4.1. Study of the survival function of the IVS

We start the discussion of the results with the investigation of the mean lifetimes of the IVS. In this process, we make the following assumption: branches arising from splitting or merging events are considered as individual IVS for the purpose of time tracking the IVS. Thus the background dynamics of the IVS, and possible interactions with other IVS, are not taken into consideration when analysing the lifetime statistics. In § 4.4 it will be shown that splits and mergers have a negligible effect on the observed lifetime.

The analysis is based on the concept of the survival function (SF) denoted as $S(\tau )$ , which is used to estimate lifetimes in different areas of science, medicine and engineering (Klein & Moeschberger Reference Klein and Moeschberger2003). If the lifetime of a structure ( $T$ ) is a continuous random variable, then the SF is the probability of the structure having a lifetime bigger than time $\tau$ , i.e. the SF is defined as

(4.1) \begin{equation} S(\tau ) = P(T \gt \tau ) = \int _{\tau }^{\infty } f(u)\, {\rm d}u, \end{equation}

where $f(\tau )$ is the probability density function (PDF) of the IVS, having a lifetime equal to $\tau$ . The cumulative distribution function (CDF) $F(\tau )$ is defined as the probability that the survival time is less than or equal to a specific time $\tau$ :

(4.2) \begin{equation} F(\tau ) = P(T \leqslant \tau ) = \int _0^{\tau } f(u)\, {\rm d}u, \end{equation}

i.e. $S(\tau ) = 1 - F(\tau )$ , since $\int _0^{\infty } f(u)\, {\rm d}u = 1$ . We measure the lifetime of each structure by subtracting its death time (when we can no longer follow the structure) from its birth time (when we start detecting it).

In the present work, all the lifetime analysis is based on the Kaplan–Meier (KM) estimator (Kaplan & Meier Reference Kaplan and Meier1958), which is a non-parametric method used to obtain the SF and to estimate lifetimes in different areas of science, medicine and engineering (Kaplan & Meier Reference Kaplan and Meier1958).

The KM estimator for the SF, denoted by $\hat {S}(\tau )$ , takes the following form when applied to the study of the lifetimes of the IVS:

(4.3) \begin{equation} \hat {S} ( \tau ) =\left \{ \begin{array}{ll} 1 , & \tau \lt \tau _{1}, \\ \displaystyle\prod _{\tau _{i}\leqslant \tau }\left [1-\dfrac {d_{i}}{r_{i}}\right ] , & \tau \geqslant \tau _{1}, \end{array} \right . \end{equation}

where $\tau _1$ is the smallest recorded lifetime, the $\tau _{i}$ form an ordered list containing the lifetimes of all tracked IVS, where $\tau _i$ is increasing with $i$ , $d_{i}$ is the number of structures that died at lifetime $\tau _{i}$ , and $r_{i}$ is the number of structures that are at risk of ‘dying’ at lifetime $\tau _i$ (that die at lifetimes greater than or equal to $\tau _{i}$ ). Naturally, at time $\tau _{1}$ , all the structures in the sample are at risk, thus $r_{1}=N_{s}$ . The actual use of this estimator is described in some detail in an example given below.

One big advantage of this estimator is that it allows one to use all the existing samples of IVS lifetimes. Recall that when the time tracking of the IVS is started (corresponding to time $t=0$ ), some of the IVS are already alive, and by the end of the simulated time, some IVS that will ‘die’ later are still alive. It follows that if these structures are considered in a ‘classical’ procedure to estimate the mean lifetime of the IVS, then the resulting estimated mean lifetime will be smaller than or equal to the ‘real’ one. The KM estimator allows one to use all the available data to obtain the SF without incurring this problem. Formally, this means that these IVS (from the start and end of the simulated time) are treated as right-censored data, while all the other IVS lifetime measurements are considered as fully observed (Kaplan & Meier Reference Kaplan and Meier1958). To clarify, in the use of (4.3) with censored data, censored structures are not classified as deaths. As an example, suppose that at lifetime $\tau _{i}$ we have five structures ( $r_{i}=5$ ), then one structure dies ( $d_{i}=1$ ) and one is censored ( $c_{i}=1$ ). Then at time $\tau _{i+1}$ , we have $r_{i+1}=r_{i}-d_{i}-c_{i} = 5-1-1=3$ , where $c_{i}$ is the number of censored structures, which are subtracted at stage $i$ .

Figure 5. Survival functions $\hat {S}(\tau ) = 1 - F(\tau )$ of the worms’ lifetimes obtained through (4.3), where data for lifetimes smaller than a given threshold value $\tau _{0}$ have been removed (filtered), for the DNS with (a) $Re_{\lambda }=54$ , (b) $Re_{\lambda }=91$ , (c) $Re_{\lambda }=143$ and (d) $Re_{\lambda }=189$ . (The DNS with $Re_{\lambda }=239$ show similar characteristics, not shown.) Here, $\tau$ is the random variable describing the lifetime, and $\tau _{\eta }$ is the Kolmogorov time scale. In the SF for $\tau _{0}/\tau _{\eta } = 0$ , all the lifetimes have been used, whereas in the other curves, lifetimes smaller than $\tau _{0}/\tau _{\eta } = 0.1, 0.2, 0.5, 1$ have been eliminated from the statistical sample.

The next two figures (figures 5 and 6) show the estimated SFs obtained by applying (4.3) to the lifetime data originated from all the DNS used in the present work, $\hat {S}(\tau ) = 1 - \hat {F}(\tau ) = 1 - \hat {P} (T/\tau _{\eta }\lt \tau /\tau _{\eta } )$ , where $\hat {P}(T \leqslant \tau )$ is the estimated CDF. As expected, all the SFs decrease as the lifetime increases. Interestingly, all the SFs are well approximated by an exponential function, provided that the number of available samples is sufficiently high. Appendix A compares the SF obtained with the KM estimator to that obtained using a ‘classical’ (straightforward) approach demonstrating the advantages of the former.

The mean lifetime of the IVS is simply the integrated SF; however, we need to address two issues, to which we now turn our attention before carrying out this integration. First, we need to address the effect of some of the thresholds used in the temporal time tracking procedure. One of these is the threshold value of the considered time itself, $\tau$ . It is plausible that data obtained for extremely small times will be ‘polluted’ by numerical artefacts, such as, for instance, when IVS oscillate very close to the detected threshold and can ‘appear’ and ‘disappear’ from the data for very small simulated times. We believe that this and similar non-physical behaviours are essentially present when too small lifetimes are considered. For that reason, we study how sensitive the lifetime distributions are to small times by eliminating them from the data, i.e. to when low time scale cut-offs are used. In short, we argue that representative IVS should live more than a certain time $\tau _{0}$ , and we investigate this effect by recomputing the CDF for our data by eliminating structures living less than a given time $\tau _{0}$ .

