Hierarchy of coherent vortices in turbulence behind a cylinder

Abstract

To draw concrete pictures of the hierarchy of multiscale coherent vortices in turbulence behind a cylinder and to reveal their sustaining mechanism, we conduct direct numerical simulations of the turbulence at the Reynolds number, which is defined by the uniform inflow velocity and the cylinder diameter, $5000$. The turbulence consists of three kinds of hierarchies of coherent vortices in three distinct regions: namely, the downstream region, the recirculation region just behind the cylinder and the separate shear layers. By tracking the temporal evolution of multiscale vortices in each of these regions, we demonstrate that, in all the three regions, smaller coherent vortices tend to align in the direction perpendicular to larger ones. This implies that smaller vortices are stretched and amplified in the strain-rate fields around larger ones. We also show the relevance of this observation to the energy cascade. Smaller-scale vortices receive the kinetic energy in the regions where they are stretched by larger-scale vortices; and, at the same time, they tend to compress larger ones, thus reducing larger-scale kinetic energy.

1. Introduction

One of the most important unsolved issues on turbulence is to elucidate the energy cascade. Many researchers have accumulated many pieces of knowledge on the energy cascade in turbulence through various analyses from multifaceted perspectives. First, it is evident that the kinetic energy cascades, on average, from large to small scales in turbulence. This was verified in studies on the energy flux in wavenumber space (e.g. Domaradzki & Rogallo 1990; Ohkitani & Kida 1992; Borue & Orszag 1998; Cerutti & Meneveau 1998; Chen et al. 2003; Aoyama et al. 2005) and in a scale space defined by the two different locations in turbulence (e.g. Cerutti & Meneveau 1998; Yasuda & Vassilicos 2018). However, there is no clear consensus on the energy transfer mechanism in the instantaneous turbulent fields. In fact, although various analyses concluded that interscale energy transfer can occur bidirectionally (e.g. Borue & Orszag 1998; Cerutti & Meneveau 1998; Tao, Katz & Meneveau 2002; Chen et al. 2003; Aoyama et al. 2005; Gomes-Fernandes, Ganapathisubramani & Vassilicos 2015; Alves Portela, Papadakis & Vassilicos 2017; Yasuda & Vassilicos 2018), we do not fully understand the relevance of these observations to the energy cascade. Here, we refer to some of the commonly used methods for quantifying energy transfer. Studies on large eddy simulations (LES) often quantify interactions between grid-resolved and subgrid scales to examine, for example, their relationship to coherent flow structures (e.g. Piomelli, Yu & Adrian 1996; Lin 1999; Natrajan & Christensen 2006; Hong et al. 2012; Watanabe, Da Silva & Nagata 2020). There are also quite a few studies (e.g. Marati, Casciola & Piva 2004; Cimarelli, De Angelis & Casciola 2013; Gomes-Fernandes et al. 2015; Hamba 2015; Cimarelli et al. 2016; Yasuda & Vassilicos 2018; Hamba 2019; Zhou et al. 2020) arguing the interscale energy transfer by using the Kármán–Howarth–Monin–Hill (KHMH) equation (Hill 2002). For instance, in turbulence behind an obstacle, which is also the target of the present study, Alves Portela et al. (2017) demonstrated that the energy transferred to larger scales in the mainstream direction. In their subsequent study (Alves Portela, Papadakis & Vassilicos 2020), they also showed the energy flux to small-scale turbulence due to the largest-scale coherent structures. In addition to studies on LES and those using the KHMH equation, there are also studies on interscale interactions of turbulence (Kawata & Alfredsson 2018, 2019; Chan, Schlatter & Chin 2021) by the spectral analyses of the scale-decomposed Reynolds stress. These studies have contributed to turbulence models and theories (e.g. Kobayashi 2005; Kobayashi, Ham & Wu 2008; Marusic & Monty 2019; Vela-Martín 2022). However, as mentioned above, it is still unclear how the energy is transferred between different scales in physical space. This is because the energy fluxes and transfers defined in most of the aforementioned studies denote the amounts of energy passing across a scale per unit time and its derivative with respect to scale, respectively. Hence, these quantities, in general, cannot capture the energy transfers between two scales. An important point of the present study is to quantify the energy flux between two scales, which may be estimated locally in space, to clarify the concrete picture of the energy cascade.

We also investigated the generation mechanism of the hierarchy of coherent vortices in turbulence. For turbulence in a periodic cube, not only our group (e.g. Goto 2008, 2012; Goto, Saito & Kawahara 2017) but also many authors (e.g. Leung, Swaminathan & Davidson 2012; Cardesa, Alberto & Jiménez 2017; Doan et al. 2018; Hirota et al. 2020) identified vortical structures with various sizes by decomposing turbulent fields obtained by the direct numerical simulations (DNS) into different scales. Such a scale decomposition gives a clear picture of the hierarchy of coherent vortices. For example, Goto et al. (2017) showed that at each level of the hierarchy, coherent tubular vortices tend to form counter-rotating pairs. Since the counter-rotating vortex pairs induce a strong strain-rate field around them, they effectively amplify smaller-scale vortices (Goto 2012; Goto et al. 2017) due to vortex stretching. It is also evident that small-scale vortices tend to be generated by being stretched by twice to eight times larger vortices (Leung et al. 2012; Goto et al. 2017). Recently, Yoneda, Goto & Tsuruhashi (2022) quantified the spatially averaged interscale energy transfer due to the vortex stretching. Under the assumptions based on the generation mechanism of the hierarchy of vortices, they also derived the $-5/3$ power law of the energy spectrum without Kolmogorov's similarity hypotheses. Thus, these results are not inconsistent with the conclusion that the scale-local vortex stretching is the cause of the energy cascade. Incidentally, the scale decomposition of turbulence also revealed the multiscale nature in near-wall turbulence (e.g. Lozano-Durán, Holzner & Jiménez 2016; Lee, Sung & Zaki 2017; Motoori & Goto 2019, 2021). In turbulent boundary layers (Motoori & Goto 2019) and turbulent channel flow (Motoori & Goto 2021), we observed similar energy cascading events for vortices at small scales, i.e. vortices smaller than the Corrsin length scale (Corrsin 1958) in the log layer.

The present study aims to elucidate (i) the concrete picture of the hierarchy of coherent vortices in the turbulent wake behind a cylinder and (ii) the sustaining mechanism of these vortices. Here, we must note that the sustaining mechanism of the large-scale vortices, which are the origins of the energy cascade, can be different from that in turbulence without flow separation, for example, periodic turbulence (Goto et al. 2017), zero-pressure-gradient turbulent boundary layers (Motoori & Goto 2019) and turbulent channel flow (Motoori & Goto 2021). We must also note that the flow state behind a cylinder is different depending on the ranges of the Reynolds number $Re_D = U_\infty D/ \nu$ (Williamson 1996; Forouzi Feshalami et al. 2022), where $U_\infty$ is the upstream uniform velocity magnitude, $D$ is the diameter and $\nu$ is the kinematic viscosity of the fluid. For $Re_D \approx 50\unicode{x2013}190$, Kármán vortices are shed alternately from the cylinder. The laminar-turbulence transition occurs for $Re_D \approx 190\unicode{x2013}260$, where streamwise rib vortices are formed between roller vortices, which correspond to the Kármán vortices at low $Re_D$. In this regime two modes are known. For $Re_D \approx 190\unicode{x2013}230$, roller vortices are deformed to tongue-shaped vortex sheets (mode A), whereas, for $Re_D \approx 230\unicode{x2013}260$, they stretch and amplify smaller rib vortices around themselves (mode B). In the turbulent regime $Re_D \approx 10^3\unicode{x2013}2\times 10^5$, small vortices emerge in the separated shear layers due to the Kelvin–Helmholtz (KH) instability. In particular, for $Re_D \lesssim 6000$, the vortices generated through the KH instability are advected into the downstream shear layer while keeping their shape. In contrast, for $Re_D \gtrsim 6000$, the KH vortices are deformed in the upstream shear layer due to the earlier transition to developed turbulence (e.g. Norberg 2003; Rajagopalan & Antonia 2005; Thompson & Hourigan 2005). For the further developed turbulence for $Re_D \approx 2\times 10^5\unicode{x2013}10^6$, the separated shear layer reattaches to the cylinder, leading to the well-known drag crisis.

