NASA Technical Publication 3555

Demonstration of Wavelet Techniques in the

Spectral Analysis of Bypass Transition Data

Jacques Lewalle

Syracuse University

David E. Ashpis

Lewis Research Center, Cleveland, Ohio

Ki-Hyeon Sohn

University of Toledo

December 1997
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NASA Technical Publication 3555

Demonstration of Wavelet Techniques in the

Spectral Analysis of Bypass Transition Data

Jacques Lewalle

Syracuse University

David E. Ashpis

Lewis Research Center, Cleveland, Ohio

Ki-Hyeon Sohn

University of Toledo

National Aeronautics and

Space Administration

Lewis Research Center

December 1997
Acknowledgments

This work was performed as part of the Multiresolution Analysis Project (Dr. Dzu K. Le, Principal

Investigator) and was supported by the NASA Lewis Research Center Director's Discretionary Fund.

Available from

NASA Center for Aerospace Information National Technical Information Service

800 Elkridge Landing Road 5287 Port Royal Road

Linthicum Heights, MD 21090-2934 Springfield, VA 22100

Price Code: A03 Price Code: A03
Demonstration of Wavelet Techniques in the Spectral Analysis

of Bypass Transition Data

Jacques Lewalle

Syracuse University

Syracuse, New York 13244

David E. Ashpis

National Aeronautics and Space Administration

Lewis Research Center

Cleveland. Ohio 44135

Ki-Hyeon Sohn

University of Toledo

Toledo, Ohio 43606

Summary

A number of wavelet-based techniques for the analysis of

experimental data are developed and illustrated. A multiscale

analysis based on the Mexican hat wavelet is demonstrated as

a tool for acquiring physical and quantitative information not

obtainable by standard signal analysis methods. Experimental

data for the analysis came from simultaneous hot-wire velocity

traces in a bypass transition of the boundary layer on a heated

flat plate. A pair of traces (two components of velocity) at one

location was excerpted. A number of ensemble and conditional

statistics related to dominant time scales for energy and

momentum transport were calculated. The analysis revealed a

lack of energy-dominant time scales inside turbulent spots but

identified transport-dominant scales inside spots that account

for the largest part of the Reynolds stress. Momentum transport

was much more intermittent than were energetic fluctuations.

This work is the first step in a continuing study of the spatial

evolution of these scale-related statistics, the goal being to

apply the multiscale analysis results to improve the modeling of

transitional and turbulent industrial flows.

Introduction

The traditional tools of signal analysis (ref. I ) work either in

the time domain, where sequential information is available at

the exclusion of spectral information, or in the frequency

domain, where the Fourier-transformed signal conceals the

time of occurrence and provides purely spectral information.

Spectral analysis is essential to describe complex phenomena

and to understand their dynamics; hence, running data through

a Fast Fourier transform (FFT) analyzer is a standard procedure. A Fourier analysis is an attempt to match infinite sinusoidal wave trains with the data. Although this approach is

rewarding and fundamentally justified when periodic components are expected in the signal, it fails to describe the complexity of the system when broadband spectra are obtained. For

example, the infinite repetition of the sine wave in Fourier

analysis is a poor match for the intermittency or modulation of

velocity traces in high Reynolds number flows. Wavelet transforms provide an alternative with a localized waveform. Wavelet methods have evolved over the past few years as a versatile

substitute for earlier time-frequency approaches (refs. 2 to 4).

Practical Relevance of Bypass Transition Study

In this study, wavelet techniques were used to analyze the

multiscale content of one pair of a-v velocity traces from the

data of Sohn and Reshotko (ref. 5). This report identifies,

demonstrates, and discusses the combination of procedures that

yield information (unavailable by other means) about the

energy spectra, momentum transport, and the flow structure

inside and between turbulent spots. This study complements

the work of Sohn, Zaman, and Reshotko (refs. 5 and 6) in

documenting a basic flow that includes some of the complex

dynamics of turbomachinery and other industrial flows.

The sequence of events leading to boundary layer transition

in laminar flows is well understood (ref. 7). However, freestream conditions in turbomachinery are practically never

laminar. If strong enough, the free-stream turbulence can induce

nonlinear perturbations in the boundary layer that bypass the

li near sequence of To]lmien-Schlichting waves and their threedimensional nonlinear offspring, leading to the development of

turbulent spots.

In bypass transition (ref. 8), the sudden appearance of spots

and their rapid growth have been measured in some detail. The

effects of isolated disturbances (ref. 9) and statistically steady

NASA TP-3555
grid turbulence have been documented in a number of papers f(t) signal

(refs.10 to 14). The complexity of the problem increases in

turbomachinery because of the unsteadiness of the ambient f^, f^(K,t) wavelet transform

turbulence associated with wake passing (refs. 15 and 16). For

engineering applications (e.g., in gas turbine design), modeling f:,(K,t) wavelet transform of f(t) with g2 wavelet

the friction and heat transfer in the transition region is an open

problem, and the importance of improving the modeling of g(t) wavelet

these phenomena has been recognized in the work of others

(refs. 17 to 21). The physical content of'practical models could Gaussian wavelet of order n

be improved with a better understanding of the active scales of

motion and their evolution ill transition region. g^ Mexican hat wavelet (Gaussian wavelet of

Bypass transition combines the characteristics of the free- order 2)

stream turbulence with those of the turbulent spots generated in

the boundary layer. Separating the multiple scales involved in j

intermittency factor

this process is important ill the physical phenomena and in improving the computational models based on /(t) indicator function

them. However, the intermittent nature of the flow field has

prevented the effective use of the conventional methods of Re6 Reynolds number based oil thickness

spectral analysis of experimental data. Therefore, the advantages of wavelet analysis over Fourier analysis are illust rated in t time

this report.

t* dimensionless time

The Data

u(t),u streamwise velocity

The experimental data analyzed herein were collected by

Sohn and Reshotko whose work is documented in reference 5 v(t),v normal velocity

and will not be reproduced here. The flow of interest is the

developing boundary layer on a heated flat plate. Free-stream (/ (x t) defined by eq. (13) for u,,

turbulence was generated by one of several grids so that

different turbulence length scales and intensity levels (in the 11 (Kt) wavelet transform of u(t) with wavelet g,

ran ge of 0.5 to 6.5 percent) could be achieved. A three-wire (eq. ( 12))

miniature probe was used to collect simultaneous records

(32 768 points) of streamwise and transverse velocities and It)( IC, 0

-

g, wavelet map

temperature at a sampling rate of 50 kHz. Some of these time

traces were saved for streamwise locations ranging from 9 to V,( IC, t) defined by eq. (13) for v2ri

20 in. from the leading edge, corresponding to Ree (Reynolds

numbers based on momentum thickness) of 470 to 1580. This V(11) gamma function

study used data collected for the grid 1 case (with a nominal

1.1 -percent free-stream turhulence level (ref. 5)) at a point 0 momentum thickness

X = 9 in. (Ree = 470). Althou g h this point is in the early

transition range, a Tollmien-Schlichting spike occurring at rc inverse duration

X = 5 in. in the Fourier power spectra is no longer visible at

X = 7 in. and beyond (ref. 5). The nominal distance to the wal I W, inverse duration of u

Y = 0.030 in. (Y/9-3.0) is within the range where reasonably

energetic spots are suitable forthe illustrations presented herein. x l, inverse duration of v

In a subsequent report, the methods demonstrated will be

applied to calculate the spatial variations of the statistics in the x-t duration of wavelet

transverse and streamwise directions.

T time integration parameter

Symbols TR Reynolds stress (eq. ( 11))

E(w) wavelet power spectrum 0 frequency

2 NASA TP-3555
Subscripts:

turbulent conditioned

order of Gaussian wavelet

1-1 nonturbulent conditioned

Wavelet Transforms

The advantage of wavelet methods over conventional techniques of spectral analysis is the ease with which local spectral

contributions (to energy, transport, or other properties of interest) can be isolated, screened according to various criteria, and

combined into conditional statistics. After a brief introduction

to wavelet transforms, several wavelet tools will be described,

the algorithms elucidated, and the results illustrated. Because

the results are quantitative, they will he interpreted from a

dynamical perspective.

