2.1 Numerical simulation set-up

We use a combined Eulerian–Lagrangian framework to compute a homogeneous isotropic turbulent (HIT) flow. The computational domain is a cube with periodic boundary conditions in all three spatial directions. The same numerical code as in Sardina et al. (Reference Sardina, Picano, Brandt and Caballero2015) is used, with the difference that a component-specific volume forcing term $f_{i}$ is used, following in Fourier space

(2.1) $$\begin{eqnarray}\hat{f}_{i}(k_{i},t)=\left\{\begin{array}{@{}ll@{}}\langle \unicode[STIX]{x1D700}\rangle \displaystyle \frac{\hat{u} _{i}(k_{i},t)}{\displaystyle \mathop{\sum }_{|k_{i}|}3|\hat{u} _{i}(k_{i},t)|^{2}} & \text{for }k_{min}\leqslant |k_{i}|\leqslant k_{max},\\ 0 & \text{otherwise},\end{array}\right.\end{eqnarray}$$

where the symbol $\hat{\ast }$ indicates the Fourier transformation of the generic real function *, $\langle \unicode[STIX]{x1D700}\rangle$ is the prescribed turbulent dissipation rate, $k_{min}$ and $k_{max}$ are the wavelengths limits influenced by the forcing. Such a forcing method, already successfully used for the computation of HIT (Carati, Ghosal & Moin Reference Carati, Ghosal and Moin1995; Watteaux, Stocker & Taylor Reference Watteaux, Stocker and Taylor2015), has the advantage of adjusting independently the three velocity components in order to achieve an accurate isotropic flow. Here, we force all Fourier components between $k_{min}=2.44$ and $k_{max}=3.09$ . Our spatial resolution consists of $1024^{3}$ grid points, chosen to give a Taylor Reynolds number of $Re_{\unicode[STIX]{x1D706}}$ of $240$ according to the constraints previously prescribed for direct numerical simulation (DNS) of HIT (Donzis, Yeung & Sreenivasan Reference Donzis, Yeung and Sreenivasan2008). While not the finest resolution used for the study of Lagrangian statistics in HIT (Yeung et al. Reference Yeung, Pope and Sawford2006b ), it still creates a flow that can largely be considered turbulent. The time step is $\unicode[STIX]{x1D70F}_{K}/40$ and therefore ensures a well-resolved temporal sampling of the Lagrangian statistics. The total simulation time corresponds to $T=245\unicode[STIX]{x1D70F}_{K}$ and to $2.6{\mathcal{T}}_{u}$ . Approximately $10^{6}$ particles were tracked, saving all the relevant observables along their trajectories at every time step. This large number of particles ensures that our one-particle one-time Lagrangian statistics are equivalent to single-point Eulerian statistics. The same simulation at $Re_{\unicode[STIX]{x1D706}}=90$ was conducted in order to assess the dependency of our results on the Reynolds number.

Figure 2.

The principal direction of $p_{2}(\unicode[STIX]{x1D700},\dot{\unicode[STIX]{x1D700}})$ can be approximated by the relation $\dot{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D700}/C_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ where $\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D700}^{3})^{1/4}$ is the local viscous scale and $C_{\unicode[STIX]{x1D700}}$ the coefficient to be determined. The Lagrangian time series are noisy due to discrepancies in the interpolation method and a weighted moving filter with characteristic time $T_{A}$ is applied to the signals. The coefficient $C_{\unicode[STIX]{x1D700}}$ is derived by taking the principal orientation of the filtered $p_{2}$ . (a) $C_{\unicode[STIX]{x1D700}}$ as a function of $T_{A}$ which stabilises at $3.6$ for $T_{A}=\unicode[STIX]{x1D70F}_{K}/3$ , and (b) $p_{2}(\unicode[STIX]{x1D700},\dot{\unicode[STIX]{x1D700}})$ using $T_{A}=\unicode[STIX]{x1D70F}_{K}/3$ with, in solid line, the corresponding scaling $\unicode[STIX]{x1D700}/C_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ using $C_{\unicode[STIX]{x1D700}}=3.6$ and in the dashed line that of $p_{2}$ when using raw data. When not filtering dissipation, fluctuations in time are larger and therefore $C_{\unicode[STIX]{x1D700}}$ decreases.

2.2 Filtering of spurious Lagrangian statistics

Physical turbulence quantities at particles position are inferred from the Eulerian grid using a tri-linear interpolation. While such a scheme represents a quick and accurate method for interpolation of the velocity, it is known to create unphysical noise in the Lagrangian time series of higher-order quantities (van Hinsberg et al. Reference van Hinsberg, ten Thije Boonkkamp, Toschi and Clercx2013) which in turn parasites the flying and diving LPDFs. To circumvent this issue, we convolute all Lagrangian time series with a weighted moving average filter of the form

(2.2) $$\begin{eqnarray}A(t)=\left\{\begin{array}{@{}ll@{}}(t+T_{A})/T_{A}^{2} & \text{if }-T_{A}<t<0,\\ -(t-T_{A})/T_{A}^{2} & \text{if }0<t<T_{A},\\ 0 & \text{elsewhere}.\end{array}\right.\end{eqnarray}$$

Filtering time $T_{A}$ was determined by studying the convergence of the local scaling of the joint distribution of dissipation and its time derivative $p_{2}(\unicode[STIX]{x1D700},\dot{\unicode[STIX]{x1D700}})$ , namely, $\dot{\unicode[STIX]{x1D700}}\approx \unicode[STIX]{x1D700}/C_{\unicode[STIX]{x1D700}}\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ (Babler et al. Reference Babler, Biferale and Lanotte2012), where $\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ is the local Kolmogorov time scale and $C_{\unicode[STIX]{x1D700}}$ is a coefficient of proportionality to define that depends on $T_{A}$ . Figure 2(a) shows values of $C_{\unicode[STIX]{x1D700}}$ with respect to $T_{A}$ . Without any filtering applied to the time series ( $T_{A}=0$ ), $C_{\unicode[STIX]{x1D700}}$ is approximately $2$ (figure 2 b). The coefficient $C_{\unicode[STIX]{x1D700}}$ stabilises at approximately $3.6$ (figure 2 a,b) for a filtering time of $T_{A}=\unicode[STIX]{x1D70F}_{K}/3$ , a time actually previously used to smooth Lagrangian signals of acceleration (Wang Reference Wang2014). We thus consider the same filtering time for all turbulence quantities computed here.

Figure 3.

(a) Maximum of diving (squares) and flying (circles) residence times detected for various dissipation thresholds, with our simulation length of $T=245\unicode[STIX]{x1D70F}_{K}$ . (b) Percentage of diving (squares) and flying (circles) residence times equal to $T$ with respect to the total number of detected diving and flying times. The numerical limitations in time affects the results in a wide range of threshold values considered here. (c) Flying LPDFs at two dissipation thresholds. The last point has a large density and corresponds the simulation length $T$ . The extrapolation method removes this point and extrapolates the remaining LPDF with reasonable success. (d) Profiles of mean flying and diving times for various dissipation thresholds inferred from the raw LPDFs (dashed black lines), the extrapolated LPDFs (in pure red circles and light red squares, respectively) and the Rice theorem (equation (3.1); pure and lighter colours, respectively). Without the extrapolation, mean statistics saturate at the numerical limits due to the time step $2\unicode[STIX]{x0394}t=\unicode[STIX]{x1D70F}_{K}/20$ and $T$ .

2.3 Extension of temporal statistics beyond numerical limits

The flying and diving LPDFs, inferred from the Lagrangian time series, are affected both by the temporal resolution of the simulation, giving a minimal residence time of $2\unicode[STIX]{x0394}t=\unicode[STIX]{x1D70F}_{K}/20$ , and by the simulation time length, $T=245\unicode[STIX]{x1D70F}_{K}$ . For instance, figure 3(a) shows the maximum value of both diving and flying residence times, found in flying and diving distributions for various dissipation thresholds. Only the range $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle =[0.2{-}1.0]$ is exempt from the influence of numerical limitations. Even the percentage of residence times equal to the numerical limits ( $2\unicode[STIX]{x0394}t$ or $T$ ) with respect to the total number of residence times gives the region of influence $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle <0.1$ for flying times and $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle >7$ for diving times (figure 3 b). The flying and diving LPDFs are consequently affected by an artificially large density at $2\unicode[STIX]{x0394}t$ or $T$ , and the corresponding mean flying and diving times saturate accordingly (figure 3 c,d).

Residence times on the scale of $2\unicode[STIX]{x0394}t$ are spurious statistics with no physical interest and are therefore not considered when constructing the LPDFs. To remove the non-physical saturation in LPDFs at $T$ , and therefore have access to mean times larger than $T$ , we first apply a weighted moving average filter on ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{D}(\ln \unicode[STIX]{x1D70F})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{F}(\ln \unicode[STIX]{x1D70F})$ (equation (2.2) with $T_{A}$ being half a decade) for times lower than $T$ . Second, a linear extrapolation of the LPDF is done for residence times larger than $T$ by considering the evolution of the LPDF over the last decade. The latter range is a trade-off between the need to have enough points to extrapolate the appropriate trend of the LPDF without depending on the noise and the need not to consider too many points so that potential variations in the trend would not be taken into account. Extrapolation is done until the integration of the LPDF reaches unity. Figure 3(c) shows the LPDFs ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{F}$ for thresholds $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle =1/100$ and $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle =1/20$ . At the smallest dissipation threshold, the trend of the LPDF is clear and the resulting extrapolation accurately reproduces the ongoing increase of the flying density for larger time residence. On the contrary, at threshold $1/20$ , one can see a slight artificial change of slope due to the fall over the last few points. However, at this stage, the extrapolated part of the LPDF does not contribute much to the mean time.

