Spatio-temporal correlations in 3D homogeneous isotropic turbulence
Anastasiia Gorbunova, Guillaume Balarac, Léonie Canet, Gregory Eyink, Vincent Rossetto
SSpatio-temporal correlations in 3D homogeneous isotropic turbulence
A. Gorbunova,
1, 2
G. Balarac,
2, 3
L. Canet,
1, 3
G. Eyink,
4, 5 and V. Rossetto Université Grenoble Alpes, Centre National de la Recherche Scientifique, Laboratoire de Physique et Modélisation des MilieuxCondensés, 38000 Grenoble, France Université Grenoble Alpes, Centre National de la Recherche Scientifique, Laboratoire des Ecoulements Géophysiques etIndustriels, 38000 Grenoble, France Institut Universitaire de France, 1 rue Descartes, 75000 Paris, France Department of Applied Mathematics & Statistics, The Johns Hopkins University, Baltimore, MD, USA,21218 Department of Physics & Astronomy, The Johns Hopkins University, Baltimore, MD, USA,21218 (Dated: 8 February 2021)
We use Direct Numerical Simulations (DNS) of the forced Navier-Stokes equation for a 3-dimensional incompressiblefluid in order to test recent theoretical predictions. We study the two- and three-point spatio-temporal correlationfunctions of the velocity field in stationary, isotropic and homogeneous turbulence. We compare our numerical resultsto the predictions from the Functional Renormalization Group (FRG) which were obtained in the large wavenumberlimit. DNS are performed at various Reynolds numbers and the correlations are analyzed in different time regimesfocusing on the large wavenumbers. At small time delays, we find that the two-point correlation function decays asa Gaussian in the variable kt where k is the wavenumber and t the time delay. The three-point correlation function,determined from the time-dependent advection-velocity correlations, also follows a Gaussian decay at small t with thesame prefactor as the one of the two-point function. These behaviors are in precise agreement with the FRG results,and can be simply understood as a consequence of sweeping. At large time delays, the FRG predicts a crossover to anexponential in k t , which we were not able to resolve in our simulations. However, we analyze the two-point spatio-temporal correlations of the modulus of the velocity, and show that they exhibit this crossover from a Gaussian to anexponential decay, although we lack of a theoretical understanding in this case. This intriguing phenomenon calls forfurther theoretical investigation. I. INTRODUCTION
Characterizing the statistical properties of a turbulent flowis one of the main challenges to achieve a complete theo-retical understanding of turbulence. Space-time correlationsare at the heart of statistical theories of turbulence, and havebeen studied and modeled for many decades, both in the Eu-lerian and Lagrangian frameworks . One of the earliest in-sights was provided by Taylor’s celebrated analysis of singleparticle dispersion by an isotropic turbulent flow . The un-derstanding of the behavior of turbulent fluctuations both inspace and time is essential for many problems in fluid me-chanics where the multiscale temporal dynamics plays a keyrole, such as particle-laden turbulence, propagation of wavesin a turbulent medium or turbulence-generated noise in com-pressible flows . Space-time correlations are also central formany closure schemes, such as the direct-interaction Approx-imation (DIA) elaborated by Kraichnan , or the eddy dampedquasi-normal Markovian (EDQNM) approximation . The ac-curate description of the spatio-temporal correlations is cru-cial for developing time-accurate large-eddy simulation (LES)turbulence models, as well as for the analysis of experimentaldata, for example, to assess the validity and corrections to theTaylor’s frozen flow model used for time-to-space conversionof measurements.A fundamental ingredient to understand the temporal be-havior of turbulent flows in the Eulerian frame is the sweepingeffect, which was early identified in References 4, 6–8. Therandom sweeping effect results from the random advection ofsmall-scale velocities by the large-scale energy-containing ed- dies, even in the absence of mean flow. This random sweep-ing was anticipated to induce a Gaussian decay in the variable tk , where k is the wavenumber and t the time delay, of thetwo-point correlations of the Eulerian velocity field, based onsimplified models of advection . However, at the theoreti-cal level, the effect of sweeping also induces, in the originalformulation of DIA, a k − / decay of the energy spectrum inthe inertial range instead of the Kolmogorov k − / scaling.This led Kraichnan to a complete reformulation of his theoryusing Lagrangian space-time correlations instead of Eulerianones. The dependence of the two-point correlation functionin k t predicted from sweeping has been observed and con-firmed in numerous numerical simulations and also in ex-periments . A notable consequence of this dependence in theproduct kt is that the frequency energy spectrum of Eulerianvelocities exhibits a ω − / decay, instead of the ω − expectedfrom K41 scaling .The random sweeping hypothesis is also a part of the ellip-tic approximation that provides a model for spatio-temporalcorrelation in turbulent shear flows combining the decor-relation effect of the sweeping by large scales and the con-vection by the mean flow, and provides a correction to Tay-lor frozen-flow model. The elliptic approximation model hasbeen tested in numerical simulations and experimental mea-surements in Rayleigh-Bénard convection flows . In Ref. 18 amodel of spatio-temporal spectrum of turbulence is proposedin the presence of a mean flow departing from the Kraich-nan’s advection problem, which is consistent with the ellip-tical model. Fewer studies address multi-point correlations,although they are used as part of closure models . An ex- a r X i v : . [ phy s i c s . f l u - dyn ] F e b pression for the three-point correlation function in a specificwave-vector and time configuration was obtained within theDIA , and multi-point correlation functions were studied nu-merically in Ref. 20.Although the random sweeping effect is phenomenologi-cally known for a long time, and the models based on it pro-vide satisfactory descriptions, the theoretical justification ofthe hypothesis of random sweeping directly from the Navier-Stokes equation has remained a challenging task. The applica-tion of the renormalization group approach to turbulence de-veloped by Yakhot et al. led to the conclusion that the sweep-ing effect on space-time correlations must be small , whichis not in agreement with the ω − / Eulerian spectrum. Thisresult and its validity are discussed in the Ref. 11. In anotherwork the effect of the random sweeping was estimated withthe use of equations of band-passed velocity advected by alarge scale velocity. This work demonstrated that the randomsweeping plays a dominant role in the Navier-Stokes dynam-ics at small scales.Recently, a theoretical progress has been achieved usingFunctional Renormalization Group (FRG), which has yieldedthe general form of any multi-point correlation (and response)function in the limit of large wave-numbers. These expres-sions are established in the Eulerian frame, in a rigorous andsystematic way. For the two-point space-time correlations,the Gaussian decay in tk is recovered for small time delays t , while a crossover to a slower exponential decay in t is pre-dicted at large time delays. Similar results are obtained for anygeneric correlations involving an arbitrary number of space-time points. While the Gaussian regime is known to originatefrom sweeping, the exponential large-delay regime was notyet predicted. We show in this work that this behavior canalso be derived from the original