Large-scale streaks in a turbulent bluff body wake

Abstract A turbulent circular disk wake database (Chongsiripinyo & Sarkar, J. Fluid Mech., vol. 885, 2020) at Reynolds number $Re = U_\infty D/\nu = 5 \times 10^{4}$ 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 wakes. The near wake is dominated by the vortex shedding (VS) mode residing at azimuthal wavenumber $m=1$ and Strouhal number $St = 0.135$. After filtering out the VS structure, conclusive evidence of large-scale streaks with frequency $St \rightarrow 0$, 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 sweeps to events of intense $-u'_xu'_r$. Spectral proper orthogonal decomposition 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 is conducted to reveal strong interaction between $m=1$ and $St = \pm 0.135$ modes to give the $m=2$, $St \rightarrow 0$ streak mode. Our results indicate that the self-interaction of the VS mode generates the $m=2$, $St \rightarrow 0$ 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.


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 2 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 < Re  < 5830).Their spanwise spacing of 100 wall units was found to be invariant with Re  .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 largescale streaky structures in the near field (until / = 8).They further demonstrated that these structures exhibit large time scales and are associated with a low frequency of St → 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  = 2 dominating by / = 30.A similar conclusion about the dominance of  = 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 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.
Streaks in a turbulent wake 3 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, St → 0, at azimuthal wavenumber  = 2 in their proper orthogonal decomposition (POD) analyses of the turbulent wake of a circular disk at Re ≈ 2.5 × 10 4 .This mode is distinct from the VS mode of a circular disk wake, which resides at  = 1 with St = 0.135 (Berger et al. 1990).In a subsequent study, Johansson & George (2006) extended the downstream distance to / = 150 and found that the  = 2 mode with St → 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 (Re = 5 × 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  ⩽ 4. Streamwise-elongated streaks, a focus of the current paper, were not considered by Nidhan et al. (2020).
None of the wake studies that report the presence and the eventual dominance at large  of the very low-frequency mode (St → 0) at  = 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 / = 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 (1967Lumley ( , 1970)), extracts a set of orthogonal modes sorted according to their energy 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. 2022Nekkanti et al. , 2023)), 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 the analysis of nonlinear interactions in the wake at select locations.The manuscript ends with discussion and conclusions in §6.

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, Re =  ∞ / = 5 × 10 4 , reported in Chongsiripinyo & Sarkar (2020).Here  ∞ is the freestream velocity,  is the disk diameter, and  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 / = 15 and a streamwise distance of / = 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 (), azimuthal () and streamwise directions () are   = 365,   = 256 and   = 4096, respectively.The simulation has high resolution with streamwise grid resolution of Δ = 10 at / = 10, where  is the Kolmogorov lengthscale.By / = 125, the resolution improves to Δ < 6 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.

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 q  = q(  ) denote the mean subtracted snapshots, where  = 1, 2, • • •   are   number of snapshots.For spectral estimation, the dataset is first segment into  blk overlapping blocks with  fft snapshots in each block.The neighbouring blocks overlap by  ovlp snapshots with  ovlp =  fft /2.The  blk blocks are then Fourier transformed in time and all Fourier realizations of the -th frequency, q (  )  , are arranged in a matrix, (2.1) The SPOD eigenvalues,   , are obtained by solving the following eigenvalue problem: Focus on Fluids articles must not exceed this page length Streaks in a turbulent wake where W is a positive-definite Hermitian matrix that accounts for the component-wise and numerical quadrature weights and (•) * denotes the complex conjugate.SPOD modes at the -th frequency are recovered as (2.3) The SPOD eigenvalues are denoted by