Figure 6. Survival functions $\hat {S}(\tau ) = 1 - F(\tau )$ of the worms’ lifetimes obtained through (4.3), with different final simulated times $t_{f}$ , for the DNS with (a) $Re_{\lambda }=54$ , (b) $Re_{\lambda }=91$ , (c) $Re_{\lambda }=143$ and (d) $Re_{\lambda }=189$ . (The DNS with $Re_{\lambda }=239$ show similar characteristics, not shown.) In these curves, times smaller than $\tau _{0} = 0.5 \tau _{\eta }$ have been removed (filtered) following the discussion in figure 5. Here, $\tau$ is the random variable describing the lifetime, and $\tau _L$ is the integral time scale ( $\tau _{\eta }$ is the Kolmogorov time scale).

Figure 5 shows the effect of $\tau _0$ on the SFs for $\tau _0$ in the range $ 0 \leqslant \tau _{0}/\tau _{\eta } \leqslant 1$ . In the SF for $\tau _{0}/\tau _{\eta } = 0$ , no removal was done, i.e. all the lifetimes have been used. The figure shows that using different cut-offs $\tau _{0}/\tau _{\eta }$ affects the SF shape; however, for each simulation, the SF converges to virtually the same function for $\tau _{0} \geqslant 0.5 \tau _{\eta }$ . For example, there is virtually no diference between the SFs for the DNS of $Re_{\lambda }=143$ (figure 6 c) corresponding to $\tau _{0}/\tau _{\eta } = 0.5$ and $\tau _{0}/\tau _{\eta } = 1$ . Similar results are observed for the other cases. We therefore proceed with our analysis by eliminating all the structures displaying lifetimes smaller than $\tau _{0}/\tau _{\eta }=0.5$ , as it is likely that events lasting less than half a Kolmogorov time are affected by artificial numerical ‘noise’ described above.

The second variable that one needs to investigate is the total time duration of the simulations, $t_f$ . In order to investigate the effect of this variable in the results obtained, we fix $\tau _{0}/\tau _{\eta }=0.5$ and repeat the SF estimation for different simulation times $t_{f}$ . Figure 6 shows the resulting SF for all the simulations carried out in this work, which can be seen as temporal evolutions of this estimation. It is clear that as the final simulated time increases, the SFs converge into the same curve (for each simulation), with an approximately constant slope, where the tail of the SF extends to longer times as $t_{f}$ increases. An implication of this observation is that very-long-living structures are relatively rare and are ‘seen’ only for sufficiently large time windows. Also, for each case, there is virtually no difference between the SFs obtained for the two biggest simulated times. For instance, for the DNS of $Re_{\lambda }=143$ (figure 6 c) , the curves corresponding to $t_{f}/\tau _L = 5.8$ and $7.4$ , are virtually the same. This shows that the total (final) simulated time ( $t_f/\tau _L \approx 6$ – $7$ ) is sufficient to capture the details of the SF for each case.

Following the results from the above discussion, we now proceed with the estimation of the mean lifetimes of the IVS by using the appropriate data for each simulation, i.e. we will be using the data from the green curves from figures 6(a–d), which comprise the entire simulated times for the DNS, $t_f/\tau _L \approx 6$ – $7$ , and by eliminating events (lifetimes) lasting less than $\tau _{0}/\tau _{\eta }=0.5$ .

4.2. Computing the characteristic lifetime of the IVS

Having now determined what data can be used to compute the SF (using only IVS with lifetimes greater that $\tau _{0}/\tau _{\eta }=0.5$ ) and that the total simulated time is sufficient to accurately compute these functions in each case, we now use three different methods to compute the characteristic lifetime of a structure, which we denote by $\tau _{ivs}$ . We will see that the three methods lead to similar values of the characteristic lifetime, and that the results allow us to establish the scaling laws associated with this quantity. In any case, whatever the method used to estimate the SF, it is important to stress that the mean lifetime is always given by the integration of the SF, i.e. for a continuous SF ( $S(\tau )$ ), we have

(4.4) \begin{equation} \tau _{ivs} = \int _0^{\infty } S(\tau )\, {\rm d}\tau = \int _0^{\infty } \tau\, f(\tau )\, {\rm d}\tau , \end{equation}

where the second equality stems from the definition of the SF in (4.1).

Formally, the restriction on $\tau _0$ discussed above means that the PDF $f ( \tau )$ for the lifetime is defined as

(4.5) \begin{equation} f ( \tau ) =\left \{ \begin{array}{ll} 0 , & 0\leqslant \tau \lt \tau _{0}, \\ h ( \tau ) , & \tau \geqslant \tau _{0}, \end{array} \right . \end{equation}

where $h ( \tau )$ is a restricted PDF (whose CDF is $F(\tau ) = \int _{\tau _0}^{\tau } h(t')\, {\rm d} t'$ ), while the SF now becomes

(4.6) \begin{equation} S ( \tau ) =\left \{ \begin{array}{ll} 1 , & 0\leqslant \tau \lt \tau _{0}, \\ 1-\displaystyle\int _{\tau _{0}}^{\tau }h(t^{\prime })\,{\rm d}t^{\prime } , & \tau \geqslant \tau _{0}, \end{array} \right . \end{equation}

since the CDF and the SF are defined by $P (T/\tau _{\eta }\lt \tau /\tau _{\eta } )$ and $S ( \tau )=1-P (T/\tau _{\eta }\lt \tau /\tau _{\eta } )$ , respectively, as before.

With these definitions, the mean lifetime of the IVS can now be estimated by

(4.7) \begin{equation} \tau _{ivs} = \int _{0}^{\infty } S ( \tau )\, {\rm d} \tau = \tau _{0}+\int _{\tau _{0}}^{\infty }\left [1-\int _{\tau _{0}}^{\tau }h (t^{\prime })\,{\rm d}t^{\prime }\right ]{\rm d}\tau . \end{equation}

Notice that in figures 5(a–d), the displayed functions are precisely $1 - P (T/\tau _{\eta }\lt \tau /\tau _{\eta } ) = 1-\int _{\tau _{0}}^{\tau }h (t^{\prime } )\,{\rm d}t$ , and the horizontal axis represents the shifted time $ (\tau - \tau _{0} )$ , which means that the integral in (4.7) is simply the area enclosed by the SFs.

As remarked above, a careful analysis of figures 5 and 6 suggests that the SFs are well approximated by an exponential function for $\tau \geqslant \tau _0$ :

(4.8) \begin{equation} \hat {S}(\tau ) = 1 - \hat {P}(T/\tau _{\eta } \lt \tau /\tau _{\eta }) \approx {\rm e}^{ - ( \tau - \tau _{0} )/\zeta }, \end{equation}

where $\zeta$ is the inverse of the decay rate associated with the SF, or the mean lifetime of the IVS ( $\tau _{ivs}$ ). The use of an exponential function greatly simplifies the computation of the mean lifetime of the IVS, which is simply

(4.9) \begin{equation} \tau _{ivs} = \tau _0 + \int _{\tau _0}^{\infty } {\rm e}^{ - ( \tau - \tau _{0})/\zeta }\, {\rm d}\tau = \tau _0 + \zeta , \end{equation}

if one considers an infinite time of recorded events.

