# Reynolds number effect on drag reduction in pipe flows by a transverse wall
oscillation

Daniel Coxe1<EMAIL_ADDRESS>Yulia Peet1  Ronald Adrian1  1Arizona State
University, School for Engineering of Matter Transport and Energy, Tempe, AZ
85281, USA

###### Abstract

Direct numerical simulations of turbulent pipe flow with transverse wall
oscillation (WWO) and with no transverse wall oscillation (NWO) are carried
out at friction Reynolds numbers $Re_{\tau}=170,360,\text{ and }720$. The
period and amplitude of the oscillation are selected to achieve high drag
reduction in this Reynolds number range, and the effect of increasing Reynolds
number on the amount of drag reduction achievable is analyzed. Of a particular
interest in this study is the identification of the scales of motion most
affected by drag reduction at different Reynolds numbers. To answer this
question, both one-dimensional and two-dimensional spectra of different
statistical quantities are analyzed with and without transverse wall
oscillation. The effect of wall oscillation is found to suppress the
intermediate- and large-scale motions in the buffer layer of the flow, while
large-scale and very-large-scale motions in the log layer and the wake region
are enhanced. While suppression of the near-wall turbulence promotes drag
reduction, enhancement of the large-scale motions in the log and the wake
region is found to oppose drag reduction. Since higher Reynolds number flows
support development of a growing range of large-scale structures, it is
suggested that their prevalence in the energy spectra combined with their
negative effect on drag reduction account for a reduced effectiveness of wall
oscillation as a drag reduction mechanism with increasing Reynolds numbers.

###### keywords:

Turbulent Pipe Flow, Transverse wall oscillation, drag reduction, energy
spectra, large-scale motions

##  1 Introduction

Reduction of skin friction drag in turbulent flows is a highly-sought outcome
of passive and active flow control techniques, especially given that skin
friction drag contributes approximately 50%, 90% and 100% of the total drag on
airplanes, submarines and pipelines, respectively (Gad-el Hak, 1994; Abbassi
et al., 2017). Most of the studies on drag reduction, both numerical and
experimental, were performed in a relatively low Reynolds number regime,
typically around $Re_{\tau}\sim 200-400$ in simulations, and $Re_{\tau}<2000$
in experiments (Gatti & Quadrio, 2016; Ricco et al., 2021; Marusic et al.,
2021), where $Re_{\tau}$ denotes the friction Reynolds number. In contrast,
typical Reynolds numbers encountered in applications attain the values of
$Re_{\tau}=4000$ on a wind turbine blade or in a long-distance pipeline,
$6000$ at the mid-span of a Boeing 787 wing, and $10^{4}-10^{5}$ along the
length of a 787 fuselage during cruise (Leschziner, 2020; Marusic et al.,
2021). Consequently, understanding the effects of Reynolds number on drag
reducing techniques becomes the matter of the utmost importance.

A decrease of effectiveness with increasing Reynolds number was reported for
drag reduction (DR) techniques that employ transversely-oscillated walls
(Touber & Leschziner, 2012; Yao et al., 2019), streamwise-traveling waves
(Gatti & Quadrio, 2013; Hurst et al., 2014; Gatti & Quadrio, 2016; Marusic et
al., 2021), opposition control (Choi et al., 1994; Chang et al., 2002; Iwamoto
et al., 2002; Deng et al., 2016), superhydrophobic surfaces (Rastegari &
Akhavan, 2019), anisotropic permeable substrates (Gómez-de Segura & García-
Mayoral, 2019), and micro-bubble injections (Ferrante & Elghobashi, 2005) as
the methods of flow control. While the trend seems to persist across a variety
of established DR methods, the physical reasons behind this loss of
performance remain elusive.

Earlier studies have attributed the decrease of performance of DR techniques
with Reynolds number to geometrical effects of “shrinking” of the near-wall
layer affected by control (which is constant in wall units with $Re_{\tau}$)
with respect to the overall domain height (which increases in wall units with
$Re_{\tau}$) (Ferrante & Elghobashi, 2005; Gatti & Quadrio, 2016). Consistent
with this reasoning, the arguments have also been proposed explaining drag
reduction via an upward shift of the mean velocity profile in the logarithmic
region (Gad-el Hak, 2000). This shift, in wall units, was postulated to be
independent of the Reynolds number. Therefore, scaled with bulk mean velocity
in wall units of uncontrolled flow, this constant shift would lead to a lower
percentage of drag reduction as Reynolds number increases (Gatti & Quadrio,
2016; Rastegari & Akhavan, 2019; Gómez-de Segura & García-Mayoral, 2019).
While these arguments offer important insights regarding the observable
outcomes of drag reduction and their trends, further elucidation is needed to
explain why the applied mechanisms of flow control affect primarily the near-
wall layer of the flow and what modifications are required to increase their
effectiveness at higher Reynolds numbers.

In recent years, attention has turned to investigating the contributions of
different scales of motion both to skin friction (Deck et al., 2014; Agostini
& Leschziner, 2019; Duan et al., 2021), and to skin friction reduction
(Agostini et al., 2014; Cormier et al., 2016; Deng et al., 2016; Zhang et al.,
2020; Agostini & Leschziner, 2021; Chan et al., 2022). It has been shown that
large-scale motions (Guala et al., 2006; Balakumar & Adrian, 2007; Hutchins &
Marusic, 2007) grow stronger in high-Reynolds number flows (Marusic et al.,
2010; Smits et al., 2011) and, consequently, their contribution to skin
friction drag increases (Yao et al., 2019; Marusic et al., 2021). In their
recent investigation, Yao et al. (2019) hypothesized that a decrease of drag
reduction effectiveness with an increase in Reynolds number in a channel flow
controlled by a transverse wall oscillation with a non-dimensional oscillation
period of $T^{+}=100$ may be attributed to a weakened effectiveness of control
in suppressing the near-wall large-scale turbulence, whose contribution to
skin friction drag progressively increases. Marusic et al. (2021) have come to
a similar conclusion for a turbulent boundary layer controlled by a
streamwise-inhomogeneous transverse wall oscillation and investigated the
effectiveness of surface motions conducted at much larger oscillation periods
of $T^{+}\approx 600-1000$ geared towards a targeted manipulation of the
large-scale structures.

The current manuscript investigates the effect of Reynolds number on drag
reduction in a turbulent pipe flow using transverse wall oscillations. This
method of flow control has received considerable attention in the literature
(Leschziner, 2020; Ricco et al., 2021), aided by the following advantages: 1)
It is relatively easy to set up in experiments and simulations; 2) It is void
of additional complicated physics as occurs, for example, with the injection
of micro-bubbles (Kodama et al., 2000; Ferrante & Elghobashi, 2005), polymers
(Warholic et al., 1999; Kim et al., 2007), or with the utilization of
compliant (Gad-el Hak, 2002; Esteghamatian et al., 2022), superhydophobic (Lee
& Kim, 2011; Rastegari & Akhavan, 2019), or porous surfaces (Gómez-de Segura &
García-Mayoral, 2019; Du et al., 2022); 3) It provides relatively high values
of drag reduction (on the order of 30%–50% (Quadrio & Ricco, 2004; Hurst et
al., 2014)). Despite a large number of studies devoted to the effect of wall
oscillations, majority of these studies have been performed in a setting of a
plane wall geometry, such as in a channel or a canonical boundary layer
(Leschziner, 2020; Ricco et al., 2021). In fact, for pipe flows, the majority
of investigations were limited to $Re_{\tau}\lessapprox 170$ (Quadrio &
Sibilla, 2000; Choi et al., 2002; Duggleby et al., 2007b; Auteri et al., 2010;
Liu et al., 2022), with the exception of an experimental study by Choi &
Graham (1998), where two cases with $Re_{\tau}=652$ and $Re_{\tau}=962$ were
reported; however, they varied the non-dimensional amplitude of wall
oscillations with the Reynolds number, which makes it hard to separate the
effect of Reynolds number from that of an increased amplitude of wall
oscillations in this setting. Turbulent pipe flow, with an obvious application
to pipeline transport, is a canonical configuration, which has some
similarities, but also significant differences with channel and boundary layer
flows (Monty et al., 2009; Smits et al., 2011). It is therefore important to
assess whether previous conclusions drawn predominantly from flat wall
configurations (channel flows and boundary layers) on the effect of Reynolds
number on drag reduction with transversely oscillated walls, hold in pipes.
The current study aims to fill this gap. We are particularly interested in
characterizing the dominant modifications that transverse wall oscillations
inflict on different scales of motion in a pipe flow, including their energy
content, shear stress spectra, net turbulent force and a turbulent
contribution to the bulk mean velocity at different Reynolds numbers, with the
ultimate goal of explaining the reason for a reduction in DR effectiveness
with Reynolds number in turbulent pipe flows.

The paper is organized as follows. Section 2 presents the problem setup and
the details of the numerical methodology. Section 3 summarizes results,
including validation and a detailed spectral analysis of turbulent quantities
in a pipe flow with and without wall oscillation at three Reynolds numbers,
$Re_{\tau}=170,360$ and $720$. Section 4 presents conclusions.

##  2 Problem Setup

###  2.1 Geometry and domain configuration

In this study, a pipe flow with an azimuthally oscillated wall (WWO case) is
considered as a prototypical configuration to achieve drag reduction, and
compared to a standard pipe flow without a wall oscillation (NWO case), each
having the same mean pressure drop over the length, $L=24R$, where $R$, is the
radius of the pipe, Figure 1. The cylindrical coordinate system $(x,r,\theta)$
represents streamwise, radial and azimuthal directions, respectively, with the
unit vectors
$\left(\vec{\mathbf{e}}_{x},\,\vec{\mathbf{e}}_{r},\,\vec{\mathbf{e}}_{\theta}\right)$
in each direction, and the corresponding velocity vector is
$\mathbf{u}=(u_{x},u_{r},u_{\theta})$. The wall oscillation is achieved in the
WWO case by specifying an azimuthal pipe wall velocity as

$W_{wall}(t)=W_{0}\sin(\frac{2\pi}{T_{0}}t),$ (1)

where $W_{0}$ is the amplitude, and $T_{0}$ is the period of the wall
oscillation.

We define the following notations for the globally averaged quantities. Angle
brackets without the indices will represent the quantities averaged over
streamwise and azimuthal directions, and in time:

$\left\langle{f}\right\rangle(r)=\frac{1}{2\pi
LT}\int_{0}^{T}\int_{0}^{L}\int_{0}^{2\pi}f(x,r,\theta,t)\,dt\,dx\,d\theta,$
(2)

with $T$ being the averaging time. For averaging in time only, angle brackets
with the subscript $t$ will be used:

$\left\langle{f}\right\rangle_{t}(x,r,\theta)=\frac{1}{T}\int_{0}^{T}f(x,r,\theta,t)\,dt.$
(3)

The functional dependencies in parentheses for the averaged quantities will be
omitted for brevity when it is clear from the context.

We define the friction Reynolds number, $Re_{\tau}=u_{\tau}R/\nu$, where
$u_{\tau}=\sqrt{\left\langle{\tau_{w}}\right\rangle/\rho}$ is the friction
velocity, $\left\langle{\tau_{w}}\right\rangle$ is the mean wall shear stress,
$\rho$ is the density, and $\nu$ is the kinematic viscosity. The bulk Reynolds
number is $Re_{bulk}=2\,U_{bulk}R/\nu$, with

$U_{bulk}=\frac{2}{R^{2}}\int_{0}^{R}\left\langle{u_{x}}\right\rangle(r)\,r\,dr$
(4)

being the bulk mean velocity.

