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# Extreme events in computational turbulence

Contributed by Katepalli R. Sreenivasan, September 4, 2015 (sent for review July 14, 2015; reviewed by Celso Grebogi, Raymond Shaw, and Harry L. Swinney)

## Significance

In the last decade or so, massive computations of turbulence performed by solving the exact equations of hydrodynamic turbulence have provided new quantitative data and enhanced our understanding. This paper presents results from the largest such computations, to date, devoted to the study of small scales. We focus on “extreme events” in energy dissipation and squared vorticity (enstrophy). For the Reynolds numbers of these simulations, events as large as

## Abstract

We have performed direct numerical simulations of homogeneous and isotropic turbulence in a periodic box with 8,192^{3} grid points. These are the largest simulations performed, to date, aimed at improving our understanding of turbulence small-scale structure. We present some basic statistical results and focus on “extreme” events (whose magnitudes are several tens of thousands the mean value). The structure of these extreme events is quite different from that of moderately large events (of the order of 10 times the mean value). In particular, intense vorticity occurs primarily in the form of tubes for moderately large events whereas it is much more “chunky” for extreme events (though probably overlaid on the traditional vortex tubes). We track the temporal evolution of extreme events and find that they are generally short-lived. Extreme magnitudes of energy dissipation rate and enstrophy occur simultaneously in space and remain nearly colocated during their evolution.

Fluid motions encountered in most circumstances are typically turbulent; therefore a good understanding of the subject is essential both for intrinsic scientific reasons and for advancing important technologies, e.g., improving jet engine performance. The difficulty of the subject (1, 2) has unfortunately consigned our present understanding to be partial at best. A milestone of turbulence theory consists of the similarity hypotheses of Kolmogorov (3, 4) and their various descendant scaling theories. In refs. 5 and 6, one can find a fair summary of the theoretical ideas as well as the considerable experimental work devoted to assessing their veracity. Rapid advances in computing power in recent decades have made computations increasingly important in advancing our understanding of the subject. Key quantities that cannot yet be measured in experiments can instead be computed by the so-called direct numerical simulation (DNS; e.g., see ref. 7), in which the exact equations of motion based on mass and momentum conservation are integrated numerically in time and space. The DNS data are capable of providing a wealth of quantitative detail (see, e.g., ref. 8) and improved qualitative understanding. In this paper, we present results from the largest DNS, to date, of isotropic turbulence aimed at the small-scale structure, rendered statistically stationary by large-scale forcing. We focus on the extreme events (to be made more precise momentarily).

Turbulent flows consist of disorderly fluctuations in all measurable properties over a range of scales in both space and time. These fluctuations produce a combination of changes in shape and orientation of an infinitesimal fluid element and can affect quantities of practical interest, such as the tendency of tiny water vapor droplets to collide and grow to millimeter-size rain drops in atmospheric clouds (9). One of the key fluctuating quantities is the energy dissipation rate, *ν* is the kinematic viscosity, *i*, and we have used Einstein’s summation convention. The second key quantity is the turbulent vorticity, *ε* and

Our focus on small-scale properties *ϵ* and ^{3} in 2003 (12). The present simulations on an 8,192^{3} box (see *Simulation and Data Processing Methodology*) push toward attaining higher Reynolds numbers. See the legend for Table 1 for specific definitions of the Reynolds number; it suffices here to say that it determines the range of excited scales. This quest to increase the Reynolds number and the need to better resolve the small scales is the crux of DNS methods today (13, 14). As the flow Reynolds number increases (which is the main reason for increasing the size of the simulations), *ϵ* and *ϵ* and *Supporting Information*.

