# Turbulent drag reduction by spanwise wall forcing. Part 2: High-Reynolds-
number experiments.

D. Chandran1 A. Zampiron2 A. Rouhi3<EMAIL_ADDRESS>M. K. Fu4 D.
Wine5
B. Holloway5 A. J. Smits6 I. Marusic1 1Department of Mechanical Engineering,
University of Melbourne, Victoria 3010, Australia
2 School of Engineering, University of Aberdeen, Aberdeen, UK
3Department of Engineering, Nottingham Trent University, Nottingham, UK
4Graduate Aerospace Laboratories (GALCIT), Caltech, Pasadena, CA, USA
5Intellectual Ventures, Bellevue, WA, USA
6Department of Mechanical and Aerospace Engineering, Princeton University,
Princeton, NJ, USA

###### Abstract

Here, we present measurements of turbulent drag reduction at high friction
Reynolds numbers in the range of $4500\leq Re_{\tau}\leq 15000$. The efficacy
of the approach, using streamwise travelling waves of spanwise wall
oscillations, is studied for two actuation regimes: (i) inner-scaled actuation
(ISA), as investigated in Part 1 of this study, which targets the relatively
high-frequency structures of the near-wall cycle, and (ii) outer-scaled
actuation (OSA), which was recently presented by Marusic et al. (Nat. Commun.,
vol. 12, 2021) for high-$Re_{\tau}$ flows, targeting the lower-frequency,
outer-scale motions. Multiple experimental techniques were used, including a
floating-element balance to directly measure the skin-friction drag force,
hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear
stress, and stereoscopic-PIV (particle image velocimetry) to measure the
turbulence statistics of all three velocity components across the boundary
layer. Under the ISA pathway, drag reduction of up to 25% was achieved, but
mostly with net power saving losses due to the high-input power cost
associated with the high-frequency actuation. The low-frequency OSA pathway,
however, with its lower input power requirements, was found to consistently
result in positive net power savings of 5 - 10%, for moderate drag reductions
of 5 - 15%. The results suggest that OSA is an attractive pathway for energy-
efficient drag reduction in high Reynolds number applications. Both ISA and
OSA strategies are found to produce complex inter-scale interactions, leading
to attenuation of the turbulent fluctuations across the boundary layer for a
broad range of length and time scales.

###### keywords:

drag reduction, turbulent boundary layers, boundary layer control

##  1 Introduction

The sensitivity of wall-bounded turbulent flows to boundary conditions means
that relatively modest adjustments made to the wall can significantly modify
fluid behaviour, including drag. This concept has driven the development of a
wide variety of flow control methods (see, for example, Kim, 2011). One such
method is the active flow control strategy of imposing streamwise travelling
waves of spanwise velocity at the wall (Quadrio et al., 2009), which is the
focus of the current study and is briefly reviewed in Part 1 (Rouhi et al.,
2022). A more extensive review is given by Ricco et al. (2021). Here, the
forcing arises due to an imposed surface movement with

$w_{s}(x,t)=A\sin{\bigg{(}\kappa_{x}x-\frac{2\pi}{T_{osc}}t\bigg{)},}$ (1)

where $w_{s}$ is the instantaneous spanwise velocity at the wall, $A$ and
$T_{osc}$ are the amplitude and time period of spanwise actuation,
respectively, and $\kappa_{x}=2\pi/\lambda$ is the streamwise wavenumber of
the travelling wave; $\lambda$ is the wavelength. In Part 1, (1) was presented
based on the angular frequency of actuation $\omega=2\pi/T_{osc}$. Streamwise,
wall-normal and spanwise coordinates are denoted by $x$, $y$, and $z$,
respectively (with corresponding instantaneous velocities $u$, $v$ and $w$),
and $t$ is time. A schematic of this forcing is shown on the right side of
figure 1.

Numerical studies indicate that this approach shows great promise (Quadrio,
2011), but applying this control strategy in practice has remained a
challenge. This is especially due to the difficulties in accurately
reproducing, predicting, or modelling the turbulent dynamics that are
encountered in high Reynolds number practical flows. Previous experimental
(Auteri et al., 2010; Bird et al., 2018) and numerical investigations (Quadrio
et al., 2009; Hurst et al., 2014; Gatti & Quadrio, 2016; Gatti et al., 2018;
Skote, 2022 etc.) of this control strategy have been restricted to lower
Reynolds numbers, $Re_{\tau}\leq 1000$. Here, the friction Reynolds number is
defined as $Re_{\tau}=\delta u_{\tau_{0}}/\nu$, where,
$u_{\tau_{0}}=\sqrt{\tau_{w_{0}}/\rho}$ is the friction velocity,
$\tau_{w_{0}}$ is the wall-shear stress, $\delta$ is the boundary layer
thickness, $\rho$ is the fluid density and $\nu$ is the fluid kinematic
viscosity. The subscript ‘$0$’ indicates parameters evaluated at the non-
actuated reference condition. The superscript ‘$+$’ indicates normalization
using viscous length ($\nu/u_{\tau_{0}}$) and velocity ($u_{\tau_{0}}$)
scales. The superscript ‘$*$’ indicates the normalization using viscous scales
where the actual friction velocity is considered, i.e., $u_{\tau}$ of the
drag-altered flow for the actuated cases and $u_{\tau}=u_{\tau_{0}}$ for the
non-actuated cases.

At relatively low Reynolds numbers, $Re_{\tau}=\mathcal{O}(10^{2})$ to
$\mathcal{O}(10^{3})$, near-wall streaks are the statistically dominant
turbulent structures close to the wall and follow inner scaling (Kline et al.,
1967; Smits et al., 2011). That is, their features scale with velocity
$u_{\tau_{0}}$ and length $\nu/u_{\tau_{0}}$. The predominant time-scale of
the near-wall streaks is found to be $T^{+}=Tu_{\tau_{0}}^{2}/\nu\approx 100$
and their characteristic streamwise and spanwise lengths are
$1000\nu/u_{\tau_{0}}$ and $100\nu/u_{\tau_{0}}$, respectively. Flow control
schemes that are implemented at the wall often prescribe a forcing of similar
time and/or length scales to couple with these features and achieve the best
result (Jung et al., 1992; Choi & Graham, 1998; Quadrio & Ricco, 2004; Choi et
al., 2011; Kim, 2011; Tomiyama & Fukagata, 2013; Lozano-Durán et al., 2020).

The actuation strategy of targeting the near-wall streaks, which we refer to
as inner-scaled actuation (ISA) was the focus of Part 1 of this study where
large-eddy simulations (LES) were used. Figure 1 shows a visualization of the
near-wall flow field from LES at $Re_{\tau}=951$ for the non-actuated case and
an actuated case where the drag reduction $DR\approx 29\%$. Here,
$DR=1-\overline{\tau_{w}}/\overline{\tau_{w_{0}}}$, where
$\overline{\tau_{w}}=\rho u_{\tau}^{2}$ is the time-averaged wall shear stress
for the actuated case. The actuation is seen to deplete the streaks and
attenuate the intensity of turbulent fluctuations near the wall. Although the
performance of ISA is reported to deteriorate with increasing Reynolds number,
it is still observed to yield significant drag reduction of $DR\approx 25\%$
at $Re_{\tau}=4000$ (Part 1) and $Re_{\tau}=6000$ (Marusic et al., 2021).
Unfortunately, however, targeting near-wall streaks typically implies high-
frequency actuation, and thus high-input power requirements, so that ISA often
ends up being energy inefficient (Marusic et al., 2021).

Figure 1: Visualization of near-wall flow features for the non-actuated case
and an actuated case with a streamwise travelling wave of spanwise velocity.
The time-scale of actuation is $T_{osc}^{+}=140$, resulting in a drag
reduction of 29%.

