J. Fluid Mech. (2013), vol. 728, R4, doi:10.1017/jfm.2013.321

Universal mechanism for saturation ofvorticity growth in fully developed fluidturbulenceChakradhar Thantanapally1, Dhiraj V. Patil1, Sauro Succi2,3

and Santosh Ansumali1,†

1Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur,Bangalore 560064, India2Instituto Applicazioni Calcolo ‘Mauro Picone’, C.N.R, Via dei Taurini, 19, 00185, Rome, Italy3Freiburg Institute for Advanced Studies, Albertstrasse, 19, D-79104, Freiburg, Germany

(Received 10 April 2013; revised 23 May 2013; accepted 17 June 2013;first published online 9 July 2013)

Numerical results from large-scale, long-time, simulations of decaying homogeneousturbulence are reported, which indicate that blow-up of inviscid flows is tamed by theemergence of collective dynamics of coherent structures. The simulations also suggestthat this collective dynamics might lead to universal behaviour during the transientevolution of turbulence. In particular, simulations with three different initial conditionsshow evidence of a k−3 log k spectrum in the transient stage, before the Kolmogorovk−5/3 asymptotic regime is attained. Such a universal transient might serve as a spectralfunnel to the time-asymptotic Kolmogorov spectrum, which is invariably observed inthe late stage of all three simulations presented in this work. The present work isentirely based on simulation evidence. However, the statistical analysis of the coherentstructures suggests an analogy with population dynamics, which might be conducive tonew mathematical models of transient decaying turbulence.

Key words: homogeneous turbulence, turbulence simulation, turbulent transition

1. Introduction

Turbulence can be defined as that state of matter characterized by a loss ofcoherence in space and time, as a result of the nonlinear interaction of a largenumber of degrees of freedom, in excess of billions even for most familiar phenomena,such as flow past an automobile (Frisch 1995; Chen et al. 2003; Davidson 2004). Thisraises a formidable computational barrier even for the most powerful computationalmethods in the foreseeable future, let alone analytical methods. It is therefore notsurprising that fluid turbulence has continued to attract attention of scientists from

† Email address for correspondence: ansumali@jncasr.ac.in

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C. Thantanapally, D. V. Patil, S. Succi and S. Ansumali

many disciplines. The research and literature on the topic is so vast that even theproblem definition is not the same for different communities of researchers (Liepmann1979; Frisch 1995; Sreenivasan 1999; Pope 2000; Lumley & Yaglom 2001). Amongothers, primary goals of turbulence research can be summarized as follows: (i) tounveil statistical universalities (Frisch 1995; Dhar, Sain & Pandit 1997; Sreenivasan& Antonia 1997; Toschi & Bodenschatz 2009), underlying the irreducible dynamicalcomplexity of different turbulent flows; (ii) to develop robust and accurate methodsto compute the (statistical) dynamics of turbulent flows (Germano 1992; Lesieur &Metais 1996; Moin & Mahesh 1998; Ansumali, Karlin & Succi 2004) in realisticgeometries, where universal and non-universal behaviour must necessarily coexist; and(iii) to understand the basic mechanisms controlling the transition from laminar toturbulent behaviour (Sreenivasan 1999; Moxey & Barkley 2010).

In the present manuscript, using large-scale computations, we investigate and reportuniversal features in the transient dynamics towards steady-state turbulence. Thisis a comparatively less studied aspect of turbulence, perhaps due to the fact thatuniversal behaviour is usually associated with time-asymptotic states, whereas transientfeatures, being driven by morphological details, are typically regarded as inherentlynon-universal.

The present simulations of transient decaying three-dimensional flows suggest thatthe saturation of vorticity growth takes place through the onset of transient, andyet universal, structures. The first is the appearance of universal k−3(log(k/k0))

n

spectra, where k0 ∼ 3 and n ∼ 1 (for notational simplicity, whenever no ambiguityarises, k−3(log(k/k0))

n, shall be referred to as k−3 log k), in the transient towardsKolmogorov’s statistical steady state, while the second is a concurrent burst ofenstrophy, which is carried by high-stretch and high-vorticity transient structures.Although rare and short-lived, such structures are sufficiently strong and long-livedto signal the transit of the system through a universal k−3 log k state, on its waytowards the asymptotic Kolmogorov spectrum.

