Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence
aa r X i v : . [ n li n . C D ] N ov Intermittency, cascades and thin sets in three-dimensionalNavier-Stokes turbulence
John D. GibbonDepartment of Mathematics,Imperial College London,London SW7 2AZ, UK
Abstract
Visual manifestations of intermittency in computations of three dimensional Navier-Stokesfluid turbulence appear as the low-dimensional or ‘thin’ filamentary sets on which vorticity &strain accumulate as energy cascades down to small scales. In order to study this phenomenon,the first task of this paper is to investigate how weak solutions of the Navier-Stokes equationscan be associated with a cascade &, as a consequence, with an infinite sequence of inverse lengthscales. It turns out that this sequence converges to a finite limit. The second task is to showhow these results scale with integer dimension D = 1 , , Introduction
The most striking visual manifestation of in-termittency in three-dimensional incompressiblefluid turbulence is the accumulation of vortic-ity & strain on ‘thin’ or low-dimensional sets.When displayed graphically as iso-surfaces in acube, these sets typically appear as spaghetti-likeentangled tubular filaments : see Fig. 1 wheresnapshots of the energy dissipation field ε , &the Q -field (defined in the caption) of a forcedNavier-Stokes flow are displayed. Although dif-fering in the fine detail from case to case, ini-tial three-dimensional vortical structures tendto flatten at intermediate times into quasi-two-dimensional pancakes which subsequently roll upinto quasi-one-dimensional tubes, with furtheriterations of flattening & filamentation result-ing in ever finer striations [1–11] : see the recentpaper by Elsingha, Ishihara & Hunt [12]. His-torically, Batchelor and Townsend [13] were thefirst to suggest that vorticity and strain are notdistributed in a Gaussian fashion across a do-main but accumulate on local, intense sets whichthey identified with intermittency in the energydissipation [14–18]. In the literature these fil-amentary structures are loosely referred to as‘fractal’ because of the roughness of the detailof their evolving fine-scale structure : see Fig. 1 for an illustration. In the last generation, vari-ous cascade models, such as the beta, bi-fractaland multi-fractal models, explicitly talk aboutaccumulation on sets of non-integer dimension D [16]. Studies in Fourier decimation have pur-sued the idea of intermittency more precisely byprojecting a three dimensional Navier-Stokes ve-locity field onto a chosen subset of Fourier modesby employing a generalized Galerkin projector[19,20]. Intermittency properties have then beeninvestigated by tuning both the restricted sub-set and the Reynolds number. Recent work hasculminated in making this restricted set fractal[21–24] : in effect, the number of degrees of free-dom are limited to a sphere of radius k growingas k D (for non-integer D ) embedded in the three-dimensional space. Ref. [24] contains an excel-lent set of references. However, from the pointof view of rigorous Navier-Stokes analysis, thephenomenon is by no means understood, mainlybecause technical tools exist to pursue analysisonly on fixed domains of integer dimension butnot on time-evolving fractal sets. In this context,many questions remain outstanding. Is there auniversal value of D or are there many disjointsets of differing dimension? How does this fit inwith the idea of a cascade to small scales andthe regularity of solutions of the Navier-Stokesequations at these scales? For instance, Biferale1igure 1: The left-hand figure is a snapshot of the energy dissipation field ε = 2 νS i,j S j,i of a forced512 Navier-Stokes flow at Re λ = 196 which is colour-coded such that yellow is 4 times the mean& blue denotes 6 times the mean. The right-hand figure shows the field Q = (cid:0) | ω | − | S | (cid:1) : thecolours correspond to − Q rms (blue) & 5 Q rms (red). Plots courtesy of J. R. Picardo & S. S. Ray.and Titi [25] have shown that a helically deci-mated version of the 3 D Navier-Stokes equationsleads to global regularity. A global theory thatanswers all these questions still remains elusive :this paper aims to build on what is known rigor-ously for three dimensional Navier-Stokes equa-tions to discuss how this fractal set might occur. Cascades, scaling & weak solutionestimates in three dimensions
