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 Reference Amsden and Harlow1970; Kajishima & Taira Reference Kajishima and Taira2017). For the global mass conservation in the computational domain, we correct $u_1$ at the exit plane (Simens et al. Reference Simens, Jiménez, Hoyas and Mizuno2009) 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 Reference Uhlmann2005; Kempe & Fröhlich Reference Kempe and Fröhlich2012) 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. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). 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. Reference Parnaudeau, Carlier, Heitz and Lamballais2008). This secondary peak was also observed in the experiment (Ong & Wallace Reference Ong and Wallace1996) and DNS (Lehmkuhl et al. Reference Lehmkuhl, Rodríguez, Borrell and Oliva2013), 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. Reference Goto, Saito and Kawahara2017), turbulent boundary layers (Motoori & Goto Reference Motoori and Goto2019) and turbulent channel flow (Motoori & Goto Reference Motoori and Goto2021). 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. Reference Lee, Lee, Choi and Sung2014; Hwang et al. Reference Hwang, Lee, Sung and Zaki2016; Lozano-Durán et al. Reference Lozano-Durán, Holzner and Jiménez2016; Lee et al. Reference Lee, Sung and Zaki2017; Motoori & Goto Reference Motoori and Goto2019, Reference Motoori and Goto2021). Here, we employ the Gaussian filtering method adopted by Motoori & Goto (Reference Motoori and Goto2021) and Motoori, Wong & Goto (Reference Motoori, Wong and Goto2022). 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 $u_i^{\prime (\sigma )}$.

3.1. 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 Reference Lin, Vorobieff and Rockwell1996) 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. Reference Watanabe, Sakai, Nagata, Ito and Hayase2014; Silva, Zecchetto & Da Silva Reference Silva, Zecchetto and Da Silva2018; Watanabe, Da Silva & Nagata Reference Watanabe, Da Silva and Nagata2019) 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. (Reference Alves Portela, Papadakis and Vassilicos2020) 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. (Reference Yasuda, Goto and Vassilicos2020) 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 Reference Norberg2003). 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. (Reference Aasland, Pettersen, Andersson and Jiang2022) 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 Reference Yao and Hussain2022; Yao, Mollicone & Papadakis Reference Yao, Mollicone and Papadakis2022) and the elliptical instability due to interactions between vortices (McKeown et al. Reference McKeown, Ostilla-Mónico, Pumir, Brenner and Rubinstein2018; Mckeown et al. Reference Mckeown, Ostilla-Mónico, Pumir, Brenner and Rubinstein2020). It was also reported that strain-rate self-amplification plays a crucial role in energy transfer (Carbone & Bragg Reference Carbone and Bragg2020; Johnson Reference Johnson2021). 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.

4.1. 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. Reference Yoneda, Goto and Tsuruhashi2022).

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. Reference Yoneda, Goto and Tsuruhashi2022). 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. Reference Gomes-Fernandes, Ganapathisubramani and Vassilicos2015; Alves Portela et al. Reference Alves Portela, Papadakis and Vassilicos2017, Reference Alves Portela, Papadakis and Vassilicos2020), 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 (Reference Motoori and Goto2019) 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. Reference Leung, Swaminathan and Davidson2012; Goto et al. Reference Goto, Saito and Kawahara2017) and turbulent boundary layers (Motoori & Goto Reference Motoori and Goto2019) to show that small-scale vorticity tends to align with the stretching direction at the four times larger scale. Goto et al. (Reference Goto, Saito and Kawahara2017) 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 Reference Miura and Kida1997; Kida & Miura Reference Kida and Miura1998).

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).