SPECIAL FEATURE: PERSPECTIVE

Turbulent mixing: A perspectiveKatepalli R. Sreenivasana,1

Edited by William A. Goddard III, California Institute of Technology, Pasadena, CA, and approved August 23, 2018 (received for reviewJanuary 9, 2018)

Mixing of initially distinct substances plays an important role in our daily lives as well as in ecological andtechnological worlds. From the continuum point of view, which we adopt here, mixing is complete whenthe substances come together across smallest flow scales determined in part by molecular mechanisms,but important stages of the process occur via the advection of substances by an underlying flow. We knowhow smooth flows enable mixing but less well the manner in which a turbulent flow influences it; but thelatter is the more common occurrence on Earth and in the universe. We focus here on turbulent mixing,with more attention paid to the postmixing state than to the transient process of initiation. In particular,we examine turbulent mixing when the substance is a scalar (i.e., characterized only by the scalar propertyof its concentration), and the mixing process does not influence the flow itself (i.e., the scalar is “passive”).This is the simplest paradigm of turbulent mixing. Within this paradigm, we discuss how a turbulentlymixed state depends on the flow Reynolds number and the Schmidt number of the scalar (the ratio offluid viscosity to the scalar diffusivity), point out some fundamental aspects of turbulent mixing thatrender it difficult to be addressed quantitatively, and summarize a set of ideas that help us appreciate itsphysics in diverse circumstances. We consider the so-called universal and anomalous features and sum-marize a few model studies that help us understand them both.

turbulent mixing | universal features | anomalous features | ramp model |model studies

From supernovae to cream in coffee, and numerouscircumstances in between, the mixing of two initiallydistinct substances plays an important role in our dailylives and in the evolution and sustenance of life itself.A simple case is the mixing of scalars (i.e., the mixingsubstances have no identifying labels other than theirconcentrations) that are chemically neutral with ini-tially flat and stationary interfaces separating them,and forces such as surface tension and gravity areabsent. Molecular diffusion causes the separatinginterface to encroach increasingly into both sub-stances by thickening with time, thus eroding theconcentration gradients and tempering the effective-ness of diffusion. One way to sustain the concentrationgradients and thus the effectiveness of diffusion is toconstantly push the substances toward each other byan external flow. A flow also destabilizes, distorts, andsharpens an interface while increasing its surface areaand enhancing diffusion effects. In the simplest case,the mixing of scalars does not affect the flow itself; thisis the so-called passive scalar mixing, which is the

subject of this article. In practice, there will bepractically important complications such as chemicalreactions, density differences, multiple species, exter-nally applied forces, and so forth, but the essence ofmixing (for small diffusivity) is the interplay betweenthe flow of substances and their intrinsic capacity todiffuse and mix molecularly. The literature on morecomplex instances of mixing is rather large but a usefulentry can be made via refs. 1–4 and referencestherein. The flow itself could be smooth or turbulent(i.e., temporally and spatially stochastic); for the mostpart, we consider the latter.

We particularly keep in mind the following situa-tion. We deposit a blob of a passive scalar such as aneutral dye within a tank of stirred fluid and observeits evolution. Experience shows that the blob will bestretched, folded, and fragmented and eventuallydispersed and diluted in the fluid volume. Thefragmentation and stretching of the blob enhancethe area of the interface and the scalar gradientsacross it, thus putting diffusion and molecular mixing

aSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540Author contributions: K.R.S. wrote the paper.The author declares no conflict of interest.This article is a PNAS Direct Submission.Published under the PNAS license.1Email: katepalli.sreenivasan@nyu.edu.Published online December 13, 2018.

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SPECIA

LFEATURE:

PERSPECTIV

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into new action. This role of diffusion is essentially secondary atthis stage. However, two thin fragments of the stretched piecesof the scalar with opposing gradients will often come togetherand coalesce into a single entity. This act changes the localtopology of the stretched fragments and allows diffusion toplay more than that secondary role. For the somewhat morecomplex case of the blob in which the scalar gradient isprescribed initially as a Gaussian distribution, Fig. 1 shows theevolution of isoconcentration contours soon after the back-ground turbulent flow launches the mixing process (5); the time-scales noted in the bottom left corners are multiples of theKolmogorov timescales (defined below). The isoconcentrationcontours acquire a fractal-like character (6) during some part oftheir evolution.

If one measures the probability density function (PDF) of thescalar concentration in a completely unmixed state, it will initiallyshow two delta functions, one corresponding to 0 of thebackground fluid with no scalar and the second corresponding to1, say, of the unmixed scalar blob. No matter how strongly theblob gets stretched and contorted by the flow, the shape of thePDF as two delta functions will remain unchanged withoutdiffusivity; it is only by diffusive action that the PDF changes inshape. It then develops an intermediate peak between 0 and 1,corresponding to the mixed fluid, and the two peaks at 0 and1 gradually diminish in magnitude even as the middle peakbecomes dominant. The area under the middle peak is thefraction of the mixed fluid. Ultimately, one observes mostlythe middle peak (see Anomalous Behaviors); this can be seenclearly in the numerical simulations of ref. 7 in a related context.The nature of the middle peak depends on the specifics ofthe flow.

Since the equations governing turbulent mixing do nothave general solutions, the path to progress has been todeduce specific results on the basis of broad paradigms andcompare their results with experiments and numerical solu-tions of the equations. There have been many insightfulexperiments but they do not always cover an adequate rangeof parameters or have the needed spatial and temporalresolutions; and they are often made in complex flow geom-etries that add new conceptual problems. On the other hand,direct numerical simulations of the equations, run on machinesthat are fast (several petabytes) and massively parallel (of theorder of 1 million cores), have allowed increasingly sharperconclusions to be drawn. We exploit the computational resultswhile also using experiments where possible. We particu-larly comment on homogeneous and isotropic turbulence ina periodic cube for which statistically stationary states havebeen computed with adequate resolution over a wide range ofReynolds and Schmidt numbers. Many of these results comefrom the frontier simulations by Yeung and coworkers (8–13),Schumacher and coworkers (14–18), Gotoh and coworkers (19–23),and others.

Equations and Classical ParadigmsThe Velocity Field. As already noted, the role of fluid flows iscentral to mixing. The flows of interest to us obey the Navier–Stokes (NS) equations

∂u∂t

+ ðu ·∇Þu=−1ρgrad  p+ ν∇2u, [1]

supplemented by the mass conservation condition for in-compressible flows, given by

∇ ·u= 0. [2]

Here, u is the velocity vector (usually in three dimensions) andp is the pressure. Eqs. 1 and 2 are equally valid for smooth andturbulent flows. Since NS equations are nonlinear and no ex-plicit turbulent solutions are known, one’s physical intuitionand understanding of turbulence is based on particular para-digms with testable outcomes. For later reference, it is usefulto summarize the most important such paradigm, named afterKolmogorov (24), although other illustrious names such asObukhov, Onsager, Heisenberg, von Weizsäcker, Fermi, andvon Neumann are associated with it. This paradigm is not validfor all commonly known flows, but it is helpful toward framingproper questions.

