Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 4, pp. 833-852, Warsaw 2007

SPACE-TIME/FREQUENCY/SCALE REPRESENTATION OF
THE TURBULENCE NEAR THE WALL

F. Sedat Tardu

LEGI BP 53 X 38041 Grenoble Cédex, France

e-mail: sedat.tardu@hmg.inpg.fr

It is proved that theWindowAverageGradient (WAG) schemedesigned
to detect singularities in the turbulence is approximately equivalent to
the Hilbert transform of the ”Mexican Hat” wavelet. Several identities
are derived between these schemes and their validity are theoretical-
ly and experimentally shown through the fluctuating wall shear stress
and velocity data taken in the turbulent boundary layer. The instanta-
neous amplitude-frequency representation of WAG at the large scale is
also considered. These results indicate that the wall turbulence is signi-
ficantly regular. It is shown that the near wall singularities involve in
the large scale frequency shift key process and that the corresponding
instantaneous phase consists of discontinuous line segments.

Key words: nearwall turbulence, singularities, wavelet transform,WAG,
instantaneous amplitude and phase

1. Introduction

1.1. Aim of the study

Wavelet analysis has become now a classical tool of multirate signal ana-
lysis in both time (space) and scale, and found several applications in the
analysis of turbulence (Farge, 1992; Akansu and Smith, 1998). One of the
major attractive features of this method is its ability of detecting localized
singularities independently of the choice of the analysing function. This pro-
perty makes this tool somewhat universal. Another scheme developed for the
same aim, and which will be detailed in Section 1.2, is the Window Average
Gradient (WAG) scheme. This paper deals mainly with the interrelationships
between these two techniques.



834 F.S. Tardu

Thepresent study is somewhat inspired by the results presented byDe Su-
za (1997) and De Suza et al. (1998). These authors analysed the interaction
of a wake generated by a cylinder placed in the inner layer, with the near
wall turbulence. They used theMexican Hat wavelet transform andWAG for
pattern recognition of thewake structures.We noticed a remarkable similarity
of the conditional pseudo-streamlines, vorticity fields andwaveforms resulting
fromboth techniques. Therefore, the first question that arises is whether these
methods are identical or not. The answer is not immediate, because of strong
(apparent) differences of the analysing functions.We clearly have two targets
here: the first one is of practical importance and deals with the clarification
of these schemes. The second is of a more general nature and handles the
interpretation of the wavelet analysing functions in general.

We aim at giving answers to these questions both theoreticaly and expe-
rimentaly in this paper.We first point at the fact thatWAG is indeed one of
the oldest version of the wavelet analysis, in 1.2.2. We subsequently establish
equivalent relationships between the WAG and Mexican Hat wavelet trans-
form bymaking use of the classical system theory approach. Furthermore, we
introduce in Section 2.3 the space (time)-frequency-scale representation of the
wavelet transform and, apply it to the singularities detected in the near wall
region of a turbulent boundary layer in Section 4.2.

1.2. Background and transfer functions

1.3. Window average gradient scheme

Thewindow average gradient schemewas first introduced byAntonia and
Fulachier (1989) and widely used in studies dealing with different flow confi-
gurations, for example in Antonia et al. (1990), Krogstad andAntonia (1994),
De Suza (1997) andDe Suza et al. (1998). This scheme has been developed to
detect discontinuities in fluctuating velocitiy signals andmainly been applied
towall boundedflows, although not exclusively. The continuous version of the
WAGdetection scheme is defined through amovingwindowof thewidth 2Tw,
and the data is transformed into

W(t,Tw)=
1

2Tw

( t+Tw∫

t

u(t) dt−
t∫

t−Tw

u(t) dt

)

(1.1)

where u(t) is the fluctuating turbulent velocity signal. It can be easily seen
that W(t,Tw) is the output of a linear systemwhose transfer function hW(t)
is defined by



Space-time/frequency/scale representation... 835

hW(t)=






1

2Tw
for Tw ¬ t< 0

−
1

2Tw
for 0¬ t<Tw

(1.2)

and hW(t) = 0 otherwise. Therefore, W(t,Tw) = u(t)⊗ hW(t), where ⊗
stands for the convolution operator. Since the input-output relationships of
the linear system are determined through the deterministic autocorrelation
function defined as the convolution ρw(t) =hw(t)⊗h∗w(−t) where the super-
script ∗ indicates the complex conjugate. The function ρw(t) and its energy
spectrum ρ̂w(ω) = ĥw(ω)ĥ

∗

w(ω) are shown in Fig.1 and Fig.2. They will be
largely used hereafter.