The mean times can also be inferred from the Rice theorem (Rice Reference Rice1945; Babler et al. Reference Babler, Biferale and Lanotte2012) described in the next section (equation (3.1)). This theorem uses distributions of the considered quantity and is therefore free of the influence of numerical temporal resolutions. Figure 3(d) compares mean flying and diving times obtained from the raw and extrapolated LPDFs and from the Rice theorem. While the aforementioned operation to clean and extrapolate the LPDFs may seem somewhat simplistic, it shows remarkable efficiency with good agreement between Lagrangian results and the Rice theorem.

Figure 4.

(a–c) Normalised mean flying and diving residence times, $T_{F}/\unicode[STIX]{x1D70F}_{K}$ and $T_{D}/\unicode[STIX]{x1D70F}_{K}$ , with respect to thresholds (a) $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle$ , (b) $a_{t}/\langle a\rangle$ and (c) $\unicode[STIX]{x1D714}_{t}/\langle \unicode[STIX]{x1D714}\rangle$ . Pure and lighter colours are flying and diving statistics, respectively. See encapsulated legend for details. The approximate global symmetry between $T_{F}$ and $T_{D}$ reflects the presence of holes, in contrast to bursts, during which magnitudes of at least one turbulent property can fall by several orders of magnitudes. Instead, local departure from this symmetry reflects a difference in the physics occurring during holes and bursts which characterises Lagrangian intermittency. Dashed black lines are the theoretical approximations (3.2) and (3.3).

4.1 The flying and diving LPDFs of dissipation at $\unicode[STIX]{x1D700}_{t}^{s}$

We first focus on the distributions of flights and dives of dissipation. Figure 5(a) shows the flying and diving LPDFs of $\unicode[STIX]{x1D700}$ at $\unicode[STIX]{x1D700}_{t}^{s}$ , ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}}^{F}(\ln \unicode[STIX]{x1D70F})$ and ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}}^{D}(\ln \unicode[STIX]{x1D70F})$ . A first result is that, although one has $T_{F}=T_{D}$ , the flying and diving LPDFs are not similar, showing once again the different dynamics governing flights and dives of a fluid parcel, even in HIT. Such asymmetry has already been detected in Eulerian statistics of turbulent boundary layers (Chamecki Reference Chamecki2013).

Figure 5.

LPDFs ${\mathcal{P}}_{C\unicode[STIX]{x1D711}_{t}^{s}}^{F}$ (pure colours) and ${\mathcal{P}}_{C\unicode[STIX]{x1D711}_{t}^{s}}^{D}$ (lighter colours) of $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$ inferred from DNS and inferred from time series computed using the stochastic model of Pope & Chen (Reference Pope and Chen1990), Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007). (a) Results for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$ from DNS at threshold $\unicode[STIX]{x1D711}_{t}^{s}$ ( $C=1$ ). The different position of the peak in ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}}^{F}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}}^{D}$ is explained by the Reynolds stress. (b) LPDFs of $\unicode[STIX]{x1D700}$ for various values of $C$ . Flying and diving LPDFs can be paired following the approximate global symmetry with axis $\unicode[STIX]{x1D711}_{t}^{s}$ giving $T_{F}(C\unicode[STIX]{x1D711}_{t}^{s})\approx T_{D}(\unicode[STIX]{x1D711}_{t}^{s}/C)$ and $T_{D}(C\unicode[STIX]{x1D711}_{t}^{s})\approx T_{F}(\unicode[STIX]{x1D711}_{t}^{s}/C)$ . (c) ${\mathcal{P}}^{D}$ of $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D709}$ and the stochastic model for $C=20$ . (d) ${\mathcal{P}}^{F}$ of $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D709}$ and the stochastic model for $C=1/20$ . The emergence of a peak in the flying LPDFs of dissipation and acceleration at $\unicode[STIX]{x1D711}_{t}^{s}$ shows the existence of a characteristic flying time at about the Taylor time scale. The flying and diving LPDFs have two regimes. The first regime (flat for $\unicode[STIX]{x1D700}$ , increasing/flat for $\unicode[STIX]{x1D709}$ and decreasing for the model) at small time scales represents the correlated events of bursts and holes and the second corresponds to temporal statistics between uncorrelated events. Hence bursts in $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$ are events mostly correlated over their respective global decorrelation time scale ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ and ${\mathcal{T}}_{\ln \unicode[STIX]{x1D709}}$ while holes are shorter events over a few $\unicode[STIX]{x1D70F}_{K}$ .

Both flying and diving LPDFs of dissipation exhibit a peak with a most probable flying time of approximately $4\unicode[STIX]{x1D70F}_{K}$ and a most probable diving time of approximately $2\unicode[STIX]{x1D70F}_{K}$ . A probability of having either short or long dives higher than that of having short or long flights explains why mean flying and diving times are equal despite a different most probable residence time. Importantly, the most probable flying time of $\unicode[STIX]{x1D700}$ at $\unicode[STIX]{x1D700}_{t}^{s}$ coincides with the Eulerian Taylor time scale, equal to $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}=\langle (\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x)^{2}\rangle ^{-1/2}=\sqrt{15}\unicode[STIX]{x1D70F}_{K}$ in HIT (Pope Reference Pope2000). Such a result is confirmed with the low Reynolds number simulation at $Re_{\unicode[STIX]{x1D706}}=90$ (see appendix D). On the contrary, neither time scale $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D700}}=1.4\unicode[STIX]{x1D70F}_{K}$ nor $T_{F}(\unicode[STIX]{x1D700}_{t}^{s})\approx \unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}\approx 7\unicode[STIX]{x1D70F}_{K}$ represent any particular characteristic time of ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}}^{F}$ (figure 5 a). Thus, the time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ associated with the Eulerian vortices in the inertial cascade seems to be a good estimator for a wide range of $Re_{\unicode[STIX]{x1D706}}$ of the Lagrangian most probable flying residence time of dissipation above $\unicode[STIX]{x1D700}_{t}^{s}$ . Instead, numerical results (confirmed theoretically below in § 6) show that the mean flying and diving times at $\unicode[STIX]{x1D700}_{t}^{s}$ do not scale with $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ (i.e. $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}/T_{F|D}(\unicode[STIX]{x1D700}_{t}^{s})$ is Reynolds dependent). This suggests that while the peak of the flying LPDFs only emerges from the physics behind the inertial cascade (characterised by $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ ), the mean flying and diving times instead also encompass the complex physics behind Lagrangian intermittency.

Interestingly, the flying LPDF of the pseudo-dissipation, a field usually considered in Lagrangian stochastic models instead of the dissipation rate as it has more favourable statistical properties such as Gaussianity, e.g. (Pope & Chen Reference Pope and Chen1990; Sawford Reference Sawford1991; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007) and derived as $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D700}-\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}u_{i}u_{j}$ (where $u_{i}u_{j}$ is the Reynolds stress which accounts for turbulent fluctuations in the flow), exhibits at $\unicode[STIX]{x1D709}_{t}^{s}$ a less prominent most probable residence time at $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ (figure 5 a). Moreover, the difference in most probable time scale between flying and diving LPDFs has vanished in the statistics of $\unicode[STIX]{x1D709}$ , with diving times of $\unicode[STIX]{x1D709}$ also equal to $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ . This result shows that Lagrangian fluctuations of the Reynolds stress (represented by the term $\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}u_{i}u_{j}$ ) play a key role in breaking the symmetry between flights and dives. Surprisingly, comparison between the statistics of $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$ shows that the Reynolds stress only influences the time associated with the most probable diving time.

4.2 Insights into dissipation bursts

As the threshold is increased beyond $\unicode[STIX]{x1D700}_{t}^{s}$ , one gradually sees appearing the signature of dissipation bursts in the LPDFs. Figure 5(b) shows ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{F}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{D}$ for various thresholds $\unicode[STIX]{x1D700}_{t}^{s}/C$ and $C\unicode[STIX]{x1D700}_{t}^{s}$ with $C=1/40$ , $1/20$ , $1$ , $20$ and $40$ . Note that due to the symmetry at $\unicode[STIX]{x1D700}_{t}^{s}$ , one has $T_{F}(C\unicode[STIX]{x1D700}_{t}^{s})\approx T_{D}(\unicode[STIX]{x1D700}_{t}^{s}/C)$ and $T_{D}(C\unicode[STIX]{x1D700}_{t}^{s})\approx T_{F}(\unicode[STIX]{x1D700}_{t}^{s}/C)$ (although the latter is less accurate as can be seen in figure 4 a). The impact of bursts is prominent on the flying LPDF at large threshold, ${\mathcal{P}}_{20\unicode[STIX]{x1D700}_{t}^{s}}^{F}$ . The most probable flying time is smaller, showing the overall brevity of bursts. The density of the most probable time has increased and, consequently, the range of possible flying times has decreased. Yet, even for a threshold ten times larger than the mean dissipation (equivalent to $20\unicode[STIX]{x1D711}_{t}^{s}$ ), particles can sometimes undergo a strong shear for up to $15\unicode[STIX]{x1D70F}_{K}$ (at a density of approximately $10^{-5}$ , not shown here). For a larger threshold ( $C=40$ ), the most probable flying time is slightly smaller but, interestingly, neither the corresponding density nor the variance changes much. Only the maximum flying time available decreases. It is worth noting that the peak of the LPDFs occurs at about the Kolmogorov time scale, however, this time scale cannot be considered here as a relevant physical time scale to describe these LPDFs since the local dissipation is much greater than the mean dissipation used to determine $\unicode[STIX]{x1D70F}_{K}$ .