The effect of the pertinent boundary conditions is replaced inthis paradigm by a steady stirring at some large scale L and acharacteristic velocity u. Since the Reynolds number Re= uL=ν,where ν is the kinematic viscosity of the fluid, is regarded aslarge, a series of instabilities generates successively smallerscales, which increasingly lose the organization of the stirringscale. When these scales are small enough, viscosity will act di-rectly on them (because of their large gradients) and dissipatethe kinetic energy transmitted and initiated by the stirring action.The dissipation occurs at scales of the order of the Kolmogorov

length scale η= ðν3=eÞ1=4, where e is the energy dissipation rate(in units of kinetic energy per unit mass and unit time). The

corresponding velocity and timescales are vη = ðeνÞ1=2 and

τη = ðν=eÞ1=2. In the intermediate range of scales called the in-ertial range, a typical scale r is small compared with L but largewith respect to η so the details of stirring and viscosity are bothirrelevant, and the only factor that controls the dynamics ofturbulence is the rate of energy transfer across its scales; in astatistically steady state, this energy transfer rate equals the rateof kinetic energy introduction by stirring at scales OðLÞ and alsothe rate of energy dissipation around scales OðηÞ. The inertialrange is thus expected to be universal because it is independentof both the stirring and dissipative mechanisms. A concrete re-sult is that the so-called longitudinal structure functions, whichare moments of velocity component differences, chosen along aprescribed direction, across a separation distance r in the samedirection in the inertial range, obey the form

CΔunr D=CnðerÞn=3 [3]

for all integers n, where Cn are constants and the angularbrackets imply a suitable average. Only for n= 3 is the resultknown to be exact (25). The Fourier representation of the casen=2 is that the energy spectral density of the formEðkÞ=CK e2=3k−5=3,   CK being a constant. The inertial rangein the wavenumber space corresponds to kL � k � kη, wherethe wavenumber k is the inverse of the length scale and the

Fig. 1. Mixing of an initially Gaussian scalar blob in a homogeneousisotropic statistically stationary turbulent flow. Reprintedwith permissionfrom ref. 5. Just one blob of the dye is introduced at t=0. The timescalecorresponding to each panel is indicated at each bottom left corner.

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subscripts L and η indicate that the wavenumbers refer tolength scales L and η, respectively.

The Advection–Diffusion Equation. The classical equation for study-ing passive-scalar mixing is the advection–diffusion (AD) equation

∂θ∂t

+ ðu ·∇Þθ= κ∇2θ, [4]

where θ is the scalar concentration and κ its molecular diffusioncoefficient, and u is the velocity advecting θ. Since the scalar ispassive, u is prescribed by the operator and defines, alongwith L and ν, the Reynolds number of the flow. Beyond that,the ratio ν=κ, called the Schmidt number, Sc, matters. In prac-tice, Sc could be as small as 10−6 in the Sun’s interior, of theorder unity when the scalar of interest is a modest amount ofheat in air and of the order of 103 for a water-soluble dye inwater. Its magnitude makes a considerable difference to de-tails of molecular mixing. Another quantity that matters is theratio of L to that at which the scalar is introduced (acting againas the surrogate for the effect of boundary conditions). The ADequation has a number of limitations as themodel for mixing, ashas been emphasized in ref. 26, sections 49, 50, 58, and 59, butit is a good model for a wide range of applications in technol-ogy, the atmosphere, and the oceans; other models with morespecific goals are discussed briefly in Anomalous Behavior ofLarge-Scale Quantities and Some Model Equations.

Eq. 4 is linear in θ and the boundary conditions, which sustain thescalar concentration from being eroded by diffusion, are also almostalways linear (although perhaps mixed), but difficulties are caused bythe turbulent nature of u. Turbulence converts the AD equation into astochastic differential equation, for which we may seek only statisticalaverages such as the mean, the mean square, etc. While linearityholds for each realization of θ, the averaging process introduces newterms because of Cðu ·∇ÞθD, and the equation becomes statisticallynonlinear. As for the velocity field, we develop paradigms andcompare their outcomes with experiment to infer their veracity.

It is helpful to recast the AD equation in Lagrangian terms. Eq.4 is equivalent to stochastic advection of Lagrangian particles

dX=u½XðtÞ; t�dt+ffiffiffiffiffi2κ

pdχ ðtÞ, [5]

where Xðt =0Þ= x0 and χðtÞ is the vectorial Brownian motion,statistically independent in its three components. Here, theparticle trajectories following the NS velocity are perturbedby the white noise of strength

ffiffiffiffiffi2κ

p.

Consideration of Lagrangian trajectories clarifies some in-teresting issues (e.g., refs. 27 and 28), but one has to treat the twinlimits on Re and Sc carefully; we will not be too formal here. TheNS velocities are analytic (i.e., space differentiable or “smooth”)for distances OðηÞ and smaller, above which, in the inertial range,they are only Hölder continuous (or “rough”). This simply meansthat a velocity increment Δur over a distance r is of the form rh,h< 1 (29). The Hölder exponent, which is 1=3 for Kolmogorovturbulence (24), assumes a distribution in practice (30). This is themultiscaling property of turbulence that sets it apart from standardscaling problems in critical phenomena. If Δur ∼ rh with h< 1, we

get rðtÞ∼ t1

1−h and two Lagrangian particles which are close at onetime separate explosively. That is, two Lagrangian fluid particles thatare advected by the fluid velocity uðx, tÞ follow wildly different pathseven if they start at the same position x0. This nonuniqueness is ap-parent in Richardson’s law of diffusion (31), which is thought to govern

how particle pairs in turbulent motion separate in time (32–36). If r isthe separation distance between a pair of particles, we have

Cr2ðtÞD=CRet3. [6]

This is the special case of h= 1=3 above. We stress again thatthe right-hand side of Eq. 6 is independent of the initial sep-aration, so this power-law separation, much more explosivethan for temporal chaos, is in principle applicable even fortwo initially coincident particles.

Universality in Second-Order Statistics of the Scalar. The spec-tral density fluctiation, EθðkÞof the scalar fluctuation, whose integralover all k yields the scalar variance, has a rich structure that dependson Sc, which is described briefly below. Denote by Lθ (whichmay bedifferent from L) the scale at which scalar fluctuations are introducedinto the flow that eventually mixes them; let ηθ be the smallest scalarscale around which diffusion evens out all fluctuations.

i) The inertial–convective range, Sc =Oð1Þ: When Sc = 1, ηθ = η.In the so-called inertial–convective region sandwiched betweenthe smaller of L and Lθ, on the one hand, and ηθ, on the other, thespectral density depends on neither the viscosity nor the scalardiffusivity; its structure, determined by the spectral flux of scalarvariance at the constant rate πθ (Fig. 2, see ref. 37), is given bydimensional arguments similar to those in ref. 24, as

EθðκÞ=COBCχDCeD−1=3k−5=3, [7]

where COB is the Obukhov–Corrsin constant (38, 39) and χ isthe rate at which the scalar variance gets smeared out bydiffusion. The cutoff occurs at the scale Oðη= ηθÞ.

