2021

# A wavelet-based detector function for characterizing intermittent velocity
signals

Satyajit De<EMAIL_ADDRESS>Aditya Anand<EMAIL_ADDRESS>Sourabh S. Diwan<EMAIL_ADDRESS>Department of Aerospace Engineering,
Indian Institute of Science, Bengaluru, 560012, India

###### Abstract

In this work, we propose a new detector function based on wavelet transform to
discriminate between turbulent and non-turbulent regions in an intermittent
velocity signal. The derivative-based detector function, which is commonly
used in intermittency calculation schemes, shows large fluctuations within
turbulent parts of the signal and requires averaging over a certain “smoothing
period” to remove the fake drop-outs, introducing subjectivity in calculating
intermittency. The new detector function proposed here is obtained by
averaging the “pre-multiplied wavelet energy” over the entire frequency range,
resulting in a function that is much smoother than the derivative-based
detector and at the same time has a good discriminatory property. This makes
the choice of the smoothing period unnecessary and removes the subjectivity
associated with it. We demonstrate the effectiveness of the wavelet detector
within the framework of the widely-used method by Hedley and Keffer (1974, J.
Fluid Mech., V64, pp625), for a range of velocity signals representing the
different stages of roughness-induced transition. The wavelet detector
function works well in detecting the edge intermittency of a canonical
turbulent boundary layer as well, thereby highlighting its generality. Our
detector can, in principle, be used with any method of specifying threshold
for obtaining an indicator function.

###### keywords:

Intermittency, Detector Function, Wavelet Transform, Transitional and
Turbulent Boundary Layers

##  1 Introduction

Intermittent velocity signals are observed in a variety of transitional and
turbulent flows. In the context of wall-bounded flows, a prototypical example
exhibiting large-scale intermittency is the transitional boundary layer which
is characterized by presence of “turbulent spots”. Turbulent spots are defined
as islands of turbulence surrounded by quasi-laminar regions (Emmons, 1951)
and are observed for both “natural” and “bypass” modes of transition (Durbin,
2017). From the onset of transition, turbulent spots grow in number and size,
and eventually merge together to form a fully turbulent flow (Emmons, 1951).
Furthermore, intermittent velocity signals are also observed in the outer
region of a turbulent boundary layer due to the effects of “edge
intermittency”, associated with its highly convoluted edge separating the
turbulent and non-turbulent fluid (Leslie et al., 1970). Quantifying the
intermittency of a velocity signal is important from both fundamental and
applied perspectives; the former can help us understand how non-turbulent
fluid acquires vorticity across a turbulent/non-turbulent interface and the
latter can lead to better engineering models for transitional and turbulent
flows.

For the transitional boundary layers, the intermittency is quantified by a
parameter called the “intermittency factor” ($\gamma$) defined as the fraction
of time for which the velocity signal is turbulent (Narasimha, 1957). Over the
decades, several methods have been developed to detect the turbulent/non-
turbulent interface and calculate the intermittency factor, many of which use
one-dimensional velocity measurements from a hotwire probe. The turbulent part
of the signal is visually discernible as a region of high frequency and high
amplitude fluctuations. This aspect of the signal is enhanced by sensitising
the signal usually by differentiating, double differentiating or high-pass
filtering (Hedley and Keffer, 1974; Fransson et al., 2005); the sensitized
signal is called the detector function. The detector function has a relatively
higher magnitude in the turbulent part than in the non-turbulent part and an
indicator function is obtained by applying a threshold on the magnitude of the
detector function. The indicator function is assigned a value of 1 when the
detector function is higher than the set threshold and 0 otherwise. The time
mean of the indicator function gives the intermittency factor, which varies
from 0 to 1 over the extent of the transition zone (Narasimha, 1985). When the
detector function is obtained by differentiation, the turbulent part of the
signal is accentuated showing large fluctuations, which results in some fake
dropouts i.e., the indicator is 0 in the turbulent part of the signal.
Similarly, the non-turbulent part can have spurious fluctuations where the
indicator is assigned a value of 1. To rectify this issue, a smoothing process
is applied on the detector function by averaging it over a fixed time
interval, $T_{s}$, called the smoothing period (Hedley and Keffer, 1974). The
signal obtained after smoothing is called the criterion function and a
threshold is now applied on the criterion function to obtain an indicator
function. Canepa et al. (2002) and Jahanmiri (2011) have summarized some of
the widely-used methods for calculating $\gamma$ proposed in the literature; a
brief account of the important developments in this field can be found in a
recent paper by Veerasamy and Atkin (2020).