In § 2, we review the main arguments behind vorticity dynamics and blow-up issues.In § 3, brief details of the simulation methodology are given. Sections 4 and 5 discussthe energy scaling laws and the statistical analysis of the numerical results. Finally, abrief summary and conclusion are presented in § 6.

2. Vorticity dynamics and blow-up

The evolution of fully developed three-dimensional homogeneous incompressibleturbulent flows is analysed in terms of the vorticity field, ω = ∇ × u, u being theflow velocity. The dynamics of vorticity is often analysed in terms of local enstrophyΩ = ω ·ω/2 as

DΩDt= 2sΩ + νω ·1ω, s= ω · (∇u) ·ω

ω2, (2.1)

where ν denotes kinematic viscosity, and s is the vortex stretching term, namely thegrowth (decay) rate of the local enstrophy. This representation elicits the two basiccontributions to the vorticity growth (decay), as it is carried along by the fluid, namelya viscous dissipative sink, and the stretching term, which vanishes in two dimensions,since vorticity is then perpendicular to the plane of motion (Tsinober, Shtilman &Vaisburd 1997; Holm & Kerr 2002; Donzis, Yeung & Sreenivasan 2008).

Leaving aside dissipation, which plays a leading role only in the late stageof the evolution, (2.1) takes the form dΩ/dt = λΩ3/2, provided the norm of the

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Universal mechanism for saturation of vorticity growth

strain tensor S ≡ (∇u + (∇u)T)/2 is loosely identified with the vorticity magnitude|ω|, times a proportionality factor λ. Such an argument is indeed loose, becausethe vorticity and strain tensors carry a different physical content, rotation versusdeformation, respectively. This is also reflected by their mathematical structure, i.e.the antisymmetric and symmetric component of the deformation tensor, respectively.For λ = const > 0, this scalar equation exhibits a finite-time singular solution,Ω(t) ∼ Ω0((t − tc)/(t0 − tc))

−2 at tc = t0 + 1/(2λ√Ω0), due to the coherent drive

of the vortex stretching term (Frisch, Matsumoto & Bec 2003). Geometrically, thiscorresponds to a vortex tube getting increasingly thinner and more intense, untilits core is annihilated, thus leading to a singular blow-up. A vast amount of workon finite-time singularities, starting from symmetric initial conditions, is available(see Kerr 1993; Pelz 2001; Cichowlas & Brachet 2005; Grafke et al. 2008; Hou& Li 2008; Bustamante & Brachet 2012). Several counter-blow-up arguments havealso been raised, which usually appeal to more detailed insights into the dynamicmorphology of turbulence (Frisch et al. 2003; Yang & Pullin 2010). Notable amongthem is the so-called nonlinear depletion mechanism, which proposes that vorticitytubes would naturally self-limit their growth, upon bending and eventually collapsinginto locally two-dimensional-like structures automatically annihilating the nonlinearstretching term.

Despite their heuristic appeal, none of the above arguments has managed to gainthe status of a rigorous proof, or theorem, whence the scope for extensive numericalsimulations. To date, these have brought up the following picture: the smooth initialconfiguration develops a fast-growing enstrophy stage, driven by a positive averagevortex stretching term. The stretched configurations then undergo a number of suddenmorphological events, such as vortex break-up and merger, during which smoothspectra, decaying with an exponent between 3 and 4, are eventually observed. In thelong term, such low-dimensional structures evolve into a tangle of vortices, supportingthe k−5/3 Kolmogorov spectrum, eventually inclusive of slight anomalous corrections,due to intermittent bursts. Even though such simulations and analytical studies aretypically performed on inviscid (Euler) flows, one can reasonably expect that suchfast-growing transients would remain virtually inviscid also in high Reynolds numbersimulations.

In line with these expectations, our analysis is based on high-Reynolds-numberNavier–Stokes simulations. Figure 1(a) reports the time evolution of energy at differentReynolds number for the case of Kida–Pelz (KP) flow, which demonstrates that,especially at high Reynolds number, in the initial stage of the evolution (t < 2.5),the system behaves like an almost inviscid fluid. In figure 1(b), we present the totalenstrophy, Ω(t), integrated over the entire volume, and the maximum vorticity, ωmax(t),as a function of time, for the KP flow at Re = 5000 (10243 resolution). This figurereveals the existence of a sharp peak in ωmax at t ∼ 2.1, marking the end of the inviscidexponential enstrophy growth. The primary peak is followed by a series of minorsub-peaks, on top of a slowly decreasing trend towards the statistical steady state. Wehave verified this scenario with Taylor–Green (TG) (figure 1c) and additional inviscid(Euler) runs, which indicate the inviscid nature of scaling. Since total dissipation isapproaching zero rapidly (table 1), while spectra are becoming closer and closer tok−3 log k, we have reason to believe that vorticity saturation is basically inviscid andeven local dissipation has not much of a role to play in this stage of the evolution.