Taylor and Kraichnan’s ar-guments, which provide a clear physical interpretation of thisresult.The aim of this work is to make precision tests of theFRG results using Direct Numerical Simulations (DNS) ofthe forced Navier-Stokes equation. We analyze the two-pointand three-point correlations and accurately confirm the FRGprediction in the small-time regime. Even though the long-time regime remains elusive in the simulations data due to theweakness of the signal amplitude in this regime and the lackof statistics, we unveil a very similar crossover from a Gaus-sian to an exponential decay in the correlations of the modulusof the velocity field. However, this observation lacks a theo-retical explanation so far.The paper is organized as follows. In Sec. II, we brieflyintroduce the functional and nonperturbative renormalizationgroup (FRG) framework and review the theoretical predic-tions stemming from it on the time dependence of multi-pointcorrelation functions. We also provide a heuristic argument al-lowing one to grasp the physical content of these results. Wepresent in Sec. III the results of our DNS analysis. We analyzethe small delay regime of the two-point correlation functionsin the Sec. III A and that of the three-point correlation func-tion in the Sec. III B. The temporal behavior of the two-pointcorrelation of the modulus of the velocity is discussed in theSec. III C. II. THEORETICAL FRAMEWORKA. Theoretical results from functional renormalization group
The FRG is a versatile method well-developed since theearly 1990’s and used in a wide range of applications, both inhigh-energy physics (quantum gravity and QCD), condensedmatter, quantum many-particle systems and statistical me-chanics, including disordered and nonequilibrium problems(see References 23–26 for reviews). This method has beenemployed in particular to study the incompressible 3D Navier-Stokes equation in several works . We here focus ona recent result concerning the spatio-temporal dependence ofmulti-point correlation functions of the turbulent velocity fieldin homogeneous, isotropic and stationary conditions. Thedetailed derivation of the theoretical results can be found inRef. 30; it relies on an expansion at large wavenumbers ofthe exact FRG flow equations. The field theory arising fromthe stochastically forced Navier-Stokes equation possesses ex-tended symmetries (in particular the time-dependent Galileansymmetry) which allow one to obtain the exact leading termof this expansion. We give below the ensuing expressions,before providing their intuitive physical interpretation in theSec. II B.We are first interested in the two-point correlation functionof the velocity expressed in the time-delay − wavevector mixedcoordinates ( t ,(cid:126) k ) , defined as C ( ) ( t ,(cid:126) k ) ≡ FT [ (cid:104) u i ( t ,(cid:126) r ) u i ( t + t ,(cid:126) r + (cid:126) r ) (cid:105) ]= (cid:68) ˆ u i ( t ,(cid:126) k ) ˆ u ∗ i ( t + t ,(cid:126) k ) (cid:69) (1)where FT denotes the spatial Fourier transform. Accordingto the FRG result, this function takes the following form forlarge wavenumbers k = | (cid:126) k | and small time delays: C ( ) S ( t ,(cid:126) k ) = C S ε / k − / exp (cid:110) − α S ( L / τ ) t k (cid:111) (2)and in the regime of large time delays: C ( ) L ( t ,(cid:126) k ) = C L ε / k − / exp (cid:8) − α L ( L / τ ) | t | k (cid:9) (3)with ε the energy dissipation rate, L the integral length scale, τ = ( L / ε ) / the eddy-turnover time at the integral scale,and α S , L and C S , L nonuniversal constants – the subscript S and L standing for ‘short time’ and ‘long time’ respectively. Thisexpression conveys that the velocity field decorrelates at smalltime delays as a Gaussian of the variable tk , whereas at largetime delays, the decay of the correlation function crosses overto an exponential in t . As mentioned in the introduction, theGaussian behavior at small t is well-known from experimen-tal data and numerical simulations and interpreted as a con-sequence of the random sweeping effect. It turns out that theexponential decay at large t can also be simply understood ina similar framework, as discussed in the Sec. II B.Let us comment on the domain of validity of these results.The factors in curly brackets in Eqs. (2) and (3) are exact inthe limit of large wavenumber k (cid:29) L − , which means that thecorrections to these terms are at most of order O ( k ) . We canquantify more precisely where this limit is reached using ourDNS data. In contrast, the terms in front of the exponentialin Eqs. (2) and (3) are not exact in these expressions, as theycan be corrected by higher-order contributions neglected inthe large wavenumber expansion. Otherwise stated, these ex-pressions do not account for intermittency corrections on theexponent 11/3, which merely corresponds to K41 scaling.The FRG theory yields a more general result: the spatio-temporal dependence of any multi-point correlation func-tions of the turbulent velocity field in the limit of largewavenumbers . We concentrate in this work on the three-point correlation function, defined as C ( ) αβγ ( t ,(cid:126) k , t ,(cid:126) k ) ≡ FT (cid:2)(cid:10) u α ( t + t ,(cid:126) r + (cid:126) r ) u β ( t + t ,(cid:126) r + (cid:126) r ) u γ ( t ,(cid:126) r ) (cid:11)(cid:3) = (cid:68) ˆ u α ( t + t ,(cid:126) k ) ˆ u β ( t + t ,(cid:126) k ) ˆ u ∗ γ ( t ,(cid:126) k + (cid:126) k ) (cid:69) (4)where translational invariances in space and time follow fromthe assumptions of homogeneity and stationarity. In the limitwhere all the wavenumbers k , k , | (cid:126) k + (cid:126) k | are large, the FRGcalculation leads to the following form at small time delays t and t C ( ) αβγ ( t ,(cid:126) k , t ,(cid:126) k ) = C ( ) αβγ ( ,(cid:126) k , ,(cid:126) k ) exp (cid:26) − α S ( L / τ ) (cid:12)(cid:12)(cid:12) (cid:126) k t + (cid:126) k t (cid:12)(cid:12)(cid:12) (cid:27) (5)with α S the same constant as in Eq. (2). Note that a similarexpression as Eq. (3) is also available for large time delays,but it is not considered here since it is out of reach of oursimulations. In this work, we consider the simplified case t = t = t , thus aiming at testing the theoretical form C ( ) αβγ ( t ,(cid:126) k , t ,(cid:126) k ) ∼ exp (cid:26) − α S ( L / τ ) (cid:12)(cid:12)(cid:12) (cid:126) k + (cid:126) k (cid:12)(cid:12)(cid:12) t (cid:27) . (6)One hence expects to observe that the three-point correlationfunctions at large wavenumbers are also Gaussian functionsof a variable | (cid:126) k + (cid:126) k | t for small time delays t , with the sameprefactor α S as in the two-point correlation functions.Let us emphasize that similar results hold for any n -pointcorrelation functions at large wavenumbers, and are valid forarbitrary time regimes, although for intermediate times theexpressions take a more complicated integral form . Theirstatus is generically the same as discussed above for the two-point correlations: The leading terms in the exponentials areexact in the limit of large wavenumbers, whereas the prefac-tors of these exponentials are not. Let us now give a simplephysical interpretation of these results. B. Physical interpretation
