proper Orthogonal Decomposition was proposed

by Lumley, 1967, [13]. The mathematical backgroundbehind POD is essentially Karhunen-Lo�eve procedureKarhunen [16] (1946), Lo�eve [17] (1955). But themethod itself is known under a variety of names in dif-ferent �elds: Principal Component or Hotelling Analy-sis (Hotelling, 1953, [45]), Empirical Component Analy-sis (Lorenz, 1956, [44]), Quasiharmonic Modes (Brooksel at., 1988, [46]), Singular Value Decomposition (Goluband Van Loan, 1983, [47]), Empirical Eigenfunction De-composition (Sirovich, 1987, [36]) and others. Closelyrelated to this technique is factor analysis, which is usedin psychology and economics (Harman, 1960, [48]).From the mathematical point of view POD or KL ex-

pansion (2) is nothing else but a transformation whichdiagonalizes a given matrixR and brings it to a canonicalform R=ULV, where L is a diagonal matrix. Thereforethe roots of KL actually go into the middle of the lastcentury. A review of the early history of KL expansioncan be found in [51]. The mathematical content of KL

procedure is therefore classical and is contained in paperby Schmidt [49] (1907).

In homogeneous directions POD reduces to Fourier-Stiltjes integral and gives Fourier modes in CS expan-sion. These function are not well-suited to describe com-pact CS. Lumley [23] proposed a noise-shot method toreconstruct CS in these homogeneous direction. RecentlyBerkooz et al. [39] have proposed a usage of wavelets onhomogeneous directions.

Because of the large amount of computations requiredto �nd the eigenvectors, POD technique was virtually un-used until the middle of the century. Radical changescame with the appearance of powerful computers and de-velopment of e�cient algorithms to compute the eigen-functions (method of snapshots, [36]). Now POD (KLexpansion) is used extensively in the �elds of detection,estimation, pattern recognition, and image processing asan e�cient tool to store random processes, in systemcontrols. Proper Orthogonal Decomposition is used inconnection with stochastic turbulence problems (Lumley,1967, [13]). In that context, the associated eigenfunc-tions can be identi�ed with the characteristic eddies ofthe turbulence �eld [12]. A really good review of PODapplication in turbulence can be found in [15]. Brief the-ory of POD, as well as other techniques of extracting CSare presented.

2.2 Application to turbulent ows

POD extracts a complete set of orthogonal modes bymaximizing the energy of the modes. Any member ofstatistical ensemble can be expanded on this set, andthis series converges as fast as possible. Also POD givesthe eigenvalues, corresponding to the kinetic energy ofeach mode. But the serious requirements to experimen-tal equipment like an availability of big data storage andautomated computer-controlled experiments to obtain adetailed second-order statistics from the ow have de-layed the wide application of POD basically until the lastdecade of the century.

Historically the �rst attempts were applied to turbulentboundary layers. One of the reasons to look at the bound-ary layer was an existence of a strong peak of a produc-tion of turbulent energy on the outer edge of the viscoussublayer. Kline et al. [37] were �rst to observe stream-wise vortex pairs in the boundary layer near the wall andbursting events associated with them. Bakewell et al.[22]using experimentally obtained 2-point correlation matrix,have sketched dominant structures within the wall regionconsisting of randomly distributed counter-rotating eddypairs elongated in streamwise direction. They showedthat the sublayer and the adjacent wall region play an ac-tive role in the generation and preservation of a turbulentshear ow. Nonlinear mechanism of vortex stretching sug-

4

gests that linearized theories cannot provide an adequatedescription of the viscous sublayer. Later these structureswere identi�ed with burst-like events in the boundarylayer. Cantwell [18] provided a detailed discussion of theresearch of turbulent boundary layer. Aubry et al. [19]were able to model a turbulent boundary layer by expand-ing the instantaneous �led in experimentally founded as astreamwise rolls eigenfunctions. By truncation of the se-ries a low-dimensional system was obtained from Navier-Stokes equation via Galerkin procedure. This model rep-resented the dynamical behavior of the rolls and was an-alyzed by methods of dynamical systems theory. Themodel captured major aspects of the ejection and burst-ing events associated with streamwise vortex pairs. Thispaper appeared to be the �rst to provide a reasonablycoherent link between low-dimensional chaotic dynamicsand realistic turbulent open ow system. Spatio-temporalthree-dimensional structures in a numerically simulatedtransitional boundary layer (Re�2 = 283) was examinedby Rempfer & Fasel [65] using POD. �-shaped vorticeswere found to be the most energetic modes in the ow.They correspond to physically observed structures andresemble bursting events in fully turbulent boundary lay-ers. POD modes was argued to correspond to physicallyexisting structures, when the most energy is collected ina �rst mode.

Bounded Flows

Because of the experimental di�culties in obtainingsecond-order statistics from the ow, a number of compu-tational simulation results were used to apply POD tech-nique to. Moin & Moser [21] made direct numerical simu-lations of N-S equations in turbulent channel for 128 x 129x 128 grid points and Re=3200. The shot-noise expan-sion proposed by Lumley [23] was used to �nd the spatialshape of the CS. A dominant eddy was found to keep 76%of the turbulent kinetic energy. Similar computational re-search was done by Ball et al. [20] for 24 x 32 x 12 gridpoints, Re=1500. Snapshot technique (Sirovich [36]) wasapplied in order to extract the modes. The eigenfunctionswere found as rolls ( the most energetic mode ) or shearingmotions. First 10 modes captured 50% of the turbulentenergy. Dynamical behavior of structures was discussed.Rolls provide the mechanism for the transport in channel ows like turbulent 'bursts', while shearing modes weresuggested to relate to the instabilities in the ow. Gavri-dakis [61] used results of numerical simulation through asquare duct with Re = 4800 to get the correlation matrix.Two di�erent structures were found after applying POD.

Sirovich & Park [25] performed numerical simulationsof Rayleigh-Benard convection in a �nite box for 17 x17 x 17 grid points. POD technique, snapshot methodand group symmetry considerations were applied to in-vestigate CS. 10 classes of eigenfunction were discovered,�rst eigenmode being captured 60% of the energy (Howes

et al [26]). Numerically simulated ow in a square duct(Re = 4800, grid 767 x 127 x 127) was analyzed by PODin [61]. Two di�erent structures were found near 'walldomain' and 'corner domain'. Near wall it was a pairof high and low-speed streaks with the �rst POD modeseizing 28% of energy. In the corner domain a vortex pairwith common ow toward the corner with 43% of the to-tal energy was discovered. It was found that the energyin this region goes from the turbulence back to the mean ow. The ow in duct was found to have a greater degreeof organization.

Turbulent jets and shear ows

Payne & Lumley [12] used experimentally obtained 2-point diagonal correlation tensor to extract CS in theturbulent wake behind a circular cylinder. O�-diagonalelements were found from a mixing layer assumptions.Glauser et al. [31] investigated a large-scale vortex ring-like structure in the mixing layer in axisymmetric jet (Re � 110; 000) at x=D = 3 by POD and shot-noise de-composition. First 3 mode were found to contain almostall energy of the ow. Kirby et al. [32] used a snap-shot method for 2-dimensional large-eddy simulation ofaxisymmetric compressible jet ow on 240 x 80 pointgrid for Re � O(104). They found �rst 10 modes, whichhold 94% energy. A similar approach was used by Kirbyet al. [33] to investigate a simulated supersonic shear ow. Sirovich et al. [24] applied POD to the analysisof digitally imaged 2-dimensional gas concentration �eldsfrom the transitional region of axisymmetric jet. HerePOD was proposed basically as a methodology for analyz-ing and treating large experimental and numerical data.Ishikawa et al [59] applied POD and wavelet techniqueto investigate coherent structures in a turbulent mixinglayer. POD procedure revealed the �rst mode to be themost energetic one, while wavelet transform found low-frequency modulation of vorticity in this mode. Hilberget al [63] investigated large-scale structures existed in amixing layer in a narrow channel using a rake of probes.They applied both classical and snapshot POD techniquesand were able to identify the largest 2-dimensional mode.Snapshot POD for a conditionally averaged velocity �eldshowed that �rst two modes contain the largest portionof coherent vorticity. Delville [64] also considered planeshear layer to apply POD. Two rakes of hot-wire probeswere used to extract correlation matrix from the ow.Vectorial version of POD was found to give a better pre-sentation of structures in the ow. Di�erent aspects andproblems applying POD procedure were also discussed.An axisymmetric jet was investigated in [53]. PODmodesfor near-�eld pressure in inhomogeneous streamwise di-rection were obtained for a �rst time and observed asgrowing, saturating and decaying waves. Phase velocityfor a �rst few modes near saturation point were foundto be the same Up = 0:58Uj. Characteristic structure

5

was reconstructed using the shot-e�ect decomposition. Itwas observed that vortex pairings do not happen peri-odically and vortex tripling occur quite often. Citriniti[66] applied POD to a mixing layer in fully turbulent(Re = 80; 000) axisymmetric jet. 138 hot wires wereused to obtain the correlation data. POD modes wereprojected into instantaneous velocity �eld and a tempo-ral dynamics of the temporal coe�cients and their corre-sponding large-scale structures were investigated. Koppet al [60] investigated large-scale structure in far �eld ofa wake behind a cylinder (x=d = 420, Re = 1200). Theyused a rake of 8 x-wire probes to get instantaneous ve-locity �eld in cross-stream and spanwise directions. Theyapplied POD technique to extract �rst modes and used�rst two of them as a template for Pattern Recognition(PR) technique. The structures found via PR and PODwere similar and represented negative u- uctuations withoutward v- uctuation in cross-stream direction. In span-wise direction a double roller structure was discovered. A ow �eld in lober mixer (enhancing mixing device) wasconsidered by Ukeiley et al [52]. A rake of the probesin streamwise direction collected data on the correlationmatrix. First POD modes were projected back to instan-taneous ow �led to get information about multifractalnature of the ow.