The diving LPDFs at large thresholds, ${\mathcal{P}}_{20\unicode[STIX]{x1D700}_{t}^{s}}^{D}$ and ${\mathcal{P}}_{40\unicode[STIX]{x1D700}_{t}^{s}}^{D}$ (figure 5 b in light red) also bear information on the physics of bursts. Indeed, at small residence times, dives are brief and can therefore belong to one specific event of burst (figure 1). One can see that the diving LPDFs have two different regimes with the first being a constant density for dives of up to $10\unicode[STIX]{x1D70F}_{K}$ and the second being a linearly increasing density with slope unity. Interestingly, pseudo-dissipation $\unicode[STIX]{x1D709}$ has the same temporal statistics except in the first regime, where density is smaller for diving times below $2\unicode[STIX]{x1D70F}_{K}$ , thus showing that the Reynolds stress adds more small-scale variations during dives in bursts of Lagrangian dissipation (figure 5 c), and potentially explaining the shift of the diving LPDFs at $\unicode[STIX]{x1D711}_{t}^{s}$ (figure 5 a). Using the Lagrangian stochastic model of dissipation (Pope & Chen Reference Pope and Chen1990; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007) based on a first-order autoregressive model, we produced a set of time series and derived from it the flying and diving LPDFs (see appendix C). Diving LPDFs at large threshold also exhibit two regimes separated at approximately $10\unicode[STIX]{x1D70F}_{K}$ (figure 5 c). The corresponding change in regime can only be dictated by the time scale prescribed to the model, namely, the decorrelation time scale of the order of magnitude of the dissipation, ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}\approx 12\unicode[STIX]{x1D70F}_{K}$ . Our results show that the first regime in temporal statistics of dissipation at large threshold is governed by the correlated fluctuations of dissipation, in part due to Reynolds stress. These correlated fluctuations reflect an ongoing burst. Hence, the diving LPDF at large threshold represents an interesting manner to detect the decorrelation time scale of a burst. The combination of results on the statistics of flights and dives at large threshold shows that, in Lagrangian turbulent dissipation, bursts are events likely to happen over approximately ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ with small-scale fluctuations composed of flying periods mainly taking place over $\unicode[STIX]{x1D70F}_{K}$ and diving periods equally occurring over times from one to ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ .

The second regime present in ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}}^{D}$ at large threshold shows the probability of the time delay between two uncorrelated events and therefore tells us about the frequency of the bursts. The most probable time period between two bursts is of several hundreds of $\unicode[STIX]{x1D70F}_{K}$ (figure 5 b), showing that in a turbulent flow, extreme events have such a low occurrence that, should the event of turbulence in a flow be finite in time, bursts would not be undergone by most particles.

4.3 Insights into dissipation holes

At small threshold, statistics of residence times are mostly dictated by the realm of holes (figure 1). Equivalently to results on bursts, the most probable diving time in ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/20}^{D}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/40}^{D}$ has also decreased down to small time scales as well as the variance of the distribution (figure 5 b). Here too, despite the extreme context, the particle can sometime undergo long quiet events for periods up to $35\unicode[STIX]{x1D70F}_{K}$ (at a density of approximately $10^{-5}$ , not shown here). However, differences exist with respect to the flying LPDFs at large threshold, ${\mathcal{P}}_{20\unicode[STIX]{x1D700}_{t}^{s}}^{F}$ and ${\mathcal{P}}_{40\unicode[STIX]{x1D700}_{t}^{s}}^{F}$ , which explain the slight break of symmetry between bursts and holes (figure 4 a).

First, the most probable time of the LPDFs is less clearly defined and the variance of the distribution is larger than that of flying times at large threshold, reflecting a larger range of possible diving times. Second, as for bursts at large threshold, events of holes below threshold $\unicode[STIX]{x1D700}_{t}^{s}/20$ are predominantly occurring on a small time scale, however, ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/20}^{F}$ does not exhibit the same two regimes as in ${\mathcal{P}}_{20\unicode[STIX]{x1D700}_{t}^{s}}^{D}$ . The first regime of ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/20}^{F}$ consists of a decrease of probability as the residence time increases, and a second regime of linearly increasing probability starting at approximately $2\unicode[STIX]{x1D70F}_{K}$ (figure 5 b, red curve). This holds true also for larger thresholds ( $C=40$ ). As for ${\mathcal{P}}_{20\unicode[STIX]{x1D700}_{t}^{s}}^{D}$ , the second linear regime in ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/20}^{F}$ is witness to dives between two uncorrelated events of holes and therefore shows that two holes are most likely to be separated by periods more than two orders of magnitude above $\unicode[STIX]{x1D70F}_{K}$ (figure 5 b).

The brevity of the first regime corresponding to the statistics of correlated events tells us that events of holes are likely to happen in one shot over small time scales of a few $\unicode[STIX]{x1D70F}_{K}$ instead of the global decorrelation time scale. Here, ${\mathcal{P}}_{\unicode[STIX]{x1D709}_{t}^{s}/20}^{F}$ behaves like ${\mathcal{P}}_{\unicode[STIX]{x1D700}_{t}^{s}/20}^{F}$ (figure 5 d), which is logically explained by the fact that, at small threshold, the corresponding Reynolds stress is small and therefore weakly influential.

Finally, the stochastic model (6.1) used here (Pope & Chen Reference Pope and Chen1990; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007) could not reproduce the flying LPDFs at large threshold (figure 5 d). This is explained by the fact that the model necessarily generates a signal continuously correlated over ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ . Such a result shows that, in turbulence, the length of memory in time of Lagrangian physics is event dependent and therefore varies across the flow. While holes belong to a physics occurring with a memory over ${\sim}2\unicode[STIX]{x1D70F}_{K}$ , bursts happen over the large scale ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ .

Figure 6.

(a) Flying (pure colours) and diving (lighter colours) LPDFs of dissipation, acceleration and enstrophy at their respective $\unicode[STIX]{x1D700}_{t}^{s}$ , $a_{t}^{s}$ and $\unicode[STIX]{x1D714}_{t}^{s}$ . As for dissipation, the most probable flying time of acceleration at $a_{t}^{s}$ is $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ . It is instead twice as large for $\unicode[STIX]{x1D714}$ . (b,c) Flying and diving LPDFs for various values of $C$ as in figure 5. Diving LPDFs of both acceleration and enstrophy at large magnitude exhibit two regimes that can be delimited by their respective decorrelation time scale.

4.4 The flying and diving LPDFs of acceleration and enstrophy

Results of temporal statistics of acceleration and enstrophy are globally equivalent to those of dissipation, showing the presence of bursts and holes in these properties as well. For both flying and diving times, statistics of acceleration at $a_{t}^{s}$ have equivalent flying and diving LPDFs as those of dissipation (figure 6 a) and also exhibit two regimes of dives and flights at large and small threshold respectively revealing the presence of bursts and holes of acceleration (figure 6 b). Such similarity with the statistics of dissipation corroborates the presumed link between intermittency of dissipation and acceleration (Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007). However, one can see that the amplitude of the fluctuations in acceleration seems more constrained and flights above $40a_{t}^{s}$ are rare, inducing small densities of the corresponding LPDFs in the observed range of time travel (figure 5 b). Moreover, while the peak of ${\mathcal{P}}_{a_{t}^{s}}^{F}$ is located at $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ in the case $Re_{\unicode[STIX]{x1D706}}=240$ , it is located at a smaller time in the low Reynolds number simulation ( $Re_{\unicode[STIX]{x1D706}}=90$ ), hence showing a different dependency on $Re_{\unicode[STIX]{x1D706}}$ than for dissipation (figure 16 b). It is therefore not possible to conclude at this stage whether $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ is a time scale of convergence for acceleration.

Instead, the statistics of enstrophy have a few differences with respect to dissipation and acceleration. The most probable flying time for enstrophy at $\unicode[STIX]{x1D714}_{t}^{s}$ is approximately $10\unicode[STIX]{x1D70F}_{K}$ (figure 6 a), showing that fluctuations of the rate-of-rotation are slower than that of the rate-of-strain. Moreover, the asymmetry between flights and dives is weaker (figure 6 a). The diving LPDFs at small thresholds ( ${\mathcal{P}}_{\unicode[STIX]{x1D714}_{t}^{s}/20}^{F}$ and ${\mathcal{P}}_{\unicode[STIX]{x1D714}_{t}^{s}/40}^{F}$ ) show that holes of enstrophy last longer up to $5\unicode[STIX]{x1D70F}_{K}$ while the flying LPDFs at large thresholds show that a particle can travel for a long period of time (i.e. few tens of $\unicode[STIX]{x1D70F}_{K}$ ) across highly enstrophic regions (figure 6 c). Finally, for both acceleration and enstrophy, the first regime is defined up to a time scale consistent with ${\mathcal{T}}_{\ln a}\approx 9.5\unicode[STIX]{x1D70F}_{K}$ and ${\mathcal{T}}_{\ln \unicode[STIX]{x1D714}}\approx 12\unicode[STIX]{x1D70F}_{K}$ , respectively.