The short-time predictions for time-dependence of two-point velocity correlations (2) and of three-point correlations(5) were both given in an early analysis of Eulerian sweepingeffects by Kraichnan . As we show now, the novel prediction of long-time exponential decay (3) and similar long-time de-cay of general multi-point correlations were implicit in thatearlier analysis, but unrecognized at the time. Both short-timeand long-time decay regimes can be obtained from the follow-ing Lagrangian expression for the Eulerian velocity field u i ( t ,(cid:126) r ) = exp → [ − (cid:126) ξ ( t ,(cid:126) r | t ) · (cid:126) ∇ ] u i ( t ,(cid:126) r )+ (cid:82) tt ds exp → [ − (cid:126) ξ ( t ,(cid:126) r | s ) · (cid:126) ∇ ] (cid:2) ν ∇ u i ( s ,(cid:126) r ) − ∇ i p ( s ,(cid:126) r ) (cid:3) . (7)This is equation (7.7) in the paper of Kraichnan when spe-cialized to s = t there (and with a minor typo corrected in thefinal term). Here p ( t ,(cid:126) r ) is the pressure, (cid:126) ξ ( t ,(cid:126) r | s ) = (cid:126) r − (cid:126) X ( t ,(cid:126) r | s ) is the Lagrangian displacement vector, where dds (cid:126) X ( t ,(cid:126) r | s ) = (cid:126) u ( s ,(cid:126) X ( t ,(cid:126) r | s )) , (cid:126) X ( t ,(cid:126) r | t ) = (cid:126) r (8)defines the position (cid:126) X ( t ,(cid:126) r | s ) at time s of the Lagrangianfluid particle located at position (cid:126) r at time t . Finallyexp → [ − (cid:126) ξ ( t ,(cid:126) r | s ) · (cid:126) ∇ ] denotes an operator-ordered exponentialwith all gradients (cid:126) ∇ ordered to the right and thus not act-ing upon the (cid:126) r -dependence in (cid:126) ξ ( t ,(cid:126) r | s ) . The intuitive mean-ing of equation (7) is that it “states that the velocity field atlater times is the result of self-convection of the initial veloc-ity field, together with convection of all of the velocity incre-ments induced at later times by viscous and pressure forces” .The formula (7) yields both of the FRG predictions (2) and(3) when some plausible statistical and dynamical assump-tions are introduced. First, the displacement field (cid:126) ξ is ex-pected to vary more slowly in space and time than the gradi-ents of velocity (cid:126) u and of pressure p that result from the actionof the exponential operator. Slowness in time allows one tofactor out the exponential as u i ( t ,(cid:126) r ) = exp → [ − (cid:126) ξ ( t ,(cid:126) r | t ) · (cid:126) ∇ ] × (cid:110) u i ( t ,(cid:126) r ) + (cid:82) tt ds (cid:2) ν ∇ u i ( s ,(cid:126) r ) − ∇ i p ( s ,(cid:126) r ) (cid:3) (cid:111) . (9)and slowness in space allows the Fourier transform to be eval-uated as ˆ u i ( t ,(cid:126) k ) = exp [ − i (cid:126) ξ ( t ,(cid:126) r | t ) · (cid:126) k ] × (cid:110) ˆ u i ( t ,(cid:126) k ) − (cid:82) tt ds (cid:104) ν k ˆ u i ( s ,(cid:126) k ) + k i ˆ p ( s ,(cid:126) k ) (cid:105) (cid:111) . (10)The next assumption is that the displacement field (cid:126) ξ is almoststatistically independent of the Fourier-transformed velocityfields at the initial time t , so that by the definition (1) C ( ) ( t ,(cid:126) k ) = (cid:104) exp [ − i (cid:126) ξ ( t ,(cid:126) r | ) · (cid:126) k ] (cid:105) (cid:110) C ( ) ( ,(cid:126) k ) + O ( | t | ) (cid:111) . (11)Finally, since the Lagrangian displacement is dominated bythe largest scales of the turbulent flow, which have nearlyGaussian statistics, it is plausible that (cid:126) ξ is also an approxi-mately normal random field, so that C ( ) ( t ,(cid:126) k ) = exp (cid:20) − (cid:104)| (cid:126) ξ ( t ,(cid:126) r | ) | (cid:105) k (cid:21) (cid:110) C ( ) ( ,(cid:126) k ) + O ( | t | ) (cid:111) . (12)According to this argument, the 2-point velocity correla-tion undergoes a rapid decay in the time-difference t whicharises from an average over rapid oscillations in the phases ofFourier modes due to sweeping, or “convective dephasing” .The variance of the Lagrangian displacement in the expo-nent of (12) was the subject of a classical study by Taylor on1-particle turbulent dispersion. Exploiting the expression (cid:126) ξ ( t ,(cid:126) r | ) = (cid:90) t (cid:126) u ( t ,(cid:126) r | s ) ds , (13)two regimes were found: (cid:104)| (cid:126) ξ ( t ,(cid:126) r | ) | (cid:105) ∼ (cid:26) u RMS t | t | (cid:28) τ D | t | | t | (cid:29) τ (14)where the early-time regime corresponds to ballistic motionwith the rms velocity u RMS and the long-time regime corre-sponds to diffusion with a turbulent diffusivity D ∝ u RMS τ . Using the relation u RMS ∝ L / τ and the result (12) for the2-point velocity correlation, these two regimes of 1-particleturbulent dispersion correspond exactly to the short-time scal-ing (2) and the long-time scaling (3) predicted by FRG, with u RMS = α S ( L / τ ) and D = α L L / τ . To make more precisecontact with the FRG analysis, one can introduce the temporalFourier transform (cid:126) v ( t ,(cid:126) r ; ω ) = (cid:90) ds e i ω s (cid:126) u ( t ,(cid:126) r | s ) (15)and the corresponding (Lagrangian) frequency spectrum (cid:104) (cid:126) v ( ω ) · (cid:126) v ( ω (cid:48) ) (cid:105) = E ( ω ) δ ( ω + ω (cid:48) ) . It is then easy to see thatthe displacement variance in (14) can be written as (cid:104)| (cid:126) ξ ( t ,(cid:126) r | ) | (cid:105) = π (cid:90) d ω − cos ( ω t ) ω E ( ω ) (16)in terms of the velocity spectrum. This formula should becompared with the leading-order FRG flow equation (30) ofTarpin et al. obtained in the limit of large wavenumber | (cid:126) k | κ∂ κ ln C ( ) κ ( t ,(cid:126) k ) = | (cid:126) k | (cid:90) d ω − cos ( ω t ) ω J κ ( ω ) (17)where the common factor inside the two frequency integralsyields identical short-time and long-time power-law asymp-totics ( ∝ t and ∝ t , resp.) in both expressions (16) and (17).The above arguments can obviously be applied to gen-eral multi-point velocity correlations, yielding similar results.They provide an intuitive physical interpretation of the twoscaling regimes of the FRG results , with time-decay corre-sponding to a convective dephasing mechanism. In particu-lar, the long-time exponential decay is suggested to arise fromthe diffusive linear growth in the position variance of a La-grangian particle advected by homogeneous turbulence. Thislong-time exponential decay regime appears to be a novel pre-diction of the FRG approach. For example, it is quite distinctfrom the instantaneous exponential decay of the 2-point ve-locity correlator predicted by Rayleigh-Ritz analysis with a K - ε closure , which occurs on very short time-scale beforeconvective dephasing can act and which is interpreted as aneddy-viscosity effect. Needless to say, the FRG derivationof (2) and (3) is considerably more systematic and controlledthan the heuristic argument presented in this section. R λ N ν u RMS τ ∆ t ∆ T w N t K c L
40 64 10 − − − − − R λ - Taylor-scaleReynolds number, N - spatial grid resolution, ν - kinematic viscosity, u RMS - root mean square velocity, τ - eddy turnover time at the inte-gral scale, ∆ t - simulation time step, ∆ T w - width of a time window ofcorrelation observation, N t - number of recorded time windows, K c L -nondimensional cut-off wavenumber of the scale decomposition. III. RESULTS OF DIRECT NUMERICAL SIMULATIONS
We perform direct numerical simulations (DNS) of a sta-tionary 3D incompressible homogeneous and isotropic tur-bulent flow. The computation domain represents a cubeof size 2 π with periodic boundary conditions. We usefive values of the Taylor-scale Reynolds number: R λ = , , , ,
250 with corresponding spatial grid size N = , , , , (see Table I). The spatial reso-lution of all simulations fulfills the condition k max η (cid:39) . k max = N / η is the Kolmogorov length scale. The incom-pressible Navier-Stokes equation is solved numerically withthe use of a pseudospectral method in space and a secondorder Runge-Kutta scheme of time advancement. To achievea statistically stationary state, the velocity field is randomlyforced at large scales . We perform a de-aliasing with theuse of the polyhedral truncation method . A. Two-point spatio-temporal correlations at small timedelays