𝑙
).By construction, represent the energies of the corresponding SPOD modes that are given by the column vectors of the matrix   = [ (1)   , represents the -th dominant coherent flow structure at the -th frequency.An useful property of the SPOD modes is their orthogonality; the weighted inner product at each frequency,  ()   , Here, we perform SPOD on various 2D streamwise planes ranging from / = 1 to 120.Thus, q contains the three velocity components at the discretized grid nodes on a streamwiseconstant plane.These planes are sampled at a spacing of 5 from / = 5 to / = 100, and five additional planes are sampled at / = 1, 2, 3, 110, and 120.The utilized time series has   = 7200 snapshots with a uniform separation of non-dimensional time of Δ ∞ / = 0.072 between two snapshots.Owing to the periodicity in the azimuthal direction, the flow field is first decomposed into azimuthal wavenumbers , and then SPOD is applied on the data at each azimuthal wavenumber.The spectral estimation parameters used here are  fft = 512 and  ovlp = 256, resulting in  blk = 27 SPOD modes for each St.Each block used for the temporal FFT spans a time duration of Δ = 36.91∞ /.

Reconstruction using convolution approach
The convolution strategy proposed by Nekkanti & Schmidt (2021) is employed for lowdimensional reconstruction of the flow field.This involves computing the expansion coefficients by convolving the SPOD modes over the data one snapshot at a time, Here, () is the Hamming windowing function and Ω is the spatial domain of interest.The data at time  is then reconstructed as (2.6) 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, () with zero mean, the bispectrum is where     ( 1 ,  2 ) =  [()( −  1 )( −  2 )] is the third moment of (), and  [•] 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 spatial integral of the point-wise bispectrum, Here, q is the temporal Fourier transform of q computed using the Welch approach (Welch 1967) and • denotes the Hadamard (or element-wise) product.
Next, as in equation (2.1), all the Fourier realizations at -th frequency are arranged into the matrix, Q .The auto-bispectral matrix is then computed as where, Q To measure the interactions between different quantities, we construct the cross-bispectral matrix (2.10) In the present application, matrices Q, R, and S comprise the time series of the field variables at the azimuthal wavenumber triad, [ 1 ,  2 ,  3 ].
Owing to the non-Hermitian nature of the bispectral matrix, the optimal expansion coefficients, a 1 are obtained by maximising the absolute value of the Rayleigh quotient of B (or B  ) (2.11) The complex mode bispectrum is then obtained as (2.12) Finally, the leading-order bispectral modes and the cross-frequency fields are recovered as and (2.13) (2.14) 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, (   ,   ,   +   ), where Q• is the cause and Q+ is the effect.The complex mode bispectrum,  1 , measures the intensity of the triadic interaction and the bispectral mode,  + , 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  fft = 512 and  ovlp = 256.Since our focus is on interactions of different azimuthal wavenumbers, the cross-BMD method, which computes the crossbispectral matrix B  , is applied to the wake database.The specific interactions among different  and their analysis using BMD will be presented and discussed in §5.1.