The first (straightforward) method to estimate the mean lifetime of the IVS consists on using the so-called restricted mean survival time (RMST) (Royston & Parmar Reference Royston and Parmar2013), whereby the SF is integrated up to a specific time point $\tau _{{max}}$ :

(4.10) \begin{equation} \tau _{ivs}^{RMST} = \tau _0 + \int _{\tau _0}^{\tau _{{max}}} \hat {S} ( \tau )\, {\rm d} \tau , \end{equation}

where $\hat {S} ( \tau )$ represents the estimated SF, and $\tau _{{max}}$ is the measured lifetime for the longest-lived structure. Specifically, this value was assigned to $(\tau _{{max}} - \tau _0) / \tau _{\eta } = 21.74, 22.52, 38.16, 37.01$ for the DNS with $Re_{\lambda }=54, 91, 143, 189$ , respectively. Using this approach, the mean lifetime of the IVS is simply given by (4.10), where the integration is performed until the longest detected time $\tau _{{max}}$ for each one of the simulations used here.

A second method to estimate the mean lifetime of the IVS consists in using the maximum likelihood estimator (MLE) (Klein & Moeschberger Reference Klein and Moeschberger2003). In this case, and again by assuming that the SF is described by an exponential function (4.8), the parameter $\zeta$ can be estimated by maximising the probability to obtain the existing/available data. Formally, for right-censored data, the so-called likelihood function for the parameter $\zeta$ is given by

(4.11) \begin{equation} L (\zeta )=\prod _{i} f ( \tau _{i})\prod _{n} S ( \tau _{n}) , \end{equation}

where $S ( \tau )=\exp [- ( \tau - \tau _{0} )/\zeta ]$ is again the SF defined by the parameter $\zeta$ , and $f ( \tau ) = \exp [- ( \tau - \tau _{0} )/\zeta ]/\zeta$ is the PDF of the IVS lifetimes, also assumed to follow an exponential distribution, consistently with the assumed form of the SF (and with the observed shape of the PDF shown in Appendix A). Notice that in (4.11), the $i$ index runs for all fully observed data, whereas the $n$ index runs for all right-censored data. With the value of $\zeta$ obtained in this way, the mean lifetime of the ‘worms’ using this estimator is $\tau _{ivs}^{{MLE}} = \tau _0 + \zeta$ .

Finally, the third method (Fit) consists in assuming that the SF obtained for the simulations represented in figures 6(a) and 6(d) is defined by an exponential function, and obtaining the parameter for each curve, namely $\zeta$ , by using the least squares regression line. As in the other methods, the mean lifetime of the IVS is simply $\tau _{ivs}^{Fit} = \tau _0 + \zeta$ .

Table 4 displays the mean lifetime of the IVS ( $\tau _{ivs}$ ) obtained by employing the three methods described above. The mean lifetimes are normalised with the Kolmogorov time ( $\tau _{\eta }$ ) and with the integral time scale ( $\tau _L$ ). It is reassuring to see that the three different methods lead to very similar mean lifetime values for all DNS data.

Table 4. Mean lifetimes of the IVS estimated with three different methods, normalised by the Kolmogorov time $\tau _{\eta }$ and by the integral time scale $\tau _{L}$ , for all the DNS used in the present work: number of collocation points ( $N^3$ ); Taylor-microscale-based Reynolds number ( $Re_{\lambda }$ ); mean lifetime using the restricted mean survival time ( $\tau _{ivs}^{{RMST}}$ ); mean lifetime using the maximum likelihood estimator considering an exponential distribution ( $\tau _{ivs}^{{MLE}}$ ); mean lifetime measured by fitting the KM estimate for the SF ( $\tau _{ivs}^{{Fit}}$ ). The last row corresponds to the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ).

The mean lifetime is typically equal to $\tau _{ivs} / \tau _{\eta } \approx 3$ – $4$ and $\tau _{ivs} / \tau _{L} \approx 0.1$ – $0.3$ . This is much smaller than the integral time scale usually claimed as being the typical lifetime of the IVS. Our explanation, supported in several animations, is that even though many of these long lifetime structures do exist, as inferred from the SFs in figures 6(a–d), they represent extreme lifetime events, which are relatively rare, even if probably these are the structures that we retain in our mind from movies and visualisations. Indeed, from figures 6(a–d), one can see that some IVS have lives that are much longer than the mean, with recorded lifetime events $\approx 15 \tau _{\eta }$ , $\approx 22 \tau _{\eta }$ , $\approx 35 \tau _{\eta }$ and $\approx 38 \tau _{\eta }$ for the DNS with $Re_{\lambda }=54, 91, 143$ and $189$ , respectively. From table 1, one can see that these extreme lifetimes correspond to $\approx 1.8 \tau _L$ , $\approx 1.9 \tau _L$ , $\approx 2.0 \tau _L$ and $\approx 1.6 \tau _L$ , respectively. This suggests that the extremes scale with the integral time scale $\tau _L$ , but not the mean.

Biferale et al. (Reference Biferale, Scagliarini and Toschi2010) also observed extreme lifetime events of the order of the integral time scale; however, their computed mean values were $\tau _{ivs} /\tau _{\eta } \approx 17$ and $25$ for DNS with $Re_{\lambda }=65$ and $180$ , respectively, which is 3–5 times higher than our values, and reveals an opposite trend with the Reynolds number. The difference can be attributed to the indirect procedure used in that work, which involved the clustering of light particles inside the vortex cores. As recognised in that paper, particle clustering requires persistent vortices, since the particles need time to move into the vortex cores. Hence their detection method may favour long-lived structures. Also, the analysis of lifetimes from particles trapped inside the IVS does exclude situations of IVS merging or splitting that are differentiated in the present study. Another difference is the chosen magnitude of the threshold used to detect the IVS. Moreover, it is unclear how the small sample size used in that work might have affected their result, and whether care was taken to exclude lives from the start and end of the simulations. Arguably, the present tracking method is the first rigorous attempt at investigating this problem because of the method being ‘direct’ (the IVS are tracked in time, not their effects) and the number of samples is very large.

Finally, from table 4 it is clear that the lifetime values show a decreasing trend with the increase of the Reynolds number, i.e. the available data suggest that the mean lifetime of the IVS at high Reynolds numbers does not scale with either $\tau _{\eta }$ or $\tau _L$ .

To allow a closer look at the effects of the Reynolds number on the shapes of the SFs, figure 7 gathers all the SFs used here, for the different Reynolds numbers available in the present work. Again, we are seeing the same SF already displayed in figures 6(a–d), together with the DNS with $Re_{\lambda }=239$ , where only $1/8$ of the computational domain is used to time track the IVS. This does not affect the mean lifetime of the IVS for this last (bigger) DNS, but it may limit the amount of extreme events available in this simulation (see Appendix C). We can see here more clearly than before that as the Reynolds number increases, the slope of each curve – and consequently the area under the curve (which represents the mean lifetime) – decreases. This is consistent with the results shown in table 4. However, we also observe another interesting result, which is that the duration (size) of the recorded lifetimes also increases with the Reynolds number. Whereas for $Re_{\lambda } =54$ no single recorded IVS lives longer than $\tau _{ivs}/\tau _{\eta } =25$ , for $Re_{\lambda } =189$ many IVS live longer than $\tau _{ivs}/\tau _{\eta } =35$ . Not only do the longest recorded lifetimes increase with the Reynolds number, following approximate $\tau _L$ scaling, but also the total ‘spectrum’ of existing lifetime events increases. This result is probably not surprising since the increase of the Reynolds number is typically associated with an increase in the multiplicity of length, velocity and time ‘events’ in turbulent flows, but it is nonetheless interesting to observe it here in the context of the analysis of the lifetimes of the IVS. (This trend is not visible in the bigger DNS probably because of the use of only $1/8$ of the computational domain.)