Basic Formulas

From the expanding body of literature on applied wavelet

techniques, only applications to fluid mechanics will be cited.

Details and alternative methods applied to experimental data

can be found in references 16 and 22 to 27. Turbulence direct

numerical simulation (DNS) data were also treated in references 3, 28, and 29. The collection of wavelets considered in

this study was limited to one type, the Mexican hat, but its value

is enhanced by the subsequent processing for selective feature

enhancement and statistical relevance. The work will concern

continuous wavelets because the presentation will be simpler.

As dictated by the data, the actual implementation of the

formulas is in discretized form.

A wavelet g(t) is a function that meets the acceptability

criterion (ref. 2): for a real wavelet, it is sufficient that the

integral over the entire time axis vanish. Practically, one wants

to combine a degree of localization in time with localization in

the frequency domain. The wavelet transform fS of a signal,f(t)

is obtained by convolving the signal with the wavelet and

repeating this process for a range of wavelet durations.

Analytically,

fg (K t)= f K I/2 f(T)g K(T-t)]d r (1)

where the asterisk denotes complex conjugation and T is the

integration parameter representing the translation of the wavelet. The localization of the wavelet is governed by the variable

tand its duration by K-I . Although the inte g ration is nominally

over the entire time axis, only the neighborhood of the center of

the wavelet will contribute significantly to the integral. The

factor K 112 is arbitrary (Wang, Q.; Brasseur, J.G.; Smith, R.W.;

Smits, A.J.: Multi-Dimensional Continuous Wavelet Transforms and Applications to Turbulence Data. Unpublished

report, 1994) and is chosen here for convenience in the Parseval

theorem (see Energy Maps section). The magnitude of the

wavelet transform fJ Kt) reflects the manner in which the shape

of the wavelet matches the local shape of the signal.

Wavelet Selection

The wavelet literature contains references to a large number

of wavelets, variously recommended for their optimal properties or specialized ability to resonate with the data. The wavelet

selection made herein was governed by flexibility and ease of

implementation and interpretation. A wavelet is characterized

by three attributes: shape, duration, and location in time relative

to the data. Time and duration become the two independent

variables, as seen in equation (1). Shape is the dominant degree

of Freedom, affecting the eventual interpretation of results and

the balance that is actually achieved between spectral and

temporal resolution.

From the Gaussian fancily of wavelets of order n,

gn(t*)=(_I)n 

di' 

e -t"h (2)

the Mexican hat wavelet g, (fig' 1) was chosen:

92(t% ) _ (t "' - I ) e

-1 *'/2

(3)

In the following, a wavelet transform of variables with the 92

wavelet will be designated by the subscript 2. The two reasons

for this choice are that ( I ) it is easy (two integrations by parts)

to show that the wavelet transform with g, is equivalent to the

smoothing of the signal with a Gaussian filter followed by

multiplication by a scale-dependent factor and double differentiation with respect to time. Thus, the92 transform will identify

maxima and minima of the smoothed signal. (2) The scale

resolution of the turbulent transport (see the sections from

Correlation Maps to Higher Order Correlations) is greatly

simplified by the fast-inverse formula (developed in ref. 30),

which applies only to g, and to higher order Gaussian wavelets

of even order.

The use of Morlet wavelets (ref. 2) to identify periodic wave

trains (and other specialized waveforms) will he attempted in

the future.

Duration and Frequency

Strictly speaking, frequency is a concept applicable only to

periodic phenomena. Even with respect to broadband signals,

repetition is implied by the decomposition into a Fourier wave

train. However, the emphasis must sometimes be placed on

NASA TP-3555
individual events for which the concept of frequency is confusing and inappropriate. Although the duration ofthe event is the

better concept in this context, there is merit in relying on

familiar terminology and numerical values. Thus, the terms

"duration" ic t and "frequency" 0 will be used herein and their

corresponding values given now. The correspondence between

duration and frequency can be calculated exactly for a cosine

signal f(t) = cos(2tcot) for which the g 2 wavelet transform is

f2 (K, t)

_ _J(2 )(2)TO) cos(27rot)K(-5/2) ex p^ -2X 1) 02 (4)

K`

The duration K-t corresponding to the largest values of' this

wavelet transform is given by

2.5 = 0.2516 (5)

K 27r

This relation was used successfully with broadband signals in

reference 23. The dependency of duration on wavelet shape is

discussed in more detail in reference 31. The conversion factor

of 0.2516 is applicable only to the Mexican hat wavelet. With

this limitation, the correspondence is

I kHz H 0.2516 ms 1 ms <--> 0.2516 kHz (6)

A Wavelet Map

In this study, the wavelet transform (eq. (1)) of each signal is

the first step in implementing all algorithms. Over the range of

times and durations selected, the wavelet coefficients (the

values of' the transform) are computed as arrays of numbers.

The resulting `wavelet map" is the color-coded plots of these

coefficients in the plane of time and duration. A short sample

will be used to illustrate this concept.

Figure 2 shows a portion of a streamwise velocity trace u(t)

and the corresponding wavelet map u,(K t). The presence of'

the turbulent spot is obvious from the raw data and is delineated

by dotted lines t i ,t,. This part ofthe raw signal is referred to as

the spot or the turbulent part. The fluctuations outside the spot

are essentially laminar and are attributed to the footprints ofthe

free-stream turbulence and/or to the spot-induced motion in the

boundary layer. These nonspot segments of' the signal are

sometimes called "laminarlike" or "pseudolaminar," but we

will refer to them as the nonturbulent part.

The wavelet map describes the temporal relationship existing in the various flow events of different sizes and quantifies

its magnitude. The time axis is common to the raw data and to

the abscissa of the wavelet map; the ordinate is usually logarithmic, and short durations (equivalent to high frequency) are

plotted at the bottom of the snap. The wavelet coefficients

(discretized values of u,) are color coded and appear as contour

lines. With signal features sorted by duration, time of occurrence, and magnitude, a quantitative picture emerges for the

succession of features at a given duration (along a horizontal

line) or for the simultaneous presence of features of different

durations (vertical line). For example, in the band of durations

of 0.3 to 2 ms (0.84 to 0.13 kHz) delineated by solid lines a t,a,,

random but apparently homogeneous fluctuations are seen and

are attributed to the footprints of the free-stream turbulence.

The small-scale activity (around 0.1 ms or 2.52 kHz) in the

lower part ofthe figure is limited to a spot located at 12 ms of

this particular sample. Also seen is the local drop in mean

velocity inside the spot (a relatively long-duration section ofthe

horizontal line a t between the dotted lines). This drop is

associated with the three largest values ofthe wavelet transform

and corresponds to the projection of the lower velocities on the

first maximum, minimum, and second maximum ofthe Mexican hat wavelet, respectively. The following sections will

present several criteria used to select sonic features from

wavelet maps. If necessary, these features will be enhanced so

that statistics can be constructed from these events.

Focus on Energy

Energy Maps

One ofthe principal analytical results obtained for continuous wavelet transforms concerns the temporal/spectral distribution of energy. As in the case of the Fourier power spectrum,

the formulas originate from Parseval's theorem (ref. 2). Thus,

for the first two Gaussian wavelets,

r 2 dt d d Kr 2

0 —

_ f 2 E(K) d K (7)

0

by which the expression [f, (K,t) 1

217r can be interpreted as the

local energy spectral density. When this energy density is color

coded in the plane of time and duration, it is called an energy

map. Figure 3 is an energy map (in logarithmic color scale) that

corresponds to the data in figure 2.

