# Large-scale streaks in a turbulent bluff body wake

Akhil Nekkanti1  Sheel Nidhan1  Oliver T. Schmidt1  Sutanu Sarkar1
<EMAIL_ADDRESS>1Department of Mechanical and Aerospace Engineering,
University of California San Diego, CA 92093, USA

###### Abstract

A turbulent circular disk wake database (Chongsiripinyo & Sarkar, J. Fluid
Mech., vol. 885, 2020) at Reynolds number $\Rey=U_{\infty}D/\nu=5\times
10^{4}$ is interrogated to identify the presence of large-scale streaks -
coherent elongated regions of streamwise velocity. The unprecedented
streamwise length - until $x/D\approx 120$ \- of the simulation enables
investigation of the near and far wake. The near wake is dominated by the
vortex shedding (VS) mode residing at azimuthal wavenumber $m=1$ and Strouhal
number $\mbox{\it St}=0.135$. After filtering out the VS structure, conclusive
evidence of large-scale streaks with frequency $\mbox{\it St}\rightarrow 0$,
equivalently streamwise wavenumber $k_{x}\rightarrow 0$ in the wake, becomes
apparent in visualizations and spectra. These streaky structures are found
throughout the simulation domain beyond $x/D\approx 10$. Conditionally
averaged streamwise vorticity fields reveal that the lift-up mechanism is
active in the near as well as the far wake, and that ejections contribute more
than sweep to events of intense $-u^{\prime}_{x}u^{\prime}_{r}$. Spectral
proper orthogonal decomposition (SPOD) is employed to extract the energy and
the spatiotemporal features of the large-scale streaks. The streak energy is
concentrated in the $m=2$ azimuthal mode over the entire domain. Finally,
bispectral mode decomposition (BMD) is conducted to reveal strong interaction
between $m=1$ and $\mbox{\it St}=\pm 0.135$ modes to give the $m=2,\mbox{\it
St}=0$ streak mode. Our results indicate that the self-interaction of the VS
mode generates the $m=2,\mbox{\it St}=0$ streamwise vortices, which leads to
streak formation through the lift-up process. To the authors’ knowledge, this
is the first study that reports and characterizes large-scale low-frequency
streaks and the associated lift-up mechanism in a turbulent wake.

###### keywords:

##  1 Introduction

Coherent structures, which are organized patterns of motion in a seemingly
random turbulent flow field, play an essential role in turbulent shear flows.
Among these structures, streaks are among the most widely discussed,
particularly in wall-bounded flows, where they were identified experimentally
(Kline et al., 1967) as elongated regions of streamwise velocity in the near-
wall region. As these streaks break up, they transfer energy from the inner to
the outer layers, thereby maintaining turbulence in the outer layers of the
boundary layer (Kim et al., 1971). This process, also known as _bursting_ ,
can account for up to 75% of Reynolds stresses (Lu & Willmarth, 1973), hence
assisting in the production of turbulent kinetic energy (TKE). Smith & Metzler
(1983) found that low-speed streaks are robust features of boundary layers,
occurring across a wide range of Reynolds numbers ($740<\Rey_{\theta}<5830$).
Their spanwise spacing of 100 wall units was found to be invariant with
$\Rey_{\theta}$. Hutchins & Marusic (2007) investigated the logarithmic region
of the boundary layer, finding that the streaks in this region are distinct
and much larger than the near-wall streaks, extending up to 20 times the
boundary layer thickness. Building upon this work, Monty et al. (2007)
reported the existence of streaks in the logarithmic region of turbulent pipe
and channel flows as well. They also found that the width of these structures
in channel and pipe flows is larger than those of the boundary layer.

The presence of a wall is not a prerequisite for the formation of streaks
(Jiménez & Pinelli, 1999; Mizuno & Jiménez, 2013). A few studies in the past
(see figure 8b Brown & Roshko, 1974; Bernal & Roshko, 1986; Liepmann & Gharib,
1992) have reported the presence of streak-like structures in the mixing
layer, with the latter two showing increasing amplification of the streaks as
the flow progresses downstream. More attention has been paid to the role of
streaks in the mixing layer and specially the jet, recently. Jiménez-González
& Brancher (2017) performed transient growth analysis in round jets finding
that, for optimal initial disturbances, the streamwise vortices evolve to
produce streamwise streaks. Marant & Cossu (2018) also reported similar
findings in a hyperbolic-tangent mixing layer. Nogueira et al. (2019) applied
spectral proper orthogonal decomposition on a PIV dataset of a circular
turbulent jet at a high Reynolds number and demonstrated the presence of
large-scale streaky structures in the near field (until $x/D=8$). They further
demonstrated that these structures exhibit large time scales and are
associated with a low frequency of $\mbox{\it St}\rightarrow 0$. The numerical
counterpart of the previous study was performed by Pickering et al. (2020).
They found that the streaky structures near the nozzle exit are dominated by
higher azimuthal wavenumbers and the dominance shifts to lower azimuthal
wavenumbers downstream, with $m=2$ dominating by $x/D=30$. A similar
conclusion about the dominance of $m=2$ was reached by Samie et al. (2022) who
utilized quadrant analysis on a low Reynolds number jet.

In wall-bounded flows, streaks are generated by the lift-up mechanism
(Ellingsen & Palm, 1975; Landahl, 1975). Streamwise vortices induce wall-
normal velocities, bringing fluid from high-speed to low-speed regions and
vice-versa to form streaks, hence the term ‘lift-up’. The subsequent
instability and breakdown of these streaks are important to the self-
sustaining cycle of wall turbulence (Hamilton et al., 1995; Waleffe, 1997).
Brandt (2014) presents a detailed review of the theory behind the lift-up
mechanism and its role in transitional and turbulent flows. Although
originally introduced as an instability mechanism that destabilizes a
streamwise-independent base flow, the lift-up mechanism has been found to be
dynamically crucial to fully turbulent wall-bounded flows as well (Jiménez,
2018; Bae et al., 2021; Farrell & Ioannou, 2012).

The lift-up mechanism is active and plays a critical role in jets too. In
their resolvent analysis of data from a turbulent jet experiment, Nogueira et
al. (2019) found that the optimal forcing modes at $\mbox{\it St}=0$ take the
form of streamwise vortices that eject high-speed fluid and sweep low-speed
fluid, depending on the orientation of these vortices. Pickering et al. (2020)
analyzed a turbulent jet LES database, finding that the response modes of
these lift-up dominated optimal forcing modes indeed take the form of
streamwise streaks. While these studies focused on circular jets, Lasagna et
al. (2021) found that the lift-up mechanism is active in the near field of
fractal jets as well.

As discussed in the foregoing, there has been a growing interest in the
investigation of streaky structures and lift-up mechanisms in free shear
flows, particularly in turbulent jets. These experimental and numerical
studies have confirmed that the presence of a wall is not necessary for the
formation of streaks, which has motivated us to explore another important
class of free shear flows, i.e., turbulent wakes. Previous wake studies have
primarily focused on the vortex shedding mechanism. Near the body and in the
intermediate wake, the vortex shedding (VS) mode emerges as the most dominant
coherent structure (Taneda, 1978; Cannon et al., 1993; Berger et al., 1990;
Yun et al., 2006). However, it is worth noting that Johansson et al. (2002)
reported the presence of a distinct very low-frequency mode, $\mbox{\it
St}\rightarrow 0$, at azimuthal wavenumber $m=2$ in their proper orthogonal
decomposition (POD) analyses of the turbulent wake of a circular disk at
$\Rey\approx 2.5\times 10^{4}$. This mode is distinct from the VS mode of a
circular disk wake, which resides at $m=1$ with $\mbox{\it St}=0.135$ (Berger
et al., 1990). In a subsequent study, Johansson & George (2006) extended the
downstream distance to $x/D=150$ and found that the $m=2$ mode with $\mbox{\it
St}\rightarrow 0$ dominated the far wake of the disk in terms of energy
content relative to the vortex shedding mode. This finding was corroborated in
the spectral POD analysis (Nidhan et al., 2020) of a disk wake at a higher
Reynolds number ($\Rey=5\times 10^{4}$). The authors further found low-rank
behavior of the SPOD modes and that almost the entire Reynolds shear stress
could be reconstructed with the leading SPOD modes of $m\leq 4$. Streamwise-
elongated streaks, a focus of the current paper, was not considered by Nidhan
et al. (2020).

None of the wake studies that report the presence and the eventual dominance
at large $x$ of the very low-frequency mode ($\mbox{\it St}\rightarrow 0$) at
$m=2$ explain the physical origins of this structure. To address this gap, we
revisit the LES database of Chongsiripinyo & Sarkar (2020), who simulated the
wake of a circular disk up to an unprecedented downstream distance of
$x/D=125$. The large streamwise domain enables us to investigate the entire
wake. Unlike the previous SPOD analysis (Nidhan et al., 2020) of this wake
database, we focus on the streaky structures. We attempt to answer the
following questions: Can streaky structures be identified in the near and far
field of the turbulent wake? Is the lift-up mechanism active in the turbulent
wake? How do the energetics and spatial structure of streaks evolve with
downstream distance? What, if any, is the link between the streak and the
well-documented and extensively studied vortex shedding mode?

In this work, besides visualizations and classical statistical analyses, we
utilize two modal techniques, spectral proper orthogonal decomposition (SPOD)
and bispectral mode decomposition (BMD), to shed light on the aforementioned
questions. SPOD, whose mathematical framework in the context of turbulent flow
was laid out by Lumley (1967, 1970), extracts a set of orthogonal modes sorted
according to their energies at each frequency. It distinguishes the different
time scales of the flow and identifies the most energetic coherent structures
at each time scale. SPOD modes are coherent in both space and time and
represent the flow structures in a statistical sense (Towne et al., 2018).
Early applications of SPOD were by Glauser et al. (1987); Glauser & George
(1992); Delville (1994) and this method has regained popularity since the work
by Towne et al. (2018); Schmidt et al. (2018). SPOD is particularly suitable
for detecting and educing modes corresponding to streaky structures in
statistically stationary flows (Nogueira et al., 2019; Pickering et al., 2020;
Abreu et al., 2020), and is hence employed in this work. Bispectral mode
decomposition, proposed by Schmidt (2020), extracts the flow structures that
are generated through triadic interactions. It identifies the most dominant
triads in the flow by maximizing the spatially-integrated bispectrum.
Furthermore, it picks out the spatial regions of nonlinear interactions
between the coherent structures. BMD has been used to characterize the triadic
interactions in various flow configurations, such as laminar-turbulent
transition on a flat plate (Goparaju & Gaitonde, 2022), forced jets (Maia et
al., 2021; Nekkanti et al., 2022), swirling flows (Moczarski et al., 2022) and
wake of an airfoil (Patel & Yeh, 2023). In this work, we will employ BMD to
investigate the presence and strength of nonlinear interactions between the VS
mode and the streak-containing modes.

