Homogeneous and Isotropic Turbulence: a short survey on recent developments
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p Roberto Benzi · Luca Biferale
Homogeneous and Isotropic Turbulence: a shortsurvey on recent developments. ⋆ Received: 29 June 2015 / Accepted: 6 July 2015
Abstract
We present a detailed review of some of the most recent developments on Eulerian andLagrangian turbulence in homogeneous and isotropic statistics. In particular, we review phenomeno-logical and numerical results concerning the issue of universality with respect to the large scale forcingand the viscous dissipative physics. We discuss the state-of-the-art of numerical versus experimentalcomparisons and we discuss the dicotomy between phenomenology based on coherent structures or onstatistical approaches. A detailed discussion of finite Reynolds effects is also presented.
Keywords
Navier-Stokes equations · tubulence · multifractal In this paper we critically review the most relevant physical features of fully developed three di-mensional turbulence. We consider the case of incompressible homogeneous and isotropic turbulence.Although this case may be considered too limited, most of our arguments can also be extended in thecase of non isotropic turbulence [1,2,3,4].The word turbulence refers to the chaotic behavior, in space and time, of fluid flow. It has beena major breakthrough to understand how the complexity of turbulent flows can be described by theNavier Stokes equations: ∂ t v + v · ∇ v = − ρ ∇ v + ν∆ v (1)where v is the velocity field , p the pressure, ν the kinematic viscosity and div v = 0 for incompressibleflows. Nowadays, eq.s (1) can be numerically simulated with high accuracy and we reproduce veryaccurately in a computer what one can measure in a laboratory experiments [5,6]. Turbulence wasthe first and more challenging physical phenomena investigated by numerical simulations. FollowingKadanoff, it is fair to say that physics is no longer divided in experimental and theoretical physics:numerical simulations or numerical experiments are new ways to understand complex phenomena.Given any external force acting in the system, we can define the rate of energy input and energydissipation in a turbulent flow. Hereafter we shall denote by v the characteristic scale of velocity fluc-tuations and by L the scale characterizing energy input. We can usually describe how much turbulent ⋆ postprint version, accepted for publication on J. Stat. Phys. DOI 10.1007/s10955-015-1323-9R. BenziDept. of Physics and INFN, University of Rome ‘Tor Vergata’, Rome, ItalyE-mail: [email protected]. BiferaleDept. of Physics and INFN, University of Rome ‘Tor Vergata’, Rome, ItalyE-mail: [email protected] a flow is, by introducing the Reynolds number Re ≡ v L/ν . Numerical simulations can be performedfor Re as large as 10 [7] which is also one of the largest Reynolds number for laboratory experiments.In the next few years, we expect to reach Re = 10 for the numerical simulations. So far, there existsno indications that eq.s (1) are wrong, i.e. that there are physical features not described by (1) fornon-relativistic, incompressible and ideal fluids. This is a highly non trivial statement which is basedfrom a detailed and careful comparison between numerical simulations and experimental data andmore than 30 years of hard work. Being a chaotic system, turbulence must be described in a statistical way, i.e. we need to introducecorrelation functions and a probability measure for the velocity field. However, turbulence is a systemstrongly out of equilibrium and the standard tools and ideas of statistical physics are useless: we needto build up a new theoretical framework. Physically, the basic and most fundamental feature of threedimensional turbulent flows is that energy dissipation is independent on the Reynolds number. Moreprecisely, energy dissipation ǫ for eq.s (1) is given by h ν ( ∇ v ) i , where h . . . i means an average in spaceand time. Since Re → ∞ is equivalent to ν →
0, we expect that ǫ → iff the velocity gradients arebounded. Experimentally as well as numerically we have that ǫ ∼ const. Re → ∞ (2)which is the so-called dissipative anomaly , implying that ∇ v grows to infinity. Kolmogorov was thefirst who clearly highlighted this basic and fundamental feature of turbulence. In his celebrated 1941paper, Kolmogorov was able to show that for homogenous and isotropic turbulence, the statisticalproperties of turbulent fluctuations must satisfy the equation [4]: h δv ( r ) i = − ǫr + 6 ν ddr h δv ( r ) i (3)where δv ( r ) ≡ ( v ( x + r ) − v ( x )) · r /r is the longitudinal difference of the velocity field between twopoints at distance r . Eq. (3) is exact and must be well verified both by numerical simulations andexperimental data if stationarity is assumed and (2) holds. In the limit ν →