## Extreme Events in Small Scales

In the turbulence literature (e.g., ref. 15), the phrase “extreme events” has referred to fluctuations that are a few times [typically

One further observation on Fig. 1 is useful. It can be seen that events of moderate intensity (of order tens to hundreds of the mean) are more likely for the enstrophy than dissipation. At high Reynolds numbers as here, the PDFs of dissipation and enstrophy tend to converge toward higher values of both variables. This result has an important consequence: It suggests that higher moments of *ϵ* and

## Simulation and Data Processing Methodology

The considerations above point to a need for advanced computing that pushes the envelope of current limits. DNS of turbulence using tens or even hundreds of billions of grid points is a demanding task that requires petascale resources in processing speed and data management. Key features of our parallel algorithms include a 2D domain decomposition*, exploiting truncation (for dealiasing) in wavenumber space to reduce communication and computational needs, remote memory addressing for interprocessor communication, and multithreading for multicored on-node parallelism. Although pseudospectral codes tend to be communication-intensive, they perform significantly better if a favorable network topology is available. Accordingly, our code has scaled reasonably well up to

In the present 8,192^{3} simulation, we have achieved stationary state by forcing the energy spectrum in the lowest wavenumber shells, as in our earlier 4,096^{3} simulations (19), and the Reynolds number is increased by reducing the viscosity. The Taylor-scale Reynolds number reached is 1,300. It is only slightly higher than in earlier 4,096^{3} simulations by others and by us, because our grid spacing of 1.5 Kolmogorov length scales is significantly better than that used earlier in such large simulations. With a dedicated 0.5-petabyte storage capacity, we were able to save many instantaneous snapshots of the Fourier velocity fields, from which many one- and two-point statistics can be computed and subsequently ensemble-averaged. Statistics such as the PDFs of dissipation rate and enstrophy can also be obtained “on the fly” within the DNS code itself. Qualitative information on spatial structure is obtained by 3D visualization, with focus on local regions of intense activity.

Table 1 lists some parameters of interest. They include information on length scales, turbulence intensity, and the scaling of the averaged dissipation rate. The nondimensional quantities behave in a manner expected of high Reynolds number turbulence based on past publications by us and by other groups in the field (8, 19).

## Spatial Structure

To understand the spatial structure of extreme events, we identify them in the DNS database and zoom in on their neighborhoods. In using a 2D domain decomposition, the 3D solution domain is divided into a number of “pencils,” which are elongated subdomains with two short (not necessarily equal) dimensions and one full-length dimension. Three-dimensional subvolumes, or “subcubes” can be formed by further subdividing each pencil along its longest direction. Larger subcubes not completely residing within a single pencil can be formed by merging neighboring pencils. The net result is a 3D domain decomposition achieved in an efficient manner. In addition to facilitating visualization, this approach allows us to compute the statistics of local averages of dissipation and enstrophy (over a subcube of linear size *r*, say), these being the key element of Kolmogorov’s (4) refined similarity theory. Local averaging smooths out extremely large values of *r* increases; extremely small values also become less extreme. Fig. 2 shows the PDFs of both *r*. The outermost line corresponding to the highest resolution has a power-law tail on the left, which has been shown (19) to arise from the fact that small strain rates and vorticity components are associated with the Gaussian core for the PDF of velocity gradient fluctuations. As *r* increases, the ranges of values, or variability, of both *r* resemble a power-law tail, which is ultimately truncated (so that the moments do exist). This power-law tail is observed only for the smallest few sizes of the subcube, corresponding to an activity that is localized in space.

The intricate detail often present in turbulence datasets makes 3D scientific visualization a useful investigative tool. Given our purpose, it is not necessary to visualize the prohibitively large full-field 8,192^{3} snapshots, but it is important to focus on local regions where extreme events reside and pick the subcubes that are large enough. We mainly use 3D color surfaces but note that the appearance of the resulting images can depend strongly on the contour threshold chosen.

Fig. 3 shows a collection of nine images, some showing magnified views of a smaller subcube to emphasize important features. In Fig. 3*A*, we begin with a low threshold of 10, and a subcube of *ϵ* and *B* and *C* shows the effect of increasing the threshold to 30 (Fig. 3*B*) and 100 (Fig. 3*C*), while zooming in on a more compact (*ϵ* and

Continuing, Fig. 3 *D–F* shows the contrasts between images where *ϵ* and *E*. In Fig. 3*F*, where both variables are shown, the chunky region appears as mainly red, which implies that the dissipation contour surface subsumes the region of peak enstrophy. This observation is consistent with the notion that high-vorticity regions at this level of intensity tend to be wrapped around by sheets of intense dissipation (and thus the strain rate) (22).