Fig. 1. Correspondance between the window average gradient scheme and the
Mexican Hat transform; (a) autocorrelation function ofWAG (full line) compared
with the modifiedmotherMexican Hat function (circles); (b) transfer function of

WAG, (c) system function betweenWAG andMexican Hat

1.3.1. Wavelet transform

The wavelet transform Ω(k,t) of the signal u(t) is defined by

Ω(k,t) =
√
k

∞∫

−∞

u(τ)g[k(τ − t)] dτ =
√
ku(t)⊗g(−kt) (1.3)

where g(t) is the mother wavelet. The wavelet transform is covariant under
time translation and scale change. It conserves the energy of the signal and



836 F.S. Tardu

Fig. 2. Fourier transform of theWAG autocorrelation function and of the scaled
Mexican Hatmother wavelet

is invertible provided that the admissibility condition is satisfied. The latter
implies

∞∫

−∞

g(t) dt=

∞∫

−∞

tng(t) dt=0 (1.4)

although the second condition of vanishing moments is not compulsory. In
the discrete version of the wavelet transform, k and t are often sampled as
k= km0 , t=nt0k

−m
0 such that

√
kg[k(τ − t)] =

√
km0 g[k

m
0 (τ−nt0)] = gmn(τ) (1.5)

The coefficients Ω(k,t) may then be mapped into Ω(m,n) and the signal is
recovered through

u(τ)=
∞∑

m=1

∞∑

n=−∞

Ω(m,n)gmn(τ) (1.6)

This summation requires an infinite number of terms because of the bandpass
character of thewavelet gmn(τ). In order to express a finite resolutionwavelet
decomposition at some finite level m = L, it is necessary to introduce a
complementary low pass scaling basis cmn(τ) in such a way that

u(τ)=
L∑

m=1

[ ∞∑

n=−∞

Ω(m,n)
1

2m
g
( τ
2m
−n
)
+

∞∑

n=−∞

Ω(L,n)
1

2L
c
( τ
2L
−n
)]
(1.7)

The scaling function cmn(τ) is generic ofwhat is left in the original signal once
the details are sorted out through Ω(m,n) up to a given scale L. It plays an



Space-time/frequency/scale representation... 837

important role in the wavelet theory but has not drawn enough attention
in the turbulence analysis. Note, for instance, that cmn(τ) may reveal the
characteristics of large scale passive eddies near the wall, when the associated
mother wavelet is chosen in some suitable way.

The window averaged gradient scheme defined in (1.1) is per se a wavelet
transform since its transfer function hw(t) is admissible. It is indeed closely
related to the Haar transform introduced in 1909 (see for example Meyer,
1992). The later is the simplest wavelet used in multiresolution analysis. Its
mother function without translation and dilatation is defined as

gH(t)=






1 for 0¬ t < 1
2

−1 for 1
2
¬ t< 1

and gH(t) = 0, otherwise. It may be easily shown that WAG is a non-causal
version of the Haar transform. For a given window time TW , one has

hW(t−TW)=
1

2TW
gH
( t
2TW

)

with the equivalent relationship in the Fourier space

ĥ(ω,TW)= e
jωTW ĝH(2TWω)

Furthermore, the complementary scaling functions ofWAG andHaar, defined
in (1.7), are related through

cW(t−2TW)=
1

2TW
cH
( t
2TW

)

which implies that

cW(t)=






1

2TW
for −2TW ¬ t< 0

0 otherwise

The low pass complementary scaling function of WAG corresponds to a time
shifted moving average transfer function. The interscale basis coefficients of
both schemes are invariant under these transformations. The users of WAG,
to the author’s knowledge, have not noticed before these equivalences.

One may imagine and construct an infinite number of admissible mother
wavelets dependinguponthe typeofmultiscale analysis onewishes toperform.



838 F.S. Tardu

One of the most widely used real wavelets in studies dealing with turbulen-
ce is the Marr wavelet (called also the ”Mexican Hat”) which is the second
derivative of Gaussian with the analysing function given by

g(t)= (t2−1)exp
(
−
t2

2

)
(1.8)

The reason of popularity of this transform may be explained by its efficiency
in detecting the discontinuities. The Marr transform is indeed equivalent to
the smoothing of the signal by a Gaussian and to its subsequent Laplacian.
It focuses, therefore, on the edges of sharp temporal variations. For more
details concerning thewavelet transform,we refer to an extensive bibliography
provided in a recent book edited by Akansu and Smith (1998).