The above procedure involves subjectivity at two levels - first, the
determination of the smoothing period and second, the determination of the
threshold. A perusal of the literature shows that there is no unique
specification of the smoothing period. Moreover, it is not clear what the
appropriate time scale should be for scaling the smoothing period. Hedley and
Keffer (1974) used a smoothing period of 4 times the sampling interval
($\Delta t$). This period was found to be of the order of the Taylor
microscale and about 28 times the Kolmogorov time scale ($t_{K}$). Other
authors have used a range for $T_{s}$ as $3-10\Delta t$ in terms of the
sampling period (Canepa et al., 2002; Veerasamy and Atkin, 2020) or
$15-200t_{K}$ in terms of the Kolmogorov scale (see Canepa et al., 2002, and
the refernecs therein). Anand and Diwan (2020) found it necessary to use a
much higher value of $T_{s}=100\Delta t$ in calculating $\gamma$ for their
measurements on roughness-induced transition. Another way of expressing
$T_{s}$ is in terms of the large-eddy time scale $T_{L}=\delta/U_{\infty}$,
where $\delta$ is the boundary layer thickness and $U_{\infty}$ is the
freestream velocity. Canepa et al. (2002) have listed studies which have used
a range of $T_{s}\approx 1-6T_{L}$; Canepa et al. (2002) themselves found
values of $T_{s}=3.7T_{L}$ and $8T_{L}$ to be more appropriate for the two
Reynolds numbers used in their exercise on hotfilm signals, which were
measured over gas-turbine blades. They found that the choice of $T_{s}$ had a
significant effect on determining the intermittency distribution along the
blade chord. Samson et al. (2021) used $T_{s}=2.5T_{L}$ in calculating
$\gamma$ for adverse-pressure-gradient boundary layers. The subjectivity in
determining the detector/criterion function is also present for methods which
use high-pass filtering of the velocity signal, as the filter cut-off
frequency can be different for different types of signal (Fransson et al.,
2005; Mamidala et al., 2022). To the best of our knowledge, there does not
exist any method that specifies the smoothing time or the filter frequency in
an objective manner.

The second level of subjectivity has been addressed by several investigators
by proposing various methods for determining the threshold on the
detector/criterion function. Hedley and Keffer (1974) proposed that the point
of maximum curvature of the cumulative-distribution-function (CDF) of the
criterion function can be chosen as the threshold for obtaining indicator
function. The dual-slope method by Kuan and Wang (1990) involves choosing the
threshold as the point of intersection of two straight lines on the
intermittency vs. threshold curve. Nolan and Zaki (2013), determine the
threshold by a method prescribed by Otsu (1979) that maximizes the contrast in
variance between the laminar and turbulent zones. More recently, Veerasamy and
Atkin (2020) have proposed a method of selecting the maximum laminar
perturbation near the onset of transition as the single threshold to be used
for all the locations within the transition zone. This choice is based on the
assumption that the laminar perturbations stay nearly constant throughout the
transition zone and Veerasamy and Atkin (2020) demonstrated the method for the
transition induced by an aerofoil wake. It is not clear whether the method can
be applied for transition induced by other types of disturbances. Some of the
other methods of choosing a threshold are due to Fransson et al. (2005) and
Mamidala et al. (2022). In most of these methods, the final determination of
the threshold is based on a visual match between the indicator function and
the signal (or criterion function) at some level; see Jahanmiri (2011).

The present work deals with the first level of subjectivity. We propose a new
detector function based on wavelet transform that does not require
specification of smoothing period. Wavelet transform is a commonly used tool
to study time-frequency behaviour of velocity signals and has been used before
for transitional boundary layers especially in the context of conditional
sampling of the turbulent and non-turbulent parts of the signal (Volino, 2005;
Zhang et al., 2018). Simoni et al. (2016) used wavelet transform to define an
intermittency measure based on the count of “events” (representing vorticies)
in two-dimensional snapshots of the velocity field; this method, however, is
specific to 2-D flow fields and cannot be applied on velocity time series. We
are not aware of any other study that has used wavelet-based method for
determining a detector function. The present idea is based on the observation
that the non-turbulent portions in an intermittent velocity signal are
associated with extended regions in the time-frequency plane where the
magnitudes of wavelet coefficients are low (Anand and Diwan, 2020). The
detector function is then obtained by simply averaging the “pre-multipied”
wavelet energy over the entire frequency axis, obviating the need for any
additional smoothing. We demonstrate the utility of this approach on
transitional velocity signals measured in our previous work on roughness-
induced transition (Anand and Diwan, 2020). Furthermore, the new detector
function works well for detecting edge intermittency for velocity signals
measured in a fully turbulent (canonical) boundary layer, suggesting its
general applicability.