Figure 7. Comparison between the several SFs obtained for the different Reynolds numbers available in the present work. Each SF shown here is the reference SF used in the discussion of the lifetime already displayed in figures 6(a–d), together with the DNS with $Re_{\lambda }=239$ . Here, $\tau$ is the random variable describing the lifetime, while $\tau _{\eta }$ is the Kolmogorov time scale.

Figure 8. Mean lifetimes of the IVS computed with three different methods ( $\tau _{ivs}^{{RMST}}$ , $\tau _{ivs}^{{MLE}}$ , $\tau _{ivs}^{{Fit}}$ ), normalised with the Kolmogorov time scale ( $\tau _{ivs}/\tau _{\eta }$ ), for all the DNS used in the present work (symbols). The data points are compared with several lines representing different scaling laws: Kolmogorov time scale ( $\tau _{\eta } = (\nu /\varepsilon )^{1/2}$ ), turnover time scale ( $\tau _{rot} = \eta /u'$ ), and effective turnover time scale ( $\tau _{eff} = R_{ivs} / U_{ivs}$ ). All the scaling curves use as reference the data point with $Re_{\lambda } = 143$ , except the curve (line) corresponding to $\tau _{\eta }$ scaling, which uses the smallest Reynolds number case for clarity.

In order to clarify the scaling law associated with the mean lifetime of the IVS, figure 8 shows the mean times obtained previously ( $\tau _{ivs}^{{RMST}}$ , $\tau _{ivs}^{{MLE}}$ , $\tau _{ivs}^{{Fit}}$ ), represented by symbols, normalised with the Kolmogorov time scale ( $\tau _{ivs}/\tau _{\eta }$ ). The data are compared with three scaling laws involving the Kolmogorov time scale ( $\tau _{\eta }$ ), turnover time scale ( $\tau _{rot}$ ), and effective turnover time scale ( $\tau _{eff}$ ). For small and intermediate Reynolds numbers ( $Re_{\lambda } \leqslant 100$ ), the results suggest that the mean lifetime of the IVS scales with the Kolmogorov time $\tau _{ivs} \sim \tau _{\eta }$ ; however, it is clear that as the Reynolds number increases, the mean lifetime of the ‘worms’ no longer scales with this law. Instead, the results show that the mean lifetime of the IVS scales with the turnover time scale or with the effective turnover time scales at higher Reynolds numbers. Notice that a slight Reynolds number effect may still be present for the case with Reynolds number $Re_{\lambda } = 143{-}189$ . Indeed, as mentioned above, Ghira et al. (Reference Ghira, Elsinga and da Silva2022) have shown that the asymptotic scaling law for $U_{ivs}$ , which influences the effective turnover time scale, is ‘settled’ only for Taylor-based Reynolds numbers greater than $Re_{\lambda } \gtrsim 200$ . The two largest available Reynolds numbers ( $Re_{\lambda } = 189$ – $239$ ) are approximately within this region, where the results suggest that the mean lifetime scales more closely with the effective time scale than with the rotation time scale, i.e. $\tau _{ivs} \sim \tau _{eff}$ .

Notice that at sufficiently high Reynolds numbers, there should be no difference between the scaling with the turnover time scale, or with the efective turnover time scale, since $\eta /u' \sim R_{ivs}/U_{ivs}$ . However, the separate assessment carried out here allows one to ‘isolate’ potential intermediate Reynolds number effects, and clearly show that the time of a single turnover of IVS is the time scale defining its lifetime. An implication of this scaling law $\tau _{ivs} \sim \tau _{rot}$ (or $\tau _{ivs} \sim \tau _{eff}$ ) is that the mean lifetime of the ‘worms’ decreases with increasing Reynolds numbers, a fact that may have implications in other areas, such as internal intermittency. Indeed, a decreasing $\tau _{ivs}/\tau _{\eta }$ is qualitatively consistent with the decreasing time scales associated with local regions of intense turbulence (e.g. Elsinga, Ishihara & Hunt Reference Elsinga, Ishihara and Hunt2020).

Figure 9. Comparison between the several SFs obtained for the different Reynolds numbers shown in figure 7, normalised by (a) the turnover time scale ( $\tau _{rot}$ ), and (b) the effective time scale ( $\tau _{eff}$ ). Also shown in (a) is the SF for the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ).

Figures 9(a) and 9(b) show the SFs where the lifetime is normalised by the turnover time scale ( $\tau _{rot}$ ) and by the effective time scale ( $\tau _{eff}$ ), instead of the Kolmogorov time scale ( $\tau _{\eta }$ ) used previously in figure 7. One can see that as the Reynolds number increases, the SFs approach each other when $\tau _{rot}$ is used (figure 9 a), but tend to colapse better when $\tau _{eff}$ is used (figure 9 b). This observation again supports the above described scaling law for the lifetime of the IVS, i.e. $\tau _{ivs} \sim \tau _{eff}$ . As discussed above, the higher Reynolds number cases used here are already sufficiently high to allow the inference of this scaling law.

An anonymous Referee noted that the exponential distribution of the SFs implies a memoryless history of the IVS. Figure 9 (as previous figures 5, 6 and 7) again shows that all the SFs exhibit an exponential distribution for most of the range. A memoryless history of the IVS is consistent with the classical role of the small scales in a turbulent flow. Note, however, that this is not true for the extreme lifetime events. Indeed, the final parts of all the SFs deviate from an exponential distribution, since the SFs decay much faster. This suggests that extreme SF events, representing IVS living much longer than average, may include some memory effects. It is plausible that bigger IVS might somehow be connected to the large scales of the flow, and therefore also to the details of the forcing. Unfortunately, we do not know at present if these extreme lifetime events are correlated with either the size or the vorticity magnitude of the IVS. Preliminary analysis of a few movies suggests that this might be the case; however, the dynamical aspects of the IVS deserve a detailed and separate investigation. These movies, carried out for the smaller Reynolds number case, show that typical IVS are born small and grow by incremental lengthening, where two or more branches emerge connected together, in a very fast process. They typically die in opposite fashion, when a long axis is broken into smaller-sized axes. It is not clear at present if this scenario is dominant (for the majority of the IVS) and still valid for high Reynolds number cases.

Finally, figure 9(a) also shows the SF for the DNS with the forcing centred at $k_p = 4$ to assess the possible effects of the forcing in these extreme lifetimes events. As can be seen, the SF for this case follows the same exponential distribution for the reference case with $k_p = 2$ (the SFs from the two cases overlap), showing that the forcing does not affect the lifetimes except for the extremely rare events.