The manuscript is organized as follows. In §2, the dataset and numerical
methodology of SPOD and BMD are discussed. §3 presents the extraction and
visualization of streaks and lift-up mechanism in the near and far wake.
Results from SPOD analysis at different downstream locations are presented in
§4 with a particular emphasis again on streaks and lift-up mechanism. §5
presents the results from BMD analysis in the wake at select locations. The
manuscript ends with discussion and conclusions in §6.

##  2  Numerical wake database and methods for its analysis

###  2.1 Dataset description

The dataset employed for the present study of wake dynamics is from the high-
resolution large eddy simulation (LES) of flow past a circular disk at
Reynolds number, $\Rey=U_{\infty}D/\nu=5\times 10^{4}$, reported in
Chongsiripinyo & Sarkar (2020). Here $U_{\infty}$ is the freestream velocity,
$D$ is the disk diameter, and $\nu$ is the kinematic viscosity. The case of a
homogeneous fluid from Chongsiripinyo & Sarkar (2020), who also simulate
stratified wakes, is selected here. The filtered Navier-Stokes equations,
subject to the condition of solenoidal velocity, were numerically solved on a
structured cylindrical grid that spans a radial distance of $r/D=15$ and a
streamwise distance of $x/D=125$. An immersed boundary method (Balaras, 2004)
is used to represent the circular disk in the simulation domain and the
dynamic Smagorinsky model (Germano et al., 1991) is used for the LES model.
The number of grid points in the radial ($r$), azimuthal ($\theta$) and
streamwise directions ($x$) are $N_{r}=365$, $N_{\theta}=256$ and
$N_{x}=4096$, respectively. The simulation has high resolution with streamwise
grid resolution of $\Delta x=10\eta$ at $x/D=10$, where $\eta$ is the
Kolmogorov lengthscale. By $x/D=125$, the resolution improves to $\Delta
x<6\eta$ so that the onus on the subgrid model progressively decreases.
Readers are referred to Chongsiripinyo & Sarkar (2020) for a detailed
description of the numerical methodology and grid quality.

###  2.2 Spectral proper orthogonal decomposition (SPOD)

SPOD is the frequency-domain variant of proper orthogonal decomposition. It
computes monochromatic modes that are optimal in terms of the energy norm of
the flow, e.g., turbulent kinetic energy (TKE) for the wake flow at hand. The
SPOD modes are the eigenvectors of the cross-spectral density matrix, which is
estimated using Welch’s approach (Welch, 1967). Here, we provide a brief
overview of the method. For a detailed mathematical derivation and algorithmic
implementation, readers are referred to Towne et al. (2018) and Schmidt &
Colonius (2020).

For a statistically stationary flow, let $\mathbf{q}_{i}=\mathbf{q}(t_{i})$
denote the mean subtracted snapshots, where $i=1,2,\cdots n_{t}$ are $n_{t}$
number of snapshots. For spectral estimation, the dataset is first segment
into $n_{\rm{blk}}$ overlapping blocks with $n_{\rm{fft}}$ snapshots in each
block. The neighbouring blocks overlap by $n_{\rm{ovlp}}$ snapshots with
$n_{\rm{ovlp}}=n_{\rm{fft}}/2$. The $n_{\rm{blk}}$ blocks are then Fourier
transformed in time and all Fourier realizations of the $l$-th frequency,
$\mathbf{q}^{(j)}_{l}$, are arranged in a matrix,

$\hat{\mathbf{Q}}_{l}=\bqty{\hat{\mathbf{q}}_{l}^{(1)},\hat{\mathbf{q}}_{l}^{(2)},\cdots,\hat{\mathbf{q}}_{l}^{(n_{\rm
blk})}}.$ (1)

The SPOD eigenvalues, ${\bf\it\Lambda}_{l}$, are obtained by solving the
following eigenvalue problem:

$\frac{1}{n_{\rm
blk}}\mathbf{\hat{Q}}_{l}^{*}\mathbf{W}\mathbf{\hat{Q}}_{l}{\bf\it\Psi}_{l}={\bf\it\Psi}_{l}{\bf\it\Lambda}_{l},$
(2)

where $\mathbf{W}$ is a positive-definite Hermitian matrix that accounts for
the component-wise and numerical quadrature weights and $(\cdot)^{*}$ denotes
the complex conjugate. SPOD modes at the $l$-th frequency are recovered as

${\bf\it\Phi}_{l}=\frac{1}{\sqrt{n_{\textrm{blk}}}}\mathbf{\hat{Q}}_{l}{\bf\it\Psi}_{l}{\bf\it\Lambda}_{l}^{-1/2}.$
(3)

The SPOD eigenvalues are denoted by
${\bf\it\Lambda}_{l}=\text{diag}(\lambda_{l}^{(1)},\lambda_{l}^{(2)},\cdots,\lambda_{l}^{(n_{\rm
blk})})$. By construction,
$\lambda_{l}^{(1)}\geq\lambda_{l}^{(2)}\geq\cdots\geq\lambda_{l}^{(n_{\rm
blk})}$ represent the energies of the corresponding SPOD modes that are given
by the column vectors of the matrix
${\bf\it\Phi}_{l}=[{\bf\it\phi}_{l}^{(1)},{\bf\it\phi}_{l}^{(2)},\cdots,{\bf\it\phi}_{l}^{(n_{\rm
blk})}]$. The SPOD mode ${\bf\it\phi}_{l}^{(j)}$ represents the $j$-th
dominant coherent flow structure at the $l$-th frequency. An useful property
of the SPOD modes is their orthogonality; the weighted inner product at each
frequency,
$\big{<}{\bf\it\phi}_{l}^{(i)},{\bf\it\phi}_{l}^{(j)}\big{>}=\pqty{{\bf\it\phi}_{l}^{(i)}}^{*}\mathbf{W}{\bf\it\phi}_{l}^{(j)}=\delta_{ij}$.

Here, we perform SPOD on various 2D streamwise planes ranging from $x/D$ = $1$
to $120$. Thus, $\mathbf{q}$ contains the three velocity components at the
discretized grid nodes on a streamwise-constant plane. These planes are
sampled at a spacing of $5D$ from $x/D=5$ to $x/D=100$, and five additional
planes are sampled at $x/D=1,2,3,110$, and $120$. The utilized time series has
$n_{t}=7200$ snapshots with a uniform separation of non-dimensional time of
$\Delta tU_{\infty}/D=0.072$ between two snapshots. Owing to the periodicity
in the azimuthal direction, the flow field is first decomposed into azimuthal
wavenumbers $m$,

$q(x,r,\theta,t)=\sum\limits_{m}\hat{q}_{m}(x,r,t),$ (4)

and then SPOD is applied on the data at each azimuthal wavenumber. The
spectral estimation parameters used here are $n_{\rm{fft}}=512$ and
$n_{\rm{ovlp}}=256$, resulting in $n_{\rm{blk}}=27$ SPOD modes for each St.
Each block used for the temporal FFT spans a time duration of $\Delta
T=36.91U_{\infty}/D$.

###  2.3 Reconstruction using convolution approach

The convolution strategy proposed by Nekkanti & Schmidt (2021) is employed for
low-dimensional reconstruction of the flow field. This involves computing the
expansion coefficients by convolving the SPOD modes over the data one snapshot
at a time,

$\mathbf{a}_{l}^{(i)}(t)=\int_{\Delta
T}\int_{\Omega}\quantity(\mathbf{\phi}_{l}^{(i)}(x))^{*}\textbf{W}(x)\textbf{q}(x,t+\tau)w(\tau)e^{-i2\pi
f_{l}\tau}\differential x\differential\tau.$ (5)

Here, $w(\tau)$ is the Hamming windowing function and $\Omega$ is the spatial
domain of interest. The data at time $t$ is then reconstructed as

$\textbf{q}(t)\approx\sum\limits_{l}\sum\limits_{i}a^{(i)}_{l}(t){\bf\it\phi}^{(i)}_{l}e^{-i2\pi
f_{l}t}.$ (6)

###  2.4 Bispectral mode decomposition

Bispectral mode decomposition (BMD) is a technique recently proposed by
Schmidt (2020), to characterize the coherent structures associated with
triadic interactions in statistically stationary flows. Here, we provide a
brief overview of the method. The reader is referred to Schmidt (2020) for
further details of the derivation and mathematical properties of the method.

BMD is an extension of classical bispectral analysis to multidimensional data.
The classical bispectrum is defined as the double Fourier transform of the
third moment of a time signal. For a time series, $y(t)$ with zero mean, the
bispectrum is

$S_{yyy}(f_{1},f_{2})=\int\int
R_{yyy}(\tau_{1},\tau_{2})e^{-i2\pi(f_{1}\tau_{1}+f_{2}\tau_{2})}d\tau_{1}d\tau_{2},$
(7)

where $R_{yyy}(\tau_{1},\tau_{2})=E[y(t)y(t-\tau_{1})y(t-\tau_{2})]$ is the
third moment of $y(t)$, and $E[\cdot]$ is the expectation operator. The
bispectrum is a signal processing tool for one-dimensional time series which
only measures the quadratic phase coupling at a single spatial point. In
contrast, BMD identifies the intensity of the triadic interactions over the
spatial domain of interest and extracts the corresponding spatially coherent
structures.