Once the simulations reach a statistically steady state, wecompute the velocity correlation functions with the followingmethod: At a chosen time t we store the spectral 3D vectorvelocity field in the memory. At the next iterations the up-dated velocity field at time t + i ∆ t is multiplied point-wiseby the velocity field at time t . Since the velocity field is sta-tistically isotropic, the two-point velocity correlation functionis computed by averaging over spherical spectral shells S n ofthickness ∆ k = (cid:126) k ∈ S n if n − < (cid:12)(cid:12) (cid:126) k (cid:12)(cid:12) < n , n = , .., N / t is redefined as the current time, and the ref-erence velocity field in the memory is updated. The resultingcorrelation function is averaged over time windows with dif-ferent reference times t , and the real part is taken:¯ C ( ) ( t , k ) = N t N t ∑ j = M n ∑ (cid:126) k ∈ S n Re (cid:104) ˆ u i ( t j ,(cid:126) k ) ˆ u ∗ i ( t j + t ,(cid:126) k ) (cid:105) (18) kL E ( k ) k / / R FIG. 1. Compensated spatial spectrum of the kinetic energy ob-tained from the averaged two-point spatio-temporal correlation func-tion C ( ) at zero time delay according to Eq. (19). ε is the energydissipation rate, L the integral length scale, and R λ the Reynoldsnumber at the Taylor microscale. where N t is the number of time windows in the simulation, M n is the number of modes in the spectral spherical shell S n , and k = n ∆ k , n ∈ Z . We hence obtain a numerical estimation of thetwo-point spatio-temporal correlation function C ( ) defined inEq. (1) with averaging in space and time.Note that at t =
0, the integration over a spherical shell inspectral space of the correlation function C ( ) in Eq. (1) givesthe spectrum of kinetic energy: E ( k ) = π k C ( ) ( t = , k ) = π C S ε / k − / (19)The compensated spatial spectra obtained from the averagedtwo-point spatio-temporal correlation function at zero timedelay are shown in the Fig. 1. The inertial regimes of thesespectra approximately conform to the Kolmogorov 5/3 power-law decay and are followed by the dissipation regime. Whilethere is no visible inertial range at the lowest R λ , it extendsover about one decade at the largest R λ . We first focus onthe behavior of the correlation function C ( ) at small time de-lay, and we normalize all data by the correlation function forcoincident times C ( ) ( t = , k ) .According to the theoretical expression Eq. (2), we expect aGaussian dependence in t at small time delays, which we pre-cisely observe in all our simulations. We show in the Fig. 2 anexample of the numerical results for C ( ) at various wavenum-bers for R λ =
90. All curves display a Gaussian behavior, thefits are analyzed in details below. Prior to this, let us commenton the scaling. When plotted as a function of the variable tk , all the curves collapse onto a single Gaussian, as expectedfrom the Eq. (2). This is illustrated in the bottom panel ofFig. 2. We emphasize that this tk scaling of the correlationfunction is different from the tk / scaling that one would ob-tain from dimensional considerations based on the standardassumptions of Kolmogorov’s theory of turbulence, taking asthe only relevant parameters the energy dissipation rate ε andthe wavenumber k . As explained in Sec. II B, the tk scal- In fact, Kolmogorov in his original 1941 paper emphasized that suchdimensional reasoning should apply to multi-time correlations only in aquasi-Lagrangian frame. t / C ( ) ( t , k ) / C ( ) ( t = , k ) k kL tkL / C ( ) ( t , k ) / C ( ) ( t = , k ) kL FIG. 2. Time dependence of the normalized two-point correlationfunction C ( ) ( t , k ) at different wavenumbers k in the simulation at R λ =
90. Upper panel: data from the numerical simulation denotedwith dots and its Gaussian fits denoted with continuous lines; bottompanel: the same data plotted as a function of the scaling variable tk , which results in the collapse of all the curves, as expected fromEq. (2). L is the integral length scale, τ is the large eddy-turnovertime scale. ing arises from dimensional analysis if the root-mean-squarevelocity u RMS ∼ L / τ is also included as a relevant parame-ter. This constitutes an explicit breaking of scale invariance,which originates in the random sweeping. u RMS is indeed thecharacteristic velocity scale of the random advection processof small-scale velocities by large vortices.We now turn to the analysis of the Gaussian fits. Thetime correlation curves at various wavenumbers and variousReynolds numbers are fitted using the nonlinear least-squaremethod (Levenberg–Marquardt algorithm), with the Gaussianfitting function: f s ( t ) = ce − ( t / τ s ) where τ s and c are the fittingparameters. Performing a nondimensionalization with param-eter L / τ ≈ u RMS renders the correlation function plots at var-ious Reynolds number comparable. The fitting range for allthe data sets corresponds to the range of nondimensional vari-able ( tkL / τ ) ∈ [ , . ] , within this range, all the correlationfunctions are accurately modelled by the Gaussian f s .The fitting parameter τ s is the characteristic time scale ofthe correlation function, its dependence on the wavenumber k is shown in Fig. 3 for various R λ . While for small wavenum-bers the dependence is not regular, at intermediate and largewavenumbers the decorrelation time clearly decays as k − .This result confirms that the collapse in the Fig. 2 occurs forthe tk -scaling. It is also in plain agreement with a similar anal-ysis performed in Ref. 13. kL s / k k R FIG. 3. Dependence of the decorrelation time τ s resulting fromthe Gaussian fit on the wavenumber in log-log scale for various R λ .Times on the vertical axis are normalized by the large eddy turnovertime scale τ , the wavenumber on the horizontal axis is normalizedby the integral length scale L . kL s R FIG. 4. Estimation of the theoretical parameter α S in the Eq. (2)from the results of the fit of numerical data. One can estimate the coefficient α S in the theoretical ex-pression Eq. (2) from the fits: α S ≈ ( τ / τ s kL ) . Plotting α S versus k as in Fig. 4 shows that the numerical estimation of α S reaches a plateau at large wavenumbers, which length in-creases with R λ .Whereas the Gaussian regime can be observed already at in- kL T ( k ) / m a x ( T ) R FIG. 5. Rate of the direct energy transfer from the forcing rangeto the wavenumber k normalized by the maximal value at variousReynolds numbers computed with the use of the shell-to-shell energytransfer method described in Ref. 36. termediate wavenumbers, the value of kL at which α S settlesto this plateau appears to be dependent on R λ . The deflectionof α S from the plateau value at the intermediate wavenumberscan be attributed to the effect of the forcing in the numericalscheme. This can be observed from the analysis of the directenergy transfers with the modes of the forcing range, as shownon the Fig. 5. The "ideal" numerical simulation would exhibita single peak of energy transfers close to the forcing rangeitself, indicating the presence of the local modal interactionsonly, when the smaller scales receive energy only through theturbulent energy cascade. However, Fig. 5 shows that directenergy transfers occur not only in the closest vicinity of theforcing range, but also at a significant level over a band ofwavenumbers, the width of which depends on R λ . This meansthat the wavenumbers from this band are subjected not only tothe local energy cascade, but also to nonlocal direct energytransfers from the forcing range. The occurrence of the non-local energy transfers in DNS can be a consequence of the ve-locity forcing concentrated in a narrow spectral band at largescales, as discussed in Ref. 37.This additional non-local interaction process slows downthe velocity decorrelation and results in lower values of α S .Matching the horizontal axes of the figures 4 and 5 shows thatthe parameter α S reaches a constant value at wavenumberswhere the direct energy exchanges with the forcing modes be-come negligible. We can draw from these observations thatthe ‘large wavenumber’ regime of the theory can be here iden-tified as the values of kL such that direct energy transfers fromthe forcing range are negligible.Let us summarize this part. The data obtained from theDNS accurately confirm the theoretical expression (2) for thetwo-point spatio-temporal correlations of the turbulent veloc-ity field for various scales at small time delays. In particu-lar, the numerical data show that the theoretical parameter α S reaches a plateau at large wavenumbers, in agreement with thetheoretical result. B. Three-point spatio-temporal correlations at small timedelays