POD also was in use in order to analyze turbulent-likemathematical equations. Chambers et al. [27] used theresults of numerical Monte-Carlo simulation of a forcedBurger's equation for di�erent Re numbers. A numberof modes were extracted from the numerical solution.The large-scale structures were found to be independentfrom Re number and exhibit viscous boundary layers nearthe walls. Sirovich & Rodriguez [28] examined Ginzburg-Landau equation in chaotic regime. Complete set of un-correlated functions was extracted and used as a basis forthe dynamical description of CS in the attractor set. First3-mode representation was found to simulate the system'sbehavior quite good. (Rodriguez & Sirovich [29])

Taylor-Couette ow: Tangborn et al. [30]

POD-based modeling

Minimal number of POD modes make it useful to buildlow-order dynamical models for di�erent applications.Using a truncated number of the POD modes as a ba-sis and projecting them to Navier-Stokes equations allowsomeone to reduce PDE down to a low-order system ofODE's. The rest of modes are modelled by a dissipativeterm. Analysis of the obtained system can shed a lightinto a dynamics of the system. Aubry et al. [19] wereable to model a turbulent boundary layer by expandingthe instantaneous �led in experimentally founded as astreamwise rolls eigenfunctions. The model captured ma-jor aspects of the ejection and bursting events associatedwith streamwise vortex pairs. Aling et al [53] proposedPOD-based technique to construct low-order dynamical

models for rapid thermal systems. Gunes et al [57] ap-plied POD technique to reconstruct a �rst six modes fornumerical time-dependent transitional free convection ina vertical channel. A low-order system of ODE's basedon the knowledge of the most energetic modes in the ow,was able to predict stable oscillations with correct ampli-tudes and frequencies. Same approach [58] was done toinvestigate a dynamics of transitional ow and heat trans-fer in periodically grooved channel. Ukeiley [67] appliedPOD technique in attempt to build a dynamical model ofa turbulent mixing layer. This mode was based on exper-imental data from two 12-probe rakes placed in the ow.Two-dimensional spanwise structures along with stream-wise vortical structures were extract form the ow. Themodel predicted correctly statistical distribution of en-ergy in cross-stream direction.It was pointed out by Lumley [23], that POD mode or

'characteristic eddy' represents coherent structure only ifit contains a dominant percentage of energy. In othercases, POD modes do not actually extract shape of CS,but rather give an optimum basis to decompose the ow.Poje & Lumley [62] proposed another technique to ob-

tain information about structures in ow. They decom-posed ow into mean, coherent modes and incoherentsmall-scale turbulence and applied energy method anal-ysis [9] to �nd the most energy containing large-scalemodes. The results compared well the with conventionalPOD technique.Both POD and wavelets were considered as new tools

in analyzing large-scale structures for wind engineering in[54]. Bienkiewicz et al [55] applied POD decomposition toanalyze pressure forces acting on a roof. For this purpose494 taps were used to obtain a spatial pressure correlationmatrix.Karhunen-Lo�eve expansion is in use in pattern recogni-

tion (Ash & Gardner [40], Fukunaga [41]), stochastic pro-cesses ( Ahmed & Goldstein [42]), meteorology; becauseof the optimality of the expansion K-L decomposition isused in data compaction and reduction.

6

References

[1] Reynolds, O., 'On the dynamical theory of turbulentincompressible viscous uids and the determinationof the criterion', Phil. Trans. R. Soc. London A, 186,1894, pp. 123-161

[2] Theodorsen T.,'Mechanism of turbulence', in Proc.2nd Midwestern Conf. on Fluid Mech. (1952), OhioState University, Columbia, OH

[3] Townsend A.,'The structure of turbulent shear ow',Cambridge University press, 1952

[4] Brown G., Roshko A.,'On density e�ects and largestructure in turbulent mixing layer', J. Fluid. Mech.(1974), 64, pp. 775-816

[5] Winant, Drowant, 'Vortex pairing: the mechanism ofturbulent mixing-layer growth at moderate Reynoldsnumber', J. Fluid Mech (1974), 63, p.237

[6] Kaplan ,'Conditional sampling technique', Turbu-lence in Fluids (1973), University of Missouri-Rolla

[7] Blackwelder, R.F., Kaplan, R.E, 'On the wall struc-ture of the turbulent boundary layer', Journal ofFluid Mechanics, 1976, 76, pp. 89-112

[8] Grossmann, A., Morlet, J., 'Decomposition of Hardyfunctions into square integrable wavelets of constantshape', SIAM J. Math Anal., 15, 1984, p. 723

[9] Orr, W.M., 'The stability or instability of steady mo-tions in a liquid', Proc. R. Irish Acad. A, 28, 1907,p. 259

[10] Antonia, R.,'Conditional sampling in turbulencemeasurements', Ann. Rev. Fluid Mech., 13, 1981,pp. 131-156

[11] Adrian R. ,'Conditional eddies in isotropic turbu-lence', Phys. Fluids (1979), 22(11), pp.2065-2070

[12] Payne, Lumley ,'Large eddy structure of the turbu-lent wake behind a circular cylinder', Phys. Fluids(1967), 10(9, part 2), pp. 194-196

[13] Lumley J.,'The structure of inhomogeneous turbu-lent ows', in Proceedings of the International Col-loquium on the Fine Scale Structure of the Atmo-sphere and its In uence on Radio Wave Propagation(ed. A.M.Yaglam, V.I.Tatarsky), Doklady AkademiiNauk SSSR, Moscow, Nauka, 1967, pp. 166-178

[14] Lumley J.,'Stochastic Tools in Turbulence ', Aca-demic Press, New York (1970)

[15] Berkooz, G., Holmes, P., Lumley, J., 'The proper or-thogonal decomposition in the analysis of turbulent ows', Ann. Rev. Fluid Mech., 25, 1993, pp. 539-575

[16] Karhunen, K., 'Zur Spektraltheorie Stochasticher',Prozessa Ann. Acad. Sci. Fennicae, (1946), 37

[17] Lo�eve , M.M., 'Probability Theory', Princeton, N.J.:VanNostrand, 1955

[18] Cantwell B. ,'Organized Motion in turbulent ow',Ann. Rev. Fluid Mech (1981), 13, pp. 457-515

[19] Aubry, Holmes, Lumley, Stone ,'The dynamics ofCoherent Structures in the wall region of turbulentshear layer', J. Fluid Mech. (1988), 192, pp. 115-175

[20] Ball K., Sirovich K., Keefe L.,'Dynamical eigenfunc-tion decomposition of turbulent channel ow', Int.J. for Num. Meth. in Fluids (1991), 12, pp. 585-604

[21] Moin, Moser ,'Characteristic-eddy decomposition ofturbulence in a channel', J. Fluid Mech. (1989), 200, pp. 471-509

[22] Bakewell H., Lumley J.,'Viscous sublayer and adja-cent wall region in turbulent pipe ows', Phys. Fluids(1967), 10(9), pp. 1880-1889

[23] Lumley J. ,'Coherent structures in turbulence', InTransition and Turbulence (ed R.E. Meyer) (1981),pp. 215-241, Academic

[24] Sirovich, Kirby, Winter ,'An eigenfunction approachto large-scale transitional structures in jet ow',Phys. Fluids, A (1990), 2(2), pp.127-136

[25] Sirovich, Park ,'Turbulent thermal convection in a�nite domain', Phys. Fluids, A (1990), 2, pp. 1649-1668

[26] Howes, Sirovich, Zang ,'Eigenfunction analysis ofRayleigh-Benard convection', Bull. Atm. Phys. Soc(1988), 33, p. 2261

[27] Chambers, Adrian, Moin, Steward, Sund,'Karhunen-Loeve expansion of Burger's model of turbulence',Phys. Fluids (1988), 31(9), pp. 2573-2582

[28] Sirovich, Rodriguez,'Coherent Structures and Chaos:a model problem', Phys. Letters A (1987), 120(5),pp. 211-214

[29] Rodriguez, Sirovich,'Low-dimensional dynamics forthe complex Ginzburg-Landau equation', Physica D,nonlinear phenomena (1990), 43, MAY, 01, n.1 p.77

[30] Tangborn, Sirovich, Strett, 'An orthogonal decom-position of �nite length chaotic Taylor-Couette ow',Bull. Atm. Phys. Soc. (1988), 33, p .2242

7

[31] Glauser, Lieb, George,'Coherent structure in ax-isymmetric jet mixing layer', Turbulent Shear Flows(1987), 5, pp. 134-145

[32] Kirby, Boris, Sirovich,'An eigenfunction Analysis ofAxisymmetric Jet Flow', J of Comp. Phys. (1990),SEP, 90, p. 98

[33] Kirby, Boris, Sirovich,'A Proper Orthogonal Decom-position of a simulated supersonic shear layer', Int.J. for Num. Meth. in Fluids (1990), MAY, 10, p. 411

[34] Hussain A.K.M.F.,'Coherent structures and turbu-lence', J Fluid Mech. (1986), 173, pp. 303-356

[35] Hussain A.K.M.F. ,'Coherent structures and studiesof perturbed and unperturbed jets', Lecture Notes inPhysics (1980), 136, pp.