BMD maximizes the spatially integral of the point-wise bispectrum,

$b(f_{k},f_{l})=E\bqty{\int_{\Omega}\hat{\mathbf{q}}^{*}(x,f_{k})\circ\hat{\mathbf{q}}^{*}(x,f_{l})\circ\hat{\mathbf{q}}(x,f_{k}+f_{l})dx}.$
(8)

Here, $\hat{\mathbf{q}}$ is the temporal Fourier transform of $\mathbf{q}$
computed using the Welch approach (Welch, 1967) and $\circ$ denotes the
Hadamard (or element-wise) product.

Next, as in equation (1), all the Fourier realizations at $l$-th frequency are
arranged into the matrix, $\hat{\mathbf{Q}}_{l}$. The auto-bispectral matrix
is then computed as

$\mathbf{B}=\frac{1}{n_{\rm{blk}}}\hat{\mathbf{Q}}_{k\circ
l}^{H}\mathbf{W}\hat{\mathbf{Q}}_{k+l}\;,$ (9)

where, $\hat{\mathbf{Q}}_{k\circ
l}^{H}=\hat{\mathbf{Q}}_{k}^{*}\circ\hat{\mathbf{Q}}_{l}^{*}$.

To measure the interactions between different quantities, we construct the
cross-bispectral matrix

$\mathbf{B}_{c}=\frac{1}{n_{\rm{blk}}}\pqty{\hat{\mathbf{Q}}_{k}^{*}\circ\hat{\mathbf{R}}_{l}^{*}}\mathbf{W}\hat{\mathbf{S}}_{k+l}.$
(10)

In the present application, matrices $\mathbf{Q}$, $\mathbf{R}$, and
$\mathbf{S}$ comprise the time series of the field variables at the azimuthal
wavenumber triad, [$m_{1}$, $m_{2}$, $m_{3}$].

Owing to the non-Hermitian nature of the bispectral matrix, the optimal
expansion coefficients, $\mathbf{a}_{1}$ are obtained by maximising the
absolute value of the Rayleigh quotient of $\mathbf{B}$ (or $\mathbf{B}_{c}$)

$\mathbf{a}_{1}=\arg\max\limits_{\|\mathbf{a}\|=1}\bigg{|}\frac{\mathbf{a}^{*}\mathbf{B}\mathbf{a}}{\mathbf{a}^{*}\mathbf{a}}\bigg{|}.$
(11)

The complex mode bispectrum is then obtained as

$\lambda_{1}(f_{k},f_{l})=\bigg{|}\frac{\mathbf{a}_{1}^{*}\mathbf{B}\mathbf{a}_{1}}{\mathbf{a}_{1}^{*}\mathbf{a}_{1}}\bigg{|}.$
(12)

Finally, the leading-order bispectral modes and the cross-frequency fields are
recovered as

$\displaystyle{\bf\it\phi}_{k+l}^{(1)}$
$\displaystyle=\hat{\mathbf{Q}}_{k+l}\mathbf{a}_{1},\quad\text{and}$ (13)
$\displaystyle{\bf\it\phi}_{k\circ l}^{(1)}$
$\displaystyle=\hat{\mathbf{Q}}_{k\circ l}\mathbf{a}_{1},$ (14)

respectively. By construction, the bispectral modes and cross-frequency fields
have the same set of expansion coefficients. This explicitly ensures the
causal relation between the resonant frequency triad, ($f_{k}$, $f_{l}$,
$f_{k}+f_{l}$), where $\hat{\mathbf{Q}}_{k\circ l}$ is the cause and
$\hat{\mathbf{Q}}_{k+l}$ is the effect. The complex mode bispectrum,
$\lambda_{1}$, measures the intensity of the triadic interaction and the
bispectral mode, ${\bf\it\phi}_{k+l}$, represents the structures that results
from the nonlinear triadic interaction.

Similar to SPOD, we perform BMD on various 2D streamwise planes and use the
same spectral estimation parameters of $n_{\rm{fft}}=512$ and
$n_{\rm{ovlp}}=256$. Since our focus is on interactions of different azimuthal
wavenumbers, the cross-BMD method, which computes the cross-bispectral matrix
$\mathbf{B}_{c}$, is applied to the wake database. The specific interactions
among different $m$ will be presented and discussed in §5.

##  3 Flow structures

###  3.1 Streaky structures in the near and far wake

Nidhan et al. (2020) showed that the near and far field of the wake of a
circular disk is dominated by two distinct modes residing at (a)
$m=1,\mbox{\it St}=0.135$ and (b) $m=2,\mbox{\it St}\rightarrow 0$. While the
former mode is the vortex shedding mode in the wake of a circular disk (Berger
et al., 1990; Fuchs et al., 1979a; Cannon et al., 1993), the physical origin
of the latter mode remains unclear. Johansson & George (2006) hint that the
$m=2$ mode is linked to ‘very’ large-scale features that twist the mean flow
slowly. Motivated by the findings of Nidhan et al. (2020) and the discussion
in Johansson & George (2006), we investigate the streamwise manifestation of
the azimuthal modes $m=1$ and $m=2$. The main result is that, different from
the $m=1$ mode, the $m=2$ mode is associated with streamwise-aligned streaky
structures.

Figure 1: Azimuthally decomposed instantaneous fields of the streamwise
velocity ($u_{x}$) for ($a$) $m=1$ and ($b$) $m=2$ azimuthal modes.
Rectangular boxes in ($b$) show large-scale streaks in the $m=2$ field. Figure
2: Instantaneous streamwise velocity on $x/D-\theta$ planes at $r/D=1.25$
($a$,$b$) and $r/D=2.5$ ($c$,$d$). Panels $a$, $c$ show the full streamwise
velocity field and panels $b$, $d$ show the field with $m=1$ contribution
removed. Rectangular boxes in ($b,d$) emphasize the large-scale streaks in the
flow field.

Figure 1 shows the azimuthal modes $m=1$ and $m=2$ of an instantaneous flow
snapshot in the wake spanning downstream distance $0<x/D<100$, obtained using
a Fourier transform in the azimuthal direction $\theta$. In the $m=1$ mode
(figure 1$a$), a wavelength of $\lambda/D=1/\mbox{\it St}_{\mathrm{VS}}$
(where the vortex shedding frequency, $\mbox{\it St}_{\mathrm{VS}}\approx
0.13-0.14$) is evident throughout the domain. This observation is in agreement
with previous studies (Johansson et al., 2002; Johansson & George, 2006;
Nidhan et al., 2020) that report the existence of the vortex shedding mode at
significantly large downstream locations $\sim O(100D)$ from the disk.

The spatial structure of the $m=2$ mode (figure 1$b$) is quite different from
that of the $m=1$ mode. In the $m=2$ mode visualization, distinct elongated
structures are present throughout the domain. Notice in particular the
structures inside the dashed rectangular boxes. The streamwise extent of these
structures can be up to $\lambda_{x}/D\approx 25$ (see $70<x/D<95$),
significantly larger than the wavelength of the vortex shedding mode
$\lambda_{x}/D\approx 7$.

Figure 2($a$,$c$) show the instantaneous streamwise velocity field in the
near-intermediate and intermediate-far wake, respectively, of the disk on a
$x/D-\theta$ plane. The $x/D-\theta$ plane is constructed by unrolling the
cylindrical surface at a constant $r/D$. For the near-intermediate field
(figure 2$a$), the plane is located at $r/D=1.25$ while for the intermediate-
far field (figure 2$c$), $r/D=2.5$ is chosen. In both figures, vortex shedding
structures are evident, spaced at an approximate wavelength of
$\lambda_{x}/D\approx 7$. Upon removing the contribution of the vortex
shedding mode that lies in $m=1$ azimuthal wavenumber (Johansson et al., 2002;
Johansson & George, 2006; Nidhan et al., 2020), the streaks become readily
apparent in figure 2($b$,$d$). See again the structures contained inside
dashed rectangular boxes in figure 2($b$,$d$). These observations in figure 1
and 2 are strong initial indications of the presence of large-scale streaks in
turbulent wakes, similar to those recently reported in the numerical and
experimental datasets of turbulent jets (Pickering et al., 2020; Nogueira et
al., 2019). In what follows, the characteristics and robustness of these
elongated structures are quantified through various statistical and spectral
measures.

Figure 3: Turbulence intensity ($K^{1/2}/U_{\infty}$) as a function of
streamwise ($x$) and radial ($r$) directions. Magenta curve demarcates the
region where $K^{1/2}/U_{\infty}$ reduces to 0.12.

To this end, we invoke the Taylor’s hypothesis, converting time $t$ at a
location $x_{0}$ to pseudo-streamwise distance from $x_{0}$:
$x_{t}=U_{\mathrm{conv}}t$, where $U_{\mathrm{conv}}=U_{\infty}-U_{d}$ is the
convective velocity. Since the defect velocity in the wake $U_{d}\ll
U_{\infty}$ (where $U_{\infty}$ is the free-stream velocity),
$U_{\mathrm{conv}}$ is approximated by $U_{\infty}$. Taylor’s hypothesis
requires velocity fluctuation to be sufficiently small compared to
$U_{\mathrm{conv}}$. Figure 3 shows that this requirement is met since
turbulence intensity ($K^{1/2}/U_{\infty}$) drops below $12\%$ beyond $x/D=10$
for all radial locations and below $4\%$ at $x/D=30$. Previous work on
turbulent wakes has shown Taylor’s hypothesis to be valid in the wake when
$K^{1/2}/U_{\infty}$ drops below $\sim 10\%$ (Antonia & Mi, 1998; Kang &
Meneveau, 2002; Dairay et al., 2015; Obligado et al., 2016). Invoking Taylor’s
hypothesis, $\mbox{\it St}\approx U_{\infty}k_{x}$. In this study we will
particularly focus on $\mbox{\it St}\rightarrow 0$ (obtained from either
temporal spectra or SPOD at different $x/D$ locations) which, by Taylor’s
hypothesis, is equivalent to large-scale streaks with $k_{x}\rightarrow 0$
($k_{x}$ being the streamwise wavenumber) at those locations.