An alternate, more energy-efficient, pathway to drag reduction at high
Reynolds numbers targets the larger scale, outer-region, structures (Marusic
et al., 2021). We refer to this relatively lower-frequency actuation strategy
as outer-scaled actuation (OSA). By outer-scaled, we refer to all motions that
scale with $y$ and/or $\delta$, corresponding to motions normally associated
with the logarithmic region and beyond (attached eddies and superstructures).
(ISA and OSA were originally referred to as small-eddy actuation and large-
eddy actuation, respectively, in Marusic et al. (2021).) The lower-frequencies
employed for OSA, as compared to ISA, means that OSA can result in positive
net power savings. Furthermore, in contrast to ISA, the performance of OSA
improves with increasing Reynolds number. Marusic et al. (2021) attributed
this trend to the difference in the turbulent drag composition at high
Reynolds numbers, where the relative contribution of large-scale (low
frequency) eddies to the wall shear stress increases.

###  1.1 Wall shear stress vs. Reynolds number

Figure 2(a) shows the pre-multiplied power spectral density (spectrum) of the
wall shear stress, $f\phi_{\tau^{+}\tau^{+}}$, obtained using the predictive
models of Marusic et al. (2010c), Mathis et al. (2013) and Chandran et al.
(2020) at $Re_{\tau}$ ranging from $10^{3}$ to $10^{6}$. Here, $f$ is the
frequency and $f^{+}=1/T^{+}=f\nu/{u^{2}_{\tau_{0}}}$. The spectra show the
relative contributions to the wall shear stress from turbulent structures of
different time-scales ($T^{+}$). The inner-scaled component, highlighted in
green, is the contribution to the wall-stress by the near-wall cycle. This
contribution is obtained by high-pass filtering the wall-shear stress signal
at a cut-off frequency $f^{+}=2.65\times 10^{3}$, following Mathis et al.
(2013), which corresponds to a cut-off timescale $T^{+}=350$. The outer-scaled
spectra, shown in red, are the contributions to the wall shear stress from the
motions centered in the logarithmic region and above. These contributions are
obtained from the phenomenological model of Chandran et al. (2020) including
hierarchies of self-similar ‘wall-attached’ eddies and VLSMs/superstructures
(Kim & Adrian, 1999; Hutchins & Marusic, 2007a).

Figure 2: (a) The contributions of near-wall, inner-scaled motions (green) and
larger, outer-scaled motions (red) to the pre-multiplied spectra of the wall
stress $\tau_{w}$, computed using predictive models (Marusic et al., 2010c;
Mathis et al., 2013; Chandran et al., 2020), at $Re_{\tau}=10^{3},\,10^{4}$
and $10^{6}$. (b) Intensity of wall-shear stress fluctuations as a function of
Reynolds number, highlighting the relative contributions from the small-scale
and large-scale structures. $\times$, data from direct numerical simulations
(DNS) by (Jiménez et al., 2010; Sillero et al., 2013).

Figure 2(a) shows that while the contributions by the inner-scaled motions to
the wall shear stress spectra are Reynolds number invariant, the contributions
by the inertial outer-scaled motions ($T^{+}\gtrsim 350$) increase with
increasing Reynolds number. This large-scale trend reflects the growing
influence of the energetic outer-region structures on the near-wall turbulence
with Reynolds number (Marusic et al., 2010c; Hutchins & Marusic, 2007b;
Agostini & Leschziner, 2018). As shown in figure 2(b), the relative
contribution of large-scales to the total wall-stress intensity
$\overline{\tau_{w}^{2}}$ increases nominally as $\mathrm{ln}(Re_{\tau})$,
from 8% at $Re_{\tau}\approx 10^{3}$ to about 35% at $Re_{\tau}\approx
10^{6}$. Therefore, at the high Reynolds numbers considered in the present
study ($Re_{\tau}\sim 10^{4})$, the outer-scaled contribution is significant,
at $18-20\%$ of the total wall shear stress intensity.

###  1.2 Present study and outline

In this paper (Part 2) we present experiments to examine streamwise travelling
waves of spanwise oscillations as a potential high Reynolds number flow
control strategy. We focus only on streamwise travelling waves of oscillations
that move in the upstream direction as they have been shown to yield
consistent drag reduction compared to downstream travelling waves at low
Reynolds numbers (Quadrio & Ricco, 2004). The combination of the high Reynolds
number boundary layer wind tunnel facility at the University of Melbourne and
a custom-made surface actuation test bed (Marusic et al., 2021) allows us to
study the actuation for a range of parameters
($A^{+},\,T_{osc}^{+},\,\kappa_{x}^{+}$) in the ISA ($T_{osc}^{+}\lesssim
350$) and OSA ($T_{osc}^{+}\gtrsim 350$) regimes, over the range of friction
Reynolds numbers $4500\leq Re_{\tau}\leq 15000$.

We use multiple experimental techniques (§2), including hot-wire anemometry, a
drag balance and stereoscopic particle image velocimetry (PIV) to (i) measure
changes in skin-friction drag due to the wall actuation and investigate their
energy-efficiency, under both ISA and OSA pathways (§3.1 to 3.2), and (ii)
examine how the wall-actuation affects turbulence statistics and the scale-
specific turbulence for a range of wall-heights (§3.3 to 3.5). We specifically
focus on the modification of turbulence in the logarithmic region of the
boundary layer as this is the major contributor to the bulk turbulence
production at high Reynolds numbers (Marusic et al., 2010a; Smits et al.,
2011).

##  2 Experimental techniques

The experiments were conducted in zero pressure gradient boundary layers in
the high Reynolds number boundary layer wind tunnel facility (Marusic et al.,
2015) at the University of Melbourne. The wind tunnel has a working section of
$27$ m length and a cross-section of $1.89$ m $\times$ $0.92$ m. All
experiments were conducted at a streamwise location of $x\approx 21$ m, where
the boundary layer attains a thickness of $\delta\approx 0.38$ m. Here,
$\delta$ is computed by fitting the mean velocity profile to the composite
profile of Chauhan et al. (2009). By varying the freestream velocities between
5 m/s $\leq U_{\infty}\leq$ 20 m/s, friction Reynolds numbers in the range
4500 $\leq Re_{\tau}\leq$ 15000 were achieved (see table 1).

Method | $\mathrm{Re}_{\tau}$ | $U_{\infty}$ | $\delta$ | $f$ | $A$ | $\kappa_{x}$ | $T_{osc}^{+}$ | $A^{+}$ | $\kappa_{x}^{+}$ | $DR\,(\%)$
---|---|---|---|---|---|---|---|---|---|---
|  | (m/s) | (m) | (Hz) | (m/s) | ($1/$m) |  |  |  |
HW, PIV | $4500$ | 5 | 0.39 | $5$ | 0.57 | 20.94 | $405$ | $3.3$ | $0.0018$ | 9 %
HW | $4500$ | 5 | 0.39 | $10$ | 1.13 | 20.94 | $203$ | $6.5$ | $0.0018$ | 16 %
HW, PIV | $4500$ | 5 | 0.39 | $15$ | 1.7 | 20.94 | $135$ | $9.8$ | $0.0018$ | 19 %
HW | $4500$ | 5 | 0.39 | $20$ | 2.26 | 20.94 | $100$ | $13$ | $0.0018$ | 22.3 %
HW | $4500$ | 5 | 0.39 | $25$ | 2.83 | 20.94 | $81$ | $16.3$ | $0.0018$ | 21.7 %
HW, PIV | $6000$ | 7 | 0.39 | $5$ | 0.57 | 20.94 | $700$ | $2.5$ | $0.0014$ | 6 %
HW | $6000$ | 7 | 0.39 | $10$ | 1.13 | 20.94 | $348$ | $4.9$ | $0.0014$ | 10 %
HW, DB, PIV | $6000$ | 7 | 0.39 | $15$ | 1.7 | 20.94 | $232$ | $7.4$ | $0.0014$ | 16 %
HW | $6000$ | 7 | 0.39 | $20$ | 2.26 | 20.94 | $174$ | $9.8$ | $0.0014$ | 20 %
HW, DB, PIV | $6000$ | 7 | 0.39 | $25$ | 2.83 | 20.94 | $140$ | $12.3$ | $0.0014$ | 24 %
HW, DB | $9700$ | 11 | 0.39 | $5$ | 0.57 | 20.94 | $1812$ | $1.5$ | $0.0008$ | 5.2 %
HW, DB | $9700$ | 11 | 0.39 | $10$ | 1.13 | 20.94 | $906$ | $3$ | $0.0008$ | 9.5 %
HW, DB | $9700$ | 11 | 0.39 | $15$ | 1.7 | 20.94 | $604$ | $4.6$ | $0.0008$ | 11.5 %
HW, DB | $9700$ | 11 | 0.39 | $20$ | 2.26 | 20.94 | $453$ | $6.2$ | $0.0008$ | 12.5 %
HW, DB | $9700$ | 11 | 0.39 | $25$ | 2.83 | 20.94 | $362$ | $7.8$ | $0.0008$ | 15 %
DB | $12800$ | 15 | 0.385 | $15$ | 1.7 | 20.94 | $1100$ | $3.5$ | $0.0006$ | 8.3 %
DB | $12800$ | 15 | 0.385 | $25$ | 2.83 | 20.94 | $653$ | $5.7$ | $0.0006$ | 13.3 %
DB | $15000$ | 20 | 0.37 | $15$ | 1.7 | 20.94 | $1975$ | $2.6$ | $0.00046$ | 4.7 %
Table 1: Summary of experimental parameters. Details of the flow conditions in
experiments with the actuation parameters adopted. The methods for the
experiments include hot-wire (HW), drag balance (DB) and stereo-PIV (PIV).