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)  = 1, St = 0.135 and (b)  = 2, St → 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  = 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  = 1 and  = 2.The main result is that, different from the  = 1 mode, the  = 2 mode is associated with streamwise-aligned streaky structures.
Figure 1 shows the azimuthal modes  = 1 and  = 2 of an instantaneous flow snapshot in the wake spanning downstream distance 0 < / < 100, obtained using a Fourier transform in the azimuthal direction .In the  = 1 mode (figure 1𝑎), a wavelength of / = 1/St VS (where the vortex shedding frequency, St VS ≈ 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 ∼  (100) from the disk.
The spatial structure of the  = 2 mode (figure 1𝑏) is quite different from that of the  = 1 mode.In the  = 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   / ≈ 25 (see 70 < / < 95), significantly larger than the wavelength of the vortex shedding mode   / ≈ 7.
Figure 2(,) show the instantaneous streamwise velocity field in the near-intermediate and intermediate-far wake, respectively, of the disk on a / −  plane.The / −  plane is constructed by unrolling the cylindrical surface at a constant /.For the near-intermediate field (figure 2𝑎), the plane is located at / = 1.25 while for the intermediate-far field (figure 2𝑐), / = 2.5 is chosen.In both figures, vortex shedding structures are evident, spaced at an approximate wavelength of   / ≈ 7.These structures and the wavelength of   / ≈ 7 become even more evident when the flow field at the respective radial locations in the near-  In what follows, the characteristics and robustness of these elongated structures are quantified through various statistical and spectral measures.
To this end, we invoke the Taylor's hypothesis, converting time  at a location  0 to pseudo-streamwise distance from  0 :   =  conv , where  conv =  ∞ −   is the convective velocity.Since the defect velocity in the wake   ≪  ∞ (where  ∞ is the free-stream velocity),  conv is approximated by  ∞ .Taylor's hypothesis requires velocity fluctuation to be sufficiently small compared to  conv .Figure 3 shows that this requirement is met since turbulence intensity ( 1/2 / ∞ ) drops below 12% beyond / = 10 for all radial locations and below 4% at / = 30.Previous work on turbulent wakes has shown Taylor's hypothesis to be valid in the wake when  1/2 / ∞ drops below ∼ 10% (Antonia & Mi 1998; Kang & 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   ≈ 1/St VS dominate the visualization in figure 4(, ).It is known a priori that these structures are contained in the  = 1 azimuthal mode.In order to assess space-time coherence other than the vortex shedding mode, the streamwise fluctuations are replotted in figure 4(,) after removing the  = 1 contribution.Once the  = 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  = 2, i.e, there are two structures over the azimuthal length of 2.Only the  = 2 component is shown in figure 4 (,  ) and, on doing so, the elongated streamwise streaks come into sharper focus.Building upon figure 1, figure 4 (,,,  ) lend support to the spatiotemporal robustness of these large-scale streaks in the turbulent wake of circular disk.
The downstream distance (0 < / < 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.the  = 2 component (figure 5𝑏 and ) highlights the streaks.Collectively, figures 4, and 5 demonstrate that the streaks span the entire wake length and that the  = 2 mode drives these streaks.
Figure 6 shows two-dimensional (2D) spectra in the streamwise wavelength (  )azimuthal mode () space at four representative streamwise locations / = 10, 20, 40, and 80. Here,   is the wavenumber of the pseudo-streamwise direction   .The  = 1 contribution is removed a priori to emphasize the large-scale streaks.At all these four locations, these streaky structures correspond to   → 0 and are found to reside in the  = 2 azimuthal mode.Note that the   = 0, St = 0 should be interpreted as   , St → 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  fft to resolve the lower frequencies and identify the frequency associated with streaks in this limit to be around St ≈ 0.006.