Finally, Fig. 3 *G–I* are images obtained at yet higher thresholds. A comparison between Fig. 3 *F* and *G* shows that some vortex filaments visible at threshold 300 are no longer present at threshold 600, while the main structure with a chunky appearance remains largely intact. In Fig. 3*H* as the threshold is raised further to 4,800, the structure becomes somewhat slimmer although not as slender as typical worm-like vortex filament of refs. 12 and 23. In Fig. 3*I* for the threshold of 9,600, the structure seems to break into two—one of which looks like a blob and and the other like a filament. The wrapping around of vortex filaments by sheet-like dissipation is now less pronounced, as the red color itself is now covering the chunky main structure less completely. Closer examination (e.g., by raising the threshold even higher) indicates that the global maximum is located inside the blob-like region in the lower half of Fig. 3*I*.

To ascertain that the snapshot used to produce Fig. 3 is representative, and to illustrate the effect of the Reynolds number, in Fig. 4, we show dissipation and enstrophy contour surfaces from three 8,192^{3} snapshots, and one 2,048^{3} at a lower Reynolds number. Fig. 4*A* is the same as Fig. 3*F* but repeated here for convenience. Fig. 4*A–C* shows a dominant chunky structure with some worm-like vortex filaments nearby. All three images are consistent with the scenario of sheets of intense dissipation being wrapped around a region of intense enstrophy. The chunky structure itself contains the grid location where both dissipation and enstrophy attain their maximum values. The near colocation of dissipation and enstrophy is apparently a very robust feature—in all of the 8,192^{3} snapshots that we examined, the maxima of dissipation and enstrophy are located just one or two grid points apart. However, Fig. 4*D*, for a lower Reynolds number, is characteristically different: The dominant structure there appears to be vortex filaments that are only partially surrounded by intense dissipation. Peak dissipation and peak enstrophy in the lower Reynolds number data are also not colocated.

## Temporal Evolution

Because small-scale motions are traditionally thought to be associated with short time scales, extreme events in dissipation and enstrophy may be expected to have short lifetimes. On the other hand, strong and organized vortex structures have long been known to be relatively persistent, visually identifiable, and coherent over a significant fraction of a large-eddy timescale (24, 25). We are thus interested in characterizing the temporal evolution (and sustenance) of the extreme events studied here, including the physical mechanism(s) that cause these extreme events to form (as addressed in a different context in ref. 15).

To obtain some concrete data, we examine a number of instantaneous velocity fields that have been saved at relatively close time intervals over a significant duration. For each instantaneous snapshot, we divide the solution domain into a number of subcubes of fixed size (of ^{3} datasets). By identifying the maxima of dissipation and enstrophy within each subcube and then repeating the procedure for all subcubes, we obtain their global maximum values for a given snapshot, and also identify the subcubes in which those maximum values reside. Only if a subcube contains the global maximum in two successive snapshots is it considered to contain the extreme event (merely for purposes of robustness). We examine the time history of the local maxima in such subcubes over the available time period, which is many times longer than the typical time scale of the extreme events.

We use data from a double-precision segment of the simulations in which 31 snapshots of velocity fields were saved over a period of about 28 Kolmogorov time scales. The maxima of

In principle, extreme dissipation or enstrophy at a point can be the result of a strong production mechanism that is localized in space and time, and/or a short-term accumulation due to turbulent transport. On the other hand, effects that tend to act against or limit extreme values include viscous diffusion (a transport effect that tends to reduce any localized spatial inhomogeneities) and viscous dissipation, which destroys the velocity gradient fluctuations that underlie both dissipation and enstrophy. To assess the relative importance of these distinct effects for intense-to-extreme enstrophy, we can compute all of the terms in the instantaneous enstrophy budget equation,**1** gives

Fig. 6 shows the conditional averages for each term in Eq. **1**, for two distinct ranges of the normalized enstrophy. Production is the dominant positive contribution, increasing strongly with the enstrophy. Although this result is not unexpected, we caution that the value of the term

## Conclusions and Implications

One of the highlights of turbulence theory is Kolmogorov’s pioneering work (3), which predicts a

The present simulations at the highest Reynolds numbers to date—with the small scales well resolved—suggest something even more complex: With increasing Reynolds numbers, the extreme events assume a form that is not characteristic of similar events at low Reynolds numbers. Our results show that, for the Reynolds numbers of these simulations, events as large as

Extreme events at the level of intensity reported in this paper are expected to have a strong influence, for instance, on the Lagrangian intermittency for infinitesimal fluid particles (34), and the local structure of the acceleration field in the neighborhood of events of extreme dissipation and/or extreme enstrophy (35).