Figure 4: Streamwise velocity fluctuations $u^{\prime}_{x}$ on a
$x_{t}/D$–$\theta$ plane at $r/D\approx 0.8$: ($a,c,e$) $x_{0}/D=10$;
($b$,$d$,$f$) $x_{0}/D=20$. Top row ($a$,$b$) include all azimuthal
components, middle row ($c$,$d$) excludes $m=1$, and the bottom row ($e$,$f$)
includes solely $m=2$.

In figure 4, the near-to-intermediate wake behavior is shown at $x_{0}/D=10$
(left column) and $20$ (right column) through plots of streamwise velocity
fluctuation ($u^{\prime}_{x}$) at $r/D=0.8$ in the $x_{t}/D-\theta$ plane.
Time $t$ is transformed to the $x_{t}$ coordinate by application of Taylor’s
hypothesis, which is valid beyond $x/D\geq 9$ in the wake, as was shown
through figure 3 and the associated discussion.

Due to the strong signature of the vortex shedding mode in the near to
intermediate wake, alternate patches of inclined positive and negative
fluctuations separated by $\lambda_{D}\approx 1/\mbox{\it St}_{\mathrm{VS}}$
dominate the visualization in figure 4($a,b$). It is known a priori that these
structures are contained in the $m=1$ azimuthal mode. In order to assess
space-time coherence other than the vortex shedding mode, the streamwise
fluctuations are replotted in figure 4($c$,$d$) after removing the $m=1$
contribution. Once the $m=1$ contribution is removed, streamwise streaks
become evident at both locations. Furthermore, one can also observe that these
streaks appear to be primarily contained in the azimuthal mode $m=2$, i.e,
there are two structures over the azimuthal length of $2\pi$. Only the $m=2$
component is shown in figure 4 ($e$,$f$) and, on doing so, the elongated
streamwise streaks come into sharper focus. Building upon figure 1, figure 4
($c$,$d$,$e$,$f$) lend support to the spatiotemporal robustness of these
large-scale streaks in the turbulent wake of circular disk.

Figure 5: The $u^{\prime}_{x}$ field on a $x_{t}/D$–$\theta$ plane at
$r/D\approx 2.0$: ($a$,$b$) $x_{0}/D=40$; ($c$,$d$) $x_{0}/D=80$. In ($a$,$c$)
$m=1$ is removed and in ($b$,$d$) only $m=2$ is shown.

The downstream distance ($0<x/D<120$) spanned in the simulations of
Chongsiripinyo & Sarkar (2020) is very large, thus enabling the far field to
be probed too for the presence (or absence) of the streamwise streaks. Figure
5($a$,$c$) show the $x_{t}/D-\theta$ plots of $u^{\prime}_{x}$, with
contribution of $m=1$ mode excluded, in regions starting at $x_{0}/D=40$ and
$80$. The $x_{t}/D-\theta$ planes are located at $r/D=2$ for these far-wake
locations since the wake width grows with $x$. Similar to the near wake plot
in figure 4($c$,$d$), large-scale streaks elongated in the streamwise
direction are found to extend into the far wake as well. Once again, isolating
the $m=2$ component (figure 5$b$ and $d$) highlights the streaks.
Collectively, figures 4, and 5 demonstrate that the streaks span the entire
wake length and that the $m=2$ mode drives these streaks.

Figure 6: 2D power spectral density of the streamwise velocity fluctuation in
($k_{x}$, $m$) space: ($a$) $x/D=10$, $r/D\approx 0.8$; ($b$) $x/D=20$,
$r/D\approx 0.8$; ($c$) $x/D=40$, $r/D\approx 2.0$; ($d$) $x/D=80$,
$r/D\approx 2.0$. The $m=1$ wavenumber is removed so as to de-emphasize the
vortex shedding structure.

Figure 6 shows two-dimensional (2D) spectra in the streamwise wavelength
($k_{x}$) – azimuthal mode ($m$) space at four representative streamwise
locations $x/D=10,20,40,$ and $80$. Here, $k_{x}$ is the wavenumber of the
pseudo-streamwise direction $x_{t}$. The $m=1$ contribution is removed a
priori to emphasize the large-scale streaks. At all these four locations,
these streaky structures correspond to $k_{x}\rightarrow 0$ and are found to
reside in the $m=2$ azimuthal mode. Note that the $k_{x}=0,\mbox{\it St}=0$
should be interpreted as $k_{x},\mbox{\it St}\rightarrow 0$ as the length of
the time series is not sufficient to resolve the large time-scale of streaks.
In appendix A, we vary the spectral estimation parameter $n_{\rm fft}$ to
resolve the lower frequencies and identify the frequency associated with
streaks in this limit to be around $\mbox{\it St}\approx 0.006$.

###  3.2 Evidence of lift-up mechanism in the wake

Figure 7: Conditionally-averaged streamwise vorticity at $x/D=10$ (left
column), $40$ (middle column) and $80$ (right column). Top row ($a,b,c$) shows
$\langle\omega^{c1}_{x}\rangle$, conditioned using $u^{\prime}_{x}$ and bottom
row ($d,e,f$) shows $\langle\omega^{c2}_{x}\rangle$, conditioned using
$u^{\prime}_{x}u^{\prime}_{r}$. Green dot shows the location of conditioning
point – $(a,d)$ $r/D\approx 0.8,\theta\approx 0^{\circ}$ and $(b,c,e,f)$
$r/D\approx 2,\theta\approx 0^{\circ}$. Radial domain extends until $r/D=2$ at
$x/D=10$ and until $r/D=5$ at $x/D=40,80$. Positive vorticity (red) is out of
the plane and negative vorticity (blue) is into the plane.

Previous work in turbulent free shear flows and wall-bounded flows often
attribute the formation of streaks in the velocity to the lift-up mechanism
(Ellingsen & Palm, 1975; Landahl, 1975). The lift-up mechanism, by sweeping
fluid from high-speed regions to low-speed regions and vice versa, leads to
the formation of high-speed and low-speed streaks, respectively. Brandt (2014)
provides a comprehensive review of the lift-up mechanism and its crucial role
in various fundamental phenomena, e.g., subcritical transition in shear flows,
self-sustaining cycle in wall bounded flows, and disturbance growth in complex
flows.

To investigate the presence of the lift-up mechanism in the wake, we plot
conditional averages of streamwise vorticity fluctuations ($\omega_{x}$) at
three representative locations in the flow – the planes, $x/D=10,40$ and $80$
in figure 7. The top row shows a conditional average,
$\langle\omega^{c1}_{x}\rangle$, designed to extract the structure of the
streamwise vorticity on a constant-$x$ plane during times of large streamwise
velocity fluctuations. Specifically, the condition is that

$u^{\prime}_{x}\geq c(u^{\prime}_{x})^{\mathrm{rms}},$ (15)

at a specified point P on that plane, and $\langle\omega^{c1}_{x}\rangle$ is
the temporal average of all $\omega_{x}(t)$ that satisfies this condition. The
conditional point P (shown as green dots in figure 7) is chosen to lie at
$\theta=0$ and radial locations of $r/D=0.8$ and $2$ for $x/D=10$ and
$x/D=40,80$, respectively. Owing to rotational invariance of statistics for an
axisymmetric wake, the condition is applied to a new point $\rm{P}_{1}$ at the
same $r/D$ but a different value of $\theta$ and the new
$\langle\omega^{c1}_{x}\rangle$ field, after a rotation to bring $\rm{P}_{1}$
to P, is included in the conditional average. Since $N_{\theta}=256$ points is
used for discretization, the ensemble used for the conditional average is
significantly expanded by exploiting rotational invariance of statistics. The
bottom row of figure 7 shows $\langle\omega^{c2}_{x}\rangle$ computed using a
different condition at point P,

$-u^{\prime}_{x}u^{\prime}_{r}\geq
c(-u^{\prime}_{x}u^{\prime}_{r})^{\mathrm{rms}}\;.$ (16)

This condition is designed to identify the structure of streamwise vorticity
at times of significant Reynolds shear stress at point P. The results exhibit
moderate sensitivity to $c\in(0,1]$ as reported in appendix B. Hence, $c$ is
set to $0.5$ as a compromise between identification of intense events and
retention of sufficient snapshots for conditional averaging.
$(u^{\prime}_{x})^{\mathrm{rms}}$ and
$(-u^{\prime}_{x}u^{\prime}_{r})^{\mathrm{rms}}$ are the r.m.s. values of the
streamwise velocity fluctuations and the r.m.s. values of the streamwise-
radial fluctuations correlation at the conditioning points, respectively.

The conditionally averaged field based on equation (15) captures the structure
of the streamwise vorticity field during events of intense positive
$u^{\prime}_{x}$. In figure 7($a$), two rolls of streamwise vorticity are
observed in the conditionally averaged field: negative on the top and positive
at the bottom of the conditioning point, respectively. These streamwise vortex
rolls push the high-speed fluid in the outer wake to the low-speed region in
the inner wake around the conditioning points (green dot), leading to
$u^{\prime}_{x}>0$. When the averaging procedure is conditioned on negative
streamwise velocity fluctuations, i.e.,
$u^{\prime}_{x}\leq-c(u^{\prime}_{x})^{\mathrm{rms}}$, the signs of the vortex
rolls in figure 7 are interchanged, as expected (figure not shown). At
$x/D=40$ and $80$ (figure 7$b,c$), two additional vortical structures are
observed in the $\theta=[90^{\circ},270^{\circ}]$ region. However, around the
conditioning point the spatial organization of vorticity remains qualitatively
similar. The size of these vortex rolls increase with $x/D$, consistent with
the radial spread of wake.