###  2.1 Surface actuation test bed

Figure 3: Schematic of the surface actuation test bed (SATB) installed in the
Melbourne wind tunnel facility. The SATB comprises four independently
controllable machines (highlighted by different colours) whose synchronous
operation generates a discrete facsimile of a long streamwise travelling wave
with a total fetch of $8\lambda$ (= 2.4 m).

The streamwise travelling waves of spanwise velocity, as defined by (1), were
implemented in the experiments using a surface actuation test bed (SATB). The
SATB was custom designed for high-Reynolds number turbulent boundary layers to
actuate in both the ISA and OSA regimes, and it measures 2.4 m x 0.7 m, as
illustrated in figure 3. It comprises a series of forty-eight, $50$ mm-wide
slats that oscillate in the spanwise direction in a synchronous manner to
produce a streamwise travelling, sinusoidal wave that underlies the turbulent
boundary layer. Spanwise oscillations within a frequency range 5 Hz $\leq
f_{osc}\leq$ 25 Hz were achieved with peak amplitudes (velocities) of
actuation equivalent to $A=2\pi f_{osc}\,d$, where $d=18$ mm being the fixed
half-stroke length. As shown in figure 3, the streamwise travelling wave is
discretized such that six slats constitute a fixed streamwise wavelength
(Auteri et al., 2010), with $\lambda=0.3$ m. As highlighted by different
colours in the schematic, the SATB was driven by four independently
controllable machines, and their selective, synchronous operation enabled a
variable streamwise fetch of actuation ($l_{act}$) of $2\lambda\leq
l_{act}\leq 8\lambda$, at $2\lambda$ increments. Marusic et al. (2021)
provides further details regarding the precision and tolerance associated with
the SATB fabrication, together with a supplementary video of its operation in
the wind tunnel.

###  2.2 Floating element drag balance

The SATB is flush mounted on a large-scale floating element drag balance
assembly (see figure 3) that is installed in the bottom wall of the wind
tunnel, between 19.5 and 22.5 m downstream of the trip. The floating element
has an exposed surface area of $3$ m $\times$ $1$ m (Baars et al., 2016).
Compressed air at 10 bar is supplied through four air bearing assemblies to
enable a nominally frictionless streamwise movement while any spanwise
movements are arrested by additional air bearings. A single-beam load cell
with a full-scale range of 6 N ($0.06\%$ accuracy of full-scale) constrains
the streamwise displacement of the floating element and thereby measures
directly the area-averaged skin-friction drag force ($F_{w}$) on the total
exposed surface of the floating element. In-situ calibrations of the load cell
with known weights were performed immediately before and after the drag
measurements, and the calibrations were performed separately for the actuated
and non-actuated cases to account for any preload induced by the mechanical
operation of the actuator.

The effective drag reduction is therefore estimated according to

$DR=(1-\overline{F_{w}}/\overline{F_{w_{0}}})/A_{ratio},$ (2)

where $\overline{F_{w_{0}}}$ and $\overline{F_{w}}$ are the time-averaged drag
forces measured by the load cell for the non-actuated and actuated cases,
respectively. The ratio of the actuated area to the total area of the floating
element is given by $A_{ratio}=0.53$, which includes about $8\%$ of the
floating element surface located immediately downstream of the SATB where
latent drag reduction effects were present (see figure 3). A sample time
series of $F_{w_{0}}$ (black) and $F_{w}$ (blue) is shown in figure 4(a), for
a case with $DR=27\%$. In all measurements, the signal from the load cell was
sampled for at least 60 seconds at 1000 Hz and the drag measurements were
repeated at least three times for both the non-actuated and actuated
conditions. About 98% of the repeated drag measurements were within $\pm 0.01$
N of the mean, which resulted in a maximum uncertainty of $\pm 4\%$ in
$DR(\%)$, as indicated by error bars in the later plots.

Figure 4: (a) Sample time series of drag measured by the load cell in the drag
balance at $Re_{\tau}=6000$ for the non-actuated and an actuated case
($A^{+}=12$, $T_{osc}^{+}=140$, $\kappa_{x}^{+}=0.0014$) as shown in solid
black and blue lines, respectively. The dashed lines denote the time-averaged
mean of the respective time signals. (b) Sample time series measurements of
wall shear stress $\tau_{w}$ obtained using hot-wires, for the cases given in
(a). (c) Sample mean velocity distributions for the non-actuated and two
actuated cases at $Re_{\tau}=6000$ obtained using hot wires. Left panel shows
profiles of the dimensional mean velocity $U$. Right panel shows the same
profiles non-dimensionalized using the actual friction velocity
($U^{*}=U/u_{\tau}$). The grey-shaded region highlights the “useful linear
region” (Hutchins & Choi, 2002).

###  2.3 Hot-wire anemometry

Local wall-shear stress measurements were obtained using hot-wire anemometry
for $4500\leq Re_{\tau}\leq 9700$. The hot-wire sensors had a diameter and
length of $d_{HW}=2.5$ µm and $l_{HW}=0.5$ mm, respectively, so that
$l_{HW}/d_{HW}=200$ and $5.7\leq l_{HW}^{+}\leq 12$ (Hutchins et al., 2009;
Smits, 2022) across the $Re_{\tau}$ range. The sensors were operated using a
Melbourne University Constant Temperature Anemometer (MUCTA) at an overheat
ratio of 1.8 and a resultant frequency response of 20 kHz. The hot-wire probes
were calibrated in-situ in the wind tunnel freestream at 15 different
velocities and subsequently fitted with a third-order polynomial.