2. Comparison between the window average gradient scheme
and the wavelet transform by Mexican Hat

2.1. Direct correspondencebetween theautocorrelation functionofWAG
and the transfer function of ”Mexican Hat”

There is a curious coincidence between the detection scheme ofWAG and
the ”Mexican Hat” abbreviated as MH hereafter. Figure 1a shows that the
correlation function ρW(α) of WAG is well approximated by

ρW(α)≈
k

3
(1−k2α2)exp

(
−k
2α2

2

)
=−k
3
g(kα) (2.1)

when the scale dilatation parameter is set as k = 3/(2TW). It is recalled
that g stands for the convolution operator occurring in theMexicanHat. The
fact that the kernel of MH compares well with the deterministic correlation
function ofWAGvia a scaling factor, may also be checked by comparing their
respective Fourier transforms

ρ̂W(ω)=
4

ω2T2W
sin4
ωTW
2

(2.2)

and

−k
3
ĝ(ω)=

4
√
2π

27
ω2T2W exp

(
−2ω

2T2W
9

)
(2.3)

where k = 3/(2TW) has been used to obtain the last relationship. Figu-
re 2 shows these distributions versus ωTW/(2π). The correspondence between



Space-time/frequency/scale representation... 839

ρ̂W(ω) and −kĝ(ω)/3 is quite satisfactory despite somedifferences, in particu-
larnear the localmaximawhere −kĝ(ω)/3 is slightly larger.Thedampedsmall
oscillations at high frequencies inherent in ρ̂W(ω) are absent in −kĝ(ω)/3 but
their effect are presumably negligible. Notice furthermore that the determini-
stic autocorrelation function ρW(α) ofWAG is admissible in terms of wavelet
terminology, i.e it has zero mean, since ρ̂W(0)= 0.

These observations lead us to conclude that the correlation function of
WAG is a satisfactorily linearized version of the function −kg(kα)/3. Conse-
quently, onemay easily relate theMH transform of a stochastic signal u(t) to
itsWAG transform. TheMH transform Ω(k,t) of u(t) is indeed nearly equal
to

Ω(k,t)=
√
ku(t)⊗g(−kt)≈−

3√
k
u(t)⊗ρW(t) (2.4)

One has therefore

Ω(k,t)≈− 3√
k
u(t)⊗ [hW(t)⊗h∗W(−t)] (2.5)

and finally

Ω(k,t)≈
3√
k
W
(
t,TW =

3

2k

)
⊗hW(t) (2.6)

since h∗W(−t) = −hW(t). It is seen that the wavelet transform by ”Mexican
Hat” is equivalent to the convolution of WAG by its own transfer function,
provided that the window averaging time is appropriately chosen. The Mexi-
can Hat transform is clearly a functional defined as the bi-convolution of the
stochastic input u(t) byWAG, i.e

Ω(k,t)≈
3√
k
W
{
W
(
t,TW =

3

2k

)}
=W(k,t) (2.7)

The function W(k,t) is called theWAG inducedwavelet and it is abbreviated
as WAGIW in the following.

2.2. Indirect correspondence between WAG and Mexican Hat

The wavelet is a bandpass function as one may easily conclude from the
admissibilty condition.Furthermore, it has tobe idealpassband inmultiresolu-
tion analysis, because the only orthonormal scaling function at full resolution
is the ideal sinc function which is nothing but the impulse response of an
ideal passband filter (Akansu and Smith, 1998; p.61). The only wavelet sa-
tisfying these conditions is the Littlewood-Paley wavelet (Meyer, 1992; p.26).



840 F.S. Tardu

In such cases, any spectral decomposition of the form ĝ(ω) = χ(k)ρ̂(ω) =
= χ(k)ĥ(ω)ĥ∗(ω) as used in the previous section and where χ(k) is the sca-
ling factor, leads to a simple conclusion that the amplitude |ĥ(ω)| is equal to
|ĝ(ω)| via only a modification in the gain, i.e.

|ĥ(ω)|=
√
|ρ̂(ω)|
χ(k)

=χ′(k)
√
|ρ̂(ω)|

Clearly, some but not all the wavelets fulfill these conditions. Yet, one may
expect approximation of the latter equality for a given wavelet provided that
the spectraldecomposition exists.Theprice topay for thisapproximation is in-
eluctably some leakage of regularity. That is the case for the inter-relationships
between theMexicanHat andWAGtransforms and this allows approximation
of indirect equivalences between these system functions.