In the current setup, the mean wall shear stress,
$\left\langle{\tau_{w}}\right\rangle$, and hence $Re_{\tau}$, are fixed
between the NWO and WWO cases, while the volumetric flow rate, hence bulk mean
velocity, is allowed to vary. Consequently, the drag reduction which arises as
a result of wall oscillation (WWO case) is manifested by an increase in
$U_{bulk}$ and a concomitant increase in $Re_{bulk}$. As is well known
(Panton, 1984), the mean wall shear stress and the mean pressure gradient are
related by a simple force balance that leads to

$\left\langle{\frac{\partial\,p}{\partial
x}}\right\rangle=-2\frac{\left\langle{\tau_{w}}\right\rangle}{R}.$ (5)

Hence, the NWO and WWO cases also have identical mean pressure gradients.

###  2.2 Test cases and flow parameters

This study considers three Reynolds numbers, $Re_{\tau}=170,360,\text{ and
}720$, leading to six total cases considered (two cases - NWO and WWO - per
Reynolds number). Viscous wall units are defined by introducing the friction
velocity $u_{\tau}$, the viscous wall length scale, $l_{\tau}=\nu/u_{\tau}$
and viscous wall time scale $t_{\tau}=\nu/u_{\tau}^{2}$, and non-
dimensionalized variables are denoted by superscript ‘+’: $W^{+}=W/u_{\tau}$,
$L^{+}=L/l_{\tau}$, and $T^{+}=T/t_{\tau}$. Whence,

$W^{+}_{wall}(t)=W_{0}^{+}\sin\left(\frac{2\pi}{T_{0}^{+}}t\right).$ (6)

We set the values of $W_{0}^{+}=10$ and $T_{0}^{+}=100$ as the non-dimensional
amplitude and period of wall oscillations, fixed across all three Reynolds
numbers. These values are chosen from a set of parametric studies (Jung et
al., 1992; Baron & Quadrio, 1995; Choi & Graham, 1998; Quadrio & Ricco, 2004)
that demonstrated high values of drag reduction with these parameters within
the range of Reynolds numbers $Re_{\tau}=200-500$.

Setting viscosity and density in all the cases as $\nu=10^{-6}\,m^{2}/s$ and
$\rho=1000\,kg/m^{3}$ (considering water as a carrier fluid) and specifying a
nominal pipe radius as $R=0.1\,m$ allows us to calculate dimensional and non-
dimensional wall scaling parameters for the cases as listed in Table 1. Note
that with these definitions, the pipe radius in viscous wall units is
$R^{+}=R/l_{\tau}=Re_{\tau}$, and $L^{+}=24Re_{\tau}$. Since
$\left\langle{\tau_{w}}\right\rangle$, hence $u_{\tau}$ is kept constant
between the NWO and WWO cases, all the values listed in Table 1 are the same
between the NWO and WWO cases for the same $Re_{\tau}$.

(a)

(b)

Figure 1: Pipe flow configuration for (a) NWO case, and (b) WWO case. Cylindrical coordinate system $(x,r,\theta)$ represents streamwise, radial and azimuthal directions, respectively. $R$ is the pipe radius, and $L$ is the pipe length.  $Re_{\tau}$ | $u_{\tau}(m/s)$ | $-\left\langle{\frac{\partial p}{\partial x}}\right\rangle(Pa)$ | $l_{\tau}(m)$ | $t_{\tau}(s)$ | $R^{+}$ | $L^{+}$
---|---|---|---|---|---|---
170 | $0.0034$ | $0.02312$ | $2.9\times 10^{-4}$ | $0.0865$ | 170 | 4080
360 | $0.0072$ | $0.10368$ | $1.4\times 10^{-4}$ | $0.0193$ | 360 | 8640
720 | $0.0144$ | $0.41472$ | $6.9\times 10^{-5}$ | $0.0048$ | 720 | 17280
Table 1: List of wall scaling parameters for a pipe with the radius $R=0.1m$,
with water flowing through it of viscosity $\nu=10^{-6}\,m^{2}/s$ and density
$\rho=1000\,kg/m^{3}$.

A turbulent flow representation in drag-reduced flow with wall oscillation
(WWO) and non-drag reduced flow with no wall oscillation (NWO) configurations
across the three Reynolds numbers is obtained via Direct Numerical Simulations
(DNS). The governing equations are the incompressible Navier-Stokes’ equations

$\nabla\cdot\mathbf{u}=0,$ (7) $\frac{\partial\mathbf{u}}{\partial
t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\frac{1}{\rho}\nabla
p\,^{\prime}+\nu\nabla^{2}\mathbf{u}-\frac{1}{\rho}\,\left\langle{\frac{\partial
p}{\partial x}}\right\rangle\vec{\mathbf{e}}_{x},$ (8)

where $\mathbf{u}$ is the velocity, $\nu$ is the kinematic viscosity, $\rho$
is the density, $p\,^{\prime}$ is the fluctuating pressure defined as
$p^{\prime}=p-\left\langle{\partial p/\partial x}\right\rangle x$, where
$\left\langle{\partial p/\partial x}\right\rangle$ is the constant mean
pressure gradient supplied externally to balance the wall shear stress (See
equation (5)). Numerically, external pressure gradient is treated as a
spatially-invariant body force.

Equations (7), (8) are numerically solved using an open-source spectral
element solver Nek5000 (Deville et al., 2002; Fischer et al., 2015). The
spectral element method (SEM), similar to a finite element methodology,
decomposes a computational domain into a collection of elements, but it
utilizes high-order basis functions within each element, specifically, high-
order Lagrange interpolating polynomials associated with the Gauss-Legendre-
Lobatto quadrature points. The method employed in the current study leverages
the chosen polynomial approximation in a tensor-product form that allows for a
fast convergence in multiple dimensions. Thus, Nek5000 solves the governing
equations (7), (8) in a Cartesian coordinate system on hexahedral grids. For
the temporal integration, a third-order backward-differentiation formula is
employed for the viscous terms, and an explicit third-order extrapolation for
the convective terms. To achieve a divergence-free solution, incompressible
Navier-Stokes equations are solved with the operator splitting technique. The
resulting Helmholtz and Poisson equations are solved with the preconditioned
conjugate gradient (PCG) method, and the generalized mean residual (GMRES)
method, respectively (Fischer, 1997; Tufo & Fischer, 2001). For an
approximation of a cylindrical geometry, an unstructured hexahedral grid is
employed with an exact curved edge treatment of a cylindrical surface (Fischer
et al., 2015). The results are subsequently transferred onto a cylindrical
grid using high-order polynomial interpolation (Fischer et al., 2015; Merrill
et al., 2016) to perform spectral analysis of the data. Spectral-element
methods provide minimal dissipation and dispersion errors and are advantageous
for DNS of turbulent flows (Kreiss & Oliger, 1972; Wang et al., 2013).
Previous application of Nek5000 to DNS simulations of turbulent pipe flows can
be found in Duggleby et al. (2007a, b); El Khoury et al. (2013); Merrill et
al. (2016).

###  2.3 Numerical grids and boundary conditions

Numerical grid parameters employed in the current study for the three
different Reynolds numbers are listed in Table 2. All simulations were
executed using ninth-order polynomials as basis functions. Periodic boundary
conditions are applied in a streamwise direction on both the velocity and the
fluctuating pressure $p^{\prime}$. No-slip boundary conditions are set up at
the pipe wall, with $\mathbf{u}_{wall}(t)=(u_{x},u_{r},u_{\theta})=0$ in the
NWO case, and
$\mathbf{u}_{wall}(t)=(u_{x},u_{r},u_{\theta})=(0,0,W_{wall}(t))$ in the WWO
case.

To initialize the simulations, the lowest Reynolds number NWO case,
$Re_{\tau}=170$, was started by superimposing wave-like perturbations onto a
mean velocity profile $(U_{mean}(r),0,0)$ (Hillewaert et al., 2017) as

$\displaystyle u_{x}(x,r,\theta,0)=U_{mean}(r)+0.01(\beta)\sin{\left(\alpha
x\right)}\sin{\left(\beta\,\theta\right)},$ $\displaystyle
u_{r}(x,r,\theta,0)=0.01\sin{\left(\alpha
x\right)}\sin{\left(\beta\,\theta\right)},$ (9) $\displaystyle
u_{\theta}(x,r,\theta,0)=-0.01(\beta)\sin{\left(\alpha
x\right)}\sin{\left(\beta\,\theta\right)},$

with $\alpha=14\,\pi R/L$, $\beta=4$, and $U_{mean}(r)$ approximated using
$\left(1/7\right)^{th}$ power law for turbulent pipe flows (Nikuradse, 1966;
Schlichting & Gersten, 2016). The simulations were run until turbulence was
fully developed, which took approximately 15$T_{flow}$, with
$T_{flow}=L/U_{bulk}$ being a flow through time. The simulation was then run
for additional $10000t_{\tau}$ to collect statistics. Each subsequent higher
Reynolds number NWO case was initialized from a fully developed lower Reynolds
number NWO case, mapped onto a corresponding finer grid. The WWO cases for
each Reynolds number were initialized from fully-developed NWO cases
corresponding to the same Reynolds number. In each case, we allowed for the
simulations to reach a statistically-steady state (which was monitored through
a time series of bulk mean velocity) and subsequently collected statistics for
additional $10000t_{\tau}$.

$\text{Re}_{\tau}$ | $N_{el}$ | $N_{gp}$ |  $\Delta x^{+}$ min/max |  $\Delta r^{+}$ min/max |  $\Delta(r\theta)^{+}$ min/max | $\Delta t^{+}$
---|---|---|---|---|---|---
170 | 36848 | 27M | 3.5/14.05 | .1/1.6 | .67/2.75 | 0.0125
360 | 238336 | 173M | 2.9/11.90 | .15/2.5 | .81/3.3 | 0.0125
720 | 860160 | 627M | 3.3/13.6 | .22/4.18 | 1.3/5.4 | 0.0125
Table 2: Numerical grid parameters for the presented DNS studies. $N_{el}$
denotes the number of elements, and $N_{gp}$ – the total number of grid points
within each grid. NWO and WWO cases utilize identical grids for each
$Re_{\tau}$.

###  2.4 Post-processing and notation

####  2.4.1 Phase averaging

For turbulent flows with a periodically varying component, the turbulent
fluctuations ($u_{i}^{\prime\prime}$) are defined to be the deviations from
the long time mean ($\left\langle{u_{i}}\right\rangle$) plus the periodically
varying component of the mean ($u_{i}^{\phi}$) (Hussain & Reynolds, 1970) as:

$u_{i}=\left\langle{u_{i}}\right\rangle+u_{i}^{\phi}+u_{i}^{\prime\prime}.$
(10)

The component $u_{i}^{\phi}$ is equal to the phase mean minus the long time
average:

$u_{i}^{\phi}=\left\langle{u_{i}\mid\phi}\right\rangle-\left\langle{u_{i}}\right\rangle,\phi=\tau+nT\>\forall(n,\tau\in\left[0,T\right)),$
(11)

with $T$ being the period of oscillation, $n$ is an integer number, and the
notation $\left\langle{u_{i}\mid\phi}\right\rangle$ denoting a conditional
average given the phase.

When we report turbulent fluctuations, they shall be reported as
$u_{i}^{\prime\prime}$ for both the WWO and NWO cases. Note, that for the NWO
case, the turbulent fluctuation $u_{i}^{\prime\prime}$ thus defined is equal
to a standard turbulent fluctuation as obtained from Reynolds decomposition,
$u_{i}^{\prime}=u_{i}-\left\langle{u_{i}}\right\rangle$, but for the WWO case,
these quantities are different.