Both simulations and experiments show that a spectral regionexists within which the slope is roughly −5=3 and that, for the 3Dspectrum, COB = 0.67 (9), consistent with experiments (40), whichgave the constant for the 1D spectrum to be about 0.40. (The 1Dconstant is three-fifths the 3D value.)

i.a) The fast roll-off part at higher wave numbers: For Sc =Oð1Þ,within the spectrum in the fast roll-off range (the part of the red curvethat drops off faster than the −5=3 power), fluid viscosity and thescalar diffusivity are both important. Only recently have we begun tounderstand the energy spectrum in the viscous range (41), and it is fairto say that we do not yet know the scalar spectrum in this region.

ii) The viscous–convective range: When Sc � 1, the scalardiffusivity becomes effective at much smaller scales than those atwhich viscosity becomes effective; there is a spectral range calledthe viscous–convective range (42) within which diffusion is notimportant but viscosity prevails; instead of appearing directly, the

Fig. 2. Schematic of scalar spectrum for Sc=1 (red), Sc < <1 (green),and Sc > >1 (purple). Reprinted with permission from ref. 37.

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viscosity appears only through the small-scale rate-of-strain ðν=eÞ1=2.This is the purple curve in Fig. 2 with the power-law scaling part.Dimensional analysis shows that the spectrum has the shape

EθðκÞ=CBχðν=κÞ1=2k−1, [8]

where CB is the Batchelor constant. Batchelor (42) derived thislaw from dynamical considerations and deduced the constantCB, as well, so this law rests on a theory. Batchelor consideredthe stretching effects of the velocity field to be stationary intime. Kraichnan (43) considered the opposite limit of infinitelyoscillating strain rate and also found the −1 power in the viscous–convective range, which probably speaks to its robustness.Past evidence for the −1 power has been mixed: while refs.44–46 found support for it, refs. 47 and 48 claimed the oppo-site. Only recently have we found convincing evidence thatthe −1 region does exist and becomes better defined with in-creasing Sc (Fig. 3). However,CB is numerically determined to beabout 5.5 (8, 11, 22), compared with Batchelor’s theoretical esti-mate of about 2, so there is still a missing piece of the puzzle.

ii.a) The rapid roll-off part at higher wave numbers: Batchelor(42) had deduced the nature of the spectrum as it rolls off in theso-called viscous–diffusion range (the part of the purple curve thatdrops off fast toward the right end, where viscosity and diffusionare both important), but ref. 11 has shown that the data fitKraichnan’s (43) formula much better.

iii) If Sc � 1, scalar diffusivity becomes important within theinertial range toward the high wavenumbers where the viscositydoes not act. The −5=3 power is truncated at a larger scale than η

at the so-called scalar dissipation length (38, 39) ηθ = ηOC = ηSc−3=4;this truncated inertial range is called the inertial convective range(49, 50), in which the wavenumbers are such that kηθ � 1, whilekη � 1. The theory of ref. 49 showed the expected power-law ex-ponent to be −17=3. This part (the green curve in Fig. 2) had notbeen observed for many years because it is hard to obtain exper-imental data for high Reynolds number and low Schmidt numbersimultaneously, and the numerical simulations also place greatdemands on the time resolution—but it has finally been confirmed(51, 52). Nothing definitive is known about how the roll-off occurs atwavenumbers to the right of ηOC.

The Yaglom Relation in the Inertial–Convective Range. Inanalogy to Eq. 3, which is exact for the special case of n= 3, thecross-correlation between velocity and scalar increments wasshown by Yaglom (53) to be exact in the inertial–convection region:

CΔur ðΔθr Þ2D=−ð2=3ÞCχDr . [9]

Here, again, the velocity increment Δur = uðx + rÞ− uðxÞ, wherethe spatial distance r is along the direction x of u and is con-tained in the inertial–convective range, and the scalar incrementΔθr = θðx + rÞ− θðxÞ also corresponds to the inertial–convectiverange. This work has been extended in ref. 54 for a refinedsimilarity hypothesis (55) and in refs. 56 and 57 for nonstationaryforcing conditions. Its verification was attempted experimen-tally in ref. 54 as well as in ref. 58, but the wait was considerablylonger (Fig. 4) before the support became clear for a variety ofReynolds and Schmidt numbers (8–23). See also ref. 59 for sim-ilar work for low Schmidt numbers.

Scalar Dissipation Rate. In the above discussion, it has been as-sumed that the flux of scalar variance across wavenumbers, equal insteady state to the scalar dissipation rate, stays fixed even as thediffusivity goes to zero. This seems counterintuitive because thescalar dissipation rate defined (from the AD equation) as

χ = 2κX3i=1

�∂θ∂xi

�2[10]

depends directly on the diffusivity. The anomaly is that χ isindependent of κ as κ→0.

The direct evidence for this behavior, which often surprises theuninitiated, can be found in ref. 10 where CχD, normalized by large-scale quantities in the flow (not involving κ), is plotted against thePeclet number, Pe=UL=κ (the Peclet number is the product of theReynolds and Schmidt numbers). For Pe>Oð100Þ, the scalardissipation rate indeed seems to be independent of Pe (and thusκ). We have already discussed, in describing the swift separationof two Lagrangian particles, a physical mechanism that may allowdiffusion not to play a direct role. As long as the diffusivity is small(i.e., Pe � 1), it is only an agent whose action appears to be slavedto the explosively dispersive effect of the particle trajectories. Wethus have χ =CχCθ2Du′=L, where Cχ is a constant of the order unity.

Fig. 3. Time-averaged 3D scalar spectrum obtained from directnumerical simulations for multiple Schmidt numbers, with the arrowin the direction of increasing Sc; sloped dotted line is proportional tok−1. Reprinted with permission from ref. 65.

Fig. 4. Mixed velocity–scalar structure function with scaling accordingto Yaglom’s relation. Various curves correspond to various values ofSc, with the arrows indicating the direction of increasing valkues. Theslope at the smallest values of r corresponds to r2, expected from thesmoothness condition at small r. Reprintedwith permission from ref. 65.

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From ref. 10, it is about 0.7 for Sc = 1; this constancy holds fornonunity Sc as well, with Cχ perhaps depending weakly on Sc.

Summary So Far. The major point of the account so far is not tocover all subregions of the spectra or the “universality” argumentsthoroughly, but to illustrate that we have come a long way inassessing past theories and are able to conclude that they performreasonably well at the level of second-order statistics; such cor-rections as may arise from intermittency, to which we return inAnomalous Behaviors, are ignored here. The broad agreement withthe theory has been tested primarily in cases where an isotropic andhomogeneous velocity field is generated in a periodic cube,maintained stationary by forcing at a few of the lowest wave-numbers, and the scalar field is maintained stationary by a meanscalar gradient. Thus, it seems that, by virtue of the work that hastaken some 70 y to complete, the physics for the second-orderquantities are founded on solid enough ground to believe thatthey hold well as a working approximation (although with a fewgaps). Concurrently, however, work of the last four decades has alsoshown that this approach, as well as the concepts behind it, breaksdown for higher-order moments and that the notion of universalityis fundamentally at peril. This is the topic in Anomalous Behaviors.