The paper is organized as follows. Section 2 briefly describes the
experimental setup used for measuring velocity in a transitional boundary
layer. The subjectivity associated with the specification of a smoothing
period is discussed in section 3. The new wavelet-based detector function is
proposed in section 4 and is demonstrated for transitional velocity signals
therein. In section 5 we test the new detector function on intermittent
velocity signals in canonical wall turbulence. Conclusions are presented in
section 6.

##  2 Experimental setup for roughness-induced transition

The velocity signals used for the present analysis were measured in our
previous experimental work on roughness-induced transition (Anand and Diwan,
2020). Here we provide a brief outline of the experimental setup for the sake
of completeness. The experiments were conducted in a low-speed wind tunnel at
Indian Institute of Science, Bengaluru. The wind tunnel has a test-section of
0.5 m x 0.5 m (and 3 m length) and has a free-stream turbulence intensity of
about 0.1% (Anand and Diwan, 2020).

Figure 1: Schematic of the flat plate used for generating a transitional
boundary layer downstream of a 40 mm wide, grade 24 (coarse) roughness strip,
shown in solid hatching. The co-ordinate system used herein is also shown.

The experiments were performed on a flat plate, 6.25 mm thick and 2.1 m long,
and having a super elliptic leading edge. A 40 mm wide spanwise strip of grade
24 emery paper was used as a randomly distributed surface roughness (overall
roughness height=1.5 mm) to induce transition and it was placed 100 mm
downstream of the leading edge as shown in Fig. 1. Dantec dynamics 55P11
miniature boundary-layer hotwire probe was used to measure instantaneous
velocity signals at a sampling rate of 20 kHz. The sampled data was low-pass
filtered, using a hardware filter, at a cut-off filter of 10 kHz to remove
aliasing errors as per the Nyquist criterion. The output voltage from the
hotwire probe was corrected for ambient temperature variations (Bruun, 1995)
and calibrated against Pitot-tube velocities using the King’s law. More
details on the experimental setup and measurement chain can be found in Anand
and Diwan (2020).

Anand and Diwan (2020) reported velocity profiles measured over the entire
transition zone downstream of the roughness at $U_{\infty}=7.5$ m/s with a
roughness Reynolds number of $U_{\infty}k/\nu=640$, where $k$ is the maximum
roughness height and $\nu$ is kinematic viscosity. For the present analysis we
have chosen five velocity signals representative of the various stages of
transition. The streamwise locations for these signals are shown schematically
in Fig. 1 and are listed in Table 1; the velocity signals are shown in Fig. 2
(a)-(e). The signal duration used for our analysis is 30 s but we show a short
stretch of time (0.6 s) in Fig. 2 to bring out the contrast between the
turbulent and non-turbulent parts. The wall-normal locations of the signals
are in the range $y/\delta\approx 0.1-0.4$, which is the typical region over
which the intermittency factor is approximately constant (Matsubara et al.,
1998). Veerasamy and Atkin (2020) sampled velocity signals at
$y/\delta^{*}=0.5$ for their analysis, which falls approximately in the above
range over the extent of the transition zone. Table 1 also lists values of
$\gamma$ calculated based on the Hedley-Keffer method (Hedley and Keffer,
1974) described in the next section. Based on the values of $\gamma$, we
denote the signals as “Laminar”, “Transition Onset”, “Mid-Transitional”,
“Nearly Turbulent” and “Turbulent” (Table 1).