Fig. 3. Amplitude of the Fourier transform of theWAG autocorrelation and transfer
functions

Figure 3 compares |ĥW |/
√
2 with the amplitude |ρ̂W |. The ripples indu-

ced by WAG are much more pronounced compared with those observed in
|ρ̂W | and |ρ̂W | ≈ |ĥW |/

√
2 only approximately. That is nothing but the con-

sequence of poorer frequency localization of the Haar wavelet and of its lack
of regularity. Both transfer functions represent approximate bandpass filters
with the common central frequency f0 ≈ 0.36/TW and half-power point ban-
dwith Br ≈ f0. In this sense, ρ̂ is a zero phase shift filter, while WAG is a
bandpass quadrature filter with 90◦ phase shift. These filters are therefore not
equivalent even though their amplitudes are similar, because their phasesdiffer



Space-time/frequency/scale representation... 841

significantly. The conversion of ĥ/
√
2 to ρ̂ may be achieved by convoluting

the former through Hilbert’s transform defined as

ĥH(ω)=−j sgn(ω)
{
−jω> 0
jω> 0

(2.8)

The transfer function related to theHilbert transformreduces for band-limited
turbulent signals to

hH(τ)=






1

2π

2πfK∫

−2πfK

−j sgn(ω)ejωτ dω= 2
πτ
sin2(πfKτ) for τ 6=0

0 for τ =0

(2.9)

where fK stands for theKolmogoroff frequency. These relationships lead to a
second equivalence between WAG andMexican Hat

Ω(k,t)=−
√
3TWW̆(t,TW) (2.10)

where W̆ denotes the Hilbert transform. Namely, the Hilbert transform of
WAG differs from the wavelet transformMH only by the scaling factor equal
to −

√
3TW (Fig.4). In other words, MH is nothing but very close to the

response of WAG to a quadrature filter. This also shows that, inversely, the
Hilbert transform of the wavelet process is related to

W(t,TW)≈
1√
3TW
Ω̆(k,t) (2.11)

Fig. 4. Hilbert transform ofWAG is approximately equivalent to its Mexican Hat
transform via a scaling factor

This correspondence is called indirect here, because it appears as a rough
estimate compared with the equivalence MH-WAGIW discussed in the pre-
vious section. It is however quite satisfactory in some practical situations. It



842 F.S. Tardu

will indeed be shown in Section 4, through analysis of real signals taken in a
turbulent boundary layer, when both sides of (2.10) results in nearly identical
traces.

2.3. Rice representation of near wall singularities

The wavelet transform is a bandpass process as we already have recalled.
Thus, it lends well itself to the representation as a randommodulated signal.
TheWAG transform, for instance, can be expressed as

W(t,TW)= i(t,TW)cos(2πfct)−q(t,TW)sin(2πfct) (2.12)

using Rice’s representation (Papoulis, 1984, p.318; Oppenheim and Schafer,
1975, p.363). In the last expression, i(t,TW) and q(t,TW) are two stochastic
processes to be defined and fc is the carrier frequency. This representation
is optimum in the sense of minimizing the average rate of the envelope of
W(t,TW), when the associated dual process equals its Hilbert transform, i.e.

W̆(t,TW)= q(t,TW)cos(2πfct)+ i(t,TW)sin(2πfct) (2.13)

The optimum carrier frequency is the center of gravity of the WAG spec-
trum. The inphase component i(t,TW) and the quadrature component
q(t,TW) are low-pass and their spectrum is constrained into approximately
−fc/2 < f < fc/2. Rewriting equation (2.12) at t±∆t with ∆t = (4fc)−1
leads to

W(t+∆t,TW)−W(t−∆t,TW)=
=−[i(t+∆t,TW)+ i(t−∆t,TW)]sin(2πfct)+ (2.14)
−[q(t+∆t,TW)+q(t−∆t,TW)]cos(2πfct)

which, of course, is exact. The processes i(t,TW) and q(t,TW) are random.
Their prediction at t+∆t may be found by mean square estimation (Papo-
ulis, 1984, Ch.13; Makhoul, 1975). The estimate of i(t+∆t,TW) in terms of
i(t,TW), its first and second time derivatives i

