Extracting turbulent spectral transfer from under-resolved velocity fields

Transcription

1 Extractng turbulent spectral transfer from under-resolved velocty felds Ru N, Greg A. Voth, and Ncholas T. Ouellette Ctaton: Physcs of Fluds (1994-present) 26, 117 (214); do: 1.163/ Vew onlne: Vew Table of Contents: Publshed by the AIP Publshng Artcles you may be nterested n Ansotropc energy transfers n quas-statc magnetohydrodynamc turbulence Phys. Plasmas 21, 1231 (214); 1.163/ Effect of shocklets on the velocty gradents n hghly compressble sotropc turbulence Phys. Fluds 23, 1213 (211); 1.163/ Rotatng helcal turbulence. I. Global evoluton and spectral behavor Phys. Fluds 22, 31 (21); 1.163/ Effects of vortex flaments on the velocty of tracers and heavy partcles n turbulence Phys. Fluds 18, 8172 (26); 1.163/ Energy spectrum n the enstrophy transfer range of two-dmensonal forced turbulence Phys. Fluds 13, 44 (21); 1.163/

2 PHYSICS OF FLUIDS 26, 117 (214) Extractng turbulent spectral transfer from under-resolved velocty felds Ru N, 1 Greg A. Voth, 2 and Ncholas T. Ouellette 1,a) 1 Department of Mechancal Engneerng and Materals Scence, Yale Unversty, New Haven, Connectcut 62, USA 2 Department of Physcs, Wesleyan Unversty, Mddletown, Connectcut 649, USA (Receved 17 August 214; accepted 8 October 214; publshed onlne 27 October 214) The strong nonlneartes n turbulent flows drve the transfer of energy and other quanttes between dfferent scales of moton. In three-dmensonal (3D) turbulence, ths transfer organzes nto the classc Rchardson Kolmogorov cascade of energy to small scales; n two-dmensonal (2D) turbulence, t leads to an nverse cascade of energy to large scales and a forward cascade of enstrophy to small scales. Drectly measurng ths spectral transfer s dffcult, partcularly n experments. Recently developed flterng technques allow spectral fluxes to be measured locally, but have been assumed to requre fnely resolved velocty felds that are typcally not avalable n 3D experments. Here we show, usng expermental data n 2D and the results of a 3D smulaton, that poorly resolved velocty felds can stll be used to extract nformaton about spectral transfer processes. Our results suggest new useful ways of analyzng data from turbulence experments wth lmted spatal resoluton. C 214 AIP Publshng LLC. [ I. INTRODUCTION In turbulent flow at hgh Reynolds numbers, the nonlnear term n the Naver Stokes equatons plays a key role n the dynamcs. Ths term ntroduces nteractons between wavenumber trads, whch spontaneously self-organze nto the Rchardson Kolmogorov energy cascade n three-dmensonal (3D) turbulence, drvng knetc energy from the large scales where t s njected nto the turbulence to small scales where t can be dsspated by vscous acton. 1 In two dmensonal (2D) turbulence, the addtonal conservaton of vortcty n the nvscd lmt produces a cascade of enstrophy from large to small scales whle reversng the drecton of the scale-to-scale flow of knetc energy. 2 4 Gross nformaton about the transfer of energy between scales s often obtaned va spectra or the velocty structure functons. 1, Nether of these quanttes, however, can easly provde the local spatotemporal detals of the spectral transfer processes at work n turbulence, as they requre averagng over tme and space. A recently developed alternatve approach s the so-called flter-space technque (FST) The essental dea of an FST s straghtforward. When one removes some degrees of freedom from a nonlnear equaton of moton by flterng (as s done n, for example, large-eddy smulaton), new terms appear that express the couplng between the degrees of freedom that are retaned and those that are suppressed. 8 By measurng these new terms, whch are related to the dfference between the fltered feld and the full feld and have prescrbed forms snce the equaton of moton s known, ths couplng can be drectly obtaned n a spatotemporally localzed way. When ths process s appled to, for example, the turbulent knetc energy, the couplng term descrbes the transfer of energy between scales. FSTs therefore allow the measurement of local spectral transfer propertes, and n prncple can be appled to any quantty n turbulence for whch the equaton of moton can be wrtten. a) Electronc mal: ncholas.ouellette@yale.edu /214/26(1)/117/9/$3. 26, C 214 AIP Publshng LLC