The hot-wire measurements were performed in the nominally linear velocity
region ($U^{*}=y^{*}$) within the viscous sublayer. The instantaneous wall
shear stress is proportional to the instantaneous velocity gradient at the
wall, and within the linear region this velocity gradient can be approximated
to within a few percent error by measuring the streamwise velocity at a single
wall-height (Hutchins & Choi, 2002). A sample time trace of this deduced wall
stress is shown in figure 4(b). In addition, for measurements at a constant
height above the stationary and actuated walls,

$DR=1-\overline{\tau_{w}}/\overline{\tau_{w_{0}}}=1-U/U_{0}.$ (3)

Figure 4(c) shows the sample mean velocity from hot-wire data at
$Re_{\tau}=6000$. In agreement with the observations of Hutchins & Choi
(2002), a “useful linear region” was found to exist only between $4\leq
y^{*}\leq 5.5$. There, the hot-wire is close enough to the wall to be in the
linear region while not close enough for the signal to be contaminated by
wall-conduction effects which were observed to come into play for $y^{*}<4$.
Consequently, in all cases, hot-wire data were acquired at two to four wall-
normal locations within the useful linear region that corresponded to about
$y\approx 400$ µm at $Re_{\tau}=4500$ and $y\approx 200$ µm at
$Re_{\tau}=9700$. The accuracy of positioning the hot-wires at such close
proximity to the wall was determined by the linear optical encoder (RENISHAW
RGH24-type, $\pm 0.5$ µm accuracy) within the stepper motor-driven vertical
traverse, supplemented by a depth measuring displacement microscope (Titan
Tool Supply, $\pm 1$ µm accuracy). The practical challenges associated with
identifying the useful linear region restricted the hot-wire measurements to
$Re_{\tau}\leq 9700$.

The signals were low-pass filtered using an 8-pole Butterworth filter
(Frequency Devices, Inc. model 9002) with the roll-off frequency set at half
the sampling frequency to minimise aliasing. The signals were sampled at $40$
kHz for $Re_{\tau}=4500$ and 6000 and $50$ kHz for $Re_{\tau}=9700$. To ensure
converged statistics, the signals were sampled for $t=60-90$ seconds,
corresponding to non-dimensional boundary layer turnover times of
$tU_{\infty}/\delta=1100-2500$.

###  2.4 Stereoscopic particle image velocimetry (PIV)

Figure 5: A schematic of the two-camera stereoscopiv-PIV arrangement for
measurements over the SATB. The red dashed lines show the field of view of the
arrangement ($150$ mm $\times$ $70$ mm) along the spanwise$-$wall-normal
plane.

A two-camera stereoscopic PIV system was used to measure the three velocity
components within a 150 mm ${\times}$ 70 mm spanwise ($y{-}z$) plane located
about the centre of the tunnel (see figure 5). Image pairs were recorded at a
constant frequency of 0.5 Hz using two Imperx GEV-B6620 CCD cameras
(6600${\times}$4400 pixels, 5.5 µm pixel pitch, 12 bits per pixel) equipped
with Tamron AF 180 mm lenses at f/8 aperture and Sigma APD 2${\times}$
teleconverters. The cameras were rotated by ${\pm}42^{\circ}$ with respect to
the laser sheet and Scheimpflug adapters were used to achieve uniform focus
across the measurement plane. A series of cylindrical lenses was used to shape
the output of a Spectra Physics Quanta Ray double-pulse Nd:Yag laser (532 nm,
400 mJ/pulse) into a 1.5 mm-thick sheet. Tracer particles of 1-2 µm diameter
were generated from a propylene glycol mixture using a fog machine and the
tracers were injected upstream of the facility’s flow conditioning section for
a homogeneous seeding density at the test section.

The acquired images were ‘dewarped’ onto a common grid and two-component
velocity fields were computed independently for each camera through cross
correlation. We used an iterative deformation method (e.g. Astarita & Cardone
(2005)) with Blackman weighted interrogation regions of size 48${\times}$48
pixels and an overlap of 75%, resulting in a vector grid spacing of 12 pixels
(${\approx}$0.25 mm). The three-component velocity fields were finally
computed by combining the velocity fields from each camera using the local
viewing angles obtained through the calibration procedure. The camera
configuration provided two estimates of the vertical velocity component, which
were used to reduce the noise contribution to the vertical velocity variance
$\langle{v^{\prime}}^{2}\rangle$ (Cameron et al., 2013).

The cases covered by the PIV measurements at $Re_{\tau}=4500$ and $6000$ are
reported in table 1. A total of 1000 independent velocity fields were captured
for each case. The spatial resolution of the PIV data was assessed by
comparing the streamwise variance distributions obtained for the non-actuated
cases with the hot-wire data from Marusic et al. (2015) at matched $Re_{\tau}$
(see figure 11). Although PIV experiments were also carried out at
$Re_{\tau}=9700$, the data were found to have insufficient spatial resolution
for the current investigation and are therefore not included in the paper.

##  3 Results and discussion

###  3.1 Drag reduction and net power savings

For the flows considered here, the wall shear stress depends on the following
parameters

$\frac{\overline{\tau_{w}}}{\rho}=u_{\tau}^{2}=g_{1}(U_{\infty},\,\delta,\,\nu,\,\kappa_{x},\,T_{osc},A).$
(4)

Dimensional analysis thereby yields

$DR=g_{2}(\kappa_{x}^{+},\,T_{\mathrm{osc}}^{+},\,A^{+},\,Re_{\tau}).$ (5)

(This is the same as equation 1.3 in Part 1.) For the SATB test-bed,
$A^{+}=(2\pi/T_{\mathrm{osc}}^{+})(d/\delta)Re_{\tau}$ and
$\kappa_{x}^{+}=(2\pi/Re_{\tau})(\delta/\lambda)$, where $d/\delta$ and
$\delta/\lambda$ are approximately constant as $\delta=0.38\pm 0.01$ m across
all measurements. Therefore, $A^{+}\propto\,Re_{\tau}/T_{osc}^{+}$ and
$\kappa_{x}^{+}\,\propto\,1/Re_{\tau}$. Consequently, $Re_{\tau}$ could not be
varied independent of the parameters of actuation
($A^{+},\,T_{osc}^{+},\,\kappa_{x}^{+}$). Similarly, at a particular
$Re_{\tau}$, one actuation parameter could not be varied independent of the
others. Therefore, in our experiments, based on the above relationship between
$A^{+},\,T_{osc}^{+}$ and $Re_{\tau}$, (5) simplifies to

$\displaystyle DR$
$\displaystyle=g_{3}(T_{\mathrm{osc}}^{+},\,Re_{\tau}),\,\mathrm{or}$ (6a)
$\displaystyle DR$ $\displaystyle~{}\approx~{}g_{4}(A^{+}).$ (6b)

Figure 6: Drag reduction ($DR$) and net power savings ($NPS$) as functions of
$T_{osc}^{+}$ and $A^{+}$ for $Re_{\tau}$ ranging from 4500 to 12800. The
green shaded regions correspond to $NPS>0$. The circles and triangles
represent the hot-wire data and drag balance data, respectively. The cross
indicates an LES data point at $Re_{\tau}=6000$ at matched actuation
conditions. The error bars indicate one standard deviation uncertainty ranges.

Figure 6 shows how the measured drag reduction ($DR$) and net power savings
($NPS$) depend on $T_{osc}^{+}$ and $A^{+}$. Here, $NPS$ is computed using the
generalised Stokes layer theory (Quadrio & Ricco, 2011; Gatti & Quadrio,
2013), $NPS$ being the difference between $DR$ and the net power required to
move the flow sideways, as in (1), to generate the Stokes layer. In other
words, it is the net input power required by an ‘ideal’ actuation system to
implement (1). Refer to § 3.6 in Part 1 for further details of how net input
power is computed using the generalised Stokes layer theory and its validation
with respect to LES data.

The results demonstrate that at $Re_{\tau}=4500$ a peak $DR\approx 24\%$ is
achieved with ISA at $T_{osc}^{+}\approx 100$ and $A^{+}\approx 13$. The time
scale of this actuation corresponds to the time scale of the near-wall streaks
that contribute to the peak in figure 2. However, by factoring in the power
input for the actuation, this ISA case actually incurs negative net power
savings ($NPS$) of $-40\%$ (i.e., a net power cost). We see that for most
cases in the ISA regime, NPS is negative, with the loss increasing sharply for
$T_{osc}^{+}<100$. In contrast, when the time period of actuation is increased
beyond $T_{osc}^{+}\gtrsim 350$ to target the OSA pathway, positive net power
savings of $+7\%$ are achieved with a moderate $DR=9\%$. A similar trend is
observed as $Re_{\tau}$ is increased to 6000 and 9700, where OSA consistently
results in positive net power savings in the range $4\%\leq NPS\leq 9\%$ for
corresponding drag reductions in the range $17\%\geq DR\geq 5\%$. It is
important to note that the positive $NPS$ are achieved here with relatively
modest amplitudes, $1.5\leq A^{+}\leq 7.8$. The LES reference point at
$Re_{\tau}=6000$ (reported in Part 1) for matched actuation parameters agrees
well with the data obtained from hot-wire and drag-balance measurements.