In this part, we estimate the three-point spatio-temporalcorrelations C ( ) of the turbulent velocity field from theDNS data. The definition of C ( ) involves a product ofFourier transforms of the velocity field ˆ u ( t ,(cid:126) k ) at three differentwavevectors: C ( ) αβγ ( t ,(cid:126) k , t ,(cid:126) k ) ≡ (cid:68) ˆ u α ( t + t ,(cid:126) k ) ˆ u β ( t + t ,(cid:126) k ) ˆ u ∗ γ ( t ,(cid:126) k + (cid:126) k ) (cid:69) (20)In contrast with the two-point correlation function C ( ) , theproduct in the expression (20) is not local in (cid:126) k . When parallelcomputation and parallel memory distribution are used, theaccess to nonlocal quantities requires the implementation ofadditional communication operations between the processorsduring the simulation. This implies a great increase of com-putation time and memory. In order to avoid these additionalimplementation difficulties and computational costs, we studyand exploit a local three-point velocity quantity naturally aris-ing from the Navier-Stokes equation and already introducedin earlier works . Advection-velocity correlation function
The Navier-Stokes equation in the spectral space can bewritten as: ∂ t ˆ u (cid:96) ( t ,(cid:126) k ) = ˆ N (cid:96) ( t ,(cid:126) k ) − ν k ˆ u (cid:96) ( t ,(cid:126) k ) + ˆ f (cid:96) ( t ,(cid:126) k ) (21)where ˆ N (cid:96) ( t ,(cid:126) k ) = − ik n P (cid:96) m ∑ k (cid:48) ˆ u m ( t ,(cid:126) k (cid:48) ) ˆ u n ( t ,(cid:126) k − (cid:126) k (cid:48) ) is theFourier transform of the advection and pressure gradient termsof the Navier-Stokes equation, P i j = δ i j − k i k j / k is the pro-jection tensor and ˆ f (cid:96) is the spectral forcing. MultiplyingEq. (21) by the conjugated velocity ˆ u ∗ (cid:96) ( t ,(cid:126) k ) at a fixed time t and performing an ensemble average leads to the followingequation for the two-point spatio-temporal correlation func-tion C ( ) ( t ,(cid:126) k ) : ( ∂ t + ν k ) C ( ) ( t ,(cid:126) k ) = ˆ T ( t ,(cid:126) k ) + ˆ F ( t ,(cid:126) k ) (22)where ˆ T ( t ,(cid:126) k ) ≡ (cid:68) ˆ N i ( t + t ,(cid:126) k ) ˆ u ∗ i ( t ,(cid:126) k ) (cid:69) is the spatio-temporalcorrelation of the advection term and velocity, and ˆ F ( t ,(cid:126) k ) = (cid:68) ˆ f i ( t + t ,(cid:126) k ) ˆ u ∗ i ( t ,(cid:126) k ) (cid:69) is the spatio-temporal correlation of thespectral forcing and velocity. Note that if the time delay isset to zero ( t = E kin ( (cid:126) k ) = C ( ) ( ,(cid:126) k ) . This energy splits into ˆ T ( ,(cid:126) k ) (the average nonlinear energy transfer between modes) and ˆ F ( ,(cid:126) k ) (the average forcing power input, which is assumedto be zero beyond the forcing range at large scales).The advection-velocity correlation function ˆ T is a three-point statistical quantity, and its link with the three-point cor-relation function C ( ) becomes clear if one develops the non-linear term in the definition of ˆ T ( t ,(cid:126) k ) :ˆ T ( t ,(cid:126) k ) ≡ (cid:68) ˆ N (cid:96) ( t + t ,(cid:126) k ) ˆ u ∗ (cid:96) ( t ,(cid:126) k ) (cid:69) = − ik n P (cid:96) m ∑ k (cid:48) (cid:68) ˆ u m ( t + t ,(cid:126) k (cid:48) ) ˆ u n ( t + t ,(cid:126) k − (cid:126) k (cid:48) ) ˆ u ∗ (cid:96) ( t ,(cid:126) k ) (cid:69) = − ik n P (cid:96) m ∑ k (cid:48) C ( ) mn (cid:96) ( t ,(cid:126) k (cid:48) , t ,(cid:126) k − (cid:126) k (cid:48) ) (23)Hence, the correlation function ˆ T actually provides a linearcombination of three-point correlation functions. The theoret-ical prediction (6) suggests that this type of sum of C ( ) mustbe Gaussian (at small times and large wavenumbers). Thus, ifthe theoretical prediction is valid, one would expect that theappropriately computed correlation function ˆ T is also a Gaus-sian of the variable tk :ˆ T ( t ,(cid:126) k ) ∼ ∑ k (cid:48) C ( ) mn (cid:96) ( t ,(cid:126) k (cid:48) , t ,(cid:126) k − (cid:126) k (cid:48) ) ∼ exp (cid:110) − α S ( L / τ ) | (cid:126) k | t (cid:111) (24)Another useful property of the correlation function ˆ T is itslink with the two-point correlation function C ( ) . Considering a small time delay t , one can use the expression of the two-point correlation function C ( ) s ( t ,(cid:126) k ) of Eq. (2). Inserting thisresult into Eq. (22) leads to an explicit expression for the func-tion ˆ T at small time delays (and for wavenumbers outside theforcing range):ˆ T ( t ,(cid:126) k ) = ν k C ( ) ( ,(cid:126) k ) (cid:16) − α S L τ ν t (cid:17) exp (cid:110) − α S ( L / τ ) k t (cid:111) == ˆ D ( (cid:126) k ) (cid:18) − α S Re t τ (cid:19) exp (cid:110) − α S ( L / τ ) k t (cid:111) (25)where ˆ D ( (cid:126) k ) = ν k C ( ) ( ,(cid:126) k ) is the spectral dissipation rate and Re = U RMS L ν is the Reynolds number. Eq. (25) indicates thatthe function ˆ T is in general not symmetric with respect to theorigin of the t -axis, and that it can have a minimum and max-imum at non-zero time delay t .To sum up, the advection-velocity correlation function ˆ T isa local quantity in spectral space, as it implies the multipli-cation of the advection and velocity fields at the same wavevector (cid:126) k , and it is related to a sum of three-point nonlocal ve-locity correlation functions. The equivalence of the function ˆ T at zero time delay to the spectral energy transfer function andits link with the two-point spatio-temporal correlation func-tion Eq. (22) facilitate the testing of the numerical method andthe interpretation of the results in the following. Note that anequation similar to Eq. (22) is also used in the Direct Interac-tion Approximation scheme (DIA) , where a time dependenttriple statistical moment similar to ˆ T is introduced. Numerical method
In the numerical simulations, we compute the correlationfunction ˆ T ( t ,(cid:126) k ) by point-wise multiplication of the Fouriertransform of the nonlinear term ˆ N ( t + t ,(cid:126) k ) by the velocityfield ˆ u ∗ ( t ,(cid:126) k ) . This quantity is local in spectral space andthe computation does not require significantly more compu-tational resources.We use the method already described in the Sec. III A tocollect and average the data. However, note that in this caseit becomes necessary to take into account the sign of the timedelay. The advection-velocity correlation function ˆ T at nega-tive time delays can be computed just by switching the timeinstants of the fields in the following way:ˆ T ( t ,(cid:126) k ) = (cid:68) ˆ N i ( t + | t | ,(cid:126) k ) ˆ u ∗ i ( t ,(cid:126) k ) (cid:69) , t > (cid:68) ˆ N i ( t ,(cid:126) k ) ˆ u ∗ i ( t + | t | ,(cid:126) k ) (cid:69) , t < T ( t ,(cid:126) k ) at negative time de-lays during the simulation one only needs to store the spectraladvection field at one reference time t . Scale decomposition