0, Eq. (3) predicts theexistence of two different range of scales r . Introducing the scale η = ( ν /ǫ ) / , we can say that for r ≫ η the velocity fluctuations are controlled by the energy dissipation ǫ and the scale r , while for r ∼ η dissipation effects become important. The first range of scale ( r ≫ η ) is called inertial range andthe second range is called the dissipation range.Eq. (3) suggests that the probability distribution of turbulent fluctuations at scale r depends onlyon ǫ and r . Using dimensional arguments one can conclude that S p ( r ) = h δv ( r ) p i ∼ ǫ p/ r p/ (4)There are two major points in the Kolmogorov theory. First, eq. (3) is the fundamental prediction ofthe theory assuming that turbulent fluctuations are statistically isotropic, isotropic and ǫ is constantfor Re → ∞ . Secondly, eq. (4) can be considered a conjecture of the theory assuming that the statisticalproperties of turbulent flows are scale invariant in the inertial range, where the notion of scale invariantshould be interpreted in the same way introduced in the theory of critical phenomena. Note that eq.(3) is true even if scale invariance does not hold.The simplest way to check (4) is to compute the quantities Γ p ( r ) ≡ S p ( r ) /S ( r ) p/ usually referred to as generalized kurtosis. Eq. (4) predicts that Γ p ( r ) ∼ const in the inertial range.This is definitively not observe both in numerical simulations and laboratory experiments where Γ p ( r )increase for r → η , see figures (1) and (2). This phenomenon is called intermittency. In order to understand intermittency in a physical way, it is possible to look at turbulent flowsalso from the Lagrangian point of view (opposed to the Eulerian one, based on measurements in afixed reference frame in the laboratory). We consider a particle (point like) which is advected by thevelocity field v and whose trajectory is described by the position x ( t ). We can compute the lagrangianvelocity difference on time τ defined as δv ( τ ) = | v ( t + τ ) − v ( t ) | and the quantities S Lp ≡ h δv ( τ ) p i ,where the superscript L denotes the lagrangian frame. The analogous of the scale dissipation η isnow the Kolomogorv time scale τ d ≡ ( ν/ǫ ) / . Dimensional arguments and scale invariance impliesthat S Lp ( τ ) ∼ ( ǫτ ) p/ . Numerical and experimental data shows clearly that Γ Lp ( τ ) ≡ S Lp / ( S L ) p/ increases when τ → τ d . Usually, the value of Γ Lp ( τ d ) is much larger than the corresponding Eulerianquantity Γ p ( η ), i.e. we observe much stronger intermittency in lagrangian framework with respect tothe eulerian framework, see fig. (1). We can also compute the acceleration a ≡ | d x /dt | . Accordingto scale invariance in the form (4), we can estimate a ∼ δv ( τ d ) /τ d and compare it against data. Itturns out that a is one of the most intermittent quantities observed in turbulent flows: in fig. (4) weshow that the probability distribution of a develops extremely long tail up to 80 times the variance!Such a spectacular behavior has a well defined physical interpretation: from time to time the particleenters a region of extremely large vorticity ω ≡ rot v , see figure (3). These regions of large vorticityusually take the form of filaments which extend in space for scales well within the inertial range andshow a cross section of order 10 η . A vortex filament is neither a stationary nor a stable structure:when a particle enters a vortex filament the value of a is of order a ∼ ω r f where ω is the vorticityof the filament and r f is the cross section. For large Re , the vorticity scales as Re / and a can easilybecomes much larger than its characteristic value outside filaments. In summary, intermittency in thelagrangian acceleration is related to the existence of vortex filaments and viceversa.Vortex filaments have been observed in all turbulent flows. In wall bounded turbulence, vortexfilaments are responsible for the drag effect near the wall and control the rate of energy productionand dissipation [8]. The peculiar property of vortex filaments is that they do not carry most of theenergy fluctuations in a turbulence flow but they organize the flow around them and the region ofthe energy dissipation. The existence of vortex filaments open up a completely different scenario forturbulent flows: although Kolmogorov theory (i.e. eq. (3)) is correct and in agreement with all