####  2.4.2 Fourier decomposition

To capture length scales of motion, we use Fourier decomposition to decompose
the flow field as:

$u_{i}(x,r,\theta,t)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\hat{u}_{i}(k_{x},r,k_{\theta},t)\exp\left(i\mathbf{k}\cdot\mathbf{x}\right)dk_{x}dk_{\theta},$
(12)

where $\hat{u}_{i}(k_{x},r,k_{\theta},t)$ is the Fourier coefficient,
$\mathbf{k}=k_{x}\vec{\mathbf{e}}_{x}+k_{\theta}\vec{\mathbf{e}}_{\theta}$ is
the vector-valued wavenumber, with ($k_{x}$,$k_{\theta}$) denoting its
streamwise and azimuthal components, and
$\mathbf{x}=x\vec{\mathbf{e}}_{x}+\theta\vec{\mathbf{e}}_{\theta}$ is a
shorthand notation for the projection of the position vector onto the axial-
azimuthal plane. Throughout the paper, we will use the bold symbols to denote
vectors, and the bold symbols with arrows to denote unit vectors.

The Fourier coefficient $\hat{u}_{i}(k_{x},r,k_{\theta},t)$ is given by:

$\hat{u}_{i}(k_{x},r,k_{\theta},t)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u_{i}(x,r,\theta,t)\exp{\left(-i(\mathbf{k}\cdot\mathbf{x})\right)}dxd\theta.$
(13)

We define a two dimensional (co-)spectrum of velocity as a time-averaged
product of the spectrum of the turbulent fluctuations of velocity:

$\Phi_{u_{i}u_{j}}(k_{x},r,k_{\theta})=\left\langle{\overline{\widehat{u^{\prime\prime}}}_{i}(k_{x},r,k_{\theta},t){\widehat{u^{\prime\prime}}}_{j}(k_{x},r,k_{\theta},t)}\right\rangle_{t},$
(14)

where $\overline{(\cdot)}$ denotes a complex conjugate. The one-dimensional
(co-)spectrum is a subset of the two dimensional (co-)spectrum taken by
integrating over the wavenumbers along the opposite direction:

$\displaystyle\Phi_{u_{i}u_{j}}(k_{x},r)=\int_{k_{\theta}}\Phi_{u_{i}u_{j}}(k_{x},r,k_{\theta})dk_{\theta},$
(15)
$\displaystyle\Phi_{u_{i}u_{j}}(r,k_{\theta})=\int_{k_{x}}\Phi_{u_{i}u_{j}}(k_{x},r,k_{\theta})dk_{x}.$
(16)

Furthermore, the Parseval’s theorem can be used to express the second-order
moments of turbulent statistics via the integration of the co-spectra in the
wavenumber space:

$\left\langle{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}\right\rangle=\int_{k_{x}}\int_{k_{\theta}}\Phi_{u_{i}u_{j}}(k_{x},r,k_{\theta})\,dk_{x}dk_{\theta}.$
(17)

To compute the corresponding Fourier transforms, we leverage periodicity of
the flow in streamwize and azimuthal directions, which allows us to replace
Fourier integral in Equation (12) with Fourier series as

$\displaystyle
u_{i}(x,r,\theta,t)=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}\hat{u}_{i}(k_{x_{n}},r,k_{\theta_{m}},t)\exp\left(i\mathbf{k}\cdot\mathbf{x}\right),$
(18) $\displaystyle\hat{u}_{i}(k_{x_{n}},r,k_{\theta_{m}},t)=\frac{1}{2\pi
L}\int_{0}^{L}\int_{0}^{2\pi}u_{i}(x,r,\theta,t)\exp{\left(-i(\mathbf{k}_{mn}\cdot\mathbf{x})\right)}dxd\theta,$
(19)

with

$k_{x_{n}}=\frac{2\pi n}{L},\>\>\>k_{\theta_{m}}=m,$ (20)

and $L$ being the domain length. Equations (15)–(17) are replaced
correspondingly with their Fourier series counterparts.

In a numerical computation with a finite number of samples, the infinite
series in (18) is approximated by a finite sum over wavenumbers as

$\displaystyle
u_{i}(x,r,\theta,t)=\sum_{n=-N_{x}/2}^{N_{x}/2}\sum_{m=-N_{\theta}/2}^{N_{\theta}/2}\hat{u}_{i}(k_{x_{n}},r,k_{\theta_{m}},t)\exp\left(i\mathbf{k}_{mn}\cdot\mathbf{x}\right),$
(21) $\displaystyle\hat{u}_{i}(k_{x_{n}},r,k_{\theta_{m}},t)=\frac{1}{2\pi
L}\int_{0}^{L}\int_{0}^{2\pi}u_{i}(x,r,\theta,t)\exp{\left(-i(\mathbf{k}_{mn}\cdot\mathbf{x})\right)}dxd\theta.$
(22)

Approximations for the co-spectra and the second-order moments are updated
accordingly.

$Re_{\tau}$ | $N_{x}$ | $N_{\theta}$ | $\lambda_{x\,\text{min}}^{+}$ | $\lambda_{s\,\text{min,wall}}^{+}$
---|---|---|---|---
170 | 384 | 80 | 21.2 | 26.8
360 | 768 | 160 | 22.6 | 28.4
720 | 2048 | 320 | 15.6 | 24.4
Table 3: Computational parameters employed for a Fourier analysis of the DNS
data. $N_{x},N_{\theta}$ – number of terms carried in the Fourier series in
streamwise and azimuthal directions, respectively;
$\lambda_{x\,\text{min}}^{+}$ \- minimum streamwise wavelength;
$\lambda_{s\,\text{min,wall}}^{+}$ – minimum azimuthal wavelength at the wall.

Fourier coefficients, spectra and co-spectra defined above can be equivalently
represented in terms of the wavelengths,
$\lambda_{x}=2\pi/k_{x},\lambda_{\theta}=2\pi/k_{\theta}$ (subscripts $n$, $m$
in a discrete representation of wavenumbers and wavelenghts will be omitted
for brevity throughout the remainder of the manuscript). Note that from the
definition of wavenumbers in Equation (20), it is seen that the wavelength
$\lambda_{x}$ has a dimension of length, but $\lambda_{\theta}$ is
adimensional. Therefore, we can define a dimensional azimuthal wavelength
$\lambda_{s}(r)=r\lambda_{\theta}$, where $r$ is the local radial location of
the variable to be considered. The number of terms, $N_{x}$ and $N_{\theta}$,
carried in a Fourier analysis of the DNS data for each Reynolds number
(equation (21)) is specified in Table 3, together with the smallest
wavelengths that are computed as a result of the Fourier analysis. It can be
seen that the smallest computed wavelengths are above the DNS grid resolution,
to avoid any spurious oscillations potentially caused by interpolation from
the DNS grid and under-resolution.

####  2.4.3 Correlation functions

Correlation functions in space are calculated using Wiener-Khinchin theorem
for a periodically extended signal and are presented in the following form:

$R_{u_{i}u_{j}}(\Delta x,\Delta\theta,r_{0},r)=\frac{1}{2\pi
LT}\int_{0}^{T}\int_{0}^{L}\int_{0}^{2\pi}u_{i}(x,r_{0},\theta,t)u_{j}(x+\Delta
x,r,\theta+\Delta\theta,t)\,dt\,dx\,d\theta.$ (23)

The correlation coefficient is defined as:

$\rho_{u_{i}u_{j}}(\Delta x,\Delta\theta,r_{0},r)=\frac{R_{u_{i}u_{j}}(\Delta
x,\Delta\theta,r_{0},r)-\left\langle{u_{i}}\right\rangle(r_{0})\left\langle{u_{j}}\right\rangle(r_{0})}{\sqrt{\left\langle{{u_{i}^{\prime\prime}}^{2}}\right\rangle(r_{0})\left\langle{{u_{j}^{\prime\prime}}^{2}}\right\rangle(r_{0})}}.$
(24)

For compactness, the correlation coefficient at zero separation along a
homogeneous direction shall be written as:

$\displaystyle\rho_{u_{i}u_{j}}(\Delta
x=0,\Delta\theta,r_{0},r)=\rho_{u_{i}u_{j}}(\Delta\theta,r_{0},r),$ (25)
$\displaystyle\rho_{u_{i}u_{j}}(\Delta
x,\Delta\theta=0,r_{0},r)=\rho_{u_{i}u_{j}}(\Delta x,r_{0},r).$ (26)

##  3 Results

###  3.1 Validation

Although validation of Nek5000 in application to DNS of turbulent pipe flows
was already documented elsewhere (El Khoury et al., 2013; Merrill et al.,
2016), Appendices A.1 and A.2 present its validation using the present
computational setup for the NWO and WWO cases, respectively, versus available
published data.

###  3.2 Effect of Reynolds number on drag reduction

Table 4 presents the global quantities for the computed drag-reduced (WWO)
cases, including the achieved bulk Reynolds number, $Re_{bulk}$, percent
increase in bulk mean velocity, $U_{bulk}$, and percent reduction in skin
friction coefficient, $C_{f}$, with respect to the corresponding standard
(NWO) cases. From Table 4, it is evident that the mechanism of drag reduction
with transverse wall oscillations becomes less effective as Reynolds number
increases.

$Re_{\tau}$ |  $(U_{bulk}^{+}$,$\bar{U}_{c})_{NWO}$ |  $(U_{bulk}^{+}$,$\bar{U}_{c})_{WWO}$ | % $\Delta U_{bulk}$ | % $\Delta C_{f}$
---|---|---|---|---
170 | (14.4,1.32) | (17.1,1.31) | 18.54 | 28.8
360 | (16.2,1.27) | (18.8,1.25) | 16.25 | 26.0
720 | (18.0,1.26) | (20.5,1.23) | 13.9 | 22.9
Table 4: Bulk quantities and their percent change for the WWO cases as
compared to the NWO cases. $\bar{U}_{c}=U_{c}/U_{bulk}$ denotes the centerline
velocity scaled with $U_{bulk}$. The percent increase in bulk mean velocity is
defined as: $\left(U_{bulk_{WWO}}-U_{bulk_{NWO}}\right)/U_{bulk_{NWO}}\cdot
100\%$. The skin friction is calculated as:
$C_{f}=2\left\langle{\tau_{w}}\right\rangle/\left(\rho\,U_{bulk}^{2}\right)$,
and the percent reduction in $C_{f}$ as:
$-\left(C_{f_{WWO}}-C_{f_{NWO}}\right)/C_{f_{NWO}}\cdot 100\%$.

It is generally accepted that drag reduction with azimuthally oscillated walls
occurs due to an interaction of turbulence with the so-called Stokes’ layer,
which refers to a layer of non-zero phase mean azimuthal velocity developed
over an oscillating wall. Figure 2 demonstrates the impact of the transverse
wall oscillations on the phase mean turbulent azimuthal velocity profile for
the highest simulated $Re_{\tau}=720$, in comparison with the corresponding
laminar solution for a pipe with its wall oscillating about its axis in
cylindrical coordinates (Song & Rau, 2020; Coxe et al., 2022). The blue
horizontal line indicates the Stokes’ layer thickness, defined as the location
where the azimuthal velocity of the laminar solution drops below 1% of the
maximum wall velocity. For the current frequency of wall oscillation, it can
be shown that the Stokes’ layer thickness over a cylindrical pipe wall is
equal to the one obtained in a classical Stokes’ second problem solution,
$\delta_{Sl}=4.6\sqrt{T_{0}\,\nu/\pi}$ (Panton, 1984; Coxe et al., 2022). In
wall units, this quantity equals to
$\delta_{Sl}^{+}=4.6\sqrt{T_{0}^{+}/\pi}\approx 25$ for the current wall
oscillation parameters, independent of the Reynolds number.