Table 1: Streamwise locations, intermittency values and stages of transition for the velocity signals chosen for the present analysis. Case |  $x$ location (mm) | Intermittency ($\gamma$) | Transition Stage
---|---|---|---
a | 250 | $\approx 0$ | Laminar
b | 400 | 0.09 | Transition Onset
c | 450 | 0.42 | Mid-Transitional
d | 650 | 0.87 | Nearly Turbulent
e | 800 | 1 | Turbulent
|  |  |
Figure 2: Velocity signals from various stages of transition along with their
corresponding $x$-locations; see Fig. 1. Figure 3: Effect of smoothing period
on the criterion function for the Mid-Transitional case (Table 1). The
detector function is averaged over the time interval $T_{s}$. The dash-dotted
lines show the threshold as calculated by the Hedley-Keffer method (i.e. the
maximum curvature of the CDF) for each criterion function $\Delta t$ is the
sampling interval and $T_{L}$ is the large-eddy time scale. Figure 4: (a,b)
Indicator functions obtained for two different smoothing periods
($T_{s}=10\Delta t$ and $T_{s}=100\Delta t$) by using the Hedley-Keffer method
(Hedley and Keffer, 1974). (c) Indicator function determined using a manual
threshold based on visual inspection. Figure 5: Laminar case: (a) Velocity
signal $(u/U_{\infty})$ vs. time (s). (b) Contour plot of (pre-multiplied)
wavelet energy ($fC_{w}^{2})$ in frequency-time plane. (c) Comparison between
the wavelet detector (red line) and the Hedley-Keffer (HK) detector (green
line). Both the detectors are scaled from 0-1 over the time interval shown.
The indicator function obtained from the wavelet detector (“Indicator-Wavelet
Detector”) is also shown (black line).

##  3 Effect of smoothing period on the calculation of intermittency factor

We first calculate intermittency factors for the signals in Fig. 2 using the
Hedley-Keffer method, and highlight the subjectivity introduced by the
smoothing time on the determination of the criterion function. We choose the
Mid-Transitional signal for outlining the step-by-step procedure. The absolute
value of the double derivative of the velocity signal is chosen as the
detector function, consistent with Hedley and Keffer (1974) and Veerasamy and
Atkin (2020). Figure 3 plots the criterion functions obtained by using
different smoothing periods: $1\Delta t$ to $100\Delta t$. As can be seen, the
criterion function gets progressively smoother with the increase in $T_{s}$.
The smoothing period is also expressed in terms of the large-eddy time,
$T_{L}$ (Fig. 3). For the present exercise, we find $T_{s}=100\Delta t$ to be
the appropriate smoothing period. This translates into $T_{s}=5.6T_{L}$ which
falls in the range $T_{s}=3.7-8T_{L}$ used by Canepa et al. (2002) for their
analysis.

To determine the threshold on the criterion function, we use the CDF method
proposed by Hedley and Keffer (1974). The maximum curvature of the CFD of the
criterion function is used as a first approximation for the threshold. This
threshold is a function of the smoothing period and is plotted in Fig. 3 as
dash-dot lines. The indicator functions obtained from these threshold values
are plotted in Fig. 4 for two smoothing periods: $T_{s}=10\Delta t$ and
$100\Delta t$, superposed on the velocity signals. It is clear from Fig. 4(a)
that there are several fake drop-outs within the turbulent regions at
$T_{s}=10\Delta t$, as a result of which the value of $\gamma$ is quite low.
At $T_{s}=100\Delta t$ there is an improved match between the indicator and
the signal, although fake drop-outs are still present (Fig. 4(b)). Finally,
the threshold was manually adjusted to get a good match between the indicator
function and the signal by visual inspection (Jahanmiri, 2011), which is shown
in Fig. 4(c). This ensures that there are no fake 0s or spurious 1s in the
indicator function. The resulting value of $\gamma(=0.48)$ is much higher than
those obtained for the CDF-based threshold (Fig. 4). (Note that in this
exercise and in what follows, $\gamma$ is calculated by averaging the
indicator function over the entire stretch of the signal, i.e., 30 s.) The
strong dependence of intermittency factor on the smoothing period seen here is
consistent with the observation in Canepa et al. (2002) for flow past a
turbine blade. It is not clear whether the sensitive dependence of $\gamma$ on
$T_{s}$ is present only for certain types of transitional flows and using
specific methods; see, for example, Veerasamy and Atkin (2020) and Samson et
al. (2021). The values of $\gamma$ listed in Table 1 for other velocity
signals have been obtained by using the same procedure, i.e. with
$T_{s}=100\Delta t$ and threshold determined by visual inspection. The above
exercise brings out the subjectivity introduced by the smoothing period and
threshold level in the accurate determination of the intermittency factor.