We provide numerical evidence that the spectra observed in connection withenstrophy saturation are convincingly close to k−3 log k, thereby pointing to anunderlying local two-dimensionalization and subsequent nonlinear depletion of the

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Ens

trop

hy, m

ax v

ortic

ity

t

10010–1 101

t

101

102

103

104

100 Ens

trop

hy, m

ax v

ortic

ity

5t

0 10

EnstrophyMax vorticity

101

102

103

100

0 2 4 6 8 10 12

EnstrophyMax vorticity

(a) (b) (c)

100

10–1

FIGURE 1. (a) Time evolution of the kinetic energy at different Reynolds numbers Re =2000, 2500, 5000, 10 000. Total enstrophy and maximum vorticity of the KP (b) and TG (c)flow at Re= 5000 with 10243 resolution, as a function of time.

Reynolds number % of initial energy Peak time Resolution

5000 95.924 2.050 10243

10 000 97.71707 2.040 15363

Infinity 99.99867 2.025 10243

Infinity 99.99973 2.025 15363

TABLE 1. Energy dissipation, in correspondence with k−3 log k spectra, for the case of KPflow

flow. Such depletion is found to be dominant, yet not exclusive, meaning that anyvorticity (enstrophy) iso-surface, apart from the extremal ones, supports coexistingsubregions of vorticity growth and decay. The statistics of these populations, tobe detailed in a later section, shows a strong prevalence of (mildly) growing ones,with a burst of strong growth, just before the vorticity peak, followed by subsequentreabsorption after the peak.

3. Simulation methods

As a primary simulation tool, we use a pseudo-spectral method (PS), which solvesthe three-dimensional Navier–Stokes (NS) equations in periodic domains. In numericalimplementations of these equations:

∂t[uj(k) exp(νk2t)] = − exp(νk2t)

(δij − kikj

k2

)F(k), (3.1)

where uj(k) denotes velocity in Fourier space, δij is the Kronecker delta, F(k) is theFourier coefficient of the nonlinear advection term and the pressure term is projectedout, time is advanced via a low-storage version of the Runge–Kutta third-order scheme(Canuto et al. 1988). De-aliasing of the nonlinear term is performed via the 2/3 rule,due to its simplicity and ease of parallel implementation (Canuto et al. 1988). Detailsof the numerical set-up are provided in the supplementary material available at http://dx.doi.org/10.1017/jfm.2013.321.

The numerical simulations were also repeated using the lattice Boltzmann method(Benzi, Succi & Vergassola 1992; Succi 2001; Ansumali, Karlin & Ottinger 2003),

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Universal mechanism for saturation of vorticity growth

105

100

10–5

(a)

10–2 10–1 100

(b)

10–2 10–1 100

100

10–2

10–4

10–6

FIGURE 2. (a) Normalized energy spectra of the KP and TG flow, at the time of peak maximumvorticity, providing clear evidence of a k−3 log k spectrum, in correspondence with the vorticitypeak. (b) Energy spectra of the KP and TG flows, at Re= 104 and Re= 5000, respectively, againproviding clear evidence of a k−3 log k spectrum, to be compared with the horizontal k−3 line.

and were found to provide similar results. We have performed simulations of twoprototypical flows: KP and TG vortices, at Reynolds numbers in the range 103–104,with resolution up to 15363. In order to further probe universality, a third set-up, theHill vortex, is also reported in the supplementary material section. The simulationscover a comparatively long time span, lasting nearly ten circulation times. In order toconfirm the inviscid nature of the results, KP and TG simulations are performed usinga PS simulation of Euler equations.

