Developing Homogeneous Isotropic Turbulence
aa r X i v : . [ n li n . C D ] N ov Developing Homogeneous Isotropic Turbulence
Wouter J.T. Bos, ∗ Colm Connaughton, † and Fabien Godeferd ‡ Laboratoire de M´ecanique des Fluides et d’Acoustique,CNRS UMR 5509, ´Ecole Centrale de Lyon, France, Universit´e de Lyon Mathematics Institute and Centre for Complexity Science,University of Warwick, Coventry CV4 7AL, UK
We investigate the self-similar evolution of the transient energy spectrum which precedes theestablishment of the Kolmogorov spectrum in homogeneous isotropic turbulence in three dimensionsusing the EDQNM closure model. The transient evolution exhibits self-similarity of the second kindand has a non-trivial dynamical scaling exponent which results in the transient spectrum havinga scaling which is steeper than the Kolmogorov k − / spectrum. Attempts to detect a similarphenomenon in DNS data are inconclusive owing to the limited range of scales available. PACS numbers: 47.27.Gs,47.27.eb
INTRODUCTION TO TRANSIENT SPECTRA INTURBULENCE
Although a large amount of work has been done char-acterising the properties of the Kolmogorov k − / spec-trum of three dimensional turbulence, rather less atten-tion has been paid to the transient evolution which leadsto its establishment. This transient evolution is essen-tially non-dissipative since it describes the cascade pro-cess before it reaches the dissipation scale. Part of thereason why this process has attracted relatively little at-tention is that this transient evolution is very fast, typ-ically taking place within a single large eddy turnovertime. It is thus of little relevance to the developed tur-bulence regime of interest in many applications. Never-theless, one may ask whether this developing turbulence,as one might call this transient regime, displays any in-teresting scaling properties. Previous studies of the de-veloping regime in weak magnetohydrodynamic (MHD)turbulence [1] suggest that this transient regime mighthave non-trivial scaling properties: in this case it wasfound that the establishment of the Kolmogorov spec-trum is preceded by a transient spectrum which is steeperthan the Kolmogorov spectrum. This latter is, in turn,set up from right to left in wavenumber space only afterthe transient spectrum has reached the end of the inertialrange and started to produce dissipation.Subsequent studies suggest that this behaviour, in par-ticular the occurence of a non-trivial dynamical scalingexponent, is typical for turbulent cascades which are fi-nite capacity - meaning that the stationary spectrum canonly contain a finite amount of energy. The Kolmogorovspectrum of three dimensional turbulence is in the classof finite capacity systems, as we shall see below. Thereare, however, examples of other turbulent cascades whichare not - infinite capacity cascades are common in waveturbulence for example [2]. In addition to the MHD cas-cade mentioned above, examples of non-trivial scalingexponents in finite capacity cascades have been found indeveloping wave turbulence [3, 4], Bose-Einstein conden- sation [5, 6] and cluster-cluster aggregation [7]. Althougha possible heuristic explanation of the transient scaling inthe MHD context has been put forward in [8], this heuris-tic relies heavily on the anisotropy of the MHD cascadeand does not seem readily generalisable to other contexts.In general, the transient exponent is associated with aself-similarity problem of the second kind [9]. From amathematical point of view, its solution requires solvinga nonlinear equation in which the exponent appears asa parameter which is fixed by requiring consistency withboundary conditions. It is probably unrealistic to expectthat there is a general heuristic argument capable of re-solving such a mathematically challenging problem. Thisis not to say, however, that particular cases may not beamenable to heuristic arguments which take into accountthe underlying physical mechanisms driving the transientevolution rather than taking a purely mathematical pointof view.This issue has not yet been studied in the contextof homogeneous isotropic turbulence. Investigations oftransient spectra in the classical Leith closure model[10] have suggested, however, that the transient spec-trum of developing homogeneous isotropic turbulence isindeed non-trivially steeper than k − / [11]. In thiswork, we investigate the transient evolution of homo-geneous isotropic turbulence using the Eddy-DampedQuasi-Normal Markovian (EDQNM) closure model anddirect numerical simulation (DNS) of the Navier-Stokesequation.The transient spectrum might be expected to evolveself-similarly. In other words there is a typical wavenum-ber, s ( t ), which grows in time, and a dynamical scalingexponent, a , such that E k ( t ) ≍ c s ( t ) a F ( ξ ) where ξ = ks ( t ) . (1)Here ≍ denotes the scaling limit: k → ∞ , s ( t ) → ∞ with ξ fixed and c is an order unity constant which ensuresthat E k ( t ) has the correct physical dimensions, L T − .As we shall see, if the exponent, (5 + a ) /