′ = ∂i/∂t and i′′ = ∂2i/∂t2 is

i(t+∆t,TW)= a1i(t,TW)+a2i
′(t,TW)+a3i

′′(t,TW) (2.15)

where the notation is not changed but it has to be kept in mind that these
are not exact but estimated values. The coefficients in the last expression are
functions of the autocorrelation function Rii(τ) of i(t,TW) and of its time



Space-time/frequency/scale representation... 843

derivatives. By making use of homogeneity and stationarity, one may show
that

a1 =
Rii(∆t)R

IV
ii (0)−R′′ii(∆t)R′′ii(0)

Rii(0)R
IV
ii (0)−R′′ii(0)R′′ii(0)

a2 =
R′ii(∆t)

R′ii(0)
(2.16)

a3 =
R′′ii(∆t)Rii(0)−Rii(∆t)R′′ii(0)
Rii(0)R

IV
ii (0)−R′′ii(0)R′′ii(0)

The Taylor series expansion of terms appearing in a1, leads to

a1 ≈ 1+
∆t4

6

RIVii (0)R
IV
ii (0)−RVIii (0)R′′ii(0)

Rii(0)R
IV
ii (0)−R′′ii(0)R′′ii(0)

which clearly shows that a1 ≈ 1 to the order ∆t4. Following the same proce-
dure, one has

i(t−∆t,TW)= a1i(t,TW)−a2i′(t,TW)+a3i′′(t,TW)

Combining with 2.15 gives:

i(t+∆t,TW)+ i(t−∆t,TW)= 2
[
a1i(t,TW)+a3

∂2i(t,TW)

∂t2

]
≈

≈ 2
[
i(t,TW)+a3

∂2i(t,TW)

∂t2

]
+O(∆t4)

In a similar manner, the mean square estimated prediction of the quadrature
component q(t,TW) results in

q(t+∆t,TW)+q(t−∆t,TW)= 2
[
b1q(t,TW)+ b3

∂2q(t,TW)

∂t2

]
≈

≈ 2
[
q(t,TW)+ b3

∂2q(t,TW)

∂t2

]
+O(∆t4)

The coefficients b1 and b3 are easily recovered through the autocorrelation
function of q(t,TW) as in (2.16). Expression (2.14) becomes therefore

W(t+∆t,TW)−W(t−∆t,TW)≈−2W̆(t,TW)+O(∆t2)

Combining with (2.10), gives

Ω(k,t)≈
√
3TW
2
[W(t+∆t,TW)−W(t−∆t,TW)]+O(∆t2)≈

(2.17)

≈
√
3TW
2
DW(t,TW)



844 F.S. Tardu

This last relationship is astonishing by its simplicity and may be generali-
zed for any process and its associated Hilbert transform. The right hand side
of (2.17) may be interpreted as the centered time derivative of WAG sam-
pled at ∆t. Consequently, theMexican Hat wavelet is simply proportional to
the smoothed time derivative ofWAG. The same result could be obtained by
using simplyTaylor series expansion, but the analysis would not be acceptable
because of the randomness of the inphase and outphase components. Repre-
sentation (2.17) is clearly an approximation of the order O(∆t2). It is however
a robust estimate because ∆t is smaller than the Nyquist sampling period of
the inphase and outphase components. Indeed, since the spectrum of low-pass
slowly varying i(t,TW) and q(t,TW) is constrained into −fc/2<f <fc/2, the
sampling period is ∆ts =1/fc and ∆t=∆ts/4. It is therefore not astonishing
that (2.17) applied to the near wall turbulent time series is quite successful,
even when ∆t is as large as the outer time scale, as wewill show in Section 4.

3. Experiments

The ensemble of measurements reported here have been realized in the low
speed wind tunnel of LEGI with a free stream velocity U∞ = 4m/s. The
boundary layer and momentum thickness at the test section are respective-
ly δ = 34mm and θ = 3.4mm. The Reynolds number based on the local
momentum thickness is Reθ =913.

Fluctuating wall shear stress and streamwise velocity are analyzed. The
wall shear stress measurements are performed by means of a the Cousteix-
Houdeville wall Hot-Wire Gauge (HWG) to avoid problems caused by the
conduction into the substrate.Awire of 4µmdiameter is set into amicrocavity
and flush mounted to the wall. The length of the sensing element is 200µm
whichcorresponds toa spanwise extent of ∆z+ =3(+denotesvariables scaled
by the viscosity and the shear velocity at X = 1.14m from the transition
point). Nice results of the statistics of the fluctuating wall shear stress up to
4th moments have been obtained by this probe. The details are available in
Tardu (1998).