4. Energy spectrum and universality

While existing theories based on nonlinear depletion concentrate on topologicalchanges of vortex tubes, our simulations suggest that blow-up may stop due to theinteractions between vortex tubes of dissimilar stretching and magnitude. This isillustrated in figure 6 below, where it is shown that each iso-surface of vorticitysupports a mix of positive and negative stretch, throughout the domain. Though wecannot offer any conclusive, direct proof of the dominance of such collision-likeevents, we have nevertheless obtained a series of indirect proofs, lending weight tosuch a conjecture. The first indirect evidence is the detection of quasi-two-dimensionalspectra, with exponent close to −3 (in fact with a logarithmic correction), whichsuggests that such collision-like events would be promoted by an underlying reverseenergy cascade (Kraichnan 1967).

In order to highlight the generality of our results, in figure 2, we report anormalized energy spectrum as a function of the dimensionless wavenumber kη, whereη = (ν3/ε)

1/4 is the Kolmogorov dissipative length, En = (εν5)1/4 the corresponding

energy and ε = 2νΩ is the energy dissipation rate. This figure highlights theoccurrence of k−3 log k spectra, which suggest a spontaneous arrangement of the flowtowards a quasi-two-dimensional structure, as an effective mechanism to tame vorticitygrowth (Kraichnan 1967).

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Max

imum

vor

ticity

Ene

rgy

Ene

rgy

1 2 3 4 5 6 7 8 9t

0 10

200

400

600

800

100

102

10–5

1.0100

102

10–5

1.5100

102

10–5

2.05

10.0100

102

10–5

2.5100

102

10–5

3.5100

102

10–5

k kk

(a)

(b)

FIGURE 3. (a) Energy spectra at t = 1, 1.5, 2.05, 2.5, 3.5, 10, corresponding to the squaresymbols in (b), reporting the time evolution of the maximum vorticity for the KP flow.Continuous and dashed straight lines indicate k−3 and k−5/3 respectively.

In figure 3(a), we present an extended sequence of energy spectra, correspondingto the square symbols in figure 3(b), where the time evolution of the maximumvorticity for the KP flow is shown. This sequence highlights a tendency towardsa k−3 log k spectrum in the initial stage of the evolution, up to about t = 2.5, andthe subsequent long-term rearrangement towards a Kolmogorov spectrum (see alsothe associated movies (movie 1, movie 2, movie 3) available at http://dx.doi.org/10.1017/jfm.2013.321). Spectral exponents between 2.68 and 3.29, just after the r.m.s.vorticity maximum, have been reported before, in spectral simulations of the TGvortex at Re = 1600 and Re = 3000 (Brachet et al. 1983; Orlandi & Pirozzoli2010). However, potential connections between two-dimensionalization of the localfluid structure and taming of enstrophy blow-up, have not received any specialattention to date. The running argument is that transients are relatively uninformative,as they depend on non-universal morphological details, as opposed to the universalstatistical features exposed by energy spectra in the time-asymptotic regime. Here, wecontend that even the morphology-driven transient dynamics can exhibit universality.On the other hand, since our spectra look convincingly closer to k−3 log k thank−3, there is clearly scope to pursue a detailed analysis of the transient, especiallyin the close vicinity of the primary peak. We wish to remind readers that thelogarithmic correction is of major conceptual importance, as it implies that theflow is not just locally smooth, but supports instead non-local effects, leading toquasi-two-dimensional physics, namely a significant scale separation between the large-

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Universal mechanism for saturation of vorticity growth

0

1

2

–1

3

5 100 15r

(a)

5 100 15r

–0.04

–0.02

0

0.02(b)

FIGURE 4. Structure function S3(r), (a) near peak, t = 2.05, and (b) near the Kolmogorov timeregion, t = 4.80, of the KP flow at Re= 2000 with resolution 7683.

scale straining structures and the small-scale strained ones. We note that the departurefrom a pure k−3 power-law and the associated log k correction, as clearly seen infigure 2, indicates that the flow is marginally non-smooth. As detailed in what follows,this observation is also backed up by the detection of a local inverse energy cascadecoinciding with the universal quasi-two-dimensional transient reported in this work.