[36] Sirovich ,L, 'Turbulence and the Dynamics of Co-herent Structure: I, II and III', Quarterly AppliedMathematics, (1987), 45, p. 561

[37] Kline S., Reynolds W., Schraub F, RundstadlerP.,'The structure of turbulent boundary layers', J.Fluid Mech. (1967), 30, pp. 741-773

[38] Hussain A.K.M.F.,'Coherent structures-reality andmyth', Phys. Fluids (1983), 26, pp. 2816-2850

[39] Berkooz, Elengaray, Holmes, Lumley, Poje,'ThePOD, wavelets and Modal Applications to the dy-namics of coherent structures', Appl. Scienti�c Re-search (1994), 53, DEC, pp.

[40] Ash. R., Gardner M.,'Topics in Stochastic Pro-cesses', Academic Press, N.Y. (1975)

[41] Fukunaga K.,'Introduction to Statistical PatternRecognition', Academic Press N.Y. (1972)

[42] Ahmed N., Goldstein H.,'Orthogonal Transformsfor Digital Signal Processing', Springer-Verlag,N.Y.(1975)

[43] V.R Algazi and D.J. Sakrison, 'On the Optimalityof the Karhunen-Lo�eve Expansion', IEEE Trans. In-form. Theory, (1969), 15, pp.319-321

[44] Lorenz, E.N., 'Empirical Orthogonal Functions andStatistical Weather Prediction', Cambridge: MIT,Department of Meteorology, Statistical ForecastingProject, 1956

[45] H. Hotelling, 'Analysis of a complex of statisticalvariables into principal components', Journal of Ed-ucational Psychology, (1933) 24,pp. 417-441, 488-520

[46] Brooks, C. L., Karplus, M., and Pettitt, B.M., 'Pro-teins: a Theoretical Perspective of Dynamics, Struc-ture and Thermodynamics', New York: Wiley, 1988

[47] Golub, G.H., Van Loan, C.F., 'Matrix Computa-tions', Oxford: North Oxford Academic, 1983

[48] Harman, H., 'Modern Factor Analysis', Chicago:University of Chicago Press, 1960

[49] Schmidt, E., 'Zur Theorie der linearen undnichtlinearen Integralgleichungen. I Teil: En-twicklung willk�ulicher Funktion nach Systemenvorgeschriebener', Mathematische Annalen, (1907),63, pp. 433-476

[50] Graham, M.D., Kevrekidis, I.G.,'Alternative ap-proaches to the Karhunen-Lo�eve decomposition formodel reduction and data analysis', Computers andchemical engineering, (1996), 20, p. 495

[51] Stewart, G.W.,'On the Early History of the Singu-lar Value Decomposition', SIAM Review, (1993), 35,pp.551-566 Structure Identi�cation in Free TurbulentShear Flows,1993 p. 239, Kluwer Academic Publish-ers

[52] Ukeiley, L., Varghese, M., Glauser, M., Valentine, D.,'Multifractal analysis of a lobed mixer ow�eld uti-lizing the proper orthogonal decomposition', AIAAJournal, 30(5), 1992, pp. 1260-1267

[53] Arndt, R.E.A., Long, D.F., Glauser, M.N., 'Properorthogonal decomposition of pressure uctuationssurrounding a turbulent jet', Journal of Fluid Me-chanics, 340, 1997, pp. 1-33

[54] Bienkiewicz, B., 'New tools in wind engineering',Journal of Wind Engineering and Industrial Aero-dynamics, 1996, 65, pp. 279-300

[55] Bienkiewicz, B., Tamura, Y., Ham, H.J., Ueda, H.,Hibi, K., 'Proper orthogonal decomposition and re-construction of multi-channel roof pressure', Journalof Wind Engineering and Industrial Aerodynamics,54-55, 1995, pp. 369-381

[56] Aling, H.; Banerjee, S.; Bangia, A.K.; Cole,V.; Ebert, J.; Emami-Naeini, A.; Jensen, K.F.;Kevrekidis, I.G.; Shvartsman, S., 'Nonlinear modelreduction for simulation and control of rapid ther-mal processing', Proceedings of the 1997 AmericanControl Conference, 4, 1997, pp. 2233-2238

[57] Gunes, H.; Sahan, R.A.; Liakopoulos, A., 'Spa-tiotemporal structures of buoyancy-induced ow ina vertical channel', Numerical Heat Transfer; A,32(1), 1997, pp. 51-62

8

[58] Sarah, R.A., Liakopoulos, A., Gunes, H., 'Re-duced dynamical models of nonisothermal transi-tional grooved-channel ow', Phys. Fluids, 9(2),1997, pp. 551-565

[59] Ishikawa, H.; Kiya, M.; Mochizuki, O.; Maekawa, H.,'POD and wavelet analysis of coherent structures ina turbulent mixing layer', Transactions of the JapanSociety of Mechanical Engineers, Part B, 63(610),1997, pp. 2001-2008 (in japanese)

[60] Kopp, G.A.; Ferre, J.A.; Giralt, Francesc, 'Useof pattern recognition and proper orthogonal de-composition in identifying the structure of fully-developed free turbulence', Journal of Fluids Engi-neering, 119(2), 1997, pp. 289-296

[61] Gavrilakis, S., 'Turbulent-�eld modes in the vicinityof the walls of a square duct', European Journal ofMechanics, B/Fluids, 16(1), 1997, pp. 121-139

[62] Poje, A.C.; Lumley, J.L., 'Model for large-scalestructures in turbulent shear ows', Journal of FluidMechanics, 285, 1995, pp. 349-369

[63] Hilberg, D.; Lazik, W.; Fiedler, H.E., 'Applicationof classical POD and snapshot POD in a turbulentshear layer with periodic structures', Applied Scien-ti�c Research, 53, 1994, pp. 283-290

[64] Delville, J., 'Characterization of the organization inshear layers via the proper orthogonal decomposi-tion', Applied Scienti�c Research, 53(3-4), 1994, pp.263-281

[65] Rempfer, D.; Fasel, H.F., 'Evolution of three-dimensional coherent structures in a at-plateboundary layer', Journal of Fluid Mechanics, 260,1994, pp. 351-375

[66] Citriniti, J., 'Experimental investigation into the dy-namics of the axisymmetric mixing layer utilizing theproper Orthogonal Decomposition', Ph.D Disserta-tion, SUNY at Bu�alo, 1996

[67] Ukeiley, L., 'Dynamics of large scale structures ina plane turbulent mixing layer', Ph.D dissertation,Clarkson University, 1995, Report # MAE-311

9

3 Linear Stochastic Estimation

3.1 Theory

Conditional sampling techniques are widely used to iden-tify and describe coherent motions or structures in tur-bulent ows. They �nd the average value of some owcharacteristics like velocity, pressure,temperature, whenthe prescribed event happens at one or several points.Below the ow characteristics will be noted as u(x; t),the event data vector ( the given vector of variables withthe associated event occurrence ) - as E, and the condi-tional average as hu j Ei. A simple example of the eventis a realization in a point (x

0

;t0), in which a ow char-acteristic lies close to a given vector c with a relativelysmall window dc,

E = u(x0

;t0) when c � u(x0

;t0) < c+ dc (9)

For this case, the conditional average selects from the en-semble of all ow realizations the subset in which u(x0; t0)is close to the prescribed vector c, and extracts a typicalstructure of the ow �eld within this subset. Mathemat-ically it is just a simple condition that assumes nothingabout the underlying structure. During the experiment,the window should be big enough to produce sampleswith reasonable frequency and small enough to resolveproperly the conditional average. In general, the eventcan include the conditions on u and any functionals of u(like the derivatives @ui

@xj) at one or more locations. Di-

rect experimental conditional average measurements arelimited to a rather simple one-dimensional ows and asmall amount of 'conditions' and becomes extremely time-consuming and di�cult for full three-dimensional owsand/or a large number of applied conditions. One of thecommon used technique to avoid this experimental prob-lem and to �nd the conditional average hu j Ei is theStochastic Estimation (SE). This method extracts struc-ture by approximating an average �eld in terms of theevent data at some given locations. In other words, SEreconstructs a ow �eld, corresponded to the structure byusing a knowledge of the ow at some selected points inspace and/or time. The approximation of u by stochasticestimation will be denoted by u:

When the elements of u and E are joint normally dis-tributed, hu j Ei is a linear function of E (Papoulis, [19]),but in general, hu j Ei is a non-linear function of E. Un-fortunately, the turbulent events are strongly non-linearprocesses, which lead to a non-normal probability distri-butions. Therefore, hu j Ei is a non-linear function of Efor most ows. Adrian in 1977 [1] introduced a techniqueto estimate conditional averages for any arbitrary condi-tions. He proposed to expand hu j Ei in a Taylor seriesabout E = 0 as

ui=hui j Ei =LijEj +NijkEjEk + ::: (10)

(repeated subscripts are summed) and truncate thisseries at some degree (Adrian, [1], [2]). The unknown co-e�cient tensors L;N ... can be determined if someonerequires the mean-square error between the approxima-tion and the conditional average to be minimal,

h[hui j Ei�LijEj �NijkEjEk � :::]2i ! min

The minimization leads to the orthogonality principle,which states that the error must be statistically uncor-related with each of the data (details)

h[hui j Ei�LijEj �NijkEjEk � :::]Eki = 0

The case where the series contains only the �rst or-der term, simple algebra leads to a set of linear algebraicequations for the estimation coe�cients Lij

hEjEkiLij = huiEki; (11)

This Stochastic Estimation then is called the LinearStochastic Estimation (LSE),

ui=; linear; estimate; ofhui j Ei = LijEj ; (12)

where Lij = Lij(x;x0) and x0 is the location of the

event data. Cross correlation tensor hEjEki between eachof the event data and between the data and the quantityto be estimated huiEki must be obtained by indepen-dent means (Adrian, [1]). If u = u(x;t) is the velocity�eld and the event data consists of velocity vectors E =u0 = u(x0; t0) at the location x0, then hEjEki = hujukiis the Reynolds stress tensor and huiEki = huiu0ki =Rik(x;x

0; t; t0) is the two-point, second-order, space-timevelocity correlation. Thus, knowledge of Rik gives theability to approximate any conditional averaged quanti-ties. Note, that Rik is the unconditionally sampled tensor.Thus, LSE o�ers a powerful tool of reconstructing condi-tional averaged estimates based on the information fromthe unconditionally sampled two-point correlation tensor.In general, conditional sampling involves di�erent

types of events like single-point vectors, two-point vec-tors, local deformation tensors, multi-point vectors,space-time vectors, space-wavenumber events and so on.In this case the correlation tensor must include all possi-ble cross-correlated values.Several Advantages of the LSE technique are:1. It requires only unconditional correlation functions,

which is much easier to measure experimentally.2. Once the estimation coe�cients have been com-

puted, they are independent of the conditional event data.Estimates for a number of di�erent events could be easilyevaluated.