existingdata, scale invariance as described by eq. (4), or its equivalent form for lagrangian dynamics, does notproperly take into account the complex non linear intermittent dynamics in the statistical propertiesof turbulence. Moreover, since vortex filaments extend well within the inertial range, their statisticalproperties may be related to the large scale forcing. This implies that there may not be a universal wayto describe the statistics of turbulence independent of the forcing mechanisms. Also, since the crosssection of a vortex filaments is not much larger than the dissipation scale, the statistical propertiesof turbulence may strongly depend on the dissipation mechanism acting at very small scales. In otherwords, turbulence may not be universal with respect to large scale forcing and small scale dissipation.This is the crucial problem we need to understand in the following. Even if vortex filaments are crucial in understanding intermittency, there is no reason to assume apriori that scale invariance, definitively in a form different from (4), does not hold. In 1983, Parisi andFrisch [10] made the following observation: the Navier-Stokes equations (1) are invariant under thescale transformation: r → λ r v → λ h v t → λ − h t ν → λ h ν (5)Note that using (5) we have ǫ → λ h − ǫ . The key observation by Parisi and Frisch is that there mayexist many different value of h each occurring with a probability P ( h ), i.e. turbulent flows can beconsidered as a superposition of many different scale invariant configurations. In order to maintainscale invariance upon averaging over P ( h ), one needs to assume that P ( h ) ∼ r F ( h ) , i.e. one needs toassume that scale invariance holds for the probability distribution of h . This conjecture is referredto as the multifractal conjecture because originally F ( h ) was written in the form 3 − D ( h ) where D ( h ) is assumed to be the fractal dimension of the scale invariant solution with exponent h . There Γ (r) ; Γ ( τ ) r; τ Eulerian FlatnessLagrangian Flatness
Fig. 1
Flatness for Eulerian and Lagrangian measurements, data are taken from a DNS at 2048 resolution[9] exists a constrain on D ( h ) since we must require, in agreement with (2), that the energy dissipation isindependent of Re , i.e. we must require that h ǫ i ∼ Z dr r − D ( h ) r h − ∼ const (6)Note that eq.(4) is now no longer valid since we must compute the average over h . Because we areinterested in the limit at small scale, we can use the saddle point technique to compute the integraland we obtain: S p ( r ) ∼ Z dr r ph +3 − D ( h ) ∼ r ζ ( p ) (7)where ζ ( p ) = inf h [ ph + 3 − D ( h )] (8)Clearly, if we know ζ ( p ) we can invert the Legendre transform (8) to compute D ( h ). At first sight themultifractal conjecture may seem rather obscure because it is not clear what is its physical meaning, i.ewhat does it mean averaging over the possible scale? invariant configurations. Next, the multifractalconjecture seem not to have any predictive power because nothing is known on the function D ( h )except the constrain (6). Finally, the discussion on vortex filaments suggests that the function D ( h )may not be universal. Using eq. (7) we can easily obtain Γ p ( r ) ∼ r ζ ( p ) − pζ (2) / . Since ζ ( p ) is a convex function of p , we obtainthat Γ p ( r ) must increase for r →
0, i.e. intermittency is consistent with the multifractal conjecture. Thenext step is to compute from experimental and numerical data ζ ( p ) for different forcing mechanism and Re and to understand whether ζ ( p ) are universal. The computation of ζ ( p ) requires the existence of a -10 -8 -6 -4 -2 -40 -30 -20 -10 0 10 20 30 40 P D F ( δ τ v ) δ τ v / <( δ τ v) > τ Fig. 2
Probability distribution functions (PDF) of the Lagrangian velocity increments at changing the timelag τ . Curves have been shifted along the y-axis for the sake of presentation.For details about the numericalsimulations see [11] clear scaling range where (7) is observed. Because any numerical simulations or laboratory experimentsare done for finite system sizes, we can expect that there must be non trivial effects in the scaling of S p ( r ). Such finite size effects are quite common in many systems and they are extremely well knownand under control in the case of critical phenomena. However, since we have no theory to compute ζ ( p )from the Navier-Stokes equation, we are unable to predict finite size effects. This is a major problemto understand whether scaling and universality is observed for turbulent flows. Fig. 3
A typical trajectory of a Lagrangian tracer in HIT inside a vortex filaments.