For completeness, table 5 lists the mean lifetime of the IVS normalised with both turnover time scales for all the DNS used in the present work. It is noteworthy that using this normalisation with the turnover time scales, the variation between the lifetime values in the last two consecutive Reynolds numbers is very mild, and much smaller in percentage than the variations listed in table 4, where the lifetimes were normalised with the Kolmogorov time scales. This observation is consistent with the mean lifetime of the IVS scaling with the turnover time scales of these structures. The results from table 5 show that the mean lifetimes for the highest Reynolds numbers attain $\tau _{ivs} \approx 26 (\eta /u')$ and $\tau _{ivs} \approx 6.5 (R_{ivs}/U_{ivs})$ , and since the perimeter of the vortex cores is $2\pi R_{ivs}$ , on average each structure turns only one time around its axis before being dissipated, i.e. before its vorticity drops below the detection threshold. Finally, table 5 shows the mean lifetime values for the DNS with different forcing ( $k_p=4$ ). The values (whatever the normalisation) are very close to those of the reference DNS ( $k_p=2$ ) at the same Reynolds number $Re_{\lambda } = 54$ , which demonstrates that modifications of the forcing location do not affect the mean lifetime statistics.

Table 5. Mean lifetimes of the IVS estimated with three different methods, normalised by the turnover time scale ( $\tau _{rot} = \eta /u'$ ) and by the ‘effective’ turnover time scale ( $\tau _{eff} = R_{ivs}/U_{ivs}$ ) for all the DNS used in the present work: number of collocation points ( $N^3$ ); Taylor-microscale-based Reynolds number ( $Re_{\lambda }$ ); mean lifetime using the restricted mean survival time ( $\tau _{ivs}^{{RMST}}$ ); mean lifetime using the maximum likelihood estimator considering an exponential distribution ( $\tau _{ivs}^{{MLE}}$ ); mean lifetime measured by fitting the KM estimate for the SF ( $\tau _{ivs}^{{Fit}}$ ). The last row corresponds to the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ).

4.3. The population of the IVS

Using the lifetime data available, it is possible to build a population model that estimates the number of existing (‘alive’) IVS, as well as the number of ‘births’ and ‘deaths’ of these structures for a given time in a simulation. For this purpose, we postulate that the population model follows a simple balance equation,

(4.12) \begin{equation} \frac {{\rm d}N}{{\rm d}t} = \frac {{\rm d}N_B}{{\rm d}t} - \frac {{\rm d}N_D}{{\rm d}t}\,, \end{equation}

where $N(t)$ is the number of existing IVS at a given time $t$ , $N_{B}$ is the accumulated number of births (thus ${\rm d}N_{B}/{\rm d}t$ represents their birth rate) and $N_D$ is the accumulated number of deaths ( ${\rm d}N_D/{\rm d}t$ is the death rate). Since the flow is statistically stationary, by assuming that the birth rate is constant, ${\rm d}N_{B}/{\rm d}t = C_B$ , and noting that the death rate can be written using the mean lifetime as ${\rm d}N_D/{\rm d}t = N/\tau _{ivs}$ , we can rewrite (4.12) as

(4.13) \begin{equation} \frac {{\rm d}N}{{\rm d}t} = C_B - \frac {N}{\tau _{ivs}}\,, \end{equation}

which yields the solution

(4.14) \begin{equation} N (t) = N_{0}\,{\rm e}^{-t/\tau _{ivs}} + C_B \tau _{ivs } [1-{\rm e}^{-t/\tau _{ivs}}] , \end{equation}

where $N_{0}=N (0 )$ is the initial number of individuals/worms at the initial time (when the time tracking starts). Note that the key step leading to (4.13), i.e. the death rate relation ${\rm d}N_D/{\rm d}t = N/\tau _{ivs}$ , stems from the observed exponential distribution of the lifetimes discussed in the previous subsection. Indeed, since the probability of a structure surviving up to time $t$ and ‘dying’ in the interval $t+{\rm d}t$ , which is ${\rm d}N_{D}/N$ , is equal to ${\rm d}t / \tau _{ivs}$ , the exponential distribution of the lifetimes naturally leads to the proposed death rate relation.

In the limit of very long simulated times $t \rightarrow \infty$ , the population (or the number) of existing IVS tends to a constant, which is equal to

(4.15) \begin{equation} N\left (t \rightarrow \infty \right ) = N_{\infty } = C_B \tau _{ivs}. \end{equation}

Thus $N_{\infty }$ , representing the statistically stationary state, is related to the birth rate $C_B$ and the mean lifetime determined in § 4.2. The birth rate is as yet unknown and is determined below. Moreover, since (4.12) is equivalent to

(4.16) \begin{equation} N(t) = N_B(t) - N_D(t) + N_{0}, \end{equation}

the number of accumulated deaths is

(4.17) \begin{equation} N_D (t) = C_B t - N_{0}\,{\rm e}^{-t/\tau _{ivs}} - C_B \tau _{ivs } [1-{\rm e}^{-t/\tau _{ivs}}] + N_{0}, \end{equation}

since

(4.18) \begin{equation} N_B (t) = C_B t \end{equation}

(due to assuming a constant birth rate). Notice that statistical stationarity implies $N_{\infty } = C_{B}\tau _{ivs} = N_{0}$ and $N_{B}=N_{D}$ .

Figure 10. Instantaneous number of existing (‘alive’) structures for the full analysed data ( $\tilde {N}(t)$ ), and for the filtered data ( $\hat {N}(t)$ ), obtained directly from the several DNS data cases, compared with the prediction obtained from the population model derived from (4.16), $N(t)$ . The black frame delimits the data from the initial ( $t_m$ ) and final ( $t_n$ ) times where the filtered data and the actual data overlap. The curves are given for the DNS with (a) $Re_{\lambda }=54$ , (b) $Re_{\lambda }=91$ , (c) $Re_{\lambda }=143$ and (d) $Re_{\lambda }=189$ . (The DNS with $Re_{\lambda }=239$ show similar characteristics, not illustrated.)

In order to test this population model, we consider a time window where we track structures that are born for times $t \gt t_{0}$ , and stop considering new structures after a time $t_{n}$ , which is defined as the largest living time of all the structures still alive by the end of the simulated time $t_{f}$ . In practice, this means that with this time window we eliminate (or filter) all structures that are alive at $t=t_{0}$ and at $t=t_{f}$ , and we also discard those structures that are born for times later than $t_{n}$ . We denote by $\tilde {N} (t )$ and $\hat {N} (t )$ the numbers of existing structures at time $t$ , considering all the simulated time ( $t_0 \lt t \lt t_f$ ) and considering only the filtered time window ( $t_{m} \lt t \lt t_n$ ), respectively, where $t_{m}$ is the largest living time for structures alive at time $t_0$ .