The information in the wavelet and energy maps comprises

every local maximum and minimum of the signal. The identification of a multitude of individual energy maxima at different

durations enables the selection of the most sivnificant events.

However, the description of each event individually is an

endless and fruitless task. The most useful applications of

wavelet transforms in the data analysis are those that reduce the

amount of information required to extract meaningful statistics.

A few applications will be presented next.

4 NASA TP-3555
In the case of figure 3, a useful procedure (ref. 24) is to

smooth each constant line icof the map with a Gaussian hI ter of

a matching duration:

f-(K, t) = ^f^^IK,Z^ K CXp — K^^ 

^_t^^dr

 

(8)

I

Then, normalize each line (fixed K) so that it has unit variance.

The result of smoothing and normalizing the energy map is

shown in figure 4, from which a number of observations can fie

made. ( I ) The plot is drastical ly simplified, as intended. (2) The

leading and trailing edges of the spot can be identified over a

range of scales extending over more than a decade (delineated

by thick, solid contour lines labeled A). (3) The small-scale

activity is not homogeneously distributed inside the spot but

appears to be structured as pulses. (4) A band of weak, intermittent activity associated with the nonturbulent part (labeled B) is

noted in the 0.2- to 0.5-ms (1.26- to 0.5-kHz) range. It may be

coincidental that this range brackets the frequency (0.9-kHz or

0.28-ms duration) of the Tollmien-Schliclitim, waves observed

upstream of the present location (ref 5). Inspection of this

segment of velocity signal reveals little evidence of periodic

oscillations, except for perhaps a few periods at the trailing

edge of the spot. However, no such wave trains are visible on

the matching transverse velocity trace. This feature, if confirmed, will be the object of subsequent studies.

Mean Power Spectra and Energetic Scales

From the energy map (fig. 3) and the Parseval relation

(eq. (7)), it is clear that a time integration of the energy map wiI I

yield a distribution of energy E(K) akin to the Fourier power

spectral density (refs. 23 and 24). A direct comparison of the

two spectra is presented in figure 5, in which the duration has

been converted to the equivalent frequency with the aid of

equation (5). Consistent with the interpretation of the wavelet

transform as a bandpass filter (Wang, Q.; Brasseur. J.G.;

Smith, R.W.; Smits, A.J.: Multi-Dimensional Continuous

Wavelet Transforms and Applications to Turbulence Data.

Unpublished report, 1994), the wavelet spectrum is considerably smoother than its Fourier counterpart. It was verified that

the factor of the order of 2 separating the two spectra is

compensated for at the very low frequencies so that both power

spectra integrate to the total energy of the signal. The spectra are

generally featureless, which is in agreement with the results of

reference 5.

The spectral energy content in a frequency band around K is

estimated by the integral

DE(K) = J E(K)d K = J KE(K) d(log K) (9)

A spectral ran ge in the vicinity of a given duration is best

specified on the logarithmic scale; indeed, on a linear scale, the

energy density per unit frequency generally exhibits no dominant scales except at infinite duration. By contrast, on a logarithmic scale, the energy density per octave de-emphasizes the

longest durations, and the presence of a maximum can serve as

the basis for a definition of dominant scales. Accordingly,

energetically dominant scales are identified as the maximum of

KE(K), as shown in figure 6. Also seen is that the energy has a

strong peak at about 1.26 ms (0.2 kHz).

The spots are easily identified on the energy maps of figures 3

and 4 at all durations shorter than 0.3 ms (frequencies higher

than 0.84 kHz). Their failure to have adominant scale signature

in figure 6 could be attributed to the relatively small fraction of

time during which the spots are present (termed the intermittency factor). The followin g subsection addresses the need for

conditional sampling.

Laminar-Turbulent Discrimination and Conditional

Spectra

The need to differentiate the contributions of nonturbulent

(nominally laminar) from turbulent portions of a signal in

intermittent flows is well known. This differentiation is based

on small-scale fluctuations present within the turbulent patches.

Hedley and Keffer (ref. 32) surveyed a number of options and

issued recommendations for the discrimination algorithms.

Variants were later suggested by Antonia (ref. 33). Applications to bypass transition can be found in references 5 and 34,

and some of the steps used in this study are summarized.

A signal such as the it-trace shown in figure 2 contains the

information necessary for making laminar-turbulent decisions

and is termed a discriminator signal. Turbulent spots can be

identified by the presence of small-scale fluctuations (fig. 4)

that, for best results, are enhanced by differentiating the discriminatorsi gnal. Squaring eliminates irrelevant sign information and results in a detector function. Smoothing over scales of

the order of the Taylor microscale tills the gaps between the

small-scale patches, yielding the criterion function. This criterion function is compared with a threshold and results in the

indicator function 1(t), which is 0 at points where the criterion

function is less than the threshold and is 1 otherwise. The

average value of the indicator function is the intermittency

factor. The indicator function is used to mask parts of the data

when the conditional statistics are being calculated (ref. 33).

lnsummary,theconsn-uctionoftheindicatorl'unction requires

choosing a discriminator signal, a level of differentiation, a

smoothin g scale, and a threshold. A good algorithm is characterized by the results being insensitive to these choices and by

the more subjective visual agreement between the indicator

function and the traces of the raw data. The systematic testing

reported in reference 32 supported the aforementioned recommendations. The present study used second derivatives of the

su-camwise velocity to construct the detector function. SmoothNASA TP-3555 5
ing and thresholding were complicated by the absence of' the

reference values of a Taylor microscale and a corresponding

velocity scale in the rapidly evolving spots. The recommendations of Sohn and Reshotko (ref. 5) were followed, and a

smoothing scale of the order of' 15 time steps was used, which

verified that the results were insensitive to the particular value.

For thresholding, a clean separation of' the detector function

into large and small values made very effective a threshold of

the order of I standard deviation of the criterion signal. Again,

the results showed little sensitivity to the selected value.

The indicator function /(t) was then used to mask the energy

map in the construction of the mean spectra. Analytically, the

time integration in equation (7) is restricted to those portions of

the sig nal where the indicator function is nonzero:

Er (K)

27t 

J f^ (K, r) ? l(r) d r (10)

This ability to select portions of the energy map to construct

conditional spectra gives wavelet methods a flexibility that

Fourier techniques do not have, as seen in references 16, 23, and

31 where it was used with different conditionin g functions.

The conditional (turbulent) power spectrum Er(K) and the

corresponding energ y content KEr(K) are compared with the

ensemble means in figures 7 and S. Figure 7 shows that

turbulence conditioning captured virtually all the small-scale

power density (durations shorter than 0.3 ins or frequencies

larger than 0.84 kHz) so that the curves nearly coincide. This is

not surprising because the intermittency criterion specifically

screens for such fluctuations. In the energy-containing range

(around 1.26 ms or 0.2 kHz), the turbulent intervals capture

about 20 percent of the energy. I1' the energy were homogeneously distr ibuted amon g the turbulent spots and the laminar

regions, this value would be in proportion to the intermittency

factor, calculated as 8 percent. The fact that a spot is associated

with large-scale fluctuations more vigorous than those in the

nonturbulent part was noted in relation to the velocity deficit

associated with the spot at the end of the Wavelet Transforms

section. The powerspectra confirm the statistical validity of the

observation.

The conditional energy content is shown in figure 8. A sharp

drop in emphasis is seen in the range of 1.26 ms (0.2 kHz) and

slight activity around 0.085 ms (3 kHz). A sharp increase in

emphasis seen around 0.0252 ms (10 kHz) is attributed to

aliasing noise. The conclusion is that no dominant scales arc

associated with the turbulent spot energy. The nearly uniform

energy content indicates that the conditional power spectrum

(fig. 7) has an extended range of 0.85 to 0.03 ms (0.3 to 8 kHz)

with an approximate slope of x` t , which appears as a nearly

constant value in the plot of KE(K) in figure 8. The same

conclusion was reached with quadrant conditioning of the

power spectra (see the section Quadrant Maps, Histo grams, and

Conditional Spectra).

