Effects of non-universal large scales on conditional structure functions in turbulence

Daniel B. Blum, Surendra Kunwar, James Johnson, and Greg A. VothDepartment of Physics, Wesleyan University, Middletown, CT 06459, U.S.A.∗

(Dated: May 14, 2019)

We report measurements of conditional Eulerian and Lagrangian structure functions in order toassess the effects of non-universal properties of the large scales on the small scales in turbulence. Westudy a 1m × 1m × 1.5m flow between oscillating grids which produces Rλ = 285 while containingregions of nearly homogeneous and highly inhomogeneous turbulence. Large data sets of three-dimensional tracer particle velocities have been collected using stereoscopic high speed cameraswith real-time image compression technology. Eulerian and Lagrangian structure functions aremeasured in both homogeneous and inhomogeneous regions of the flow. We condition the structurefunctions on the instantaneous large scale velocity or on the grid phase. At all scales, the structurefunctions depend strongly on the large scale velocity, but are independent of the grid phase. Wesee clear signatures of inhomogeneity near the oscillating grids, but even in the homogeneous regionin the center we see a surprisingly strong dependence on the large scale velocity that remainsat all scales. Previous work has shown that similar correlations extend to very high Reynoldsnumbers. Comprehensive measurements of these effects in a laboratory flow provide a powerful toolfor assessing the effects of shear, inhomogeneity and intermittency of the large scales on the smallscales in turbulence.

I. INTRODUCTION

Many of the most powerful insights in the study offluid turbulence are rooted in the idea of an energy cas-cade where the chaotic process of transferring energy tosmaller scales allows the small scales to become universaland independent of the details of the forcing mechanismat the large scales. However, careful examination of manysmall scale statistics in different flows has shown thatthe reality of turbulence is quite a bit more complicated.Some statistics such as the scaling exponents of Eulerianand Lagrangian structure functions are nearly identicalin different flows1,2,3. But other small scale statisticssuch as the coefficients in scaling laws4,5 or the scalarderivative skewness6 show dependence on the propertiesof the large scales up to the largest Reynolds numbersmeasured.

A traditional approach to deal with dependence on thelarge scales has been to classify flows, and allow thatthere might be differences between categories of flowssuch as free shear flows (jets, mixing layers, etc), wallbounded shear flows (boundary layers, channel flows,etc), or isotropic turbulence (wind tunnel grid turbulenceor numerical simulations in a box with periodic bound-ary conditions, etc). A careful empirical comparison ofstatistics between different flows could then show whichproperties of the small scales are truly independent of thelarge scales.

Recent developments in experimental tools have al-lowed for the measurement of small scale Lagrangianstatistics7. In contrast to flows designed for Eulerianmeasurements such as hot-wire anemometry, Lagrangianmeasurements do not require large mean velocity in or-der to use Taylor’s frozen flow hypothesis. In fact theopposite is desired since small mean velocity allows a

particle to be tracked in an observation volume for thelongest possible time. This has led to significantly dif-ferent flow designs for Lagrangian measurements. Manynew flows have been introduced which have complex largescales and are difficult to place into traditional categorieswhich were originally meant for flows designed for hot-wire anemometry. Two widely used examples of flowswith low mean velocity are counter rotating disks8,9,10and oscillating grids11,12,13. A yet newer generation offlows are currently under study including the randomjet array14, corner stirred tank15, Lagrangian explorationmodule (LEM), and radial acoustic jets which each haveunique large scales.

Initial work on Lagrangian measurements in flows withcomplex large scales and a small mean velocity has pri-marily assumed that the small scale statistics of inter-est are independent of the large scale forcing of the flow.This assumption has been tested in several cases by thor-ough quantitative comparison of small scale statistics indifferent flows. Comparison of acceleration probabilitydistribution functions (pdf) in direct numerical simula-tions (DNS) and counter rotating disk experiments arefound to be flow independent 9,16,17 . In addition, thescaling exponents of the Lagrangian structure functionshave been compared between DNS and experiment, andfound to be in close agreement3. Much more work isneeded to determine the degree to which the large scalesof different flows affect various Lagrangian statistics.