A systematic investigation of the scaling properties of Eulerian turbulence required several years ofwork and the introduction of new ideas to overcome finite size scaling (see the work on Extended SelfSimilarity). The overall conclusion [5] was that scaling is observed and that ζ ( p ) are independent on Re and the forcing mechanism, see e.g. figure (7). This conclusion is obtained by accurate data analysis ofmany different laboratory and numerical data and it does not imply anything about the multifractalconjecture. In order to make progress, one needs to understand in a deeper way the physical mechanismof intermittency and its relation, if any, with the function D ( h ).A major breakthrough in this direction was provided by the exact non trivial solution of the socalled Kraichnan model of a passive scalar [12]. We consider the following problem: let θ a scalar fieldadvcted by a velocity field v and forced by some large scale mechanism F θ : ∂ t θ + v · ∇ θ = χ∆θ + F θ (9) -11 -10 -9 -8 -7 -6 -5 -4 -3 -2
0 20 40 60 80 a/ σ a Fig. 4
Acceleration PDF for HIT. The dashed line represents the dimensional K41 prediction. The blackcontinuos line is the multifratcal prediction [13]. Inset: Acceleration PDF multiplied by fourth order ponwer ofthe acceleration, a P ( a ) and the corresponding prediction from the multifratcal formalism Following Kraichnan we assume: (i) the velocity field is gaussian random field delta-correlated intime and with prescribed correlation function g ( r ) ∼ r ξ ; (ii) the large scale forcing is random, gaussian,and statistically isotropic. Given (9) we are interested to compute the behavior of the correlationfunctions C n ( x , x , ..., x n ) = h θ ( x ) θ ( x ) ..θ ( x n ) i as a function of ξ in the limit χ →
0. For thisproblem, we can rephrase the scale transformation (5) as: r → λ r θ → λ h θ t → λ − ξ t ν → λ ξ ν (10)The analogous of (3) can be easily derived upon assuming that χ h ( ∇ θ ) i ∼ const. , which is equivalentto ξ + 2 h = 1 (for delta-correlated random field v a subtle difference apperas that is not important in the following discussion). Since eq. (9) is linear and because the random field is delta-correlated intime, we can obtain a closed equation for C n which can be formally written as L n [ v ] C n + D n C n + F n = 0 (11)where L [ v ] is a linear operator depending on the velocity statistics, D n is the n-dimensional Laplacianproportional to the diffusivity χ and F n is the term due to the forcing. The exact form of the operatorsappearing in (11) is irrelevant for our argument. The general soution of (11) is given by C ( i ) n + C ( d,f ) n where the first term is one of the zero mode solutions of L [ v ] C ( i ) n = 0 while the second term isthe (particular) solution depending on χ and the forcing, i.e. C ( i ) n is the correlation function in theinertial range while C ( d,f ) n is the non universal part of the correlation function which depends onforcing and dissipation. The important result is that in the limit ξ → C ( i ) n , is the relevant contribution to the correlation function. Moreover, the n − ordercorrelation function display anomalous scaling, i,e C n ( λx , ...λx n ) ∼ λ z ( n ) C n ( x , ..x n ) where z ( n ) is anon linear function of n . In summary, for the passive scalar problem described by (9), one is able toshow that scaling occurs and that the scaling properties of the correlation functions are universal. L o ca l S ca li ng E xpon e n t τ/τ η Re λ = 400 tracerRe λ = 200 tracerSt = 0.5 heavySt = 0.5 light Fig. 5
Comparison of local scaling exponents for the 4th order Lagrangian Flatnees between: (i) tracersparticles at two different Reynolds number (ii) one light and (iii) one heavy particle. Notice the ehnancement(depletion) of the bottleneck around τ /τ η ∼ One can wonder whether there exists something analogous of vortex filaments in the passive scalar.Numerical simulations show that in the case of passive scalar there are many fronts in θ (abrupt changesin θ over a very small distance) [14]. A front like structure corresponds to h = 0 in the multifractallanguage. Therefore, if the multifractal conjecture is correct, we should expect z ( n ) → = z ∞ ≡ − D (0)for n → ∞ , where D (0) is the fractal dimension of the front. This relation can be checked and it appearsto be consistent with the numerical simulations. The consistency by itself is not surprising since D (0) is defined as the set of point where h = 0, i.e. where we observe a front. The interesting point is that theresults on the correlation functions and the universality of z ( n ) imply that the statistical properties ofthe fronts are independent on the dissipation mechanism and the large scale forcing. In other words,the statistical properties of θ can be described using the multifractal conjecture independently onthe existence of well defined coherent structures (fronts) carrying the strongest singularities in ∇ θ .Moreover, the general solution of eq.