Anomalous BehaviorsAnomaly Due to Shear. The first major anomalous behaviorconcerns the persistence of anisotropy at high Reynolds numbersin scales that are nominally regarded as small. This would suggestthat the physics of universal behavior is incorrect at the basic level;the fact that it may work well for second-order quantities does notnegate this thought. We now discuss an example of such prop-erties and their place in scalar mixing.

Recall that the vector ∂θ=∂xi, being a derivative, is regarded asa small-scale property. If small scales are universal, they are alsostatistically isotropic; rotational symmetry of an isotropic vectorwould need its odd moments, in particular the third one, to vanishat high Reynolds number. However, one finds that the derivativeskewness, the ratio of the third derivative moment to the 3=2power of the mean-square derivative, assumes values of the orderunity with no tendency to decrease with Reynolds number (60)and has been reaffirmed multiple times in the last 40 y (61, 62). If itremains a constant of about unity at all moderate Reynoldsnumbers, small-scale isotropy appears to be violated, with thepossible implication that we should not expect any universality toemerge. Does it mean that we have to discard the paradigm thatseemed quite successful for second-order quantities?

The ostensible reason for this anomalous behavior is the ramp–cliffstructure of Fig. 5, which shows the time trace of temperature fluc-tuations in a heated jet (60). The heating is only slight so that thetemperature is passive. The so-called Taylor’s hypothesis (the equiv-alence of the transformation x =−Ut, where U is the velocity withwhich turbulent features are advected) implies that these are also 1Dspatial cuts of the temperature field. The ramp–cliff structure showsthat the temperature builds up rather sharply but falls rather gradually.

A plausible physical reason for the ramp–cliff signature is this:Imagine large structures in the flow that are moving at a speedthat is different from the local value. For example, in a boundarylayer, it has been shown in ref. 63 that large structures travel nearlyeverywhere in the outer boundary layer with approximately thespeed of the free stream. One can imagine then that there will besome stagnation regions toward the front of such structures wherethe jump of a property such as its temperature will be rather steep,but the fall-off toward the back of the structure will be rathergradual because of the diffuse nature in the wake; see ref. 64 foran average of streamline patterns around such structures. The

basic notion is that the temperature jump across the stagnation re-gion will be on the order of the maximum value possible, that is, themaximumoverheat in a jet. Let us denote this maximum temperaturejump by T. The longer leg of the ramp is on the order of L, and theshorter leg is on the order of the shortest scalar scale; what is trueof the temperature trace is also true of any scalar field, so theshortest scale will be on the order of the Batchelor scale ηB whenSc ≤ 1. On each ramp are the superimposed fluctuations comingfrom all other scales of the temperature. These are the parts essen-tially unaffected by the mean gradients and other inhomogeneitiesin the flow, contributing most to low-order even moments of thescalar derivative such as its mean square; and the contributions of thecliffs to even-order statistics are intrinsically small, but are importantfor odd and high-order moments, both odd and even.

The ramp–cliff model can be used to calculate the skewness ofthe scalar derivative in a straightforward manner. The nonvanishingpart of the third moment of the scalar derivative within a ramp ison the order of ð∂T=∂xÞ3, and its average value over the ramp is

ð∂T=∂xÞ3 × ðηB=LÞ. This forms the numerator for the skewness.Noting that the even moments are largely coming from fluctuationssuperimposed on the ramp, the denominator for the skewness isthe 3=2 power of 1

3 χ (by isotropy, valid for even-order moments)

and can be substituted using dissipative anomaly as 13CχCθ

2Du′=L.It follows that Sθ ∝Re0Sc−1=2—that is, the skewness is independentof Re but varies as the −1=2 power of Sc. In arriving at the aboveresult, we have used only standard relations valid in turbulence.

We can indeed verify the dependence on Sc. Fig. 6, repro-duced from the thesis of Clay (65), shows the skewness as afunction of Sc. Its variation is only slightly smaller than the −1=2power (but this minor difference can be addressed by a modestlymore sophisticated analysis). These considerations can be usedto predict the behavior of the normalized fifth moment, orthe hyperskewness, Hθ, of the scalar derivative. The result for

Fig. 5. The ramp–cliff structure in the temperature trace in a slightlyheated jet. Reproduced with permission from ref. 60. Below the traceis shown a schematic of the large eddy structure that could create theramp–cliff. The scales are indicated on the schematic.

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Hθ ∝Re1=2Sc−1=2. Data bear out these predictions. The nonvanishingof odd-order structure functions occurs in the inertial–convectiverange and can be understood the same way.

What we have shown is that the skewness and the hyper-skewness, which are ostensibly small-scale properties, are actuallylarge-scale properties, and so their anomalous behavior need notin itself negate the likelihood that small scales are universal at low-order even moments. Thus, our earlier conclusion that small-scaleproperties at the second-order level are essentially consistent withthe paradigm of universal theories is quite reasonable. In reality,second-order statistics are also influenced by the large scale“somewhat” and are thus only approximately universal, becausethe large scale is somehow imprinted on the small scale. Thus, allmoments of the small scale are susceptible to the influence of thelarge scale, the ramp–cliff structure being a manifestation of thiseffect. This structure has the important consequence that themixing in shear flows is never complete (which means, in principle,that the PDFs of the scalar discussed in the Introducton will neverattain a state at which the two delta functions at 0 and 1 disappearentirely). We discuss this feature in the next subsection by con-sidering structure functions of the scalar.

Anomalous Scaling of Structure Functions. For simplicity, wediscuss Sc = 1. Define the structure functions of order n for thescalar as CΔθnr D≡ C½θðx + rÞ− θðxÞ�nD. For r < L, these objects areindependent of the large scale; they are also independent ofdiffusivity when r > ηθ = η. In the intermediate scales η< r < L, theproperties of CΔθnr D depend only on the energy and scalar fluxesacross the scale range. This is the inertial–convective region wehave already encountered. Dimensional reasoning says that

CΔθnr D=Cnθχn=2e−n=6rn=3. [11]

This n=3 dependence on r is the normal scaling. Althoughexperiments and simulations are susceptible to many short-comings (such as inadequate scaling range, uncertain conver-gence of statistics, inadequate repeatability from one flow toanother, dependence on the method of forcing the largescale, etc.), new developments have given us confidence thatthe measured scaling exponents ξn ≠ n=3 and, in fact, saturatewith respect to n (66) for moment orders of the order 10. Thissaturation is the extreme case of anomalous scaling and

implies that there are always some regions in the flow wherethe scalar concentration varies from the maximum possible tothe minimum possible, or between 0 and 1, in quick succes-sion, as the ramp–cliff model has already suggested.

The Kraichnan Model. All this strong evidence for anomalous scal-ing suggests the need for a good theoretical understanding. Since ithas not yet been possible to obtain this for the AD equation, it is help-ful to have a nontrivial model. Kraichnan’s model (43, 67) is one such.