3 117-2 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) In prncple, the result of an FST gves the net couplng between all the retaned scales and all the suppressed scales; thus, t would seem that an accurate measurement of the spectral transfer would requre a very hghly resolved velocty feld as an nput. Such hgh resolutons are typcally unavalable n experments, partcularly for 3D, hghly turbulent flows. In a turbulent cascade, however, we expect that spectral dynamcs are approxmately local, 14, 1 so that only modes near a gven length scale contrbute to spectral transfer across that scale. If localty holds, then as long we lmt our queres to scales somewhat larger than the resoluton of the velocty feld, FSTs should produce accurate measurements of spectral transfer, just as other tools such as velocty structure functons are expected to do. The queston remans, however, as to exactly how much resoluton we need to measure the spectral transfer across a gven scale wth suffcent accuracy. Prevous studes usng FSTs have prmarly nvestgated the double cascades of energy and enstrophy n 2D turbulence, 1, 11, 16, 17 where hgh spatal resoluton s much easer to obtan n experments. Here, we show n both 2D (usng expermental data) and 3D (usng numercal smulaton) turbulence that ndeed the localty assumpton holds, and that FSTs can gve accurate measurements of spectral transfer even when the velocty feld s poorly resolved. We quantfy the accuracy of the spectral energy transfer rates as a functon of the spatal resoluton of the velocty feld, and show that accurate measurements can be made for scales larger than about 6 tmes the resoluton of the velocty feld. For 3D turbulence, we also compare our FST results wth measurements of the secondand thrd-order Euleran velocty structure functons, and show that the FST gves comparable f not better estmates of the energy transfer rate. Our results have useful mplcatons for 3D turbulence experments, where hghly resolved velocty felds are typcally not avalable; even n these cases, FSTs can be appled and can provde good measurements of spectral processes. We begn n Sec. II wth a bref descrpton of our data sets and of the mechancs of applyng an FST. In Sec. III, we present our results for both 2D and 3D turbulence, and compare them wth more tradtonal estmates of spectral transfer processes. Fnally, n Sec. IV, we summarze our results and brefly dscuss ther mplcatons. II. METHODS A. Data sets To generate (weak) quas-2d turbulence, we drve a thn layer of electrolytc flud wth electromagnetc Lorentz forces. The detals of our experments are reported elsewhere; 18 2 here, we only brefly descrbe the setup. The workng flud s a thn layer of salt water (16% NaCl by mass n deonzed water), measurng cm 3, that rests on a glass plate. Beneath the glass les an array of permanent neodymum-ron-boron magnets arranged n a checkerboard pattern of alternatng polarty. To remove the possblty of long-range couplng between the partcles we use to measure the flow (see below) due to surface-tenson effects, we float a second layer of pure water (also. cm deep) above the salt water. To force the flud, we run d.c. electrc current laterally through the salt-water layer between a par of copper bar electrodes. The combnaton of electrc current and magnetc feld produces a Lorentz force that drves flud moton and njects energy nto the system. Thus, the energy njecton scale s approxmately set by the spacng between magnets L m, equal to 2.4 cm n our apparatus. We defne a Reynolds number Re = UL m /ν based on ths length scale, the root-mean-square (rms) velocty U, and the knematc vscosty ν of the workng flud. For the data shown here, Re = 18, suffcent to keep the flow nearly two-dmensonal. 