In describing the remainder of results, we will frequently be referring to
different regions commonly identified within the turbulent boundary layer
(Schlichting & Gersten, 2016), in addition to the Stokes’ layer, as summarized
in Table 5, where we use the notation $y^{+}=y/l_{\tau}$, with $y=R-r$. We
will be using the asterisk to denote the quantities scaled with the outer
dimension, such that $r^{\star}=r/R$ and $y^{\star}=1-r/R$.

Region | Range
---|---
Viscous layer | $y^{+}\leq 5$
Buffer layer | $5<y^{+}\leq 30$
Stokes’ layer | $y^{+}\leq 25$
Log layer | $30<y^{+}\leq 0.2Re_{\tau}$
Wake region | $y^{+}>0.2Re_{\tau}$
Table 5: Regions within the turbulent boundary layer, including the Stokes’
layer, referred to throughout this work. The inner layer is the composite of
the viscous, buffer and log layers while the outer layer is the composite of
the log layer and the wake region of the flow (Panton, 1984). Figure 2:
Comparison between phase mean azimuthal velocity profile (black solid lines)
for $Re_{\tau}=720$ and a laminar solution (red dashed lines) for a pipe with
its wall oscillating about its axis in cylindrical coordinates (Song & Rau,
2020; Coxe et al., 2022) at phases $0$, $\pi/4$, $\pi/2$, and $\pi$. Blue
horizontal dashed line indicates the Stokes’ layer thickness,
$\delta_{Sl}^{+}\approx 25$.

###  3.3 Effect of Reynolds number on single point statistics

A well-known consequence of drag reduction is an upward shift of the mean
velocity profile in the log region of the flow (Hurst et al., 2014; Gatti &
Quadrio, 2016). Figure 3 documents this shift for the three Reynolds numbers
considered in this study. The upward shift is approximately
$\Delta\left\langle{u_{x}^{+}}\right\rangle=3.4,2.5$ and 2.3 for
$Re_{\tau}=170,360$ and 720, respectively. Consistent with the observations in
Hurst et al. (2014), the shift decreases with $Re_{\tau}$, but the amount of
decrease is diminishing. Gatti & Quadrio (2016) argue that the shift becomes
independent from the Reynolds number once it reaches a high enough value, and
the observed trend supports this argument. Figure 3(d) documents the change in
the mean velocity as a result of transverse wall oscillation for the three
Reynolds numbers. The change is defined as $\Delta f=f(WWO)-f(NWO)$, i.e.
quantity evaluated with transverse wall oscillations minus quantity evaluated
with no wall oscillation. This convention will be maintained throughout the
remainder of the work. For all three Reynolds numbers the maximum change in
mean streamwise velocity occurs around $y^{+}\approx 100$. This location
happens to be in the top half of the log layer for the highest Reynolds
number, above the log layer for $Re_{\tau}=360$ and approaching the centerline
of the pipe for $Re_{\tau}=170$. Consistent with the reduction in the value of
log-layer shift, as the Reynolds number increases, the peak change in the mean
streamwise velocity reduces. Figure 4 subsequently documents the change in the
second-order statistics as a result of transverse wall oscillation. Wall
oscillations suppress the streamwise turbulent kinetic energy within the
buffer layer for all three Reynolds numbers. Above the buffer layer,
streamwise turbulent kinetic energy is slightly increased. Radial turbulent
kinetic energy changes are two orders of magnitude smaller than the changes in
streamwise turbulent kinetic energy. Its trends in the buffer and the log-
layer are the reverse of those of the streamwise turbulence kinetic energy.
The change in the Reynolds shear stress is one order of magnitude smaller than
the change in streamwise kinetic energy. The result of wall oscillations is to
suppress the Reynolds shear stress through the top of the log layer for all
three Reynolds numbers.

(a) $Re_{\tau}=170$

(b) $Re_{\tau}=360$

(c) $Re_{\tau}=720$

(d) Change in $\left\langle{u_{x}^{+}}\right\rangle$

Figure 3: Mean streamwise velocity profiles at (a) $Re_{\tau}=170$, (b)
$Re_{\tau}=360$, and (c) $Re_{\tau}=720$. Black solid line, NWO; blue dashed
line, WWO. (d) Change in mean velocity between NWO and WWO. The red dotted
line indicates the top of the Stokes’ layer, the purple dotted line is the top
of the buffer layer, and the location of the enlarged blue markers indicates
the top of the log layer with respect to each Reynolds number.

(a) Streamwise velocity

(b) Radial velocity

(c) Shear stresses

Figure 4: Comparison of the change in the second-order statistics as
influenced by the Reynolds number. The red dotted line indicates the top of
the Stokes’ layer, the purple dotted line is the top of the buffer layer, and
the location of the enlarged blue markers indicates the top of the log layer
with respect to each Reynolds number. The location of changes to the
statistical profiles shows a reasonable collapse with viscous units indicating
that the wall oscillations impact turbulence scales within the log layer of
the flow. While the Stokes’ layer is confined to a region bounded by the
buffer layer of the flow, effects on statistics are also observed above the
log layer.

###  3.4 Effect of Reynolds number on energy spectra

Figure 5 documents the streamwise spectra of streamwise kinetic energy in the
NWO and WWO cases, as well as its change, as a result of wall oscillations.
Wall oscillations enhance the energy in large streamwise wavelengths in and
above the log layer of the flow. This is consistent with drag reduction
mechanisms which suppress near-wall turbulence (Kim et al., 2008; Ricco et
al., 2021). It is worth noting that at the low Reynolds number
($Re_{\tau}=170$) and moderately low Reynolds number ($Re_{\tau}=360$), these
large outer-layer structures are restricted by the vertical height of the
domain. However, across all Reynolds numbers wall oscillation suppresses the
energy of the streamwise structures in the buffer layer having wavelengths
less than $\lambda_{x}^{+}\approx 10,000$. These large-scale structures in the
buffer layer are thought to correspond to the concatenation of hairpin packets
(Adrian, 2007; Lee et al., 2019), whose legs produce a signature of the well-
known streamwise streaks in the boundary layer (Jiménez, 2022). The trend of
large scales of motion being suppressed versus enhanced exhibits a clearly
defined boundary located between the buffer and the log layer, at
$y^{+}\approx 30$. Of interest is a subtle amplification of very short
streamwise scales in the buffer layer ($\lambda_{x}^{+}<500$) by wall
oscillations.

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=170$, WWO

(c) $Re_{\tau}=170$, WWO-NWO

(d) $Re_{\tau}=360$, NWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=360$, WWO-NWO

(g) $Re_{\tau}=720$, NWO

(h) $Re_{\tau}=720$, WWO

(i) $Re_{\tau}=720$, WWO-NWO

Figure 5: Streamwise kinetic energy as a function of wall normal location and
streamwise wavelength,
$\Phi_{u_{x}u_{x}}(\lambda_{x}^{+},y^{+})/u_{\tau}^{2}$: (a,d,g) NWO spectra;
(b,e,h) WWO spectra; (c,f,i) change in spectra,
$\Delta\Phi_{u_{x}u_{x}}(\lambda_{x}^{+},y^{+})/u_{\tau}^{2}$. From top to
bottom: (a,b,c) $Re_{\tau}=170$, (d,e,f) $Re_{\tau}=360$, and (g,h,i)
$Re_{\tau}=720$.

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=170$, WWO

(c) $Re_{\tau}=170$, WWO-NWO

(d) $Re_{\tau}=360$, NWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=360$, WWO-NWO

(g) $Re_{\tau}=720$, NWO

(h) $Re_{\tau}=720$, WWO

(i) $Re_{\tau}=720$, WWO-NWO

Figure 6:  Streamwise kinetic energy as a function of wall normal location and
arclength, $\Phi_{u_{x}u_{x}}(\lambda_{s}^{+},y^{+})/u_{\tau}^{2}$: (a,d,g)
NWO spectra; (b,e,h) WWO spectra; (c,f,i) change in spectra,
$\Delta\Phi_{u_{x}u_{x}}(\lambda_{s}^{+},y^{+})/u_{\tau}^{2}$. From top to
bottom: (a,b,c) $Re_{\tau}=170$; (d,e,f) $Re_{\tau}=360$; (g,h,i)
$Re_{\tau}=720$.

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=360$, NWO

(c) $Re_{\tau}=720$, NWO

(d) $Re_{\tau}=170$, WWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=720$, WWO

Figure 7: Azimuthal correlation coefficient of streamwise velocity at a fixed
radial location, $\rho_{u_{x}u_{x}}(\Delta\,\theta,{r^{\star}},{r^{\star}})$,
for (a,d) $Re_{\tau}=170$; (b,e) $Re_{\tau}=360$; (c,f) $Re_{\tau}=720$. Top,
NWO cases; bottom, WWO cases. Black contour lines indicate a level of zero
correlation.

Wall oscillations impact the azimuthal energy spectrum of streamwise velocity
similarly across the Reynolds numbers when scaled with viscous units, Figure
6. The energy in the azimuthal wavelengths associated with the near-wall
streak spacing, $\lambda_{s}^{+}\approx 100-200$, is reduced in the buffer
layer between ($y^{+}\approx 5-30$). Since streaks and quasi-streamwise
vortices are closely related, wall oscillation presumably weakens the quasi-
streamwise near-wall vortices, thereby reducing their transport of streamwise
momentum into the streaks.

At the two highest Reynolds numbers wall oscillations enhance the spectral
energy density at azimuthal wavelengths greater than
$\lambda_{s}^{*}\approx\pi/4$. The maxima in Figures 6(f), 6(i) suggest
periodicity of about $\pi/2$. To visualize the structure more explicitly,
Figure 7 plots the azimuthal correlation coefficient of streamwise velocity
$\rho_{u_{x}u_{x}}(\Delta\theta,{r^{\star}},{r^{\star}})$ between a fixed
point at $({r^{\star}},\Delta\theta=0)$ and all other points on the circle
${r^{\star}}=\text{constant}$, while varying ${r^{\star}}$ from $0$ to $R$
(note the difference between this set of one-point correlations and the full
two-point correlation
$\rho_{u_{x}u_{x}}(\Delta\theta,{r^{\star}},r_{0}^{\star})$ between velocities
at a fixed point $(r_{0}^{\star},\Delta\theta=0)$ and all other points in the
pipe cross-section $({r^{\star}},\Delta\theta)$, as presented, e.g., in
Baltzer et al. (2013). Strengthening of the correlations for the two highest
Reynolds numbers is a maximum at the separation angles of
$\Delta\,\theta\approx 4\pi/5$ at $Re_{\tau}=360$ and $\Delta\,\theta\approx
2\pi/3$ at $Re_{\tau}=720$, consistent with the energy spectra enhancement in
Figures 6(f), 6(i) at the wavelengths corresponding to these separation
angles. As expected, this mode is not observed in the NWO cases. The origin of
this large-scale mode in a wall-oscillated pipe is unknown and requires
further investigation.

Figure 8 further elucidates on the structure of this enhanced large-scale mode
in the WWO pipe by plotting the two-dimensional streamwise-azimuthal spectrum
of the streamwise kinetic energy,
$\Phi_{u_{x}u_{x}}(\lambda_{x}^{+},y^{+},s^{+})$, as well as its change, at
two wall-normal locations of $y^{+}=20$ and $y^{+}=200$ for the highest
Reynolds numbers, $Re_{\tau}=720$. The absence of the large-scale mode in the
buffer layer ($y^{+}=20$) in the NWO case and its presence in the WWO case is
clearly seen. The structures energized as a result of the developed large-
scale mode are very long in the streamwise direction ($\lambda_{x}^{+}\geq
10,000$) and their spanwise scale grows proportionally to the streamwise
scale. While, consistent with the previous observations (Guala et al., 2006;
Lee et al., 2019), large-scale motions do preside in the NWO pipe in the outer
layer ($y^{+}=200$), they are significantly energized by wall oscillations.