The crucial element of this model is that it replaces the velocityu in the AD equation by a stochastic Gaussian field with a timecorrelation that decays infinitely rapidly (“white in time”) and aspatial correlation that has the expected power law with a pre-scribed scaling exponent, 0< ζ< 2. That is,

Cuiðx, tÞujðy, t′ÞD=Dijðx− yÞδðt− t′Þ [12]

with

DijðrÞ=D0δij −dijðrÞ=D0δij −D1�ð2+ ζÞδij

�− ζ

�rirjr2

�rζ , [13]

where r= x− y, r = jrj and i, j= 1, 2, 3, Dij is a diffusivity and D0

and D1 are constants, and ζ= 0 corresponds to advection in avery rough flow and ζ= 2 to a smooth flow (see ref. 27 for anintroduction). This power-law scaling on Dij is similar to the NScase (although the scaling exponent ζ here assumes a moregeneral value between 0 and 2), but the temporal scaling isconceptually different by being a delta function. For statisticalstationarity, a random forcing fθðx, tÞ has to be added to theright-hand side of the AD equation, with the property that

Cfθðx, tÞfθðy, t′ÞD=Cðr=LÞδðt− t′Þ. [14]

The function Cðr=LÞ decays rapidly to zero for small scales.Kraichnan’s insight was that this model possesses the essentialelements of the scalar mixing while retaining analytical tracta-bility. Indeed, it has been possible to establish anomalousscaling (see ref. 68 for a review) for this model even thoughthe idealized advecting flow itself does not exhibit any anom-aly. In other words, the scaling exponents for the Eulerianstructure functions of the scalar increments differ from theclassical form (with ξ2n =2ξn) and depend on the order of themoment. We come to the conclusion that the anomalous be-havior is imprinted on it by the scalar itself and does not needan anomalous velocity field.

We must stress, however, that no explicit solution has beenfound for all values of ζ, but only expansions in the limits of ζ→ 0and ζ→ 2 have been found (e.g., refs. 69–72). However, accuratenumerical solutions have been obtained for the general case (73–75). One important feature of these calculations is that the scalingexponents asymptote to a constant instead of increasing withmoment order (76), just as happens for real flows (66).

The physical picture that has emerged from the Kraichnanmodel is this: To understand the third-order scaling exponent ξ3, itis obvious that one needs to study the properties of objectsgenerated from the scalar at three different positions in space.The triangle formed by these positions is described by the lengthscale R—which, for specificity, can be taken as the geometricmean of the lengths of the sides of the triangle—and two of thethree angles, say ψ and ϕ (77, 78). As the three particles advect,the triangles change in shape and size. If we rescale the trianglesto the same R at each time step, the dynamics reduce to the

Fig. 6. The decay of the scalar derivative skewness as a function ofthe Schmidt number. Most data from ref. 65. The line corresponds toa slope close to the prediction of the ramp–cliff model.

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evolution of shapes of triangles, or to a suitable function f ðψ ,ϕÞ ofthe two angles ψ and ϕ. The important result obtained for theKraichnan model is that the three-point statistics are governed bythose trajectories for which the change in the length scale R iscompensated by the change in shape of the triangles such that theproduct Rξ3 f ðψ ,ϕÞ is a constant. In general, as particles move in theKraichnan flow, an n-particle cloud grows in size but fluctuations inthe cloud shape decrease in magnitude. This happens because thecorrelation between particles—which arises, in the first place, be-cause they are contained within the integral scale of the velocityfield—weakens with the separation distance. Therefore, as men-tioned above, one looks for suitable functions of size and shape thathave the property of being conserved via the balance between thegrowth in size and the decrease of shape fluctuations.

An important qualitative lesson from this work is that certaintypes of Lagrangian characteristics, conserved only on the aver-age, determine the statistical scaling of Eulerian structure func-tions. This may mean that there are certain statistically conservedlaws which determine the anomalous exponents (79). Clearly, inthis instance the Lagrangian point of view has given a better un-derstanding of Eulerian quantities—a conclusion that may havebroad validity in systems with strong fluctuations. While there areseveral differences between the predictions of the Kraichnanmodel and the behavior of a passive scalar in NS turbulence (5), ithas advanced our understanding of anomalous scaling.

An interesting observation concerns the ramp–cliff structurediscussed earlier. One may think that the ramp–cliff structurearises because the shear persistently acts on the scalar, which isnot the feature of the Kraichnan model because it reshuffles theflow at every instant. Despite this, simulations in ref. 75 suggestthat ramp–cliff features are possible even in the Kraichnan model,leading to the saturation of the scaling exponents: The scalar willoccasionally undergo the largest possible fluctuation over a veryshort length scale, these being T and ηB in our ramp model.

A Consequence of Anomalous Scaling: Length-Scale Fluctuations.We pointed out earlier that velocity increments in the inertial–con-vective range are Hölder continuous; for small r, the magnitudes ofvelocity increments change greatly from one spatial position to an-other. This intermittent feature shows up evenmore conspicuously inenergy and scalar dissipation rates, e and χ, for which fluctuationsincrease with increasing Reynolds number. These intermittent prop-erties of e and χ have been studied extensively—in refs. 10, 30, and80–84 among others.

One property of intermittency is that the magnitude fluctua-tions are accompanied by length-scale fluctuations as well. Sincethe instantaneous and local values of e and χ fluctuate rather wildly(tens of thousands of times the mean value are likely even atmodest Reynolds numbers), it can be expected that, if one de-

fined local Kolmogorov scale η= ðν3=eÞ1=4 based on the localvalues of e, that scale would fluctuate as well; ηB and τ wouldfluctuate too, and both smaller and larger values than the meanvalues would result (6). The result would be similar if one de-fined (85, 86) the fluctuating dissipation scale through the in-termittency of the structure functions via η as ηΔuη=ν= 1 (i.e., byrequiring the corresponding Reynolds number to be unity). Thetwo definitions, although not identical, illustrate the same basicphenomenon.

These length-scale fluctuations imply, for example, that

ν3=CeD1=4 will be significantly different from the definition ν3=Ce1=4D.The same situation would arise with the timescale. Data show

that the mean timescale CτD= Cη2BD=κ≈ 10CηBD2=κ. This means that

a measure of the eddy diffusive time/molecular diffusive time

could well be Re1=2=100, which exceeds unity only when Re> 104

[the so-called mixing transition (86)].