Figure 10 shows $\tilde {N} (t )$ and $\hat {N} (t )$ obtained for the several DNS data cases, where the black frame represents the time interval $t_{m}\lt t\lt t_{n}$ , considered in the filtering described above. As can be seen, the filtering essentially makes $\tilde {N} (t )=\hat {N} (t )$ for $t_{m}\lt t\lt t_{n}$ , and contains a large population history, with several tens of Kolmogorov simulated times, and thus many birth and death events, making it possible to test the population model proposed here. The figure shows also the number of structures ‘alive’ for any given time, predicted by the population model $N(t)$ , as discussed below.

Figure 11. Accumulated number of births ( $\hat {N}_{B}(t)$ ) and deaths ( $\hat {N}_{D}(t)$ ) obtained directly from the several DNS simulations, using data from the time window $t_0 \lt t \lt t_n$ . These curves are used to compute estimates for $N_B(t)$ and $N_D(t)$ that allow the estimation of the number $N(t)$ of existing structures for a given time in each simulation. The plots also show the birth rate constants $C_B$ obtained for each case, using the least squares method. The curves are given for the DNS with (a) $Re_{\lambda }=54$ , (b) $Re_{\lambda }=91$ , (c) $Re_{\lambda }=143$ and (d) $Re_{\lambda }=189$ . (The DNS with $Re_{\lambda }=239$ show similar characteristics, not illustrated.)

Figure 11 shows the number of accumulated births ( $\hat {N}_{B} (t )$ ) and deaths ( $\hat {N}_{D} (t )$ ) obtained directly from the simulations in the interval $t_{0}\lt t\lt t_{n}$ . As expected, both $\hat {N}_{B} (t )$ and $\hat {N}_{D} (t )$ increase with time during the simulation, with a time lag between the two quantities where for any given time we have $\hat {N}_{B} (t ) \gt \hat {N}_{D} (t )$ , as required by (4.16). From these DNS curves for $\hat {N}_{B} (t )$ , and since $N_B (t ) = C_B t$ from (4.18), it is possible to estimate the slopes $C_B$ for each simulation by using the least squares method. The normalised values of $C_B$ for each curve are shown in the plots, e.g. for the highest Reynolds number case, we have a birth rate of $\approx 823$ new structures per Kolmogorov time (figure 11 d). Finally, using (4.17), we compute the modelled value of $N_D(t)$ , where we have set $N_{0}=N (0 )=0$ , since at $t=t_{0}$ no existing (‘alive’) structures are considered. These curves, predicted by the present model, are also shown in the figure for each DNS case, and show good agreement with the existing data, thus validating the model.

Table 6 shows the normalised birth rates obtained for each simulation $ C_B \tau _{\eta }$ , computed as described before, together with the resulting number of existing structures in the limit of very long simulated times, $N_{\infty }$ . This quantity was computed using the available mean lifetime of the structures for each case (listed in table 4) with the birth rates given here (table 6), i.e. $N_{\infty } = C_B \tau _{ivs} = (C_B \tau _{\eta } ) \tau _{ivs} /\tau _{\eta }$ . The temporal average of the instantaneous number of structures alive during the simulation, obtained directly from the DNS, is also shown ( $\langle \tilde {N} (t )\rangle$ ) for comparison. Notice that in the DNS with $Re_{\lambda } =239$ only 1/8 of the computational domain is used, with $512^3$ grid points. Therefore, the number of time tracked IVS in this simulation is closer to the DNS with $Re_{\lambda } = 143$ than to the DNS with $Re_{\lambda } =189$ .

Table 6. Population statistics for all the DNS used in the present work: number of collocation points ( $N^3$ ); Taylor-based Reynolds number ( $Re_{\lambda }$ ); total number of structures tracked during the simulation time window ( $N_{ivs}$ ); birth rate normalised by the Kolmogorov time scale ( $C_B \tau _{\eta }$ ); number of structures predicted by the population model at saturated conditions ( $N_{\infty }=C_B \tau _{ivs}$ ); temporal average of the instantaneous number of structures alive during the simulation ( $\langle\tilde {N}(t)\rangle$ ); accumulated number of births ( $\tilde {N}_{B}(t_{f})$ ); accumulated number of deaths ( $\tilde {N}_{D}(t_{f})$ ). The last row corresponds to the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ). The $\ast$ in front of the row for the largest DNS is a reminder that, in this case, the IVS were followed in a subdomain consisting of only $1/8$ of the total box size.

The results show that the modelled number of existing structures ( $N_{\infty }$ ) agrees well with the one recovered from the DNS data ( $\langle \tilde {N} (t )\rangle$ ), which supports the population model. The same conclusion can be obtained in figure 10 by comparing the number of existing structures $N(t)$ computed through (4.16), with the curves of $\tilde {N} (t )$ and $\hat {N} (t )$ already discussed. Naturally, our deterministic model is not able to predict the stochastic character of the fluctuating DNS data, but it is noteworthy that those fluctuations occur around the values predicted by the model, which thus represents the averaged behaviour of these variables.

The results show also that the number of structures alive increases with the Reynolds number, from roughly $18$ structures for $Re_{\lambda } = 54$ , to more than $2700$ structures for $Re_{\lambda } = 189$ . The same is true for the birth rates and total numbers of births and deaths. For instance, for the highest Reynolds number simulation more than 130 000 new structures are born (and a similar number of structures die), with a total of more than $1.3$ million structures tracked, and the birth rate attains $C_B = 823.1$ newly generated structures per Kolmogorov time. Table 6 shows that the values of $C_B \tau _{\eta }$ , $N_{\infty }$ and $\langle \tilde {N} (t )\rangle$ for the DNS with the different forcing ( $k_p=4$ ) are much bigger than those for the reference DNS at the same Reynolds number. However, if we take into account the larger domain size of these new DNS ( $N^3 = 256^3$ ), which is $8$ times bigger than the reference DNS with the same Reynolds number ( $Re_{\lambda } = 54$ ), and if we multiply by $8$ the values of these variables for that DNS, then we get $C_B \tau _{\eta } = 36.7$ , $N_{\infty } = 144$ and $\langle \tilde {N} (t )\rangle = 136$ , which are very close to the values $C_B \tau _{\eta } = 35.67$ , $N_{\infty } = 147$ and $\langle \tilde {N} (t )\rangle = 143$ obtained in the DNS with the different $k_p$ . This fact indicates that the forcing location does not affect these statistics. The values of the other quantities involve much larger simulated time than the reference DNS, and involve many more samples, thus cannot be compared in this form, but require a proper normalisation that will be discussed below.