Evidence of lift-up mechanism in the wake
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 lowspeed 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 (  ) at three representative locations in the flow -the planes, / = 10, 40 and 80 in figure 7. The top row shows a conditional average, ⟨ 1  ⟩, designed to extract the structure of the streamwise vorticity on a constant- plane during times of large streamwise velocity fluctuations.Specifically, the condition is that at a specified point P on that plane, and ⟨ 1  ⟩ is the temporal average of all   () that satisfies this condition.The conditional point P (shown as green dots in figure 7) is chosen to lie at  = 0 and radial locations of / = 0.8 and 2 for / = 10 and / = 40, 80, respectively.The selection of different radial locations at / = 10 and / = 40, 80 is based on the approximate values of mean wake half-widths at the respective / locations (Chongsiripinyo & Sarkar 2020).Owing to rotational invariance of statistics for an axisymmetric wake, the condition is applied to a new point P 1 at the same / but a different value of  and the new ⟨ 1  ⟩ field, after a rotation to bring P 1 to P, is included in the conditional average.Since   = 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 ⟨ 2  ⟩ computed using a different condition at point P, 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  ∈ (0, 1] as reported in appendix B. Hence,  is set to 0.5 as a compromise between identification of intense events and retention of sufficient snapshots for conditional averaging. ( ′  ) rms and (− ′   ′  ) 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 (3.1) captures the structure of the streamwise vorticity field during events of intense positive  ′  .In figure 7(), 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  ′  > 0. When the averaging procedure is conditioned on negative streamwise velocity fluctuations, i.e.,  ′  ⩽ −( ′  ) rms , the signs of the vortex rolls in figure 7 are interchanged, as expected (figure not shown).At / = 40 and 80 (figure 7𝑏, ), two additional vortical structures are observed in the  = [90 • , 270 • ] region.However, around the conditioning point the spatial organization of vorticity remains qualitatively similar.The size of these vortex rolls increase with /, consistent with the radial spread of wake.
The conditionally averaged field based on equation (3.2) captures the vorticity field corresponding to intense positive − ′   ′  values.In a turbulent wake, − ′   ′  is predominantly positive such that the dominant production term in the wake,   = ⟨− ′   ′  ⟩/ > 0, acts to transfer energy from the mean flow to turbulence.Now turning to equation (3.2), positive − ′   ′  can result from two scenarios: (a) ( ′  > 0,  ′  < 0), i.e., 'ejection' of low-speed fluid from the inner wake to the outer wake and (b) ( ′  < 0,  ′  > 0), i.e., 'sweep' of highspeed 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, ⟨ 2  ⟩ ≈ 0 due to the opposite spatial distribution of vortices during ejection and sweep events.However, ⟨ 2  ⟩ obtained from − ′   ′  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 ( ′  > 0), pushing low-speed fluid from the inner wake to the outer wake.In short, ⟨ 2  ⟩ fields in the bottom row of figure 7 correspond to the ejection events at the conditioning point.A similar spatial organization of ⟨ 2  ⟩ 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 − ′   ′  , 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  ′  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  ′   ′  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 .