Anomalous Behavior of Large-Scale Quantities and SomeModel EquationsInstances exist when the observed large-scale behaviors cannotbe explained from the AD equation directly, so various modelstudies have been proposed. As in the Kraichnan model, thesemodels replace the NS velocity by an artificial field satisfying thecondition ∇ ·u= 0. A collection of such models is discussed in ref.87. A broad-brush summary of the large-scale, long-time results isgiven below.

i) For smooth velocity fields (e.g., periodic and deterministic), ho-mogenization is possible. That is, Cuðx; tÞ ·∇θD=∇ κT ·∇ðð θðx; tÞÞ,where the effective diffusivity κT can be determined. See, forexample, ref. 88.

ii) Velocity is a homogeneous random field, but a scale separa-tion exists between the velocity and scalar fields: Lu=Lθ � 1(i.e., the scalar is injected at a much larger scale than thekinetic energy). Homogenization is possible here as well.

iii) Velocity is a homogeneous random field but delta correlatedin time, Lu=Lθ =Oð1Þ; eddy diffusivity can be computed hereas well.

iv) For the special case of shearing velocity (with and withouttransverse drift), the problem can be solved essentially com-pletely: Eddy diffusivity and anomalous diffusion can be cal-culated without any scale separation (89–91).

We now cite two specific examples where such models havebeen helpful:

i) Decay rates: For the case of turbulent flow behind grids,there is no source of turbulent energy except the grid itself; theturbulent energy thus decays with distance from the grid, and therate of decay for the energy follows ∼ −1.2 power of the distance(ref. 92 and references therein). The turbulence-generating griditself, or a secondary grid just downstream, can be heated togenerate temperature fluctuations [for which Sc =Oð1Þ]. Initialstudies (93, 94) showed that the scalar variance decays as a powerlaw with nonunique exponents. It was shown (95) by a simplemodel that all of the decay exponents collapse when the length-scale ratio Lθ=Lu is taken into account. This insight would not havebeen possible without the model.

ii) The PDF of the scalar variable: The decaying scalar de-scribed in the previous paragraph possesses a self-similarGaussian PDF (94); the PDF in simulations of the stationary casewith stochastic scalar driving is Gaussian or only slightly sub-Gaussian (19). Also when driven by a mean scalar gradient, thePDF assumes a nearly Gaussian form (96–98). However, someexperiments have yielded exponential tails (99), consistent withthe Kraichnan model, for which the tails of the PDF are super-Gaussian or exponential (100). At present, we do not fully un-derstand the circumstances for which the passive scalar mixed byNS turbulence assumes an exponential PDF, but some insightcomes from model studies (101) which suggest that the PDF isexponential when Lu=Lθ > 1 and Gaussian (when Lu=Lθ > 1).

The shape of the PDF in inhomogeneous shear flows is un-derstood even less (102).

Concluding RemarksThis article has attempted to provide a perspective on the state ofturbulent mixing essentially after the process is complete, focusingon passive scalars. Many basic aspects of mixing, relating to thetransient process, are omitted. Also omitted, by necessity, are

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aspects such as mixing time, efficiency of mixing, and other bulkmeasures (103, 104). We did not likewise explore the effect of ro-tation (105), compressibility (106), buoyancy (107), chemical re-actions (108), chaotic mixing (109), and other external effectsand the mixing of non-Newtonian fluids. The views taken hereare entirely continuum in character. Nevertheless, we hope thatthis article is a useful background for understanding these richercontexts.

While we need not deny the usefulness of universality at thespectral level for scalar variance, it is clearly the case that noteverything falls within the cloak of universality; indeed, we dis-cussed several examples of anomaly. Our view is that a goodunderstanding of turbulent mixing is possible only if we un-derstand the value and limitations of the universality paradigms

while being aware of the causes of anomalies. Our attempt in anutshell has been to show that there are underpinnings of decenttheory but they have yet to be strung together to create a full-fledged narrative. To go deeper than the continuum, using mo-lecular dynamics, and establish specific results, for example, theSchmidt number effects, would be spectacular.

AcknowledgmentsThis report has been inspired by many colleagues over the years, and Iparticularly cite P. K. Yeung, D. A. Donzis, and J. Schumacher from whosecomputer simulations I have learned much of the material presented; G. Eyink,V. Yakhot, T. Gotoh, and S. I. Abarzhi for many insightful comments anddiscussions; K. P. Iyer for his help in preparing the LaTeX file; and M. Clay forpermission to use his figures.

1 Abarzhi SI, Sreenivasan KR (2010) Turbulent mixing and beyond. Phil Trans Roy Soc Lond A 368:1539–1546.2 Abarzhi SI (2010) Review of theoretical modeling approaches of Rayleigh-Taylor instabilities and turbulent mixing. Phil Trans Roy Soc Lond A 368:1809–1828.3 Abarzhi SI, Gauthier S, Sreenivasan KR (2013) Turbulent mixing and beyond: Non-equilibrium processes from atomistic to astrophysical scales I. Phil Trans RoySoc A 371:20120435.

4 Abarzhi SI, Gauthier S, Sreenivasan KR (2013) Turbulent mixing and beyond: Non-equilibrium processes from atomistic to astrophysical scales II. Phil Trans RoySoc Lond A 371:20130268.

5 Sreenivasan KR, Schumacher J (2010) Lagrangian views on turbulent mixing of passive scalars. Philos Trans R Soc Lond A 368:1561–1577.6 Sreenivasan KR (1991) Fractals and multifractals in fluid turbulence. Annu Rev Fluid Mech 23:539–600.7 Yeung PK, Xu J (2014) Effects of rotation on turbulent mixing: Nonpremixed passive scalars. Phys Fluids 16:93–103.8 Yeung PK, Xu S, Donzis DA, Sreenivasan KR (2004) Simulations of three-dimensional turbulent mixing for Schmidt numbers of the order 1000. Flow TurbulCombust 72:333–347.

9 Yeung PK, Donzis DA, Sreenivasan KR (2005) High-Reynolds-number simulation of turbulent mixing. Phys Fluids 17:081703.10 Donzis DA, Sreenivasan KR, Yeung PK (2005) Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J Fluid Mech 532:199–216.11 Donzis DA, Sreenivasan KR, Yeung PK (2010) The Batchelor spectrum for mixing of passive scalars in isotropic turbulence. Flow Turbul Combust 85:549–566.12 Donzis DA, Yeung PK (2010) Resolution effects and scaling in numerical simulations of passive scalar mixing in turbulence. Phys D 239:1278–1287.13 Donzis DA, Aditya K, Sreenivasan KR, Yeung PK (2014) The turbulent Schmidt number. J Fluids Eng 136:060912.14 Schumacher J, Sreenivasan KR, Yeung PK (2003) Schmidt number dependence of derivative moments for quasi-static straining motion. J Fluid Mech 479:221–230.15 Schumacher J, Sreenivasan KR (2003) Geometric aspects of the passive scalar mixing at high Schmidt number. Phys Rev Lett 91:174501.16 Schumacher J, Sreenivasan KR (2005) Statistics and geometry of passive scalar fields in turbulence. Phys Fluids 17:125107.17 Schumacher J, Sreenivasan KR, Yeung PK (2005) Very fine structures in scalar mixing. J Fluid Mech 531:113–122.18 Kumar B, Schumacher J, Shaw RA (2014) Lagrangian mixing dynamics at the cloudy-clear air interface. J Atmos Sci 71:2564–2580.19 Watanabe T, Gotoh T (2004) Statistics of a passive scalar in homogeneous turbulence. New J Phys 6:40.20 Watanabe T, Gotoh T (2006) Intermittency in passive scalar turbulence under the uniform mean scalar gradient. Phys Fluids 18:058105.21 Watanabe T, Gotoh T (2007) Inertial-range intermittency and accuracy of direct numerical simulation for turbulence and passive scalar turbulence. J Fluid Mech