Although the advection-velocity correlation function ˆ T pro-vides a three-point statistical quantity that can be easily ac-cessed in the numerical simulations, it contains a summa-tion coming from the convolution in the advection termEq. (23). Contributions from all possible wavevector triads { (cid:126) k (cid:48) ,(cid:126) k − (cid:126) k (cid:48) ,(cid:126) k } of any scale are thus summed up. However, theFRG prediction Eq. (5) is valid in the limit where all threewavenumbers are large. One hence needs to refine this sum inorder to eliminate contributions from the small wavenumbers.The simplest way to solve this issue is to perform a scaledecomposition of the velocity fields. We choose a thresh-old wavenumber K c , so that all wavevectors of smaller norm | (cid:126) k | < K c are considered as "large" scales and are denoted witha superscript L , whereas the modes with higher wavenumbersare considered as "small scales" and denoted with S . The ve-locity field is decomposed into small- and large-scale parts (cid:126) u = (cid:126) u L + (cid:126) u S . In the spectral domain the decomposition is per-formed by a simple box-filtering operation:ˆ u Li ( (cid:126) k , t ) = (cid:40) ˆ u i ( (cid:126) k , t ) , | (cid:126) k | < K c , | (cid:126) k | ≥ K c ˆ u Si ( (cid:126) k , t ) = (cid:40) , | (cid:126) k | < K c ˆ u i ( (cid:126) k , t ) , | (cid:126) k | ≥ K c (27)The velocity field scale decomposition leads to a decom-position of the advection-velocity correlation function ˆ T intofour terms (here written as an example for a wavevector (cid:126) k be-longing to the "small" scales):ˆ T ( (cid:126) k , t ) = (cid:2) ˆ T SSS + ˆ T SLS + ˆ T SSL + ˆ T SLL (cid:3) ( (cid:126) k , t ) (28)with ˆ T XYZ ( (cid:126) k , t , t ) = − [ ˆ u Xi ] ∗ ( (cid:126) k , t ) FT [ u Yj ∂ j u Zi ]( (cid:126) k , t + t ) where X , Y , Z stand for S or L .A similar decomposition at equal times has been used instudies of the energy transfer function (Ref. 36 and 38). Us-ing the terminology of Ref. 36 for energy transfers, the firstsuperscript of a decomposition term is related to the mode re-ceiving energy in a triadic interaction process (it is actually themode (cid:126) k for which the equation (28) is written setting t = T SSS gathers all triadic interac-tions where the three modes belong to the small scales. Theterm ˆ T SLS is related to the energy transfers between two smallscales mediated by large scale modes. Both energy transfersˆ T SSS and ˆ T SLS occur between small scales, and are thus sup-posed to be local in spectral space, so they form the turbulentenergy cascade. The terms ˆ T SSL and ˆ T SLL denote the directenergy transfers from large scale modes to small scale modes,thus non-local interactions that we expect to be small com-pared to the local interactions. Let us now focus on the all-small scale term ˆ T SSS , which corresponds to the limit of largewavenumbers on which the theoretical prediction relies.The cut-off wavenumber K c of the filter in the Eq. (27) ischosen in such a way that at k (cid:38) K c the direct energy trans-fer between small scale modes and those of the forcing range t / T ( t , k ) / D ( k ) T total k / K c t / T SSS k / K c FIG. 6. The advection-velocity spatio-temporal correlation functionˆ T ( t ,(cid:126) k ) versus time at selected values of wavenumbers k / K c : totalone (top panel), small scale one ˆ T SSS (bottom panel). The curves arenormalized by the spectral dissipation rate ˆ D ( k ) = ν k C ( ) ( , k ) . (shown in the Fig. 5) becomes negligible. We expect that thedynamics of the modes at k (cid:38) K c does not depend directly onthe forcing mechanism and we should observe an approachto the universal behavior predicted by the theory. The valueof K c L used for each simulation is provided in Table I. Thewavenumbers k (cid:38) K c approximately correspond to the rangeof validity of the theoretical prediction for the two-point cor-relation function at large wavenumbers, as discussed in theSec. III A. Results for the temporal correlations
The data presented in this section are obtained from thesame set of simulations used for the analysis of the two-pointcorrelation function at small time delays in Sec. III A and de-scribed in Table I. The results for the time dependence of ˆ T atdifferent wavenumbers k (cid:38) K c are shown in Fig. 6. One ob-serves that the total advection-velocity correlation function ˆ T (top panel of Fig. 6) is not symmetric with respect to the timeorigin and takes negative values, in qualitative agreement withthe form of the Eq. (25). However, the term ˆ T SSS , which onlycontains contributions from small scale modes to the correla-tion function ˆ T , significantly changes shape (bottom panel ofFig. 6).For the wavenumbers close to the cut-off wavenumber K c ,the curves are affected by the filter. To explain this, one shouldrecall that at zero time delay ˆ T SSS ( t = ,(cid:126) k ) is equal to the tkL / T ( t , k ) / T ( , k ) k / K c FIG. 7. The small scale advection-velocity correlation function ˆ T SSS versus tk in semilog scale at various wavenumbers, for R λ = N = t = local nonlinear energy transfer between small scales modes.At wavenumbers close to the filter cut-off K c , some spec-tral modes participating in the local energy transfers are sup-pressed by the filter. Thus, the modes close to the filter cut-offtransmit the energy to smaller scales, but they do not receiveenergy from the nullified larger scales, which results in a neg-ative energy balance. For the larger wavenumbers k (cid:38) K c ,the curves deform towards the expected Gaussian shape. Thisis further illustrated on Fig. 7, where the correlation functionˆ T SSS is plotted versus the scaling variable tk in semi logarith-mic scale, inducing a collapse of all the curves onto a singleGaussian. This is in plain agreement with the theoretical re-sult (5) for the three-point correlation function. This behavioris very similar to the one for the two-point correlation functionpresented in Fig. 2.We fit the curves obtained for the advection-correlationfunction ˆ T with a function of the form of Eq. (25): f ( t ) = c (cid:18) − t τ b (cid:19) e − ( t / τ a ) . (29)where τ a , τ b and c are the parameters.We find that both correlation functions ˆ T and ˆ T SSS accu-rately fit (29). Moreover, we verify that the fitting parameter τ a for both functions is proportional to k − , as displayed inFig. 8 (upper panel), and in agreement with Eq. (5). We es-timate from this parameter the value of the coefficient α S inEq. (6) as α S = τ / ( τ a k L ) . The result is shown in Fig. 8.At sufficiently large wavenumbers, the values of α S extractedfrom the small scale function ˆ T SSS and from the total ˆ T arecomparable. They also match with the value obtained from C ( ) , as predicted by the theory. The small discrepancy visi-ble between the values of α S from C ( ) and from ˆ T SSS couldbe attributed to a loss of accuracy due to the decomposition:The magnitude of the filtered signal is much weaker, so it ismore sensitive to the noise due to numerical errors.Lastly, we examine the role of the parameter τ b in the fit-ting function (29). To do this, we can refer to the Eq. (25) forthe total correlation function ˆ T , which was obtained from theNavier-Stokes equation assuming that the two-point correla-tion function C ( ) has a Gaussian shape. Therefore, the fitting kL a / k k T SSS T total C (2) R R kL s T SSS C (2) T total R R FIG. 8. Numerical estimation of the parameter τ a (upper panel) and α S (bottom panel) obtained from the small scale advection-velocitycorrelation ˆ T SSS (continuous lines), compared with the result for thetwo-point correlation function C ( ) from Fig. 4 (dash-dotted lines).Both estimations converge to a similar value, as expected from thetheory. The result of the fitting for the total advection-velocity corre-lation ˆ T is also indicated with dotted lines for completeness. kL s R e b / T SSS T total R FIG. 9. Dependence of the parameter τ b