The conditionally averaged field based on equation (16) captures the vorticity
field corresponding to intense positive $-u^{\prime}_{x}u^{\prime}_{r}$
values. In a turbulent wake, $-u^{\prime}_{x}u^{\prime}_{r}$ is predominantly
positive such that the dominant production term in the wake,
$P_{xr}=\langle-u^{\prime}_{x}u^{\prime}_{r}\rangle\partial U/\partial r>0$,
acts to transfer energy from the mean flow to turbulence. Now turning to
equation (16), positive $-u^{\prime}_{x}u^{\prime}_{r}$ can result from two
scenarios: (a) ($u^{\prime}_{r}>0$, $u^{\prime}_{x}<0$), i.e., ‘ejection’ of
low-speed fluid from the inner wake to the outer wake and (b)
($u^{\prime}_{r}<0$, $u^{\prime}_{x}>0$), i.e., ‘sweep’ of high-speed fluid
from outer wake to the inner wake. Both of the above-mentioned scenarios are
consistent with the lift-up mechanism. If ejection and sweep events were
equally probable, $\langle\omega^{c2}_{x}\rangle\approx 0$ due to the opposite
spatial distribution of vortices during ejection and sweep events. However,
$\langle\omega_{x}^{c2}\rangle$ obtained from $-u^{\prime}_{x}u^{\prime}_{r}$
based conditioning (bottom row of figure 7) shows that the positive and
negative vortices are spatially organized such that the flow induced by these
vortices at the conditioning point is outward ($u^{\prime}_{r}>0$), pushing
low-speed fluid from the inner wake to the outer wake. In short,
$\langle\omega^{c2}_{x}\rangle$ fields in the bottom row of figure 7
correspond to the ejection events at the conditioning point. A similar spatial
organization of $\langle\omega^{c2}_{x}\rangle$ is observed across the wake
cross-section when the radial location of the conditioning point is varied
(plots not shown for brevity). This observation establishes that ejections are
the dominant contributors to intense positive $-u^{\prime}_{x}u^{\prime}_{r}$,
as opposed to sweep events, and therefore, ejections are more instrumental in
the energy transfer from mean to turbulence. Previous studies (Wallace, 2016;
Kline et al., 1967; Corino & Brodkey, 1969) of the turbulent boundary layer
have also reported that ejection events are the primary contributors to
Reynolds shear stress.

Figure 7 has two important implications. First, the top row demonstrates
strong correlation between intense $u^{\prime}_{x}$ fluctuations and distinct
streamwise vortical structures, indicating that the lift-up mechanism is
active in the turbulent wake, both in the near field as well as the far field.
Second, the conditionally averaged fields obtained using
$u^{\prime}_{x}u^{\prime}_{r}$ inform us that the lift-up mechanism
corresponding to the ejection of low-speed fluid from the inner wake to the
outer wake is more dominant than the sweep of high-speed fluid from the outer
wake to the inner wake. To the best of authors’ knowledge, both these
observations constitute the first numerical evidence in the near and far field
of a canonical bluff-body turbulent wake of (a) the lift-up mechanism and (b)
the dominance of ejection events .

##  4 SPOD analysis of streaks in the wake

§3 reveals the presence of large-scale streaks and also that the lift-up
mechanism is active in the wake. Furthermore, the $m=2$ azimuthal wavenumber
visually appears to be the dominant streak-containing mode. In this section,
SPOD is employed to quantify the energetics and educe the dominant structures
of the dominant features at $\mbox{\it St}\rightarrow 0$ in the wake. We
particularly focus on the $m=2$ mode, providing further evidence that these
modes exhibit properties of streaks and are formed due to the lift-up
mechanism. Direct comparison with the VS mode ($m=1$, $St=0.135$) is provided
as appropriate to differentiate the role of streaks from that of the VS mode.

###  4.1 Energetics of streaky structures using SPOD analysis

Figure 8: Percentage of energy contained in the leading SPOD modes,
$\lambda^{(1)}$, as a function of frequency St and streamwise location $x/D$
at different azimuthal wavenumbers: ($a$) $m=0$; ($b$) $m=1$; ($c$) $m=2$;
($d$) $m=3$; ($e$) $m=4$; ($f$) $m=5$. The leading SPOD eigenvalue,
$\lambda^{(1)}(St,x/D)$, is normalized with the total TKE $E_{k}^{T}(x/D)$ at
the corresponding streamwise location to calculate the percentage
contribution.

SPOD is performed at different streamwise locations ($x/D$) in $1\leq x/D\leq
120$. By definition, the leading SPOD modes at a given $x/D$ represents the
most energetic coherent structures at the associated frequency ($St$) and
azimuthal wave number ($m$). Figure 8 shows the percentage of energy in the
leading SPOD modes ($\lambda^{(1)}$) as a function of frequency and streamwise
distance for the first six azimuthal wavenumbers $m=0$ to $m=5$. The
percentage of energy at each streamwise location is obtained by normalizing
the leading eigenvalue with the total turbulent kinetic energy,
$E^{T}_{k}(x/D)$ at the corresponding location. Overall, the most significant
contributors to the TKE are the vortex shedding mode ($m=1$, $\mbox{\it
St}=0.135$) and the mode corresponding to streaks ($m=2$, $\mbox{\it
St}\rightarrow 0$), as reported in Nidhan et al. (2020). The leading vortex
shedding SPOD mode contains about 10% energy in the near wake region
($5\lesssim x/D\lesssim 15$) and decreases thereafter. The leading SPOD mode
corresponding to the streaks in the $m=2$ mode contains approximately 3%
energy from $x/D=10$ onward. The axisymmetric component ($m=0$) exhibits a
peak at $\mbox{\it St}\rightarrow 0.054$ at $x/D=1$ and a much smaller peak at
$\mbox{\it St}\approx 0.19$ between $10\lesssim x\lesssim 120$. The former is
associated with the pumping of the recirculation bubble (Berger et al., 1990),
whereas the latter was observed in previous studies (see figure 12 in Berger
et al. (1990) and figure 7 in Fuchs et al. (1979b)) but was not investigated
further. The $m=0$ mode is not the focus of this study.

The energy, $\lambda^{(1)}$, contained in the higher azimuthal wavenumbers
($m=3-5$) is shown in figures 8($d$-$f$), respectively. Although
$\lambda^{(1)}$ is smaller than at $m=1$ or $m=2$, the higher modes also
exhibit temporal structure. The $m=3$ component shows energy concentration at
the vortex shedding frequency $\mbox{\it St}=0.135$ for $15\lesssim
x/D\lesssim 70$ and at $\mbox{\it St}\rightarrow 0$ for $x/D\gtrsim 50$. For
$m=4$, energy is concentrated near $\mbox{\it St}\rightarrow 0$ for
$x/D\gtrsim 60$. For the $m=5$ component, traces of the vortex shedding mode
and streaks ($\mbox{\it St}\rightarrow 0$) are observed at the streamwise
locations $x/D\gtrsim 60$ and $x/D\gtrsim 80$, respectively. Figure 8
indicates that the peaks at the vortex shedding frequency are present only at
the odd azimuthal wavenumbers ($m=1,3,5$), whereas the peaks corresponding to
the large-scale streaks, i.e., $\mbox{\it St}\rightarrow 0$, can be found at
both the odd and even $m$. It is also interesting to note that, for higher
$m$, both the vortex shedding modes and streaks do not appear until larger
vales of $x/D$. This suggests nonlinear interactions among different
frequencies and azimuthal wavenumbers as the wake evolves, as will be
elaborated in §5.

Figure 9: ($a$) Contribution of the leading SPOD modes $\lambda^{(1)}$ at
different azimuthal wavenumbers to $\mbox{\it St}\rightarrow 0$. ($b$)
Percentage energy contained in the frequencies $\mbox{\it St}=0.135$ and
$\mbox{\it St}\rightarrow 0$.

The $\mbox{\it St}\rightarrow 0$ streaks are dominated by the $m=2$ azimuthal
wavenumber as demonstrated by figure 9($a$), which shows the contribution of
different $m$ at $\mbox{\it St}\rightarrow 0$. The leading eigenvalues of each
azimuthal wavenumber is normalized by the total energy at $\mbox{\it
St}\rightarrow 0$, i.e., $\sum_{m}\sum_{i}\lambda^{(i)}(St\rightarrow 0,m)$.
Streaks are azimuthally non-uniform structures and are not present in the
axisymmetric $m=0$ component. Hence, we focus on $m\geq 1$. The azimuthal
wavenumber $m=2$ is energetically dominant at the $\mbox{\it St}\rightarrow 0$
frequency, containing about $40-50\%$ of the total energy at $\mbox{\it
St}\rightarrow 0$. The sub-optimal wavenumber is $m=3$ for $5\leq x\leq 80$
and switches between $m=3$ and $m=4$, thereafter. However, the difference in
energy between the $m=2$ and $m=3$ wavenumbers is always large, $>$$30\%$.
This dominance of the $m=2$ wavenumber at $\mbox{\it St}\rightarrow 0$ is also
consistent with the visualizations of the streamwise velocity fluctuations in
figure 4 and figure 5 where one can even see the presence of $m=2$ with the
naked eye.

The energetic contribution of the two dominant frequencies, $\mbox{\it
St}=0.135$ (vortex shedding) and $\mbox{\it St}\rightarrow 0$ (streaks), is
compared by computing the percentage of total energy (across all $m$ and $i$)
at a given frequency St as

$\xi(\mbox{\it
St};x/D)=\frac{\sum\limits_{m}\sum\limits_{i}\lambda^{(i)}(m,\mbox{\it
St};x/D)}{E^{T}_{k}(x/D)}\times 100.$ (17)

Figure 9($b$) shows that the vortex shedding frequency is more dominant in the
region $x\leq 70$, whereas the zeroth frequency dominates for $x\geq 70$. This
implies that although the streaks are present throughout, they are
energetically more prominent in the far wake region. It is interesting to note
that beyond $x/D\approx 65$, the defect velocity decay rate changes from
$x^{-1}$ to $x^{-2/3}$ in the wake (Chongsiripinyo & Sarkar, 2020). For
comparison, the most dominant component at the zeroth frequency, i.e., $m=2$
is also shown in figure 9($b$), which exhibits a similar trend of increasing
prominence in the downstream direction.

Figure 10: Component-wise instantaneous energy reconstructed from the $m=2$
streaky structures in the top row ($a$-$c$) and vortex shedding structures
($d$-$f$) in the bottom row. The plane integral of the energy is shown at
various streamwise locations: ($a,d$) $x/D=10$, ($b,e$) $x/D=40$ and ($c,f$)
$x/D=80$. Flow reconstruction uses the leading five SPOD modes.