590:117–146.22 Gotoh T, Watanabe T, Miura H (2014) Spectrum of passive scalar at very high Schmidt number in turbulence. Plasma Fusion Res 9:3401019.23 Gotoh T, Watanabe T (2015) Power and non-power laws of passive scalar moments convected by isotropic turbulence. Phys Rev Lett 115:114502.24 Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30:9–13.25 Kolmogorov AN (1941b) Dissipation of energy in the locally isotropic turbulence. Dokl Akad Nauk SSSR 32:16–18.26 Landau LD, Lifshitz EM (1987) Hydrodynamics by Fluid Mechanics (Pergamon Press, Burlington, MA, Vol 6.27 Bernard D (2000) Turbulence for (and by) amateurs. arXiv:cond-mat/0007106. Preprint, posted July 6, 2000.28 Eyink GL, Drivas TD (2015) Spontaneous stochasticity and anomalous dissipation for Burgers equation. J Stat Phys 158:386–432.29 Parisi G, Frish U (1983) On the singularity structure of fully developed turbulence. Turbulence and Predictability in Geophysical Fluid Dynamics, eds Ghil M,

Benzi R, Parisi G (North-Holland, Amsterdam), pp 84–87.30 Chen SY, Dhruva B, Kurien S, Sreenivasan KR, Taylor MA (2005) Anomalous scaling of low-order structure functions of turbulent velocity. J Fluid Mech 533:183–192.31 Richardson LF (1926) Atmospheric diffusion shown on a distance-neighbor graph. Proc R Soc Lond A 110:709–733.32 La Porta A, Voth GA, Crawford AM, Alexander J, Bodenschatz E (2001) Fluid particle accelerations in fully developed turbulence. Nature 409:1017–1019.33 Mordant N, Metz P, Michel O, Pinton JF (2001) Measurement of Lagrangian velocity in fully developed turbulence. Phys Rev Lett 87:214501.34 Boffetta G, Sokoloff IM (2002) Relative dispersion in fully developed turbulence: The Richardsons law and intermittency corrections. Phys Rev Lett 88:094501.35 Biferale L, et al. (2004) Multifractal statistics of Lagrangian velocity and acceleration in turbulence. Phys Rev Lett 93:064502.36 Bourgoin M, Ouellette NT, Xu H, Berg J, Bodenschatz E (2006) The role of pair dispersion in turbulent flow. Science 311:835–838.37 Gotoh T, Yeung PK (2012) Passive scalar transport in turbulence: A computational perspective. Ten Chapters in Turbulence, eds Davidson PA, Kaneda Y,

Sreenivasan KR (Cambridge Univ Press, Cambridge, UK).38 Obukhov AM (1949) Structure of the temperature field in turbulent flows. Izv Geogr Geophys 13:58–69.39 Corrsin S (1951) On the spectrum of isotropic temperature fluctuations in isotropic turbulence. J Appl Phys 22:469–473.40 Sreenivasan KR (1996) The passive scalar spectrum and the Obukhov-Corrsin constant. Phys Fluids 8:189–196.41 Khurshid S, Donzis DA, Sreenivasan KR (2018) Energy spectrum in the dissipative range. Phys Rev Fluids 3:082601(R).42 Batchelor GK (1959) Small scale variation of convected quantities like temperature in a turbulent fluid. Part 1. General discussion and the case of small

conductivity. J Fluid Mech 5:113–133.43 Kraichnan RH (1968) Small-scale structure of a scalar field convected by turbulence. Phys Fluids 11:945–953.44 Gibson CH, Schwarz WH (1963) The universal equilibrium spectra of turbulent velocity and scalar fields. J Fluid Mech 16:365–384.45 Sreenivasan KR, Prasad RR (1989) New results on the fractal and multifractal structure of the large Schmidt number passive scalars in fully turbulent flows. Phys D

38:322–329.46 Holzer M, Siggia ED (1994) Turbulent mixing of a passive scalar in two dimensions. Phys Fluids 6:1820–1837.47 Miller PL, Dimtakis PE (1996) Measurements of scalar power spectra in high Schmidt number turbulent jets. J Fluid Mech 308:129–146.48 Williams BS, Marteau D, Gollub JP (1997) Mixing of a passive scalar in magnetically forced two-dimensional turbulence. Phys Fluids 9:2061–2080.

18182 | www.pnas.org/cgi/doi/10.1073/pnas.1800463115 Sreenivasan

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49 Batchelor GK, Howells ID, Townsend AA (1959) Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of largeconductivity. J Fluid Mech 5:134–139.

50 Howells ID (1960) An approximate equation for the spectrum of a conserved scalar quantity in a turbulent fluid. J Fluid Mech 9:104–106.51 Yeung PK, Sreenivasan KR (2013) Spectrum of passive scalars of high molecular diffusivity in turbulent mixing. J Fluid Mech 716:R14.52 Yeung PK, Sreenivasan KR (2014) Direct numerical simulation of turbulent mixing at very low Schmidt number with a uniform mean gradient. Phys Fluids 26:015107.53 Yaglom AM (1949) Local structure of the temperature field in a turbulent field. Dokl Akad Nauk SSSR 69:743–746.54 Stolovitzky G, Kailasnath P, Sreenivasan KR (1995) Refined similarity hypotheses for passive scalars mixed by turbulence. J Fluid Mech 297:275–291.55 Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds

number. J Fluid Mech 13:82–85.56 Danaila L, Anselmet F, Zhou T, Antonia RA (1999) A generalization of Yaglom’s equation which accounts for the large-scale forcing in heated decaying

turbulence. J Fluid Mech 391:359–372.57 Orlando P, Antonia RA (2002) Dependence of the non-stationary form of Yaglom’s equation on the Schmidt number. J Fluid Mech 451:99–108.58 Mydlarsky L (2003) Mixed velocity-passive scalar statistics in high-Reynolds-number turbulence. J Fluid Mech 475:173–203.59 Iyer KP, Yeung PK (2014) Structure functions and applicability of Yaglom’s relation in passive-scalar turbulent mixing at low Schmidt numbers with uniform mean

shear. Phys Fluids 26:085107.60 Sreenivasan KR, Antonia RA, Britz D (1979) Local isotropy and large structures in a heated jet. J Fluid Mech 94:745–775.61 Sreenivasan KR, Tavoularis S (1980) On the skewness of the temperature fluctuation in turbulent flows. J Fluid Mech 101:783–795.62 Sreenivasan KR (1991) On local isotropy of passive scalars in turbulent shear flows. Proc R Soc Lond A 434:165–182.63 Kovasznay LSG, Kibens V, Blackwelder RF (1970) Large-scale motion in the intermittent region of a turbulent boundary layer. J Fluid Mech 41:283–325.64 Antonia RA, Chambers AJ, Britz D, Browne LW (1986) Organized structures in a turbulent plane jet: Topology and contribution to momentum and heat transport.