of the fitting functionEq. (29) on the wavenumber for the small scale advection-velocitycorrelation ˆ T SSS (continuous) and for the total one ˆ T (dotted lines).The values are normalized by 2 α s Re / τ to enable comparison withthe Eq. (25). parameter time scale τ b can be estimated as: τ b = τ α s Re (30)In the Fig. 9, we show the dependence of the nondimensionalparameter 2 α S Re τ b / τ on the wavenumber. The values of α s for the normalization are taken from the fit of the two-pointcorrelation function C ( ) . As expected, for the total advection-0correlation function ˆ T the values from all simulations are inthe vicinity of unity independently from the wavenumber,which is consistent with the Eq. (30). Besides, one can ob-serve from the Fig. 9 that for ˆ T SSS the non-dimensionalizedparameter τ b is at least one order of magnitude larger than forthe total ˆ T . This means that for ˆ T SSS the time scale of the lin-ear part τ b of the function (29) becomes much larger than thetime scale τ a of the Gaussian part. In other words, the Gaus-sian part decays fast and the function already approaches zerobefore the slower linear part comes into play, which results inthe Gaussian-like shapes of ˆ T SSS in the figures 6 and 7. On thecontrary, for the total function ˆ T the time scale τ b is smallerthan τ a and the shape of the total ˆ T is dominated by the linearpart at short times, resulting in a non-symmetric shape.An interpretation of this result can be proposed based onthe identification of the advection-velocity correlation func-tion ˆ T at t = T SLS in the decomposition (28)). The same conclusioncan be found in Ref. 36, 39, and 40. However, as discussedin Ref. 41, although these triads have significant individualcontributions to energy transfer, they are much less numer-ous than the fully local triads formed of small-scale modes(the term ˆ T SSS in the decomposition), because there are fewerlarge-scale modes. In the limit of large Reynolds numbers,the fully local triads become numerous and dominate in theturbulent energy cascade.In addition, the detailed analysis of the contributions in thedecomposition (28) shows that the nonsymmetric behavior intime of the total correlation ˆ T is also determined by the con-tribution of ˆ T SLS . The occurence of the maximal and mini-mal values of the advection-velocity correlation ˆ T at non-zerotime delays (see the top panel of the Fig. 6) implies that thereis some coherence between two small scale vortices simul-taneously advected by a large scale, slowly varying, vortex.The origin of this coherence can be through an alignment ofturbulent stress and large scale strain rate. The dynamics ofthe alignment between time-delayed filtered strain rate and thestress tensors, as well as its link with the energy flux betweenscales, has been recently analyzed in the Ref. 42, where thealignment also displays an asymmetrical behavior in time andis peaked at scale dependent time delays. As the energy flux,which could be expressed as a product of stress and strain rate,also represents a triple statistical moment of the velocity field,it would be natural to expect that it exhibits a temporal behav-ior similar to the advection-velocity correlation ˆ T .In the case of the purely small scale correlation functionˆ T SSS , the characteristic time scales of all modes in the triadare comparable, and the mediator mode cannot impose anycoherence on the interacting modes, as all three modes decor-relate faster before any alignment could occur. This results inthe symmetric, close to Gaussian form of the small scale cor-relation functions ˆ T SSS . Note that all the three modes in ˆ T SSS are still transported simultaneously by the random large scalevelocity field. This mechanism is the same random sweepingeffect that is reponsible for the Gaussian time dependence of ˆ T SSS and of C ( ) .To conclude, the spatio-temporal correlation between thevelocity and advection fields constitutes a triple statistical mo-ment easily accessible in numerical simulations. The applica-tion of the scale decomposition to this correlation is a neces-sary refinement to approach the regime of large wavenumbersof the theoretical result and gives an insight into the statis-tics of the three-point spatio-temporal correlation functions.We observe a Gaussian with the same time and wavenumberdependence as in the theoretical prediction. Moreover, thisanalysis provides also a nontrivial validation of the theoreticalresult, which predicts the parameter α S to be the same for thetwo-point and the three-point correlations. C. Two-point spatio-temporal correlation of the modulus ofthe velocity
The numerical analysis of the two-point correlation func-tion at large time delays represents a more challenging task,as the values of the correlation functions become very lowand are drowned into noise and numerical errors. Moreover,it requires larger observation times, and thus longer simula-tions and more computational resources. We did not succeedin resolving the large time regime from our numerical datafor the two-point correlation function, due to both the lack ofstatistics in the time averaging and the weakness of the signal,comparable with numerical errors.However, in order to increase the amplitude of the sig-nal, we study the correlation function of the velocity modulusrather than the real part of the complex correlation function.For this quantity, the large time regime indeed turns out tobe observable, as we now report. We thus introduce the con-nected two-point correlation function of velocity modulus inspectral space¯ C ( ) n ( t , k ) = (cid:68) (cid:107) ˆ (cid:126) u ( t ,(cid:126) k ) (cid:107) (cid:107) ˆ (cid:126) u ( t + t ,(cid:126) k ) (cid:107) (cid:69) − (cid:68) (cid:107) ˆ (cid:126) u ( t ,(cid:126) k ) (cid:107) (cid:69) (cid:68) (cid:107) ˆ (cid:126) u ( t + t ,(cid:126) k ) (cid:107) (cid:69) (31)with spatial and time averaging identical to Eq. (18) (cid:104) ... (cid:105) = N t M n ∑ N t j = ∑ (cid:126) k ∈ S n ( ... ) . This correlation function was computedin another set of simulations with larger width of the time win-dow.An example of the correlation function computed accord-ing to Eq. (31) for R λ =
60 is presented in Fig. 10. Similarlyto the two-point correlation studied in Sec. III A, one observesat short time delays the Gaussian decay in time and the curvesat different wavenumbers collapse in the tk -scaling. However,Fig. 10 reveals a crossover to another regime at larger timedelays: a slower decorrelation in time, that can visually beestimated as exponential. The curves at various wavenum-bers do not collapse anymore in the horizontal scaling tk , andthe slope of the correlation function appears to be steeper forlarger wavenumbers.We compute the normalized time derivative of ¯ C ( ) n ( t , k ) inorder to study the transition between the two time regimes of1 tkL / C ( ) ( t , k ) / C ( ) ( t = , k ) kL FIG. 10. Time dependence of the normalized two-point correla-tion function of the velocity norms ¯ C ( ) n ( t , k ) at R λ =
60 for differ-ent wavenumbers k in semi-logarithmic scaling. The numerical dataare denoted with dots, the exponential fit is denoted with the dashedlines. tkL / D ( t , k ) Gaussian
ExponentialCrossover kL FIG. 11. The normalized time derivative D defined in Eq. (32)calculated numerically with the data from the simulation at R λ =
60. The linear part of D , highlighted by green shades, correspondsto the Gaussian decay at small time delays, and the approximatelyconstant part of D , highlighted by purple shades, corresponds to theexponential time correlation at large time delays. the correlation function: D ( t , k ) = ∂ t ¯ C ( ) n ( t , k ) ¯ C ( ) n ( t , k ) . (32)If the correlation function ¯ C ( ) n is a Gaussian, the time deriva-tive D is simply a line with a slope equal to − / τ s , and if thecorrelation function is an exponential function, the function D becomes a constant. The derivative D is represented inFig. 11 for R λ =