A cascade is a sequential process that involvesvorticity & strain being driven down to eversmaller length scales in the flow & has long beenclosely associated with intermittency [16,26–28].For sufficiently long times a cascade to smallerscales should show up in estimates of bothspatially & temporally averaged gradients of adivergence-free velocity field u ( x , t ) that evolveaccording to the Navier-Stokes equations( ∂ t + u · ∇ ) u + ∇ p = ν ∆ u + f ( x ) . (1)The domain V = [0 , L ] per is chosen to be three-dimensional & periodic. ν is the viscosity & f is an L -bounded forcing. Here we showthat Leray’s weak solutions of the Navier-Stokesequations [29] can be interpreted in terms of acascade. We define a doubly-labelled set of norms indimensionless form F n,m = ν − L /α n,m k∇ n u k m , (2)where α n,m is defined by α n,m = 2 m m ( n + 1) − . (3)The norm notation k · k m in (2) is defined by k∇ n u k m = (cid:18)Z V |∇ n u | m dV (cid:19) / m . (4)Higher values of n allow the detection of smallerscales, while higher values of m account forstronger deviations from the mean, with m = ∞ representing the maximum norm.The Navier-Stokes equations are well-knownto possess the scale-invariance property u ( x , t ) → λ − u (cid:0) x /λ, t/λ (cid:1) , (5)for any value of the dimensionless parameter λ .Under this scaling the F n,m in (2) are invari-ant in λ & are thus invariant at every length& time scale in the flow. This makes them in-valuable as a tool for investigating a cascade of2nergy through the system. This is further illus-trated by the fact that there exists a bounded,weighted, double hierarchy of their time averages (cid:10) F α n,m n,m (cid:11) T ≤ c n,m Re (cid:27) n ≥ ≤ m ≤ ∞ ,n = 0 3 < m ≤ ∞ , (6)as demonstrated in [30]. The angled brackets h·i T are defined by h·i T = T − Z T · dt , (7)& the Reynolds number Re by Re = LU/ν with U = L − (cid:10) k u k (cid:11) T . (8)The physical meaning of the set of inequalities in(6) can be illustrated thus. Consider the time-averaged energy dissipation rate defined in theconventional manner as ε av = νL − (cid:10) k∇ u k (cid:11) T .Then in the case n = m = 1, (6) becomes ε av ≤ c , ν L − Re . (9)The upper bound is recognizable as the same re-sult derived by Kolmogorov’s theory [16] &, aswe shall see below, leads to the well-known Re / estimate for the inverse Kolmogorov length. Thedouble hierarchy displayed in (6) furnishes uswith bounds which generalize (9) to all deriva-tives & in every L m -norm. It is valid for Leray’sweak solutions & encapsulates all the knownweak solution results in Navier-Stokes analysis[30]. These are distributional in nature & arenot unique & thus the result in (6) falls short ofa full regularity proof ; i.e. existence & unique-ness of solutions. It was shown in [30] that toachieve this would require D F α n,m n,m E T < ∞ . (10)While it remains an open problem, there is noevidence that any bounds with the factor of 2in the exponent exist. Indeed it is possible thatweak solutions are all that are available. Whathas been deduced is that (2) & (3) lead to a def-inition of a set of inverse length scales ℓ − n,m (cid:0) Lℓ − n,m (cid:1) n +1 := F n,m , (11) whose estimated time averages are [30] (cid:10) Lℓ − n,m (cid:11) T ≤ c n,m Re n +1) αn,m + O (cid:0) T − (cid:1) . (12)The exponents of Re in the two cases n = m = 1& n, m → ∞ are3( n + 1) α n,m (cid:12)(cid:12)(cid:12)(cid:12) n,m =1 = 3 / , (13)lim n,m →∞ n + 1) α n,m = 3 . (14)The first