2, is greaterthan 1, then the characteristic wavenumber diverges infinite time corresponding to a cascade which accelerates“explosively”. The direct cascade in 3D turbulence is ofthis type. The characteristic wavenumber is most easilydefined as a ratio of moments of the energy spectrum.Let us define M n ( t ) = Z ∞ k n E k ( t ) dk. (2)Eq. (1) suggests that the ratio M n +1 ( t ) /M n ( t ) is pro-portional to s ( t ) so that we may define a typical scaleintrinsically by s n ( t ) = M n +1 ( t ) M n ( t ) . (3)A little caution is required: we must take n sufficientlyhigh to ensure that the moments M n ( t ) used in defin-ing the typical scale, converge at zero. Otherwise, theintegral is dominated by the initial condition or forcingscale and does not capture the scaling behaviour. In thispaper, we mostly take n = 2, which turns out to be suf-ficient for our purposes, although we will compare thebehaviour obtained for n = 2 and n = 3 in our numer-ical simulations to assure the reader that the picture isconsistent.We would like to emphasise that the self-similar tran-sient dynamics which we study in this paper occur before the onset of dissipation. This is in contrast to the tran-sient dynamics describing the long time decay of homoge-neous isotropic turbulence after the onset of dissipationwhich are also believed to exhibit self-similarity. See [12]for recent experiments and a review of previous work.Some numerical results on the long time transient dy-namics of the EDQNM model can be found in [13]. Thepre-dissipation transient occurs very quickly. Indeed, aswe shall see, the typical scale, s ( t ), in this regime di-verges as s ( t ) ∼ ( t ∗ − t ) b where t ∗ is the time at whichthe onset of dissipation occurs (typically less than a sin-gle turnover time) and b <
0. For finite Reynolds num-ber, this singularity is regularised by the finiteness of thedissipation scale. The fact that, in the limit of infiniteReynolds number, the typical scale can grow by an arbi-trary amount in an arbitrarily small time interval as t ∗ is approached explains the statement often found in theliterature that the Kolmogorov spectrum is establishedquasi-instantaneously in the limit of large Reynolds num-ber. THE EDQNM MODEL
In this section we examine the self-similar solutions ofthe EDQNM model [14]. The structure of the EDQNMmodel can be obtained in different ways. One way isstarting from the Quasi-Normal assumption [15]. An-other way is by simplifying the Direct Interaction Ap-proximation [16] which was obtained by applying a renor-malized perturbation procedure to the Navier-Stokes equation. It is thus directly related to the Navier-Stokesequation, unlike the Leith model which was heuristicallyproposed to capture some features of the nonlinear trans-fer in isotropic turbulence. However, recent work [17]showed that the structure of the Leith model can be ob-tained by retaining a subset of triad interactions involv-ing elongated triads from closures like EDQNM. SinceEDQNM contains a wider variety of triad interactions,it is able to capture more details of the actual dynamicsof Navier-Stokes turbulence, as for example illustrated in[18]. At the same time it has the advantage over DNSthat much higher Reynolds numbers can be obtained.The EDQNM model closes the Lin-equation by ex-pressing the nonlinear triple correlations as a functionof the energy spectrum, ∂E k ∂t = T [ E k ] − ν k E k (4)where ν is the viscosity and T [ E k ] represents the nonlin-ear interactions between different scales. T [ E k ] has theformT [ E k ] = Z ∆ dk dk T k,k ,k k ( k k ) − E k ( k E k − k E k ) , (5)where ∆ signifies that the region of integration is overall values of k and k for which the triad ( k, k , k ) canform the sides of a triangle and the interaction strengthof each triad, T k,k ,k , is given by T k,k ,k = k k ( θ k θ k + θ k ) 1 − exp [ − ( µ k + µ k + µ k ) t ] µ k + µ k + µ k . (6)where θ , θ and θ are the cosines of the angles oppositeto k , k and k respectively in the triangle formed by thetriad ( k, k , k ) and µ k = ν k + λ sZ k p E p dp, (7)is the timescale associated with an eddy at wavenumber k , parameterised by the EDQNM parameter, λ , which ischosen equal to 0 .