J Fluid Mech 172:211–229.65 Clay M (2017) Strained turbulence and low-diffusivity turbulent mixing using high performance computing. PhD thesis (School of Aerospace Engineering,

Georgia Institute of Technology, Atlanta).66 Iyer KP, Schumacher J, Sreenivasan KR, Yeung PK (2018) Steep cliffs and saturated exponents in three dimensional scalar turbulence. arXiv:1807.01651v1.

Preprint, posted July 4, 2018.67 Kraichnan RH (1994) Anomalous scaling of a randomly advected scalar. Phys Rev Lett 72:1016–1019.68 Falkovich G, Gawedzki K, Vergassola M (2001) Particles and fields in fluid turbulence. Rev Mod Phys 73:913–975.69 Bernard D, Gawedzki K, Kupiainen A (1996) Anomalous scaling of the N-point functions of a passive scalar. Phys Rev E 54:2564–2572.70 Chertkov M, Falkovich G (1996) Anomalous scaling exponents of a white-advected passive scalar. Phys Rev Lett 76:2706–2709.71 Shraiman BI, Siggia ED (1995) Anomalous scaling of a passive scalar in turbulent flow. C R Acad Sci Paris IIB 321:279–284.72 Arad I, Biferale L, Celani A, Procaccia I, Vergassola M (2001) Statistical conservation laws in turbulent transport. Phys Rev Lett 87:164502.73 Frisch U, Mazzino A, Vergassola M (1998) Intermittency in passive scalar advection. Phys Rev Lett 80:5532–5535.74 Gat O, Procaccia I, Zeitak R (1998) Anomalous scaling in passive scalar advection: Monte Carlo Lagrangian trajectories. Phys Rev Lett 80:5536–5539.75 Chen SY, Kraichnan RH (1998) Simulations of a randomly advected passive scalar field. Phys Fluids 10:2867–2884.76 Balkovsky E, Lebedev V (1998) Instanton for the Kraichnan passive scalar problem. Phys Rev E 58:5776–5795.77 Pumir A, Shraiman BI, Chertkov M (2000) Geometry of Lagrangian dispersion in turbulence. Phys Rev Lett 85:5324–5327.78 Celani A, Vergassola M (2001) Statistical geometry in scalar turbulence. Phys Rev Lett 86:424–427.79 Falkovich G, Sreenivasan KR (2006) Lessons from hydrodynamic turbulence. Phys Today 59:43–49.80 Meneveau C, Sreenivasan KR (1991) The multifractal nature of turbulent energy dissipation. J Fluid Mech 224:429–484.81 Yeung PK, Zhai XM, Sreenivasan KR (2015) Extreme events in computational turbulence. Proc Natl Acad Sci USA 112:12633–12638.82 Sreenivasan KR, Antonia RA (1997) The phenomenology of small-scale turbulence. Annu Rev Fluid Mech 29:435–472.83 Yeung PK, Zhai XM, Sreenivasan KR (2016) Extreme events in computational turbulence. Proc Natl Acad Sci USA 112:12633–12638.84 Schumacher J, et al. (2014) Small-scale universality in fluid turbulence. Proc Natl Acad Sci USA 111:10961–10965.85 Paladin G, Vulpiani A (1987) Degrees of freedom of turbulence. Phys Rev A 35:1971–1973.86 Yakhot V (2008) Dissipation-scale fluctuations and mixing transition in turbulent flows. J Fluid Mech 606:325–337.87 Majda AJ, Kramer PR (1999) Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena. Phys Rep 314:237–574.88 Stroock DW, Varadhan SRS (1969) Diffusion processes with continuous coefficients I. Commun Pure Appl Math 22:345–400.89 Glimm G, Lundquist B, Pereira F, Peierls R (1992) The multifractal hypothesis and anomalous diffusion. Mat Aplic Comput 11:189–207.90 Avellaneda M, Majda AJ (1994) Simple examples with features of renormalization for turbulent transport. Philos Trans R Soc Lond A 346:205–233.91 Ben Arous G, Owhadi H (2002) Super-diffusivity in a shear flow model from perpetual homogenization. Commun Math Phys 227:281–302.92 Sinhuber M, Bodenschatz E, Bewley GP (2015) Decay of turbulence at high Reynolds numbers. Phys Rev Lett 114:034501.93 Warhaft Z, Lumley JL (1978) An experimental study of the decay of temperature fluctuations in grid-generated turbulence. J Fluid Mech 88:659–684.94 Sreenivasan KR, Tavoularis S, Henry R, Corrsin S (1980) Temperature fluctuations and scales in grid-generated turbulence. J Fluid Mech 100:783–795.95 Durbin P (1982) Analysis of the decay of temperature fluctuations in isotropic turbulence. Phys Fluids 25:1328–1332.96 Overholt MR, Pope SB (1999) Direct numerical simulation of a statistically stable turbulent reacting flow. Combust Theor Model 3:371–408.97 Ferchichi S, Tavoularis S (2002) Scalar probability density function and fine structure in uniformly sheared turbulence. J Fluid Mech 461:155–182.98 Schumacher J, Sreenivasan KR (2005) Statistics and geometry of passive scalars in turbulence. Phys Fluids 17:125107.99 Jayesh WZ (1992) Probability distribution, conditional dissipation and transport of passive temperature fluctuations in grid-generated turbulence. Phys Fluids A

4:2292–2307.100 Balkovsky E, Fouxon A (1999) Universal long-time properties of Lagrangian statistics in the Batchelor regime and their application to the passive scalar problem.

Phys Rev E 60:4164–4174.101 Bourlioux A, Madja AJ (2002) Elementarymodels with probability distribution function intermittency for passive scalars with amean gradient. Phys Fluids 14:881–897.102 Kailasnath P, Sreenivasan KR, Saylor J (1993) Conditional scalar dissipation rates in turbulent wakes, jets and boundary layers. Phys Fluids A 5:3207–3215.103 Doering CR, Thiffeault J-L (2006) Multiscale mixing efficiencies for steady sources. Phys Rev E 74:025301.104 Thiffeault J-L (2012) Using multiscale norms to quantify mixing and transport. Nonlinearity 25:R1–R44.105 Pouquet A, Rosenberg D, Marino R, Herbert C (2018) Scaling laws for mixing and dissipation in unforced rotating stratified turbulence. J Fluid Mech 844:519–545.106 Jagannathan S, Donzis DA (2016) Reynolds and Mach number scaling in solenoidally-forced compressible turbulence using high-resolution direct numerical

simulations. J Fluid Mech 789:669–707.107 Gibson CH (1981) Buoyancy effects in turbulent mixing: Sampling turbulence in the stratified ocean. AIAA J 19:1394–1400.108 Baldyga J, Makowski L, Orciuchi W (2015) Interaction between mixing, chemical reactions, and precipitation. Chem Eng Res Des 85:745–752.109 Aref H, et al. (2017) Frontiers of chaotic advection. Rev Mod Phys 89:025007.

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