In summary, this section focused on ener gy considerations

available from single signals. The advantage oftime-frequency

resolutions, displayed graphically by the energy maps, provides greater flexibility than that offered by similar Fourier

methods, particularly in terms of conditional sampling. However, it was expected that turbulent spots would be particularly

active and would yield dominant internal scales relevant to the

parameterization of turbulent transport. Theconclusion reached

in the present study is that either the internal structure of the

spots is comparatively inactive at this location, or that the

techniques presented in this section failed to identify them. The

next section will provide a definitive answer.

Focus on Momentum Transport

In this section, the emphasis shifts from the energy of a single

signal to the Reynolds stress TR

TR = —ptrn (11)

which is obtainable from pairs of signals u and v and where p

is the density.

Quadrant Maps, Histograms, and Conditional Spectra

One of the methods of isolating the Reynolds-stressproducing events in boundary layers is the quadrant-splitting

method from reference 35. Streamwise and transverse velocity

fluctuations can be positive or negative, dividing the a-v plane

into four quadrants. The first quadrant is identified by a positive

it (fluid faster than average) and positive v (fluid moving away

from the wall). Counter clockwise from the first quadrant, the

second quadrant is associated with slower fluid moving away

from the wall (ejection), the third, with slower fluid moving

toward the wall, and the fourth, with fast fluid coming toward

the wall (sweep). In terms of the magnitude rather than the

intermittency factor, the dominance of second- and fourthquadrant events results in a positive Reynolds stress.

A variant of the quadrant-splitting method is scaledependent quadrant splitting that is made possible by the scale

resolution displayed in the wavelet map of figure 2. A quadrant

indicator map can be constructed by scanning quadrant indicator tests at all successive durations. However, the interpretation

of' the result obtained by that method is difficult because the

wavelet coefficients are not the local contributions to the signal.

The inverse transform formula (ref. 2),

u(t) = f )c trl f [,V (t n (K,r)dTdK (12)

0 —^

shows that the local value of a si g nal includes wavelet coefficients from all nearby times, a convolution with the wavelet

NASA TP-3555
similar to the transform formula (eq. (3)). Equation (12) is

greatly simplified in the case of even-indexed Gaussian wavelets (ref. 30) for which

1)11 7r- 1/22 -11 +1/4 -1/2 r K 1

0

= f U11 (K,t)dK (13)

0

Accordingly, the amplitude of the signal u(t) can be reconstructed by a summation of the amplitude transform U,r (K,t) at

instant t (rather than by a convolution involving neighboring

times). Note that U 1 corresponds to the Mexican hat transform

92 , which is used in all the results presented.

The amplitude transform maps lend themselves to scaledependent quadrant splitting. At each time and duration, a

quadrant is identified from the signs of the amplitude transforms of the u- and v-signals. Figure 9 shows the result and a

v-trace from which the spot can be identified easily. The plots

map the islands of quadrant identification (value 1) over the

reference value 0 in the time-duration domain. In the second

quadrant, note that the long-duration ejections extending over

at least a decade from 0.6 ms and larger (0.42 kHz and lower

frequencies) seem to be associated with turbulent spots, an

observation that will be confirmed in the next section. The

observation that certain ranges ol'durations appearmore crowded

in some quadrants (e.g., that the third quadrant contains more

events in the 1.0-ms (0.25-kHz) range than does the second or

fourth) is not supported by the calculation of the intermittency

factor as a function of duration in each quadrant (plotted in

fig. 10). Except for the noise-related increase in intermittency

in the first and third quadrants (same-sign fluctuations of it and

v) at very short durations, the intermittency factors do not

depart much from the equidistribution value of 25 percent.

Minor differences of the order of 5 percent call observed in

the range of 0. I to 3 ms (2.5 to 0.084 kHz) with the larger values

occurring, in the second and fourth quadrants. This equidistribution amona the scale-dependent quadrants contrasts with the

quadrant intermittency factors based on the raw signals for

which the intermittency factors are calculated to be 0.34, 0.16,

0.20 and 0.30 for quadrants 1 to 4, respectively.

In the section Conditional Stress Maps, it will be shown that

this nearly uniform distribution of intermittency factors

between quadrants is not reflected by the corresponding stress

distribution. The magnitude of the contributions to the stress

will be the subject of the following subsections.

Correlation Maps

It is well known (ref. 36) that in turbulent flows, correlations

between velocity components or between velocity and scalar

fields such as temperature indicate the presence of turbulent

fluxes of momentum or concentration. These terms are important to the flow dynamics, and their modeling relies extensively

on the adequate characterization of the length and time scales

associated with the transport process.

The energy spectra provided one definition of dominant time

scales, which have the advantages of being easy to compute and

being, available from single-wire data (as in ref. 16). However,

the failure of the time scales to identify a significant role of the

turbulent spots in figure 8 provides motivation to identify

alternatives. This section uses one of the most innovative and

successful wavelet tools introduced in references 24 and 25.

Consider the Reynolds stress component —puv. . By making

use of the amplitude transform (eq. (13)), it can be rewritten in

the form

TR = —pim = —p p

f ii(t)v(t) d t

0

T-

=—p T f f U,(K,,,t)dK„ f V„(K,,,t)dK,,dt

00 0

--Pf f Un(K,,,t)V(K,.,t)dKr,dK,,

00

--P f f Ku KV Un

(Kill t)V,,(K,,,t)•d(logKrr)d(logK,,)

(14)

Therefore, the Reynolds stress call decomposed into contributions from the plane of the logarithm of the durations of each

component signal. As in the case of the power spectrum,

multiplying by the corresponding inverse durations yields a

quantity that is proportional to the transport accomplished

within a pair of scale ranges. The expression — pK„K,. U„V„ is

called the spectral Reynolds stress content ( for short, Reynolds

stress map).

In the Reynolds stress map of figure 11, the axes are

logarithmic and span the durations of a (abscissa) and v

(ordinate). The color contour lines represent lines of constant

Reynolds stress density as color coded in the scale. If one

thinks of the Reynolds stress map as a surface, the volume

under that surface is equal to the Reynolds stress. The diagonal

structure of this map is notable because it expresses quantitaNASA TP-3555
lively the commonly held belief that velocity components can

be well correlated only if they are in the same range of time

scales (or frequencies). Figure 1 I shows several groups of

dominant active scales separated by factors of the order of 4 to

5. Because modeling has to capture the correct physics, this

separation of active scales should be reflected in the model as

discussed in the Conclusions.

The contrast between the dominant scales identified from

energy considerations (fig. 6) and from momentum transport

(fig. 11) is striking. The unique energy peak at a wavelet

duration of about 1.25 ms (corresponding to 0.2 kHz) is divided

into two active scales on the Reynolds stress map: one at 2.5 ms

(0.1 kHz and labeled A) and one at about 0.6 ins (0.42 kHz and

labeled B). These scales are associated with the nonturbulent

part of the signal. The frequency ratio of 4 between these

nontur bulent scales is unexplained. Note that none of these

scales is in the correct range (0.28 ms or 0.9 kHz) to be

associated with Tolltnien-Schlichtin g waves. Furthermore, the

frequency range typical of turbulent spots and conspicuously

missing from figure 6 is clearly visible in the 0.05-ms (5-kHz)

range on the Reynolds stress map (labeled Q. The combination

of low energy content and large cont-ihution to t ransport led to

the hypothesis that the turbulent spot fluctuations are highly

correlated (addressed in the next section). Finally, note that the

dark circular area in the upper right corner of fi gure I I is

associated with aliasing noise.