10

3. The tensor information contained in the correlationfunctions is presented in the form of simple scalar or vec-tor �elds, which are more appropriate for an analysis.4.It can be proved that the estimated �elds satisfy the

continuity equation, and they possess the correct lengthand/or time scales.5. The procedure is applicable to all turbulent owsThus, the linear stochastic estimation exploits a

second-order correlation functions between the givenevent data and the ow �eld at some selected points. Thisprocedure establishes a simple link between conditionalaverages, the coherent structure that they may represent,and correlation functions.One of the shortcomings of LSE is that the structure

of the estimated ow pattern is independent of the mag-nitude of u(x0; t0) (see (11)). It might be argued fromthe physical grounds that the ow structures obtainedthrough LSE are associated with weak uctuations onlyand for the case of strong uctuations they may be dif-ferent.LSE, as well as POD, uses the cross-correlation ten-

sor R to extract structures from the ow. The connec-tion between LSE and POD is straightforward to �nd([5]). Recall, that POD decomposes the ow into an in-�nite number of orthogonal modes �n(x; t), for them tobe found from the integral equationZ Z

R(x0;x;t0; t)�n(x0

; t0)dx0

dt0 = �n�n(x; t)

The eigenmodes are used to decompose the ow �eldas

u(x; t) =X

an�n(x; t); (13)

The cross-correlation tensor itself can be presented interms of the orthogonal functions as

R(x0;x;t0; t) =X

�n�n(x0

; t0)�n(x; t); (14)

Using(11), (13) and (14), the equation (12) can berewritten as

u(x0; t0) = u(x; t)X

�n(x; t)f (x0; t) (15)

where f(x0; t) = �n�n(x0; t)=

P�n�

2n(x

0

; t0) can beviewed as relative contribution or weight of each mode,�n(x; t), to the conditional average. Therefore, LSE canbe treated as a weighted sum of an in�nite number ofPOD modes. From here another potential weakness ofLSE comes. LSE gives the representation of the coher-ent structures, associated with conditional estimates interms of one single ow pattern only. It could lead tophysically wrong conclusions when two or more distinc-tive structures exist in the ow. On the contrary, PODconsidered all the orthogonal modes.

The ow �elds obtained from the type of conditionalaverages were referred by Adrian [2] to as 'conditional ow patterns', or, more brie y, 'conditional eddies' inan e�ort to distinguish them clearly from physical coher-ent structures and characteristic modes �n(x; t) obtainedfrom POD technique.

3.2 Application of LSE and SE in turbu-lent ows

The validity and accuracy of the approximation of con-ditional averages of turbulent ows by LSE method havebeen investigated by numerous studies. First in 1979,Adrian [2] investigated the existence of structures inisotropic turbulence. The accuracy of LSE was evalu-ated by adding the quadratic term into (10). The resultsof LSE and second-order SE were found almost identical.The estimate predicted conditional eddies in the shape oflarge-scale vortex rings. Physical interpretation of theserings was discussed.

In [3] the validity and accuracy of LSE was investigatedin details by comparing stochastic estimates to direct ex-perimental conditional averages (9) measured in four dif-ferent turbulent ows:

a) Axisymmetric shear layer with ReD = 390; 000. Theconditional average hu1(x; t+ �) j u1(x; t)i was measuredon the centerline of the shear layer at x=D = 2 and 3.According to LSE, the coe�cient L11(�) = R11(�)=�

21

(�21 should be independent of the value of the conditionalevent. This LSE prediction was veri�ed for several di�er-ent values of u1(x; t). Systematic di�erences was discov-ered for small values of � , which questioned the accuracyof the linear estimates for small scales.

b) Plane shear layer (Re�! = �U1�!=� = 45; 000).The linear estimate of hu1(x; t+ �) j u1(x; t); u2(x; t)i forthe case of two conditional components of velocity at xbecame

u(x+ r; t) = Li1u1(x; t) + Li2(x; t)u2(x; t); i = 1; 2

Measurements of the conditional averages of u1 andu2 were compared to their linear estimates for the caseu1 = 1:0�1 and u2 = �1:0�2 (�1;2 are RMS's of u1;2respectively). The linear estimate represented the largescale structure of the conditional averages with accuracywithin the limits of measurement uncertainty.

c) Turbulent pipe ow (Re = 50; 000). The quantityhui(x; t + �) j u(x; t)i was measured experimentally andcalculated by SE. Two velocity components were mea-sured by X-�lm probes. The accuracy of LSE was eval-uated by including the nonlinear terms in the stochasticestimation

ui(x; t+ �)=Lij(x; �)uj(x; t) +Nijk(x; �)uj(x; t)uk(x; t)

11

Also the conditional average hui(x+ r; t) j u(x; t)i wasmeasured using the two X probes. Again, the nonlin-ear estimate including a second-order term was found. Inboth cases, LSE estimates matched the experimental con-ditional averages really well. Second-order SE indicatedrelatively little improvement in accuracy versus the LSE,.

d) Grid turbulence with Reynolds number of 57,000based on the free-stream velocity and the mesh spac-ing. The conditional average with time delay hu1(x; t +�) j u1(x; t)i and the autocorrelation function R11(�)were measured using a single hot-wire probe. The linearestimate and the experimental conditional average com-pared well for values of the event velocity �1:5�1, �1:0�1,1:0�1and 1:5�1, where �1 is RMS of u1.

The authors concluded that, although all consideredturbulent ows were non-normal joint random processes,LSE worked well in these ows.

In [5] the optimal choice of a model of stochastic es-timation was discussed. It was shown that the choiceis strongly dependent on the event upon which the av-erage is conditioned. The series of tests based on one-point velocity measurements in a turbulent boundarylayer (Re� = 3; 100) demonstrated how the stochastic es-timation may be re�ned to give more accurate descrip-tions of particular coherent motions. Selection of the or-der of the stochastic estimate was also discussed and itwas shown that merely increasing the order of a poly-nomial model of the event vector would not necessarilyincrease the accuracy of the stochastic estimation.

Conditional eddies in isotropic turbulence were the fo-cus of [9]. The velocity measurements for isotropic gridturbulence were used to approximate the stochastic esti-mate in a higher-order (up to a fourth-order) expansion.It was concluded the LSE is a acceptable method for in-vestigating the qualitative large scale structures.

Fully turbulent boundary layer (Re� = 4900) wasunder investigation in [14]. Measurements consisted offour simultaneously sampled hot-wire signals on a 13x 11 x 8 three-dimensional grid. The results of linearand nonlinear estimates for two di�erent events (fourth(u > 0; v < 0) and second (u < 0; v > 0) quadrant events)were compared to the corresponding experimental condi-tional averages and the agreement was found to be ex-cellent. Extensions of the technique to space-time esti-mates of one- and two-point conditional averages werepresented. It was pointed out that multipoint condi-tional averages focus on a particular scale based on theseparation between the points at which conditions wereimposed. Again, the stochastic estimation was found toallow one to implement averaging technique easily for avariety of conditions.

Multipoint conditional averages and spatio-temporalevolution of the three-dimensional structures of the tur-bulent wake of a cylinder was investigated in [17] (at

x=D = 100 for Re = 5000). The measurements of thefull 3-D correlation tensor for all components across thewidth of the wake were used to identify events contribut-ing most to the Reynolds stress. The most likely ensembleaveraged structure corresponding to these events were re-constructed using the stochastic estimation procedure. Anew technique, the pseudo-dynamic reconstruction wasdeveloped to estimate the evolution of 3-D velocity �eldsfrom the experimental data.

In [6] multi-point conditional analysis obtained fromLSE was applied to investigate dominant structures in ajet mixing layer. Since a single-point estimates did notgive an adequate presentation of the dynamics of underly-ing structures, pseudo-dynamical evolution of the struc-tures based on two-point velocity measurements was com-pared with the instantaneous velocity �eld. The agree-ment was found qualitatively good. It was noticed thatin inhomogeneous ows a conditional sampling is prettymuch a function of the location of the probes.

In [13] the results of various investigations based onstochastic estimation of the structures in turbulent shear ow were discussed. They supported a picture in whichvortex rings and/or hairpin vortices dominate in the outerlayer of the shear ow, whereas structures elongated inthe streamwise direction with vorticity in the streamwisedirection dominate inside the wall layer.