Having diagnosed the streaky structure ($m=2,\mbox{\it St}=0$) and the vortex
shedding structure ($m=1,\mbox{\it St}=0.135$) using SPOD, we shift focus to
their imprint on the flow in physical space by reconstructing the flow field
using their leading five SPOD modes. The $m=2$ wavenumber is selected because
it is energetically dominant at $\mbox{\it St}\rightarrow 0$. The
reconstruction is performed using the convolution strategy described in
section §2. Figure 10($a$,$b$,$c$) shows the instantaneous energy in the three
fluctuation components, $u^{\prime}_{x}$, $u^{\prime}_{r}$, and
$u^{\prime}_{\theta}$ at $x/D=10$, 40, and 80, after reconstruction with the
streaky-structure SPOD modes. The energy of $u^{\prime}_{x}$ is significantly
higher than that of $u^{\prime}_{r}$ and $u^{\prime}_{\theta}$. The dominance
of the streamwise component over its radial and azimuthal counterparts is one
salient feature of streaks. The finding of $u^{\prime}_{x}$ dominance is in
agreement with those of Boronin et al. (2013, see figure 11) and Pickering et
al. (2020, see figure 10) in the context of round jets, indicating similarity
in streak formation between jets and wakes. The lift-up mechanism, in which
the streamwise vortices lift up and push down the low-speed and high-speed
fluid, respectively, is responsible for elongated streaks accompanied by an
amplification of $u^{\prime}_{x}$. The analog of panels in figure 10($a$-$c$)
is shown for the VS mode in panels of figure 10($d$-$f$). Here,
$u^{\prime}_{x}$ and $u^{\prime}_{r}$ components have comparable energy,
showing a fundamental differences between the vortex shedding mode and the
streaky-structures mode as to how each mode contributes to velocity
fluctuations in the wake.

Figure 11: TKE and production due to shear at different downstream locations
for the flow fields reconstructed from the streaky-structures mode in the top
row ($a$-$c$) and VS mode in the bottom row ($d$-$f$). Left column ($a,d$)
corresponds to $x/D=10$, middle column ($b,e$) to $x/D=40$ and right column
($c,f$) to $x/D=80$. Flow reconstruction uses the leading five SPOD modes.

Figure 11 shows the TKE ($K=\big{<}u^{\prime}_{i}u^{\prime}_{i}\big{>}/2$) and
its shear production
($P_{xr}=-\big{<}u^{\prime}_{x}u^{\prime}_{r}\big{>}\partial U/\partial r$)
corresponding to the large-scale streaks and vortex shedding structures at
$x/D=10$, $40$ and $80$. As in figure 10, TKE and $P_{xr}$ are computed from
the leading five SPOD modes. Figure 11($a$-$c$) show that the TKE and the
production peak at the similar radial location for $m=2$, $St\rightarrow 0$.
This is not the case for $m=1$, $\mbox{\it St}=0.135$ (figure 11$b$-$f$) where
the peak TKE occurs close to the centerline while the production peaks away
from the centerline. This difference in the locations of peak $K$ and $P_{xr}$
indicates that turbulent transport plays an important role in distributing the
TKE in the vortex shedding mode, similar to its importance in the full TKE
budget of an axisymmetric wake (Uberoi & Freymuth, 1970). The difference in
radial locations of peak $K$ and $P_{xr}$, as demonstrated in figure 11, is
another crucial distinction between the large-scale streaky mode and the
vortex shedding mode. The presence of streaks is associated with high TKE
around the region of high production/mean shear indicating their important
role in the energy transfer from mean to fluctuation velocity in the turbulent
wake, similar to other shear flows (Gualtieri et al., 2002; Brandt, 2007;
Jiménez-González & Brancher, 2017). As a result, the TKE achieves its global
maximum around the same location as that of $P_{xr}$ for streaks.

###  4.2 Lift-up mechanism through the lens of SPOD analysis

Figure 12: Leading SPOD mode of $m=2$ and $St\rightarrow 0$: ($a$) $x/D=10$;
($b$) $x/D=40$; ($c$) $x/D=80$. The false colors represent the real part of
the streamwise velocity component
$\mathrm{Re}\bqty{\phi^{(1)}_{u_{x}}(St\rightarrow 0)}$ and the grey and
green-dotted lines correspond to positive and negative streamwise vorticity
$\mathrm{Re}\bqty{\phi^{(1)}_{\omega_{x}}(St\rightarrow 0)}$, respectively.

§4.1 demonstrated that the structures associated with $\mbox{\it
St}\rightarrow 0$ exhibit the characteristics of streaks and their
significance increases from the near to far wake, with the azimuthal
wavenumber $m=2$ being the most significant, energetically, to the streaks.
Figure 12 shows the leading SPOD mode of the streamwise velocity fluctuation
($u^{\prime}_{x}$) corresponding to $m=2$ and $\mbox{\it St}\rightarrow 0$ at
three streamwise locations $x/D=10$, $40$, and $80$. Overlaid on the
$u^{\prime}_{x}$ contour is the streamwise vorticity ($\omega^{\prime}_{x}$)
of the corresponding mode. Both $u^{\prime}_{x}$ and $\omega^{\prime}_{x}$ are
characterized by four lobes of alternate sign, the size of which increase
monotonically with $x/D$. Importantly, the set of $\omega^{\prime}_{x}$ lobes
is shifted with respect to the $u^{\prime}_{x}$ lobes by a clockwise rotation
of approximately $45^{\circ}$. As a result of the shift, the maximum of
$u^{\prime}_{x}$ in the mode appears at the location where the vortices bring
in high-speed fluid from the outer to the inner wake, and vice versa. This
observation further confirms the presence of lift-up mechanism in the wake.

Figure 13: ($a$) Normalized mean defect velocity ($U_{d}=U_{\infty}-U$)
profiles. ($b$) Normalized leading SPOD mode of the streamwise velocity
($u_{x}$) at $m=2$ and $\mbox{\it St}\rightarrow 0$. Dashed lines in ($a$) and
dotted-dashed lines in ($b$) correspond to the radial location of the maximum
of SPOD mode and the maximum of mean shear ($\partial U/\partial r$),
respectively.

Figure 13($a$) shows the normalized radial profiles of mean defect velocity
($U_{d}$) at $x/D=10,40$ and $80$. Figure 13($b$) shows the radial profile of
the normalized leading SPOD mode’s streamwise component for $m=2,\mbox{\it
St}\rightarrow 0$, at the same streamwise locations as in figure 13($a$). As
the wake develops in the $x$ direction, the location of mode maximum shifts
away from the centerline. A visual comparison shows that the location of
amplitude maximum (dashed lines in figure 13$b$) of the dominating streak-
containing mode lies in close proximity to the location of the maximum mean
shear (dotted-dashed lines in figure 13$a$). The large radial gradient of the
streamwise velocity induces a positive mean vorticity, lifting up the low-
speed fluid from the inner wake to form streaks. Hence, an extremum in
$u^{\prime}_{x}$ appears in the SPOD mode. The seminal work of Ellingsen &
Palm (1975) demonstrates that , for linearized disturbances in an inviscid
flow, $\partial u_{x}/\partial t\;\propto\;-u_{r}\partial U(r)/\partial r$,
and the lift-up mechanism is most active in the region closest to the largest
mean shear. So is the case in the present turbulent wake.

Figure 14: Leading SPOD mode at $x/D=40$ and $St\rightarrow 0$ for higher
azimuthal wavenumbers: ($a$) $m=3$; ($b$) $m=4$; ($c$) $m=5$. The false colors
represent $\mathrm{Re}\bqty{\phi^{(1)}_{u_{x}}(St\rightarrow 0)}$ and the grey
and green-dotted lines correspond to the positive and negative values of
streamwise vorticity $\mathrm{Re}\bqty{\phi^{(1)}_{\omega_{x}}(St\rightarrow
0)}$.

The lift-up mechanism is also active at higher azimuthal wavenumbers as
demonstrated by figure 14, which shows the leading SPOD modes at $x/D=40$ and
frequency $\mbox{\it St}\rightarrow 0$ for the higher modes: $m=3$, $4$ and
$5$. Similar to figure 12, positive and negative streamwise velocity contours
are encompassed by counter-rotating vortices that move the fluid from the
fast- to slow-speed regions and vice-versa. The radial spread of the
streamwise velocity lobes increases with $m$ and the number of lobes scales as
$2m$. Also, the vortices are shifted by $30^{\circ}$, $22.5^{\circ}$, and
$18^{\circ}$, for $m=3$, 4, and 5, respectively. In other words, the set of
streamwise velocity lobes for wavenumber $m$ is shifted by an angle of
$\pi/2m$ radians with respect to the streamwise vortices.

##  5 BMD analysis at select locations in the wake

Figure 15: SPOD spectra at $x/D=10$: ($a$) $m=1$; ($b$) $m=2$. Dashed red and
blue lines in ($a$) and ($b$) correspond to $\mbox{\it St}=0.135$ and
$\mbox{\it St}=0.27$, respectively.

Figure 15 shows the SPOD spectra for the azimuthal wavenumbers, $m=1$ and
$m=2$, at $x/D=10$. Both SPOD spectra exhibit a large difference between the
first and second eigenvalues for $\mbox{\it St}\lesssim 0.5$, thus
demonstrating a _low-rank_ behaviour. The leading eigenvalue of the $m=1$
azimuthal mode peaks at the vortex shedding (VS) frequency, $\mbox{\it
St}=0.135$. On the other hand, the leading eigenvalue of the $m=2$ azimuthal
mode exhibits a global peak at $\mbox{\it St}=0$, and an additional local peak
at $\mbox{\it St}=0.27$ (blue dashed line in figure 15). Furthermore, Nidhan
et al. (2020, see their figure 20) find that the VS mode gains prominence at
$x/D\approx 1$ while the peak corresponding to $m=2,\mbox{\it St}\rightarrow
0$ appears further downstream at $x/D\approx 5$. These observations
collectively point towards different sets of triadic interactions involving
the VS mode. For example, $m=1,\mbox{\it St}=0.135$ can interact with
$m=1,\mbox{\it St}=-0.135$ to give rise to $m=2,\mbox{\it St}\rightarrow 0$
that appears further downstream. Similarly, the self-interaction of
$m=1,\mbox{\it St}=0.135$ can generate $m=2,\mbox{\it St}=0.27$ (local peak
denoted by dashed line in figure 15b). In what follows, we quantitatively
demonstrate that the presence of these triadic interactions at select $x/D$
locations using bispectral mode decomposition (BMD) (Schmidt, 2020).