49, [19]. For a full discussion of theorigins and properties of the EDQNM model see [20, 21].We concern ourselves here only with the inviscid limitwhere ν → ν = 0 then, in the scaling limit, the nonlinear trans-fer term becomes homogeneous of degree a in s andone finds dsdt = √ c s a (8) a F − ξ dFdξ = T [ F ] . (9)Scaling alone does not determine the dynamical exponent a . To determine a we may attempt to impose conserva-tion of energy on the scaling solution to obtain a secondconstraint which will fix a . Let us go down this path,at first naively, and then reconsider our argument morecarefully:1. Forced case
If we consider forced turbulence, then energy isinjected into the system in a narrow band of lowwavenumbers (which necessarily lie outside of theregion of applicability of the scaling solution). Thetotal energy grows linearly in time (remember weare interested in the dynamics before the onset ofdissipation): R ∞ E k ( t ) dk = ǫ t . If we use the scal-ing ansatz, Eq. (1), differentiate with respect totime and rearrange we obtain dsdt = ǫ (cid:20) ( a + 1) c Z ∞ F ( ξ ) dξ (cid:21) − s − a . (10)Taken together with Eq. (8) we are led to expect a = −
53 for forced turbulence. (11)The same conclusion would be reached by dimen-sional analysis of Eq. (1) under the assumption thatthe sole parameter available is the energy flux, ǫ ,(having physical dimension L T − ).2. Unforced case
In unforced turbulence, the energy is suppliedsolely through the initial condition which is takento be supported in a narrow band of low wavenum-bers (which, again, lie outside of the region of ap-plicability of the scaling solution). In extremis,one could take E k (0) = E δ ( k ). In the timewindow of interest ( before the onset of dissipa-tion), the total energy remains constant in time: R ∞ E k ( t ) dk = E . In this case, the scaling ansatz,Eq. (1), immediately yields: a = − E (having physical dimension L T − ).Note that upon subsitution into Eq. (8) both cases,Eq. (11) and Eq. (12), predict explosive growth of thecharacteristic wavenumber. This is in line with expecta-tions: it is widely believed that onset of dissipation inthe direct cascade is set by the large scale eddy turnovertime rather than the Reynolds number. This explosivegrowth is the key to understanding why these argumentsfor the value of the exponent a are flawed. In both caseswe assumed implicitly that the integral, R ∞ F ( ξ ) dξ doesnot diverge at its lower limit (it does not diverge at its upper limit since F ( ξ ) decays exponentially for large val-ues of ξ ). In order to study this issue, let us assume that F ( ξ ) has power law asymptotics near 0: F ( ξ ) ∼ A ξ − x as ξ →
0. (13)The exponent x is the spectral exponent of the tran-sient spectrum. In the case that s ( t ) diverges in finitetime, then this assumption of power law asymptotics for F ( ξ ) taken together with the scaling ansatz requires that x = − a . To choose otherwise would result in the largescale part of the energy spectrum either diverging or van-ishing at the onset of dissipation, neither of which is ac-ceptable. Both values of a = − / a = − R ∞ F ( ξ ) dξ rendering our argu-ments inconsistent. In the latter (unforced) case, this di-vergence is only logarithmic allowing us, perhaps, to hopethat it does not ruin the scaling argument completely. Weshall see from numerical measurements however, that theunforced case looks much more like the forced case (seeFigs.1 and 2) from the point of view of the scaling partof the spectrum and the exponent a = − second kind [9] so the dynamical scaling exponent, a , cannot therefore be determined from dimensional con-siderations and we must either try to solve Eq. (9) asa nonlinear eigenvalue problem and hope that it deter-mines a or return to trying to solve the original kineticequation. We do the latter, necessarily numerically. NUMERICAL MEASUREMENTS OFTRANSIENT SPECTRA