18 The flow feld was measured usng partcle trackng velocmetry (PTV) by seedng the flud wth around 3 fluorescent polystyrene partcles wth dameters of 1 μm. The partcles have a specfc gravty of 1., between that of the salt water and the fresh water; they are therefore gravtatonally pnned at the nterface between the two flud layers. Images of the partcles n the central cm 2 of the apparatus were captured at a rate of 6 frames/s usng an IDT MotonPro M camera wth a resoluton of pxels. The hgh seedng densty allows us to measure very hghly resolved velocty felds. To ensure that the measured felds are as 2D as possble and to remove nose, we projected the measured veloctes onto a bass of numercally constructed streamfuncton egenmodes. 18

4 117-3 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) The 3D turbulence data set we used was obtaned from drect numercal smulaton (DNS) results hosted at the Johns Hopkns Turbulence Database. 21 Ths DNS of forced sotropc turbulence was performed on a grd n a 2π 2π 2π doman; the Reynolds number based on the Taylor mcroscale s R λ = 433. Detals of the smulaton can be found n Ref. 21. B. Flter-space technques Implementng an FST requres suppressng small-scale varaton whle retanng the large scales. We accomplsh ths task by convolvng the quantty of nterest wth a spectral low-pass flter. As a concrete example, we defne the th component of the fltered velocty feld as u (r) (x) G(x x ; r)u (x )dx. (1) G here s the flter kernel, and the superscrpt (r) denotes a quantty fltered at a length scale r (where spatal scales smaller than r have been removed). In prncple, any functon that suppresses hghfrequency varaton wll work as a kernel for an FST. We choose to base our kernel on a Gaussan wth a Fourer-space wdth of 2π/r, followng prevous studes, 9 11 snce t has reasonable propertes n both real and Fourer space. Wth ths defnton, the equaton of moton for the fltered knetc energy E (r) = (1/2)[u (r) ] 2 s gven by 8 E (r) = J (r) ν u(r) u (r) (r), (2) t x x j x j where summaton s mpled over repeated ndces. The frst two terms on the rght-hand sde of ths equaton are analogous to terms n the equaton of moton for the full knetc energy, and represent spatal transport of knetc energy (the current J (r) ncludes contrbutons from flud advecton, pressure, and vscous dffuson) and drect vscous dsspaton. But the thrd term, gven by (r) = 1 [ ] [ (u u j ) (r) u (r) u (r) u (r) j + u ] (r) j, (3) 2 x j x s new. Ths term acts as a source or snk of knetc energy for the fltered velocty feld, and s expressed as the nner product of the fltered rate of stran wth a quantty that plays the role of a stress and that accounts for the momentum couplng between the fltered velocty feld and the small-scale part that we have removed. Wth our sgn conventon, (r) > mples energy flux from scales larger than r to those that are smaller, whle (r) < ndcates the opposte. C. Spatal resoluton To study the dependence of FSTs on the spatal resoluton of the velocty feld, we must be able to change ths resoluton. But for both the 2D and 3D data sets, the resoluton s fxed and we cannot ncrease t past ts gven value. What we can do, however, s artfcally reduce the resoluton by down-samplng the data sets. Here, we descrbe how we acheve ths down-samplng for both the expermental and numercal data. In a PTV experment lke ours, we obtan at each tme step