In [20] two-point space-time pressure-velocity correla-tions have been measured in the near �eld of a roundjet ow with a stationary velocity probe and a movingpressure transducer. The correlations revealed the down-stream translation of the vortices, and the physical mech-anisms responsible for this behavior of the correlationswere identi�ed. Conditional averages of pressure uctua-tions given the state of the velocity uctuation was alsoobtained. A linear estimation mode agreed very well withthe measured data.

[21] The properties of conditional averages have beenstudied for isotropic turbulence and for anisotropic tur-bulence in a plane high Reynolds number shear layer with2:1 velocity ratio. It was shown that in isotropic turbu-lence a linear estimate was the dominant term. It gavea good approximation for estimated values of the veloc-ity uctuations, and it predicted a vortex ring structurein the ow. In the shear layer, conditional averages ofthe velocity component u(x + r; t) and v(x + r; t) andthe Reynolds stresses uv(x+ r; t), u2(x+ r; t), v2(x+ r; t)have been calculated by measuring the values of u(x; t)and v(x; t) using two X-wire probes. Comparison of theconditional velocity �eld with its linear estimate showeda good agreement.

Adrian and Moin in [8] applied LSE to homoheneousturbulent shear ow data generated by a direct numer-ical simulation. Events contributing to the most of theReynolds shear stress were thoroughly investigated. The

12

conditional event under consideration was the value ofthe velocity u and the deformation tensor dij(x; t) =

@ui@xj

.

The including of d was shown to have a signi�cant in u-ence on the conditional eddy �eld. A rational method ofselecting events was proposed. It was based on determin-ing the values of (u, d) at which the greatest contributionto some mean quantities occurred. It was found that oneconditional-eddy structure was a hairpin vortex. Over-all, LSE gave a good approximation of the conditionalaverage on large scale.In [4] LSE technique was applied to turbulent plasma to

investigate the formation, propagation and decay of neg-ative potential wells, which corresponded to ion phase-space vortices. Additional conditions as a sign of thederivatives of the velocity were imposed on the signal toimprove the representation of extracted coherent struc-tures in the turbulence. The results of LSE were foundto be useful as a guideline to analyze the physics of tur-bulence.Adrian in 1994 [24] made a detailed review of imple-

mentation of the stochastic estimation in a search of con-ditional structures. The mathematical method of SE waspresented and the selected results obtained from di�erentexperiments in isotropic turbulence, turbulent boundarylayer and channel ow were discussed and summarized.

3.3 Comparison of LSE (SE) with othertechniques

Number of papers compared LSE (and/or SE) with theother existing techniques, as Proper Orthogonal Decom-position (POD), which as also based on the analysis ofunconditional correlation tensor.Sullivan and Pollard in [25] discussed methods for the

eduction of coherent structures using multipoint measure-ments. They considered a three-dimensional wall jet us-ing four di�erent techniques: POD, LSE, Gram-CharlierEstimation (GCE) and Wavelet Decomposition (WD).The analysis of the obtained data revealed that the �rstPOD mode clearly captured the local peaks in the veloc-ity �eld. The comparison of both POD and LSE resultsrevealed very good agreement, because LSE is actually aweighted sum of a in�nite number of POD eigenmodes[5]. It was noticed that, unless the �rst mode contains asigni�cant amount of the total energy of the data, a owstructure may not be identi�ed. The GCE was used todevelop a spatially dense mapping of the instantaneousvelocity �eld. Combination of GCE and LSE helped toreconstruct evolution and interaction between structuresdownstream in the ow. These structures were shown tobe associated with the exiting structure from the nozzle.Finally, wavelet decomposition was used to identify scalesimportant within the ow. WD was found to provide amore general method for investigating a ow �eld.

In [16] both POD and LSE were used to identify struc-ture in the axisymmetric jet shear layer (Re = 110; 000)and in the 2-D mixing layer. The new complementarytechnique was introduced. This technique took projec-tions of the estimated velocity �eld obtained from LSEonto the POD eigenmodes to obtain random coe�cientsan in (13). These estimated random coe�cients were thenused together with the POD eigenmodes to reconstructthe estimated velocity �eld. It was shown this comple-mentary technique allows one to obtain time dependentinformation about the velocity �eld while greatly reduc-ing the amount of instantaneous data. This approach canbe useful to build and verify low dimensional dynamicalsystems models based on POD coe�cients.In [26] POD, and conditional sampling techniques were

applied to a simple test functions and the results werecompared to determine the strength and weaknesses ofeach approach. POD was found to be useful only forthe signals where the energy consisted in a few modes.The physical interpretation of POD results was also dis-cussed. Conditional sampling was found to be sensitiveto the conditions chosen. None of the above methodswas found of capable of analyzing or identifying the lo-cal small-scale structure of the ow. New methods forinvestigating complex structures based on fractural andwavelet analysis were presented. The application of thewavelet transform was shown to provide a simple func-tional description in terms of local length scales and thedistribution of velocity and vorticity within turbulencestructures.

13

References

[1] Adrian, R.J., 'On the role of conditional averages inturbulent theory', In: Turbulence in Liquids: Pro-ceedings of the 4th Biennial Symposium on Turbu-lence in Liquids, ed. G. Patteson and J. Zakin, Sci-ence Press, Princeton, 1977, pp. 322-332

[2] Adrian, R.J., 'Conditional eddies in isotropic turbu-lence', Physics of Fluids, (1979), 22, pp. 2065-2070

[3] Adrian, R.J.,Jones, B.G., Chung, M.K., Hassan, Y., Nithiandam,C.K., and Tung, A.T.C.,'Approximation of turbulentconditional averages by stochastic estimation, Phys.Fluids, A, (1989), 1, pp. 992-998

[4] Pecseli, H.L., Trulsen, J., 'A statistical analysisof numerically simulated plasma turbulence', Phys.Fluids, B, (1989), 1, pp. 1616-1636

[5] Breteton, G.J., 'Stochastic estimation as a statisti-cal tool for approximating turbulent conditional av-erages', Phys. Fluids, A, (1992), 4, pp. 2046-2054

[6] Cole, D.R., Glauser, M.N., Guezennec, Y.G., 'Anapplication of the stochastic estimation to the jetmixing layer', Phys. Fluids, A, (1992), 4, pp. 192

[7] Bagwell, T.G., Adrian, R.J., Moser, R.D., Kim, J.,'Improved approximation of wall shear stress bound-ary conditions for large eddy simulation', In: NearWall Turbulent Flows, ed. R.M.C. So, C.G. Spezialeand B. E. Launder, (1993), pp. 243-257

[8] Adrian R.J., Moin, P., 'Stochastic estimation of orga-nized turbulent structure: Homogeneous shear ow',J. Fluid Mech., (1988), 190, pp. 531-559

[9] Tung, T.C., Adrian, R.J., 'Higher-order estimatesof conditional eddies in isotropic turbulence', Phys.Fluids, (1980), 23, pp. 1469-1470

[10] Moin, P., Adrian, R.J., Kim, J., 'Stochastic esti-mation of organized structures in turbulent chan-nel ow', Proc. 6th Turbulent Shear Flow Symp.,Toulouse, (1987), pp. 16.9.1-16.9.8

[11] Moser, R.D., 'Statistical analysis of near-wall struc-tures in turbulent channel ow', In: Near Wall Tur-bulence, ed. S. Kline and N. Afgan, New York: Hemi-sphere Publ. (1990), pp. 45-62

[12] Adrian, R.J., 'Stochastic estimation of sub-grid scalemotions', Appl. Mech. Rev., (1990a), 43, pp. S218

[13] Adrian, R.J., 'Linking correlations and structure:stochastic estimation and conditional averaging',In:Proceedings of the International Centre for Heat andMass Transfer. ed. S. Kline and N. Afgan, Hemi-sphere Publ Corp, New York, (1990b), p 420-436

[14] Guezennec, Y.G., 'Stochastic estimation of coherentstructure in turbulent boundary layers', Phys. Flu-ids, A, 1(1), (1989), pp. 1054-1060

[15] Balachandar, S., Adrian, R.J., 'Structure extractionby stochastic estimation with adaptive events', J.Theor. & Comp. Fluid Dyn., (1993), 5, pp. 243-257

[16] Bonnet, J.P., Cole, D.R., Delville, J., Glauser, M.N.,Ukeiley, L.S., 'Stochastic estimation and proper or-thogonal decomposition: Complementary techniquesfor identifying structure', Exp. Fluids, (1994), 17(5),pp. 307-314

[17] Gieseke, T.J., Guezennec, Y.G., 'Stochastic estima-tion of multipoint conditional averages and theirspatio-temporal evolution', Applied Scienti�c Re-search, (1994), 53, pp. 305-320

[18] C.-H. Chen, R.F Blackwelder, JFM, 89, 1, (1978)

[19] Papoulis, A., 'Probability, random Variables andStochastic Theory', McGraw-Hill, New York, 1984,2nd ed

[20] Chang, P.H.-P., Adrian, R.J., Jones, B.G., 'Two-point pressure-velocity correlations, conditional av-erage and linear estimation in a round jet ow', In:Fluid Control and Measurement, Tokyo, v 1. Perga-mon Press, Oxford. p 481-491.

[21] Tung, A.T -C.,Adrian, R.J., Jones, B.G.,'Conditional velocity andReynolds stresses in a plane turbulent shear layer',T&AM Rep Univ Ill Urbana Champaign Dep TheorAppl Mech., 483, (1987), p. 162.