(a) $y^{+}=20$, NWO

(b) $y^{+}=20$, WWO

(c) $y^{+}=20$, WWO-NWO

(d) $y^{+}=200$, NWO

(e) $y^{+}=200$, WWO

(f) $y^{+}=200$, WWO-NWO

Figure 8:  Two-dimensional spectra of streamwise kinetic energy,
$\Phi_{u_{x}u_{x}}(\lambda_{x}^{+},y^{+},\lambda_{s}^{+})/u_{\tau}^{2}$, at a
wall normal location of (a,b,c) $y^{+}=20$; (d,e,f) $y^{+}=200$ for
$Re_{\tau}=720$. (a,d) NWO spectra; (b,e) WWO spectra; (c,f) change in
spectra,
$\Delta\Phi_{u_{x}u_{x}}(\lambda_{x}^{+},y^{+},\lambda_{s}^{+})/u_{\tau}^{2}$.

###  3.5 Effect of Reynolds number on the net turbulent force

To introduce the net turbulent force, consider the streamwise momentum
equation,

$\frac{\partial u_{x}}{\partial t}+\frac{1}{r}\frac{\partial
ru_{x}u_{r}}{\partial r}+\frac{1}{r^{2}}\frac{\partial
u_{x}u_{\theta}}{\partial\theta}+\frac{\partial u_{x}u_{x}}{\partial
x}=-\frac{1}{\rho}\frac{\partial\,p^{\prime}}{\partial
x}+\nu\,\nabla^{2}u_{x}-\frac{1}{\rho}\left\langle{\frac{\partial\,p}{\partial\,x}}\right\rangle,$
(27)

where $\nabla^{2}=\left\\{\frac{1}{r}\frac{\partial}{\partial
r}\left(r\frac{\partial}{\partial
r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\partial^{2}}{\partial
x^{2}}\right\\}$. Averaging equation (27) in the stationary time and along the
homogeneous (streamwise and azimuthal) directions yields:

$0=-\frac{1}{r}\frac{d\\{r\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle\\}}{dr}+\frac{\nu}{r}\frac{d}{dr}\left(r\frac{d}{dr}\right)\left\langle{u_{x}}\right\rangle-\frac{1}{\rho}\left\langle{\frac{\partial\,p}{\partial\,x}}\right\rangle.$
(28)

The net turbulent force (per unit mass) reduces to the first term on the
right-hand side of equation (28):

$F_{turb}(r)=-\frac{1}{r}\frac{d\\{r\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle\\}}{dr}.$
(29)

While equation (29) defines the net turbulent force per unit mass, we will be
referring to it as “the net turbulent force” for brevity. The net turbulent
force is made non-dimensional by scaling the radial coordinate with
$r=r^{+}l_{\tau}$ and velocities with $u_{i}=u_{i}^{+}u_{\tau}$, such that:

$F_{turb}^{+}(r^{+})=-\frac{1}{r^{+}}\frac{d\\{r^{+}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle\\}}{dr^{+}},$
(30)

where $F_{turb}^{+}=F_{turb}\,l_{\tau}/u_{\tau}^{2}$.

The net turbulent force determines the local acceleration or deceleration of
the mean flow due to turbulent shear stresses. When the flow is stationary, it
balances the contributions from the mean pressure gradient and the viscous
stress. The net turbulent force accelerates the mean flow below the region of
maximum Reynolds shear stress (premultiplied Reynolds shear stress,
$r\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle$,
for a pipe) and decelerates the flow above it. At the location of the maximum
(premultiplied) Reynolds shear stress, the net turbulent force equals zero.
Utilizing Parseval’s theorem (See equation (17)), the net turbulent force can
be decomposed into a sum of contributions from the streamwise and azimuthal
Fourier modes as:

$F_{turb}^{+}(r^{+})=-\sum_{k_{x}}\sum_{k_{\theta}}\frac{1}{r^{+}}\frac{\partial\\{r^{+}\Phi_{u_{x}u_{r}}(k_{x},r,k_{\theta})\\}}{\partial
r^{+}}.$ (31)

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=360$, NWO

(c) $Re_{\tau}=720$, NWO

(d) $Re_{\tau}=170$, WWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=720$, WWO

Figure 9: Streamwise spectra of the net turbulent force as a function of the
wall normal coordinate and streamwise wavelength,
$F_{turb}^{+}(\lambda_{x}^{+},y^{+})$, for (a,d) $Re_{\tau}=170$; (b,e)
$Re_{\tau}=360$; (c,f) $Re_{\tau}=720$. Top, NWO cases; bottom, WWO cases.

Figure 9 shows the streamwise spectra of the net turbulent force,
$F_{turb}^{+}(\lambda_{x}^{+},y^{+})$, as a function of the wall normal
coordinate and the streamwise wavelength for the NWO and WWO cases. We remark
that Figure 9 agrees well with the data presented in Wu et al. (2012) (Figure
17) for the NWO pipe flow if replotted on a log-log scale (not shown here for
the sake of brevity). Consistent with previous observations (Guala et al.,
2006; Balakumar & Adrian, 2007; Wu et al., 2012), we find that the net force
is positive across all the scales of motion $\lambda_{x}^{+}\geq 100$ in the
buffer layer ($y^{+}<20$), amounting to an acceleration of the mean flow, and
negative above it, implying retardation. While all scales of motion experience
the aforementioned acceleration and deceleration, the effect is larger for
large scales at all Reynolds numbers, consistent with the works of Guala et
al. (2006); Wu et al. (2012). The impact of the wall oscillations is to reduce
this effect, diminishing both the acceleration of turbulent structures near
the wall and their deceleration in the outer layer. In general these changes
are conducive to drag reduction, since they bring the mean velocity profile
closer to its laminar shape.

To quantify the modification of the net turbulent force by wall oscillations
and to assess the contributions of different scales of motion, we apply a
Gaussian low-pass filter (Guala et al., 2006; Lee et al., 2019),

$\hat{g}_{lpf}(k_{x})=\exp\left(-\frac{k_{x}^{2}}{2\sigma^{2}}\right),$ (32)

with $\sigma$ being the filter width, and $k_{x}$ the streamwise wavenumber.
We set the filter width such that the strength of the filter is at 50% of its
peak at a filter cutoff location, $k_{x,cutoff}=2\pi/\lambda_{x,cutoff}$. This
gives the value of $\sigma^{2}=k_{x,cutoff}^{2}/(2\ln 2)$. The cutoff
wavelength $\lambda_{x,cutoff}^{+}=1000$ is chosen such that the scales
smaller than this value are attenuated by the low-pass filter. Conversely, its
high-pass filter counterpart, $\hat{g}_{hpf}(k_{x})=1-\hat{g}_{lpf}(k_{x})$,
attenuates the scales with $\lambda_{x}^{+}\geq\lambda_{x,cutoff}^{+}$. We
apply both filters to the net turbulent force spectra. The filtered net
turbulent force is defined as

$\tilde{F}_{turb,\\{lpf,hpf\\}}^{+}(r^{+})=-\sum_{k_{x}}\sum_{k_{\theta}}\frac{1}{r^{+}}\frac{\partial\\{r^{+}\Phi_{u_{x}u_{r}}(k_{x},r,k_{\theta})\\}}{\partial
r^{+}}\hat{g}_{\\{lpf,hpf\\}}(k_{x}).$ (33)

The cumulative low-pass and high-pass filtered contributions are documented in
Figure 10. Additionally, Table 6 records the position of the zero net
turbulent force for the total (unfiltered) and filtered quantities for the NWO
and WWO cases, together with the difference between the NWO and WWO locations
($\Delta$). We refer to the location of the zero net turbulent force based on
the total (unfiltered) quantities as $y_{f0}^{+}$.

(a) Low-pass filtered

(b) High-pass filtered

Figure 10: Filtered net turbulent force profile as a function of the wall
normal coordinate: (a) low-pass filtered net turbulent force, and (b) high-
pass filtered net turbulent force. Black solid lines indicate NWO and blue
dashed lines indicate WWO. The marker closest to the wall (smallest) indicates
the location of the maximum accelerating turbulent force, the next marker
indicates the location of the zero turbulent force and the last (largest)
marker indicates the top of the log layer for the given Reynolds number.

Figure 10 shows that the wall oscillations significantly attenuate the
magnitude of the low-pass filtered net turbulent force while leaving its high-
pass filtered counterpart relatively unchanged, i.e. the major effect of the
net turbulent force modification is coming from relatively large scales of
motion ($\lambda_{x}^{+}>1000$). A considerable net force reduction in large
scales (low-pass filtered contribution) is found all the way from the wall and
throughout the top of the log layer of the flow. Another important effect is
the shift of both the maximum and the zero net force locations upwards by wall
oscillations, which is primarily seen in its low-pass filtered contribution.
As can be judged from the Table 6, the major effect on the shift indeed comes
from the large scales of motion, with the small and intermediate scales (high-
pass filtered) contributing less than 20% of the total shift. Overall, the
small and intermediate scales, shown in Figure 10(b), promote a slightly
higher acceleration of the mean velocity profile in the WWO case near the
wall, with the decreasing effect as the distance from the wall increases.
Between the top of the buffer layer and the top of the log layer, the small
and intermediate scales exhibit a stronger deceleration of the mean velocity
profile in the WWO case. Overall, the results indicate that the acceleration
due to large and very large scales is the most impacted by the wall
oscillations. To the contrary, large-scale net turbulent force is enhanced
above the log layer of the flow for the two highest Reynolds numbers. This is
consistent with a reduction of the normalized centerline velocity
$\bar{U}_{c}=U_{c}/U_{bulk}$, despite the growth of the bulk mean velocity
$U_{bulk}$ in the WWO as compared to the NWO cases observed in Table 4:
increased (negative) turbulent force in this region acts to decelerate the
large-scale structures more significantly in the center of the pipe flow with
wall oscillation. This effect indicates a decreased effectiveness of the drag
reduction mechanism in the outer layer. The centerline retardation increases
with the Reynolds number, pointing once again towards a reduced effectiveness
of the current drag reduction mechanism at higher Reynolds numbers.

$Re_{\tau}$ | (NWO, WWO, $\Delta$) | (NWO, WWO, $\Delta$) | (NWO, WWO, $\Delta$)
---|---|---|---
| unfiltered | low-pass filtered | high-pass filtered
170 | (25.8, 31.1,5.3) | (24.9, 38.0, 13.1) | (45.3, 29.2, -16.1)
360 | (31.7, 37.1,5.4) | (30.1, 55.9, 25.8) | (47.4, 29.9, -17.5)
720 | (39.1, 44.8,5.7) | (37.7, 110.1, 72.4) | (46.7, 30.2, -16.5)
Table 6: Wall normal location of the zero net turbulent force in plus units
for (NWO,WWO) based on unfiltered, low-pass filtered, and high-pass filtered
quantities, along with the difference between the WWO and NWO locations
($\Delta$).

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=360$, NWO

(c) $Re_{\tau}=720$, NWO

(d) $Re_{\tau}=170$, WWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=720$, WWO

Figure 11: Azimuthal spectra of the net turbulent force as a function of the
wall normal coordinate and azimuthal wavelength,
$F_{turb}^{+},(\lambda_{s}^{+},y^{+})$, for (a,d) $Re_{\tau}=170$; (b,e)
$Re_{\tau}=360$; (c,f) $Re_{\tau}=720$. Top, NWO cases; bottom, WWO cases.