The question that arises is whether this increase in the number of structures with the Reynolds number can be explained simply by the increasing size of the computational domain, which increases also with the Reynolds number of our simulations. Note that the average number of existing structures, $\langle \tilde {N} (t )\rangle$ , can be seen as a ‘density’ (of structures) since it represents the number of existing structures in a box of volume $V_{box} = ( 2\pi )^3$ ( $V_{box} = ( 2\pi )^3/8$ for the DNS with $Re_{\lambda } = 239$ ). To compute the average number of structures $\langle \tilde {N} (t )\rangle _{V}$ living in any other given volume $V$ , one simply needs to evaluate

(4.19) \begin{equation} \langle \tilde {N} (t )\rangle _{V} = \langle \tilde {N} (t )\rangle\, \frac {V}{V_{box}} \end{equation}

using the values of $\langle \tilde {N} (t )\rangle$ in table 6. We use two different volumes: (i) a cubic volume defined by the size of the integral scale $V_L=L_{11}^3$ , and (ii) a volume defined by the Kolmogorov microscale $V_{\eta } = (100\, \eta )^3$ (where the arbitrary constant $100$ was chosen for convenience but does not modify the conclusions). In order to see if the increase in the number of structures with the Reynolds number can be simply explained by the increasing size of the computational domain, table 7 shows both the average number of existing IVS and their birth rates, normalised by a fixed amount of volume ( $V_L$ and $V_{\eta }$ ) using (4.19). When normalised by the integral scale of motion, both the number of structures and the birth rate increase with the Reynolds number. In a cube with the size of the integral scale $V_L = L_{11}^3$ , more than 66 structures exist, and close to 20 new structures are created per Kolmogorov time, for the highest Reynolds number considered in this study. However, when the normalisation is done in a volume defined by the Kolmogorov microscale, these numbers are approximately constant (but not exactly, as discussed below). Specifically, for a volume defined by $V_{\eta } = (100\, \eta )^3$ , we have approximately 6 living structures and a birth rate of approximately 1.7 new structures per Kolmogorov time. An anonymous Referee remarked that if the mean lifetime of the IVS normalised by the Kolmogorov time ( $\tau _{ivs}/\tau _{\eta }$ ) varies with the Reynolds number (as observed above), then this necessarily implies that the birth rate density cannot be Reynolds-number-independent. The arguments from the Referee are derived from (4.15) and can be summarised as follows. From this equation, we can write

(4.20) \begin{equation} N_{\infty } = C_B \tau _{\eta } \left ( \frac {\tau _{ivs} }{\tau _{\eta }} \right ), \end{equation}

thus

(4.21) \begin{equation} N_{\infty } \left ( \frac { V_{\eta } }{ V_{box} } \right ) = C_B \tau _{\eta } \left ( \frac {\tau _{ivs} }{\tau _{\eta }} \right ) \left ( \frac { V_{\eta } }{ V_{box}} \right ). \end{equation}

Now table 7 shows the density of the IVS $N_{\infty } ({ V_{\eta } }/{ V_{box} } )$ , and the density of the birth rate $C_B \tau _{\eta } ({ V_{\eta } }/{ V_{box}} )$ , as functions of the Reynolds number. It is obvious that since $ {\tau _{ivs} }/{\tau _{\eta }} $ varies with the Reynolds number, it is not possible that both $N_{\infty } ( { V_{\eta } }/{ V_{box}} )$ and $C_B \tau _{\eta } ({ V_{\eta } }/{ V_{box}} )$ can be Reynolds-number-independent. Indeed, the results from the new DNS seem to show a mild Reynolds number increase of the birth rate density, that somehow compensates for the decrease of the lifetime normalised by the Kolmogorov time, so that the density of the IVS stays approximately constant (notice that this quantity seems to be oscillating as the Reynolds number increases; see table 7). In other words, $N_{\infty } ( { V_{\eta } }/{ V_{box} } )$ becomes approximately Reynolds-number-independent thanks to the increase of $C_B \tau _{\eta } ( { V_{\eta } }/{ V_{box}} )$ with the Reynolds number (see table 7). This increase of the birth rate density with the Reynolds number may be easily explained if the birth events are associated with extreme events of the flow field, because of the well-known increase of the intermittency with the Reynolds number that is typical of high Reynolds turbulence. For the time being, these statements need to be taken cautiously, until similar simulations are carried out at higher Reynolds numbers, or when the dynamics of the ‘birth’ and ‘death’ events is further investigated.

Table 7. Spatially averaged population statistics for all the DNS used in the present work: number of collocation points ( $N^3$ ); Taylor-based Reynolds number ( $Re_{\lambda }$ ); temporal average of the instantaneous number of structures alive at a given time per unit volume defined with the integral scale ( $\langle\tilde {N}(t)\rangle V_L / V_{box}$ ); temporal average of the instantaneous number of structures alive at a given time per unit volume defined with the Kolmogorov microscale ( $\langle \tilde {N}(t)\rangle V_{\eta } / V_{box}$ ); birth rate $C_B$ per unit volume, where the volume is defined with the integral scale ( $C_B \tau _{\eta } V_L / V_{box}$ ); birth rate $C_B$ per unit volume, where the volume is defined with the Kolmogorov microscale ( $C_B \tau _{\eta } V_{\eta } / V_{box}$ ); average number of structures within the computational domain using the estimate defined by (4.23) ( $\overline {N}$ ). The last row corresponds to the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ). The $\ast$ in front of the row for largest DNS is a reminder that in this case, the IVS were followed in a subdomain consisting of only $1/8$ of the total box size.

The Kolmogorov scaling for the density described above suggests another way to observe the density of the IVS. The volume occupied by one single structure can be estimated as $V_{ivs} = \pi R_{ivs}^2 L_{ivs}$ , and since the normalised radius and length of these structures both scale with the Kolmogorov microscale (Ghira et al. Reference Ghira, Elsinga and da Silva2022), one can write

(4.22) \begin{equation} V_{ivs} = \pi \left ( \frac {R_{ivs}}{\eta } \right )^2 \left ( \frac {L_{ivs}}{\eta } \right ) \eta ^3, \end{equation}

where, at sufficiently high Reynolds numbers, the quantities in brackets are constants equal to $R_{ivs}/ \eta \approx 4.0$ and $L_{ivs} / \eta \approx 55$ , respectively, so that $V_{ivs} \sim \eta ^3$ .

Using these values, the volume of a single ‘worm’ is approximately equal to $V_{ivs} \approx 2765\, \eta ^3$ (at high Reynolds numbers). Therefore, the average number of structures in a given simulation can be estimated by considering the volume occupied by the IVS, which occupy $1\, \%$ of the entire volume $(2\pi )^3 / 100$ , i.e.

(4.23) \begin{equation} \overline {N} = \frac {(2\pi )^3 }{100 V_{ivs}} = \frac {2 \pi ^2}{ 25 \left ( \dfrac {R_{ivs}}{\eta } \right )^2 \left ( \dfrac {L_{ivs}}{\eta } \right ) \eta ^3}, \end{equation}

where the bar is used to name this new estimate. Table 7 displays the values obtained for each simulation used in the present work, where one can see that the crude estimate given by (4.23) recovers values for $\overline {N}$ that are of the same order of those for $\langle \tilde {N} (t )\rangle$ (see table 6). Also, it is interesting to note that when properly normalised (as in table 7), the statistics for the DNS with different forcing ( $k_p=4$ ) are virtually equal to those of the reference DNS ( $k_p=2$ ) at the same Reynolds number ( $Re_{\lambda } = 54$ ). This again demonstrates that modifications of the forcing location do not affect the birth rate and the number of existing structures. Note that the value $\overline {N} = 76$ for this simulation is also consistent with $\overline {N} \approx 10$ for the first simulation in the list since the domain size of this last simulation is $8$ times bigger.