We performed simulations of the EDQNM model in theunforced case by integrating numerically Eq. (4), startingfrom an initial spectrum, E k (0) = Bk exp (cid:2) − ( k/k L ) (cid:3) , (14)with B chosen to normalize the energy to unity and k L = 0 .
01. The initial Taylor-scale-Reynolds number isof order 10 and the resolution is chosen 24 gridpoints perdecade, logarithmically spaced. A sequence of snapshotsof E k ( t ) before the viscous dissipation became apprecia-ble are shown in Fig. 1. To find the value of the dynam-ical exponent we should find the value of a which givesthe best data collapse under the scaling ansatz, Eq. (1).We defined the typical wavenumber, s ( t ), to be the ra-tio, M ( t ) /M ( t ) of the third to the second moments ofthe energy spectrum. To find the value of a giving thebest data collapse we used the minimization proceduresuggested in [22]. Only data with s ( t ) > . -12 -10 -8 -6 -4 -2 -3 -2 -1 E ( k ,t ) ks(t)=0.02s(t)=0.10s(t)=1.00s(t)=10.2s(t)=91.4s(t)=842 -6 -2 -4 -2 F ( ξ ) ξ Data rescaled by Eq.(1) x -1.88 FIG. 1: Time evolution of the energy spectrum, E ( k, t ), ofthe EDQNM model in the decay case. The main panel showssnapshots of E ( k, t ) at a succession of times. The inset showsthe data collapsed according to Eq. (1) with a = 1 .
88 and s ( t ) = M ( t ) /M ( t ). -12 -10 -8 -6 -4 -2 -3 -2 -1 E ( k ,t ) ks(t)=0.02s(t)=0.10s(t)=1.00s(t)=10.4s(t)=102s(t)=1005 -6 -2 -4 -2 F ( ξ ) ξ Data rescaled by Eq.(1) x -1.90 FIG. 2: Time evolution of the energy spectrum, E ( k, t ), ofthe EDQNM model in the forced case. The main panel showssnapshots of E ( k, t ) at a succession of times. The inset showsthe data collapsed according to Eq. (1) with a = 1 .
90 and s ( t ) = M ( t ) /M ( t ). decade of scales to forget the initial condition (which had s (0) ≈ . . ± .
04 where theerror estimate is the standard deviation of the distribu-tion of minima obtained by bootstrapping the minimiza-tion procedure on randomly selected subsets of the totalset of snapshots obtained from the numerical simulation.The data collapse thus obtained, shown in the inset ofFig. 1, is of high quality thereby supporting the scalingansatz. -3 -2 -1
0 500 1000 1500 2000 2500 3000 3500 s n ( t ) = M n + ( t ) / M n ( t ) t n=2n=3 s ( t ) / s ( t ) s (t) Scaling with higher moments
FIG. 3: Time evolution of the typical scale, s n ( t ), as definedby Eq.(3), of the EDQNM model in the forced case for dif-ferent choices of n . The main panel demonstrates that s ( t )and s ( t ) show the same qualitative behaviour with a finitetime singularity which is regularised by the onset of dissipa-tion. The inset illustrates that the ratio s ( t ) /s ( t ) is approx-imately constant as the typical scale (as measured by s ( t ))grows over several decades. Corresponding results for the case of forced turbulenceare presented in Fig. 2. The simulation was forced bykeeping the energy in the first two wavenumber shellsfixed in time. Performing the same analysis on the dataas for the unforced case, the optimal data collapse (shownin the inset of Fig. 2) occurs for a value of the dynamicalscaling exponent of 1 . ± .