velocty vectors for each tracked partcle. These partcles, however, are randomly seeded n the measurement area, and so the velocty feld s typcally sampled nonunformly. Ths nonunform samplng s undesrable when applyng Fourer flters as we do n the FST; thus, before flterng, we frst nterpolate our measured velocty vectors onto a regular grd. The spacng d of ths grd wll then set the effectve spatal resoluton of the velocty feld. Because the seedng densty s nhomogeneous, the fnest resoluton of the feld cannot be unquely defned; t should be, however, on the order of the typcal separaton between pars of partcles. To estmate the smallest d, we therefore calculated the dstrbuton of the nearest-neghbor dstance between partcle pars. More than 99.7% of pars were separated by dstances smaller than.11l m ; thus, we take d mn =.11L m to be the fnest spatal resoluton of the

5 117-4 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) velocty feld, and down-sample the feld by nterpolatng the measured velocty vectors onto grds wth d > d mn. For the 3D DNS results, the data are already grdded, and so nterpolaton s not necessary; down-samplng, however, s. The orgnal smulaton was performed wth grd ponts spaced by d = 2.1η, where η s the Kolmogorov length scale. To down-sample the resoluton of the DNS data, we smply removed every nth grd pont, coarsenng the grd by a factor of n. We chose ths method rather than an nterpolaton-based scheme because t mmcs to some degree how data would be obtaned from a poorly sampled 3D partcle-trackng experment, where veloctes are known only at dscrete ponts and the velocty at other ponts can typcally not be obtaned by nterpolaton. We dd, however, only choose values of n that were commensurate wth 124, so that the boundary grd ponts remaned. Had we not done so, removng a few grd ponts near the boundares of the perodc box would have the effect of applyng a hgh-pass flter at a scale close to the box sze, whch would ntroduce complcated alasng effects. Here, we present data for fve values of n:4,8, 16, 32, and 64. Note that the way we ncrease d n both data sets s smlar to applyng a low-pass box flter n real space, whch wll certanly nduce some rngng at other scales n Fourer space. Ths effect, however, usually affects hgh frequences (small scale) much more strongly than t does low frequences (large scales). In Sec. III below, we use relatve error to provde emprcal, a posteror evdence that ths effect s not sgnfcant for scales larger than a few multples of d. III. RESULTS AND DISCUSSION A. 2D weak turbulence In Fg. 1(a), we plot the spatally averaged spectral energy flux (r) for a sngle velocty feld as measured n our quas-2d experment, for a range of dfferent spatal resolutons d. The closed symbol at the begnnng of each curve shows the smallest flter scale, chosen here to be r = 4d. Although the curves vary somewhat n detal, the overall trend s smlar. In partcular, all the resolutons capture the strong negatve flux for r 1.8L m, ndcatve of the nverse energy transfer that s expected for 2D turbulence. 1. (a) 3 (b) Π (r) (mm 2 /s 3 ) d=.11l m d=.16l m d=.22l m d=.27l m d=.32l m d=.38l m d=.43l m d=.48l m ξ (%) r/l m r ref /d FIG. 1. (a) Spatally averaged energy flux (r) as a functon of length scale r (scaled by the magnet spacng L m ) for one nstantaneous velocty feld n the quas-2d experment. Each curve shows data for a dfferent spatal resoluton d. The frst data pont on each curve, marked wth a symbol, shows the result for a resoluton of d = r/4. (b) Relatve error ξ,defnedby Eq. (4), for the same data as n (a) as a functon of the relatve spatal resoluton r/d. Each data pont shows results averaged over the range 2 < r/l m < 3, gvng an estmate of the value of ξ at r ref = 2.L m (see the text), and the error bars show the standard error n ξ.