result in (13) is consistent with the in-verse Kolmogorov length while the second resultin (14) implies that there exists a finite limit tothe cascade process. However, when Re is largeit does so at a level below molecular scales wherethe Navier-Stokes equations are not valid. Nev-ertheless it validates Richardson’s original asser-tion that viscosity eventually terminates the cas-cade process [16, 34]. Estimates & scaling in D -dimensions Inequalities (6) & (12) are true for weak solu-tions in a D = 3 domain. For integer values of D = 2 or D = 3 on a periodic domain V D thedefinition of (2) can be generalized to F n,m,D = ν − L /α n,m,D k∇ n u k m , (15)where α n,m in (3) & (6) is replaced by α n,m,D = 2 m m ( n + 1) − D . (16)The F n,m,D in (15) possess the same invarianceproperties as F n,m in (2). The details of theproof of (6) has been generalized for the inte-ger D -dimensional case using the same methods& results as in three dimensions [30], althoughthe calculation is far from straightforward : seethe Appendix. When D = 1 the Navier-Stokes equations make no sense unless the pressure & divergence-free terms are removed,in which case we have Burgers’ equation. The results expressed in D -dimensions with D = 1 are valid for this. heorem 1 For D = 2 , and for n ≥ & ≤ m ≤ ∞ , the equivalent of (6) is D F (4 − D ) α n,m,D n,m,D E T ≤ c n,m,D Re . (17) For D = 1 the same result holds for Burgers’equation. More than 40 years ago Fournier & Frisch [35]introduced the idea of turbulence in D dimen-sions where D is no longer an integer but is re-stricted to the range D ≥
2. They achieved thisby analytically continuing the Taylor expansionin time of the energy spectrum E k ( t ), assum-ing Gaussian initial conditions. Since then theidea of a non-integer dimension has taken root inthe many papers on the beta, bi-fractal & multi-fractal models [16, 17]. Can the Navier-Stokesestimates in (17) be performed on a domain ofnon-integer dimension? In a fully rigorous sense,the answer is in the negative. For instance, thereare no proofs of the Divergence Theorem or theSobolev inequalities on fractal domains. Thuswe can only claim the validity of Theorem 1 forinteger values of D . What the result does do,however, is show how the exponent of F n,m,D scales with integer values D . The surprising butcrucial factor of 4 − D in the exponent multiply-ing α n,m,D deserves some remarks :1. When D = 3, the factor of 4 − D is simplyunity & (17) reduces to (6) ;2. When n = m = 1 this factor cancels tomake (4 − D ) α , ,D = 2 for every value of D , as it should. It also furnishes us with the correct bound on the averaged energydissipation rate ε av .3. When D = 2 we achieve the 2 α n,m, boundrequired for full regularity, as in (10). Thusthe case D = 2 is critical for regularity, asis well-known [37–39].As in Fig. 1, computations in [7–12] haveshown that the process of flattening & filamen-tation results in ever finer striations as the flowprogresses. This would indicate that the set(s)on which vorticity or strain are concentrated hasa non-integer & decreasing dimension. While wehave no rigorous methods for proving the validityof (17) when D takes non-integer values, it raisesthe intriguing possibility that this may neverthe-less be true. Certainly it is clear that when D de-creases in (17) then the exponent (4 − D ) α n,m,D of F n,m,D increases, which is the direction ofmore, not less, regularity. This suggests that aflow may adjust itself to find the smoothest, mostdissipative set, not the most singular, on whichto operate.
This runs counter to the traditionallyheld theory of viscous turbulence in which sin-gularities have been long-st&ing c&idates as theunderlying cause of turbulent dynamics [31–33],even though they must be rare events [36–39].Adjustment to find the smoothest, most dissipa-tive set could be a way of the flow re-organizing& regularizing itself to avoid singularities.