Although the transition mechanisms in a wide range of $Re_D$ have been extensively investigated by experiments (see the reviews by Williamson 1996; Forouzi Feshalami et al. 2022) and LES (e.g. Breuer 1998; Lehmkuhl et al. 2014; Rodríguez et al. 2015), there are only a few studies of DNS that focus on energy cascade and its relation to multiscale flow structures. For example, Aljure et al. (2017) conducted the DNS of turbulence behind a cylinder for $Re_D = 5000$ to demonstrate that the shed vortices are distorted in the axial direction (i.e. the so-called vortex dislocation). Aasland et al. (2022) also conducted DNS of turbulence behind a single cylinder and tandem cylinders for $Re_D=10^4$ to investigate the reattachment mechanism of the separated shear layers. In the present study we investigate developed turbulence behind a cylinder at $Re_D=5000$, where vortices emerge in the shear layer but they are hardly deformed. Here, we regard the present turbulence as a developed one because, as will be shown below, the power spectrum obeys a power law (see figure 2 in § 2), which implies a sufficiently large separation in scales.

In the rest of the paper we first describe the numerical methods in § 2. Then, we investigate qualitatively (§ 3) and quantitatively (§ 4) the generation mechanism of the hierarchy of multiscale coherent vortices in the three distinct regions behind the cylinder. In § 5 we investigate energy transfer in instantaneous fields to show the relevance of the generation mechanism of the hierarchy of vortices to the energy cascade.

2. Numerical methods

2.1. Direct numerical simulations

We conduct DNS of turbulence behind a cylinder at $Re_D=5000$ by numerically integrating the non-dimensional forms of the Navier–Stoke equation,

(2.1)\begin{equation} \frac{\partial\boldsymbol{u}}{\partial t} + \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\boldsymbol{u} ={-}\boldsymbol{\nabla}p +\frac{1}{Re _D} \nabla^2 \boldsymbol{u}, \end{equation}

and the continuity equation,

(2.2)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0, \end{equation}

of an incompressible fluid. Here, $\boldsymbol {u}(\boldsymbol {x},t)$ and $p(\boldsymbol {x},t)$ are the non-dimensional fluid velocity and pressure fields at position $\boldsymbol {x}$ and time $t$, where we use $D$, $D/U_\infty$ and $\rho D^3$ ($\rho$ is the mass density of the fluid) as the length, time and mass units, respectively.

Figure 1 shows the numerical configuration. We define the $x_1$, $x_2$ and $x_3$ axes in the streamwise, the normal to the cylinder and the axial directions, respectively. We list numerical parameters of the DNS in table 1, where $N_i$ and $L_i$ denote the number of grid points and the side of the numerical domain in the $x_i$ direction, respectively.

Figure 1. Numerical configuration. We set the origin of the coordinate at the cross-point of a centreline in the $x_2$ direction and the axis of the cylinder.

Table 1. Numerical parameters. Here $N_i$ is the number of grid points in the $x_i$ direction, $L_i$ is the domain size in the $x_i$ direction, $\Delta t$ is the time per step and $\varDelta$ is the grid width.

We impose the uniform flow, $\boldsymbol {u}=(1,0,0)$, at the inlet and the convective condition, ${\partial u_{i}}/{ \partial t}+U_{m} {\partial u_{i}}/{\partial x}=0$, at the exit, where $U_{m}(t)$ is the spatial average of the $u_1$ at the exit plane. Under these boundary conditions, we numerically integrate (2.1) and (2.2) by the simplified marker and cell method (see Amsden & Harlow 1970; Kajishima & Taira 2017). For the global mass conservation in the computational domain, we correct $u_1$ at the exit plane (Simens et al. 2009) before we solve the Poisson equation for the pressure. For the temporal integration, we use the second-order Adams–Bashforth method for the advection term and the second-order Crank–Nicolson method for the viscous term. We set a uniform Cartesian staggered grid, and employ the second-order central finite difference to evaluate the spatial derivatives. The grid width $\varDelta$ is set to be smaller than $2.6\eta (x_1,x_2)$, and the time $\Delta t$ per step is smaller than $0.013 \tau _\eta (x_1,x_2)$. Here, $\eta (x_1,x_2)$ and $\tau _\eta (x_1,x_2)$ are the Kolmogorov length and time, which are estimated by $(\nu ^3/\bar {\epsilon })^{1/4}$ and $(\nu /\bar {\epsilon })^{1/2}$, respectively. Here, $\bar {\epsilon }(x_1,x_2)$ is the energy dissipation rate averaged over $t$ and $x_3$. We express the cylinder using the immersed boundary method (Uhlmann 2005; Kempe & Fröhlich 2012) to impose the non-slip boundary condition on the surface of the cylinder. In this method we distribute Lagrangian force points uniformly on the cylinder surface with the same spacing as the grid width of the DNS. We then impose body force on the fluid grid points around the force points so that the fluid velocity at the force points satisfies the non-slip boundary condition.

In the following, we show results in the statistically steady state. We define the origin ($t=0$) of time by the moment when about $3.42$ washout times elapse since roller vortices start shedding. We take statistics for the duration $0\leqslant t\leqslant T$ with $T=81.24$, during which about $16$ pairs of roller vortices are shed. We validate the present DNS in Appendix A.

2.2. Scale decomposition

We evaluate the power spectrum density,

(2.3)\begin{equation} E_2(\boldsymbol{x},f) = \frac{|\widehat{u_2^{\prime}}(\boldsymbol{x},f)|^2}{T}, \end{equation}

for the $x_2$ component of the fluctuating turbulent velocity. Here,

(2.4)\begin{equation} \widehat{u_i^{\prime}}(\boldsymbol{x},f)=\int_0^T\:u_i^{\prime}(\boldsymbol{x},t)\exp({-2{\rm \pi}\text{i}ft}) \,\text{d}t \end{equation}

denotes the Fourier transform of $u_i^{\prime }(\boldsymbol {x},t)$. Figure 2(a) shows the mean power spectrum density of $E_2(x_1,x_2=0,f)$, which is averaged over $x_3$, at eight streamwise locations ($1\leqslant x_1\leqslant 8$). We see that the power spectrum densities have a peak at $f=0.20$. This means that the Strouhal number $St$ is $0.20$ for this $Re_D$ ($=5000$), which is in good agreement with the value $0.203$ of the DNS results at $Re=3900$ (Parnaudeau et al. 2008). Incidentally, the secondary peak at $f=3St$ observed for $x_1\geqslant 2$ is also in good agreement with the LES results (Parnaudeau et al. 2008). This secondary peak was also observed in the experiment (Ong & Wallace 1996) and DNS (Lehmkuhl et al. 2013), though, to the best of our knowledge, its origin is unknown. We also observe an approximate $-5/3$ power law at all locations between $x_1 = 1$ and $8$. We can confirm the establishment of the power law by looking at figure 2(b), where the compensated spectra $E_2 f^{5/3}$ show plateaus within a factor of $2$–$3$. This implies that there exists a hierarchy of vortices with various sizes in this region. We cannot, however, observe such a hierarchy by using the raw data of the velocity gradients. This is demonstrated in figure 3(a), which shows the isosurfaces of the second invariant $Q$ of the velocity gradient tensor. In this figure we can only observe fine tubular vortices with the diameter approximately $10\eta$ (see figure 8 in § 3.2). The observation that the raw values of $Q$ only capture the smallest-scale vortices is common in many kinds of turbulence; e.g. turbulence in a periodic cube (Goto et al. 2017), turbulent boundary layers (Motoori & Goto 2019) and turbulent channel flow (Motoori & Goto 2021). This is because the smallest-scale vortices determine the velocity gradients. Therefore, we need a scale decomposition to observe the hierarchy of vortices.

Figure 2. Power spectrum density ($a$) $E_2(x_1, x_2 = 0, f)$ and ($b$) compensated by $f^{5/3}$: $E_2(x_1, x_2 = 0,f) f^{5/3}$, both averaged over $x_3$ at $x_1 = 1, 2, \ldots, 8$. We have vertically shifted the curves for $x_2\ge 2$. Blue dotted line indicates the $-5/3$ power law.