A brief speculation is worthwhile: What might one see in well-resolved simulations at even higher Reynolds numbers? Will new types of extreme events (or singular structures) emerge? The current understanding is that, in the singularity, exponents saturate in the limit of infinite Reynolds number (see ref. 36). One does not know if that asymptotic state has been reached in these simulations.

## Resolution, Sampling, and Machine Precision

The amplitudes of extreme events studied in this paper are much higher than those usually characterized as intense vorticity in the literature. The local Kolmogorov length and time scales associated with dissipation rate fluctuations at thousands or tens of thousands of times of the mean value obviously cannot be fully resolved by finite grid spacings and time steps deemed adequate for conventional DNS work. The inherent rarity of extreme events and their rapid evolution require vigilance in ensuring that the qualitative and quantitative results are not affected by limitations in sampling (see ref. 1). Given the demanding nature of the numerics, one may also ask if machine precision has a significant effect. It is important to separate these artificial effects from real Reynolds number dependencies.

The cleanest but also most expensive approach to address such concerns about accuracy is to repeat the simulation at higher resolution, allow more (independent) samples, increase the level of machine precision, etc., and then compare the results. For our 8,192^{3} simulation, although double-precision results have been obtained for a limited time span (Fig. 5), not all of these strategies are feasible. However, the key property of convergence of dissipation and enstrophy PDFs (Fig. 1) had previously been seen in DNS at ^{3} and 4,096^{3} grids at substantially less cost. We keep the Reynolds number the same but increase machine precision and grid resolution.

To prevent statistical uncertainties from contaminating deterministic effects due to changes of resolution, it is useful to perform separate simulations starting from the same initial conditions. Fig. S1 shows time traces from four such simulations of the global maximum (in space) enstrophy used as a marker of extreme events. The time span covered is similar to that in Fig. 5, but here we have collected data at hundreds of time instants economically by computing dissipation and enstrophy on the fly within the DNS code instead of using postprocessing. Clearly, spatial resolution has a substantial effect, with the line in magenta (

Fig. S2 compares the PDFs of dissipation and enstrophy corresponding to the different scale resolutions of Fig. S1. In Fig. S2*A*, the apparent convergence between these two PDFs at normalized value of about 1,000 is in very good agreement with previous work (2) that averaged over about 20 realizations covering a much longer time span. However, Fig. S2*B* shows that extreme enstrophy is still more likely than extreme dissipation. Higher-resolution data (dashed lines) display similar features—but at higher values of both variables, because improved spatial resolution allows sharper velocity gradients contributing to both dissipation and enstrophy to develop.

The comparisons given here show that the extreme events observed in this work are not an artifact of limited resolution (in fact, they become stronger if resolution is improved), and statistical convergence can be obtained in a relatively short physical time span if samples are taken sufficiently frequently. The highest peak values of

## Acknowledgments

The authors gratefully acknowledge dedicated assistance from consultants and staff on the Blue Waters Project, which is supported by National Science Foundation (NSF) at the National Center for Supercomputing Applications, University of Illinois at Urbana–Champaign. This work is supported by NSF Grant ACI-1036170 under the Petascale Resource Allocations program.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: krs3{at}nyu.edu.

Author contributions: P.K.Y. and K.R.S. designed research; P.K.Y. and X.M.Z. performed research; P.K.Y. and X.M.Z. analyzed data; and P.K.Y. and K.R.S. wrote the paper.

Reviewers: C.G., University of Aberdeen; R.S., Michigan Technological University; and H.L.S., University of Texas at Austin.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1517368112/-/DCSupplemental.

↵*Donzis DA, Yeung PK, Pekurovsky D, 2008 TeraGrid Conference, June 9--13, 2008, Las Vegas, NV.

Freely available online through the PNAS open access option.

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