Figure 11 shows the spectral decomposition of the net turbulent force as a
function of the azimuthal wavelength. In the NWO cases, we observe a clear
peak in the net turbulent force with the azimuthal wavelength
$\lambda_{s}^{+}\approx 100$, associated with the near-wall streak spacing, at
the bottom of the buffer layer ($y^{+}\approx 10$) in all three Reynolds
numbers. Consistent with the previous observations, the turbulent motions are
generally accelerated in and below the buffer layer and are decelerated above
it. The effect of wall oscillations, as in the case of a streamwise spectra,
is to diminish these accelerating and decelerating motions. We observe that
the major reduction in the net turbulent force comes from large azimuthal
scales in the buffer layer ($\lambda_{s}^{+}>1000$), whose acceleration is
retarded as a result of wall oscillations. The suppression of the net
turbulent force in large streamwise and azimuthal scales due to wall
oscillations is consistent with the hypothesis that wall oscillations inhibit
the growth of the hairpin packets. The lack of growth in the large-scale
structures forming the packets prevents steepening of the velocity gradient in
the buffer layer, which leads to a lower net turbulent force in this region in
the WWO cases. This inhibition of growth of the hairpin packets may be a
consequence of suppression of turbulent auto-generation mechanisms with wall
oscillation.

The budget of the net turbulent force describes the contribution of the
velocity-vorticity correlations to the turbulent force (Klewicki, 1989). In
cylindrical coordinates, such a decomposition applied to equation (30) can be
shown to be:

$F_{turb}(r)^{+}=\langle{u_{r}^{{\prime\prime}}}^{+}{\omega_{\theta}^{\prime\prime}}^{+}\rangle-\langle{u_{\theta}^{{\prime\prime}}}^{+}{\omega_{r}^{\prime\prime}}^{+}\rangle-\frac{\langle{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}\rangle}{r^{+}}.$
(34)

The first term on the right-hand side,
$\langle{u_{r}^{{\prime\prime}}}^{+}{\omega_{\theta}^{\prime\prime}}^{+}\rangle$,
is referred to as the advective vorticity transport, the second term,
$-\langle{u_{\theta}^{{\prime\prime}}}^{+}{\omega_{r}^{\prime\prime}}^{+}\rangle$,
is the vortex stretching term (Yoon et al., 2016), and the last term,
$-\langle{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}\rangle/r^{+}$
arises due to a cylindrical geometry of the problem. This allows for a
physical interpretation of the effects causing the reduction in the net
turbulent force.

(a) $Re_{\tau}=170$, NWO

(b) $Re_{\tau}=360$, NWO

(c) $Re_{\tau}=720$, NWO

(d) $Re_{\tau}=170$, WWO

(e) $Re_{\tau}=360$, WWO

(f) $Re_{\tau}=720$, WWO

(g) $Re_{\tau}=170$, WWO-NWO

(h) $Re_{\tau}=360$, WWO-NWO

(i) $Re_{\tau}=720$, WWO-NWO

Figure 12: Budget of the net turbulent force through the velocity-vorticity
correlations. From left to right, $Re_{\tau}=170,360,\text{ and }720$, with
the top row corresponding to $NWO$, middle row to $WWO$, and bottom row to
their difference (WWO-NWO). The lines with no marks is the total net turbulent
force; lines with squares indicate
$-\left\langle{{u_{\theta}^{{\prime\prime}}}^{+}{\omega_{r}^{\prime\prime}}^{+}}\right\rangle$
term; lines with circles
$\left\langle{{u_{r}^{{\prime\prime}}}^{+}{\omega_{\theta}^{\prime\prime}}^{+}}\right\rangle$
term; and the thin dotted lines indicate
$\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle/r^{+}$
term. The top inset in each subfigure plots a zoom-in of the region of
acceleration, $y^{+}<y_{f0}^{+}$, while the bottom inset shows a zoom-in of
the region of deceleration, $y^{+}>y_{f0}^{+}$. The bottom axis in each
subfigure also marks a location of $y_{f0}^{+}$ for each Reynolds number.
$y_{f0}^{+}$ is taken from the NWO case. Pink dotted horizontal line indicates
the zero value.

Figure 12 shows the decomposition of the net turbulent force into its
corresponding velocity-vorticity components. It can be seen that the the
vortex stretching term,
$-\left\langle{{u_{\theta}^{{\prime\prime}}}^{+}{\omega_{r}^{\prime\prime}}^{+}}\right\rangle$
(Chen et al., 2014) is the primary contributor to the flow acceleration
(positive $F_{turb}^{+}$) in both NWO and WWO flows. This term however is
markedly reduced in the WWO as compared to the NWO flows in the region of
$y^{+}<20$ promoting reduction of the near-wall velocity gradient, while it is
increased in the buffer, logarithmic and wake region with wall oscillation.
The amount of reduction in the vortex stretching between WWO and NWO cases
diminishes with the Reynolds number. The location at which the change switches
from negative to positive (thus accelerating the flow) appears to saturate at
$y^{+}\approx 20$ with $Re$, which is driven closer and closer to the wall as
a fraction of $y_{p}^{+}$. The second budget term,
$\left\langle{{u_{r}^{{\prime\prime}}}^{+}{\omega_{\theta}^{\prime\prime}}^{+}}\right\rangle$,
the advective vorticity transport, has a smaller effect on the acceleration of
the near-wall streamwise velocity, and in fact mostly acts to decelerate the
flow, as compared to the vortex stretching term. Its contribution to a
difference between the NWO and WWO cases is also less significant as compared
to the vortex stretching term. It is interesting to note the overall effect of
the pipe flow geometry (the third budget term,
$-\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle/r^{+}$)
to diminish $F_{turb}^{+}$ and thus to decelerate the mean flow at all wall-
normal locations. Its value is however significantly smaller than that of the
other two terms, and, conceivably, its contribution with the Reynolds number
also reduces.

###  3.6 Effect of Reynolds number on turbulent contribution to bulk mean
velocity

The Fukagata-Iwamoto-Kasagi (FIK) identity relates the wall shear stress to a
componentwise contribution of different dynamical effects in a turbulent flow
(Fukagata et al., 2002). To assess the effect on drag reduction in the current
setup, where the wall shear stress is fixed but the volumetric flow rate is
allowed to vary, it is more appropriate to express the FIK identity in terms
of the bulk mean velocity (Marusic et al., 2007; Yakeno et al., 2014). Such an
expression for the bulk mean velocity in a pipe flow (expressed in wall units)
can be derived as

$U_{bulk}^{+}=\frac{Re_{\tau}}{4}-Re_{\tau}\int_{0}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{r^{\star}}^{2}d{r^{\star}},$
(35)

where, recall, $r^{\star}=r/R$ is a radial coordinate scaled with the outer
units (the reader is referred to the Appendix B for the derivation of this
equation). It is shown in the Appendix B that the presence of spanwise wall
oscillations does not change this expression, which is valid in both NWO and
WWO cases.

The first term in equation (35) corresponds to the laminar contribution to the
bulk mean velocity (i.e. a contribution from a corresponding parabolic flow
profile which were to develop under the same mean pressure gradient in a
laminar flow), and the second term corresponds to the turbulent contribution.
Since laminar contribution scaled with $Re_{\tau}$ is identical between the
NWO and WWO cases (the spanwise Stokes’ layer due to the transverse wall
motion is decoupled from the streamwise boundary layer in a laminar solution
(Panton, 1984; Coxe et al., 2022)), we turn our attention to the turbulent
contribution, which is the only term responsible for the difference in the
bulk mean velocity between the two flows. We can represent a turbulent
contribution as a limiting value of the cumulative distribution function
evaluated at the pipe centerline $r^{\star}=0$ as:

$U_{bulk}^{t\,+}={U_{bulk}^{t,cum}}^{+}(r^{\star}=0)=-Re_{\tau}\int_{0}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{r^{\star}}^{2}d{r^{\star}},$
(36)

with the cumulative distribution function (termed as a “cumulative turbulent
contribution”) defined as shown in Appendix B:

${U_{bulk}^{t,cum}}^{+}(r^{\star})=-Re_{\tau}\int_{r^{\star}}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{r^{\star}}^{2}d{r^{\star}}.$
(37)

With this definition, the value of the cumulative turbulent contribution at
the pipe wall is zero, ${U_{bulk}^{t,cum}}^{+}(r^{\star}=1)=0$, consistent
with the physical meaning of this term. This integral can be written
equivalently as a function of the normalized wall normal coordinate
$y^{\star}=1-{r^{\star}}$ (Fukagata et al., 2002) as:

${U_{bulk}^{t,cum}}^{+}(y^{\star})=-Re_{\tau}\int_{1-y^{\star}}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{{r^{\star}}}^{2}d{r^{\star}}.$
(38)

Applying Parseval’s theorem (equation (17)) to the equations (36) and (38), we
can express the turbulent total and cumulative contributions to the bulk mean
velocity through the sums of their corresponding spectral contributions:

$U_{bulk}^{t\,+}=-Re_{\tau}\int_{0}^{1}\sum_{k_{x}}\sum_{k_{\theta}}\Phi_{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}(k_{x},{r^{\star}},k_{\theta}){{r^{\star}}}^{2}d{r^{\star}},$
(39)
${U_{bulk}^{t,cum}}^{+}(y^{\star})=-Re_{\tau}\int_{1-y^{\star}}^{1}\sum_{k_{x}}\sum_{k_{\theta}}\Phi_{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}(k_{x},{r^{\star}},k_{\theta}){{r^{\star}}}^{2}d{r^{\star}}.$
(40)

Figure 13 shows the cumulative turbulent contribution to the bulk mean
velocity, as well as the spectra of the total turbulent contribution for the
NWO and WWO cases, together with their change. From Figure 13(d), one can
observe that the major increase in the bulk mean velocity in a controlled flow
as compared to the uncontrolled flow comes from the buffer and the log layer
of the flow, with the peak around the top of the log layer, and the cumulative
contribution decreasing in the outer layer. Figure 13(b) shows an increased
contribution of large streamwise scales to the mean flow retardation by
turbulence, with the effect of wall oscillation to suppress this retardation
in the intermediate scales and increase it in larger scales. Interestingly, an
azimuthal spectra presented in Figure 13(c) shows a clear peak in
$-U_{bulk}^{t\,+}$ at azimuthal scales around $\lambda_{s}^{+}\approx 1000$
which increase with $Re_{\tau}$. These scales, representative of the hairpin
packet organization (Adrian et al., 2000; Adrian, 2007), contribute the most
to the turbulent drag. Figure 13(e) shows a remarkable collapse of the
$\Delta\,U_{bulk}^{t\,+}$ streamwise spectra across all three Reynolds
numbers, indicating that these are the same scales of motion (in wall units)
that are responsible for drag reduction, irrespective of the Reynolds number.
Specifically, length scales in the range of $500\leq\lambda_{x}^{+}\leq 5000$
reduce drag with wall oscillation. An important result is that larger scales
of motion in WWO flows act to increase drag. This explains the decreased
effectiveness of the wall oscillation mechanism (at least with the chosen
oscillation parameters) at higher Reynolds numbers: there are more large-scale
motions that develop at higher $Re$, and they are the ones which negatively
effect drag reduction. The azimuthal spectra of $\Delta\,U_{bulk}^{t\,+}$
shown in Figure 13(f) shows a reasonable amount of collapse but not to the
same extent as found in the streamwise spectra. The lowest Reynolds number,
$Re_{\tau}=170$, seems to be very effective at reducing drag in azimuthal
scales corresponding to the individual hairpins ($\lambda_{s}^{+}\approx
100-200$), perhaps because the Reynolds number is too low to effectively form
larger structures composed from the agglomeration of hairpins. The two higher
Reynolds numbers, $Re_{\tau}=360$ and $Re_{\tau}=720$, affect a larger range
of azimuthal wavenumbers ($100\leq\lambda_{s}^{+}\leq 700$). Larger azimuthal
scales, $\lambda_{s}^{+}>1000$, negatively contribute to drag reduction.