Figure 3. Coherent vortices with various sizes in the turbulent wake. (a) Isosurfaces of the second invariant $Q^\prime$ of the velocity gradient tensor. (b–d) Isosurfaces of the second invariant ${Q}^{\prime (\sigma )}$ of the scale-decomposed velocity gradient tensor with (b) filter scale $\sigma =\sigma _{max}$, (c) $\sigma _{max}/4$ and (d) $\sigma _{max}/16$. We set the thresholds in terms of the root mean square of $Q^\prime _{rms}$ (or $Q^{\prime (\sigma )}_{rms}$) at $(x_1,x_2)=(3,0)$: (a) $1.53Q^\prime _{rms}(=75)$, (b) $0.60Q^{\prime (\sigma )}_{rms}(=5.0\times 10^{-3})$,(c) $0.50Q^{\prime (\sigma )}_{rms}(=1.0\times 10^{-1})$ and (d) $1.42Q^{\prime (\sigma )}_{rms}(=2.0)$.

There are many studies using scale decompositions for inhomogeneous turbulence (e.g. Lee et al. 2014; Hwang et al. 2016; Lozano-Durán et al. 2016; Lee et al. 2017; Motoori & Goto 2019, 2021). Here, we employ the Gaussian filtering method adopted by Motoori & Goto (2021) and Motoori, Wong & Goto (2022). First, we apply the three-dimensional Gaussian filter to the velocity $u_i$,

(2.5)\begin{equation} u_i^{[\sigma]} ({\boldsymbol{x}},t) = \frac{1}{ (\sqrt{2{\rm \pi}}\sigma)^3 } \int u_i ({\boldsymbol{x}}^\prime,t) \exp \left(\frac{-(\boldsymbol{x}-\boldsymbol{x}^\prime)^2}{2\sigma^2}\right) \text{d}\,\boldsymbol{x}^\prime, \end{equation}

to obtain low-pass filtered velocity components $u_i^{[\sigma ]}$. Here, $\sigma$ denotes the filter width. We choose a constant $\sigma$ irrespective of $\boldsymbol {x}$ because we will investigate Lagrangian tracking of vortices on given scales in § 3.1. Note that $u_i^{[\sigma ]}$ only contains the velocity fields with scales larger than $\sigma$. Note also that the Fourier transform of (2.5) leads to

(2.6)\begin{equation} \widehat{u_i^{[\sigma]}} (\boldsymbol{k},t)= \widehat{u_i} (\boldsymbol{k},t) \exp \left(\frac{-|\boldsymbol{k}|^2\sigma^2}{2} \right) , \end{equation}

where $\hat {X}(\boldsymbol {k})$ denotes the Fourier transform of $X(\boldsymbol {x})$ and $\boldsymbol {k}$ is the wavenumber vector. In the present DNS, we use for the $x_1$ direction the one-dimensional Gaussian filter in a discrete form,

(2.7)\begin{equation} u_i^{[\sigma]_{1D}}(x_1,x_2,x_3,t) = \frac{\sum_{x_1^{\prime}}u_i(x_1^{\prime},x_2,x_3,t)\exp\left(\dfrac{-(x_1-x_1^{\prime})^2}{2\sigma^2}\right)\varDelta} {\sum_{x_1^{\prime}}\exp\left(\dfrac{-(x_1-x_1^{\prime})^2}{2\sigma^2}\right)\varDelta} , \end{equation}

and the Fourier representation (2.6) for the other (i.e. $x_2$ and $x_3$) periodic directions to implement the three-dimensional Gaussian filter (2.5). In (2.7) we take the summation for $x_1^\prime$ over the whole domain; that is, the filter length is $L_x$.

Next, we take the difference between the filtered velocities with two different filter widths $\sigma$ and $2\sigma$,

(2.8)\begin{equation} {u_i}^{(\sigma)}(\boldsymbol{x},t)={u_i^{[\sigma]}}(\boldsymbol{x},t)-{u_i^{[2\sigma]}}(\boldsymbol{x},t) , \end{equation}

which corresponds to a band-pass filter, since it contains information only in the band $[\sigma, 2\sigma )$ of length scales. Hereafter, the superscript $\,{\cdot }\,^{(\sigma )}$ denotes the filter band of $[\sigma, 2\sigma )$.

In the following analyses we first compute ${u}_i^{(\sigma )}(\boldsymbol {x},t)$, and then evaluate scale- decomposed quantities, which are simply called quantities at scale $\sigma$, from $u_i^{(\sigma )}(\boldsymbol {x},t)$. For example, we define the second invariant,

(2.9)\begin{equation} {Q}^{(\sigma)} ={-}\tfrac{1}{2}{D}_{ij}^{(\sigma)}{D}_{ji}^{(\sigma)} = \tfrac{1}{2}\left( {W}^{(\sigma)}_{ij}{W}^{(\sigma)}_{ij} - {S}^{(\sigma)}_{ij}{S}^{(\sigma)}_{ij} \right), \end{equation}

at scale $\sigma$ of the scale-decomposed velocity gradient tensor,

(2.10)\begin{equation} {D}_{ij}^{(\sigma)} = \frac{\partial{u}_i^{(\sigma)}}{\partial x_j} = {S}_{ij}^{(\sigma)}+{W}_{ij}^{(\sigma)}. \end{equation}

Here, $W_{ij}=(D_{ij}-D_{ji})/2$ and $S_{ij}=(D_{ij}+D_{ji})/2$.

Recall that the present system has a mean flow. It is therefore important, in some cases, to consider the scale decomposition of the fluctuation velocity $u_i^\prime (=u_i-\overline {u_i})$ obtained by using the Reynolds decomposition. The band-passed fluctuation velocity $u_i^{\prime (\sigma )}(\boldsymbol {x},t)$ is estimated by the difference between $u_i^{\prime [\sigma ]}$ and $u_i^{\prime [2\sigma ]}$ that are defined similarly to $u_i^{[\sigma ]}$ and $u_i^{[2\sigma ]}$, respectively. Accordingly, we also define ${Q}^{\prime (\sigma )}$ by using $u_i^{\prime (\sigma )}$.

3. Hierarchy of coherent vortices

In this section we demonstrate that the scale decomposition, which is introduced above, captures the hierarchy of coherent vortices. First, we visualize vortices at different scales (figure 3b–d) by using the isosurface of ${Q}^{\prime (\sigma )}$. Here, we choose thresholds in terms of root-mean-square values at $(x_1,x_2)=(3,0)$ so that we can identify coherent vortices in the three subdomains examined below. We set the filter scales as ($b$) $\sigma =\sigma _{max}$, ($c$) $\sigma _{max}/4$ and ($d$) $\sigma _{max}/16$. Here, we define the largest length scale $\sigma _{\rm max}$ by $D/(2{\rm \pi} St)$ (Yasuda, Goto & Vassilicos 2020) with $St=0.20$ (see § 2.2), which corresponds to the diameter ($\sigma _{max} = 0.8D$) of the shed roller vortices. On the other hand, the smallest scale $\sigma =\sigma _{max}/16$ corresponds to $\sigma = 6.4\varDelta$, which means that the three filter scales are well resolved by the DNS grid width. The scale ratio, four, was also used in our previous studies (Goto et al. 2017; Motoori & Goto 2019, 2021) because we can identify distinct coherent vortices with this scale ratio. Incidentally, this is consistent with the observation that small vortices are most likely to be perpendicular to four times larger ones (see figure 14 in § 4.4). Figure 3(b) for $\sigma =\sigma _{max}$ shows that roller vortices are coherent even in high-Reynolds-number turbulence, though they cannot be observed without the scale decomposition (figure 3a). Figures 3(c) and 3(d) show that smaller vortices, which are larger than the Kolmogorov-scale eddies (figure 3a), are also coherent.

Figure 4 simultaneously shows the vortices at four length scales shown in figure 3(a–d). We see that these vortices show clear coherence; namely, blue rib vortices are located between grey roller vortices and the axis of ribs is perpendicular to rollers; similarly, further smaller yellow vortices are located around blue rib vortices and the axes of these small vortices tend to be perpendicular to rib vortices. As will be shown in the following sections, these tendencies are due to the creation mechanism that smaller-scale vortices are amplified by being stretched around larger-scale ones. The supplementary movies 1–4 visualize the detailed dynamics.