At higher Reynolds number, the energy-efficiency offered by the OSA pathway
improves further (Marusic et al., 2021). For example, at $Re_{\tau}=12800$ the
OSA pathway results in $NPS=8.7\%$ and 7%, corresponding to $DR=13.3\%$ and
8.3%, respectively. At the highest Reynolds number achieved here,
$Re_{\tau}=15000$, a very low-amplitude ($A^{+}=2.6$), low-frequency
($T_{osc}^{+}=1975$), low power actuation yielded $NPS=4\%$ even with a
relatively low amount of drag reduction, $DR=4.6\%$.

Figure 7: Drag reduction $DR$ and net power savings $NPS$ versus $T_{osc}^{+}$
and $A^{+}$ across the full range of Reynolds numbers and actuation
parameters. The LES results are from Part 1.

The complete $DR$ and $NPS$ results are presented in figure 7, along with
select LES data points from Part 1. Figure 7(a) shows the distribution of $DR$
as a function of $T_{osc}^{+}$ and $Re_{\tau}$, as given by (6a). Again, we
see that the experimental data agree well with the two LES data points at
similar operating conditions, i.e., at $Re_{\tau}\approx 4500$,
$T_{osc}^{+}\approx 100$ and at $Re_{\tau}=6000$, $T_{osc}^{+}=140$. Our data
show that $DR$ varies only moderately with Reynolds number under both ISA and
OSA strategies, for the range investigated. However, as noted earlier, in our
experiments, as $T^{+}_{osc}$ increases $A^{+}$ decreases and therefore limits
$A^{+}$ to relatively smaller values at high Reynolds numbers (see table 1).

Figure 7(a) shows that the drag reduction peaks at $DR\approx 25\%$ at
$T^{+}_{osc}\approx 100$ in the ISA regime, beyond which $DR$ decreases
approximately semi-logarithmically with increasing $T^{+}_{osc}$. The $DR$ in
LES decreases by a lesser extent than in the experiments because $A^{+}$ in
LES is fixed at $12$, while in the experiments it decreases from about 12 at
$T^{+}_{osc}\simeq 100$ to $2.6$ at $T^{+}_{osc}=1975$. However, these high-
frequency ISA actuations mostly yield negative $NPS$, as shown in figure 7(c),
with $NPS$ decreasing rapidly as $T_{osc}^{+}$ falls below 100. $NPS$ is found
to be slightly positive (up to $+4\%$) for two data points at
$T_{osc}^{+}\approx 200$. In contrast, the OSA pathway consistently yields
positive $NPS$ in the range $5-10\%$ even with a moderate $DR$ of $5-15\%$.

The variation of $DR$ and $NPS$ with $A^{+}$ is shown in figures 7(b,d). For
the current actuation system, $DR$ when expressed as a function of $A^{+}$ is
nearly independent of Reynolds number (6b). The results support this
conclusion, in that for a particular $A^{+}$, $DR$ obtained at various
$Re_{\tau}$ collapse well. Further, $DR$ increases almost linearly with
$A^{+}$ up to $A^{+}\approx 12$, beyond which it appears to saturate. Although
the value of $A^{+}$ to achieve maximum $DR$ appears to be $\geq 12$, the plot
of $NPS$ in figure 7(d) suggests lower amplitudes of actuation
($A^{+}\lessapprox 7$) are necessary to yield positive $NPS$. These lower
amplitudes correspond to larger time periods of actuation as $A^{+}\propto
Re_{\tau}/T_{osc}^{+}$ in our experiments. We believe these trends demonstrate
the potential of an OSA pathway, with its low-frequency, low-amplitude
actuation strategy, for energy-efficient drag reduction at high Reynolds
numbers.

###  3.2 Spatial modification of turbulent drag and its recovery

Figure 8: (a) Variation of drag reduction ($DR$) along the length of the
actuator, $0<l_{act}<2.4$ m (yellow shaded region) at $Re_{\tau}=6000$ for two
cases: $A^{+}=7.4,\,T_{osc}^{+}=232$ and $A^{+}=12,\,T_{osc}^{+}=140$. For the
first case, the recovery of skin friction drag downstream of the actuator
$l_{act}>2.4$ m (grey shaded region) is also plotted. (b) The effect of the
streamwise length of actuation on the spectra of $\tau_{w}$ for $l_{act}$ up
to 2.4 m ($\approx 6\,\delta$) for the case $A^{+}=7.4,\,T_{osc}^{+}=232$. (c)
The downstream recovery of $\tau_{w}-$spectra to the reference non-actuated
state for the case in (b).

The variation of the drag reduction with downstream distance, obtained using
hot wire anemometry, are shown figure 8(a) for two ISA cases. The majority
(85%) of the net $DR$ is achieved within $l_{act}\approx 0.1$ m, which
corresponds to $l_{act}/\lambda=0.33$ (or $l_{act}/\delta=0.25$), where
$l_{act}$ is the length of actuation. In viscous units, this is equivalent to
$l_{act}^{+}\approx 1500$ and therefore 50% greater than the nominal length of
the near-wall streaks. This trend of rapidly rising $DR$ with actuation length
agrees with the observations from the temporal wall-oscillation studies of
Ricco & Wu (2004) and Skote et al. (2019). However, the localised $DR$ is
observed to saturate by $l_{act}=2\lambda=0.6$ m, beyond which the drag
reduction is almost constant with $DR\approx 16\%$ for $T_{osc}^{+}=232$ and
$DR\approx 24\%$ for $T_{osc}^{+}=140$. The wall shear stress spectra also
collapse for 0.075 m $\leq l_{act}\leq$ 2.325 m (see figure 8(b)), which
suggests that in addition to the time-averaged $DR$, the broadband time-scales
of turbulent fluctuations also saturate within $l_{act}\approx 0.1$ m.
Therefore, all hot-wire data that were used to compute $DR$ (§3.1) were
acquired with an actuation length of at least $l_{act}=2\lambda$.

Similarly, the recovery of skin-friction drag downstream of the actuator is
very rapid. For the $T_{osc}^{+}=232$ case shown in figures 8(a) and (c), a
complete recovery of both mean drag and the spectra of wall-stress is observed
within 0.2 m ($\approx$ 3000 viscous units) downstream of the actuator. This
recovery length is similar to that observed by Ricco & Wu (2004) and Skote et
al. (2019) for $T_{osc}^{+}=67$ in the ISA regime.

The spatial transients could not be investigated for OSA. It was not possible
to achieve a matched $A^{+}\,(\approx 10)$ for OSA at similar $Re_{\tau}$, due
to the limitations of the current experimental setup. We consider this to be
an important subject for a future study.

###  3.3 Mean velocity

Figure 9: (a,d) Wall-normal profiles of mean streamwise velocity, $U$, for the
non-actuated and actuated cases at $Re_{\tau}=4500$ and 6000, respectively .
(b,e) Profiles in (a,d) normalized using the local $u_{\tau}$:
$U^{*}=U/u_{\tau}$ and $y^{*}=yu_{\tau}/\nu$. (c,f) Diagnostic function
($y^{*}\,\mathrm{d}U^{*}/\mathrm{d}y^{*}$) for the non-actuated and actuated
cases. The LES data are from Part 1 at $Re_{\tau}=6000$ (Rouhi et al., 2022,
Appendix A).