[22] Liu, Z -C., Landreth, C C., Adrian, R J. Hanratty, TJ., 'High resolution measurement of turbulent struc-ture in a channel with particle image velocimetry',Experiments in Fluids, 10, (1991), pp. 301-312.

[23] Zhang, Y., Huang, T S., Adrian, R J., 'Recon-struction of vector �eld in uid measurement Imageand Multi-Dimensional Signal Processing', ICASSP,IEEE International Conference on Acoustics, Speechand Signal Processing, v 4, (1995), p 2615-2618

[24] Adrian, R.J., 'Stochastic estimation of conditionalstructure: a review', Applied Scienti�c Research, 53,(1994), p 291-303

14

[25] Sullivan, P., Pollard, A.,'Coherent structure identi�-cation from the analysis of hot-wire data', Measure-ment Science & Technology. 7 (10), 1996. pp. 1498-1516

[26] Kevlaham, N.K.-R., Hunt, J.C.R., Vassilicos, J.C.,'A comparison of di�erent analytical techniques foridentifying structures in turbulence', Applied Scien-ti�c Research', 53, (1994), pp. 339-355

15

4 Wavelet Analysis

4.1 Theory

The wavelet transformation of the continuous signalf(t) 2 L2(R) is de�ned the following way:

Fg(�; a) =1pa

Z +1

�1

f(t)g��t� �

a

�dt (16)

where parameter a is called dilatation parameter, � iscalled shift parameter, asteric denotes a complex conju-gate and the complex valued function g(x) is called awavelet mother function and satis�es the following condi-tions: Z

g(x)g�(x)dx <1 (17)

C(g) = 2�

Z +1

�1

jbg(!)j2!

d! <1 (18)

Here and below the hat sign over a function denotes aFourier transformation of the function,

bg(!) = 1p2�

Z +1

�1

g(x)e�i!xdx

The condition (18) is called the admissibility condi-tion and in case of integrable function g(x) implies thatR +1�1

g(x)dx = 0. The admissibility condition guaranteesthe existence of the inverse wavelet transformation,

f(t) =1

C(g)

Z +1

0

Z +1

�1

Fg(�; a)pa

g

�t� �

a

�da d�

a2

Because of the local support in physical domain (17),the integration in (16) is evaluated over a �nite domain,proportional to a and centered near � . This propertyof the wavelet transformation allows someone to analyzelocal characteristics of the signal at time � and scale a.The wavelet transformation (16) can be rewritten in

the Fourier space as follows

Fg(�; a) =pa

Z +1

�1

bf(!)bg�(a!)ei�!d!and gives another highly useful interpretation of the

wavelet transformation as a multiple band-bass �lteringacting on the signal f(t).Parseval theorem - the energy of the signal can be de-

composed in terms of the wavelet coe�cients Fg(�; a) asZ +1

�1

f(t)f�(t)dt =1

C(g)

Z +1

0

Z +1

�1

Fg(�; a)F�

g (�; a)da d�

a2

Energy at the scale a is de�ned as

E(a) =1

C(g)

1

a2

Z +1

�1

Fg(�; a)F�

g (�; a)d�

4.2 Properties of wavelet transformation

1. Linear operator:

Fg(f1 + f2)(�; a) = Fg(f1)(�; a) + Fg(f2)(�; a)

2. Commutative with di�erentiation:

dn

dtnfFg(f)(�; a)g = Fg

�dnf

dtn

�(�; a)

3. For any function f(t) of homogeneous degree � att = t0, t.e. f(�t) = ��f(t) near t = t0:

Fg(f)(t0; a) = a�+1=2Fg(f)

�t0a; 1

�or Fg(f)(t0; a) � a�+1=2 as a �! 0

This property could be really useful when analyzinga local regularity of the function f(t). For instance,if the function is discontinuous at t0, � = �1.

4. The information provided by the complete set of thecoe�cients is redundant, t.e. there is a strong corre-lation between the wavelet coe�cients,

Fg(�0; a0) =

Z Zp

��0 � �

a;a0a

�Fg(�; a)

da d�

a2;

where

p(�; a) =1

C(g)

1pa

Zg��t� �

a

�g(t) d�

Last property is not desirable when working with tur-bulence modelling, because the number of coe�cients tobe is very big. One can construct the orthonormal basisof the functions f ij(x)g, complete in L2(R) and orthog-onal to themselves when translated by a discrete step anddilatated by a power of 2:

ij(x) = 2j=2 (2jx� i)Z ij(x)

�

kl(x)dx = �ik�jl

The discrete version of the wavelet decomposition

f(x) =Xi

Xj

Fij ij(x) ; where

Fij =

Zf(x) (x � 2�ji)dx

removes the redundancy and thus minimizes the num-ber of the wavelet coe�cients Fij in L

2-norm to describea given function f(x). This transformation is attractive

16

from computational point of view as an alternative candi-date for the Fourier transformation of Navier-Stokes equa-tions. Complete theory of discrete wavelet transform canbe found in [5], [6]

From a variety of wavelet mother functions constructedby now, a few important wavelets are worth of mention-ing:

Continuous wavelet transform

1. Morlet wavelet - good for an analysis of local period-icity of the signal:

g(x) = exp(ibx) exp(�x2=2); b � 5

2. 'Mexican Hat' wavelet (Maar wavelet) - good for asearch of localized structures:

g(x) =d2

dx2exp(�x2=2)

3. First derivative of a Gaussian: investigation of localgradients:

g(x) =d

dxexp(�x2=2)

Discrete wavelet transform

4. Lemarie-Meyer-Battle (LMB) wavelet - explicit def-inition [7]

5. Daubechies compactly supported wavelet - recurrentde�nition [5]

Generalization of the wavelet transformation to amulti-dimensional case is pretty straightforward ([30], forinstance) and will be not discussed here.

Historically the idea of generating the basis which pos-sesses a locality property in both physical and Fourierspaces goes to quantum mechanics [2] and signal process-ing [1]. In aerodynamics the similar constructions canbe found in work by Siggia (1977) [32] and Zimin (1981)[18]. In attempt to build a model capable of predict-ing intermittent properties of small-scale turbulence, theyproposed an algorithm of constructing a basis of wavepackets which is local in Fourier and physical space. In-dependently Morlet applied a wavelet decomposition asa modi�cation of Gabor elementary wavelets [1] in seis-mology [8], [37]. The �rst rigorous theory of wavelet de-composition was done by Morlet and Grossmann in 1984[4]. Discrete version of the wavelet transform was devel-oped by Daubechies [5], [6]. An excellent presentation ofwavelet theory as well as it's application in turbulenceresearch was done by Farge in [30]. Another good sourceis IEEE issue on wavelets [42].

4.3 Applications to turbulent ows

It is well-known that turbulence has both energy spec-trum cascade, which is well-described in Fourier spaceand spacial intermittent events, localized in space andtime ([38], e.g.). The wavelet property to be local inboth physical and Fourier spaces looks really attractiveto apply it in turbulence research. Original Kolmogorov'stheory of turbulence [39] was modi�ed to incorporate ex-perimentally discovered properties of real turbulence asintermittency e�ects and energy backscatter (transfer ofenergy from small scales to large scales). A number of socalled hierarchical models of 2-D and 3-D isotropic tur-bulence, based on the wavelet decomposition was built.One of he �rst attempt for 2-D turbulence was made in1977 [32]. It utilized a wave-packet decomposition, whichis essentially a wavelet transform. It was pointed outthat the convection term in N-S equations is essentiallylocal in physical space and the pressure term is local inwavenumber space and guarantees the incompressibility.Thus, the Fourier space was divided in number of shellsbn < jkj < bn+1; n = 1:: lnb �K : Here b � 2, k is awavenumber in Fourier space and �K is Kolmogorov dis-sipation scale. A wave-packets were introduced as a func-tions 'n(r) with their Fourier transform �n(k) to havea non-zero value only within a corresponding shell. Thefunctions 'n are local in both Fourier and physical space.The Fourier decomposition in physical space within a shellwas used to describe the functions. After projecting N-Sequations into this basis the system of non-linear ODE'swas obtained. After several simplifying assumptions, theinteraction between adjacent shells was examined numer-ically for 4 shells x 26 Fourier mode decomposition withina shell. The energy cascade and temporal intermittencywas investigated.

Independently in 1984 [18] essentially the same ap-proach was taken to build a model of 3-D turbulence. Abasis functions with a non-zero value within a shell wereintroduced. After the inverse transformation the physi-cal representation was obtained. These eddies were dis-tributed in physical space randomly and were allowed tomove. After projecting N-S equations into this basis a sys-tem of non-linear ODE's was investigated. The model wasfound to have a qualitative agreement with experimentaldata. Later this model was modi�ed using eddy represen-tation in wavelet form in [19]. This model does not involveany empirical assumptions and possesses some importantfeatures of the turbulence, including k�5=3 Kolmogorovspectrum. non-local interactions between the scales andback-scatter of energy from small scales to large scales.It can be used as a subgrid non-equilibrium turbulencemodelling.

In [20] a hierarchical-tree model of 2-D turbulence waspresented. The cascade of vortices was arranged in a tree-

17

like structure, with smaller 'children' vortices connectedto a bigger 'parent' vortex from the previous level. Themain di�erence from shell-based models was that the dis-tances between 'parent' and 'children' physical vorticeswere kept �xed. Thus a new variables for this Hamilto-nian model were the amplitudes of vortices and the rela-tive angles between adjacent vortices. Almost orthogonalbasis of discrete wavelets was used. The model exhibitsthe spatial intermittency and fractal properties of it wereobtained. The model showed good agreement with theo-retically predicted energy cascade for 2-D turbulence.