05. This is consistent withthe value obtained for the unforced case.As a final set of checks on the consistency of our nu-merical simulations with the scaling hypothesis, Eq. (1),Fig. 3 shows the evolution in time (for the forced case)of the typical scale, s n ( t ) as defined in Eq. (3), for n = 2and n = 3. The fact that s ( t ) diverges in finite time isclearly evident from the main panel as is the fact that thequalitative behaviour is the same regardless of the choiceof n . More quantitatively, the inset of Fig. 3 shows thatthe ratio of the typical scales obtained by taking n = 3and n = 2 is approximately constant over a large rangeof values of s ( t ). The typical scales obtained for differ-ent values of n are therefore proportional to each otherin the scaling regime, s ( t ) → ∞ (the subsequent decreaseafter s ( t ) ≈
100 is due to the onset of dissipation). Theseresults justify our earlier comment that the scaling anal-ysis is insensitive to the choice of ratio of moments usedto define the typical scale provided these moments are ofsufficiently high order.Several remarks may be made. Firstly, although thereis no a-priori reason why this should be so, the transientexponents measured for the forced and unforced cases are -4 -3 -2 -1 E ( k ,t ) ks(t)=5.78s(t)=8.71s(t)=28.0s(t)=99.4s(t)=176 -2 -2 F ( ξ ) ξ Data rescaled by Eq.(1)
FIG. 4: Time evolution of the energy spectrum, E ( k, t ), in adirect numerical simulation of the decay case. The main panelshows snapshots of E ( k, t ) at a succession of times. The insetshows the data collapsed according Eq. (1) with a = 1 .
47 and s ( t ) = M ( t ) /M ( t ). the same within our estimated range of uncertainty. Thisis quite different from infinite capacity cascades whereconstraints imposed by conservation laws result in dif-ferent transient scaling exponents for the forced and un-forced cases [23]. Secondly, the measured transient ex-ponents are discernibly different from either of the naivevalues argued in Eq. (12) or Eq. (11). This confirms ourexpectation that the transient scaling is different fromKolmogorov. Thirdly, the fact that a is larger than 5 / steeper than the Kolmogorov spectrum. The latter is then setup from right to left in wavenumber space after the on-set of dissipation. This transition from the steeper spec-trum to k − / also evolves quasi-instantaneously in thesame sense as the pre-dissipation transient does. It veryquickly sets up the usual Kolmogorov spectrum over allscales once the onset of dissipation has occured. Thisspectrum then decays globally for all subsequent time asdetailed, for example, in [13]. The EDQNM equation istherefore no different to any of the other finite capac-ity cascades which have been investigated to date, all ofwhich showed this behaviour. The measured value of thedynamical exponent is remarkably close to the value of1 .
86 measured for the Leith model [11]. This is consis-tent with recent arguments of Clark et al. [17] suggestingthat the Leith model can be obtained from rational clo-sure models by keeping only a subset of the wavenumbertriads.Given that we expect this kind of transient behaviourto be generic, we close this study with an attempt to mea-sure the corresponding dynamical scaling in a DNS of thefull Navier-Stokes equation. A classical Fourier pseudo- spectral method is used to solve the semi-implicit form ofthe Navier-Stokes equations with tri-periodic boundaryconditions, at a resolution of 1024 [24]. Full de-aliasingis performed to remove spurious Fourier coefficients, timemarching is done with a third-order Adams-Bashforth ex-plicit scheme, while the viscous term is solved implicitly.The initial velocity conditions consist of a random gaus-sian field whose energy spectrum is of the form of (14) al-though with a peak at k L = 4 .