Spectral Correlation Coefficients

This section focuses on the correlation coefficients for the

spectral contr ibutions to a and v. The correlation coefficients

arederived by normalizing U„ (Kit I t)V„ (K,., t) by the respective

standard deviations U„ and V„ of U„ (Kt,,t) and V„(Kc „ t), respectively. Recall that wavelet transforms have a zero mean as a

result of the admissibility condition (ref. 2). Thus,

11 U12

Then the spectral correlation map is simply the plot of

U„ ( Ku 1 0 V” (Kr't )

U„ V1'

as a function of Kit and Kv and is shown in figure 12. The laraescale transport is associated with correlation coefficients of the

order of-0.3(2.5 ins or 0.1 kHz) and –0.2 (0.6 ms or 0.42 kHz.).

At the shorter durations associated with the turbulent spots

(fig. 13), the correlation coefficients are much larger, reaching

–0.45. This high level ofcorrelation is indicative of very strong

interactions that organize the velocity field. Although the

energy content of these fluctuations is relatively low, the

combination with high levels of correlation yields one of the

dominant contributions to the transport of momentum (fig. 11).

Conditional Stress Maps

Stress maps similar to that shown in figure I 1 can be constructed by masking out unwanted parts of the data to calculate

conditional stress maps. This selective process was implemented with the maps in the section Energy Maps and will now

be used to focus on the different classes of events and their

contributions to the momentum transport. Four types of conditioning criteria have been considered:

(1) Laminar-turbulent

(2) Quadrant

(3) Scale-dependent quadrant

(4) Event-based (local maxima of energy, as in ref. 23).

In the case of event-based conditioning (4), the results were

rather uninteresting, which indicated a poor match between

energetic fluctuations and momentum transport. The results

obtained from the first three types ofcriteria are presented in the

followin g sections.

Loininar-turbulent conditional scales of niomentuni trartstnort .—The conditional Reynolds stress maps are built from

sampling portions of the signals that are simultaneous with the

presence or absence of turbulent spots, as indicated by the

function 1(t) in

– p ryv^

–Pf ^ Kt, Kv U„O Ktt, t) Vi, (K,.,t)1(t)-d(logK,r)d(logic,,)

(17)

and the result is shown in figure 13. The figure shows that the

large-scale transport islands (2.5 and 0.6 in or 0.1 and 0.42 kHz,

labeled A and B) appear only in the laminar-conditioned map,

whereas the small-scale island (labeled C) appears only in the

turbulent-conditioned map. The isolation of the large-scale

transport islands to be mutually exclusive with the presence of

spots is striking in view of the considerable energy content of

the spots (as shown in fi( T . 7) in this range of durations. This

implies thatonly a fraction of the energetic fluctuations is active

in momentum transport, and this only between spots. This idea

is developed further in the next section.

The lam i nar-conditioned stress map also shows two peaks of

positive a-v at the off-diagonal locations (labeled D and E),

indicating interactions between the two largest active scales.

The reason these peaks do not appear on the ensemble stress

map (fig. I I) is that it is not simply the sum of the two

complementary conditional maps. This is easily seen from the

following relations in which a and v stand for the instantaneous

velocities rather than for the fluctuations:

lctt K 1, (16)

NASA TP-3555
(it - u)(v - V) _ (,t - U)(v - V)i + (it - u)(v - V)(1- 1)

signs. Off-diagonal components in quadrants 2 and 3 show that

a negative a is weakly correlated with v-fluctuations of either

sign at equal or longer durations.

=(n-Ut)(v-Vt)I+ (it - (11-t)(''-VI-t)(I -I)

+i(I - l)(Ut -Ui-t)( V1 -Vi-t)

= ttv t +uv [-t +l(I - l)(UI - Ut - t)( Vi - VI-t)

(18)

In this relation, Ut = U/f/ is the conditional mean estimated

over the turbulent portion of the signal. From equation (18),

note that the ensemble stress map results from the addition of

the turbulent- and laminar-conditioned stress maps and an extra

term dependent on the differences between the conditional

means and the intermittency factor 1 .

Quadrant-conditioned stress nta .—According to this conditioning criterion, the only contributions sampled are those

associated with the quadrant decomposition of' the velocity

signals (i.e., conditioning with the quadrant indicator function).

The results are presented in figure 14 where the magnitudes of

Reynolds stress densities are plotted oil same scale to

facilitate comparisons. The figure shows that quadrants 1 and

4 are practically inactive in the transport of momentum, with all

dominant contributions taking place in quadrants 2 and 3, that

is, with the streamwise velocity below average values. These

two quadrants show a combination of symmetry and

antisymmetry. Along the dia g onal, all contributions are negative (wallward transport), with the second quadrant dominating. The off-diagonal components indicate the coexistence of

long-duration streamwise fluctuations it (abscissa) with shortduration transverse fluctuations v (ordinate). The correlation is

negative in quadrant 2 and positive in quadrant 3. Therefore, it

appears as though, over a range of durations of a decade and a

half, the sign of long-duration transverse velocity fluctuations

responsible for the quadrant labeling applies on average to

shorter durations as well. These observations contrast the near

uniformity in intermittency factors discussed in the Quadrant

Maps section.

Snectrcrl-quadrant-cortditiorted stress nutp.r.—A variant of

quadrant conditioning is sampling points in the time-duration

plane of each amplitude transform map so that the corresponding spectral contribution to the u- and v-signals is in the same

quadrant. This is accomplished by maskin g both it- and v-maps

with the same spectral quadrant indicator map shown in

figure 9. The results are displayed in figure 15. These subsets

of the ori;inal data show that most of the stress is associated

with the turbulent spots in quadrants 2 and 4; shorter duration

laminarcontributions (0.6-ms or0.42-kHzrange) have opposite

Intermittency of Reynolds-Stress -Producing Events

From the previous sections and figures 11 to 15, spectral

contributions to the Reynolds stress were identified. The various plots reveal an overall consistency, particularly for the

contributions from the turbulent spots on the upper right portion

of the diagonals and a puzzling scatter of active islands on and

off thediagonal, particularly in the lower left corner of theplots.

The need to understand the significance of these islands led to

a study of the history of the associated correlations, and the

result is shown in figure 16.

Three representative durations close to the peaks of the stress

map were chosen: 2.24, 1.2, and 0.08 ms (0. 11, 0.2 1, and

3.1 kHz). These durations were plotted in figure 16 with the

matching histories of (1) the turbulence indicator function (spot

identifier); (2) the instantaneous contribution to the energy; (3)

the instantaneous contribution to the Reynolds stress; and (4)

the instantaneous contribution to the energy flux (see the next

section). The figure presents the entire length of the available

records, which makes plotting the velocity traces impractical in

this instance. The indicator function is shown instead, highlighting the locations of the spots. Values were normalized to

unit variance (except for the indicator functions); then values in

excess of three standard deviations were truncated to provide an

adequate depiction of the smaller values, eliminating from the

plots the clutter of large spikes. Also plotted (dashed lines) were

the cumulative energy, stress, and energy flux so that the contributions from very large events could be measured according

to their relative contributions (the cumulative curves are normalized by their extreme value).

In fi g ure 16 (a) at the longer duration of 2.24 ms (0.1 1 kHz),

the energy is only loosely associated with the presence of'spots.

The nonnegative values of instantaneous ener gy build up the

cumulative value gradually, with an occasional small jump

such as that seen at a time of 300 ms. This relatively smooth

evolution contrasts with that of the Reynolds stress curve for

which energetic packets (e.g., at 150 and 200 ms) do not always

contribute to the cumulative stress (or contribute negatively)

and the largest part of the cumulative spectral stress is accounted for by individual jumps. The number of these events at

large duration scales is too small to produce convergent statistics with the present signal length and therefore are called rare

events. This observation may explain the "lumpy" appearance

of some of the stress maps and conditional stress maps at the

longer duration ranges in that a few large events are likely to

determine the magnitude of the cumulative value. An example

of a dominant rare contributor can be seen in fi gure 16(b) at a

time of 350 ms. The gradual Reynolds stress buildup leading to

this (spot-related) event would be canceled by the subsequent

decrease in cumulative stress.