One of the �rst application of the wavelets in turbu-lence was done by Farge et al [22], where an optimally con-structed wavelet decomposition was proposed to reduce anumber of basis functions (modes) to describe turbulenceevolution. Investigating an evolution of a truncated sub-set of important modes, they have shown better perfor-mance of the wavelet decomposition against the standardFourier decomposition. Another classical paper in thisarea is [30], where a brief theory and the basic proper-ties of the wavelets and their application in turbulence iswell-presented.

In [11] the wavelet transformation was applied foranalysis of turbulent ows. It was pointed out thatthe wavelets closely resemble the 'eddy' structures intro-duced by Lumley [3] as a building blocks for turbulence.The brief theory and basic properties of the continuouswavelet transformation was presented. In an example ofshock wave/free turbulence it has been clearly showedthe advantage of wavelet transform to detect discontinu-ities of a signal (front of a shock-wave). Thus, intermit-tent events can be easily localized using 'Mexican hat'wavelets. Another example concerning the wall turbu-lence was considered. The wavelet technique was com-pared with VITA technique [29] for a detection of sweepand ejection events taking places in the boundary layer.It was found that VITA technique could give false or nodetections of these burst-like events and requires someknowledge about the characteristic scales of the events.It comes from the fact that VITA technique is a singlescale �ltered technique. Because of variable scaled �lteredproperty of the wavelet transformation it detects all theevents well and does not require any a priory informationof the events to be detected. Also a discretized versionof the wavelet transformation was proposed as a compet-itive technique for CFD versus Fourier based techniquefor ows with large local gradients or discontinuities.

In [12] coherent structures (ejections and sweeps) in aheated turbulent boundary layer (Re = 5124 based onthe momentum thickness � = 6:1 mm) were investigated.Regions near the viscous sublayer y+ = 7 and a fullyturbulent central region y+ = 180 were considered andboth velocity and thermal measurements were taken inthese regions using cold and hot wires. Several techniques

like VITA, WAG (Window Averaged Gradients)[28] andwavelet analysis based on Morlet mother function wereused to analyze the intermittent events. Conditionalspectra for cooling (ejection) and heating (sweeps) eventswere obtained. The ejections were found to be relativelyslow more localized events with comparison to more rapidsweep events.

In [15], [16] a discretized version of the three-dimensional wavelet transformation using LMB waveletwas applied for turbulent ows, both experimental (wakebehind a cylinder) and numerically simulated ones.Spatially dependent quantities like total kinetic energyE(k; x), net transfer to the wavenumber k; T (k; x) andtotal ux through the wavenumber k to all smaller scales,�(k; x) were derived using wavelet analysis. Analysis ofthese quantities from experimental data have shown a sig-ni�cant level of spatial intermittency with large variationsfrom the mean values, essentially at a smaller scales. Spa-tial pdf's for energy distribution and dissipation were ob-tained and revealed an existence of long exponential tails.Several versions of the wavelet mother function were usedto verify a robustness of the results. The wavelet resultshas revealed a multifractal nature of the turbulence atsmall scales and some fractal statistics (the generalized di-mensions) of the multifractal turbulence were calculated.An additive mixed multifractal cascade model was builtand was shown to duplicate all the essential results of theexperiments. Also a multifractal behavior of turbulencewas explored in [38].

Similar approach of analyzing generalized dimensionsof turbulence has been taken in [33]. Again, turbulencewas treated as a multifractal process and the continuouswavelet transformation was applied to the equations ofmotion. The author was able to come up with the systemof dynamical equations for an evolution of the generalizeddimensions based on the dynamics of Navier-Stokes equa-tions. Thus, the model which possesses both multifractalthermodynamical properties and the dynamics inherentto N-S equations was proposed.

An application of the discrete wavelet transform toexplore the intermittency in turbulent ows was a pur-pose of the work [34]. Authors have been establisheda �rm connection between the intermittent events andunderlying coherent structures. Using the experimentaldata for isotropic turbulence behind a grid, they foundthat the ow reveals a big degree of intermittency forRe� � 10. Also applying a velocity phase averagedtechnique based on the wavelet transform, they havebeen able to reconstruct the 'signature' of the coher-ent structures corresponding to the intermittency events.For an isotropic turbulence a typical size of the small-scale �lamentary-like structures have been found as of(4� 5) Kolmogorov scales , with these structures captur-ing about 1% of the ow energy. For a jet-generated tur-

18

bulence (Re� = 250� 800) large-scale structures (vortexrings) from a longitudinal component of velocity as wellas small-scale structures (�lamentary structures) from atransverse component were found. A strong phase corre-lation between the large and small scale structures wasobserved. The universality of a scaling exponents �(p) inKolmogorov's scaling law jU(x+ r)� U(x)jp � r �(p) forp = 2::6 was veri�ed for a grid and jet turbulent ows formoderate Re�.

A transition to turbulence in a shear layer was inves-tigated using the continuous wavelet transform in [24].A pairing process was found to be intermittent in theregion where the subharmonic dominates over the funda-mental mode, with the intermittency being stronger forlarge scales of motion.

In [17] ow behind a sphere in a strati�ed uid wasanalyzed for a range of Re and Fr numbers. The DPIVtechnique was used to get the velocity �eld. The two-dimensional version of Morlet wavelet was applied to ob-tain spatially local length scales, as well as local Re andFr numbers. Spatial distributions of Re and Fr werefound to be similar. Vortex core centers were marked byvery low Re and Fr numbers.

The wavelet decomposition turns out to be an opti-mum tool to analyze this object. In [14] 2-D turbulencefrom an axisymmetric jet with ReD = 4; 000 was in-vestigated. A two-dimensional version of 'Mexican Hat'wavelet was used to decompose dye concentration pic-tures. Two di�erent structures, large-scale beads andsmall-scale strings were observed. Strings were found tobe a strongly anisotropic structures with essential lack ofself-similarly across the scales.

In [24] a continuous wavelet analysis was applied toinvestigate transition in turbulent mixing layer. It re-vealed that the paring process is intermittent. In [26] in-termittent events on di�erent scales in atmospheric windwere reported by applying the wavelet technique. In [40],[41] several techniques like wavelet transform, VITA andWAG were applied to wind velocity data . Intermittentcoherent large and small structures were discovered andinvestigated. Dynamics of sweep-ejection process was dis-cussed and the obtained results were found in agreementwith previous results. In [43] a shear layer in a regionof strong interaction between a fundamental and subhar-monic modes was investigated by a continuous Morletwavelet transform. An intermittent �-shifts in subhar-monic phase were discovered. Motivated by the exper-imental data, a dynamical Hamiltonian model based onstructural interactions of vortices was proposed and it wasshown that the model have a very good agreement withthe experiment.

A multiple acoustic modes in an underexpended su-personic rectangular jet were investigated by the wavelettransform. in [44]. They found to exist simultaneously

and don't exhibit a mode switching. Rather detailedwavelet-based analysis of acoustic mode behavior in su-personic jets can be found in [45]In [27] new methods for investigating complex struc-

tures based on fractural and wavelet analysis were pre-sented. The application of the wavelet transform wasshown to provide a simple functional description in termsof local length scales and the distribution of velocity andvorticity within turbulence structures. In [25] severaltechniques for analysis of turbulent ows were discussed.The wavelet transform was found useful to detect inter-mittent events and turbulent structures in ows.Wavelet transform was successfully used to diagnostics

in seismology [37], cardiology [21], in diagnostics of enginecylinders [23]. In optics Optical version of Wavelet Trans-form (OWT) was introduced in [10]. Also the wavelettransform is actively used in multifractal analysis [35],[36] and information theory [7].

4.4 Comparison with other techniques

4.4.1 Windowed Fourier transformation

The main problem with Fourier transform is that the ba-sis functions exp(i!t) do not belong to L2(R). In otherwords, they have an in�nite span in physical space andprovide no information about spacial localization of thesignal. This problem could be overcome, if the signal iswindowed or multiplied by a function with local physicalsupport w(t). This idea was introduced by Gabor in 1946[1]. The windowed transformation is as follows

L(�; !) =

Zf(t)w(t� �)e�i!(t��)dt (19)

In practice, two kinds of w(t) are widely used, co-sine window cos(�=2 t); t 2 [�1; 1] and Gabor packetexp(�t2=2�2). Transform (19) resembles (16), but pro-vides a �xed time window for any !. This time win-dow can be varied by an appropriate dilatation of w(t),though.

4.4.2 VITA (Variable Integral Time Averaging)

VITA technique was introduced in 1976 [29]. The idea isto look for a short-time averaging version of RMS of thesignal,

var(t; T ) =1

T

Z t+T=2

t�T=2

f(t)2dt� 1

T

Z t+T=2

t�T=2

f(t)dt

!2

�f = limT!1

var(t; T ) � RMS of the signal

The event is said to be detected if var(t; T ) > K � �fThe technique introduce two constants to play with,

T and K. This technique is valid only for second-order

19

stationary signals. It uses a �xed time window, whichmeans low-pass single scale �ltering.Problems with VITA

1. Choice of T and K is intricate

2. Does NOT detect events separated by time less thanT

3. Possible false detections

4. Smoothing of some events

4.4.3 WAG (Window Averaged Gradients)

Another technique to detect sudden changes in velocitysignal was proposed in [28] and it is based on the analysisof the average gradients.