Acknowledgments :
I thank J. R. Picardo (IITMumbai) & S. S. Ray (ICTS Bangalore) for theplots in Fig.1 from their Navier-Stokes data. Appendix : proof of Theorem 1
The aim of Theorem 1 is to roll together estimates for the Navier-Stokes equations that are alreadyknown individually in both the D = 2 & D = 3 cases. In addition, Burgers’ equation is included,which is appropriate for D = 1 when the pressure term & the incompressibility condition have beendropped. The main foundation of the proof of Theorem 1 is the original result of Foias, Guillop´e &Temam (FGT) in 3-dimensions [40]. Given that all three results are known separately, we are ableto formally manipulate & differentiate the H n , defined below in (18) below, on a periodic domainof integer dimension D . 4 The FGT result in integer D dimensions We require the definition H n = Z V D |∇ n u | dV , (18)from which we can write [37] ˙ H n ≤ − νH n +1 + c n k∇ u k ∞ H n . (19)For simplicity, we have omitted the forcing. An integer- D -dimensional Gagliardo-Nirenberg in-equality gives k∇ u k ∞ ≤ c n H a/ n +1 H (1 − a ) / (20)with a = D/ n & n > D/
2. After re-arrangement, (19) becomes ˙ H n ≤ − ν (1 − a ) H n +1 + c n ν − a − a H − a n H − a − a ≤ − ν (1 − a ) H n +1 + c n ν − a − a H n n − D n H n − D n − D . (21)Divide by H nα n, n & time average to give (cid:28) H n +1 H nα n, n (cid:29) T ≤ c n ν − − a * H n (4 − D ) αn, ,D n − D n H n − D n − D + T ≤ c n ν − − a (cid:28) H (4 − D ) α n, ,D n (cid:29) n n − D T h H i n − D n − D T . (22)Then a Holder inequality gives (cid:28) H (4 − D ) α n +1 , ,D n +1 (cid:29) ≤ (cid:28) H n +1 H nα n, n (cid:29) (4 − D ) α n +1 , ,D (23) × * H
12 (4 − D ) nαn, ,Dαn +1 , ,D −
12 (4 − D ) αn +1 , ,D n + − (4 − D ) α n +1 , ,D T . It is then easy to show that the exponent of H n within the average can be simplified to (4 − D ) nα n, ,D α n +1 , ,D − (4 − D ) α n +1 , ,D = (4 − D ) α n, ,D . (24)Taking (23) & (24) together & using the dimensionless notation of F n,m,D , we end up with D F (4 − D ) α n +1 , ,D n +1 , E T ≤ c n, D F (4 − D ) α n, ,D n, E T + c n, (cid:10) F , ,D (cid:11) T . (25)To begin an iteration procedure it is necessary to have a bound in the n = 2 case because n > D/ D = 2 ,
3. We repeat the argument above for n = 2 only ˙ H ≤ − νH + k ω k k ω k (26)We note that in D -dimensions k ω k ≤ c k∇ ω k a k ω k − a were a = D/
4. Thus we have ˙ H ≤ − ν (1 − D ) H + c ν − D − D H − D − D (27)5irstly we consider (cid:28) H α , ,D (cid:29) T = * H H β ! α , ,D H βα , ,D + T ≤ * H H β + α , ,D T * H βα , ,D − α , ,D + − α , ,D T (28)Thus we must choose β to make the exponent of H equal to unity : β = 2 α − , ,D − − D − D − − D (29)To see about the ratio we look at (26) & divide by H β to obtain * H H β + T ≤ c ν − − D (cid:28) H − D − D − β (cid:29) T = ν − − D h H i T . (30)Thus D F (4 − D ) α , ,D , E T < ∞ . Then, from (25), the result follows for all n ≥ D F (4 − D ) α n, ,D n, ,D E T ≤ c n, Re . (31)Formally this is the equivalent of the result in [40] when D = 3. The result for ≤ m ≤ ∞ The bound in (31) is true for m = 1 only. To move up to the m > D -dimensions k A k m ≤ c k∇ N A k a k A k − a (32)where 2 aN = D ( m − /m . Therefore, with A ≡ ∇ n u , we use the F n,m,D -notation. We also keepin mind the result (31) above to find D F (4 − D ) α n,m,D n,m,D E T ≤ c D F a (4 − D ) α n,m,D N + n, ,D F (4 − D ) α n,m,D (1 − a ) n, ,D E T = c *(cid:16) F (4 − D ) α N + n, ,D N + n, ,D (cid:17) aαn,m,DαN + n, ,D F (4 − D )(1 − a ) α n,m,D n, ,D + T (33) ≤ D F (4 − D ) α N + n, ,D N + n, E aαn,m,DαN + n, ,D T * F (4 − D ) αn,m,D (1 − a ) αN + n, ,DαN + n, ,D − aαn,m,D n, ,D + − aαn,m,DαN + n, ,D T . Using the fact that 2 aN = D ( m − /m & the expression for α n,m,D given in (16), we can thenshow that the exponent of F n, ,D in the time average satisfies(4 − D ) α n,m,D (1 − a ) α N + n, ,D α N + n, ,D − aα n,m,D = (4 − D ) α n, ,D . (34)Using (31), we see that both factors on the right hand side of (33) are bounded & give (17). (cid:4) eferences [1] Douady S., Couder Y. & Brachet M. E., Phys. Rev. Lett. , 983, (1991)[2] Tanaka M. & Kida S., Phys. Fluids
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