5.1 The various structures of HIT

The velocity field surrounding a fluid parcel is quantified by the velocity gradient tensor $\unicode[STIX]{x1D608}_{ij}$ , which encodes information about the first-order near-future directions taken by the parcel. In turbulence, this information holds for a time of the order of the local viscous time scale $\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})=(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D700}^{3})^{1/4}$ (Meneveau Reference Meneveau2011). The eigenvectors and eigenvalues of $\unicode[STIX]{x1D608}_{ij}$ define the principal directions and magnitudes of influence of velocity gradients, respectively. Each eigenvalue of $\unicode[STIX]{x1D608}_{ij}$ satisfies a characteristic equation whose coefficients $Q_{A}$ and $R_{A}$ are the velocity invariants and indicate whether the fluid parcel is in a vortex or straining structure predominantly stretching or compressing the flow (figure 7 b). Similarly, the study of the non-zero invariants of the tensors of rate-of-strain, $\unicode[STIX]{x1D61A}_{ij}$ , and the rate-of-rotation, $\unicode[STIX]{x1D61E}_{ij}$ , namely $Q_{S}$ , $R_{S}$ and $Q_{W}$ , indicate the strain and rotation configuration of the flow structure (figure 7 a,c). Invariants $Q_{S}$ and $Q_{W}$ are directly related to dissipation and enstrophy such that $Q_{S}/\langle Q_{W}\rangle =-\unicode[STIX]{x1D700}/\langle \unicode[STIX]{x1D700}\rangle$ and $Q_{W}/\langle Q_{W}\rangle =\unicode[STIX]{x1D714}/\langle \unicode[STIX]{x1D714}\rangle$ . Details on the theoretical background linking the invariants and the flow geometries in the context of HIT are given in appendix B. Thus, the set of five invariants ( $Q_{S}$ , $Q_{W}$ , $R_{S}$ , $Q_{A}$ and $R_{A}$ ) represents a first-order description of the local flow structures. Specifically, the study of the joint distributions ( $Q_{W}$ , $Q_{S}$ ), ( $R_{S}$ , $Q_{S}$ ) and ( $R_{A}$ , $Q_{A}$ ) tells whether the particle is within a straining or vortical, and stretching or compressive flow structure (figure 7).

Total joint distributions of invariants obtained with our DNS (figure 7; see appendix B for details) resemble those of previous studies (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994; Martin et al. Reference Martin, Ooi, Chong and Soria1998; Ooi et al. Reference Ooi, Martin, Soria and Chong1999). The joint distribution of ( $Q_{A}$ , $R_{A}$ ) has a teardrop shape highlighting the predominance of vortices and sheet-like structures in the flow. It has been shown (Martin et al. Reference Martin, Ooi, Chong and Soria1998) that, in homogeneous isotropic turbulence, a fluid parcel spends approximately 20 % of the time as part of a vortex stretching flow structure, 20 % of the time as a vortex compressing topology, more than half the time as a sheet-like straining structure and, finally, the remaining time as a tube-like straining structure (figure 7 b). Instead, irrotational dissipations, vortex sheets and vortex tubes have a somewhat equal chance of existing in the flow (figure 7 a). Statistics of residence time in each region tend to follow exponentially decaying tails (Bhatnagar et al. Reference Bhatnagar, Gupta, Mitra, Pandit and Perlekar2016). The slight inclination toward the axis of $Q_{W}$ suggests that highly enstrophic geometries occur with small straining and are possibly long lived (Ooi et al. Reference Ooi, Martin, Soria and Chong1999). Finally, configurations of rate-of-strain can be differentiated with the parameter $s^{\star }=-3\sqrt{6}\unicode[STIX]{x1D6FC}_{S}\unicode[STIX]{x1D6FD}_{S}\unicode[STIX]{x1D6FE}_{S}/(\unicode[STIX]{x1D6FC}_{S}^{2}+\unicode[STIX]{x1D6FD}_{S}^{2}+\unicode[STIX]{x1D6FE}_{S}^{2})^{3/2}$ (Lund & Rogers Reference Lund and Rogers1994), where $\unicode[STIX]{x1D6FC}_{S},\unicode[STIX]{x1D6FD}_{S}$ and $\unicode[STIX]{x1D6FE}_{S}$ are the principal strain rates of $\unicode[STIX]{x1D61A}_{ij}$ (see appendix B). Isolines of $s^{\star }$ are drawn in the distribution ( $Q_{S}$ , $R_{S}$ ) (figure 7 c). Our results show that the preferred straining geometry at large rate of strain is a sheet-like structure with a configuration $\unicode[STIX]{x1D6FC}_{S}:\unicode[STIX]{x1D6FD}_{S}:\unicode[STIX]{x1D6FE}_{S}=2:1:-3$ (figure 7 a). Note that although this preferred configuration is different than that found in the first studies on this topic (Ashurst et al. Reference Ashurst, Kerstein, Kerr and Gibson1987; Lund & Rogers Reference Lund and Rogers1994) i.e. $3:1:-4$ , it is consistent with more recent work (Ooi et al. Reference Ooi, Martin, Soria and Chong1999). As the Reynolds number of the simulations conducted in these aforementioned studies was similar, the difference in small-scale configuration could be imputed to different initial and forcing configurations as already suggested (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994).

Figure 8.

Normalised joint distributions of the velocity gradient invariants during bursts (main figure) and holes (inset) using (a–c) $X=Q_{W}/\langle Q_{W}\rangle ,Y=-Q_{S}/\langle Q_{W}\rangle$ , (d–f) $R_{A}/\langle Q_{W}\rangle ^{3/2},Q_{A}/\langle Q_{W}\rangle$ and (g–i) $R_{S}/\langle Q_{W}\rangle ^{3/2},Q_{S}/\langle Q_{W}\rangle$ for (a,d,g) dissipation ( $C=20$ and $C=1/10$ ), (b,e,h) acceleration ( $C=10$ and $C=1/5$ ) and (c,f,i) enstrophy ( $C=20$ and $C=1/10$ ). Black, pure colour, grey and lighter colour distributions are obtained by considering invariants at initial time of flights, times of maximum magnitude during flights, initial time of dives and times of minimum magnitude during dives, respectively (represented in figure 1). Distributions are inferred by only considering flights and dives of less than $3\unicode[STIX]{x1D70F}_{K}$ , hence capturing bursts events (large $C$ ) and holes (small $C$ ). Each joint distribution is normalised by the maximum density and isolines $0.90$ and $0.05$ are shown here. (a–c) In contrast to the global distributions shown in figure 7(b), bursts of dissipation and acceleration occur mainly in irrotational structures. Flights and dives of enstrophy are often correlated with an increase and decrease of $Q_{S}$ . The joint distributions are globally self-similar. (d–f) For dissipation and acceleration, there are no preferred structures relative to those in HIT (figure 7 a). Dives tend however to equally happen in stretching and compressing vortices. Finally, bursts of enstrophy occur mainly in vortices where flights and dives correspond to travel from stretching to compressing vortical structures. (g–i) For all three quantities, whether during bursts or holes, dives are associated with two-dimensional (2-D) flow structures and flights to bi-axial straining structures, showing the existence of a self-similar behaviour of Lagrangian intermittency.

5.2 Flow structures characteristics associated with Lagrangian bursts and holes

We now analyse the evolution of the joint distributions of invariants during events of Lagrangian bursts or holes for dissipation, acceleration and enstrophy (figure 8). For a given threshold, we derive the distributions at four different key times: (i) the initial time of flight at $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{t}$ , (ii) the time of maximum value during the flight, (iii) the initial time of dive and (iv) the time of minimum value during the dive (figure 1). The time delay between each key time is sufficiently small so that the reconstructed evolution of the flow characteristics can be considered accurate enough for our analysis (see § B.3). Events of bursts and holes are isolated by taking $C=20$ and $1/10$ respectively for $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D714}$ , and $C=10$ and $1/5$ , for $a$ . These coefficients were chosen in order to have enough data for the distributions (see appendix B for details). Since we are only interested in the characteristics of the local flow structure during bursts or holes, we need to remove the statistics of invariants corresponding to transition between two uncorrelated events. As previously shown (§ 4.2), flying and diving LPDFs describe the temporal statistics of holes as mostly characterised by flights and dives of up to a few Kolmogorov time scales (i.e. $2\unicode[STIX]{x1D70F}_{K}$ for $\unicode[STIX]{x1D700}$ and $a$ and $4\unicode[STIX]{x1D70F}_{K}$ for $\unicode[STIX]{x1D714}$ ; figures 5 d and 6 b,c). Instead, temporal statistics of bursts are also mostly composed of flights shorter than a few $\unicode[STIX]{x1D70F}_{K}$ but dives can be longer with events up to the decorrelation time scale (i.e. approximately $10\unicode[STIX]{x1D70F}_{K}$ ; figures 5 d and 6 b,c). Thus, we computed the joint distributions of invariants considering, for holes, an upper bound residence time of $3\unicode[STIX]{x1D70F}_{K}$ both for flights and dives and, for bursts, a limit of $3\unicode[STIX]{x1D70F}_{K}$ for flights and $10\unicode[STIX]{x1D70F}_{K}$ for dives. Each of the resulting joint distributions was normalised by the maximum density to allow comparison, and isolines $0.90$ and $0.05$ are drawn in all subsequent figures. Finally, given our large number of long Lagrangian trajectories (one million trajectories over $245\unicode[STIX]{x1D70F}_{K}$ ), we believe we have statistically captured all possible flow characteristics occurring during Lagrangian bursts and holes.