The profiles of mean streamwise velocity are shown in figure 9. The profiles
were obtained from the stereo-PIV experiments, with a streamwise actuation
length of $l_{act}=4\lambda$, where the modified skin friction due to the
actuation was found to be invariant with downstream distance (see section
3.2). As displayed in figure 9(a), the dimensional mean velocity reduces
systematically near the wall for the drag-reduced cases, consistent with
previous studies on streamwise-travelling waves (Gatti & Quadrio, 2016; Bird
et al., 2018).

In figures 9(b,e) the velocity profiles are normalized with $u_{\tau}$, i.e.,
$U^{*}=U/u_{\tau}$ and $y^{*}=yu_{\tau}/\nu$, where the local $u_{\tau}$ is
estimated by fitting the PIV sublayer profiles to $U^{*}=y^{*}$ (the values
matched the hot-wire values to within 3%). As a consequence, the normalized
velocity profiles are forced to agree near the wall. In the outer region,
however, the profiles for the drag-reduced cases are shifted upwards
systematically, with the shift proportional to the amount of drag reduction.
These shifted profiles have a nominally constant slope of $\kappa^{-1}$ for
all cases considered here. Skote (2014) reported that the slope of the mean
velocity profile varies as $(\kappa\,\sqrt{1-DR})^{-1}$ in the initial non-
equilibrium state and subsequently reverts back to $\kappa^{-1}$ when the
whole of the boundary layer has adjusted to the wall actuation and hence
reached an equilibrium state. This is further verified in figures 9(c,f) using
a log-law diagnostic function, $y^{*}\,\mathrm{d}U^{*}/\mathrm{d}y^{*}$ (Nagib
et al., 2007; Skote et al., 2019). The profiles of both non-actuated and
actuated cases collapse in the logarithmic region ($200\lesssim y^{*}\lesssim
0.1Re_{\tau}$), with a nearly constant $\kappa^{-1}$. Furthermore, the shift
in the peak of the diagnostic function to higher $y^{*}$ indicates the
thickening of the viscous sublayer with increasing drag reduction, as reported
in Part 1. In figures 9(e,f), the reduced-domain LES data at $Re_{\tau}=6000$
(Part 1, Appendix A) are included for the non-actuated case and an actuated
case at matched conditions ($A^{+}=12$, $T_{osc}^{+}=140$) where $DR=23.6\%$.
These profiles agree well with the PIV data.

###  3.4 Turbulence statistics

Figure 10: Spanwise$-$wall-normal planes showing the fluctuating streamwise
velocity, $u^{\prime}$, for the (a) non-actuated and (b) an actuated case,
$A^{+}=9.8,\,T_{osc}^{+}=135,\,DR=19\%$ at $Re_{\tau}=4500$.

Figure 10 shows a sample snapshot of the fluctuating streamwise velocity from
the PIV measurements. For the non-actuated case, we see the alternating
regions of positive and negative velocity fluctuations (red and blue,
respectively) that are characteristic of the near-wall streaks. These well-
documented streaks are centered around $y^{*}=15$ with a mean spanwise
separation of about 100 viscous units, as seen in the figure. For the actuated
case the near-wall streaks are clearly weakened, and might be taken as a
symptom of reduced drag. This effect of the actuation on the near-wall
turbulence was discussed in Part 1, and with respect to the LES data shown
earlier in figure 1.

Figure 11: (a-c) Normal stresses and (d) Reynolds shear stress for the non-
actuated and actuated cases at $Re_{\tau}=4500$ and 6000. The axes are
normalized using the reference $u_{\tau_{0}}$. The grey-shaded region refers
to the logarithmic region. The profiles of
$\langle{w^{\prime}}^{2}\rangle^{+}$ are plotted in a log-log scale in (b) to
highlight the extent of its near-wall amplification due to the Stokes layer
(as represented in Part 1). The $\langle{u^{\prime}}^{2}\rangle^{+}$ profiles
from the hot-wire measurements of Marusic et al. (2015) at matched $Re_{\tau}$
are included in (a) as reference for the non-actuated cases and to highlight
the spatial resolution of PIV data.

Figure 11 shows the corresponding Reynolds stress statistics from the PIV
experiments, averaged across both the spanwise direction and time. These
statistics are often scaled either with the reference non-actuated
$u_{\tau_{0}}$ (‘$+$’ superscript) or with the actual $u_{\tau}$ (‘$*$’
superscript). Scaling with $u_{\tau_{0}}$ highlights the absolute response of
the boundary layer to the wall-actuation (Laadhari et al., 1994; Baron &
Quadrio, 1995; Ricco & Wu, 2004; Touber & Leschziner, 2012; Agostini et al.,
2014), while scaling with the actual $u_{\tau}$ tests the universal nature of
these drag-modified turbulence statistics (Baron & Quadrio, 1995; Choi, 2002;
Touber & Leschziner, 2012; Quadrio & Ricco, 2011; Gatti & Quadrio, 2016). §3.3
in Part 1 presents a detailed discussion on these different scaling choices
and their key differences. Here, in Part 2, our priority is to understand the
extent of the high-Reynolds-number boundary layer that is impacted by the
current wall-actuation strategy, in an absolute sense. We find this especially
interesting as high-Reynolds-number boundary layers are characterized by large
separation of scales and strong inner-outer interactions. Therefore, here we
choose $u_{\tau_{0}}$ to normalize the turbulence statistics. However, for
reference, a plot of turbulent stresses normalized by the actual $u_{\tau}$ is
provided in Appendix A.

As shown in figures 11(a-d), all components of the Reynolds stress are
affected by the actuation, with the strongest attenuation observed for the
highest drag reducing case. In particular, the peak in
$\langle{u^{\prime}}^{2}\rangle^{+}$ is increasingly attenuated and shifted to
higher $y^{+}$ as $DR$ increases, in accord with previous studies at lower
$Re_{\tau}$ (Jung et al. (1992); Choi & Clayton (2001); Ricco & Wu (2004);
Quadrio & Ricco (2011); Touber & Leschziner (2012) etc.). Similarly,
significant attenuation is observed in the profiles of
$\langle{v^{\prime}}^{2}\rangle^{+}$ and $-\langle
u^{\prime}v^{\prime}\rangle^{+}$, indicating a reduced momentum flux towards
the wall (Ricco et al., 2021). For $\langle{w^{\prime}}^{2}\rangle^{+}$,
however, the periodic oscillations of the Stokes layer produce a sharp
increase near the wall ($y^{+}\lesssim 25$). These trends in the Reynolds
stresses are consistent with the discussion in Part 1 (§3.3) where we also
subsequently decomposed $\langle{w^{\prime}}^{2}\rangle^{+}$ into the phase-
averaged and the stochastic components, and investigated the interaction
between the Stokes layer and the background turbulence. A similar analysis
could not be performed with the PIV data as the number of images per phase
were inadequate to generate converged phase-averaged statistics.

Figure 11 demonstrates that the effects of the actuation persist away from the
wall into the logarithmic region (grey shaded region). To highlight these
effects, we show in figure 12 the pre-multiplied turbulent kinetic energy
production $y^{+}P^{+}$, where the production term $P^{+}=-\langle
u^{\prime}v^{\prime}\rangle^{+}\,dU^{+}/dy^{+}$. Here, equal areas under the
curve (on this log-linear plot) indicate equal contributions to the
production. The logarithmic region is known to be an important contributor at
high Reynolds numbers (Marusic et al., 2010b). Interestingly, the wall
actuation reduces the turbulence production everywhere, near the wall and in
the logarithmic region (consistent with figure 5e in Part 1). The attenuation
in the logarithmic region shows the efficacy of the current wall actuation
strategy to influence the inertial-scale dynamics at high Reynolds numbers.
The reduction of turbulence production in this outer region occurs for both
ISA ($T_{osc}^{+}\lesssim 350$) and OSA ($T_{osc}^{+}\gtrsim 350$) cases and
the reduction is stronger with increasing $DR$.