NASA TP-3555
2.5 ms (0.31 and 0.1 kHz) of u2 with I ms (0.25 kHz) of v, 2.5

and 0.1 ms (0.1 and 2.5 kHz) of u'- with 0.15 ms (1.7 kHz) of v,

and finally 0.08 ms (3.1 kHz) of u 2 with 0.06 ms (4.2 kHz) of

v in addition to short-duration noise.

When the flux map is conditioned by sampling only turbulent

or laminar portions of the signals, the result is shown in

figure 18. For turbulent conditioning, the map is approximately

diagonal, with three of the islands identified above dominating

the energy flux. The laminar statistics are predominantly at

0.5 ms (0.5 kHz) of u'- and 1.0 ms (0.25 kHz) of v but are

negative and have a ma g nitude comparable to that of the

turbulent-conditioned fluxes. Hence, it may be concluded that

all off-diagonal features of' figure 17 can be att r ibuted to

differences in the conditional means of u2 and v in the laminar

and turbulent regions.

Figure 19 shows the quadrant distribution of the energy

fluxes. Again, quadrants 1 and 4 are basically inactive whereas

quadrants 2 and 3 (i.e., streamwise velocity smaller than average)

have a lower diagonal structure. It is surmised that the offdiagonal contrihutions are again the result of different sample

means.

The quadrant-conditioned correlation coefficients are reproduced in figure 20. A lumpy diagonal structure is visible in all

quadrants, with large correlation coefficients in quadrant 2 and

in the short durations characteristic of the spots. The lumpy

structure of these maps is clearly due to the few rare events that

account for a significant part of the cumulative value of u2 v at

all durations, as seen in fi g ures 16 (a) to (c). It is concluded that

the flux maps presented here are mostly illustrative of the

potential of the method but are based on records too short to

ensure statistical convergence.

At shorter durations (0.08 ms or 3.1 kHz) typical of spotgenerated turbulence shown in figure 16(c), the spots themselves become typical contributors, as would be expected, but

the larger number of them available in our records leads to a

smoother and more nearly monotonous buildup of the cumulative spectral stress. Departures from the linear growth of the

cumulative energy and Reynolds stress are smaller, indicating

that individual events have less impact on the cumulative value

than they have at longer durations.

Higher Order Correlations

The correlation methods presented in the previous section

can be applied to higher order moments. For example, the

kinetic energy fluxes can be mapped by correlating an energy

map (u'- in this section) with the amplitude wavelet map of

another velocity component (v in this instance). The interpre2

tation (fig. 3) of d K u„ (K, t) l27cas instantaneous energycan A

0

be used to estimate the spectral content energy fluxes of the type

u - v. Then,

11V VW - 271 J 

d Kit J

0 0

1

= d^log Ku) 

J 

Kil K l. III ^Kil , ' I I VI 

(

Kv , t) d(log K,)

27r f

Conclusions

!!^^ K ,t) VI(KI.,t)-dKi,

u -

(19)

Figures 17 to 20 show the spectral energy flux content measured by the integrand in equation (19). The ensemble Ilux map

in figure 17 is remarkable for several features. First, there is a

positive correlation between excess energy and motion away

from the wall for all pairs of' durations. Second, the upper

diagonal structure of the map shows that energy fluctuations at

one given duration seem con-elated with a positive v at all

durations comparable or smaller. In particular, a vertical band

on the left side of the plot shows that energ y in the 2.5-ms

(0.1-kHz) range is correlated with normal velocities at all

durations, and the horizontal band in the middle of the plot

shows that transverse velocities in the 0.15-ms (1.7-kHz) range

are strongly correlated with energy fluctuations at the larger

durations. This may indicate a coupling between the activity of'

the spot in the 0.1-ms (2.5-kHz) ran ge and the energ y fluctuations in the 2.5-ms (0.1-kHz) range. Finally, the lumpiness of

the map seems to indicate active frequency bands: 0.8 and

The conclusions are presented at four levels:

I . At a general level,

(a) Experimental and numerical data contain much

more information than that made available by conventional methods of signal analysis. In the wavelet

representation, much of' that information is made

more accessible by the time-frequency resolution.

(b) Our procedure consists of two major steps: mapping the data in the time-frequency domain and

reducing the volume of information by statistical

means. By using only the Mexican hat wavelet (a

relatively simple wavelet), a large number of

ensemble and conditional statistics have been

devised. Many more can conceivably be explored,

and other wavelets will emphasize other features in

the data.

10 NASA TP-3555
(c) As in the cited cases of other data explored by similar

methods, new algorithms were tested in this project

and provided answers to new questions. Selecting

the more useful algorithms will remain a considerable task until experience reveals those that show

the most promise.

(d) The new statistics calculated are quantitative and

can be used to compare related data sets. With more

diagnostic tools available, significant differences in

the underlying physics are more likely to be exposed.

2. With regard to the specific flow physics of this data set,

(a) The dynamical role played by the internal structure

of the spots is highlighted differently by energyrelated and Reynolds-stress-related methods. Freestream turbulence and spot-related fluctuations in

the nonturbulent portions of the flow (between

spots) are energetically dominant whereas the fluctuations internal to the spots are much less energetic

but dominate the momentum transport.

(b) This conclusion has major implications for data

collection and filtering in transitional flows because

transport is at the core of all modeling efforts. Preservation of the energetic events cannot be viewed

as a satisfactory criterion.

(c) Contributions to the momentum transport were

much more intermittent than the contributions to

the fluctuating energy. At the smaller time scales,

a large collection of events is nonetheless obtained,

and statistics can be considered stable. However, at

the Ion gertime scales, the rare major events included

in the records dominate the magnitude and timescale of the statistics.

(d) The use of longer records, or a large number of

smaller independent samples, is required to yield

stable statistics of the shear stress and the higher

moments. This has been known for a long time, but

the rates of convergence are likely to be much

slower with rare events than they would be with a

stationary signal.

3. With respect to the reduction of the data and the extraction

of time scales,

(a) The ultimate reduction of the data consists of the

extraction of a few simple parameters. In the case of

the pair of anemometry signals examined in this

report, the focus was deliberately on the extraction

of meaningful time scales from the data. The reason

is that once dominant energy and time scales are

defined and quantified, they can be used for engineering modeling. The modeling of boundary layer

transition in noisy and/or transient industrial flows

could benefit from a systematic documentation of

eddy viscosities by these methods.

(b) The time scales are constructed in various ways by

sorting the information according to scale using

wavelet transforms. In contrast to Fourier-based

spectral discrimination, it was possible to identify

individual events, screen them according to various

criteria, and construct meaningful time scales

attached to different classes of events.

(c) The contrast in the time scales obtained from singlesignal, energy-based statistics and two-signal, stressbased statistics is relevant not only to the physical

understanding of transitional flows but also to the

modeling of these flows.

4. With regard to future work,

(a) The techniques presented here provide the basis for

abundant future work that will be pursued in several

directions. The spatial evolution of the statistics

developed here is reported in a separate paper.

Spectral information about the internal structure of

spots may result in a better understanding of receptivity, formation, and evolution.

(b) The application of other wavelets may enhance

different features in the signals and yield valuable

statistics to complement the current results.

(c) Finally, other statistical quantities such as scaledependent probability distributions and moments

are being developed.

In conclusion, wavelet tools were used to gain insight into the

physics and dynamics of the flow. The full impact of this work

will be seen in the potential future use of these results for

engineering modeling. Scale-dependent physics, particularly

scale-dependent dynamics and transport, must be incorporated

in the models if great improvements in them are to be realized.

Lewis Research Center

National Aeronautics and Space Administration

Cleveland, Ohio, July, 1997

NASA TP-3555
References

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June 1972, pp. 39-48.

36. Tennekes H.; and Lumley, J.L.: A First Course in Turbulence. M.I.T.

Press, Cambridge, MA, 1972.