Figure 8(a–c) shows the joint distributions of $Q_{W}$ and $-Q_{S}$ during bursts (main plots) and holes (insets) of dissipation, acceleration and enstrophy, respectively. Recall that $Q_{W}\sim \unicode[STIX]{x1D714}$ and $-Q_{S}\sim \unicode[STIX]{x1D700}$ . Both bursts of dissipation and acceleration are mainly linked to relatively small $Q_{S}$ (isoline 0.9), reflecting irrotational geometries, although some happen in highly enstrophic local regions (figure 8 a,b). Interestingly, flights and dives of acceleration during both bursts and holes are systematically correlated with an increase and decrease of $Q_{S}$ (figure 8 b). This confirms again the link between intermittency of dissipation and acceleration (Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007). Finally, bursts of enstrophy mainly occur in the invariant space characterising vortex tubes, also called the worm-like vortical structures (Jimenez et al. Reference Jimenez, Wray, Saffman and Rogallo1993), and, the increase and decrease of enstrophy are significantly correlated with $Q_{S}$ i.e. dissipation (figure 8 c).

For all quantities $\unicode[STIX]{x1D700}$ , $a$ and $\unicode[STIX]{x1D714}$ , the joint distributions ( $Q_{W},-Q_{S}$ ) during holes resemble those during bursts but occur at small magnitude, suggesting self-similarity (figure 8; insets). Importantly, results show that the region of invariants scanned by the particles during a hole of any of the three quantity is close to zero. Complementary joint distributions between acceleration and $Q_{W}$ or $-Q_{S}$ during holes of dissipation or enstrophy, respectively, show that acceleration is also bound to be low in magnitude when dissipation and enstrophy are (figure 9). Since $Q_{S}$ and $Q_{W}$ scale with $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D714}$ , respectively, this yields the conclusion that a hole of dissipation, enstrophy or acceleration entails low magnitudes of dissipation and enstrophy. Thus, Lagrangian holes mostly occur in quiescent regions far from turbulence. Moreover, since the particles followed here are point-like and passive, a Lagrangian hole is necessarily linked with an Eulerian hole and vice versa. Similar results have therefore been found using the Eulerian dataset of our control simulation at $Re_{\unicode[STIX]{x1D706}}=90$ . Hence, our results seem to contradict previous suggestions that Eulerian holes of dissipation were linked to highly enstrophic filamentary vortex-like structures (She & Leveque Reference She and Leveque1994; Gledzer et al. Reference Gledzer, Villermaux, Kahalerras and Gagne1996).

Figure 9.

Joint distributions of (a) $X=Q_{W}/\langle Q_{W}\rangle$ , $Y=a/\langle a\rangle$ during holes of dissipation ( $C=1/10$ ) and (b) $-Q_{S}/\langle Q_{W}\rangle$ , $a/\langle a\rangle$ during holes of enstrophy ( $C=1/10$ ). In both events of holes of dissipation or enstrophy, the three quantities $\unicode[STIX]{x1D700}$ , $\unicode[STIX]{x1D714}$ and $a$ are bound to be small.

Visualisation of the joint distributions of $R_{A}$ and $Q_{A}$ during bursts of dissipation and acceleration (figure 8 d,e) shows no preferential flow characteristics relative to those generally preferred in HIT (figure 7 b). Differences in joint distributions show that bursts, when located in vortex local structures, have flights associated with stretching local structures and dives associated with compressive ones. Such behaviour corroborates previous findings on Lagrangian evolution of the invariants (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998) where it was shown that fluid particles initially located in a stretching vortex local structure generally evolve towards the origin of $(R_{A},Q_{A})$ while the local topology changes from stretching to a compressive vortex structure. This clockwise spiralling motion of particles, also captured here by the joint distributions, was found to appear with cycles on the scale of the eddy turnover time (Martin et al. Reference Martin, Ooi, Chong and Soria1998) and was suggested to be induced by a change between small-scale- and large-scale-dominated regions of the flow during travel (Ooi et al. Reference Ooi, Martin, Soria and Chong1999). This results is also confirmed with the joint distribution associated with bursts of $\unicode[STIX]{x1D714}$ (figure 8 f), where bursts are necessarily associated with enstrophic local structures, and flights and dives mainly correspond to stretching and compressive structures, respectively, with a passage across the more quiescent region near the origin during dives.

The behaviour captured for bursts of enstrophy calls for a closer look. Indeed, flying LPDFs during bursts (figure 6 c) exhibit two different regimes for residence times up to ${\mathcal{T}}_{\ln \unicode[STIX]{x1D714}}$ , one up to $3\unicode[STIX]{x1D70F}_{K}$ corresponding to increasing density and one up to ${\mathcal{T}}_{\ln \unicode[STIX]{x1D714}}$ corresponding to a flat density. Complementary joint distributions of ( $R_{A}$ , $Q_{A}$ ) during bursts of enstrophy by considering either $\unicode[STIX]{x1D70F}<3\unicode[STIX]{x1D70F}_{K}$ or $3\unicode[STIX]{x1D70F}_{K}<\unicode[STIX]{x1D70F}<10\unicode[STIX]{x1D70F}_{K}$ shows two different behaviours during dives, namely, dives with residence time $3\unicode[STIX]{x1D70F}_{K}<\unicode[STIX]{x1D70F}<10\unicode[STIX]{x1D70F}_{K}$ mostly correspond to the passage across zero structure (figure 10 b) while dives shorter than $3\unicode[STIX]{x1D70F}_{K}$ mostly correspond to a direct passage from a stretching to a compressive vortex local structure (figure 10 a), suggesting two different phenomena occurring during bursts. Following previous studies (Chong et al. Reference Chong, Soria, Perry, Chacin, Cantwell and Na1998; Martin et al. Reference Martin, Ooi, Chong and Soria1998; Ooi et al. Reference Ooi, Martin, Soria and Chong1999), we speculate that the first regime is associated with transition between small-scale stretching and compressing vortices while the second regime highlights the effect of the large-scale flow structure.

Distributions of $R_{A}$ and $Q_{A}$ during holes show that dissipation logically prefers vortices while enstrophy prefers straining structures (figure 8 d,f; insets). Distributions of acceleration holes have instead a similar shape to that of bursts (figure 8 e; inset). Here too, holes are always associated with regions of small magnitudes of invariants i.e. where no specific local flow geometries exist and all turbulent quantities are small (figure 8 d–f; insets).

Figure 10.

Joint distributions $X=R_{A}/\langle Q_{W}\rangle ^{3/2}$ , $Y=Q_{A}/\langle Q_{W}\rangle$ ) during bursts of enstrophy ( $C=20$ ) by considering flights shorter than $3\unicode[STIX]{x1D70F}_{K}$ and dives (a) shorter than $3\unicode[STIX]{x1D70F}_{K}$ and (b) larger than $3\unicode[STIX]{x1D70F}_{K}$ and shorter than $10\unicode[STIX]{x1D70F}_{K}$ . Dives shorter than $3\unicode[STIX]{x1D70F}_{K}$ correspond to direct passage from stretching to vortex structure while dives larger than $3\unicode[STIX]{x1D70F}_{K}$ imply passage across a quiescent region.

Finally, distributions of bursts and holes of $\unicode[STIX]{x1D700}$ in the space ( $R_{S}$ , $Q_{S}$ ) show a self-similar behaviour (figure 8 g–i). For both events, there exists a common preferential Lagrangian path in structures of turbulence where particles dive toward regions of 2-D local straining structures $1:0:-1$ and fly toward regions of bi-axial stretching configurations $2:1:-3$ . Since bursts and holes of acceleration and enstrophy are globally correlated with those of dissipation, their distributions also present a self-similar behaviour with the preferential Lagrangian path. The fact that the configuration $2:1:-3$ is also associated with the largest value of dissipation in the corresponding global (and therefore also Eulerian) distribution (figure 7 b) shows that the detected path is not an artefact emerging from the Lagrangian referential. Although there exists a common preferential path toward sheet-like expanding structures at large dissipation, particles can nonetheless cross tube-like contractions during bursts and holes, specially for acceleration or enstrophy (figure 8 h,i).

6.1 Model of $T_{F}(\unicode[STIX]{x1D711})$ and $T_{D}(\unicode[STIX]{x1D711})$ using a stochastic approach

Two models of $T_{D}$ (equations (C 1) and (C 2), see § C.1 for details) have already been developed and based on the scaling $\dot{\unicode[STIX]{x1D700}}\sim \unicode[STIX]{x1D700}/\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ to approximate the Rice theorem (Babler, Morbidelli & Baldiga Reference Babler, Morbidelli and Baldiga2008; Babler et al. Reference Babler, Biferale and Lanotte2012). However, as shown below, both models cannot capture accurately the mean residence times at low to moderate thresholds. Besides, these models are not symmetrical and therefore fail for all thresholds to capture $T_{F}$ . These issues led us to consider another approach.