6 117- N, Voth, and Ouellette Phys. Fluds 26, 117 (214) We can quantfy the effects of the lmted spatal resoluton by defnng a relatve error ξ = (r) (d) (r) (d mn ), (4) (r) (d mn ) where (r) (d mn ) s the mean energy flux for the velocty feld wth the fnest spatal resoluton d mn =.11L m. As wrtten, ξ s a functon of the flter scale r; thus, we must choose a value or range of r over whch to consder ξ. If our 2D turbulence were fully developed and we observed an extended nertal range (whch would be ndcated by a plateau over an apprecable range of r n Fg. 1(a)), a better choce would be to average ξ over ths plateau, as we do for the 3D case below. But because we (unsurprsngly, gven our Reynolds number) do not observe an nertal range, we cannot take ths approach. If we consdered ξ over the full range of r, we would ntroduce large fluctuatons snce ξ wll tend to become very large when (r) (d mn ). Thus, for the 2D data we report values of ξ averaged over the range 2 < r/l m < 3, where we observe net nverse energy transfer and where we emprcally observe that ξ fluctuates wthout a systematc dependence on r; ths procedure gves us an estmate of ξ(r = 2.L m ). The result s shown n Fg. 1(b) as a functon of r/d, the rato of the flter scale to the underlyng resoluton of the feld. Although ξ fluctuates, the trend of ts dependence on resoluton s clear. Unsurprsngly, ξ s large for very poorly resolved felds; but once r/d 7, ξ saturates to a value of about %. Ths result suggests that an FST can accurately provde an estmate of spectral transfer even for a poorly resolved data set, as long as one asks about scales larger than about 7 tmes the spatal resoluton of the data. In addton to average values, FSTs allow the determnaton of the spectral energy flux locally n both tme and space. It s thus natural to ask how reducng the resoluton affects the local values of the spectral flux. Here, we choose two dfferent spatal resolutons (d =.22 L m and.32 L m )to llustrate the effect of the resoluton d on the local spectral flux. In Fgs. 2(a) and 2(b), we show the same velocty feld but sampled at these two dfferent resolutons. The correspondng maps of the spectral flux calculated for the same flter length (r = 2.2L m, where the mean nverse energy flux s the most ntense) are shown n Fgs. 2(c) and 2(d). Despte the larger and somewhat fuzzer patches n Fg. 2(d), the spatal dstrbuton and magntude of the energy flux n the two fgures are nearly the same. Smlar effects are seen for other flter scales. In Fgs. 2(e) and 2(f), for example, we show the spectral energy flux for the same velocty feld at r = 3.2L m, where the mean spectral energy flux s much smaller. Comparng the results for the two values of r, t s clear that because the typcal (a) (c) 2 (e) 1 (b) 1 (d) (f) FIG. 2. (a) and (b) A sngle velocty feld, shown wth arrows, n the quas-2d flow wth a spatal resoluton of (a) d =.22L m and (b) d =.32L m. (c) and (d) Spatally resolved spectral energy flux for a flter length of r = 2.2L m and spatal resolutons of (c) d =.22L m and (d) d =.32L m. (e) and (f) Spatally resolved spectral energy flux for a flter length of r = 3.2L m and spatal resolutons of (e) d =.22L m and (f) d =.32L m. The color bars n (c) (f) show the magntude of the energy flux n unts of mm 2 /s 3.

7 117-6 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) sze of a patch of energy flux of the same sgn s larger for r = 3.2L m (snce more of the small scales have been removed), these patches are well resolved for both resolutons (panels (e) and (f)). For the smaller flter length and lower resoluton (panel (d)), though, some patches are smlar n sze to d, and thus are poorly resolved. For Fg. 2(d), r = 6.8d, close to the pont below whch the relatve error ξ grows sgnfcantly. Thus, Fg. 2(d) gves a vsual confrmaton of the result seen n Fg. 1(b). To quantfy the effect of resoluton further, we plotted the ponts n panels (c) and (d) and panels (e) and (f) aganst each other and measured the R 2 coeffcents, whch would be unty f the data were dentcal. For panels (c) and (d), we found R 2 =.91, and for panels (e) and (f), we found R 2 =.97. B. 3D turbulence In Fg. 3(a), we show the spatally averaged spectral energy flux computed by applyng an FST to a sngle snapshot from the 3D DNS data at fve reduced resolutons d. As wth the 2D data, the curves agree well wth each other for all resolutons, agan (unsurprsngly) confrmng the quas-localty of the energy transfer n turbulence. In 3D turbulence, the mean energy transfer rate n the nertal range s expected to be equal to the mean energy dsspaton rate ɛ. Thus, t s natural to compare the spatally averaged energy flux as measured usng an FST to ɛ. InFg.3(a), we therefore also show the results of three tradtonal ways used to determne the energy dsspaton rate 22 for the fully resolved DNS data. By defnton, ɛ 2ν S j S j, where ν s the knematc