Figure 4. Simultaneous plot of the vortices with different scales in figure 3(a–d). The subdomains (i)–(iii) indicate the three distinct regions examined in §§ 3 and 4. See supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.824.

We also emphasize that, although some smaller vortices exist within a larger vortex, they do not cluster to form the vortex; in other words, even if we lower the threshold of the isosurface for smaller-scale vortices, the isosurface gets connected but they do not form a larger vortex. These observations are similar to turbulence in a periodic cube (Goto 2008; Goto et al. 2017), turbulent boundary layers (Motoori & Goto 2019, 2020) and turbulent channel flow (Motoori & Goto 2021). However, the turbulent wake is more complex than these flows because the largest-scale vortices are different depending on regions. To clarify this feature, we examine in the following subsections the sustaining mechanism of the hierarchy of vortices in three distinct regions: (i) the downstream region, (ii) the recirculation region just behind the cylinder and (iii) the separated shear layer. We show the magnified visualizations in each region (i.e. the regions labelled by (i)–(iii) in figure 4) in figures 5–8, respectively.

Incidentally, in figures 3 and 4 we identify coherent vortices by using ${Q}^{\prime (\sigma )}$, which is estimated with the Reynolds decomposition. Appendix B shows that ${Q}^{(\sigma )}$, without the Reynolds decomposition, differs from ${Q}^{\prime (\sigma )}$ for larger scales because ${Q}^{(\sigma )}$ can capture not only these vortices identified by ${Q}^{\prime (\sigma )}$ but also a pair of quasi-stationary vortices just behind the cylinder. As will be discussed in § 4.2, these quasi-stationary vortices play crucial roles in the recirculation region. However, in this section we use ${Q}^{\prime (\sigma )}$ to identify vortices since the pair of large vortices identified by ${Q}^{(\sigma )}$ prevent us from observing small vortices near the cylinder.

3.1. Downstream region of the wake

First, we examine the downstream region of the wake where roller vortices exist; see figure 5(a), which corresponds to the subdomain labelled by (i) in figure 4. To see the temporal evolution of coherent vortices in this subdomain, we track the subdomain with $0.9U_\infty$ so that a pair of grey rollers remain in the frame. The subdomain is located in $1.2\leqslant x_1 \leqslant 4.6$ (figure 5a) at $t=0$, and it is in $3.4\leqslant x_1 \leqslant 6.8$ (figure 5b) at $t=2.4$. We observe in figure 5(a) that blue rib vortices tend to align to the stretching direction around the grey pairs of roller vortices. Figure 6 shows the direction of the vorticity $\boldsymbol {\omega }^{(\sigma )}$ of (a) roller and (b) rib vortices, and figure 6(c) shows the eigenvector $\boldsymbol {e}_1^{\prime (\sigma )}$ corresponding to the maximum eigenvalue of $S_{ij}^{\prime (\sigma )}$. We can see that the axes of rib vortices are indeed aligned to the stretching direction induced by the roller vortices. This implies that the rib vortices are amplified by vortex stretching, which will be quantitatively verified in the next section. Similar events were also observed in experiments (Lin, Vorobieff & Rockwell 1996) of the turbulent wake at $Re_D=10^4$. In the downstream region (figure 5b) blue rib vortices also tend to form counter-rotating pairs (see also figure 6b), while the roller vortices are weakened (see also figure 4), and further smaller vortices (yellow ones) are stretched and amplified in the straining regions around these blue vortex pairs. In fact, yellow vortices tend to perpendicularly align to blue ones, i.e. the stretching direction around counter-rotating pairs of rib vortices. Thus, the scale-by-scale cascading events originating from the shed roller vortices are evident in this downstream region. Supplemental movie 2 is also available online to observe these events.

Figure 5. Lagrangian tracking (with the velocity $0.82\overline {u_1}$ at $(x_1,x_2)=(5,0.6)$) of subdomain (i) that is framed with black lines in figure 4. The subdomain is located in (a) $1.2\leqslant x_1 \leqslant 4.6$ at $t=0$ and in (b) $3.4 \leqslant x_1 \leqslant 6.8$ at $t=2.4$. See supplementary movie 2.

Figure 6. Vorticity direction at (a) $\sigma = \sigma _{max}$ and (b) $\sigma _{max}/4$. (c) Stretching direction ${\boldsymbol {e}}_1^{(\sigma )}$ at $\sigma = \sigma _{max}$. We show the arrows only in vortices at scales (a) $\sigma = \sigma _{max}$ and (b,c) $\sigma _{max}/4$ shown in figure 5(a).

The above observation may imply that the origin of the $-5/3$ power law of the power spectrum (figure 2) in the downstream of the wake is similar to that in homogeneous turbulence. Since vortex-stretching events at different scales simultaneously occur in statistically steady turbulence, vortices at various ages coexist. This makes the observation of each elementary process obscure in turbulence in a periodic cube. In contrast, in a turbulent wake, since smaller-scale vortices are created successively as the flow develops downstream, the elementary process of the cascade is rather clear (figure 5 and movie 2). It is also important that, since the successive cascading events occur outwardly toward the turbulent/non-turbulent interface, Kolmogorov-scale vortices are dominant at the interface. This explains observation (Watanabe et al. 2014; Silva, Zecchetto & Da Silva 2018; Watanabe, Da Silva & Nagata 2019) that the interface is accompanied by a sharp change of the vorticity.

3.2. Recirculation region

Next, we examine the recirculation region (i.e. $-0.5\lesssim x_2\lesssim 0.5$) just behind the cylinder. Recall that the power spectrum $E_2(f)$ obeys the $-5/3$ power law even in this region (figure 2).

In figure 7(a), which is a magnification of subdomain (ii) of figure 4, we visualize the isosurface of $Q^\prime$ (black objects) and ${Q}^{\prime (\sigma )}$ with $\sigma =\sigma _{max}/4$ (blue) and $\sigma _{max}/16$ (yellow) in the same region. In this region blue vortices with $\sigma _{max}/4$ are quasi-stationary, which are shedding as roller vortices with $\sigma _{max}$ in the downstream. It is evident in figure 7(a) and its supplementary movie 3 that smaller yellow vortices tend to align in the $x_2$ direction, that is, the stretching direction in the velocity field induced by the quasi-stationary blue vortices. This implies that the yellow vortices are stretched and amplified by them.

Figure 7. (a) Magnification of subdomain (ii) that is framed with blue lines in figure 4. (b) Further magnification of the subdomain that is framed with blue lines in (a). See supplementary movie 3.

In figure 7(b) we further magnify a subdomain of figure 7(a). Further smaller (i.e. the Kolmogorov-scale) black vortices tend to be perpendicular to yellow ones. This implies that these vortices are also amplified by the vortex stretching. Incidentally, since the size of yellow vortices is $10$–$20\eta$, some yellow vortices are identical to black ones.

These observations in figure 7 are, again, similar to turbulence in a periodic cube except that the origin of the cascading events is the blue paired vortices in the wake instead of the largest vortices sustained by some external force in periodic turbulence. In other words, flow in the recirculation region is statistically stationary turbulence, which explains the immediate appearance of the $-5/3$ power law just behind a cylinder. More concretely, the quasi-stationary counter-rotating roller vortices are the source of the energy cascade to create the spectrum. This is consistent with the results obtained by the analysis of the KHMH equation; for example, Alves Portela et al. (2020) demonstrated the coherent roller vortices feed energy to stochastic motions, at smaller scales, in turbulence just behind the square prism ($x_1 = 2$). Incidentally, although Yasuda et al. (2020) suggested the possibility that the interaction between the shed roller vortices and the smaller-scale vortices in the shear layers (see the next subsection) led to the establishment of the $-5/3$ power-law spectrum, the present visualizations (figure 7) suggest a simpler process of the small-scale vortex generations.