60. At small time delays, D is a linear func-tion with a negative slope. It then displays a non-monotonoustransition before approximately reaching a constant value atlarge time delays. We can define the crossover time delay t as the location of the minimum of the derivative D . Thiscrossover time at different Reynolds numbers is shown in theFig. 12. It depends on the wavenumber as τ c ∼ k − . Wechecked that this k − behavior does not depend on the precisedefinition chosen for the crossover time.Let us emphasize that the correlation function of the veloc-ity norms introduced in Eq. (31) is not related in any simple kL c / k R FIG. 12. Crossover time for the two-point correlations of the veloc-ity norms ¯ C ( ) n between the small time and large time regimes as afunction of the wavenumber k , estimated from the minimum of D . way to the standard real part of the correlation function (1)computed theoretically in the FRG approach. Moreover, asthe phases play no role for these correlations, the sweepingargument proposed in Sec. II B cannot explain this behavior.The decorrelation must ensue a priori from another physicalmechanism, yet to be identified. However, the results of thenumerical simulation show that the correlation of the velocitymodulus and the real part of the complex velocity correlationfunction at small time delays (the Gaussian decay) are simi-lar, and exhibit close values for the characteristic decorrelationtime. In addition, at large time delays the correlations of thevelocity modulus demonstrate a crossover to an exponentialdecay in time, analogous to the one expected for the real partof the correlation function.While a complete understanding of these intriguing obser-vations is lacking, some insight into the mechanisms at playin the regime of small time delays can be obtained from theexpression valid to first-order in t (cid:126) u ( t + t ,(cid:126) r ) = (cid:126) u ( t ,(cid:126) r − (cid:126) u ( t ,(cid:126) r ) t ) − (cid:126) ∇ p ( t ,(cid:126) r ) t + O ( t ) , where the second term which is required to enforce incom-pressibility involves the pressure satisfying the Poisson equa-tion −(cid:52) p ( t ,(cid:126) r ) = tr [( (cid:126) ∇ (cid:126) u ( t ,(cid:126) r )) ] . If one assumes that the (cid:126) r -dependence can be ignored for the inner velocity field multi-plied by t , then this expression simplifies to (cid:126) u ( t + t ,(cid:126) r ) = (cid:126) u ( t ,(cid:126) r − (cid:126) u ( t ,(cid:126) ) t ) + O ( t ) and one obtains ˆ (cid:126) u ( t + t ,(cid:126) k ) = e − it (cid:126) k · (cid:126) u ( t ,(cid:126) ) t ˆ (cid:126) u ( t ,(cid:126) k ) , so thatsweeping is represented by a pure change of phase of theFourier mode. However, it is clearly inconsistent to neglectthe (cid:126) r -dependence of (cid:126) u ( t ,(cid:126) r ) in one instance and not in theother. Thus, the effects observed in Fig. 11 must presum-ably be due to the spatial inhomogeneity of sweeping andthe associated long-range pressure forces arising from incom-pressibility, which decorrelate the moduli of the Fourier ve-locity amplitudes. If one furthermore plausibly assumes thatthe correlation ¯ C ( ) n ( t , k ) is a maximum at t =
0, then ana-lyticity in t requires in the regime of small time delays that¯ C ( ) n ( t , k ) . = ¯ C ( ) n ( , k )( − t / τ k ) for some parameter τ k with2units of time and then immediately D ( t , k ) . = − t τ k , as observed in Fig 11. These considerations do not explain thedetailed observations, neither the k -dependence of τ k nor theexponential decay in the regime of long time lags, but they dosuggest some possible relevant physics for future theoreticaland empirical exploration.Interestingly, a very similar behavior has been observed inthe air jet experiments described in Ref. 15. In these exper-iments, the temporal decay of the two-point correlation func-tion of the amplitude of the vorticity field is measured, and itdisplays a crossover from a tk Gaussian decay to a slower ex-ponential one. The crossover time between these two regimesis found to scale as k − as observed in our simulations . IV. SUMMARY AND PERSPECTIVE
In this paper, we use DNS to study the spatio-temporal de-pendence of two-point and three-point correlations of the ve-locity field in stationary, homogeneous and isotropic turbu-lence. The motivation underlying this work is to test a theo-retical result obtained within the FRG framework, which givesthe exact leading term at large wavenumbers of the spatio-temporal dependence of any n -point correlation function ofthe velocity field . This result establishes that the two-pointcorrelation function decays as a Gaussian in the variable tk (or | ∑ i t i (cid:126) k i | for a n -point correlation) at small time delays t i ,while at large time delays, the decorrelation slows down to asimple exponential in t i . While these results can in fact be in-terpreted quite simply by extending the analysis of the randomsweeping effect, following the original arguments by Kraich-nan, they are endowed through the FRG calculation with arigorous and very general expression. In particular, these ex-pressions show that for any fixed time delays, the correlationfunction as a function of any wavenumber is always Gaussian.Furthermore, the multiplicative constant in the exponential isthe same for all the Gaussian decays and all the exponentialdecays as well, independently from the order n .In the small time regime that we could access via DNS ofthe two-point and three-point correlation functions with anequal time delay, our numerical data confirm the theoreticalprediction with great accuracy. In particular, we verify thatthe prefactors of time are proportional to k (or | (cid:126) k + (cid:126) k | ) andthe numerical constants at small time delays are indeed equalfor the two-point and three-point correlations. Furthermore,our analysis provides a deeper insight into the range of valid-ity of the theory. All the theoretical results discussed here arederived under the assumption that all the wavenumbers (andtheir partial sums) are large. From the DNS data, we estimatethe range of k where this condition is fulfilled and show thatit corresponds to the range where the direct energy transferfrom the forcing modes is negligible. For the three-point cor-relations, we show that once the small wavenumbers k < K c are removed through an appropriate decomposition, the theo-retical prediction is precisely recovered. Our analysis of the correlation function of the modulus ofthe velocity shows a very similar behavior as the one expectedfor the velocity itself, although the theoretical results do notapply in this case. It would be desirable to understand themain physical mechanism at play for the decorrelation of themodulus, which cannot be attributed to convective dephasing.This calls for further theoretical developments. On the nu-merical side, it would be interesting to extend this analysisto higher-order correlations, and for more general configura-tions in time (since our approach restricts to equal and shorttime delays for the three-point correlations). A particularlychallenging task is the access to the long-time regime. Thiswould of course require important computing resources. Theunderstanding of the temporal correlations for passive scalarsin turbulent flows is also very important for many applica-tions. This is work in progress. ACKNOWLEDGMENTS
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