(11) tells us that fronts are formed due to the complex non linearinteraction between the velocity field and the passive scalar in the inertial range and scale invarianceis not affected by the existence of fronts. Going back to the Navier-Stokes equation we can imaginethat vortex filaments are consistent with the asymptotic form of ζ ( p ) → ph + 3 − D ( h ) and thatthe statistical properties of vortex filaments do not affect the scaling properties of the inertial range(universality). This is a delicate statement which we now investigate in details. Fig. 6
Comparison between Eulerian scaling exponents for longitudinal, ζ l ( p ), and transverse, ζ tr ( p ), StructureFunction [11] together with two different multifractal predictions (MF) obtained with two different choices of D ( h ). Let us assume for the time being that the multifractal conjecture is correct, although we do notknow how to compute D ( h ) in the Navier-Stokes equation. We already said that any numerical and/orexperimental observation shows finite size effects. Let us focus on the dissipation effects due to theviscosity. We cannot blindly assume that the multifractal picture holds in the dissipation range ( r ≪ η ).We can speculate that when δv ( r ) r/ν ∼
1, the velocity fluctuations are damped by viscous effects. Theproblem is that we must give a reliable meaning to δv ( r ). The question is: when we write δv ( r ) ∼ r h what we really mean? ( Unfortunately the solution of the passive scalar does not help in this case.) Itturns out that one can give a well defined meaning to scaling δv ( r ) ∼ r h using the theory of random -1 Lo c a l S c a li ng E x ponen t τ / τ η E1 Re λ =124E2 Re λ =690E3 Re λ =740 D1 Re λ =140D2 Re λ =320D3 Re λ =400 D4 Re λ =600D5 Re λ =650 Fig. 7
Comparison between experimental and numerical data of local scaling exponents of Flatness (from [6]).The horizontal line corresponds to the K41 non-intermittent prediction.The yellow band is the multifractalprediction with uncertainty estimated out of the two different D ( h ) curves estimatred from either longitudinalor transverse Eulerian increments. Data are taken from [15,16,17,18,19,20,21] multifractal fields. For our purpose, it is enough to say that the statistical properties of velocityfluctuations at scales r and R are linked by the relation (with r < R ): δv ( r ) = δV ( R ) h rR i h (12)with propability P h ( r/R ) ∼ ( r/R ) − D ( h ) . It is possible to construct explicit examples of randomfields obeying eq. (12) with a prescribed function D ( h ) [22]. Using (12) we can write that dissipationeffects are relevant when δv ( r ) rν = δv ( R ) Rν h rR i h ∼ R = L and denoting the dissipation scale η ( h ) we obtain η ( h ) = Re − h L (14)The above relation is true with probability ( η ( h ) /L ) − D ( h ) = Re − − D ( h )1+ h . Using (13) and (14) , we cancompute the average energy dissipation ǫ : ǫ = Z dhν δv ( η ( h )) η ( h ) ∼ Z dhRe − h +2 − D ( h )1+ h (15)It is easy to show that the saddle point computation of the above integral is equivalent to the condition ζ (3) = 1 which is a constrain on D ( h ). Using this result, we obtain ǫ independent of Re , which showsthe consistency of (13) with the eq. (3). We can generalize eq.(15) to compute the scaling behavior in Re of moments of velocity gradients [23]: h ( ∇ v ) p i = Z dh δv ( h ) p η ( h ) p Re − D ( h )1+ h ∼ (cid:20) δv ( L ) L (cid:21) p Z dhRe − p ( h − − D ( h )1+ h ∼ (cid:20) δv ( L ) L (cid:21) p Re χ ( p ) (16) where χ ( p ) = sup h [ − ( p ( h −
1) + 3 − D ( h )) / (1 + h )]. From (16) it is possible to show that for p > γ p ≡ h ( ∇ v ) p ih ( ∇ v ) i p/ = Re χ ( p ) − p/ > lim r → η Γ p ( r ) ∼ Re ( ζ ( p ) − pζ (2) / (17)The above equation tells us that intermittency in the dissipation range Re χ ( p ) is greater than intermit-tency extrapolated from the inertial range, l.h.s of (17). This is a highly non trivial prediction givenby multifractal conjecture and consistent with the experimental and numerical data.One can take advantage of (13) to generalize the multifractal conjecture for finite Re numbers andintroducing the effect of dissipative contribution. Instead of (12) we have the following result: δv ( r ) = g ( rL , η ( h ) L ) f ( rL , η ( h ) L ) h (18) P h ( r ) = f ( rL , η ( h ) L ) − D ( h ) (19)where the two functions g ( x, y ) and f ( x, y ) satisfy the following asymptotic conditions: lim y → f ( x, y ) = x, (20) im y → g ( x, y ) = const. (21)and lim x → f ( x, y ) = const. (22) lim x → g ( x, y ) = x. (23)In the range η ( h ) ≫ r , we assume that no intermittent fluctuations occur and the velocity fluctuationare smooth according to (23) and (18), while in the inertial range r ≫ η ( h ) the dissipation effects areirrelevant. The precise shape of the function f and g are supposed to be universal. Presently, we arenot able to compute f and g theoretically but we can provide very accurate fit of both functions usingexperimental and numerical data. At any rate, we shall see in the following that the detailed shape ofthe two functions do not play a crucial role in the theory. It is interesting to remark that, if we neglectthe fluctuation of the dissipative scale by assuming η ( h ) = η = Re − / L , eq.s (18,19) predict S p ( r ) ∼ g p ( r ) f ( r ) ζ ( p ) (24)In the range of scales where g ( r ) ∼ const we obtain: S p ( r ) ∼ S ( r ) ζ ( p ) /ζ (3) ∼ S ( r ) ζ ( p ) (25)which is known in literature as Extended Self Similarity (ESS) [24,25]. ESS is useful to extract accuratevalues of the scaling exponents ζ ( p ) even at relatively low Re number because it does not require anyknowledge on the function f and because, at low Reynolds, intermittent fluctuations of the dissipativescale are relatively small. Clearly, the range of scale where ESS is useful must be outside the dissipationrange. The size of the dissipation range depends on the minimum and maximum value of h . Theoreti-cally, we know that h ∈ [0 , Re − , Re − / ]. E.g.for Re = 10 and a characteristic value of L ∼ m , the dissipation range is [30 µ, mm ].We can now use (13) to predict the probability distribution of the acceleration. Let us define τ η ( h ) = η ( h ) /δv η ( h ) the dissipative time scale associated to the dissipation scale η ( h ). Then, we cancompute the acceleration a ( h ) as δ v η ( h ) /τ η ( h ): a ( h, δV ( L )) = δV ( L ) L (cid:20) νLδV ( L ) (cid:21) h − h (26) which holds with probability ( η ( h ) /L ) − D ( h ) . Therefore, given the probability of the large scale fluc-tuation δV ( L ), using (26) we can compute the probability P ( a ) to observe in a given point a value a for the acceleration and we can compare our findings against experimental results. In most cases, thefluctuations at very large scale L are observed to be distributed in a gaussian way and we can providean analytical prediction for P ( a ). It turns out that the multifractal prediction of P ( a ) is very accuratecompared against experimental and numerical data, see figure (4) and [13]. The word prediction inthis case should be interpreted in the following way: given the probability distribution of large scalefluctuations and the function D ( h ), we can predict P ( a ). The function D ( h ) can be computed from theknowledge of the scaling exponent ζ ( p ) which can be obtained from the available data. Therefore, wecan say that from the knowledge of the intermittent fluctuations in the inertial range we can predictthe probability distribution of the lagrangian acceleration. Having saying that, the prediction of P ( a )is a highly non trivial result because it shows that the multifractal framework, based on the scaleinvariance of the Navier-Stokes equations, correctly describes the statistical properties of turbulentfluctuations over the whole range of scales, from large scale to dissipative scales. The prediction of P ( a ), in the sense previously discussed, represents a major achievement of our ability to provide auniversal and consistent description of turbulent flows. As we previously discussed, strong fluctuations in the lagrangian acceleration are due to vortex fila-ments and are correctly described by the multifractal framework. In the multifractal approach, however,there is no point whatsoever where we introduced any physical informations concerning the existenceand the relevance of coherent structures or vortex filaments. Clearly, it seems relevant to investigatethis question in more details. To do so, we need to look at turbulence from the largrangian point ofview. It is relatively easy to rewrite the multifractal approach in terms of lagrangian variables. Weneed to consider velocity difference between two points on the lagrangian trajectory at time interval τ . The scaling property of the velocity field in terms of τ can be obtained by using the scaling relationbetween r and τ , namely τ ∼ r − h or equivalently r ∼ τ / (1 − h ) . Then eq. (7) is generalized as follows: S Lp ( τ ) ≡ h δv ( τ ) p i ∼ Z dhτ ph − h τ − D ( h )1 − h ∼ τ ξ ( p ) (27)Eq. (27) enables us to compute the lagrangian scaling exponents ξ ( p ) in terms of the same multifractalfunction D ( h ) used for the computation of ζ ( p ). In other words, we can compute the lagrangian scalingexponents by the eulerian scaling properties. Before comparing the ”prediction” given by (27) againstavailable data, we need to discuss a subtle but non trivial question concerning isotropy. So far we haveassumed that in the limit Re → ∞ , small scale turbulent fluctuations are isotropic. This assumptionis based upon the fact that the Navier-Stokes equations are invariant under SO (3) rotation group [3].However, real experimental data and/or numerical simulations are done neither in the limit Re → ∞ nor with perfect isotropic forcing. Even a small anisotropic on the large scale can introduce, at finite Re ,non isotropic effects at small scales. To be more quantitative, we can compute the structure functionsin the Eulerian frame for longitudinal velocity difference and transverse velocity difference. Let usindicate with ζ l ( p ) and ζ tr ( p ) the corresponding scaling exponents. Isotropy implies that ζ l ( p ) = ζ tr ( p )is true for any p . Careful investigations, using high resolution numerical simulations, have shown thatisotropy is verified for p ≤ p one observes ζ tr ( p ) < ζ l ( p ), see figure (6). In principleit should possible to formulate the multifractal conjecture by introducing isotropic and non isotropicsectors. where the non isotropic contributions are subleading with respect to the isotropic ones atsmall scales and large Re . Therefore the discrepancy between ζ l ( p ) and ζ tr ( p ) is a measure of thefinite size Re effects. In the Eulerian farmework it is possible to disentangle isotropic contributionfrom the non isotropic ones. However, lagrangian structure functions are mixing both contributionsand, consequently, for latge p we can predict the lagragian scaling exponent ξ ( p ) from the eulearianones ζ l,tr ( p ) with error bars increasing with increasing p . The non trivial result is that, within errorbars (careful computed following the previous discussion), eq. (27) is consistent with experimental andnumerical data up to p = 10 [11]. The validity of (27) allows us to investigate the dissipative effects in the lagrangian dynamics. Thefundamental advantage of the lagrangian point of view can be understood by considering (13) in thetime domain: the dissipation time τ d ( h ) can be defined by the relation: δv ( τ d ( h )) τ d ( h ) ν ∼ τ d ( h ) (3 − D ( h )) / (1 − h ) . A simple computation shows that τ d ( h ) ∼ Re h − h (29)Eq. (29) shows that the dissipation effects in the lagrangian framework cover a range [ Re − ,
1] muchlarger the dissipation range in the Eulerian framework. Thus, in the lagrangian turbulence dissipationeffects are magnified. Using (29), we can generalize (18) to obtain δv ( τ ) = G ( τT , τ d ( h ) T ) F ( τL , τ d ( h ) T ) h − h (30) P h ( τ ) = F ( τL , ττ d ( h ) ) − D ( h )1 − h (31)where the functions G ( x, y ) and F ( x, y ) satisfy the same asymptotic behavior of the Eulerian case. Wenow consider the local scaling exponents ruling the scale-behaviour of generalised Flatness: K p ( τ ) ≡ dlogS Lp ( τ ) dlogS L ( τ ) (32)These quantities can be directly measured on data, they do not need any fitting and can be consideredan estimate of the intermittent fluctuations at changing the reference scale: for large τ (inertial range)we have K p ( τ ) → ξ ( p ) − ξ (2) / K p ( τ ) → p (dissipative range), Using K p ( τ ) we can quantitatively measure the effect of vortex filaments for the intermittent fluctuations.The smart idea is to compute K p ( τ ) for lagrangian particles and for inertial particles. The latterscan be heavy or light particles: heavy particles are concentrated outside vortex filaments while lightparticles are concentrated inside vortex filaments. It is found that K p ( τ ) show a well defined deepin the dissipation range for lagrangian particles, which disappears for heavy particles and becomesdeeper for light particles. Vortex filaments are clearly associated to the increase of intermittency in thedissipation range, see figure (5).It is important to understand that even for p = 4 the value of K p depends on the whole function D ( h ) and not from the value of ζ ( p ) or ξ ( p ). In other words, if we consider K , the increase ofintermittency (i.e. K < ξ (4) − ξ (2) depends on the whole structure D ( h ) and the shapes of the twofunction G and F . Using K we can assess the universality of our results. Recently, a major effort wasundertaken to compute K in a number of experimental simulations and laboratory experiments. Allthe results, within error bars, collapse on the same universal curve. The effect of vortex filaments, ifany, is hidden in the shape of the two functions G and F , which interpolate the inertial range scaling G ∼ const, F ∼ τ and the dissipative scaling G ∼ τ, F ∼ const . For instance the choice G = (cid:20) x c x c + y c (cid:21) /c (33) F = [ x c + y a ] /c (34)provides a good fit to the data with c ∼