vscosty and S j s the rate of stran. Ths defnton, however, s dffcult to apply n experments, snce resolvng the velocty gradents accurately requres sub-kolmogorov-scale resoluton of the velocty feld. Kolmogorov s scalng theory 1 provdes an alternatve way of determnng ɛ by relatng t to the Euleran velocty structure functons n the nertal range (that s, the range of scales that are far from both the scale of energy dsspaton and the scale of energy njecton). The velocty structure functons are defned to be the statstcal moments of the dfference n velocty between two ponts separated by a length scale r. The second-order structure functon s gven by D 2 (r) = [u(x + r) u(x)] 2. Projectng t along the separaton drecton gves the longtudnal second-order velocty structure functon D LL. In the nertal range, 1 D LL = C 2 (ɛr) 2/3, where C s expected to be a unversal constant, although wth some uncertanty Thus, the dsspaton rate s often estmated n experments by measurng D LL and scalng t approprately. In Fg. 3(a),we.1 (a) (b) 3D, N=128 3, d=17.2η 3D, N=64 3, d=34.3η 3D, N=32 3, d=68.7η ε, Π (r) (a.u.) , d=8.6η 128 3, d=17.2η 64 3, d=34.3η 32 3, d=68.7η 16 3, d=137.4η [D LL /(C 2 r 2/3 )] 3/2 D LLL /4r 2ν S j S j ξ (%) r/η 1 1 r/d FIG. 3. (a) The energy dsspaton rate ɛ andenergyflux (r) calculated va dfferent methods as a functon of length scale r, normalzed by the Kolmogorov length scale η. The symbols show (r) calculated from the same velocty feld wth dfferent spatal resolutons d, as shown n the legend. The legend also gves the number of DNS grd ponts used. The dashed and sold lnes show the compensated second- and thrd-order longtudnal Euleran structure functons, and the dashed-dotted lne shows the energy dsspaton rate obtaned drectly from the velocty gradent. (b) Relatve error ξ for (r) as shown n (a) for three representatve spatal resolutons d.

8 117-7 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) plot the compensated second-order structure functon (that s, (D LL /C 2 ) 3/2 /r) computed from the DNS data, whch should be equal to ɛ n the nertal range. And ndeed, there s a reasonably extended range (though less than one decade) over whch the compensated structure functon has a constant value that agrees well wth the exact computaton of ɛ from the velocty gradents. The thrd-order longtudnal structure D LLL s also often used to extract ɛ; t s appealng gven that the so-called 4/ law, that D LLL = 4/ɛr n the nertal range, s exact. We therefore also show the compensated thrd-order structure functon n Fg. 3(a). However, as s often the case at fnte Reynolds numbers, the plateau regon s short, makng the estmate of ɛ somewhat unrelable. We note that ths structure functon s computed from only a sngle snapshot of the full DNS data set; usng more tme steps would gve a longer plateau regon. In general, though, the thrd-order structure functon s often dffcult to converge n experments, as t requres very large numbers of samples, and ts scalng regon can be qute small at low Reynolds numbers. Comparng our FST values to the three more tradtonal measurements of ɛ n Fg. 3(a), we can see that the FST performs qute well, even for reduced resoluton. Ths result s partcularly appealng snce, unlke the structure functons, there are no free parameters and no scalng assumptons necessary for nterpretng the FST results. To make the comparson quanttatve, we can agan compute the error ξ defned n Eq. (4); here, we use d mn = 8.6η as our baselne resoluton. As compared wth the 2D experment, the 3D DNS contans enough data ponts that we do not need to average over multple flter values to obtan smooth curves; thus, we do not do any averagng n computng ξ for the 3D case, and we can vary both r and d ndependently. Our results are shown n Fg. 3(b). The trend we observe s very smlar to the 2D case: the relatve error decreases wth r/d, but saturates to a constant, small value for r/d 6. For that reason, the symbols n Fg. 3(a) for the FST results wth dfferent resolutons d are computed only for flter lengths r 6d. In ths range, curves for dfferent d collapse well both wth each other and wth D LLL. In partcular, the estmate of ɛ from the plateau values of the FST s very accurate for d = 17.2η, and s adequate even for d = 34.3η, resolutons that are accessble n experments. As wth the 2D data set, we close by consderng the local spectral flux nformaton provded by FSTs at low resoluton n the 3D DNS. Fgure 4 shows slces through a sngle 3D spectral flux FIG. 4. Slces from the 3D feld of spectral energy flux (r) taken from an nstantaneous snapshot of the 3D DNS data. The color bar ndcates the magntude of the nondmensonal energy flux. Data are shown for four combnatons of flter length r and spatal resoluton d: (a)r = 2.2η and d = 8.6η (26 3 ); (b) r = 2.2η and d = 34.2η (64 3 ); (c) r = 342.1η and d = 8.6η (26 3 ); and (d) r = 342.1η and d = 34.2η (64 3 ).