Figure 16: cross-BMD spectra for the azimuthal triad
[$m_{1}$,$m_{2}$,$m_{3}=m_{1}+m_{2}$]=[1,1,2]: ($a$) $x/D=10$, ($b$) $x/D=40$,
and ($c$) $x/D=80$. The black dotted line denotes $St_{1}+St_{2}=St_{3}=0$.

Figure 16 shows the cross-mode bispectra for the azimuthal wavenumber triad,
[$m_{1}$,$m_{2}$,$m_{3}$] = [1,1,2], at three axial locations, $x/D=10$, 40,
and 80. These spectra provide a measure of the interaction among the three
azimuthal components. The abscissa ($St_{1}$) and ordinate ($St_{2}$) of the
BMD spectra correspond to the frequencies of the $m_{1}$ mode and $m_{2}$
mode, respectively. Here, $m_{1}=1$, $m_{2}=1$ and $m_{3}=2$. The high-
intensity regions in the spectra represent the energetically dominant triads.
In figure 16, the mode bispectra are symmetric about the diagonal
$St_{1}=St_{2}$. At $x/D=10$ (figure 16($a$)), the most dominant triad is
($0.135,-0.135,0.0$) and other significant triads are ($0.135,0.135,0.27$) and
($-0.135,-0.135,-0.27$). This observation confirms the presence of triadic
interactions hypothesized in the context of figure 15 and emphasizes that the
strongest triadic nonlinear interaction is between the VS mode ($m=1$,
$\mbox{\it St}=0.135$), its conjugate ($m=1$, $\mbox{\it St}=-0.135$), and
streaks ($m=2$, $\mbox{\it St}\rightarrow 0$). A similar observation is made
from the mode bispectra at $x/D=40$. At $x/D=80$, the most dominant triad
occurs at ($0.09,-0.09,0.0$) and along the $\mbox{\it St}_{1}=-\mbox{\it
St}_{2}$ line, but a local maximum (marked by a circle in magenta) is still
present at ($0.135,-0.135,0$), which indicates that the triadic interaction
between the VS mode and streak is still active, albeit subdominant. From
figure 16, we infer that the triadic interaction between [$1,1,2$] at
($0.135,-0.135,0.0$) is prominent in the near wake and its intensity
progressively decreases downstream. It is important to note that when all
interactions with $[m_{1},\mbox{\it St}_{1},m_{2},\mbox{\it St}_{2}]$ such
that $\mbox{\it St}_{1}+\mbox{\it St}_{2}=0$ were quantified (not shown here
for brevity), that between $m_{1}=m_{2}=1$ and $\mbox{\it St}_{1,2}=\pm 0.135$
was found to be the strongest.

Figure 17: cross-BMD modes for the triad, $[1,1,2]$ in azimuth and
$(0.135,-0.135,0.0)$ in frequency, at three streamwise locations: ($a$)
$x/D=10$; ($b$) $x/D=40$; ($c$) $x/D=80$. The false colors represent the real
part of streamwise velocity component.The absolute value of the BMD mode and
the corresponding leading SPOD mode are compared in ($d$)-($f$).

Next, we visualize the structures associated with the triadic interaction of
the vortex shedding mode, its conjugate, and streaks in figure 17. The real
part of the streamwise velocity component of the corresponding cross-
bispectral modes is shown at $x/D=10$, 40, and 80. Similar to figure 12, the
radial extent of these modes increases downstream and they exhibit four lobes
of alternate signs. The difference between the cross-BMD and the SPOD modes is
that the cross-BMD modes are tilted by an angle. For a more quantitative
comparison, figure 17 ($d$-$f$) shows the magnitude of the modes normalized by
their maximum value. The corresponding curves are coincident, indicating that
the spatial structures generated by the triadic interactions are also the most
energetic coherent structures. This further confirms that the $m=2,\mbox{\it
St}=0$ mode is indeed generated through the interaction of $\mbox{\it St}=\pm
0.135$ at the $m=1$ azimuthal mode.

##  6 Discussions and Conclusions

Streaks, which are coherent elongated regions of streamwise velocity, have
been found in a variety of turbulent shear flows. However, they have not
received attention in turbulent wakes motivating the present examination of a
LES dataset of flow past a disk at $\Rey=50,000$ (Chongsiripinyo & Sarkar,
2020). Visualizations and spectral proper orthogonal decomposition (SPOD) are
employed and they reveal the presence of streaks from the near wake to the
outflow at $x/D\approx 120$. Until now, most of the wake literature has
understandably focused on the vortex shedding (VS) structure ($m=1$,
$\mbox{\it St}=0.135$ for the circular disk), since it the energetically
dominant coherent structure in the near and intermediate wake. Upon removing
the contribution of the $m=1$ azimuthal wavenumber a priori in visualizations,
the streaks become evident, even in the near and intermediate wake. Moreover,
in the far wake ($x/D\geq 70$), it is the streaks that become the
energetically dominant coherent structure. To the best of our knowledge, this
is the first study that reports the existence of streaks in turbulent wakes.
These results re-emphasize that mean shear, not a wall boundary condition, is
a necessary condition for the existence of streaks (Jiménez & Pinelli, 1999;
Mizuno & Jiménez, 2013; Nogueira et al., 2019).

Streaks differ from VS structures in three key ways: ($i$) they exhibit a much
larger wavelength and time scale; ($ii$) VS structures are tilted at an obtuse
angle to the downstream direction whereas streaks are almost parallel; ($iii$)
the streamwise velocity ($u^{\prime}_{x}$) significantly exceeds the other two
velocity components ($u^{\prime}_{r},u^{\prime}_{\theta}$) in magnitude for
streaks whereas $u^{\prime}_{x}$ and $u^{\prime}_{r}$ are comparable for VS
structures. The streaky structures are associated with a frequency of
$\mbox{\it St}\rightarrow 0$ and hence, by Taylor’s hypothesis (validated here
for the wake) to a wavenumber of $k_{x}\rightarrow 0$. While streaky
structures are observed for all non-zero azimuthal wavenumbers, $m=2$
dominates in the near-to-far wake. In particular, SPOD analysis reveals that
$m=2$ contains about 55% (far wake) to 40% (far wake) of the total energy of
streaks. This is in contrast to turbulent jets, where Pickering et al. (2020)
show that the dominant azimuthal wavenumber ($m_{dom}$) at $\mbox{\it
St}\rightarrow 0$ varies as $m_{dom}\sim 1/x$ implying that higher $m$ and not
$m=2$ would be dominant near the jet nozzle. It is worth noting that only two
studies (Johansson & George, 2006; Nidhan et al., 2020) have reported the
importance of $m=2$, $\mbox{\it St}\rightarrow 0$ in turbulent wakes, however
they do not link this mode to streaks.

We find that the lift-up mechanism is active in turbulent wakes, similar to
wall-bounded shear flows (Abreu et al., 2020) and turbulent jets (Lasagna et
al., 2021; Nogueira et al., 2019). Conditional averaging and SPOD analysis
clearly demonstrate that streamwise vortices lift up low-speed fluid from the
wake’s core and push down high-speed fluid from the outer wake. It is also
observed that the lift-up mechanism is spatially most active in the vicinity
of the largest mean shear and TKE production, indicating that energy is
directly transferred from the mean flow to the velocity fluctuations in the
streaks. The lift-up process triggered by the streamwise vortices shows a
similar energy transfer mechanism in turbulent pipe flow (Hellström et al.,
2016) and homogeneous shear flow (Gualtieri et al., 2002; Brandt, 2014).

The lift-up mechanism results in the formation of both low-speed and high-
speed streaks. These low- and high-speed streaks exhibit large negative values
of Reynolds shear stress. Conditional averaging of streamwise vorticity
fluctuations, performed based on peak negative Reynolds shear stress, show
that the ejection of low-speed fluid from the wake’s core is more dominant
than the sweep of the high-speed fluid from the outer wake. The boundary
layers also exhibit a similar phenomenon where ejections are a greater
contributor to Reynolds shear stress than sweeps (Kline et al., 1967; Lu &
Willmarth, 1973).

Beyond identification of streaks, we also explore the role of nonlinear
interactions in the context of wake streaks. Specifically, bispectral mode
decomposition (BMD) is used to investigate the nonlinear interactions between
the $m=1$, $\mbox{\it St}=\pm 0.135$ VS mode and the $m=2$, $\mbox{\it
St}\rightarrow 0$ streak mode. The $m=1$, $\mbox{\it St}=\pm 0.135$ vortices
are found to interact and generate the $m=2$, $\mbox{\it St}\rightarrow 0$
vortices. These streamwise vortices of the $m=2$, $\mbox{\it St}\rightarrow 0$
mode then lift up low-speed fluid from the inner wake and push down the high-
speed fluid from the outer wake (figure 12) resulting in the formation of
streaks. This suggests the wake has a phenomenon analogous to the
‘regeneration cycle’ (Hamilton et al., 1995; Farrell & Ioannou, 2012) of wall-
bounded flows, which involves the generation of streamwise vortices through
nonlinear interactions and the formation of streaks through linear advection
by these streamwise vortices. Recently, Bae et al. (2021) have shown that the
nonlinear interactions between spanwise rolls and oblique streaks regenerate
streamwise vortices, which then amplify streaks through the lift-up mechanism
in wall-bounded flows.

Figure 18: Nonlinear transfer ($\mathcal{T}_{nl}$) from the VS mode and the
shear production ($\mathcal{P}_{xr}$) at different streamwise locations for
the $m=2,\mbox{\it St}\rightarrow 0$ mode: ($a$) $x/D=2$; ($b$) $x/D=3$; ($c$)
$x/D=5$; ($d$) $x/D=10$; ($e$) $x/D=40$; ($f$) $x/D=80$.