3.3. Separated shear layer

Next, we examine the region ($0.2 \leqslant x_1 \leqslant 1.5$, $0\leqslant x_2\leqslant 1$) around one of the shear layers attached to the cylinder; see figure 8, which is a subdomain labelled by (iii) in figure 4. The transparent object in the figure is the cylinder. Figure 8(a) shows that upstream yellow vortices are parallel to the cylinder. We also show in figure 8(b) the rotational direction of these vortices identified by the vorticity. The black arrows show the same vorticity direction as the mean shear layer, while the grey ones show the opposite. We can see that these yellow vortices tend to form counter-rotating pairs. There also exist smaller black vortices, whose diameter is approximately $10\eta$, around these yellow vortices. Note that the grid width drawn on the bottom of the subdomain is $20\eta$ at $(x_1,x_2)=(0.8,0.5)$. The events of the amplification of smaller vortices are more evident in supplementary movie 4. The observation suggests that the yellow shear layer vortices are parents and they stretch and amplify smaller ones.

Figure 8. (a) Magnification of subdomain (iii) that is framed with red lines in figure 4. The width of the black grid indicates $20\eta$ at $(x_1,x_2)=(0.8,0.5)$. In (b) we show the vorticity direction of yellow vortices. Black arrows indicate negative vorticity in the spanwise component (same as the vorticity of the shear layer), whereas grey arrows indicate the positive one. See supplementary movie 4.

For $Re_D\gtrsim 6000$, vortices are deformed in the shear layer to interact with the shed vortices in the recirculation region (Norberg 2003). In such a case, although these parent vortices in the two regions can interact, the generation mechanism of sufficiently smaller vortices may not be altered; that is, they are generated through the successive vortex stretching. In fact, Aasland et al. (2022) demonstrated that smaller-scale streamwise vortices are generated by being stretched by the shear layer vortices even for $Re_D=10^4$.

3.4. Summary of observations

We have unraveled the generation mechanism of the hierarchies of coherent vortices in three distinct regions. In all of the regions, smaller vortices are generated by vortex stretching due to about four times (recall that the scale ratio between the successive filter scales is four in figures 3–8) larger vortices, but the origin of the cascading process is different in these three regions: (i) the shed roller vortices in the downstream of the wake, (ii) the counter-rotating pair of the quasi-stationary vortices in the recirculation region, and (iii) the shear layer vortices in the shear layer.

Since the three regions overlap, around their boundaries, we observe different generation mechanisms of vortices. For example, in figure 7 (i.e. recirculation region) yellow vortices in the bulk are stretched by quasi-stationary pairs (around $x_2 \approx 0$), while we also observe those are parallel to the cylinder in the shear layer (around $x_2 \approx \pm 0.5$). In figure 8 (i.e. the separated shear layer), while the upstream yellow vortices are generated in the shear layer, we also observe the downstream vortices aligned in the streamwise direction, which are stretched by quasi-stationary vortices and/or shed rollers.

In addition, in figure 4 we can see some of downstream yellow vortices ($\sigma =\sigma _{max}/16$) stretched by grey roller vortices ($\sigma =\sigma _{max}$). This is because, as will be shown quantitatively in the next section, the vortex stretching contributes to a wide range of scales, though the contribution is strongest with the scale ratio being about four.

Before closing this section, it is worth mentioning that many authors recently described the energy cascade in terms of different mechanisms: for example, the reconnection of anti-parallel vortices (Yao & Hussain 2022; Yao, Mollicone & Papadakis 2022) and the elliptical instability due to interactions between vortices (McKeown et al. 2018; Mckeown et al. 2020). It was also reported that strain-rate self-amplification plays a crucial role in energy transfer (Carbone & Bragg 2020; Johnson 2021). Although the other mechanisms can contribute to the energy transfer, in the following analyses we quantify the energy transfer due to the vortex stretching to verify that the events shown in the present section contribute to the transfer.

4. Scale-dependent energy transfer due to vortex stretching

To quantify the energy cascading events observed in the previous section, we define the energy transfer rates due to the vortex stretching as

(4.1)\begin{equation} \overline{\mathcal{T^\prime}}(\sigma_{S}\to\sigma_{\omega}) = \overline{( \omega_i^{\prime(\sigma_{\omega})} S_{ij}^{\prime(\sigma_S)} \omega_j^{\prime(\sigma_{\omega})} ) } \sigma_\omega^2 . \end{equation}

Here, $\,\bar{\cdot }\,$ denotes the average over $x_3$ and $t$. Since $\omega _i^{\prime (\sigma _{\omega })}$ is the vorticity evaluated from $u_i^{\prime (\sigma _\omega )}$ and since $S_{ij}^{\prime (\sigma _{S})}$ is the strain-rate tensor evaluated from $u_i^{\prime (\sigma _S)}$, the quantity $\omega _i^{\prime (\sigma _{\omega })} S_{ij}^{\prime (\sigma _{S})} \omega _j^{\prime (\sigma _{\omega })}$ in (4.1) indicates the contribution to the stretching of vorticity at $\sigma _{\omega }$ from strain-rate fields at $\sigma _{S}$ on average. Here, we have assumed that the off-diagonal terms $\omega _i^{\prime (\sigma _{\omega }^\prime )} S_{ij}^{\prime (\sigma _S)} \omega _j^{\prime (\sigma _{\omega })}$ with $\sigma _{\omega }^\prime \neq \sigma _{\omega }$ and the advection term have smaller contributions to the enstrophy production rate. The former assumption may be justified by the observation that vortices at different scales are located in distinct regions (Yoneda et al. 2022). Although the amplification of enstrophy due to vorticity advection seems weaker in three-dimensional turbulence, quantitative evidence is needed to justify the latter assumption. We have also assumed that the viscous effect is negligible in the inertial range ($\sigma _\omega \gg \eta$). This scale-dependent enstrophy production rate $\overline { \omega _i^{\prime (\sigma _{\omega })} S_{ij}^{\prime (\sigma _S)} \omega _j^{\prime (\sigma _{\omega })} }$ due to vortex stretching was also used to investigate the generation mechanism of the hierarchy of vortices in periodic turbulence (Goto et al. 2017), turbulent boundary layers (Motoori & Goto 2019) and turbulent channel flow (Motoori & Goto 2021). In the present study we multiply the enstrophy production rate by $\sigma _{\omega }^2$ to quantify the production rate of the energy at $\sigma _\omega$. Therefore, $\overline {\mathcal {T}^\prime }$ indicates the interscale energy transfer rate due to vortex stretching. When $\overline {\mathcal {T}^\prime }(\sigma _{S} \to \sigma _{\omega })$ is positive, the energy of vortices at $\sigma _{\omega }$ is amplified due to vortex stretching by strain rate at $\sigma _{S}$ on average; otherwise, the energy contained by vortices at $\sigma _{\omega }$ is weakened due to the vortex compression by vortices at $\sigma _S$. The definition of the energy transfer (4.1) is similar to the quantity defined by Yoneda et al. (2022) for homogeneous isotropic turbulence.

Note that the mean strain rate $\overline {S_{ij}}$ can also contribute to the enstrophy production in addition to (4.1). To quantify this contribution, we define

(4.2)\begin{equation} \overline{\mathcal{T}_{M}}\left(M\to\sigma_{\omega}\right) = \overline{( \omega_{i}^{\prime(\sigma_{\omega})} \overline {S_{ij}} \omega_{j}^{\prime(\sigma_{\omega})} )} \sigma_{\omega}^2 . \end{equation}

Note that, since $x_3$ direction is homogeneous ($\overline {S_{13}} = \overline {S_{31}} = \overline {S_{23}} = \overline {S_{32}} = \overline {S_{33}}=0$), only the components of $\overline {S_{12}}(=\overline {S_{21}})$ and $\overline {S_{22}}(=-\overline {S_{11}})$ are non-zero. Figure 9 shows the distribution of ($a$) $\overline {S_{12}}$ and ($b$) $\overline {S_{22}}$. In the downstream along the centreline ($x_2 \approx 0$), the mean strain rates are small; whereas, around $x_2\approx \pm 0.5$, the mean shear $|\overline {S_{12}}|$ can be as large as ${O}(1)$. Hence, in the free shear layer in the downstream regions (i) and separated shear layer (iii), $\overline {\mathcal {T}_M}$ indicates the contribution to the energy transfer from the mean shear $\overline {S_{12}}$.