For our study, the experimental restrictions imposed by the SATB hardware and
the PIV technique, limit the $A^{+}$ values and $Re_{\tau}$ range that can
examined. In figure 12, the OSA cases have $A^{+}=2.5$ and 3.3 while ISA cases
are at $A^{+}\geq 7.4$. Given that $DR$ was observed to increase almost
linearly with $A^{+}$ in the range $A^{+}\lesssim 12$ (see figure 7), we
expect that OSA would be even more effective at larger amplitudes of
actuation.

Figure 12: Premultiplied turbulence production for the non-actuated and
actuated cases at $Re_{\tau}=4500$ and 6000. Here, the normalization is using
the reference $u_{\tau_{0}}$ and the grey-shaded region refers to the
logarithmic region. The DNS data of Lozano-Durán & Jiménez (2014) at a similar
Reynolds number is included as a reference for the non-actuated case at
$Re_{\tau}=4500$.

###  3.5 Energy spectra

Figure 13: Premultiplied spectrograms of streamwise velocity
$k_{z}^{+}\phi_{uu}^{+}$, computed from PIV data, as functions of spanwise
wavelength ($\lambda_{z}$) and wall-height. The left panels show the spectra
for the non-actuated (red) and actuated (black) contour conditions. The middle
panels show the difference between the non-actuated and actuated spectra (the
red-shaded contours indicate energy attenuation), and the right-most panels
show the difference normalized by the respective amplitude of actuation
$A^{+}$. Here, the spectra and the axes are normalized using the reference
$u_{\tau_{0}}$.

Figure 14: Premultiplied co-spectra $k_{z}^{+}\phi_{-uv}^{+}$, computed with
the PIV data, as functions of spanwise wavelength ($\lambda_{z}$) and wall-
height. Here, the spectra and the axes are normalized using the reference
$u_{\tau_{0}}$. The figure is panelized as in figure 13.

The spanwise/wall-normal PIV fields were used to compute the spanwise-
premultiplied energy spectra of the streamwise velocity component,
$k_{z}^{+}\phi_{uu}^{+}$, and the co-spectra, $k_{z}^{+}\phi_{-uv}^{+}$ as a
function of spanwise wavelength ($\lambda_{z}$) and wall-height ($y^{+}$).
Here, $k_{z}^{+}\phi_{uu}^{+}=k_{z}\phi_{uu}/u_{\tau_{0}}^{2}$ and
$k_{z}^{+}\phi_{-uv}^{+}=k_{z}\phi_{-uv}/u_{\tau_{0}}^{2}$. The streamwise
turbulence intensity $\langle{u^{\prime}}^{2}\rangle^{+}$ and the Reynolds
shear stress $-\langle u^{\prime}v^{\prime}\rangle^{+}$ are the integral of
$k_{z}^{+}\phi_{uu}^{+}$ and $k_{z}^{+}\phi_{-uv}^{+}$, respectively, as given
by,

$\langle{u^{\prime}}^{2}\rangle^{+}=\int_{-\infty}^{\infty}k_{z}^{+}\phi_{uu}^{+}\,\text{d}(\mathrm{ln}\,\lambda_{z}^{+})\,\,\,\,\mathrm{and}\,\,\,\,-\langle
u^{\prime}v^{\prime}\rangle^{+}=\int_{-\infty}^{\infty}k_{z}^{+}\phi_{-uv}^{+}\,\text{d}(\mathrm{ln}\,\lambda_{z}^{+}).$
(7)

The energy spectra allow us to study the scale-specific response to the wall-
actuation, revealing the range of scales that contribute to the attenuation of
turbulence intensities observed in §3.4.

The spectrograms of $k_{z}^{+}\phi_{uu}^{+}$ are displayed in the left-hand
panels of figure 13. For the non-actuated cases, the spectrograms show the
expected near-wall peak at $y^{+}\approx 15$ and $\lambda_{z}^{+}\approx 100$
of the near-wall cycle. The spectrograms also show the emergence of a
secondary peak in the logarithmic region, typically observed at high Reynolds
numbers (Hutchins & Marusic, 2007a; Mathis et al., 2009; Vallikivi et al.,
2015; Lee & Moser, 2015; Samie et al., 2018). With actuation, the near-wall
peak is attenuated and its location shifts to slightly higher wall-heights,
corresponding with the shift in the peak in
$\langle{u^{\prime}}^{2}\rangle^{+}$ seen earlier (figure 11). A similar trend
was observed in Part 1 in the spectrograms of streamwise velocity at
$Re_{\tau}=4000$, and by Gatti et al. (2018) at lower Reynolds numbers.

To highlight changes due to actuation, the spectrograms from the actuated
cases are subtracted from the corresponding non-actuated spectrograms and
plotted in the middle panels in figure 13. The red-shaded contours therefore
indicate energy attenuation. We see that the actuation results in the
attenuation of a broad range of turbulent scales and importantly, that the
effect of actuation at the wall is also felt by the large-scales
($\lambda_{z}\gg 100$) in the outer region. The extent of this effect appears
to be directly proportional to $DR$. Since $DR$ was found to vary almost
linearly with $A^{+}$ for $A^{+}\leq 12$ (figure 7), we divide the difference
spectra by $A^{+}$ in the right-most panels in the figure. (Note, for the SATB
the values of $A^{+}$ decrease with increasing $T_{osc}^{+}$ as,
$A^{+}=(2\pi/T_{osc}^{+})(d/\delta)Re_{\tau}$.) In these scaled plots, the
level of attenuation, along with the range of turbulent scales ($\lambda_{z}$)
and wall-heights that are attenuated, look more similar among the different
cases at a given Reynolds number.

Similar trends are observed in the co-spectra, $k_{z}^{+}\phi_{-uv}^{+}$, as
shown in figure 14. The wall actuation attenuates the energy of the spectral
peaks while their wall-normal locations are shifted to slightly higher wall-
heights. These trends support the notion that the actuation reduces the
momentum transfer to the wall (Luchini, 1996; Gatti et al., 2018). As in
figure 13, when the difference co-spectra are divided by $A^{+}$ (right-most
panels), the level and the range of scales attenuated look more similar across
the different cases at a particular $Re_{\tau}$. If this conjecture of $A^{+}$
scaling holds, then at matched amplitudes ISA and OSA could potentially have a
similar impact on the range of turbulent scales attenuated. This would make
the low-frequency OSA a good deal more attractive due to its significantly
lower input power requirements compared to ISA. However, testing this
conjecture would require varying $A^{+}$ independent of $T_{osc}^{+}$, which
is not possible with the current actuation system.

Figure 15: Difference between the actuated and the corresponding non-actuated
(reference) $\tau_{w}-$spectra at $Re_{\tau}=6000$ (inner-scaled actuation
with $T_{osc}^{+}\lesssim 350$) and $Re_{\tau}=9700$ (outer-scaled actuation
with $T_{osc}^{+}\gtrsim 350$). The negative regions represent the energy
attenuated due to the actuation, and vice versa. The reference $u_{\tau_{0}}$
is used for the normalization. The plots demonstrate that ISA and OSA both
affect a broad range of scales.

The $u$-spectra and $uv$ co-spectra results indicate that a broad range of
turbulent scales is affected, in both the ISA and OSA cases. To investigate
this further, we compute the spectra of wall shear stress fluctuations
obtained using hot-wire measurements. Figure 15 shows the difference between
the actuated and the corresponding non-actuated cases. The negative regions
represent the scales where energy is attenuated. The actuation time-scales are
highlighted by the red line. Regardless of the time-scale of actuation, the
effect of actuation is not localised to $T^{+}=T_{osc}^{+}$ but it is felt
across a broad range of scales. That is, the ISA strategy targeting the near-
wall motions with $T_{osc}^{+}=140$ reduces the turbulent energy in the scale-
range $20\lesssim T^{+}\lesssim 10^{3}$. Similarly, an OSA strategy targeting
the larger, outer-scale motions with $T_{osc}^{+}=906$ significantly impacts
the small, near-wall motions at $T^{+}\approx 100$.