The sampling frequency is f+s =2.Thesignals areprefilteredby theKrohn
Hite filter at adequate cut off frequencies. The total duration of each record is
Ttot =5000T∞ where T∞ = δ/U∞ is the outer time scale. This is long enough
to ensure the convergence of statistics up to the 4th order moments including
those of the time derivative of fluctuating signals. Bucking amplifiers are used
to suppress the DC anemometer output at zero velocity, so that the signal



Space-time/frequency/scale representation... 845

could be amplified before A/D conversion. This conversion is performed with
anAnalog-Device RTI-800 board (accuracy: 11 bit+sign; 8 channels) installed
in aPC.TheHWGis calibrated in situ bydetermining thevelocity gradient at
thewallwith a single hot filmprobe (TSI 1276-10W, sensing length ∆z+ =4).
TheHilbert Transform is obtained by using a 64 points digital Finite Impulse
Response filter (FIR).

4. Results

4.1. WAG-Mexican Hat equivalence

Figure 5a shows a sample of the instantaneous wall shear stress τ ′ and its
corresponding WAG transform Wτ′ versus time. The variables scaled by the
inner parameters, i.e. viscosity ν and shear velocity uτ =

√
τ/ρ are indexed

by (+). Therefore, t+ = tu2τ/ν, τ
′+ = τ ′/τ, etc. The WAG integration time

in Fig.5a is T+W = 13 and that corresponds to half of the outer time scale
δ+/U+

∞
. For the sake of clarity, we opted for a direct presentation of traces

instead of usingusual contours of thewavelet coefficients in the time-frequency
domain.

The intenseWAG events are shown by arrows in Fig.5a. They were iden-
tified in a way similar to Antonia and Fulachier (1989) and Krogstad and
Antonia, (1994). Thus, an event occurs when W+

τ′
exceeds a given threshold,

i.e. W+
τ′
­±β

√
τ ′τ ′/τ. The sign ± is selectively used to detect strong trans-

itions in the signal from negative to positive values and vice versa, and they
are indicated separately by open and bold arrows in Fig.5a. The threshold
is taken as β = 0.4. It is clearly seen that WAG detects strong accelerations
or decelerations occurring within the given window time and associated with
rapid changes of the sign in the signal.

Figure 5b shows the Mexican Hat (MH) and the WAG induced wavelet

(WAGIW) transformsof τ ′ denoted respectively inwall units as Ω+
τ′
and W

+

τ′.
It is recalled here that the scale dilatation parameter in Ω+

τ′
is k+ =3/(2T+W).

It is seen in Fig.5b that the waveforms of MH merge completely into those
of WAGIW except for some slight differences occurring from time to time,
in particular near the local maxima. A close inspection of different groups of
data of total record lengths exceeding several thousand times the outer time
scale, revealed that the residual Ω+

τ′
−W+τ′ is rarely larger than 15% of Ω+τ′.

Clearly thewavelet transform throughMHand its linearized versionWAGIW
are equivalent. Onemay therefore easily interpret theMH transformas strong
transitions from negative to positive values of WAG.



846 F.S. Tardu

Fig. 5. Sample of instantaneous wall shear stress fluctuations and corresponding
wavelet andWAG transforms; (a) instantaneous wall shear stress and its WAG
transform in wall units, (b)Mexican Hat transform comparedwith theWAG
induced wavelet transform of the signal shown in (a). The window averaging time

and the scale parameter in wall units are respectively T+
W
=13 and

k+ =3/(2T+
W
)=0.11

This is also confirmed by the analysis of the fluctuating velocity signal
u′(t) in the whole boundary layer. Figure 6a shows for instance u′

+
and its

WAG transform W+
u′
measured in the low buffer layer at y+ = 10 from the

wall where the production is nearly maximum. The window average time is
now equal to the outer time scale and T+W = δ

+/U+
∞
= 26. It is clearly seen

that the coincidence between Ω+
u′
and W

+

u′ is almost perfect (Fig.6b).