In the next section, we propose a new wavelet-based detector function that
largely eliminates the uncertainty associated with the specification of the
smoothing period. We use the Hedley-Keffer method as the framework for
demonstrating effectiveness of the new detector function.

##  4 New detector function based on wavelet transform

Figure 6: Transition Onset case: (a) Velocity signal $(u/U_{\infty})$ vs. time
(s). (b) Contour plot of (pre-multiplied) wavelet energy ($fC_{w}^{2})$ in
frequency-time plane. (c) Comparison between the wavelet detector (red line)
and the Hedley-Keffer (HK) detector (green line). Both the detectors are
scaled from 0-1 over the time interval shown. The indicator function obtained
from the wavelet detector (“Indicator-Wavelet Detector”) is also shown (black
line). Figure 7: Mid-Transitional case: (a) Velocity signal $(u/U_{\infty})$
vs. time (s). (b) Contour plot of (pre-multiplied) wavelet energy
($fC_{w}^{2})$ in frequency-time plane. (c) Comparison between the wavelet
detector (red line) and the Hedley-Keffer (HK) detector (green line). Both the
detectors are scaled from 0-1 over the time interval shown. The indicator
function obtained from the wavelet detector (“Indicator-Wavelet Detector”) is
also shown (black line). (d) Comparison between the wavelet detector function
(red line) and the HK criterion function (blue line) for the smoothing period
of $100\Delta t$; see Fig.3.

Wavelet transform maps a signal in time, onto a time-frequency plane using
functions called “mother wavelet” and their scaled and translated versions, as
a basis for the mapping (Farge, 1992). Since wavelet functions are band-
limited both in time and frequency, the wavelet transform is ideal for time-
frequency analysis of an intermittent signal, and its ability to detect
turbulent spots in a transitional signal has already been demonstrated (Anand
and Diwan, 2020; Samson et al., 2021; Zhang et al., 2018); see Anand and Diwan
(2020) and Anand (2020) for more details. In this work we have used the MATLAB
command “cwt” to calculate the wavelet coefficients ($C_{w}$) for our
analysis; “cwt” uses convolution algorithm to compute the continuous wavelet
transform coefficients. The analytical Morlet wavelet is used as the mother
wavelet (Anand and Diwan, 2020; Zhang et al., 2018).

The results from wavelet analysis for the five cases listed in Table 1 are
plotted in Fig. 5-9. In each figure, window (a) plots the non-dimensional
velocity signal $(u/U_{\infty})$ as a function of time, window (b) plots the
corresponding squared wavelet coefficients multiplied with frequency
($fC_{w}^{2}=e$) (Anand and Diwan, 2020), hereafter referred to as wavelet
energy, as a function of time and frequency and window (c) plots wavelet-based
detector function (described in the following) compared with the Hedley-Keffer
detector function. Also plotted in window (c) is the indicator function
obtained from the wavelet detector. The use of the pre-multiplied form of
wavelet energy enables an even representation of wavelet energy on the
logarithmic frequency axis, akin to the pre-multiplied Fourier spectrum
commonly used in the wall turbulence literature (Smits et al., 2011). The time
window of 6.3 s to 6.9 s is chosen as it gives a better visual representation
of these signals.

Figure 8: Nearly Turbulent case: (a) Velocity signal $(u/U_{\infty})$ vs. time
(s). (b) Contour plot of (pre-multiplied) wavelet energy ($fC_{w}^{2})$ in
frequency-time plane. (c) Comparison between the wavelet detector (red line)
and the Hedley-Keffer (HK) detector (green line). Both the detectors are
scaled from 0-1 over the time interval shown. The indicator function obtained
from the wavelet detector (“Indicator-Wavelet Detector”) is also shown (black
line). Figure 9: Turbulent case: (a) Velocity signal $(u/U_{\infty})$ vs. time
(s). (b) Contour plot of (pre-multiplied) wavelet energy ($fC_{w}^{2})$ in
frequency-time plane. (c) Comparison between the wavelet detector (red line)
and the Hedley-Keffer (HK) detector (green line). Both the detectors are
scaled from 0-1 over the time interval shown. The indicator function obtained
from the wavelet detector (“Indicator-Wavelet Detector”) is also shown (black
line).