Figure 4(a), near the peak (t = 2.05), clearly shows that the third-order structurefunction S3(r) is positive at sufficiently large separations, r > 10, thereby pointing toan inverse energy cascade at those scales. On the other hand, around the Kolmogorovtime zone (t = 4.8) (figure 4b), S3(r) is mostly negative, in the small-scale region(r < 10), thereby signalling a direct energy cascade. Finally, the spectra tend toadjust towards the long-term −5/3 Kolmogorov shape. Inspection of the spectralmovies (see supplementary material) provides evidence of this oscillation betweenthe k−3 log k and k−5/3 power spectra, the residence time in k−3 log k being shorterand shorter as time unfolds. The same qualitative picture is observed for both KPand TG vortices, the latter exhibiting more persistent multiple peaks (at a given Re).This lends further weight to the conjecture that freely decaying three-dimensionalhomogeneous incompressible fluid turbulence, exhibits universality not only in itsasymptotic, Kolmogorov, steady state, but also in the transient stage towards such anasymptotic steady state. This universality is realized through local two-dimensionalflattening (vortex mergers) and ensuing substantial nonlinear depletion. It seems thatthe long-term Kolmogorov state cannot be attained, unless the system goes firstthrough the k−3 log k transient, configuring the latter as a sort of funnel to the former.

Here, it should be noted that, though figure 2 shows less than half a decade ofk−3 log k range, the trend is generally towards k−3 log k, the relatively narrow rangebeing due to limited numerical power for high-Reynolds-number flows. On the otherhand, from Euler simulation results, as plotted in figure 5, it is apparent that thetrend is towards k−3 log k spectra, which are then visible for over a decade. Sensitivityanalysis of the spectra, via plotting k3E(k) as a function of k in figure 5(a), clearlyindicates the requirement for a logarithmic correction. This is indeed confirmed withthe curve fit to the simulation data, yielding k0 = 3.2 ± 0.6 and n = 1.33 ± 0.09 as

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C. Thantanapally, D. V. Patil, S. Succi and S. Ansumali

(a)

k

k3E

(k)

104

102

100

102 103101

(b)

log(k)

104

102

100

101100

Datak0

Data

FIGURE 5. (a) Compensated energy spectra at t = 2.0 for Euler simulations of KP flow,showing the necessity for logarithmic correction. (b) Compensated energy spectra as a functionof log k in order to obtain the exact form of logarithmic correction.

shown in figure 5(b), which is clearly a better fit to the simulation data than a plaink−3 spectrum.

5. Statistical analysis

Next, we proceed to a statistical analysis of the correlation between the stretchterm s, as defined in (2.1), and the local enstrophy. To this purpose, we introducefour basic types of structures, based on low (high) vorticity and positive (negative)stretch. As demonstrated by the iso-surfaces shown in figure 6, these four classesstrongly intermix as a consequence of the complex structural dynamics of vorticitytubes (see supplementary movie 4). LN (low-vorticity–negative-stretch) are the weakeststructures, the ones most in danger of disappearing, while HP (high-vorticity–positive-stretch) show the opposite trend; they are high in vorticity and still growing. Theywould absorb the entire enstrophy budget upon untamed growth. The other two,low-vorticity–positive-stretch (LP) and high-vorticity–negative-stretch (HN), are, inprinciple, the carriers of stable behaviour. As we shall see, however, the HNsare very short-lived, as they quickly decay into LNs. If only in descriptive terms,this population-dynamics schematization indicates that the blow-up scenario wouldcorrespond to a complete decoupling of the HP population from the other three, aliteral form of enstrophic runaway.

In figure 7, we report the probability distribution function (p.d.f.) of the stretchs in the low-vorticity (LV) and high-vorticity (HV) regions for the KP flow.The figure demonstrates a bell-shaped distribution of both positive- and negative-growth populations in the LV and HV sectors, with a mild prevalence of positiveones, realizing a positive average stretch. The bell-shaped form of the distributiondemonstrates that nonlinear depletion, s= 0, is indeed taking place in the system. Thedistribution in the HV sector is very rough, due to the much smaller number of HVcounts, roughly 0.01 %, as compared to the LV ones. The substantial shift towardspositive growth is nevertheless quite visible in both cases.

We have observed that at any value of the enstrophy, the LV region contains bothpositive and negative growth rates, indicating a coexistence of growing and decayingsubregions (LP and LN), within the same iso-vorticity surface (figure 6). The bulk

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Universal mechanism for saturation of vorticity growth

(a) (b)

FIGURE 6. Iso-contours of vorticity in KP flow for Re = 2000 with 7683 resolution at t = 1.15.(a) High vorticity at ω = 58, with positive (orange) and negative (black) stretching. (b) Lowvorticity at ω = 29 with positive (red) and negative (blue) stretching.