The validity of equation (2.10) relating the Mexican Hat to the Hilbert
transform W̆ ofWAG is now debated through the traces shown in Fig.7.We
only present results concerning the fluctuatingwall shear stress as reported in
Fig.5awith T+W =13 for the sake of brevity. Themain conclusion drawn from

Fig.7 is that there is an excellent agreement between −
√
3T+WW̆

+
τ′
and Ω+

τ′
.

We closely inspected, compared and confirmed the validity of equivalences
discussed inSection2, for both thefluctuatingwall shear stress and streamwise
velocity signals, obtained at different positions y+ < 300 and within a large



Space-time/frequency/scale representation... 847

Fig. 6. Sample of instantaneous streamwise velocity fluctuations at y+ =10 and
corresponding wavelet andWAG transforms; (a) instantaneous streamwise velocity
and itsWAG transform in wall units, (b)MexicanHat transform compared with the
WAG induced wavelet transform of the signal shown in (a). The window averaging
time and the scale parameter in wall units are respectively T+

W
=26 and

k+ =3/(2T+
W
)= 0.057

Fig. 7. Comparison of theMexican Hat transformwith the Hilbert transform of
WAG and its smoothed time derivative.The raw data is those given in Fig.5a with

T+
W
=13



848 F.S. Tardu

Fig. 8. Comparison of theMexican Hat transform of the wall shear stress with the
smoothed time derivative ofWAG given by (2.17) for the largest outer scale

integration time T+
W
=26

range of the window averaging time 3 ¬ T+W ¬ 26. The upper limit in the
last expression is the outer time scale, and the inner limit is roughly the
Kolmogoroff time scale in the inner layer.

4.2. Time-frequency-scale representation

Any signal, moreover, the wavelet coefficients Ω(k,t) may be expressed as

Ω(k,t) = r(k,t)cos
[ t∫

0

ωi(k,t)
]

(4.1)

where r(k,t) stands for the instantaneous amplitude and ωi(k,t) is the instan-
taneous angular frequency at the scale k. This representation is not unique
and different characterizations are possible, depending upon the choice of the
dual processes. In the Rice canonical representation, that is optimum in the
sense of minimizing the average rate of the signal envelope, one has

r2(k,t)=Ω2(k,t)+ Ω̆2(k,t)
(4.2)

ωi(k,t)=
Ω(k,t)⌣Ω′(k,t)−Ω′(k,t)Ω̆(k,t)

r2(k,t)

where (·)′ denotes the time derivative. The corresponding optimum carrier
frequency equals

ωc(k)=
r2(k,t)ωi(k,t)

r2(k,t)
(4.3)



Space-time/frequency/scale representation... 849

Moreover, the dual of Ω(k,t) is obviously its Hilbert transform

Ω̆(k,t) = r(k,t)sin
[ t∫

0

ωi(k,t) dt
]

(4.4)

The instantaneous frequencymay also be written as

ωi(k,t)=ωc(k)+
dϕ(k,t)

dt
(4.5)

where ϕ(k,t) is the random phase at the scale k. It is straightforward that
ωi(k,t) governs directly the behaviour of Ω(k,t) near the zero-crossings. The
transitions fromejections to sweeps detected by theHaarwavelet at large eddy
scales behave as the discontinuous phase frequency shift keying process (Papo-
ulis, 1983; Aulin and Sundberg, 1981), with random, yet somewhat coherent
and regular periodicity.

Fig. 9. Samples of the instantaneous amplitude and phase in radians of the Haar
wavelet coefficients of the fluctuating streamwise velocity at y+ =10 versus time.
The scale parameter of the wavelet transform is k+ =0.057 in wall units.
CD – constant phase zone wherein the instantaneous frequency is equal to the

carrier frequency: singularities are smoothly oscillating due to the meandering of the
streaks. AB – the phase increases while the amplitude decreases: apparition of
small-scale structures. EF – the phase and amplitude increase simultaneously:

active but smaller structures. DE – phase jump

Figure 9 shows some traces of the phase ϕ(k+, t+) and amplitude
r+(k+, t+) of the fluctuating streamwise velocity signal u′ at y+ =10, for the
wavelet scale parameter k+ =0.057 (corresponding to thewavelet windowdu-
ration T+W =26). The optimumangular carrier frequency is ω

+
c =0.08 in this

case. It is seen that the instantaneous phase ϕ(k,t) consists of line segments
that are discontinuous at points B and D where random phase jumps occur.