To motivate the choice of the new detector function, we first discuss the Mid-
Transitional signal (Fig. 7) which was used for the detailed analysis of the
Hedley-Keffer method in the previous section. It can be seen from Fig. 7(b)
that the presence of turbulent spots is associated with elevated levels of
wavelet energy, localized in time, for frequencies higher than 100 Hz, whereas
the non-turbulent regions are associated low wavelet energy as expected. The
important observation from Fig. 7(b) is that the non-turbulent parts of the
velocity signal are represented as extended regions of low activity in the
wavelet space over the entire “high” frequency band, i.e., above $\approx 100$
Hz; this was also reported by Anand and Diwan (2020). The energy contained in
the “low” frequency band, i.e., $<100$ Hz is more spread out along the time
axis indicating that low-frequency motions are not as time-localized.
Moreover, the wavelet energy associated with them is much smaller than that
contained in the turbulent spots. This suggests that averaging the wavelet
energy (in the pre-multiplied form) over the entire frequency band (Eq. 1)
should result in a time signal that brings out the contrast between the
turbulent and non-turbulent regions fairly well and therefore serves as a
discriminator between the two. The averaged wavelet energy is given as

$D(t)=\frac{1}{n}\ \sum^{n}_{i=1}e(t,f_{i})$ (1)

where, $f_{i}$ is the set of discrete frequencies and $n$ is the total number
of frequencies in the wavelet transform. We propose to use $D(t)$ in Eq. 1 as
the new detector function, which is plotted in Fig. 7(c) labelled as “Wavelet
Detector”. A comparison of the wavelet detector with the Hedley-Keffer
detector (labelled as “HK Detector”; Fig. 7) shows that the former resembles
the latter in overall behaviour but is much smoother and exhibits far less
fluctuations than the latter. (Note that the detector functions shown in Fig.
7(c) are scaled by their maximum values within the time interval 6.3-6.9 s.)
In fact the wavelet detector seems to provide an approximate envelope for the
HK detector in a way that a clear distinction between turbulent and non-
turbulent parts can be made. This is also seen in Fig. 7(d) where we compare
the wavelet detector function and the Hedley-Keffer criterion function (“HK
Criterion”) obtained for $T_{s}=100\Delta t$ (Fig. 3); a fairly good
resemblance is seen between the two functions. More importantly, the wavelet
detector and HK criterion functions match well in the region close to the
threshold level set for determining the indicator function, shown in Figs.
7(c) and (d) by a dashed line. This threshold is the same as that obtained in
the previous section using the Hedley-Keffer method, followed by a visual
comparison between the HK criterion and the velocity signal (Fig. 3c). Since
the HK criterion and wavelet detector functions cross the threshold at nearly
the same points (Fig. 7c), one can expect that the indicator functions
obtained from these two functions should be nearly identical; see Fig. 10(b).
The indicator function determined by using the wavelet detector (labelled as
“Indicator-Wavelet Detector”) is shown in Fig. 7(c) by a black solid line.

Figure 10: Indicator functions obtained from the wavelet detector and HK
criterion functions, superposed on the velocity signal. This shows the “fit”
of the indicator on the signal. Cases: (a) Transition Onset (b) Mid-
Transitional (c) Nearly Turbulent. Figure 11: Comparison between intermittency
factors calculated based on the Hedley-Keffer criterion function and the
wavelet detector function. A common threshold is employed for both the methods
for a given stage of transition.

Next we discuss how the wavelet detector fairs for the velocity signals at
other stages of transition (Fig. 2). For the laminar signal, the wavelet
energy is low and is predominantly present around 100 Hz (Fig. 5(b)). The HK
detector is noisy whereas the wavelet detector is much smoother (Fig. 5(c)).
For this case, however, the latter does not act as an envelope over the
former. This is because it is difficult to calculate intermittency for near-
laminar signals accurately, which is a well-recognized fact in the literature
(Jahanmiri, 2011). The indicator function for this signal is zero at all times
(Fig. 5(c)). For the Transition Onset case we see emergence of localized
wavelet energy for frequencies higher than 100 Hz (Fig. 6) and a clear
distinction between the turbulent and non-turbulent parts; see Fig. 6(c). As
we move downstream in the transition zone, the duration of turbulent spots
increases which is also reflected in the wavelet-energy distribution; however
the extended high-frequency regions associated with non-turbulent parts in the
velocity signals are clearly seen at all stages of transition (Figs. 6(b) -
8(b)). Consequently, the wavelet detector acts as a smoother version of the
HK-detector at every stage of transition (after its onset) and is a better
discriminator between the turbulent spots and non-turbulent regions (Figs.
6(c) and 8(c)). This is also seen for the fully turbulent case for which non-
turbulent parts are absent; the wavelet detector still picks up the variation
of the HK-detector quite well (Fig. 9). The indicator functions plotted in
Figs. 5(c)-9(c) are obtained by using the same threshold as used in the
Hedley-Keffer method for each case(section 3), which is applied after scaling
the wavelet detector from 0 to 1. It is important that the indicator function
shows a good fit on the detector function as the latter is the actual
discriminator between the laminar and turbulent zones; Figs. 6(c)-9(c) show
that this is indeed the case for the wavelet detector.