12 NASA TP-3555
2

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ro -4

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3

-6

ro -8

1.0

N

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a) 0.5

a^

ro

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ro 0.0

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ro

X -0.5

-1.0

-4 -2 0 2 4

Dimensionless time, t*

Figure 1.—Mexican hat wavelet, 92.

-10

5 10 15 20 25

I II E

l Y I

C

U

ro

^ 1.0 I

N

3 T

^ I

d

CO

3 I ^ I

= I

0.1 I

C

roUI I X I

U I I

I

5 10 15 20 25

Elapsed time, t, ms

U2

-0.15 0 0.15

Figure 2.—Excerpt from streamwise velocity trace u(t) and its Mexican hat 92 wavelet map u 2(K,t) in ms-112

ai

a2

NASA TP-3555 13
U)

E

iY

C

O

roL

1.0

N

f^

3

ro

0.1

Cz

O

X

O

2

2

C 0

E

-2

ro -4

ro

3

-6

mc° -8

-10

5 10 15 20 25

5 10 15 20 25

Elapsed time, t, ms

Iu212/ (2r,)

0 0.01

Figure 3.—Energy map u 2 (K,t)I 2/ (2r,) for streamwise velocity trace excerpt u(t) (same as fig. 2).

14 NASA TP-3555
U)

E

Y

C

O

.ro

1.0

N

w

1

ro

3

ro

L 0.1

c

ro

U_

X

d

2

U 0

E

c -2

cv —4

ro

3

-6

Ir -8

10 5 10 15 20 25

5 10 15 20 25

Elapsed time, t, ms

r

W2

0 5

Figure 4.—Smoothed and normalized version of energy map of figure 3. Note hints of internal spot structure, which

could be Tollmien-Schlichting packets.

v 10

w

10

c

a

3 10 Wavelet

Fourier

O

a

m 10

U

U

a

N 10

0.1 1.0 10.0

Frequency, (b, kHz

Figure 5.—Mean wavelet spectrum and Fourier power

spectrum are generally featureless.

0.8

w

c 0.6 a^

C

O

0.4

Fourier

mc

0.2

Wavelet

U

a 00

UD 0.1 1.0 10.0

Frequency, (^, kHz

Figure 6.—Wavelet mean spectral energy content

at various frequencies and Fourier counterpart.

(Multiplying the power spectra in fig. 5 by the

corresponding frequencies yields the wavelet

mean spectral energy content.)

NASA TP-3555 15
v 10-3

W

N

10-4

a

3

n 10-5

ro

U

cn 10-6

Ensemble

Turbulent- ,. spectrum

conditional (fig. 5)

0.1 1.0 10.0

Frequency, c1,, kHz

Figure 7.—Conditional wavelet spectrum obtained by

including only contributions to the total energy that

are simultaneous with the presence of small-scale

turbulence.

v 0.4

w

c

Ensemble

o

T 0.2

/conditional m 

entc ^

ro..................

U _

a 0.0

cn 0.1 1.0 10.0

Frequency, (^, kHz

Figure 8.—Conditional energy content in turbulent spots

(Except for noisy high-frequency fluctuations, no

dominant scales are associated with turbulent spot

energy.)

16 NASA TP-3555
cd 4

-0 F0

rr

-4

q

P

df

p

^^ ^ I I I

E

C

1.0

a

> 0.1

ro

(a)

^Il^hl^l^^ll^a^^^I i^ ^ II ' I dIr^VV^d^i4 I I

^

II^

c 1.0

O

.ro

> 0.1

ro

(b)

E

C 1.0

0

0

0.1

(c)

E

C 1.0

0

0 0.1

IV^I^{^

5 10 15 10 25

(d) Elapsed time, t, ms

Figure 9.—Quadrant maps indicating that spots may be associated with large scale

ejections. (a) Quadrant 1. (b) Quadrant 2. (c) Quadrant 3. (d) Quadrant 4.

NASA TP-3555 17
1.0 1.0

(b) (a)

0 0.8 0 0.8

U U

ro ro

U 0.6 0.6 C

N U

0.4 0.4

a^

c c

c 0.2 c 0.2

ro ro

-a

ro ro

0.0

0.1 1.0

0.0

0.1 1.0

Wavelet 92 duration, K-1 , ms Wavelet 92 duration, K- 1 , ms

1.0

(c)

0 0.8

U

ro

c

0.6

m

0.4 a^

c

ro

0.2

ro

' 00

1.0

(d)

0 0.8

U

ro

= 0.6

m

F

a 

0.4

^

ro

0.2

-o

ro

00

0.1 1.0 0.1 1.0

Wavelet 92 duration, K-1 , ms Wavelet 92 duration, K-1 , ms

Figure 10.-Spectral quadrant intermittency factors measure fraction of the time at each duration that spectral

contributions to fluctuations are in given quadrant. (a) Quadrant 1. (b) Quadrant 2. (c) Quadrant 3. (d) Quadrant 4.

18 NASA TP-3555
U)

E

C

B

U)

E

^Y> 0.1

O

C

O

ro

N

0)

1.0

i

>0.1

Y

O

CO

ro

N

m 1.0

AlpA

1.0 0.1

Wavelet 92 duration Of u, Ku 1 , ms

I ]

-0.001 0 0.001

Figure 11.—Reynolds stress map showing filled contours

Of K U K V UI (K U ,t) V1 (KV ,t) in m 2/sec 3 as function of

durations K 1 and KU 1 . Reynolds stress is associated

with three narrow ranges of scales at durations of 2.5 ms

(A), 0.6 ms (B), and around 0.05 ms (C). Noise is also in

evidence in the 0.03-ms range.

1.0 0.1

Wavelet 92 duration Of u, K U 1 , ms

-0.5 0 0.5

Figure 12.—Spectral correlation coefficients showing filled

contours Of K U KvU1 (K U ,t) V1 (KV , t)/(U1 V^ ) as function of

KU1 and Kv1

NASA TP-3555 19
O

C

O

.ro

N

CM 1

1.0

m it

E

Y 

> 0.1

C

1.0 0.1

Wavelet 9 2 duration of u, K U 1 , ms

IP_^

-0.001 0 0.001

(a)

B

E

D

A (b)

V)

E

i > 0.1

Y

O

c

0

Y

ro

L

N

m

0 1.0

ro

1.0 0.1

Wavelet 92 duration of u, KU 1 , ms

-0.0005 0 0.0005

Figure 13.—Stress maps. (a) Turbulent-conditioned.

(b) Laminar-conditioned. Three dominant scales noted

in figure 11 are cleanly separated into a contribution

from turbulent spots at small scales (C) and a double

contribution from nonturbulent fluctuations at larger

scales (A and B). Note regions of positive u-v

correlations (D and E).

20 NASA TP-3555
E

^>

Y 0.1

0

c

0

N

1.0

c0

1

U)

E

Y> 0.1

0

c

0

ro

c

N

d

^ 1.0 AP

V)

E

:e 0.1

0

c

0

ro

a

N

m 1.0

(Q

En

E

^>

Y 0.1

0

c

0

N

N

1.0

cu

(b) (a)

L III II I I

1.0 0.1 1.0 0.1

Wavelet 92 duration of u, Ku l , ms Wavelet 92 duration of u, Ku ms

4

I

iV

-0.001 0 0.001 -0.001 0 0.001

4 (c) (d)

1.0 0.1

Wavelet 9 2 duration of u, Ku 1 , ms

I 1^

-0.001 0 0.001

Figure 14.—Quadrant-conditioned stress maps. (a) Quadrant 1

1.0 0.1

Wavelet 92 duration of u, Ku 1 , ms

-0.001 0 0.001

(b) Quadrant 2. (c) Quadrant 3. (d) Quadrant 4.