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  = 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 St → 0 in the wake.We particularly focus on the  = 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 ( = 1,  = 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 SPOD is performed at different streamwise locations (/) in 1 ⩽ / ⩽ 120.By definition, the leading SPOD modes at a given / represents the most energetic coherent structures at the associated frequency () and azimuthal wave number (). Figure 8 shows the percentage of energy in the leading SPOD modes ( (1) ) as a function of frequency and streamwise distance for the first six azimuthal wavenumbers  = 0 to  = 5.The percentage of energy at each streamwise location is obtained by normalizing the leading eigenvalue with the total turbulent kinetic energy,    (/) at the corresponding location.Overall, the most significant contributors to the TKE are the vortex shedding mode ( = 1, St = 0.135) and the mode corresponding to streaks ( = 2, St → 0), as reported in Nidhan et al. (2020).The leading vortex shedding SPOD mode contains about 10% energy in the near wake region (5 ≲ / ≲ 15) and decreases thereafter.The leading SPOD mode corresponding to the streaks in the  = 2 mode contains approximately 3% energy from / = 10 onward.The axisymmetric component ( = 0) exhibits a peak at St → 0.054 at / = 1 and a much smaller peak at St ≈ 0.19 between 10 ≲  ≲ 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  = 0 mode is not the focus of this study.The energy,  (1) , contained in the higher azimuthal wavenumbers ( = 3 − 5) is shown in figures 8(- ), respectively.Although  (1) is smaller than at  = 1 or  = 2, the higher modes also exhibit temporal structure.The  = 3 component shows energy concentration at the vortex shedding frequency St = 0.135 for 15 ≲ / ≲ 70 and at St → 0 for / ≳ 50.For  = 4, energy is concentrated near St → 0 for / ≳ 60.For the  = 5 component, traces of the vortex shedding mode and streaks (St → 0) are observed at the streamwise locations / ≳ 60 and / ≳ 80, respectively.Figure 8 indicates that the peaks at the vortex shedding frequency are present only at the odd azimuthal wavenumbers ( = 1, 3, 5), whereas the peaks corresponding to the large-scale streaks, i.e., St → 0, can be found at both the odd and even .It is also interesting to note that, for higher , both the vortex shedding modes and streaks do not appear until larger values of /.This suggests nonlinear interactions among different frequencies and azimuthal wavenumbers as the wake evolves, as will be elaborated in §5.
The St → 0 streaks are dominated by the  = 2 azimuthal wavenumber as demonstrated by figure 9(), which shows the contribution of different  at St → 0. The leading eigenvalues of each azimuthal wavenumber is normalized by the total energy at St → 0, i.e.,    () ( → 0, ).Streaks are azimuthally non-uniform structures and are not present in the axisymmetric  = 0 component.Hence, we focus on  ⩾ 1.The azimuthal wavenumber  = 2 is energetically dominant at the St → 0 frequency, containing about 40 − 50% of the total energy at St → 0. The sub-optimal wavenumber is  = 3 for 5 ⩽  ⩽ 80 and switches between  = 3 and  = 4, thereafter.However, the difference in energy between the  = 2 and  = 3 wavenumbers is always large, >30%.This dominance of the  = 2 wavenumber at St → 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  = 2 with the naked eye. Figure 9() shows that the vortex shedding frequency is more dominant in the region  ⩽ 70, whereas the zeroth frequency dominates for  ⩾ 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 / ≈ 65, the defect velocity decay rate changes from  −1 to  −2/3 in the wake (Chongsiripinyo & Sarkar 2020).For comparison, the most dominant component at the zeroth frequency, i.e.,  = 2 is also shown in figure 9(), which exhibits a similar trend of increasing prominence in the downstream direction.
Having diagnosed the streaky structure ( = 2, St → 0) and the vortex shedding structure ( = 1, 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  = 2 wavenumber is selected because it is energetically dominant at St → 0. The reconstruction is performed using the convolution strategy described in section §2. is significantly higher than that of  ′  and  ′  .dominance of the streamwise over radial and azimuthal counterparts is one salient feature of  components have comparable energy, showing a fundamental difference between the vortex shedding mode and the streaky-structures mode as to how each mode contributes to velocity fluctuations in the wake.Furthermore, the instantaneous energy of the flow field reconstructed from the VS mode has a much smaller time scale in comparison to that of the large-scale streaks.
Figure 11 shows the TKE ( =  ′   ′  /2) and its shear production (  = −  ′   ′  /) corresponding to the large-scale streaks and vortex shedding structures at / = 10, 40 and 80.As in figure 10, TKE and   are computed from the leading five SPOD modes.Figure 11(-) show that the TKE and the production peak at the similar radial location for  = 2,  → 0. This is not the case for  = 1, St = 0.135 (figure 11𝑏- ) where the peak TKE occurs close to the centerline while the production peaks away from the centerline.This difference in the locations of peak  and   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  and   , 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   ( → 0) and the grey and green-dotted lines correspond to positive and negative streamwise vorticity Re (1) ( → 0) , respectively.
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   for streaks.
4.2.Lift-up mechanism through the lens of SPOD analysis §4.1 demonstrated that the structures associated with St → 0 exhibit the characteristics of streaks and their significance increases from the near to far wake, with the azimuthal wavenumber  = 2 being the most significant, energetically, to the streaks.Figure 12 shows the leading SPOD mode of the streamwise velocity fluctuation ( ′  ) corresponding to  = 2 and St → 0 at three streamwise locations / = 10, 40, and 80. Overlaid on the  ′  contour is the streamwise vorticity ( ′  ) of the corresponding mode.Both  ′  and  ′  are characterized by four lobes of alternate sign, the size of which increase monotonically with /.Importantly, the set of  ′  lobes is shifted with respect to the  ′  lobes by a clockwise rotation of approximately 45 • .As a result of the shift, the maximum of  ′  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() shows the normalized radial profiles of mean defect velocity (  ) at = 10, 40 and 80. Figure 13() shows the radial profile of the normalized leading SPOD mode's streamwise component for  = 2, St → 0, at the same streamwise locations as in figure 13().As the wake develops in the  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 13𝑏) of the dominating streak-containing mode lies in close proximity to the location of the maximum mean shear (dotted-dashed lines in figure 13𝑎).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  ′  appears in the SPOD mode.The seminal work of Ellingsen & Palm (1975) demonstrates that, for linearized disturbances in an inviscid flow,   / ∝ −   ()/, 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.
The lift-up mechanism is also active at higher azimuthal wavenumbers as demonstrated  by figure 14, which shows the leading SPOD modes at / = 40 and frequency St → 0 for the higher modes:  = 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  and the number of lobes scales as 2.Also, the vortices are shifted by 30 • , 22.5 • , and 18 • , for  = 3, 4, and 5, respectively.In other words, the set of streamwise velocity lobes for wavenumber  is shifted by an angle of /2 radians with respect to the streamwise vortices.As in the case of  = 2, the peak of the leading SPOD modes for  = 3, 4, 5 lies in the vicinity of the maximum mean shear.Even for higher , lift-up effect occurs nears the region of the largest mean shear.This further confirms that the lift-up mechanism is active for higher azimuthal wavenumbers.