The above exercise shows that the wavelet detector function provides an
effective substitute for the Hedley-Keffer criterion function. This eliminates
the need to introduce an additional step of averaging the detector function
over an arbitrary value of the smoothing period that can only be determined
empirically. Although there is averaging involved for the wavelet detector
function also, it is performed over the frequency axis (instead of time axis)
and the averaging interval is the entire frequency range thereby eliminating
any arbitrariness.

Figure 12: Canonical Turbulent Boundary Layer; $y/\delta$=0.49. (a) Velocity
signal $(u/U_{\infty})$ vs. time (s). (b) Contour plot of (pre-multiplied)
wavelet energy ($fC_{w}^{2})$ in frequency-time plane. (c) Comparison between
the wavelet detector (red line) and the Hedley-Keffer (HK) detector (green
line). Both the detectors are scaled from 0-1 over the time interval shown.
Figure 13: Canonical Turbulent Boundary Layer; $y/\delta$=0.9. (a) Velocity
signal $(u/U_{\infty})$ vs. time (s). (b) Contour plot of (pre-multiplied)
wavelet energy ($fC_{w}^{2})$ in frequency-time plane. (c) Comparison between
the wavelet detector (red line) and the Hedley-Keffer (HK) detector (green
line). Both the detectors are scaled from 0-1 over the time interval shown.
The effects of edge intermittency are apparent in the signal.

Figure 10 shows a comparison between the indicator functions obtained from the
HK criterion and wavelet detector functions using a common threshold as
discussed above. There is a reasonably good match between the two indicator
functions; the small deviations are expected due to the different types of
detector function involved. This is also evident in the variation of the
intermittency factor plotted in Fig. 11 for different stages of transition.
There is a good match between $\gamma$ values calculated based on the wavelet
detector and HK criterion functions. The maximum difference is about 2$\%$
which is small compared to the typical uncertainty involved in accurate
determination of intermittency factors (Canepa et al., 2002).

Note that in the above we have used the Hedley-Keffer method (along with
visual inspection) for determining the indicator function for illustrative
purposes. The wavelet detector function proposed herein should work with any
other method used for setting the threshold and determining the indicator
function.

##  5 Wavelet detector for intermittent signals in canonical wall turbulence

To test the generality of the wavelet detector function, we apply it to a
totally different scenario involving intermittent velocity signals
representing the edge intermittency in a canonical turbulent boundary layer
(TBL). The experimental setup used for this purpose is different from that
used above for measuring transitional velocity signals (although the wind
tunnel facility is the same). The TBL was generated on a flat plate with a
sharp leading edge and velocity profile was measured at a distance of 1 m
downstream of the leading edge, where conditions corresponding to canonical
wall turbulence were established. Measurements were carried out at
$U_{\infty}=17.5$ m/s corresponding to a Reynolds number of
$Re_{\theta}=U_{\infty}\theta/\nu=2600$, where $\theta$ is the momentum
thickness. More details on the experimental setup and the mean and root-mean-
square profiles for the TBL can be found in Tyagi et al. (2022).