Figure 9. Mean strain-rate fields (a) $\overline {S_{12}}$ and (b) $\overline {S_{22}}$. Only the region with $\overline {u_1}<1$ is shown.

In the recirculation region (ii) we do not use the Reynolds decomposition. This is because $\overline {\mathcal {T}_{M}}$ in this region does not quantify the contribution from the mean shear $\overline {S_{12}}$ but from the quasi-stationary vortices behind the cylinder. We therefore evaluate

(4.3)\begin{equation} \bar{\mathcal{T}}\left(\sigma_S \to\sigma_{\omega}\right) = \overline{ ( \omega_i^{(\sigma_{\omega})} S_{ij}^{(\sigma_{S})} \omega_j^{(\sigma_{\omega})} )} \sigma_{\omega}^2 \end{equation}

in the recirculation region (ii) instead of $\overline {\mathcal {T}^\prime }$ and $\overline {\mathcal {T}_M}$. Here, $\omega _i^{(\sigma _{\omega })}$ is the vorticity evaluated from the total velocity $u_i^{(\sigma _\omega )}$ at $\sigma _\omega$ without the Reynolds decomposition and $S_{ij}^{(\sigma _{S})}$ is the strain-rate tensor evaluated from $u_i^{(\sigma _S)}$. See Appendix B for the visualizations without the Reynolds decomposition.

In the following subsections, to quantify the energy transfer in the events observed in figures 4–8, we evaluate $\overline {\mathcal {T}^\prime }$, $\overline {\mathcal {T}_M}$ and $\bar {\mathcal {T}}$ under the condition of ${Q}^{\prime (\sigma _\omega )}>0$ (or ${Q}^{(\sigma _\omega )}>0$). We examine six scales ($\sigma _{max}$, $\sigma _{max}/2$, $\sigma _{max}/4$, $\sigma _{max}/8$, $\sigma _{max}/16$ and $\sigma _{max}/32$) at several positions in the downstream region (§ 4.1), the recirculation region (§ 4.2) and the separated shear layer (§ 4.3).

4.1. Downstream region of the wake

We show, in figure 10(a), $\overline {\mathcal {T}^\prime }$ (grey) at the location $(x_1,x_2)=(3,0)$, which is on the centreline in the downstream region (figure 5a). Note that the vertical axis denotes $\overline {\mathcal {T}^\prime }$ normalized by the mean energy dissipation rate $\bar {\epsilon }$ and the horizontal one denotes $\sigma _S / \sigma _\omega$. Looking at the black open squares ($\sigma _\omega = \sigma _{max}/32$), we see that the twice-larger-scale ($\sigma _S/\sigma _\omega = 2$) strain-rate fields contribute most to the energy transfer due to vortex stretching. We also see that the four times larger-scale ($\sigma _S/\sigma _\omega =4$) strain rates contribute secondly. These trends are also the case for the smaller scales $\sigma _\omega = \sigma _{max}/16$ (closed squares), $\sigma _{max}/8$ (open triangles) and $\sigma _{max}/4$ (closed triangles). Therefore, vortices smaller than the roller vortices ($\sigma _\omega \ll \sigma _{max}$) are stretched in the twice and four times larger-scale strain-rate fields, and through this process, they receive the energy. This is consistent with the visualizations (figure 5); namely, blue rib vortices with size $\sigma _{max}/4$ are stretched by the roller vortices, and further smaller vortices (yellow ones with size $\sigma _{max}/16$) are amplified by rib vortices.

Figure 10. Average energy transfer rate $\overline {\mathcal {T}^\prime }$ due to the vortex stretching by fluctuation strain rates at scale $\sigma _{S}$ to fluctuation vorticity at $\sigma _{\omega }=\sigma _{max}/32$ (open squares), $\sigma _{max}/16$ (closed squares), $\sigma _{max}/8$ (open triangles), $\sigma _{max}/4$ (closed triangles), $\sigma _{max}/2$ (open circles) and $\sigma _{max}$ (closed circles) at locations (a) $(x_1,x_2)=(3,0)$ and (b) $(3,0.6)$. Blue symbols indicate the transfer rate $\overline {\mathcal {T}_M}$ from the mean strain rates.

It is also important that $\overline {\mathcal {T}^\prime }(\sigma _S\to \sigma _\omega )$ for $\sigma _\omega \leqslant \sigma _{max} / 8$ collapse. This implies that, through a few steps of the scale-by-scale vortex stretching, small-scale statistics attain self-similarity. This observation is similar to that in turbulence in a periodic cube (Yoneda et al. 2022).

In addition, figure 10(a) shows that $\overline {\mathcal {T}^\prime }$ is negative for $\sigma _S < \sigma _\omega$. This implies that smaller vortices, which are created by being stretched around larger vortices, are likely to compress their parent vortices on average. The vortex compression by smaller vortices is also consistent with the observations in figure 5 and movie 2; for example, pairs of rib vortices induce strong straining fields, where smaller vortices are generated; whereas they also induce compressing fields on the other side of the pairs, where their parent (roller) vortices are weakened. Note that the vortex compression by smaller vortices implies the forward energy cascade. It is also important to see that, for small scales $\sigma _\omega (\leqslant \sigma _{max} / 8)$, the total energy transfer from larger scales ($\sigma _S> \sigma _\omega$) due to vortex stretching is approximately balanced with the negative transfer from small scales ($\sigma _S < \sigma _\omega$) due to the vortex compression; more precisely, although there is imbalance because of the viscous effect for the present $Re_D$, the energy transfer caused by vortex stretching and compression is balanced in the inertial range for higher Reynolds numbers (see figure 1(b) in Yoneda et al. 2022). We will develop further arguments on this point in § 5.

Next, we see, in figure 10(b), $\overline {\mathcal {T}^\prime }$ and $\overline {\mathcal {T}_M}$ at $(x_1,x_2)=(3,0.6)$ in the free shear layer in the downstream region. Similarly to the downstream core region (figure 10a), vortices at the small scales $\sigma _\omega \leqslant \sigma _{max}/8$ receive the energy by being stretched by twice and four times larger-scale strain rates. The difference is in $\overline {\mathcal {T}_M}$; more concretely, $\overline {\mathcal {T}_M}$ for vortices at $\sigma _\omega = \sigma _{max}/4$ (closed triangles) is the largest among the six scales and this value is comparable to $\overline {\mathcal {T}^\prime }(2\sigma _\omega \to \sigma _\omega )$. This means that rib vortices, which are stretched between counter-rotating pairs of the roller vortices in the core region, are also stretched by the mean shear around $x_2=\pm 0.6$. We emphasize that, however, rib vortices are initially stretched around $x_2\approx 0$ by the roller vortices (see movie 2).

4.2. Recirculation region

Next, we turn to the recirculation region (figure 7). Here, we evaluate $\mathcal {T}$ defined as (4.3), which contains the contribution from the quasi-stationary vortices. Figure 11 shows $\bar {\mathcal {T}}$ at $(x_1,x_2)=(0.65,0)$. Looking at the black open squares ($\sigma _\omega = \sigma _{max}/32$), we can see that the twice-larger-scale strain-rate fields contribute most to the energy transfer, and the four times larger-scale ones are next. These are also the case for $\sigma _\omega = \sigma _{max}/16$ (closed squares). These results are consistent with the events shown in figure 7 and supplementary movie 3; namely, yellow vortices with size $\sigma _{max}/16$ are stretched by the counter-rotating quasi-stationary vortices and further smaller black vortices are stretched by yellow ones.

Figure 11. Average energy transfer rate $\bar {\mathcal {T}}$ due to the vortex stretching by total strain rates at scale $\sigma _{S}$ to fluctuation vorticity at $\sigma _{\omega }=\sigma _{max}/32$ (open squares), $\sigma _{max}/16$ (closed squares), $\sigma _{max}/8$ (open triangles), $\sigma _{max}/4$ (closed triangles), $\sigma _{max}/2$ (open circles) and $\sigma _{max}$ (closed circles) at location $(x_1,x_2)=(0.65,0)$.