Analysis of triadic interactions in the wake
Previous sections show that the streaks are predominantly present in the  = 2 azimuthal mode.To shed light on the possible dynamics behind the formation of streaks in a turbulent wake, we focus on the nonlinear interactions between the VS-containing  = 1 mode and the streak-containing  = 2 mode.
5.1.Bispectral mode decomposition at select locations Figure 15 shows the SPOD spectra for the azimuthal wavenumbers,  = 1 and  = 2, at / = 10.Both SPOD spectra exhibit a large difference between the first and second eigenvalues for St ≲ 0.5, thus demonstrating a low-rank behaviour.The leading eigenvalue of the  = 1 azimuthal mode peaks at the vortex shedding (VS) frequency, St = 0.135.On the other hand, the leading eigenvalue of the  = 2 azimuthal mode exhibits a global peak at St → 0, and an additional local peak at 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 / ≈ 1 while the peak corresponding to  = 2, St → 0 appears further downstream at / ≈ 5.These observations collectively point towards different sets of triadic interactions involving the VS mode.For example,  = 1, St = 0.135 can interact with  = 1, St = −0.135 to give rise to  = 2, St → 0 that appears further downstream.Similarly, the self-interaction of  = 1, St = 0.135 can generate  = 2, 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 / locations using bispectral mode decomposition (BMD) (Schmidt 2020).