To characterize both the length scales and the wall normal location of
turbulent motions contributing to drag reduction, Figure 14 plots the
streamwise and azimuthal spectra of the cumulative turbulent contribution as a
function of wall normal coordinate. It can be clearly seen that the streamwise
scales of motion with the wavelengths between $500\leq\lambda_{x}^{+}\leq
5000$ contribute to drag reduction (positive
$\Delta\,{U_{bulk}^{t,cum}}^{+}(\lambda_{x}^{+},y^{+})$) all throughout the
vertical extent of the pipe for all three Reynolds numbers. It can also be
noted that larger wavelengths, while still acting to cumulatively reduce drag
in the log layer, overtake and lead to a drag increase in the outer layer.
This effect is absent at the lowest Reynolds number, $Re_{\tau}=170$, and is
the strongest at the highest Reynolds number, $Re_{\tau}=720$.

(a) Cumulative turbulent contribution

(b) Streamwise spectra of total turbulent contribution

(c) Azimuthal spectra of total turbulent contribution

(d) Change in cumulative turbulent contribution

(e) Change in streamwise spectra

(f) Change in azimuthal spectra

Figure 13: Turbulent contribution to the bulk mean velocity. (a,b,c)
Contributions for NWO (black solid lines) and WWO (blue dashed lines) cases;
(d,e,f) change between NWO and WWO cases. (a,d) Cumulative turbulent
contribution as a function of wall normal coordinate; (b,e) streamwise spectra
of the total turbulent contribution; (c,f) azimuthal spectra of the total
turbulent contribution. The notation $\lambda_{s,wall}^{+}$ refers to
$\lambda_{s}^{+}$ evaluated at the wall. The red dashed line in (e,f) denotes
the zero level.

(a) $Re_{\tau}=170$, Streamwise

(b) $Re_{\tau}=360$, Streamwise

(c) $Re_{\tau}=720$, Streamwise

(d) $Re_{\tau}=170$, Azimuthal

(e) $Re_{\tau}=360$, Azimuthal

(f) $Re_{\tau}=720$, Azimuthal

Figure 14: Change in the cumulative turbulent contribution spectra (normalized
by the laminar contribution) as a function of wall normal location and the
wavelength: (a,b,c) streamwise spectra,
$\Delta\,4{U_{bulk}^{t,cum}}^{+}(\lambda_{x}^{+},y^{+})/Re_{\tau}$; (d,e,f)
azimuthal spectra,
$\Delta\,4{U_{bulk}^{t,cum}}^{+}(\lambda_{s}^{+},y^{+})/Re_{\tau}$. From top
to bottom: (a,d) $Re_{\tau}=170$, (b,e) $Re_{\tau}=360$, and (c,f)
$Re_{\tau}=720$.

For the azimuthal spectra, it is observed that the azimuthal length scales
between $50\leq\lambda_{s}^{+}\leq 500$ act to increase the flow rate (reduce
drag) while larger structures reduce the flow rate (increase drag),
essentially independent of a wall normal location. Interestingly, drag-
reducing azimuthal scales organize themselves into a fractal-like pattern (red
“fingers”) visible in Figures 14(d), 14(e), 14(f). A fractal-like pattern is
consistent with the attached eddy hypothesis of the near-wall turbulence
(Townsend, 1951; Hwang, 2015) and the trends in hairpin packet organization
(Adrian, 2007); the fact that the drag-reducing motions adhere to this pattern
suggests a link between the drag reduction mechanisms and a weakening of the
hairpin packets. Similarly to the streamwise spectra, we observe that
increasing Reynolds number introduces larger azimuthal scales of motion
(relative to the viscous scale) which hinder the effectiveness of the selected
oscillation parameters to reduce turbulent drag.

To compare with the analysis performed by Hurst et al. (2014); Yao et al.
(2019) in a turbulent channel flow with wall oscillation, we decompose the
turbulent contribution to the bulk mean velocity into the corresponding
“inner” (accelerating layer) and “outer” (decelerating layer) components, such
that

$\displaystyle
U_{bulk,"inner"}^{t\,+}=-Re_{\tau}\int_{1-y_{f0}^{\star}}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{{r^{\star}}}^{2}d{r^{\star}}$
(41) $\displaystyle
U_{bulk,"outer"}^{t\,+}=-Re_{\tau}\int_{0}^{1-y_{f0}^{\star}}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{{r^{\star}}}^{2}d{r^{\star}},$
(42)

where $y_{f0}^{\star}=y_{f0}/R$ is the location of the zero net turbulent
force. We remark that the zero net turbulent force location coincides with the
peak Reynolds shear stress location in a channel flow (Chen et al., 2018; Yao
et al., 2019), however it is slightly different in a pipe flow due to a
curvature effect (Wu et al., 2012). In a pipe flow, the net turbulent force
attains zero when
$r\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle$
reaches its maximum, and not
$\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle$,
see Eq. (29). The classification of the “inner” and “outer” layers based on
the location of the zero net turbulent force is different from the classical
demarcation of the inner and outer layers as being directly affected and
unaffected by viscosity, respectively (Sreenivasan & Sahay, 1997; Adrian et
al., 2000; Jiménez, 2018) (see also Table 5). Figure 15(a) presents the
corresponding component contributions to the bulk mean velocity (normalized by
the laminar component) for the NWO and WWO pipe flows. The “inner” and “outer”
turbulent contributions are computed from Eqs. (41)–(42), taking the NWO value
for $y_{f0}^{\star}$ as a reference for both the NWO and WWO cases at each
Reynolds number. Consistent with the previous observations, we see that the
“outer” layer contributes more significantly to the mean flow retardation from
the laminar flow, and this contribution increases with the Reynolds number.
The effect of wall oscillations is to reduce the contributions to $U_{bulk}$
from both the “inner” and the “outer” layers. Figure 15(b) presents the
difference of the component contributions between the NWO and WWO cases
normalized by $U_{bulk,NWO}$. We observe that the reduction of effectiveness
of wall oscillations in increasing the bulk mean velocity of controlled flow
as compared to the uncontrolled flow with $Re_{\tau}$ mostly comes from the
“outer” (decelerating) layer, consistent with Figure 13(d). The analysis
presented here is different from Hurst et al. (2014); Yao et al. (2019) in
that we keep the wall shear stress (or pressure gradient) constant, while they
keep the bulk mean velocity (or bulk flow rate) constant. To adhere to their
analysis, Figures 15(c), 15(d) present the corresponding component
contributions to the skin friction coefficient,
$C_{f}=2\left\langle{\tau_{w}}\right\rangle/\rho U_{bulk}^{2}$ (equation 47 in
Appendix B), together with the change between the NWO and WWO cases.
Concerning the contribution to skin friction, these results are very close to
the data presented in Hurst et al. (2014); Yao et al. (2019) for channel
flows, albeit we also see a change in a laminar contribution to skin friction
between the NWO and WWO cases due to a change in the bulk flow rate with and
without wall oscillation in our setup. Interestingly, while the loss of
effectiveness mostly comes from the “outer” (decelerating) layer when the
change in bulk mean velocity is concerned, it mostly comes from the “inner”
(accelerating) layer when the change in skin friction coefficient is
considered, the latter conclusion drawn by Yao et al. (2019) in their channel
flow study. In both cases, loss of effectiveness is attributed to the large
scales, which are either not effectively suppressed (as found in channel
flows) or even energized (as found here in regard to pipe flows), both in the
inner and outer layers of the flow.

(a) $U_{bulk}^{comp}/U_{bulk}^{lam}$

(b) Change in $U_{bulk}^{comp}$

(c) $C_{f}^{comp}/C_{f,NWO}$

(d) Change in $C_{f}^{comp}$

Figure 15: Component contributions to (a,b) bulk mean velocity; (c,d) skin
friction coefficient. (a,c) The normalized component contributions (filled
symbols, NWO; open symbols, WWO); (b,d) change between the NWO and WWO cases.
Line colors indicate: red, “outer” layer turbulent contribution; blue, “inner”
layer turbulent contribution; orange, laminar contribution; black, total
contribution.

##  4 Conclusions

The current study documents the results of direct numerical simulation of a
turbulent pipe flow with and without transverse wall oscillation for three
Reynolds numbers, $Re_{\tau}=170,360$ and $720$. It is found that wall
oscillation results in an increase of a flow rate by almost 20% and,
consequently, achieves a drag reduction of approximately 30% at the lowest
Reynolds number; however, this effect decreases as the Reynolds number
increases. One-dimensional and two-dimensional spectra of streamwise kinetic
energy, net turbulent force and the turbulent contribution to the bulk mean
velocity are analyzed to explain this effect.

It is found that the primary effect of wall oscillation is to reduce the
energy and the net turbulent force in the intermediate- to large- streamwise
and azimuthal scales of motion in the buffer layer of the flow. To the
contrary, energy is increased in the large-scale structures in the log layer
and the wake region. At the lowest Reynolds number, $Re_{\tau}=170$, the inner
layer extends through $\approx 65\%$ of the domain while it comprises $\approx
15\%$ of the domain at $Re_{\tau}=720$. Since the overall attenuation of
energetic structures is limited to the inner layer of the flow, this explains
the reduced effectiveness of the wall oscillation as a drag reduction
mechanism as the Reynolds number increases. This effect is well illustrated by
Figure 10, where low-pass filtered and high-pass filtered net turbulent force
is plotted as a function of wall-normal coordinate. This figure shows that (a)
most of the attenuation comes from the effect of the scales of
$\lambda_{x}^{+}>1000$ (above the filter cut-off length), and (b) the
reduction of the magnitude of the net turbulent force is confined to the
buffer layer and the log layer of the flow. The reduction of the magnitude of
the net turbulent force by wall oscillation results in a reduced flow
acceleration in the near-wall layer and an increased acceleration in the log
layer, making the velocity profile less blunt and more “laminar-like”. This is
offset by an increase of the net turbulent force magnitude (promoting flow
deceleration) above the log layer, leading to a lower ratio of the centerline
velocity to the bulk velocity in WWO cases, which reduces effectiveness of
drag reduction at higher Reynolds number flows. From the velocity-vorticity
budget decomposition of the net turbulent force, it is observed that this
effect mostly comes from a suppression of the vortex stretching within the
Stokes’ layer of the flow and its augmentation above the Stokes’ layer and
below $y^{+}\approx 100$. A reduced vortex stretching in the Stokes’ layer
inhibits a lift-up of the hairpins and formation of their necks, thus
suppressing the hairpin auto-generation.

From superimposing the analysis of the streamwise and azimuthal spectra,
together with the wall normal location of the effected length scales, one can
deduce the shape of the structures most affected by drag reduction. It can be
seen that a significant energy reduction occurs at streamwise scales at and
slightly above $\lambda_{x}^{+}\approx 1000$ and azimuthal scales of at and
slightly above $\lambda_{s}^{+}\approx 100$, which corresponds to the scales
of motions typically associated with the hairpin packets (Adrian et al., 2000;
Adrian, 2007). Since streaks and quasi-streamwise vortices are closely
related, wall oscillation presumably weakens the quasi-streamwise near-wall
vortices, thereby reducing their transport of streamwise momentum into the
streaks. This is consistent with the observations of Yao et al. (2019) who
reported a suppression of Reynolds shear stresses at $\lambda_{s}^{+}<400$ in
a turbulent channel flow with wall oscillation. This points towards a
suppression of hairpin packets by wall oscillation being one of the main
mechanisms of drag reduction. It is hypothesized that the auto-generation
mechanism of turbulence (Zhou et al., 1999; Kim et al., 2008; Kempaiah et al.,
2020) is suppressed by the wall oscillation, thus attenuating the formation
and growth of the hairpin packets. Interestingly, the shorter streamwise
scales of motion, $\lambda_{x}^{+}<500$, are amplified, which suggests that
the wall oscillation mechanism does not suppress the energy in the individual
hairpins but rather hinders their regeneration abilities. Large streamwise
scales, $\lambda_{x}^{+}>5000$, and large azimuthal scales,
$\lambda_{s}^{+}>1000$, are also found to be amplified by wall oscillation,
both in the buffer layer and above. Such amplification of the large-scale
azimuthal energy in the buffer layer may be associated with the large-scale
mode observed in the current study for the wall-oscillated pipe flow cases
with $Re_{\tau}=360$ and 720, potentially created by the sloshing motions
spurred by the wall oscillation. In the outer layer, the amplified structures
correspond to the very-large-scale motions (Guala et al., 2006; Balakumar &
Adrian, 2007) of the high Reynolds number flows. Interestingly, while Yao et
al. (2019) reported a reduced effectiveness of WWO control in suppressing
large-scale azimuthal motions ($\lambda_{s}^{+}>1000$) in channel flows, they
did not observe an amplification of such scales, as the current study does,
which might point towards a particular influence of the large-scale mode,
potentially distinct to pipe flows.