Finally, the relative role of nonlinear interaction and the linear lift-up
mechanism in streak energetics is examined. In figure 18, $P_{xr}$ denotes
shear production in the $m=2,\mbox{\it St}\rightarrow 0$ mode and
$\mathcal{T}_{nl}$ denotes nonlinear energy transfer from $m=1,\mbox{\it
St}=\pm 0.135$ modes to the $m=2,\mbox{\it St}\rightarrow 0$ mode. These two
terms are defined in appendix C. Figure 18 reveals that both terms are
comparable in magnitude and of the same sign for $x/D\leq 5$, whereas $P_{xr}$
dominates beyond $x/D=10$. Thus, near the wake generator, both (a) the
nonlinear interaction of the vortex-shedding mode with its conjugate and (b)
the linear production due to the lift-up process are of similar importance to
streak energetics. Beyond the near wake, the linear mechanism is responsible
for maintaining the streaks.

This work demonstrates that streaks and the associated lift-up mechanism are
operative in the turbulent disk wake. The results also open directions for
future research. Previous work by Ortiz-Tarin et al. (2021, 2023) on the wake
of a slender 6:1 prolate spheroid found that the wake differs significantly
from its bluff-body counterpart. Therefore, one possible direction is to
investigate how the shape of the wake generator affects the streaky structures
and the lift-up mechanism. Moreover, the influence of the angle of attack and
surface properties (roughness, porosity) on the development of these
structures could also be explored. Second, it is worth investigating how
density stratification, often found in the natural environment, affects the
lift-up mechanism and formation of streaks. Lastly, a direct comparison of the
characteristics of streaks in wakes, such as length scales, intermittency and
life cycle, with those in other turbulent flows such as channel flows and jets
can inform us on potentially universal behavior of streaks in turbulent flows.

[Acknowledgments]We thank Dr. K. Chongsiripinyo for the disk wake database.
[Funding]We acknowledge the support of Office of Naval Research (ONR) grant
N00014-20-1-2253. [Declaration of interests] The authors report no conflict of
interest. [Author ORCIDs]

A. Nekkanti, https://orcid.org/0000-0002-2173-8704;

S. Nidhan, https://orcid.org/0000-0003-0433-6129;

O.T. Schmidt, https://orcid.org/0000-0002-7097-0235;

S. Sarkar, https://orcid.org/0000-0002-9006-3173. [Author contributions] A.N.
and S.N. contributed equally to this paper and are co-first authors.

##  Appendix A Effect of parameter, $n_{\rm{fft}}$, on the frequency of the
large-scale streaks

Figure 19: Effect of $n_{\rm{fft}}$ and tapers on the PSD ($a$-$d$) at
$r/D=2$, $x/D=40$ and SPOD ($e$-$h$) at $x/D=40$: ($a,e$) $n_{\rm{fft}}=1024$,
$N_{\rm{win}}=10$; ($b,f$) $n_{\rm{fft}}=2048$, $N_{\rm{win}}=10$; ($c,g$)
$n_{\rm{fft}}=4096$, $N_{\rm{win}}=10$; ($d,h$) $n_{\rm{fft}}=7132$,
$N_{\rm{win}}=10$. The white and red lines in the figure correspond to the
peak in PSD and SPOD spectra. The PSD and SPOD spectra are plotted for
$\mbox{\it St}>0$.

Here, the parameter $n_{\rm{fft}}$ is varied to identify the _true_ frequency
of the streaks in the wake. Due to the constraint of the time-series length,
the PSD and SPOD spectra corresponding to the streaks peak in the first
frequency bin and are hence interpreted as $\mbox{\it St}\rightarrow 0$. For
better resolution, one can increase $n_{\rm{fft}}$. However, this results in
an increase in variance. Hence, we employ the multitaper-Welch based PSD
(Thomson, 1982) and SPOD (Schmidt, 2022) as it outperforms the standard Welch
estimator in terms of variance and resolution (Bronez, 1992). Recently,
Nekkanti & Schmidt (2023) have shown that the parameters, $n_{\rm fft}$ and
$n_{\rm ovlp}$ can significantly affect the spectrum and reconstruction of
data. Figure 19 shows the PSD and SPOD performed using $n_{\rm{fft}}=1024$,
$2048$, $4096$ and $7132$ with $N_{\rm{win}}=10$ Slepian tapers as windows.
The PSD is computed at $x/D=40,r/D=2$ and SPOD spectra at $x/D=40$. As the
goal is to identify the true frequency, we plot the spectra for $\mbox{\it
St}>0$. For figure 19 ($a$,$b$,$e$,$f$), as the $n_{\rm fft}$ is not large
enough, the peak in the PSD and SPOD is at $\mbox{\it St}=0$. On increasing
the $n_{\rm{fft}}$, two-dimensional PSD and SPOD spectra can resolve the lower
frequencies and their peak approaches $\mbox{\it St}\approx 0.006$ at the
highest possible $n_{\rm{fft}}$ (figure 19$d,h$). Using this approach, the
frequency associated with the large-scale streaky structures can be
identified, which is $St\approx 0.006$. Note that, the availability of more
snapshots will result in a better convergence of the peak frequency associated
with streaks.

##  Appendix B Conditional averaging: sensitivity of parameter, $c$

Figure 20: Effect of changing $c$ on the conditionally averaged streamwise
vorticity $\langle\omega_{x}^{c1}\rangle$ obtained from $u^{\prime}_{x}$ based
conditioning at $x/D=40$. Figure 21: Effect of changing $c$ on the
conditionally averaged streamwise vorticity $\langle\omega_{x}^{c2}\rangle$
obtained from $u^{\prime}_{x}u^{\prime}_{r}$ based conditioning at $x/D=40$.

Figures 20 and 21 show the effect of varying $c$ in equations 15 and 16 on the
obtained $\langle\omega_{x}^{c1}\rangle$ and $\langle\omega_{x}^{c2}\rangle$
fields, respectively, at a representative location of $x/D=40$. In both
figures, changing $c$ between 0 to 1 has little qualitative effect on the
distribution of negative and positive vortices around the conditioning point.
It is interesting to note, however, that the $\langle\omega_{x}^{c2}\rangle$
field is more robust to changes in $c$ than the
$\langle\omega_{x}^{c1}\rangle$ field. Increasing the value of $c$ leads to
the capture of more intense events; but, it also reduces the number of
realizations available for averaging. Hence, based on figure 7, $c=0.5$ is
selected as a compromise between capturing intense events and having
sufficient realizations for temporal averaging.

##  Appendix C Scale specific production and non-linear transfer terms in the
disk wake for $m=2,\mbox{\it St}\rightarrow 0$ mode

Fourier modes and Welch estimation are employed to compute the scale-specific
production and nonlinear transfer terms. The mathematical form of these two
terms is as follows.

The scale-specific production term is

$\mathcal{P}\pqty{m,\mbox{\it
St}}=-\mathrm{Re}\bqty{\hat{u}_{j}^{*}\pqty{m,\mbox{\it
St}}\hat{u}_{i}\pqty{m,\mbox{\it St}}\frac{\partial U_{j}}{\partial x_{i}}},$
(18)

where, $\mathrm{Re}\bqty{\cdot}$ denotes the real part. We are specifically
interested in the production term by lateral mean shear operating on the
$m=2,\mbox{\it St}\rightarrow 0$ mode:

$\mathcal{P}_{xr}\pqty{m=2,\mbox{\it St}\rightarrow
0}=-\mathrm{Re}\bqty{\hat{u}_{x}^{*}\hat{u}_{r}\frac{\partial U}{\partial
r}}_{\pqty{m=2,\mbox{\it St}\rightarrow 0}}.$ (19)

The scale-specific nonlinear transfer is

$\mathcal{T}_{nl}\pqty{m,\mbox{\it
St}}=-\mathrm{Re}\bqty{\hat{u}_{j}^{*}\pqty{m,\mbox{\it
St}}\widehat{u_{i}\frac{\partial u_{j}}{\partial x_{i}}}\pqty{m,\mbox{\it
St}}}$ (20)

This term represents the total nonlinear transfer term that generates the
frequency St and wavenumber $m$. For identifying the triadic energy transfer,
i.e., $m_{1}+m_{2}=m_{3}$ and $\mbox{\it St}_{1}+\mbox{\it St}_{2}=\mbox{\it
St}_{3}$, following Cho et al. (2018), we write this term in discretized
convolution:

$\mathcal{T}_{nl}\pqty{m_{3},\mbox{\it
St}_{3}}=-\mathrm{Re}\bqty{{\hat{u}_{j}^{*}\pqty{m_{3},\mbox{\it
St}_{3}}\sum\limits_{\begin{subarray}{c}\mbox{\it St}_{1}+\mbox{\it
St}_{2}=\mbox{\it St}_{3}\\\
m_{1}+m_{2}=m_{3}\end{subarray}}\hat{u}_{i}\pqty{m_{1},\mbox{\it
St}_{1}}\frac{\partial\hat{u}_{j}}{\partial x_{i}}\pqty{m_{2},\mbox{\it
St}_{2}}}}$ (21)

.

For a single triad [$m_{1},m_{2},m_{3}$], [$\mbox{\it St}_{1},\mbox{\it
St}_{2},\mbox{\it St}_{3}$] this is expanded as follows:

$\mathcal{T}_{nl}\bqty{(m_{1},m_{2},m_{3}),(\mbox{\it St}_{1},\mbox{\it
St}_{2},\mbox{\it
St}_{3})}=-\mathrm{Re}\bqty{{\hat{u}_{j}^{*}\pqty{m_{3},\mbox{\it
St}_{3}}\hat{u}_{i}\pqty{m_{1},\mbox{\it
St}_{1}}\frac{\partial\hat{u}_{j}}{\partial x_{i}}\pqty{m_{2},\mbox{\it
St}_{2}}}}$ (22)

The nonlinear term of our interest is the one responsible for transfer of
energy from the VS mode to the $m=2,\mbox{\it St}\rightarrow 0$ mode:

$\mathcal{T}_{nl}\bqty{(1,1,2),(0.135,-0.135,0)}=-\mathrm{Re}\bqty{{\hat{u}_{j}^{*}\pqty{2,0}\hat{u}_{i}\pqty{1,0.135}\frac{\partial\hat{u}_{j}}{\partial
x_{i}}\pqty{1,-0.135}}}.$ (23)

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