Comparison between nonlinear interactions and linear lift-up mechanism
Finally, the relative role of nonlinear interaction and the linear lift-up mechanism in streak energetics is examined.In figure 18,   denotes shear production in the  = 2, St → 0 mode and T  denotes nonlinear energy transfer from  = 1, St = ±0.135modes to the  = 2, St → 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 / ⩽ 5, whereas   dominates beyond / = 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 liftup process are of similar importance to streak energetics.Beyond the near wake, the linear mechanism is responsible for maintaining the streaks.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 Re = 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 / ≈ 120.Until now, most of the wake literature has understandably focused on the vortex shedding (VS) structure ( = 1, St = 0.135 for the circular disk), since it is the energetically dominant coherent structure in the near and intermediate wake.Upon removing the contribution of the  = 1 azimuthal wavenumber a priori in visualizations, the streaks become evident, even in the near and intermediate wake.Moreover, in the far wake (/ ⩾ 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: () they exhibit a much larger wavelength and time scale; () VS structures are tilted with respect to the downstream direction whereas streaks are almost parallel; () the streamwise velocity ( ′  ) significantly exceeds the other two velocity components ( ′  ,  ′  ) in magnitude for streaks whereas  ′  and  ′  are comparable for VS structures.The streaky structures are associated with a frequency of St → 0 and hence, by Taylor's hypothesis (validated here for the wake) to a wavenumber of   → 0. While streaky structures are observed for all non-zero azimuthal wavenumbers,  = 2 dominates in the near-to-far wake.In particular, SPOD analysis reveals that  = 2 contains about 55% (near 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 (  ) at St → 0 varies as   ∼ 1/ implying that higher  and not  = 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  = 2, St → 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  = 1, St = ±0.135VS mode and the  = 2, St → 0 streak mode.The  = 1, St = ±0.135vortices are found to interact and generate the  = 2, St → 0 vortices.These streamwise vortices of the  = 2, St → 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.
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 (Nidhan et al. 2022;Gola et al. 2023), often found in the natural environment, affects the lift-up mechanism and formation of streaks in turbulent disk wakes.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.In the same vein, it will be also interesting to build reduced-order models to isolate and understand the spatiotemporal features of interaction between the VS mode and streaks as was done with in a problem with KH-like instabilities and streaks (Nogueira & Cavalieri 2021;Cavalieri et al. 2022).which is  ≈ 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
Figures 20 and 21 show the effect of varying  in equations 3.1 and 3.2 on the obtained ⟨ 1  ⟩ and ⟨ 2  ⟩ fields, respectively, at a representative location of / = 40.In both figures, changing  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 ⟨ 2  ⟩ field is more robust to changes in  than the ⟨ 1  ⟩ field.Increasing the value of  leads to the capture of more intense events; but, it also reduces the number of realizations available for averaging.Hence, based on figure 7,  = 0.5 is selected as a compromise between capturing intense events and having sufficient realizations for temporal averaging.
Figure 5(,) show the   / −  plots of  ′  , with contribution of  = 1 mode excluded, in regions starting at  0 / = 40 and 80.The   / −  planes are located at / = 2 for these far-wake locations since the wake width grows with .Interested readers can refer to figures 5, 15, 16 of Chongsiripinyo & Sarkar (2020) and figure 5 of Nidhan et al. (2020) for an in-depth discussion about the mean and turbulence statistics.Similar to the near wake plot in figure 4(,), large-scale streaks elongated in the streamwise direction are found to extend into the far wake as well.Once again, isolating

Figure 10 :
Figure 10: Component-wise instantaneous energy reconstructed from the  = 2 streaky structures in the top row (-) and vortex shedding structures (- ) in the bottom row.The plane integral of the energy is shown at various streamwise locations: (, ) / = 10, (, ) / = 40 and (,  ) / = 80.Flow reconstruction uses the leading five SPOD modes.The oscillation frequency of the reconstructed energy in (- ) is twice that of the VS frequency.
Figure9() shows that the vortex shedding frequency is more dominant in the region  ⩽ 70, whereas the zeroth frequency dominates for  ⩾ 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 / ≈ 65, the defect velocity decay rate changes from  −1 to  −2/3 in the wake(Chongsiripinyo & Sarkar 2020).For comparison, the most dominant component at the zeroth frequency, i.e.,  = 2 is also shown in figure9(), which exhibits a similar trend of increasing prominence in the downstream direction.Having diagnosed the streaky structure ( = 2, St → 0) and the vortex shedding structure ( = 1, 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  = 2 wavenumber is selected because it is energetically dominant at St → 0. The reconstruction is performed using the convolution strategy described in section §2.Figure10(,,) shows the instantaneous energy in the three fluctuation components,  ′  ,  ′  , and  ′  at / = 10, 40, and 80, after reconstruction with the streaky-structure SPOD modes.The instantaneous energy is depicted within the time interval  ∈ [0, 110], which corresponds to the first five blocks used for SPOD, and is representative of the entire reconstructed flow fields.The energy of  ′  is significantly higher than that of  ′  and  ′  .dominance of the streamwise over radial and azimuthal counterparts is one salient feature of

Figure 11 :
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 (-) and VS mode in the bottom row (- ).Left column (, ) corresponds to / = 10, middle column (, ) to / = 40 and right column (,  ) to / = 80.Flow reconstruction uses the leading five SPOD modes.

Figure 13
Figure 13: () Normalized mean defect velocity (  =  ∞ − ) profiles.() Normalized leading SPOD mode of the streamwise velocity (  ) at  = 2 and St → 0. Dashed lines in () and dotted-dashed lines in () correspond to the radial location of the maximum of SPOD mode and the maximum of mean shear (/), respectively.