A convincing evidence of the effect of different scales of motion on drag
reduction comes from the spectral analysis of the Fukagata-Iwamoto-Kasagi
(FIK) identity (Fukagata et al., 2002); specifically, of the turbulent
contribution to the bulk mean velocity. To this end, Figure 13 demonstrates a
remarkable collapse of the difference in its spectra between the NWO and WWO
pipes across all three Reynolds numbers, showing that the drag reduction is
limited to the streamwise wavelengths of $\lambda_{x}^{+}<5000$ independent of
the Reynolds number. The wavelengths with $\lambda_{x}^{+}>5000$, exerting
increasingly larger dominance in higher Reynolds number flows, act to increase
drag. This brings us to a conclusion that the wall oscillation mechanism with
the parameters investigated in the current paper, which are optimized for
controlling the near-wall turbulent cycle (Jung et al., 1992; Choi & Graham,
1998; Quadrio & Ricco, 2004), may not be effective for drag reduction in high
Reynolds number flows. Possibly, new drag reduction mechanisms that
specifically target large- and very-large scales of motions need to be
developed. This can perhaps be achieved by reducing the frequency of the wall
oscillation as suggested by Marusic et al. (2021). It is also possible that
completely new drag reduction mechanisms need to be devised to target high
Reynolds number flows.

## Acknowledgements

This research was supported by the NSF CAREER award # CBET-1944568 and by the
Ira A. Fulton Professorship endowment. The computational time has been
provided by NSF ACCESS supercomputing resources.

## Declaration of Interests

The authors report no conflict of interest.

##  Appendix A Validation

This section presents a validation of the spectral-element code Nek5000 in
application to DNS of turbulent pipe flows within the current computational
setup. Additional validation is available in previous studies in Duggleby et
al. (2007b); El Khoury et al. (2013); Merrill et al. (2016).

###  A.1 Turbulent pipe flow with no wall oscillation (NWO)

In this section, validation of the current DNS results for a turbulent pipe
flow with no wall oscillation (NWO case) is presented. Figure 16 illustrates a
comparison of statistical quantities (mean streamwise velocity and fluctuating
Reynolds stresses) with the previously available computational (El Khoury et
al., 2013) and experimental (Eggels et al., 1994; Chin et al., 2015) data.
Good agreement of statistics with the previously published data is observed.
Figure 17 compares a calculated pre-multiplied energy spectra of the
streamwise, radial, and azimuthal velocity fluctuations for $Re_{\tau}=720$
case with the DNS data of Wu et al. (2012). Again, a favorable agreement is
demonstrated.

(a) Mean streamwise velocity

(b) Streamwise velocity fluctuations

(c) Radial velocity fluctuations

(d) Reynolds shear stress.

Figure 16: Validation of statistical quantities for the DNS of turbulent pipe
flow with Nek5000 (no wall oscillation): (a) mean streamwise velocity, (b)
streamwise velocity fluctuations, (c) radial velocity fluctuations, and (d)
Reynolds shear stress. Lines, current DNS: $Re_{\tau}=170$, dotted line;
$Re_{\tau}=360$, dash-dotted line; $Re_{\tau}=720$, solid lines;. Symbols, El
Khoury et al. (2013) (DNS): $Re_{\tau}=180,\vartriangle$;
$Re_{\tau}=360,\triangledown$; $Re_{\tau}=550,\Delta$;
$Re_{\tau}=1000,\lozenge$ (LES); Eggels et al. (1994) (PIV):
$Re_{\tau}=200,\times$; Chin et al. (2015) (hot wire): $Re_{\tau}=1000,+$.

(a) Streamwise velocity, $k_{\theta}\,\Phi_{u_{x}u_{x}}/u_{\tau}^{2}$ ($j=x$)

(b) Radial velocity, $k_{\theta}\,\Phi_{u_{r}u_{r}}/u_{\tau}^{2}$ ($j=r$)

(c) Azimuthal velocity, $k_{\theta}\,\Phi_{u_{\theta}u_{\theta}}/u_{\tau}^{2}$
($j=\theta$)

Figure 17: Premultiplied azimuthal spectrum for the $Re_{\tau}=720$ case as
compared with the work of Wu et al. (2012) for (a) streamwise velocity,
$k_{\theta}\,\Phi_{u_{x}u_{x}}/u_{\tau}^{2}$; (b) radial velocity,
$k_{\theta}\,\Phi_{u_{r}u_{r}}/u_{\tau}^{2}$; and (c) azimuthal velocity,
$k_{\theta}\,\Phi_{u_{\theta}u_{\theta}}/u_{\tau}^{2}$. Color snapshots,
current DNS; black contour lines, data of Wu et al. (2012). Black contour
lines are spaced by 0.1 starting from the minimum value for the radial and
azimuthal velocity spectra, and by 0.4 for the streamwise velocity spectra.

###  A.2 Turbulent pipe flow with wall oscillation (WWO)

This section presents a validation of the current DNS simulations for the case
of a turbulent pipe flow with wall oscillation (WWO). Figure 18 documents a
comparison of the single-point velocity statistics with the previously
available data. In particular, we compare the present DNS results with the DNS
of a turbulent pipe flow with wall oscillation at $Re_{\tau}=150$ (Duggleby et
al., 2007a), and DNS of a turbulent channel flow with wall oscillation at
$Re_{\tau}=1000$ (Agostini et al., 2014). The latter dataset is chosen for
comparison, since no data for a turbulent pipe flow with wall oscillations is
available past $Re_{\tau}=180$ (Ricco et al., 2021). Figure 18 shows that the
computed statistics in the WWO cases is within the range of the previously
published data.

(a) Mean streamwise velocity

(b) Streamwise velocity fluctuations

(c) Radial velocity fluctuations.

(d) Reynolds shear stress

Figure 18: Validation of statistical quantities for the DNS of turbulent pipe
flow with Nek5000 (with wall oscillation): (a) mean streamwise velocity, (b)
streamwise velocity fluctuations, (c) radial velocity fluctuations, and (d)
Reynolds shear stress. Black lines with symbols, current DNS at
$Re_{\tau}=170$, 360 and 720 (See the legend); blue dashed line, DNS of pipe
flow with wall oscillation at $Re_{\tau}=150$ (Duggleby et al., 2007a); red
dashed line, DNS of channel flow with wall oscillation at $Re_{\tau}=1000$
(Agostini et al., 2014).

##  Appendix B FIK identity for the bulk mean velocity in a turbulent pipe
flow with oscillating walls

This appendix derives an analogue of the Fukagata-Iwamoto-Kasagi (FIK)
identity (Fukagata et al., 2002) for the bulk mean velocity in a turbulent
pipe flow with and without the oscillating walls. We start with the ensemble-
averaged streamwise momentum equation (28), multiply it by $r$ and integrate
across the vertical coordinate as $\int_{r}^{R}(\cdot)\,r\,dr$ to yield:

$\nu\,r\frac{d\left\langle{u_{x}}\right\rangle}{dr}=r\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle+\frac{r^{2}}{2\rho}\left\langle{\frac{\partial\,p}{\partial\,x}}\right\rangle.$
(43)

Application of the boundary conditions at $r=R$, together with the equation
(5) and the definition of the mean wall shear stress
$\left\langle{\tau_{w}}\right\rangle=\left(-\rho\,\nu\,d\left\langle{u_{x}}\right\rangle/dr\right)\big{|}r=R$
was used to arrive at (43).

We proceed in the same way, multiplying equation (43) by $r$ and integrating
it as $\int_{r}^{R}(\cdot)\,r\,dr$ again. We obtain the following relation:

$2\nu\int_{r}^{R}\left\langle{u_{x}}\right\rangle
rdr+\nu\left\langle{u_{x}}\right\rangle
r^{2}=-\frac{R^{4}}{8\rho}\left\langle{\frac{\partial\,p}{\partial\,x}}\right\rangle\left(1-\left(\frac{r}{R}\right)^{4}\right)-\int_{r}^{R}\left\langle{{u_{x}^{{\prime\prime}}}{u_{r}^{{\prime\prime}}}}\right\rangle
r^{2}dr.$ (44)

Recasting equation (44) into the non-dimensional coordinates
${r^{\star}}=r/R$, $u_{i}^{+}=u_{i}/u_{\tau}$, and utilizing equation (5) once
again yields:

$2\int_{r^{\star}}^{1}\left\langle{u_{x}^{+}}\right\rangle{r^{\star}}d{r^{\star}}+\left\langle{u_{x}^{+}}\right\rangle{r^{\star}}^{2}=\frac{Re_{\tau}}{4}(1-{r^{\star}}^{4})-Re_{\tau}\int_{r^{\star}}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{r^{\star}}^{2}d{r^{\star}}.$
(45)

The first term on the right-hand side of equation (45) represents the
cumulative laminar contribution to the bulk mean velocity, while the last term
corresponds to the cumulative turbulent contribution, already presented in
equation (37). Evaluating equation (45) at ${r^{\star}}=0$ and using the
definition of bulk mean velocity (4) cast into a non-dimensional form as
$U_{bulk}^{+}=2\int_{0}^{1}\left\langle{u_{x}^{+}}\right\rangle{r^{\star}}d{r^{\star}}$
gives the FIK identity:

$U_{bulk}^{+}=\frac{Re_{\tau}}{4}-Re_{\tau}\int_{0}^{1}\left\langle{{u_{x}^{{\prime\prime}}}^{+}{u_{r}^{{\prime\prime}}}^{+}}\right\rangle{r^{\star}}^{2}d{r^{\star}},$
(46)

which is the same as equation (35) shown previously. Note that, since only
streamwise mean momentum equation is used in the derivation of (46), while
non-zero boundary conditions for the oscillating pipe wall are set on the
azimuthal velocity, the FIK identity (46) applies to both NWO and WWO cases.

The presented derivation can be easily extended to the skin friction
coefficient, $C_{f}=2\left\langle{\tau_{w}}\right\rangle/\rho\,U_{bulk}^{2}$,
by evaluating equation (44) at $r=0$, utilizing equation (5) to relate mean
pressure gradient to the wall shear stress, and the definition of
$Re_{bulk}=2U_{bulk}R/\nu$, to yield

$C_{f}=\frac{16}{Re_{bulk}}+8\int_{0}^{1}\left\langle{u_{x}^{\star}u_{r}^{\star}}\right\rangle{r^{\star}}^{2}d{r^{\star}},$
(47)

where we used the definitions $u_{x}^{\star}=u_{x}/U_{bulk}$,
$u_{r}^{\star}=u_{r}/U_{bulk}$ for velocities scaled in the outer units.

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