NASA TP-3555 21
if N

E

Y> 0.1

0

C

0

L

N

Q1

1.0

ro

N

E

'Y 0.1

0

c

ro0

N

O)

1.0

ro

(b)

1.0 0.1

Wavelet 9 2 duration of u, KU 

1, ms

-0.001 0 0.001

(a)

1.0 0.1

Wavelet 92 duration of u, K 1 , ms

i V

-0.001 0 0.001

V)

E

Y > 0.1

O

C

O

roL

N

a)

1.0

>

ro

(c)

Cn

E

Y > 0.1

O

C

O

ro

N

0)

1.0

>

ro

(d)

1.0 0.1

Wavelet 92 duration of u, KU 1 , ms

I _M 1, ^_

-0.001 0 0.001

Figure 15.—Spectral quadrant-conditional stress maps. (a) Quadrant 1

1.0 0.1

Wavelet 9 2 duration of u, KU 1 ms

I Ij

-0.001 0 0.001

(b) Quadrant 2. (c) Quadrant 3. (d) Quadrant 4.

22 NASA TP-3555
4 1 1, 11 1 U I..... [4 ---------h------i,...f ---A I .1- -^

(3)

(4)

(3)

(a) 1.0 -

..,... ;

0.0 (4)

-1.0

1.0

0.0 (3)

-1.0

1.0

0.0 aL ^VVV1^ (2)

0.0 LT 

I I I Ill I_l 111 1 II 11 I I III I I I I II I I ^^ lfl (1)

0 100 200 300 400 500 600

Elapsed time, t, ms

(b) 1.0

0.0

-1.0

1.0

0.0

-1.0

1.0

(4)

0.0 Wt ^^^ _ *r^ianlrrlr ^irll^^ ^^l^ira^lw^lnll^ ^dr^ull^ ^q (2)

00. L^ I I I fll I I I^ fl III I I III I I II II I ^I 111 (^)

0 100 200 300 400 500 600

Elapsed time, t, ms

M 1.0

0.0

-1.0

1.0

0.0

-1.0 N n r i i i ii w iii i i i i i i i i r.._• r•• 1•• 1I

1.0

01" . '.

-

( 2)

0.0

0.0rn II II III^I III II^III III II II III I I

(1)

0 100 200 300 400 500 600

Elapsed time, t, ms

Figure 16.-Histories and spectral contributions to (1) turbulence indicator function, (2) energy, (3) Reynolds stress, and (4)

energy flux. (a) Duration, 2.24 ms (0.11 kHz); wavelet mode n = 5. (b) Duration, 1.24 ms (0.2 kHz); wavelet mode n = 8.

(c) Duration, 0.08 ms (3.2 kHz); wavelet mode n = 22. Note: solid lines are normalized histories (truncated at 3 standard

deviations and normalized to unit variance); dashed lines are cumulative values (integrals of normalized history).

NASA TP-3555 23
V)

E

iy 0.1

0

C

0

N

CD

1.0

V)

E

7>

Y 0.1

0

c

0

a

N

1.0

ro

E

>

Y 01

0

C

0.ro

N

Q1

1.0

ro

1.0 0.1

Wavelet 9 2 duration of u, K U 1 , ms

-0115 0 0.15

1.0 0.1

Wavelet 9 2 duration of u, K

u

1 , ms

I V

-0.002 0 0.002

(a)

Figure 17.—Spectral distribution of transverse

energy fluxes showin g contour levels of

K u K v lu2(K U , t) 1 2 V1 (KV ,t) as function of K 

1 and KV1

(b)

1.0 0.1

Wavelet 9 2 duration of u, K u

1 , ms

5i

-0.002 0 0.002

Figure 18.—Turbulent and laminar contributions

to spectral flux map in figure 17. (a) Turbulent

conditioned. (b) Laminar conditioned.

24 NASA TP-3555
(b) (a)

U)

E

7>

Y 0.1

0

c

O

c}Lo

N

> 1.0

ro

U)

E

y 0.1

0

c

O

ro

a

N

Q7

> 1.0

M

V)

E

' 2> 0.1

0

c

0

N

a) 1.0

cd ^ ^ 1

V)

E

Y 0.1

0

c

0

a

N

Q1

m 1.0

1.0 0.1 1.0 0.1

Wavelet 9 2 duration of u, K U 1 , ms Wavelet 92 duration of u, K u

1 , ms

-0.5 0 0.5 -0.5 0 0.5

1.0 0.1 1.0 0.1

Wavelet 92 duration of u, Ku1 , ms Wavelet 92 duration of u, K u

1 , ms

i

-0.5 0 0.5 -0.5 0 0.5

Figure 19.—Quadrant contributions to spectral flux map in figure 17. (a) Quadrant 1. (b) Quadrant 2. (c) Quadrant 3.

(d) Quadrant 4.

NASA TP-3555 25
U)

E

^Y> 0.1

0

c

0

a

N

1.0

T-- , ; -T1.

E

;>Y 0.1

o ^

c

0

N

1.0

ro

(b)

1.0 0.1

Wavelet 9 2 duration of u, KU 

1, ms

-0.002 0 0.002

E

T>

Y 0.1

0

C

0

m

N

1.0

1.0 0.1

Wavelet 9 2 duration of u, KU t , ms

-0.002 0 0.002

(a)

E

:e 0.1

0

c

0

a

N

U)

N

1.0 r

M

(d)

1.0 0.1 1.0 0.1

Wavelet 92 duration of u, KU 1 , ms Wavelet 92 duration of u, KU^, ms

-0.002 0 0.002 -0.002 0 0.002

Figure 20.—Correlation coefficients corresponding to energy fluxes in figure 19. (a) Quadrant 1. (b) Quadrant 2.

(c) Quadrant 3. (d) Quadrant 4.

26 NASA TP-3555
REPORT DOCUMENTATION PAGE _T Form Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

Decemher 1997 Technical Paper

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Demonstration of Wavelet Techniques in the Spectral Analysis of

Bypass Transition Data

WU-522-31-23 6. AUTHOR(S)

Jacques Lewalle, David E. Ashpis, and Ki-Hyeon Sohn

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSES) 8. PERFORMING ORGANIZATION

REPORT NUMBER

National Aeronautics and Space Administration

Lewis Research Center E-9675

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORfNG AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

National Aeronautics and Space Administration

Washington, DC 20546-0001 NASA TP-3555

11. SUPPLEMENTARY NOTES

Jacques Lewalle, Syracuse University, Syracuse, New York 13244-0003; David E. Ashpis, NASA Lewis Research Center;

and Ki-Hyeon Sohn, University of Toledo, Toledo, Ohio 43606. Responsible person, David E. Ashpis, organization code

5820, (216) 433-8317.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited

Subject Categories: 34 and 02 Distribution: Standard

This publication is available from the NASA Center for AcroSpace Information, (301) 621-0390.

13. ABSTRACT (Maximum 200 words)

A number of wavelet-based techniques for the analysis of experimental data are developed and illustrated. A multiscale

analysis based on the Mexican hat wavelet is demonstrated as a tool for acquiring physical and quantitative information

not obtainable by standard signal analysis methods. Experimental data for the analysis came from simultaneous hot-wire

velocity traces in a bypass transition of the boundary layer on a heated flat plate. A pair of traces (two components of

velocity) at one location was excerpted. A number of ensemble and conditional statistics related to dominant time scales

for energy and momentum transport were calculated. The analysis revealed a lack of energy-dominant time scales inside

turbulent spots but identified transport-dominant scales inside spots that account for the largest part of the Reynolds stress.

Momentum transport was much more intermittent than were encr aetic fluctuations. This work is the first step in a continuing study of the spatial evolution of these scale-related statistics, the goal heing to apply the multiscale analysis results to

improve the modeling of transitional and tllrhLIlCI1t industrial flows.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Wavelets; Multiresolution; Bypass transition; Transition; Boundary layers; TLIrhU1CPCe: 32

16. PRICE CODE Conditional-statistics; Mexican hat wavelet; Intermittency

AW

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACT

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