By using the definition of the non-dimensional dissipation $C_{\epsilon }$ , i.e.

(4.24) \begin{equation} \epsilon = C_{\epsilon }\, \frac {u'{}^{3}}{L_{11}}, \end{equation}

where $u'{}^{3} = ( \overline {u'{}^2} )^{3/2}$ , and $\overline {u'{}^2}$ is the velocity variance in the $x$ direction, one can write (4.23) in the equivalent form

(4.25) \begin{equation} \overline {N} = \frac {2 \pi ^2}{ 25 \left ( \dfrac {R_{ivs}}{\eta } \right )^2 \left ( \dfrac {L_{ivs}}{\eta } \right ) \left ( 15^{3/4} \dfrac { L_{11} }{C_{\epsilon }} \right )^3 }\, Re_{\lambda }^{9/2} = C^{\ast }\, Re_{\lambda }^{9/2}, \end{equation}

where $C^{\ast }$ is a constant approximately equal to $1.1 \times 10^{-7}$ , obtained by averaging the data from all DNS.

Figure 12 shows the comparison between the real (obtained from the DNS) number of ‘worms’ inside the computational domain $\langle \tilde {N} (t )\rangle$ (listed in table 6), and the estimate described above, $\overline {N}$ (listed in table 7). The power law described above is also shown. It is clear that the two quantities evolve in a similar way, and are reasonably well approximated from the mentioned power law. The results and expressions from the text above are useful for estimating the number of IVS present inside a given HIT flow region. A similar analysis can be extended to estimate the density of ‘births’ and ‘deaths’ per unit time and volume.

Figure 12. Mean number of ‘worms’ inside the computational domain $\langle \tilde {N}(t)\rangle$ as obtained from the DNS (from table 6), compared with its estimate $\overline {N}$ , from table 7. The dashed line represents the scaling law defined in (4.25).

4.4. Splits and merger between the ‘worms’ and their statistical (ir)relevance

The lifetime maps described in § 3.2 can be used to analyse the life history of the IVS. Specifically, the number of mergings (when two or more ‘worms’ come together) or splits (when one ‘worm’ breaks into two or more structures), as well as the number of isolated (solitary) structures, can be obtained from the life maps for the six simulations, such as the one shown in figure 3 for the smallest Reynolds number. This was analysed here by assessing all the identified structures for the duration of their life span, from ‘birth’ to ‘death’, where the merging or splitting events have been marked as instantaneous events.

Table 8 presents a summary of the main results obtained from the post-processing of the life maps, showing the total number of tracked structures ( $N_{ivs}$ ), the number of solitary/isolated structures ( $\tilde {N}_{I} (t_{f} )$ ), and the number of accumulated mergings ( $N_{M} (t_{f} )$ ) and splits ( $N_{S} (t_{f} )$ ), together with their corresponding fractions. The numbers of splits and mergers increase with the Reynolds number, but represent always a very small fraction of the total number of recorded lives. For the highest Reynolds numbers, the fractions of isolated, merged and split structures are approximately $84\, \%$ , $2\, \%$ and $14\, \%$ of the total, respectively. Thus the great majority of lives correspond to solitary/isolated structures that emerge from the background vorticity and live a solitary life, dying again in the background, without any visible interaction with the other structures. We want to stress that the fact that we do not see many splits/mergings does not necessarily imply that the interaction between a structure and its neighbours is negligible. Our data do not allow us to assess the possible dynamical interactions between the IVS. Our algorithm is able to detect only interactions that result in splits/mergings. Finally, the normalised values for the DNS with different forcing ( $k_p=4$ ) shown in the last three columns are again close to those of the reference DNS with the smallest Reynolds number.

Table 8. Statistics of merging and splitting events for all the DNS used in the present work: number of collocation points ( $N^3$ ); Taylor-based Reynolds number ( $Re_{\lambda }$ ); total number of structures tracked during the simulation time window ( $N_{ivs}$ ); total number of isolated or solitary (non-interacting) structures ( $\tilde {N}_{I}(t_{f})$ ); accumulated number of merging events ( $N_{M}(t_{f})$ ); accumulated number of splitting events ( $N_{S}(t_{f})$ ); fraction of isolated or solitary (non-interacting) structures ( $\tilde {N}_{I}(t_{f}) / N_{ivs}$ ); fraction of merging events ( $N_{M}(t_{f}) / N_{ivs}$ ); fraction of splitting events ( $N_{S}(t_{f}) / N_{ivs}$ ). The last row corresponds to the DNS where the forcing is centred at $k_p = 4$ (instead of $k_p=2$ ). The $\ast$ in front of the row for largest DNS is a reminder that in this case, the IVS were followed in a subdomain consisting of only $1/8$ of the total box size.

Figure 13 shows histograms of the numbers of branches undergone by the structures tracked in the several simulations. In these plots, solitary/isolated structures are represented by one ( $1$ ) branch since they do not experience any merging or splitting event, i.e. they are born and die without interactions. Structures with 3 branches could represent structures that split into two branches and die without further interactions (pure splitting or pure merging events will always involve the interaction between at least $3$ branches). Two branches are due to events where one of the branches lives less than $t_{0}$ . It is clear that solitary structures are by far the most frequent in all cases, in agreement with the results in table 8, and also that the small numbers of branches are always far more common than large numbers. However, the number of possible branches increases with the Reynolds number. Whereas for $Re_{\lambda } = 54$ only three branches have been detected, up to $16$ branches can be observed for $Re_{\lambda } = 189$ . Again, the increasing complexity of existing structures is consistent with the increasing number of degrees of freedom that is characteristic of high Reynolds numbers.

Even if the number of merging and splitting structures is much smaller than the total, it is important to assess how the lifetime data are affected by including in the analysis all individual branches, regardless of the way in which they appear, i.e. to consider all the detected branches in the lifetime computation. The results show that the inclusion of these splitting and merging events does not affect the results. For instance, for the highest Reynolds number case, the mean lifetime of the IVS using the MLE changes from $\tau _{ivs}^{\textrm {MLE}} / \tau _{\eta } = 3.38$ (table 4) to $\tau _{ivs}^{\textrm {MLE}} / \tau _{\eta } = 3.26$ for including in the analysis the detected merging and splitting events, i.e. a variation of only $3.5 \, \%$ . Thus we conclude that the interaction mechanisms not only are very rare but also have a negligible impact in the mean lifetime statistics.

Finally, Appendix B shows that the main results from this work are relatively independent of the values chosen for the two parameters used to define the IVS.

Figure 13. Histograms of the numbers of branches undergone by the structures tracked in the several simulations. A solitary structure is represented by one branch since it does not experience any merging or splitting event, i.e. is born and dies without interactions, while a structure with three branches could represent a single structure that splits into two branches that will die with no further interactions. The histograms are given for the DNS with (a) $Re_{\lambda }=54$ , (b) $Re_{\lambda }=91$ , (c) $Re_{\lambda }=143$ and (d) $Re_{\lambda }=189$ . (The DNS with $Re_{\lambda }=239$ show similar characteristics.)