6.1.1 Derivation of the model

We present here another model of $T_{F}$ and $T_{D}$ , which, rather than considering a physical scaling of the turbulent dissipation, uses the log-normal characteristic of turbulence quantities. Recalling that the flying and diving LPDFs, and therefore the mean times, are transformation invariant (e.g. $T_{F}(\unicode[STIX]{x1D711}_{t}/\langle \unicode[STIX]{x1D711}\rangle )$ for $\unicode[STIX]{x1D711}/\langle \unicode[STIX]{x1D711}\rangle$ equals $T_{D}(\ln \unicode[STIX]{x1D711}_{t}/\langle \unicode[STIX]{x1D711}\rangle )$ for $\ln \unicode[STIX]{x1D711}/\langle \unicode[STIX]{x1D711}\rangle$ ), we assume that for a given turbulence quantity $\unicode[STIX]{x1D711}$ the related quantity $\ln \unicode[STIX]{x1D711}/\langle \unicode[STIX]{x1D711}\rangle$ can be approximated with a Gaussian distribution, $\unicode[STIX]{x1D712}$ , and can therefore be modelled by an Ornstein–Ulhenbeck stochastic process and used to compute mean residence times from the Rice theorem. Relying on the Lagrangian stochastic model of Pope & Chen (Reference Pope and Chen1990), originally developed to model pseudo-dissipation $\unicode[STIX]{x1D709}$ , and the theory on Gaussian distributions, one can then define the distributions $p(\unicode[STIX]{x1D712})$ and $p_{2}(\unicode[STIX]{x1D712},\dot{\unicode[STIX]{x1D712}})$ needed by the Rice theorem (3.1), and therefore derive an analytical approximation of $T_{F}$ and $T_{D}$ .

The Lagrangian stochastic model of Pope & Chen (Reference Pope and Chen1990) (details in § C.2) generates a Gaussian stochastic process $\tilde{\unicode[STIX]{x1D712}}$ following

(6.1) $$\begin{eqnarray}\text{d}\tilde{\unicode[STIX]{x1D712}}=-(\tilde{\unicode[STIX]{x1D712}}-\unicode[STIX]{x03BC}_{\tilde{\unicode[STIX]{x1D712}}})\frac{\text{d}t}{{\mathcal{T}}_{\tilde{\unicode[STIX]{x1D712}}}}+\left(\frac{2\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}}{{\mathcal{T}}_{\tilde{\unicode[STIX]{x1D712}}}}\right)^{1/2}\,\text{d}{\mathcal{W}},\end{eqnarray}$$

where ${\mathcal{T}}_{\tilde{\unicode[STIX]{x1D712}}}=\int \langle \tilde{\unicode[STIX]{x1D712}}^{\prime }(t)\tilde{\unicode[STIX]{x1D712}}^{\prime }(t+\unicode[STIX]{x1D70F})\rangle /\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}\,\text{d}\unicode[STIX]{x1D70F}$ is the decorrelation time scale of $\tilde{\unicode[STIX]{x1D712}}$ , with $\tilde{\unicode[STIX]{x1D712}}^{\prime }=\tilde{\unicode[STIX]{x1D712}}-\langle \tilde{\unicode[STIX]{x1D712}}\rangle$ , $\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}$ is the variance of $\tilde{\unicode[STIX]{x1D712}}$ , $\unicode[STIX]{x03BC}_{\tilde{\unicode[STIX]{x1D712}}}$ is the average of $\tilde{\unicode[STIX]{x1D712}}$ and ${\mathcal{W}}$ is a Wiener process. Since $\unicode[STIX]{x1D711}$ is log-normal, the corresponding $\tilde{\unicode[STIX]{x1D712}}$ is calibrated so that $\langle \text{e}^{\tilde{\unicode[STIX]{x1D712}}}\rangle =1$ , hence giving $\unicode[STIX]{x03BC}_{\tilde{\unicode[STIX]{x1D712}}}=-\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}/2$ (e.g. Pope & Chen Reference Pope and Chen1990). The process $\text{d}\tilde{\unicode[STIX]{x1D712}}$ , needed for our derivations, is also a Gaussian process with mean and variance inferred from (6.1) as (considering that $\mathbb{E}[(\text{d}{\mathcal{W}})^{2}]=\text{d}t$ )

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x03BC}_{\text{d}\tilde{\unicode[STIX]{x1D712}}}=0, & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{\text{d}\tilde{\unicode[STIX]{x1D712}}}^{2}=\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}\frac{\text{d}t^{2}}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}^{2}}+2\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}\frac{\text{d}t}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}}=\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}^{2}\frac{\text{d}t^{2}}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}^{2}}\left(1+2\frac{{\mathcal{T}}_{\unicode[STIX]{x1D712}}}{\text{d}t}\right). & \displaystyle\end{eqnarray}$$

Due to the Wiener process, the model (6.1) introduces unphysical fluctuations which influence the variance of $\text{d}\tilde{\unicode[STIX]{x1D712}}$ on the scale of $\text{d}t$ (equation (6.3)) instead of $\text{d}t^{2}$ for any continuous process, therefore making the Lagrangian Taylor time scale of $\tilde{\unicode[STIX]{x1D712}}$ , $\unicode[STIX]{x1D70F}_{L,\tilde{\unicode[STIX]{x1D712}}}=\unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}\,\text{d}t/\unicode[STIX]{x1D70E}_{\text{d}\tilde{\unicode[STIX]{x1D712}}}$ , dependent on $\text{d}t$ (or numerically on the chosen time step). To remove this dependency we convolute the stochastic process $\tilde{\unicode[STIX]{x1D712}}$ with a weighted average filtering function $A$ (equation (2.2)), following $\unicode[STIX]{x1D712}=A\star \tilde{\unicode[STIX]{x1D712}}$ and $\text{d}\unicode[STIX]{x1D712}=A\star \text{d}\tilde{\unicode[STIX]{x1D712}}$ , and where the time scale of the filtering, $T_{A}$ , is defined below in order to remove spurious fluctuations shorter than $\unicode[STIX]{x1D70F}_{K}$ . Since $A$ is a deterministic and filter function, $\unicode[STIX]{x1D712}$ and $\text{d}\unicode[STIX]{x1D712}$ also have a Gaussian distribution. Since the filter only acts on very small scales, we consider ${\mathcal{T}}_{\unicode[STIX]{x1D712}}\approx {\mathcal{T}}_{\tilde{\unicode[STIX]{x1D712}}}$ and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}\approx \unicode[STIX]{x1D70E}_{\tilde{\unicode[STIX]{x1D712}}}$ , giving $\unicode[STIX]{x03BC}_{\unicode[STIX]{x1D712}}=-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}/2$ . Finally, as for $\text{d}\tilde{\unicode[STIX]{x1D712}}$ , the mean of $\text{d}\unicode[STIX]{x1D712}$ is zero and its variance is inferred from (6.1) using Itô isometry and considering again that $\mathbb{E}[(\text{d}{\mathcal{W}})^{2}]=\text{d}t$ , giving (in its differentiated form since $\unicode[STIX]{x1D712}$ is continuous)

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x03BC}_{\dot{\unicode[STIX]{x1D712}}}=0, & \displaystyle\end{eqnarray}$$

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D712}}}^{2}=\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}^{2}}\left(1+2{\mathcal{T}}_{\unicode[STIX]{x1D712}}\int _{-\infty }^{\infty }A^{2}\,\text{d}t\right)=\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}^{2}}\left(1+\frac{4}{3}\frac{{\mathcal{T}}_{\unicode[STIX]{x1D712}}}{T_{A}}\right). & \displaystyle\end{eqnarray}$$

We choose $T_{A}$ so that the smallest time scale of fluctuations characterising the variance of $\text{d}\unicode[STIX]{x1D712}$ is no longer $2\text{d}t$ but instead the Kolmogorov time scale $\unicode[STIX]{x1D70F}_{K}$ . From (6.3) and (6.5), this implies considering $T_{A}=\unicode[STIX]{x1D70F}_{K}/3$ , therefore leading to

(6.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D712}}}^{2}=\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}}{{\mathcal{T}}_{\unicode[STIX]{x1D712}}^{2}}\left(1+4\frac{{\mathcal{T}}_{\unicode[STIX]{x1D712}}}{\unicode[STIX]{x1D70F}_{K}}\right). & & \displaystyle\end{eqnarray}$$

Finally, because of the Wiener process in (6.1), the cross-correlation between $\unicode[STIX]{x1D712}$ and $\dot{\unicode[STIX]{x1D712}}$ can be neglected and therefore $p_{2}(\unicode[STIX]{x1D712},\dot{\unicode[STIX]{x1D712}})=p(\unicode[STIX]{x1D712})p(\dot{\unicode[STIX]{x1D712}})$ . Thus, using the Gaussian distributions $p(\unicode[STIX]{x1D712})$ and $p_{2}(\unicode[STIX]{x1D712},\dot{\unicode[STIX]{x1D712}})$ in (3.1) leads to (recalling that $T_{F|D}(\unicode[STIX]{x1D711}_{t}/\langle \unicode[STIX]{x1D711}\rangle )=T_{F|D}(\ln \unicode[STIX]{x1D711}_{t}/\langle \unicode[STIX]{x1D711}\rangle )$ and $\ln \unicode[STIX]{x1D711}/\langle \unicode[STIX]{x1D711}\rangle \approx \unicode[STIX]{x1D712}$ )

(6.7) $$\begin{eqnarray}T_{F|D}\left(\frac{\unicode[STIX]{x1D711}_{t}}{\langle \unicode[STIX]{x1D711}\rangle }\right)\approx C_{T\unicode[STIX]{x1D711}}\unicode[STIX]{x03C0}\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}}{\unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D712}}}}\frac{\displaystyle 1\mp \text{erf}\left(\frac{\unicode[STIX]{x1D712}_{t}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}/2}{\sqrt{2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}}}\right)}{\text{e}^{-(\unicode[STIX]{x1D712}_{t}+\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}/2)^{2}/2\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}}}\end{eqnarray}$$

with, from (1.1) and (6.6),

(6.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}}{\unicode[STIX]{x1D70E}_{\dot{\unicode[STIX]{x1D712}}}}=\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}={\mathcal{T}}_{\unicode[STIX]{x1D712}}\left(1+4\frac{{\mathcal{T}}_{\unicode[STIX]{x1D712}}}{\unicode[STIX]{x1D70F}_{K}}\right)^{-1/2}.\end{eqnarray}$$