9 117-8 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) feld calculated for two dfferent flter lengths r and spatal resolutons d. For both flter scales, the spectral flux felds agree reasonably well despte the reduced resoluton, just as they dd n 2D. As above, we also computed the R 2 coeffcents for these data; for panels (a) and (b), we found R 2 =.7, and for panels (c) and (d) we found R 2 =.93. Even though the agreement s somewhat weaker than n the 2D case, our results ndcate that FSTs can be usefully appled to 3D turbulence even wth sgnfcantly under-resolved velocty felds, and that they are therefore potentally a very useful tool for analyzng partcle-trackng data. IV. CONCLUSIONS We have demonstrated that FSTs can be appled to study the average and local dynamcs of scale-to-scale energy transfer even when the velocty feld s not well resolved n both 2D and 3D turbulence. Our results show that as long as one restrcts the analyss to scales larger than about 6 tmes the spatal resoluton of the velocty feld, the spectral fluxes provded by an FST wll be accurate to wthn about %. Ths result confrms the expectaton that the energy cascade n turbulence s approxmately scale-local, snce the scales that are not present due to under-resoluton of the velocty feld do not contrbute strongly to the energy transfer processes at dstant scales. These results have useful mplcatons for partcle-trackng experments, where hghly resolved velocty felds are typcally not avalable. In partcular, snce they do not requre any assumptons about extended nertal ranges or scalng n order to estmate spectral transfer, FSTs may be a better way to estmate the energy dsspaton rate n turbulence experments than the more tradtonal velocty structure functons. FSTs are also lkely to be valuable for studyng spectral transfer n systems such as turbulence n non-newtonan fluds 26, 27 where hghly resolved numercal smulatons are very dffcult and expensve. ACKNOWLEDGMENTS Ths work was supported by the U.S. Natonal Scence Foundaton Grant Nos. DMR and DMS to Yale Unversty and DMR to Wesleyan Unversty. 1 A. N. Kolmogorov, The local structure of turbulence n ncompressble vscous flud for very large Reynolds numbers, Dokl. Akad. Nauk SSSR 3, (1941). 2 R. H. Krachnan, Inertal ranges n two-dmensonal turbulence, Phys. Fluds 1, (1967). 3 C. E. Leth, Dffuson approxmaton for two-dmensonal turbulence, Phys. Fluds 11, (1968). 4 G. K. Batchelor, Computaton of the energy spectrum n homegeneous two-dmensonal turbulence, Phys. Fluds 12, II-233 II-239 (1969). S. B. Pope, Turbulent Flows (Cambrdge Unversty Press, Cambrdge, England, 2). 6 M. Germano, Turbulence: The flterng approach, J. Flud Mech. 238, (1992). 7 S. Lu, C. Meneveau, and J. Katz, On the propertes of smlarty subgrd-scale models as deduced from measurements n a turbulent jet, J. Flud Mech. 27, (1994). 