Here, we emphasize that the cascading events occur locally in the recirculation region. We show in figure 12 the ratio of the advection time scale $T_A=D/U$ ($U=\sqrt { \overline {u_i}^2 }$) to the cascading time scale $T_C = 1/S$ ($S=\sqrt { \overline {S_{ij}}^2 }$). In the darker colour region with $T_A/T_C > 1$, which is surrounded by the blue line ($T_A/T_C = 1$), the cascade is faster than the advection. The spatial locality of the energy cascade explains that the $-5/3$ power law, which was also observed in other wakes (e.g. Gomes-Fernandes et al. 2015; Alves Portela et al. 2017, 2020), is established immediately behind an obstacle. In contrast, in a downstream region ($x_1\gtrsim 2$), advection is faster than the cascade. This is the reason why we have tracked in § 3.1 vortices with the advection velocity.

Figure 12. Ratio between the cascading time ($T_c$) and the advection time ($T_A$) by the mean flow in the recirculation region. The blue line indicates $T_A/T_C=1$.

4.3. Separated shear layer

Figure 13 shows $\overline {\mathcal {T}^\prime }$ (grey) and $\overline {\mathcal {T}_M}$ (blue) at $(x_1,x_2)=(0.8,0.65)$ in the separated shear layer (figure 8). Similarly to the downstream and recirculation regions, to $\overline {\mathcal {T}^\prime }$ for small vortices (the open squares, $\sigma _\omega = \sigma _{max}/32$), the contribution from the twice-larger-scale strain-rate fields is most significant. This verifies the observation (figure 8 and movie 4) that the shear layer vortices stretch and amplify black smaller vortices. However, it is evident that the transfer $\overline {\mathcal {T}_M}$ from the mean flow is also significant even for the smallest-scale ($\sigma _{max}/32$) vortices. This is because $\sigma _\omega =\sigma _{max}/32\approx 7.6\eta$ is comparable with the Corrsin length scale $L_c= \bar {\epsilon }^{1/2} {\overline {S_{12}}}^{-3/2}$ ($\sigma _\omega =0.24L_c$). Motoori & Goto (2019) showed that vortices larger than $L_c$ are stretched mainly by the mean shear in turbulent boundary layers, and this is also the case in the separated shear layer. Incidentally, if the Reynolds number is higher, further smaller-scale ($\sigma _\omega \ll L_c$) vortices can be created by twice-larger-scale vortices.

Figure 13. Same as in figure 10 but the results for $(x_1,x_2)=(0.8,0.65)$.

4.4. Alignment between stretching direction and vorticity

In §§ 4.1–4.3 we have demonstrated that smaller vortices are generated by being stretched by larger (more, precisely, twice larger) vortices in all the three regions of the wake. Here, we show that this is consistent with the observation in § 3 that small vortices tend to be aligned in the straining regions induced by four times larger vortices.

To this end, we rewrite the scale-dependent enstrophy production rate in (4.1) as

(4.4)\begin{equation} \omega_i^{\prime(\sigma_\omega)} S_{ij}^{\prime(\sigma_S)} \omega_j^{\prime(\sigma_\omega)} = s_i^{\prime(\sigma_S)} (\boldsymbol{e}_i^{\prime (\sigma_S)} \boldsymbol{\cdot} \boldsymbol{\omega}^{\prime(\sigma_\omega)})^2 , \end{equation}

where $s_i^{\prime (\sigma _S)}$ ($s_1^{\prime (\sigma _S)} \geqslant s_2^{\prime (\sigma _S)} \geqslant s_3^{\prime (\sigma _S)}$ ) are the eigenvalues of $S_{ij}^{\prime (\sigma _S)}$ and $\boldsymbol {e}_i^{\prime (\sigma _S)}$ are the corresponding eigenvectors. Note that $s_1^{\prime (\sigma _S)}+s_2^{\prime (\sigma _S)}+s_3^{\prime (\sigma _S)}=0$ for incompressible fluids. We here define the alignment between the vorticity and stretching direction as

(4.5)\begin{equation} \cos \theta_{S,\omega} (\sigma_S, \sigma_\omega) =\boldsymbol{e}_1^{\prime (\sigma_S)} \boldsymbol{\cdot} \tilde{\boldsymbol{\omega}}^{\prime(\sigma_\omega)} , \end{equation}

where $\tilde {\boldsymbol {\omega }}^{\prime (\sigma _\omega )}$ is the normalized vorticity $\boldsymbol {\omega }^{\prime (\sigma _\omega )} / |{\omega }^{\prime (\sigma _\omega )}|$. A similar quantity was evaluated in periodic turbulence (Leung et al. 2012; Goto et al. 2017) and turbulent boundary layers (Motoori & Goto 2019) to show that small-scale vorticity tends to align with the stretching direction at the four times larger scale. Goto et al. (2017) also quantitatively showed that smaller vortices tend to be perpendicular to larger ones by using vortex axes identified with the low-pressure method (Miura & Kida 1997; Kida & Miura 1998).

To demonstrate this tendency in the turbulent wake, we plot in figure 14(a) the probability density function (p.d.f.) of $|\cos \theta _{S,\omega }|$ for $\sigma _\omega = \sigma _{max}/16$ at $(x_1,x_2)=(3,0.6)$. The probability $P$ of $|\cos \theta _{S,\omega }| > 0.9$ for $\sigma _S/\sigma _\omega \geqslant 2$ (darker lines) is larger than unity. This implies that smaller-scale vortices tend to align with the stretching direction at larger scales. Then, we show in figure 14(b) the probability $P$ of $0.9 < |\cos \theta _{S,\omega }| \leqslant 1$ for vortices at several scales ($\sigma _\omega = \sigma _{max}/4$, $\sigma _{max}/8$, $\sigma _{max}/16$ and $\sigma _{max}/32$). Looking at the results (closed triangles, $\sigma _\omega = \sigma _{max}/4$) for rib vortices, we see that these vortices are most likely to be aligned to the stretching direction at the scale four times as large as themselves. We also see vortices at scale $\sigma _{max}/16$ align most to the stretching direction of rib vortices ($\sigma _{max}/4$). These observations are consistent with those in § 3.1, that is, blue rib vortices are perpendicular to grey rollers; similarly, further smaller yellow vortices tend to be perpendicular to rib vortices. We can also see a similar tendency of the alignment for smaller-scale vortices (other symbols in figure 14b) and for other locations. In summary, as observed in the visualizations (figures 5–8) and in the quantification (figure 14), smaller vortices always tend to align perpendicular to four times larger vortices. However, this is not inconsistent with the result that the twice larger vortices contribute most to the energy transfer (figures 10, 11 and 13), since $s_i^{\prime (\sigma _S)}$ is larger for smaller scales. In other words, as a result of the competition between the magnitude of $s_i^{\prime (\sigma _S)}$ and the alignment of $\boldsymbol {e}_i^{\prime (\sigma _S)}$ and $\boldsymbol {\omega }^{\prime (\sigma _\omega )}$, the strain rate at scales twice as large as the vortices is dominant in the energy transfer due to vortex stretching.

Figure 14. (a) The p.d.f. of $|\cos \theta _{S,\omega }|$ for $\sigma _\omega = \sigma _{max}/16$ in the region $Q^{\prime (\sigma _\omega )} > 0$ at $(x_1,x_2)=(3,0.6)$. The six lines show, from the thinner (and darker) to thicker (and lighter), $\sigma _S/\sigma _\omega = 0.5$, $1$, $2$, $4$, $8$ and $16$. The dashed line shows the self-alignment (i.e. $\sigma _S/\sigma _\omega = 1$). (b) Probability of $0.9 < |\cos \theta _{S,\omega }| \leqslant 1$ for $\sigma _{\omega }=\sigma _{max}/32$ (open squares), $\sigma _{max}/16$ (closed squares), $\sigma _{max}/8$ (open triangles) and $\sigma _{max}/4$ (closed triangles).

5. Relevance to the energy cascade

As shown in § 4, vortices are stretched by larger ones and compressed by smaller ones on average. If this vortex stretching process corresponds to the energy cascade, the parent vortices must lose their energy as smaller-scale vortices acquire it. The energy reduction might be due to the vortex compression by smaller vortices, though our previous studies (Goto 2008; Goto et al. 2017; Motoori & Goto 2019, 2021; Yoneda et al. 2022) did not show clear evidence. Here, to describe the energy cascading process in instantaneous fields in terms of the vortex stretching and compression, we examine the downstream region where the multilevel hierarchy of coherent vortices exists.