52 instead of 0 .
01. The re-sults are shown in Fig. 4. Proceeding as described above,we obtained a = 1 . ± .
24. The result is therefore incon-clusive as one might expect given the very short scalingrange available in DNS data (as compared to numericalsolutions of the EDQNM equation).
CONCLUSION
To summarise, we have investigated the self-similarevolution of transient spectra in three dimensional turbu-lence using numerical solutions of the EDQNM equationand full DNS data. These transients develop before theonset of dissipation and lead to the establishment of theKolmogorov spectrum. We argued that the self-similarityis of the second kind allowing the transient scaling tobe anomalous in the sense that it cannot be determinedfrom dimensional considerations. This is supported bynumerical data for the EDQNM equation which gave atransient exponent of 1 .
88 compared to the Kolmogorovvalue of 5 /
3. Corresponding measurements for the DNSdata were inconclusive owing to the relatively short scal-ing range available. Nevertheless we would expect, basedon our results, that a DNS at sufficiently high Reynoldsnumber would see a steeper transient spectrum. Themost relevant message from this work for turbulence re-search is probably not the value of the transient exponentitself, since few applications care about this early stageregime. Rather it is the fact that such a non-Kolmogorovscaling exists in the first place which serves as a reminderthat, while the k − / scaling is quite robust when the en-ergy flux through the inertial range is constant, it is notthe sole scaling law consistent with the transfer of en-ergy to small scales in turbulence when the constant fluxrequirement is relaxed. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] S. Galtier, S. Nazarenko, A. Newell, and A. Pouquet, J.Plasma Phys. , 447 (2000).[2] A. Newell, S. Nazarenko, and L. Biven, Physica D , 520 (2001).[3] C. Connaughton, A. Newell, and Y. Pomeau, Physica D , 64 (2003). [4] C. Connaughton and A. C. Newell, Phys. Rev. E ,036303 (2010).[5] R. Lacaze, P. Lallemand, Y. Pomeau, and S. Rica, Phys-ica D , 779 (2001).[6] C. Connaughton and Y. Pomeau, Comptes RendusPhysique , 91 (2004).[7] M. Lee, J. Phys. A: Math. Gen. , 10219 (2001).[8] S. Galtier, A. Pouquet, and A. Mangeney, Phys. Plasmas , 092310 (2005).[9] G. Barenblatt, Scaling, self-similarity, and intermediateasymptotics (CUP, Cambridge, 1996).[10] C. E. Leith, Phys. Fluids , 1409 (1967).[11] C. Connaughton and S. Nazarenko, Phys. Rev. Lett. ,044501 (2004).[12] P. Lavoie, L. Djenidi, and R. A. Antonia, J. Fluid Mech. , 395 (2007).[13] M. Lesieur, O. M´etais, and P. Comte, Large Eddy Simu-lations of Turbulence (CUP, Cambridge, 2005).[14] S. Orszag, J. Fluid Mech. , 363 (1970).[15] A. Monin and A. Yaglom, Statistical fluid mechanics (MIT press, Cambridge, 1975).[16] R. Kraichnan, J. Fluid Mech. , 497 (1959).[17] T. Clark, R. Rubinstein, and J. Weinstock, J. Turbulence , 1 (2010).[18] W. J. T. Bos and J.-P. Bertoglio, Phys. Fluids , 071701(2006).[19] W. J. T. Bos and J.-P. Bertoglio, Phys. Fluids , 031706(2006).[20] M. Lesieur, Turbulence in Fluids (Springer, Heidelberg,2008).[21] P. Sagaut and C. Cambon,
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