4, see figure (7).The above analysis tells us something extremely interesting. First of all intermettincy and scalingin small scale turbulent fluctuations are universal and independent of the large scale mechanisms.Second, the effects of coherent structures sum up to the same statistical probability distribution for theturbulent fluctuations. Third and more important, The small scale velocity fluctuations are consistentwith the scaling properties of the system. The latter can be described as the superposition of all the possible scaling exponents h with the weight r − D ( h ) . The physical meaning of 3 − D ( h ) correspondsto that of entropy in equilibrium statistical mechanics: the mutlifractal conjecture can be rephrasedby saying that the statistical properties of turbulence can be obtained by summing all possible flowconfigurations with exponent h (energy) with the number of available configurations being r − D ( h ) . Thedescription of turbulence in terms of coherent structures is not in contradiction with the multifractalconjecture providing that scaling is satisfied. The advantage of the multifractal description is that theknowledge of D ( h ) is sufficient to derive all the statistical properties of turbulent fluctuations at allscales and times, at least for observables which are invariant respect to the same group of symmetriesof the Navier-Stokes equations. Turbulence is the classical prototype of a complex system: we exactly know the equation of motions butwe are or were unable to describe the macroscopic behavior of the system. In the case of homogenousand isotropic turbulence, the major effort performed in the last 20 years provides us of a well definedand relatively simple picture of turbulence: the statistical properties are scale invariant and universalcharacterized by strong intermittency at all scales. The only technical point left to be done is a way tocompute the function D ( h ) from the Navier-Stokes equations. We know how to compute D ( h ) is somerelatively simple although non trivial case (the passive scalar and the Burgers’ equation) and there areseveral hopes that the computation of D ( h ) in the Navier-Stokes equations may eventually be donefollowing similar ideas. In summary, we believe that the computation of D ( h ) is not linked to a newphysical ideas, although it represents a challenging problem to be solved.So far we discussed the case of homogenous isotropic turbulence. However, there exist many differentturbulent problems which are worthwhile to be investigated. In particular, it is interesting to considercases where there are new physical space scale and/or time scale which appear in the system such thateq. (3) should be reconsidered. A non exhaustive list includes: Rayleigh Benard convection, turbulentflows with the dilute polymers, spinodal decomposition, MHD turbulence, shear flows. In some cases,the effect of non isotropic contribution should be considered and a number of new challenging questionsmust be answered. It is not clear whether the same arguments reviewed in this paper can be appliedto all turbulent flows. In some cases (shear turbulence) it appears that scaling argument and themultifractal conjecture are still valid. In other cases, for instance turbulent in MHD, the question isstill controversial. Also, there exists the special case of turbulence in superfluids where the dissipationmechanism is definitively not captured by the standard Navier-Stokes equations.We authors thank long lasting and useful collaboration with J. Bec, G. Boffetta, E. Calzavarini,A. Celani, M. Cencini, A.S. Lanotte and F. Toschi. One of us (RB) thanks the organizing committeeof DFSD-2014, and in particular Francois Dubois and Stephan Fauve for the kind invitation to theconference in Paris where this paper has been presented. The work has been supported by the Euro-pean Research Council under the European Community’s Seventh Framework Program, ERC GrantAgreement No 339032. References
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