First, to examine the energy transfer due to the vortex stretching by four times larger-scale vortices, we define

(5.1)\begin{equation} \mathcal{T}_{L\to S} = \left( \omega_i^{\prime(\sigma/4)} S_{ij}^{\prime(\sigma)} \omega_j^{\prime(\sigma/4)} \right)\sigma^2 . \end{equation}

This is the unaveraged quantity in (4.1) with the fixed scale of strain rates as $\sigma _S=4\sigma _\omega$; recall that vortices tend to align in the straining regions induced by the four times larger vortices (§ 4.4). When $\mathcal {T}_{L\to S}$ is positive, the larger vortices amplify the energy of smaller vortices through the vortex stretching; otherwise, they reduce the energy of smaller ones through the vortex compression.

We plot in figure 15(a) the active regions of positive and negative energy transfers due to vortex stretching and compression by the isosurfaces of positive (yellow) and negative (red) $\mathcal {T}_{L\to S}(=\pm 6.4\times 10^{-3})$ for $\sigma = \sigma _{max}$, respectively. For visibility, we do not display in panel (b) the yellow objects. Comparing these panels, we see that yellow isosurfaces coincide with most of the blue rib vortices. We can also see that the red regions are fewer than the yellow ones and they do not correlate with rib vortices. The observations are the direct evidence that rib vortices are created by being stretched by roller vortices.

Figure 15. Positive (yellow, shown only in panel $a$) and negative (red) isosurfaces of the energy transfer $\mathcal {T}_{L\to S}$ for $\sigma = \sigma _{max}$ through the vortex stretching and compression, respectively. The thresholds are $\mathcal {T}_{L\to S}= \pm 6.4 \times 10^{-3}$. Grey and blue objects are the isosurfaces of $Q^{\prime (\sigma _{max})}$ and $Q^{\prime (\sigma _{max}/4)}$, respectively. Their thresholds are the same as in figure 3.

To show the relevance of the scale-local vortex stretching to the energy cascade, it is necessary to demonstrate that the energy of larger vortices is reduced through the vortex interactions. For this purpose, we define the energy transfer from smaller to larger vortices as

(5.2)\begin{equation} \mathcal{T}_{S\to L} = \left( \omega_i^{\prime(\sigma)} S_{ij}^{\prime(\sigma/4)} \omega_j^{\prime(\sigma)} \right)\sigma^2 , \end{equation}

fixing the scale of strain rates to a quarter as large as that of vorticity for the unaveraged quantity in (4.1). When $\mathcal {T}_{S\to L}$ is negative, the energy of larger vortices is reduced by smaller ones through the vortex compression; otherwise, the energy of larger vortices is amplified by smaller ones.

Figure 16 shows the positive (yellow) and negative (red) isosurfaces of $\mathcal {T}_{S\to L}$ for $\sigma = \sigma _{max}$. Both compression (red) and stretching (yellow) regions are alternatively observed along the axis of roller vortices. This observation is quantified in figure 17(a) by the p.d.f. of $\mathcal {T}_{S\to L}$ inside the roller vortices at $\sigma _{max}$ (see the lightest grey line). We can see that the probabilities of both of $\mathcal {T}_{S\to L}>0$ and $\mathcal {T}_{S\to L}<0$ are not small. A rib vortex forms a counter-rotating pair with adjacent vortices, and therefore, they stretch and compress the roller vortices as observed in figure 16. However, we emphasize that, on average, the energy of a single roller vortex is reduced by rib vortices. To show this, we evaluate the spatially local average, $\langle \mathcal {T}_{S\to L} \rangle _Q$, of $\mathcal {T}_{S\to L}$ inside each of the individual vortices identified by the isosurface of $Q^{\prime (\sigma )}$, and we plot in figure 17(b) the p.d.f. of $\langle \mathcal {T}_{S\to L} \rangle _Q$. The lightest grey line shows the results for $\sigma =\sigma _{max}$. It is evident that negative values of $\langle \mathcal {T}_{S\to L} \rangle _Q$ are more dominant. This implies that rib vortices tend to compress roller vortices; consequently, roller vortices are weakened in total even in instantaneous fields. This tendency is also observed irrespective of the scale $\sigma$ of vortices (see the other lines in figure 17b).

Figure 16. Positive (yellow) and negative (red) isosurfaces of the energy transfer $\mathcal {T}_{S\to L}$ for $\sigma = \sigma _{max}$ through the vortex stretching and compression, respectively. The thresholds are $\mathcal {T}_{L\to S}= \pm 6.3 \times 10^{-3}$. Grey objects are the isosurfaces of $Q^{\prime (\sigma _{max})}$, whose threshold is the same as in figure 3.

Figure 17. The p.d.f. of (a) $\mathcal {T}_{S\to L} / \bar {\epsilon }$ and (b) $\langle \mathcal {T}_{S\to L} \rangle _Q / \langle \epsilon \rangle _Q$ for scales $\sigma =\sigma _{max}$ (light grey), $\sigma _{max}/2$ (grey), $\sigma _{max}/4$ (dark grey) and $\sigma _{max}/8$ (black). We normalize the values of the horizontal axis by (a) $\bar {\epsilon }$ at $(x_1,x_2)=(3,0)$ and by (b) the energy dissipation rate $\langle \epsilon \rangle _Q$ averaged inside each single vortex. In (b) we exclude the results for too small vortices (the volume of the identified vortices is smaller than $(2\sigma )^3$) from the p.d.f.

We have observed that larger vortices are not only compressed by smaller ones but also stretched. The latter phenomenon is reminiscent of the backscatter of energy. However, the present observation can be different from the backscatter observed in studies by LES or the KHMH equation because we examine the energy transfer due to vortex stretching and compression between two different scales. Nevertheless, since the stretching of larger vortices by smaller ones (yellow regions in figure 16) and the compression of smaller vortices by larger ones (red regions in figure 15) are physically interpretable, these events may provide clues about the physical mechanism of the backscatter of energy.

6. Conclusions

To investigate the hierarchy of coherent vortices in turbulent wake sustained behind a cylinder, we have conducted DNS of fully developed turbulence for $Re_D=5000$. By decomposing the obtained velocity fields with the band-pass filter (2.8), we extract vortices at different scales. The isosurfaces of the second invariant of the scale-decomposed velocity gradient tensor capture the three kinds of hierarchies of coherent vortices. To elucidate the generation mechanism of these vortices, we track them (see figures 4–8 and the supplemental movies) in the three distinct regions: (i) the downstream region in the wake, (ii) the recirculation region just behind the cylinder and (iii) the separated shear layers. In all the three regions, the generation mechanism of smaller vortices is common. They are generated in the direction perpendicular to four times larger vortices, being stretched by them (figures 5–8); however, the origin of the cascading process is different in these regions: (i) the shed roller vortices in the downstream region, (ii) a quasi-stationary pair of vortices in the recirculation region, and (iii) the large vortices in the separated shear layers. We have quantitatively verified these observations by evaluating the energy transfers (4.1) or (4.3) due to the vortex stretching (§ 4). More concretely, the twice and four times larger strain-rate fields contribute most to the vortex stretching on average. In conclusion, the generation mechanism of small-scale vortices in turbulence behind a cylinder is similar to that in periodic turbulence (Goto et al. 2017), turbulent boundary layers (Motoori & Goto 2019) and turbulent channel flow (Motoori & Goto 2021).

Although counter-rotating smaller vortices, which are created around larger ones, compress and stretch their parent vortices locally in space (figure 16), the total contribution to each parent vortex is dominated by the compression (figure 17b). Through this vortex compression of larger vortices by smaller ones, the energy at large scales is reduced and its energy is transferred to small scales due to the stretching of smaller vortices by larger ones. Although we cannot deny the possibility that another mechanism also leads to energy cascade, the present results, in particular the observation of the space-local energy transfer (figures 15–17), may suggest that the vortex stretching/compression processes can explain the energy cascade in turbulence.