0

0.01

0.02

0.03

0

0.05

0.10

0.15

–0.02 0 0.02

(a)

0

0.05

0.10

0.15

–0.02 0 0.02

(b)

0

0.05

0.10

0.15

–0.02 0 0.02

(c)

00

0.01

0.02

0.03

–0.5 0 0.50

0.01

0.02

0.03

–0.5 0 0.5

Stretch Stretch Stretch–0.5 0.5

p.d.

f.p.

d.f.

FIGURE 7. Probability distribution of the local stretch in the LVregion(<ωmax/2, normalizedby the total number of LV events) (top row), and the HVregion(>ωmax/2) (bottom row), at(a) t = 2.0, (b) t = 2.1 and (c) t = 2.2, of KP flow at Re= 104, with 15363 resolution.

population in the LV region is centred around the depletion line, s = 0, with aprevalence of mildly positive stretch. A progressive shift towards larger positive sis observed while moving towards the HV region. By and large, the region s ∼ 0.1,contains the bulk of the events, thereby supporting the picture of nonlinear depletion.We have also observed that at all times, the LV sector is vastly more populated thanthe HV sector. The latter is populated only around the outburst of the enstrophypeak, through both HP and HN populations, contributing nonetheless four orders ofmagnitude fewer events than the LV populations. In spite of their paucity in number,these events carry a sizable weight of the overall enstrophy budget, and a dominantone in the local enstrophy peak as shown in figure 1, which de facto carries their

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signature. Both HV populations are short-lived, i.e. they vanish away from the peakinterval. However, their lifetime is nonetheless appreciable on the overall time scale ofthe flow, as it lasts ∼0.3 global transit times.

6. Summary and conclusions

High-resolution pseudo-spectral simulations strongly suggest that the saturationof vorticity growth in non-equilibrium transients of homogeneous incompressibleturbulence might be associated with a universal mechanism of local two-dimensionalization. Such a conjecture is supported by evidence of two-dimensional-like k−3 log k spectra in the peak stage of vorticity growth, and further corroboratedby visual inspection of the flow morphology and statistical analysis of the localgrowth rates on each iso-vorticity surface. However, our data indicate local two-dimensionalization and the third-order structure function suggests an inverse energycascade. The attainment of such a universal transient state is signalled by an intenseburst of high-vorticity and high-stretch structures, whose signature is manifested in theform of a clearcut peak in the maximum vorticity. As the statistical distribution ofthe growth rates indicates coexistence of decaying and growing subregions on eachgiven iso-vorticity surface, we are led to speculate that the aforementioned universalmechanism might be the result of a built-in vorticity redistribution, whereby thehigh-vorticity structures are prevented from uncontrolled growth by direct interactionwith the low-vorticity ones. Based on the present simulations, we believe that viscositywould not play any significant role on the physical picture portrayed in this paper. Thepicture presented here is entirely based on simulation evidence yet to be developedinto a quantitative mathematical model. Finally, while the occurrence of similarbehaviour across three different types of flow is a strong pointer to universality, furtherexplorations are certainly needed to verify the conjecture of universality.

Acknowledgements

The project is funded by Department of Science and Technology via Ramanujanfellowship grant. We also thank Intel India for providing high end workstation wherepresent simulations were performed. Authors, particularly S.S., are grateful to theIndian Academy of Sciences for supporting visits to India under the Raman Chairprogram and the kind hospitality at JNCASR, where this work was performed.

Supplementary movies/materials

Supplementary movies/materials are available at http://dx.doi.org/10.1017/jfm.2013.321.

References

ANSUMALI, S., KARLIN, I. V. & OTTINGER, H. C. 2003 Minimal entropic kinetic models forsimulating hydrodynamics. Europhys. Lett. 63, 798–804.

ANSUMALI, S., KARLIN, I. V. & SUCCI, S. 2004 Kinetic theory of turbulence modelling: smallnessparameter, scaling and microscopic derivation of smagorinsky model. Physica A 338 (3),379–394.

BENZI, R., SUCCI, S. & VERGASSOLA, M. 1992 The lattice Boltzmann equation: theory andapplications. Phys. Rep. 222 (3), 145–197.

BRACHET, M. E., MEIRON, D. I., ORSZAG, S. A., NICKEL, B. G., MORF, R. H. & FRISCH, U.1983 Small-scale structure of the Taylor–Green vortex. J. Fluid Mech. 130, 411–452.