850 F.S. Tardu

The phase increases first at AB, remains constant during large period CD,
jumps again and increases at DF. The constancy of the phase indicates that
the instantaneous frequency is sensibly equal to the carrier frequency. The
periods like CD wherein ωi ≈ ωc coincide generally with large amplitudes
r+(k+, t+). Strong ejection-sweeps transitions marking the arrival of coherent
structures are, therefore, merely constant phase events. The time intervals as
AB wherein ϕ(k+, t+) increases while r+(k+, t+) decreases are reminiscent of
apparition of small scales. The slope of AB is dϕ+/dt+ = ω+c /3 indicating
that ωi(k,t) is jumped by a factor 4/3. The jumps in the frequency with the
same fraction of ω+c are often and repetitively observed. It is asked whether
this behaviour can be partly explained by themultifractal nature of the casca-
de process or not (Argoul et al., 1989). The epochs as EF, wherein both the
instantaneous phase and the amplitude increase from small values, are pre-
sumably related to the arrival of smaller scale active structures. Note finally
that, duration of the segments is about 100-200 wall units that is close to the
ejection (bursting) period.The occurrence of these long periods is particularly
interesting. These characteristics may eventually be used in the decision loop
of some drag-reduction control schemes.

Thenearwall singularities identified by theHaarwavelet (and forwavelets
of any kind such as the classical Mexican Hat) at large integration time may
consequently be modeled as the discontinuous frequency shift keying process
with random phase discontinuities.

5. Discussion, generalization and conclusion

The relationships deduced in Section 2were based on the piecewise continuous
linearized version of theMexican Hat mother wavelet. The principal aim was
to point at the close similarity between the Haar wavelet designed to detect
singularities and the Marr wavelet identifying local maxima or minima. The-
refore, the present contribution has to be considered at a first glance, as a
”warning” for the users of these schemes. The first of these transforms is the
simplest yet the poorest one in terms of frequency localization. The second,
on the other hand, stays one of the most popular tools used in the analysis of
turbulence data conducted so far. The fact that onemay approximately relate
both transforms through a simpleHilbert filter and furthermore the smoothed
time derivative of the Hilbert transform is quite striking. The analysis leading
to this last conclusion, i.e. equation (2.17) is somewhat original and may be
generalized for any stochastic process.



Space-time/frequency/scale representation... 851

On a more general basis, the relationships obtained in Section 2 resul-
ted from the spectral factorization of the linearized Mexican Hat transform.
This process can eventualy be generalized at least to the Daubechies’s com-
pactly supported wavelets (1992). The transfer function of this family at the
Nth iteration may be put in the form of the convolution of N-times WAG
with a regularizing polynomial filter. The N-times convolution strengthens
the regularity of the wavelet to the order N. The spectral decomposition as
conductedhere,will then lead to eitherWAG itself (oddwavelet) or itsHilbert
transform (even wavelet) via a scaling parameter. Lewalle (1994) has pointed
out that ”the exploration of options reveals a large (and rapidly growing)
number of wavelets, possibly giving the reader the impression that wavelet
selection is arbitrary and the ensuing results biased”. We clearly claim out
here that the simplestWAG (Haar) wavelet and its Hilbert transform, despite
their bad frequency localizations, seem to be adequate enough to characterize
the (sufficiently regular) near wall turbulence in the inner region, and more
information can hardly be extracted by refining or redefining any other one
directional multiscale wavelet which, after all, can always be linked to WAG
in some way.

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Czasoprzestrzenno-częstościowo-skalowy opis turbulencji przyściennej

Streszczenie

W pracy wykazano, że metoda WAG (ang. Window Average Gradient) przewi-
dziana do wyszukiwania osobliwości w turbulencji jest w przybliżeniu równoważna
transformacjiHilberta falki typu ”meksykański kapelusz”.Na podstawie badań teore-
tycznych i doświadczalnychdotyczących fluktuacji naprężeń na ściance oraz rozkładu
prędkościwwarstwie przyściennejwyprowadzonokilka tożsamości pomiędzy schema-
tami WAG i udowodniono ich prawdziwość. Rozważono także chwilową charaktery-
stykę amplitudowo-częstościowąWAGdla dużej skali. Rezultaty analizy wskazały na
znaczącą regularność turbulencji przyściennej. Pokazano, że osobliwości obserwowane
przy brzeguw dużej skali związane są z procesemprzesuwania częstości i odpowiada-
jącymmu nieciągłym zmianom chwilowej fazy.

Manuscript received February 21, 2007; accepted for print April 4, 2007