For a fully turbulent boundary layer, the intermittency factor is expected to
be unity (Table 1) over the bulk of the flow. However, as the edge of the
boundary layer is approached the effects of edge intermittency start becoming
apparent (Leslie et al., 1970), which is reflected as intermittent laminar
patches in an otherwise turbulent velocity signal. Therefore the detector
function designed to pick out turbulent spots in a transitional boundary layer
should also be able to pick out laminar patches associated with the edge
intermittency. (Incidentally, Hedley and Keffer (1974) originally demonstrated
their method on the intermittent velocity signals near the edge of a TBL.) To
test this expectation we have chosen velocity signals at two heights within
the canonical TBL measured by Tyagi et al. (2022): $y/\delta=0.49$ and
$y/\delta=0.9$. The first location corresponds to a fully turbulent signal and
the second corresponds to an intermittent signal; see Figs. 12(a) and 13(a).
The distribution of (pre-multiplied) wavelet energy for $y/\delta=0.49$ (Fig.
12(b)) is similar to that for the “turbulent” case discussed in section 4
(Fig. 9). On the other hand, the wavelet energy for $y/\delta=0.9$ shows
extended regions of low activity corresponding to the presence of laminar
patches in the signal; Fig. 13(b). This behaviour is qualitatively similar to
that for the transitional velocity signals seen in Figs. 6-8. The only
difference is that the frequency range over which the wavelet energy is
intermittent is shifted to higher frequencies for the TBL, as the TBL
measurement is performed at a higher freestream velocity ($U_{\infty}=17.5$
m/s) than that used for the transitional boundary layer ($U_{\infty}=7.5$
m/s). The wavelet and Hedley-Kefer detector functions for the two signals are
compared in Figs. 12(c) and 13(c). For both the signals, the wavelet detector
is seen to smooth out the fluctuations in the HK detector and provides an
approximate envelope for the latter. The wavelet detector shows large values
corresponding to the turbulent regions and extended regions of low magnitude
when the laminar patches are present (Fig. 13c). These are desirable qualities
of a detector function which the proposed detector adequately satisfies.

##  6 Conclusion

We have proposed a wavelet-transform based detector function as a substitute
for the derivative-based criterion function commonly used for calculating
intermittency for transitional velocity signals. The subjectivity involved in
specifying smoothing period in determining criterion function is brought out
using the absolute value of second derivative of velocity as the detector
function (consistent with the Hedley-Keffer method) and averaging it over
different smoothing periods. Continuous wavelet transform of the velocity
signals is carried out and a new detector function is obtained by averaging
the pre-multiplied wavelet energy over all frequencies. This choice of the
detector is motivated by the observation that the non-turbulent parts of an
intermittent velocity signal are represented as extended regions of low
wavelet energy in the frequency-time plane. The wavelet detector shows much
less fluctuations than the Hedley-Keffer detector and acts as an approximate
envelope over the latter. This makes the wavelet detector sufficiently smooth
thereby minimizing fake dropouts and obviating the need to specify a smoothing
period. At the same time the detector provides a good contrast between the
turbulent and non-turbulent regions, enabling easier discrimination between
the two. The distinguishing features of the proposed detector are as follows.

  * •

The averaging is performed along the frequency axis and not the time axis.

  * •

The specification of averaging interval is not arbitrary as averaging is done
over the entire frequency range.

The effectiveness of the wavelet detector is demonstrated by calculating
intermittency factors for transitional velocity signals measured in our
previous experiment on roughness-induced transition. The indicator functions
obtained from our detector compares well with those obtained from the Hedley-
Keffer criterion function (with $T_{s}=100\Delta t$) based on common threshold
levels; the resulting intermittency factors also match well between the two
methods. Next, we have used the wavelet detector to characterize the
intermittent velocity signals observed in canonical TBL measured in a
different experimental setup. The wavelet detector picks out the laminar
patches near the edge of the TBL fairly well suggesting that the proposed
detector should work in any general situation.

The wavelet detector function can, in principle, be used with any method that
prescribes threshold for obtaining an indicator function. We believe this
feature will prove useful in engineering situations (e.g. wind- and gas-
turbine blades) for an accurate calculation of intermittency factors.

## Declarations

## Ethical approval

Not applicable.

## Competing interests

The authors declare no competing interests.

## Authors’ contributions

SSD conceived of the study. SD and AA carried out the data processing and
analysis. SD prepared the figures. SD, AA and SSD wrote the manuscript. SSD
supervised the work.

## Funding

We acknowledge financial support from Science and Engineering Research Board
(SERB), India (Grant No.: ECR/2018/002417) towards conducting experiments on
the turbulent boundary layer.

## Availability of data and materials

Data are available from the corresponding author (SSD) on request.

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