These observations suggest a complex inter-scale interaction between the
(viscous) inner-scaled and the (inertial) outer-scaled motions. At higher
Reynolds numbers, when the attached eddies and very-large-scale motions emerge
and contribute meaningfully to the drag composition of the flow, this inter-
scale coupling is expected to be significant. A clear understanding of the
associated dynamics requires further study and is outside the scope of the
current paper. Nevertheless, due to its significantly lower input power
requirements, leveraging this coupling would make OSA a potential candidate
for energy-efficient drag reduction at high Reynolds numbers (Marusic et al.,
2021).

##  4 Summary and conclusions

We have reported on the efficacy of upstream travelling waves of spanwise
oscillations as an active drag reduction strategy for high-Reynolds number
turbulent flows. In Part 1, we described a numerical study of this control
strategy at $Re_{\tau}=950$ and $4000$. Here in Part 2, we studied its
performance experimentally over a range of Reynolds numbers an order of
magnitude higher than that previously investigated in the literature. At these
high Reynolds numbers, the drag composition is no longer dominated by the
high-frequency turbulent motions that are universally encountered very close
to the wall, and it includes a significant contribution from the inertial low-
frequency motions that are centered in the logarithmic region and have a
signature at the wall. The relative contribution of these outer-scaled motions
to skin-friction drag increases nominally as $\mathrm{ln}(Re_{\tau})$ and is
here found to be about 20% at $Re_{\tau}\sim 10^{4}$.

Following Marusic et al. (2021), we pursued two pathways to drag reduction:
inner-scaled actuation (ISA) that targets the universal inner-scaled, near-
wall features with characteristic time-scales of $T^{+}\lesssim 350$, and
outer-scaled actuation (OSA) that targets the inertial, outer-scaled
high-$Re_{\tau}$ features with time-scales of $T^{+}\gtrsim 350$.

When the flow is actuated with ISA parameters ($81\leq T_{osc}^{+}\leq 348$
and $6.5\leq A^{+}\leq 16.3$), we find substantial drag reduction up to a
maximum of $DR\approx 25\%$ at $Re_{\tau}=6000$. Consistent with Marusic et
al. (2021), these results imply that spanwise wall oscillation strategies
designed to target the near-wall structures are effective for reducing drag at
higher Reynolds numbers, as predicted by the formulation of Gatti & Quadrio
(2016). We note, however, that ISA mostly incurred negative net power savings
between $-85\%\leq NPS\leq+4\%$ for the range of parameters evaluated here. On
the other hand, following the OSA pathway, characterized by much lower
actuation frequencies ($362\leq T_{osc}^{+}\leq 1975$) and relatively low-
amplitudes ($1.5\leq A^{+}\leq 7.8$), we demonstrated that considerable drag
reduction could still be achieved, contrary to previous predictions (Ricco et
al., 2021), and with much lower input power. Actuation under OSA reduced drag
in the range $5\%\leq DR\leq 15\%$ but consistently resulted in positive net
power savings with $5\%\leq NPS\leq 10\%$, across a wide range of Reynolds
numbers ($4500\leq Re_{\tau}\leq 15000$). For the current study, however, the
actuation parameters are inter-dependent as
$A^{+}\propto\,Re_{\tau}/T_{osc}^{+}$. This limited the large$-T_{osc}$ OSA
strategy to relatively lower amplitudes of $A^{+}\leq 7.8$. The relationship
also resulted in $DR$ having a nearly $Re_{\tau}-$independent functional form
with $A^{+}$, with $DR$ nominally increasing linearly with $A^{+}$ up to
$A^{+}=12$.

For the drag reduced cases, the boundary layer is modified by the thickening
of the viscous sublayer as observed in the profiles of mean streamwise
velocity and a log-law diagnostic function,
$y^{*}\,\mathrm{d}U^{*}/\mathrm{d}y^{*}$. The reduction in mean turbulent drag
is accompanied by attenuation of the Reynolds normal and shear stresses and
the production of turbulent kinetic energy. Both ISA and OSA impact the
boundary layer close to the wall and in the outer region. Although the
attenuation is most significant around the near-wall peak ($y^{+}\approx 15$),
the impact extends up to the end of the logarithmic region. This attenuation
of turbulent intensities, under both ISA and OSA pathways, is associated with
the attenuation of a broad spectrum of turbulent time scales
$\mathcal{O}(10)<T^{+}<\mathcal{O}(10^{3})$, i.e., the low-frequency OSA
strategy ($T_{osc}^{+}\gtrsim 350$) reduces drag by also attenuating the high-
frequency, near-wall cycle ($T^{+}\approx 100$). These findings suggest that
inter-scale interactions may play a central role in the ability of OSA to
considerably reduce drag at high Reynolds numbers. These inter-scale
interactions therefore warrant closer inspection.

## Acknowledgments

The research was funded through the Deep Science Fund of Intellectual
Ventures. We thank the Scientific Instrument Manufacturing and Test Group at
Intellectual Ventures Laboratory for the fabrication of SATB, Mr. Geoff Duke
for the technical support in the laboratory, and Drs. Kevin, Charitha de Silva
and Stuart Cameron for providing helpful suggestions for the PIV experiments.

Declaration of interests. The authors report no conflict of interest.

##  Appendix A Turbulence statistics scaled by actual $u_{\tau}$

In §3.4, discussion of turbulence statistics computed using the PIV data was
based on the statistics being scaled by the reference non-actuated
$u_{\tau_{0}}$ (‘$+$’ superscript). This scaling highlighted the absolute
attenuation in the turbulence statistics for the drag-reduced cases (figures
11 and 12). An alternate ‘inner-scaling’ is generally adopted to investigate
the degree of universality of the turbulence statistics where the statistics
are normalized with the actual $u_{\tau}$ (‘$*$’ superscript). This scaling is
shown here as figure 16. Here, we note that $\delta^{*}$
($=\delta^{+}\sqrt{1-DR}$) for the drag reduced cases would be lower than that
of the corresponding non-actuated case. Similarly, the spatial resolution (in
terms of $u_{\tau}$) of the PIV data for the drag-reduced cases would be
correspondingly slightly higher than that of the non-actuated case.

For the prescribed inner-scaling, we see from figure 16 that the Reynolds
stress profiles of the actuated cases shift closer to the non-actuated
profile. For example, the magnitude of the peaks in the profiles of
$\langle{u^{\prime}}^{2}\rangle^{*}$ appear to be roughly similar, even though
still being shifted slightly to higher wall-heights. A similar trend was
observed with the LES data in Part 1 (see figure 5 in Part 1). However, we do
not see the profiles collapsing away from the wall and towards the logarithmic
region (shaded in grey). Instead, they seem to be systematically shifted
upwards with increasing $DR$. This trend could be expected because when scaled
by reference $u_{\tau_{0}}$ (figure 11), the profiles were observed to start
merging towards the end of the logarithmic region, indicating that the
absolute effects of wall-actuation were minimal at those higher wall
locations. Figure 16(e) show the premultiplied turbulence production
$y^{*}P^{*}$ and its trend with $DR$. This trend is observed to be consistent
with similar profiles from the LES as shown in figure 5(f) in Part 1.

Figure 16: Turbulence statistics normalized using the actual $u_{\tau}$. (a-c)
Normal stresses, (d) Reynolds shear stress and (e) premultiplied turbulence
production for the non-actuated and actuated cases at $Re_{\tau}=4500$ and
6000. The $\langle{u^{\prime}}^{2}\rangle^{*}$ profiles from the hot-wire
measurements of Marusic et al. (2015) at matched $Re_{\tau}$ are included in
(a) as reference for the non-actuated cases.

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