8 G. L. Eynk, Local energy flux and the refned smlarty hypothess, J. Stat. Phys. 78, (199). 9 M. K. Rvera, W. B. Danel, S. Y. Chen, and R. E. Ecke, Energy and enstrophy transfer n decayng two-dmensonal turbulence, Phys. Rev. Lett. 9, 142 (23). 1 S. Chen, R. E. Ecke, G. L. Eynk, X. Wang, and Z. Xao, Physcal mechansm of the two-dmensonal enstrophy cascade, Phys. Rev. Lett. 91, 2141 (23). 11 S. Chen, R. E. Ecke, G. L. Eynk, M. Rvera, M. Wan, and Z. Xao, Physcal mechansm of the two-dmensonal nverse energy cascade, Phys. Rev. Lett. 96, 842 (26). 12 M. Wan, Z. Xao, C. Meneveau, G. L. Eynk, and S. Chen, Dsspaton-energy flux correlatons as evdence for the Lagrangan energy cascade n turbulence, Phys. Fluds 22, 6172 (21). 13 Y. Lao and N. T. Ouellette, Spatal structure of spectral transport n two-dmensonal flow, J. Flud Mech. 72, (213). 14 G. Eynk, Localty of turbulent cascades, Physca D 27, (2). 1 J. A. Domaradzk and D. Carat, An analyss of the energy transfer and the localty of nonlnear nteractons n turbulence, Phys. Fluds 19, 8112 (27). 16 Z. Xao, M. Wan, S. Chen, and G. L. Eynk, Physcal mechansm of the nverse energy cascade of two-dmensonal turbulence: A numercal nvestgaton, J. Flud Mech. 619, 1 44 (29). 17 Y. Lao and N. T. Ouellette, Geometry of scale-to-scale energy and enstrophy transport n two-dmensonal flow, Phys. Fluds 26, 413 (214). 18 D. H. Kelley and N. T. Ouellette, Onset of three-dmensonalty n electromagnetcally drven thn-layer flows, Phys. Fluds 23, 413 (211).

10 117-9 N, Voth, and Ouellette Phys. Fluds 26, 117 (214) 19 D. H. Kelley and N. T. Ouellette, Spatotemporal persstence of spectral fluxes n two-dmensonal weak turbulence, Phys. Fluds 23, 1111 (211). 2 Y. Lao, D. H. Kelley, and N. T. Ouellette, Effects of forcng geometry on two-dmensonal weak turbulence, Phys. Rev. E 86, 3636 (212). 21 Y. L, E. Perlman, M. Wan, Y. Yang, C. Meneveau, R. Burns, S. Chen, A. Szalay, and G. Eynk, A publc turbulence database cluster and applcatons to study Lagrangan evoluton of velocty ncrements n turbulence, J. Turbul. 9, N31 (28). 22 M. Gbert, H. Xu, and E. Bodenschatz, Inertal effects on two-partcle relatve dsperson n turbulent flows, EPL 9, 64 (21). 23 K. R. Sreenvasan, On the unversalty of the Kolmogorov constant, Phys. Fluds 7, (199). 24 R. N and K.-Q. Xa, Kolmogorov constants for the second-order structure functon and the energy spectrum, Phys. Rev. E 87, 232 (213). 2 C.-C. Chen, D. B. Blum, and G. A. Voth, Effects of fluctuatng energy nput on the small scales n turbulence, J. Flud Mech. 737, 27 1 (213). 26 N. T. Ouellette, H. Xu, and E. Bodenschatz, Bulk turbulence n dlute polymer solutons, J. Flud Mech. 629, (29). 27 H.-D. X, E. Bodenschatz, and H. Xu, Elastc energy flux by flexble polymers n flud turbulence, Phys. Rev. Lett. 111, 241 (213).