Interestingly, since $\unicode[STIX]{x1D712}$ and $\dot{\unicode[STIX]{x1D712}}$ are uncorrelated, one has $p_{2}(\unicode[STIX]{x1D712},\dot{\unicode[STIX]{x1D712}})=p(\unicode[STIX]{x1D712})p(\dot{\unicode[STIX]{x1D712}})$ , hence giving that the integration of $p(\dot{\unicode[STIX]{x1D712}})$ in (3.1) becomes a constant independent of $\unicode[STIX]{x1D712}_{t}$ . This entails that the accuracy of the function $T_{F|D}/\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}$ in predicting variations of $T_{F|D}(\unicode[STIX]{x1D711}_{t})$ mainly depends on the degree of Gaussianity of $\ln \unicode[STIX]{x1D700}$ . On the contrary, the ratio $\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D711}}/\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}$ is dependent on the degree of Gaussianity of both $\ln \unicode[STIX]{x1D700}$ and its variation rate, hence making (6.8) an interesting equation for testing the Gaussianity of a quantity and its first-order derivative. Departure from Gaussianity of $\ln \unicode[STIX]{x1D711}$ and $\dot{\ln \unicode[STIX]{x1D711}}$ can be partially compensated by an adjustment coefficient $C_{T\unicode[STIX]{x1D711}}$ (which is equal to 1 for purely Gaussian distributions). Values of $C_{T\unicode[STIX]{x1D711}}$ are obtained using the DNS dataset and discussed below. Equation (6.7) shows that $C_{T\unicode[STIX]{x1D711}}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}$ is the mean flying and diving time at $\unicode[STIX]{x1D712}_{t}^{s}=-\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D712}}^{2}/2=\langle \unicode[STIX]{x1D712}\rangle$ . This result is consistent both with (3.2) and (3.3). In the case of dissipation, both $\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}$ and ${\mathcal{T}}_{\ln \unicode[STIX]{x1D700}}$ have been approximated empirically as a function of $Re_{\unicode[STIX]{x1D706}}$ in HIT (Yeung et al. Reference Yeung, Pope, Lamorgese and Donzis2006a ; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007 see (C 4) and (C 5) in § C.2), hence making (6.7) entirely analytical. Last, note that this model also works for Eulerian flying and diving time scales by using $\unicode[STIX]{x1D70F}_{E,\unicode[STIX]{x1D712}}$ instead of $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}$ .

6.1.2 Results for $\unicode[STIX]{x1D711}=\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$

We first compare in the case of dissipation the mean times inferred from our DNS with those derived from the models of Babler et al. (Reference Babler, Morbidelli and Baldiga2008, Reference Babler, Biferale and Lanotte2012). Both previous models consider for their scaling (C 1) and (C 2) a coefficient of proportionality $C_{\unicode[STIX]{x1D700}}$ equal to $1$ , however, as previously mentioned in § 2, the unity approximation for the scaling $\unicode[STIX]{x1D700}/\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ is somewhat accurate only when considering unfiltered Lagrangian data, and $C_{\unicode[STIX]{x1D700}}\approx 3.6$ is found to be more accurate when removing numerical discrepancies (figure 2). Figure 11(a) shows the mean flying and diving times obtained from the DNS using the Rice theorem, and using the corrected models. Results from DNS are only recovered accurately for $T_{D}$ at large thresholds, hence missing the small-scale characteristics of dissipation and the existing approximate global symmetry between $T_{F}$ and $T_{D}$ . This can be explained by the fact that the scaling $\dot{\unicode[STIX]{x1D700}}\sim \unicode[STIX]{x1D700}/\unicode[STIX]{x1D70F}_{K}(\unicode[STIX]{x1D700})$ does not capture the multi-scale properties of the fluctuations of dissipation where a single value of dissipation can correspond to various values of time variation and vice versa.

Figure 11.

(a,b) Normalised mean flying and diving residence times, $T_{F}/\unicode[STIX]{x1D70F}_{K}$ (pure colours) and $T_{D}/\unicode[STIX]{x1D70F}_{K}$ (lighter colours), with respect to (a) $\unicode[STIX]{x1D700}_{t}/\langle \unicode[STIX]{x1D700}\rangle$ and (b) $\unicode[STIX]{x1D709}_{t}/\langle \unicode[STIX]{x1D709}\rangle$ derived from the Rice theorem with DNS data and from two previous models (Babler et al. Reference Babler, Morbidelli and Baldiga2008, Reference Babler, Biferale and Lanotte2012) and our model (6.7) (see inset legends). The previous models only capture accurately the mean diving times for large thresholds of dissipation. The new model derived from Gaussian theory here globally captures both mean flying and diving statistics with discrepancies at extreme thresholds where Lagrangian statistics are known to depart from Gaussianity. This departure from Gaussianity can be removed (completely for diving statistics during holes and partially for flying statistics during bursts) when not considering the correlation between dissipation and its time derivative. Although pseudo-dissipation is usually considered more suitable for Gaussian models, our model also cannot capture Lagrangian statistics of $\unicode[STIX]{x1D709}$ for extreme thresholds. However, departure from Gaussianity is completely removed for both flying and diving times when not considering the correlation between $\unicode[STIX]{x1D709}$ and $\dot{\unicode[STIX]{x1D709}}$ .

The mean flying and diving times obtained using our model (6.7) more accurately match those obtained from DNS (figure 11 a). Note however that our model logically does not capture the existing asymmetry between $T_{F}$ and $T_{D}$ since it uses a symmetrical stochastic model, and results at extreme thresholds are impacted by the fact that bursts and holes cannot be accurately modelled by log-normal statistics (figures 1 and 14). Interestingly, results of the DNS when considering uncorrelated $\unicode[STIX]{x1D711}$ and $\dot{\unicode[STIX]{x1D711}}$ in the Rice theorem were well recovered by our theoretical results of mean diving times during holes and partially captured for flying times during bursts. Hence, a better characterisation of the link between $\unicode[STIX]{x1D700}$ and its temporal variation during bursts and holes would lead to better predictions.

Results of mean residence times from the DNS were recovered by our model with a non-unity coefficient $C_{T\unicode[STIX]{x1D700}}$ of $1.3$ , and ratios $T_{F|D}/\unicode[STIX]{x03C0}(\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D700}}})$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}$ of $1.1$ and $1.2$ , respectively, hence highlighting the known departure from Gaussianity of $\ln \unicode[STIX]{x1D700}$ (figure 14 a) and its time derivative (figure 14 b). Results of mean times obtained from the control simulation at $Re_{\unicode[STIX]{x1D706}}=90$ were also recovered with our model, using $C_{T\unicode[STIX]{x1D700}}=1.4$ with $T_{F|D}/\unicode[STIX]{x03C0}(\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D700}}})\approx 1.1$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}\approx 1.3$ (see appendix D). The fact of non-unity and Reynolds number dependency of $C_{T\unicode[STIX]{x1D700}}$ confirms that a Gaussian-distribution-based model is not sufficient to capture Lagrangian intermittency (Chevillard et al. Reference Chevillard, Castaing, Arneodo, Lévêque, Pinton and Roux2012).

Interestingly, although pseudo-dissipation is known to have better Gaussianity than dissipation (Yeung & Pope Reference Yeung and Pope1989; Yeung et al. Reference Yeung, Pope, Lamorgese and Donzis2006a ), results of our model fitted the Lagrangian statistics from DNS using $C_{T\unicode[STIX]{x1D709}}=1.7$ , hence highlighting the impact of the non-Gaussianity of $\unicode[STIX]{x1D709}$ . Ratios $T_{F|D}/\unicode[STIX]{x03C0}(\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D709}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D709}}})\approx 1.1$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D709}}\approx 1.5$ show that the discrepancy of our model essentially comes from the non-Gaussianity of $\dot{\ln \unicode[STIX]{x1D709}}$ . Moreover, our model does not capture accurately the flying and diving temporal statistics at extreme thresholds of $\unicode[STIX]{x1D709}$ . Considering uncorrelated $\unicode[STIX]{x1D709}$ and $\dot{\unicode[STIX]{x1D709}}$  in the Rice theorem entirely removes this departure of the DNS results from our model. Thus, the Lagrangian intermittency of $\unicode[STIX]{x1D709}$ can be described by the link between pseudo-dissipation and its temporal variation during bursts and holes. Since $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D709}$ only differ by the Reynolds stress tensor, one can conclude that the Lagrangian intermittency of $\unicode[STIX]{x1D700}$ during bursts and holes is characterised both by the correlation between $\unicode[STIX]{x1D700}$ and $\dot{\unicode[STIX]{x1D700}}$ and the Reynolds stress, the latter being negligible during holes.

6.1.3 Results for $\unicode[STIX]{x1D711}=a$ , $\unicode[STIX]{x1D714}$ and $u$

Results of our model (6.7) for acceleration are less accurate with substantial discrepancies in the temporal statistics during holes and also deviation at very large thresholds during bursts (figure 12 a). This was expected as acceleration is normally modelled by a more complex stochastic process (Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007). Note that the deviation during bursts might again be simply due to numerical discrepancies (van Hinsberg et al. Reference van Hinsberg, ten Thije Boonkkamp, Toschi and Clercx2013). Here too, a main portion of the deviation can be explained by the correlation between the magnitude of the acceleration and its variation rate, however, contrarily to dissipation, it is the events of holes which cannot be explained only by such correlation and during which other underlying physics appear to control the Lagrangian intermittency of acceleration. Results from DNS were fitted using $C_{Ta}=0.9$ , a value close to unity, however, ratios $T_{F|D}/\unicode[STIX]{x03C0}(\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D700}}})\approx 1.4$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}\approx 0.7$ show large deviations from Gaussianity of both $\ln a$ and $\dot{\ln a}$ but that compensate each other.