728 R4-10

Universal mechanism for saturation of vorticity growth

BUSTAMANTE, M. D. & BRACHET, M. 2012 Interplay between the Beale–Kato–Majda theorem andthe analyticity-strip method to investigate numerically the incompressible Euler singularityproblem. Phys. Rev. E 86, 066302.

CANUTO, C., HUSSAINI, M. Y., QUARTERONI, A. & ZANG, T. A. 1988 Spectral Methods in FluidDynamics. Springer.

CHEN, H., KANDASAMY, S., ORSZAG, S., SHOCK, R., SUCCI, S. & YAKHOT, V. 2003 ExtendedBoltzmann kinetic equation for turbulent flows. Science 301 (5633), 633–636.

CICHOWLAS, C. & BRACHET, M. E. 2005 Evolution of complex singularities in Kida–Pelz andTaylor–Green inviscid flows. Fluid Dyn. Res. 36 (4), 239–248.

DAVIDSON, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford UniversityPress.

DHAR, S. K., SAIN, A. & PANDIT, R. 1997 Inertial- and dissipation-range asymptotics in fluidturbulence. Phys. Rev. Lett. 78 (15), 2964–2967.

DONZIS, D. A., YEUNG, P. K. & SREENIVASAN, K. R. 2008 Dissipation and enstrophy in isotropicturbulence: resolution effects and scaling in direct numerical simulations. Phys. Fluids 20 (4),045108.

FRISCH, U. 1995 Turbulence: The Legacy of AN Kolmogorov. Cambridge University Press.FRISCH, U., MATSUMOTO, T. & BEC, J. 2003 Singularities of Euler flow? not out of the blue!.

J. Stat. Phys. 113 (5), 761–781.GERMANO, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.GRAFKE, T., HOMANN, H., DREHER, J. & GRAUER, R. 2008 Numerical simulations of possible

finite time singularities in the incompressible Euler equations: comparison of numericalmethods. Physica D 237 (14), 1932–1936.

HOLM, D. D. & KERR, R. 2002 Transient vortex events in the initial value problem for turbulence.Phys. Rev. Lett. 88 (24), 244501.

HOU, T. Y. & LI, R. 2008 Blowup or no blowup? the interplay between theory and numerics.Physica D 237 (14), 1937–1944.

KERR, R. M. 1993 Evidence for a singularity of the three-dimensional, incompressible Eulerequations. Phys. Fluids 5 (7), 1725–1746.

KRAICHNAN, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7),1417–1423.

LESIEUR, M. & METAIS, O. 1996 New trends in large-eddy simulations of turbulence.Annu. Rev. Fluid Mech. 28, 45–82.

LIEPMANN, H. W. 1979 The rise and fall of ideas in turbulence. Am. Sci. 67 (2), 221–228.LUMLEY, J. L. & YAGLOM, A. M. 2001 A century of turbulence. Flow Turbul. Combust. 66 (3),

241–286.MOIN, P. & MAHESH, K. 1998 Direct numerical simulation: a tool in turbulence research. Annu.

Rev. Fluid Mech. 30, 539–578.MOXEY, D. & BARKLEY, D. 2010 Distinct large-scale turbulent-laminar states in transitional pipe

flow. Proc. Natl Acad Sci. 107 (18), 8091–8096.ORLANDI, P. & PIROZZOLI, S. 2010 Vorticity dynamics in turbulence growth. Theor. Comput. Fluid

Dyn. 24 (1), 247–251.PELZ, R. B. 2001 Symmetry and the hydrodynamic blow-up problem. J. Fluid Mech. 444, 299–320.POPE, S. B. 2000 Turbulent Flows. Cambridge University Press.SREENIVASAN, K. R. 1999 Fluid turbulence. Rev. Mod. Phys. 71 (2), 383–395.SREENIVASAN, K. R. & ANTONIA, R. A. 1997 The phenomenology of small-scale turbulence.

Annu. Rev. Fluid Mech. 29, 435–472.SUCCI, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University

Press.TOSCHI, F. & BODENSCHATZ, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev.

Fluid Mech. 41, 375–404.TSINOBER, A., SHTILMAN, L. & VAISBURD, H. 1997 A study of properties of vortex stretching

and enstrophy generation in numerical and laboratory turbulence. Fluid. Dyn. Res. 21 (6),477–494.

YANG, Y. & PULLIN, D. I. 2010 On Lagrangian and vortex-surface fields for flows withTaylor–Green and Kida–Pelz initial conditions. J. Fluid Mech. 661, 446–481.

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