WAG(t; T ) =1

�f

1

T

"Z t

t�T=2

f(t)dt�Z t+T=2

t

f(t)dt

#

The event is detected when WAG(t; T ) > � � u0RMS .Similar problems as with VITA technique: single scale�ltering, stationary signals only.

References

[1] Gabor, D., 'Theory of communication', J. Inst.Electr. Engin., London, 93(III), 1946, pp. 429-457

[2] Bargmann, V., 'On a Hilbert space of analytic func-tions and an associated integral transform', Part I,Comm. Pure Appl. Math, 14, 1961, pp. 187-214:Part II, Comm. Pure Appl. Math, 20, 1967, pp. 1-101

[3] Tennekes, H., Lumley, J.L., 'A First Course in Tur-bulence', Cambridge: MIT Press, 1972

[4] Grossmann, A., Morlet, J., 'Decomposition of Hardyfunctions into square integrable wavelets of constantshape', SIAM J. Math Anal., 15, 1984, p. 723

[5] Daubechies, I., 'Orthonormal bases of compactlysupported wavelets', Communications on Pure andApplied Mathematics, 41, 1988, pp. 909-996

[6] Daubechies, I., 'Ten lectures on wavelets', CBMSLectures Notes Series, SIAM, 1991

[7] Mallat, S., 'A theory for multiresolution signal de-composition: the wavelet representation', IEEE,Trans. on Pattern Anal. Machine Intell., 11, 1989,p. 674

[8] Morlet, J., 'Sampling theory and wave propagation',Proc. 51st Annual International Meeting of the Soci-ety of Exploration Geophysicists, 1981, Los Angeles.

[9] Argoul, F., Arneodo, A., Grasseau, G., Gagne,Y., Hop�nder, E., Frisch, U., 'Wavelet Analysis ofTurbulence Reveals the Multifractal Nature of theRichardson Cascade', Nature, 338(6210), 1989, pp.51-53

[10] Pouligny, B., Gabriej G., Muzy, J F., Arneodo, A.,Argoul, F., Freysz, E., 'Optical wavelet transformand local scaling properties of fractals', Journal ofApplied Crystallography, 24(5), 1991, pp. 526-530

[11] Liandrat, J., Moret-Bailly, F., 'Wavelet transform.Some applications to uid dynamics and turbulence',European Journal of Mechanics B-Fluids, 9(1), 1990pp. 1-19

[12] Benaissa, A., Liandrat, J., Anselmet, F,. 'Spec-tral contribution of coherent motions in a turbulentboundary layer', European Journal of Mechanics B-Fluids, 14(6), 1995. pp. 697-718

[13] Otaguro, T., Takagi, S., Sato, H., 'Pattern Search ina turbulent signal using wavelet analysis', Proc. 21stJapan Symp. on Turbulence, Tokyo, Japan, 1989

20

[14] Everson, R., Sirovich, L., Sreenivasan, K.R.,'Wavelet analysis of the turbulent jet', Phys. Lett.A, 145, 1990, pp. 314-322

[15] Meneveau, Charles., 'Analysis of turbulence inthe orthonormal wavelet representation', Journal ofFluid Mechanics, 232, 1991, pp. 469-520

[16] Meneveau, C., 'Dual spectra and mixed energy cas-cade of turbulence in the wavelet representation',Physical Review Letters , 66, 1991, pp. 1450-1453

[17] Spedding, G R., Browand, F K., Fincham, A M.,'Turbulence, similarity scaling and vortex geometryin the wake of a towed sphere in a stably strati�ed uid' Journal of Fluid Mechanics, 314(10), 1996, pp.53-103

[18] Zimin, V.D., 'Hierarchical model of turbulence',Izvestia Academy of Science USSR, Atmospheric andOcean Physics, 17(12), 1981, pp. 941-949

[19] Zimin, V., Hussain, F.,'Wavelet based model forsmall-scale turbulence', Physics of Fluids, 7(12),1995, pp. 2925-2927

[20] Aurell, E., Frick, P., Shaidurov, V., 'Hierarchicaltree-model of 2D-turbulence', Physica D, 72, 1994,pp. 95-109

[21] Morlet, D., Peyrin, F., Desseigne, P., Touboul, P.,Rubel, P., 'Time-scale analysis of high-resolutionsignal-averaged surface ECG using wavelet transfor-mation' Computers in Cardiology, Publ. by IEEE,Computer Society, Los Alamitos, CA, USA (IEEEcat n 92CH3116-1). pp. 393-396

[22] Marie Farge, Eric Goirand, Yves Meyer, FredericPascal, Mladen Victor Wickerhauser, 'Improved pre-dictability of two-dimensional turbulent ows usingwavelet packet compression', Fluid Dynamics Re-search, 10(4-6), 1992, pp. 229-250

[23] Wiktorsson, M., Lindo�, B., Johansson, B., Soder-berg, F., 'Wavelet analysis of in-cylinder LDV veloc-ity measurements', Diagnostics and Modeling in SIEngines SAE Special Publications, 1212, 1996. SAE,Warrendale, PA, USA. pp. 1-10

[24] Bokde, A L W., Jordan, D A Jr., Miksad, R W.,'Experimental investigation of the temporal inter-mittency in the transition to turbulence of a planemixing layer', Proceedings of Engineering Mechanics,11th Conference on Engineering Mechanics, 2, 1996,ASCE, New York, NY, USA. pp. 1070-1073

[25] Hajj, M R., 'Analysis tools of transitioning and tur-bulent ows', Proceedings of Engineering Mechanics,

11th Conference on Engineering Mechanics, 1, 1996,ASCE, New York, NY, USA. pp. 434-437

[26] Hajj, M.R., Tieleman, H.W., 'Application of waveletanalysis to incident wind in relevance to wind loadson low-rise structures', Journal of Fluid Engineering,118(4), 1996, pp. 874-876

[27] Kevlahan, N K -R., Hunt, J C R., Vassilicos, JC., 'Comparison of di�erent analytical techniques foridentifying structures in turbulence', Applied Scien-ti�c Research, 53(3-4), 1994, pp. 339-355

[28] Bisset, D.K., Antonia, R.A., Browne, L.W.B., 'Spa-tial organization of large structures in the turbulentfar wake of a cylinder', Journal of Fluid Mechanics,218, 1990, pp. 439-461

[29] Blackwelder, R.F., Kaplan, R.E, 'On the wall struc-ture of the turbulent boundary layer', Journal ofFluid Mechanics, 1976, 76, pp. 89-112

[30] Farge, M., 'Wavelet transforms and their applica-tions to turbulence', Annual Rev. Fluid Mech., 24,1992, pp. 395-457

[31] Kronland-Martinet, R., Morlet, J., Grossman, A.,Int. J. Pattern Recognition and Arti�cial Intelli-gence, Special Issue, Expert System and PatternAnalysis, 1, 1987, pp. 273-302

[32] Siggia, E., 'Original intermittency in fully developedturbulence', Physical Review A, 15(4), pp. 1730-1750

[33] Lewalle, J., 'Wavelet Transforms of the Navier-Stokes Equations and the Generalized Dimensions ofTurbulence', Applied Scienti�c Research, 1993, 51,pp. 109-113

[34] Camussi, R., Guj, G., 'Orthonormal wavelet decom-position of turbulent ows: intermittency and coher-ent structures', Journal of Fluid Mechanics, 1997,348, pp. 177-199

[35] Arneodo, A., Grasseau, G., Holschneider, M.,'Wavelet transform of multifractals', Physical ReviewLetters, 61, 1988, pp. 2281-2284

[36] Argoul, F., Arneodo, A., Elezgaray, J., Grasseau,G., Murenzi,'Wavelet analysis of the self-similarityof di�usion-limited aggregates and electrodepositioncluster', Physical Review A, 41, 1990, pp. 5537-5560

[37] Goupilland P., Grossmann, A., Morlet, J., 'Cycle oc-tave and related transforms in seismic signal analy-sis', Geoexploration, 23, 1984, pp. 85-102

21

[38] Meneveau, C., Sreenivasan, K.R., 'The multifractalnature of turbulent energy dissipation', Journal ofFluid Mechanics, 224, 1991, pp. 429-484

[39] Kolmogorov, A.N., 'Dissipation of energy in locallyisotropic turbulence', Doklady Akademii Nauk SSSR,32, 1941, pp. 16-18 (reprinted in Proc. R. Soc. Lond.A, 434, 1991, pp. 15-17)

[40] Collineau, S., Brunet, Y., 'Detection of turbulentmotions in a forest canopy, Part I: Wavelet analysis',Boundary-Layer Meteorology, 65, 1993, pp. 357-379

[41] Collineau, S., Brunet, Y., 'Detection of turbulentmotions in a forest canopy, Part II: Time-scale andconditional averages', Boundary-Layer Meteorology,66, 1993, pp. 49-73

[42] Scanning the special issue on wavelets, Proceedingsof the IEEE, 84(4), 1996

[43] Gordeyev, S., Thomas, F.O., Chu, H.C., 'Experi-mental investigation of unsteady jet shear layer dy-namics using a wavelet decomposition', ASME, FluidEngineering Division, Unsteady Flows, 216, 1995,pp. 167-172

[44] Walker, S.W., Gordeyev, S.V., Thomas, F.O., 'Awavelet transform analysis applied to unsteady as-pects of supersonic jet screech resonance', Experi-ments in Fluids, 22, 1997, pp. 229-238

[45] Gordeyev, S.V, Thomas, F.O., 'Characterization ofunsteady aspects of jet screech using a wavelet anal-ysis technique', Final Project Report for Boeing Cor-poration, PO#Z70423, 1997

22