Instead, results on mean statistics of enstrophy are similar to those of dissipation with deviation of our model from our numerical results being explained by the correlation between $\unicode[STIX]{x1D714}$ and $\dot{\unicode[STIX]{x1D714}}$ entirely during holes and partially during bursts, highlighting here too the effects of other driving forces on enstrophy. Theoretical results were fitted using $C_{T\unicode[STIX]{x1D714}}=1.8$ with ratios $T_{F|D}/\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D700}}}\approx 1.2$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}\approx 1.5$ , therefore showing a large deviation from Gaussianity of $\dot{\ln \unicode[STIX]{x1D714}}$ .

Figure 12.

Normalised mean flying and diving residence times, $T_{F}/\unicode[STIX]{x1D70F}_{K}$ and $T_{D}/\unicode[STIX]{x1D70F}_{K}$ , with respect to (a) $a_{t}/\langle a\rangle$ , (b) $\unicode[STIX]{x1D714}_{t}/\langle \unicode[STIX]{x1D714}\rangle$ and (c) $u_{t}/\langle u\rangle$ derived from the Rice theorem with DNS data and from our model (6.7) (see inset legends). While the mean time statistics of enstrophy is well recovered by (6.7), especially when not considering the correlation between $\unicode[STIX]{x1D714}$ and $\dot{\unicode[STIX]{x1D714}}$ in the Rice theorem, it is less accurate in the case of acceleration. Finally, results of our model show that Lagrangian intermittency of velocity i.e. the mean time statistics during bursts and holes could be theoretically recovered should one model the correlation between $u$ and $\dot{u}$ .

Finally, it is noteworthy that, when applied to the Lagrangian velocity, our model fits numerical results using $C_{Tu}=2.5$ with ratios $T_{F|D}/\unicode[STIX]{x03C0}(\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}/\unicode[STIX]{x1D70E}_{\dot{\ln \unicode[STIX]{x1D700}}})\approx 1.3$ and $\unicode[STIX]{x1D70F}_{L,\unicode[STIX]{x1D712}}/\unicode[STIX]{x1D70F}_{L,\ln \unicode[STIX]{x1D700}}\approx 1.9$ . Thus, even though the Lagrangian velocity is known to be almost Gaussian, such large values of $C_{Tu}$ and corresponding ratios highlight the strong impact of the known deviation from Gaussianity of the Lagrangian variation rate of velocity (Chevillard et al. Reference Chevillard, Castaing, Arneodo, Lévêque, Pinton and Roux2012). Surprisingly, even though our results are inferred from HIT, previous results of $C_{Tu}$ obtained for the Eulerian velocity in turbulent boundary layers were closer to unity ( $1.1\pm 10\,\%$ , see Sreenivasan et al. Reference Sreenivasan, Prabhu and Narasimha1983). Further study will be necessary to compare the Lagrangian and Eulerian statistics as well as assess the effect of the anisotropy introduced by the boundary. Our model could entirely recover the mean time statistics of the Lagrangian velocity during holes and bursts when considering uncorrelated $u$ and $\dot{u}$ , showing that the Lagrangian intermittency of velocity could be recovered with a theoretical modelling of the correlation between the velocity and its variation rate.

Thus, our model represents an overall accurate first approximation of the Lagrangian time scales of turbulence quantities in HIT. Interestingly, equations (6.7) and (6.8) state that the mean temporal statistics of turbulence quantities can be approximated knowing only their variance and those of their variation rates. In terms of time scales, this means that the mean temporal statistics of turbulent quantities can be approximated from their decorrelation time scales along with the Kolmogorov time scale. This defines $[\unicode[STIX]{x1D70F}_{K}-{\mathcal{T}}_{\ln \unicode[STIX]{x1D712}}]$ (with ${\mathcal{T}}_{\ln \unicode[STIX]{x1D712}}\approx 12\unicode[STIX]{x1D70F}_{K}$ at $Re_{\unicode[STIX]{x1D706}}=240$ ) as the core range of time scales in turbulence.

Figure 13.

(a,b) Profiles of mean flying (pure colours) and diving (lighter colours) times and LPDFs at $\unicode[STIX]{x1D700}_{t}^{s}$ derived from our DNS at $Re_{\unicode[STIX]{x1D706}}=240$ and Lagrangian stochastic models of Pope & Chen (Reference Pope and Chen1990), Chevillard & Meneveau (Reference Chevillard and Meneveau2006) and Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007) presented in appendix C. (b,c) Same as in (a,b) but for acceleration using data from our DNS and from the model of Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007).

6.2 Flying and diving time statistics from current Lagrangian stochastic models

Here, we assess the accuracy of current Lagrangian stochastic models in predicting the mean residence flying and diving times as well as the corresponding LPDFs in HIT. Lagrangian stochastic models have received particular interest these past decades as they represent an efficient way to compute the temporal variation of turbulence quantities. Models either focus on a specific turbulence quantity, whether velocity (Sawford Reference Sawford1991), dissipation (Pope & Chen Reference Pope and Chen1990; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007) or acceleration (Sawford Reference Sawford1991; Lamorgese et al. Reference Lamorgese, Pope, Yeung and Sawford2007), or on velocity gradients (Girimaji & Pope Reference Girimaji and Pope1990; Chevillard & Meneveau Reference Chevillard and Meneveau2006; Meneveau Reference Meneveau2011) from which can be inferred various quantities. We consider two Lagrangian stochastic models for dissipation (the models of Pope & Chen (Reference Pope and Chen1990), Chevillard & Meneveau (Reference Chevillard and Meneveau2006) and Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007)) and one for acceleration (the model of Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007)). Details on the models as well as the values of the various parameters taken for comparison with results of our DNS are given in appendix C.

Figure 13(a) shows the profiles of mean flying and diving times obtained from time series of dissipation computed with the model of Pope & Chen (Reference Pope and Chen1990), Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007), (C 3), and of Chevillard & Meneveau (Reference Chevillard and Meneveau2006), (C 6). Since all models are based on a stochastic process (i.e. the Wiener process) which, as previously explained, introduces unphysical fluctuations on scales smaller than $\unicode[STIX]{x1D70F}_{K}$ , the same filtering function (2.2) with $T_{A}=\unicode[STIX]{x1D70F}_{K}/3$ was applied to all computed time series. The model of Pope & Chen (Reference Pope and Chen1990), Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007) globally reproduces profiles of mean flying and diving times found using our DNS and captures the correct $\unicode[STIX]{x1D700}_{t}^{s}$ , although the corresponding mean time $T_{F|D}$ is slightly slower by a factor $1.5$ which can be explained both by the Gaussian approximation (as in our theoretical derivation) and the empirical approximations of $\unicode[STIX]{x1D70E}_{\ln \unicode[STIX]{x1D700}}$ and $T_{\ln \unicode[STIX]{x1D700}}$ (C 4) and (C 5). On the contrary, results from the model of Chevillard & Meneveau (Reference Chevillard and Meneveau2006) are not as accurate. While the statistics of mean flying times are recovered, although the model does not reach the large values obtained with DNS, with a maximum of $10\langle \unicode[STIX]{x1D700}\rangle$ , mean flying times are critically different than those obtained from DNS. The existing asymmetry between mean flying and diving statistics is much more accentuated than for DNS. Finally, $\unicode[STIX]{x1D711}_{t}^{s}$ is not recovered, however, the corresponding mean time $T_{F|D}$ is about the same value as that obtained with the model of Pope & Chen (Reference Pope and Chen1990), Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007).

Figure 13(b) shows the flying and diving LPDFs obtained from both models at $\unicode[STIX]{x1D700}_{t}^{s}$ . Both models fail to capture the distributions obtained from DNS, especially the peak at $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D706}}$ . Interestingly, both models produce the same diving LPDFs. The flying LPDFs produced by the symmetrical model of Pope & Chen (Reference Pope and Chen1990), Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007) are by definition equal to the diving LPDFs. Instead, the model of Chevillard & Meneveau (Reference Chevillard and Meneveau2006) creates a slight asymmetry between flying and diving statistics but it is not consistent with that of DNS. Finally, the predominance of short flights and dives shows that the LPDFs are mainly governed by the Wiener process at numerical time step $\unicode[STIX]{x0394}t$ which has been filtered at $\unicode[STIX]{x1D70F}_{K}$ . This explains why the mean residence time $T_{F|D}$ at $\unicode[STIX]{x1D711}_{t}^{s}$ could not be recovered and is equivalent for both models since the same $\unicode[STIX]{x0394}t$ and filtering were used.

Figure 13(a) shows the profiles of mean flying and diving times obtained from time series of acceleration computed with the model of Lamorgese et al. (Reference Lamorgese, Pope, Yeung and Sawford2007), equation (C 9). Results are not conclusive. While flying times somewhat follow the same trend as that from DNS, mean diving times are not accurate. The magnitude $a_{t}^{s}$ is captured but not its corresponding residence time. This is explained here too by the predominance of small-scale fluctuations in the signals which induce peaks of the LPDFs at $a_{t}^{s}$ corresponding to the filtering time scale instead of that obtained from DNS (figure 13 b).