However, there is a more direct way to evaluatewhether the small scales in a flow are independent of thelarge scales: the small scale measurements can be condi-tioned on a measurement of the state of the large scales.Two previous studies have shown strong dependence ofthe small scales of Eulerian structure functions on theinstantaneous velocity, which is dominated by the largescales. Praskovsky et al.18 extensively study interactions

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between the large scales and inertial range scales in twohigh Reynolds number wind tunnel flows. Strong correla-tions between the large scales and the velocity structurefunctions are found at all length scales. They interpretthis as being consistent with the correct application ofKolmogorov theory with a fluctuating energy injectionat the large scales. Sreenivasan and Dhruva19 measuredEulerian velocity structure functions from atmosphericboundary layer data for Rλ > 104, some of the largestReynolds numbers ever measured. They find that thestructure functions conditioned on the large scale veloc-ity show a strong dependence, and they show that DNSand grid turbulence measurements show almost no de-pendence. They attribute the dependence to large scaleshear, and show how to remove the effect to improvepower law scaling. There is evidence that other smallscale statistics show conditional dependence on the largescales. The acceleration variance shows a strong depen-dence when it is conditioned on the large scale veloc-ity20,21.

One challenge in discussing interactions between largescales and small scales is the very non-universal natureof the large scales. Each flow has a unique set of largescales, which may depend on time, geometry, or drivingparameters. So it has been difficult to isolate the aspectsof the large scale flow that are affecting the small scales.Anisotropy is the aspect that is best understood. Ex-tensive work has identified persistent anisotropy at smallscales even at very high Reynolds numbers6,22, and anal-ysis using spherical tensor decomposition has placed thisproblem on solid footing 23,24. However, this is not theonly effect of the large scales. Here we wish to distinguishtwo additional aspects of the large scales that are partic-ularly important. Inhomogeneity is the spatial variationof statistics. Large scale intermittency is temporal fluc-tuations on time scales longer than the eddy turnovertime, L/u. Both inhomogeneity and large scale intermit-tency often occur together in real flows, but are distinctproperties since flows can be conceived that have eachwithout the other. For example, a homogeneous turbu-lent flow in DNS can have large scale intermittency byhaving the energy injection varied in time.

In this paper we present a comprehensive set of mea-surements of the dependence of Eulerian and Lagrangianvelocity structure functions conditioned on the large scalevelocity. We use a flow between two oscillating gridswhich is relatively homogeneous in the central region, buthas large inhomogeneity near the grids. This allows us toisolate the signatures of different properties of the largescales. We find clear signatures of inhomogeneity, but asignificant part of the dependence of the structure func-tions on the large scale velocity seems to be the result oflarge scale intermittency. A better understanding of thisdependence on non-universal large scales will help in theidentification of universal statistics, and the comparisonof different flows.

II. EXPERIMENT

This work is based on optically tracking passive tracerparticles seeded in a turbulent flow agitated by two oscil-lating grids as shown in Fig. 1. For clear measurements of1-D inhomogeneity a large system is needed to create sig-nificant separation for locally homogeneous and inhomo-geneous regions, and to create sufficiently high Reynoldsnumbers. In order to study large scale effects conditionalstatistics were analyzed, which required large data sets(> 109 particle pairs). Storage, speed, and budget con-cerns led to the development of real time image compres-sion circuits25. These devices enabled nearly endless dataacquisition for a nominal cost.

A. Experimental Apparatus

Turbulence was generated between two identical octag-onal grids oscillating in phase in an octagonal Plexiglastank that is 1m × 1m × 1.5m and filled with approxi-mately 1,100 liters (300 gallons) of filtered, degassed wa-ter. The grids have 8 cm mesh size, 36% solidity, andwere evenly spaced from the top and bottom of the tankwith a 56.2 cm spacing between grids and a 1 cm gapbetween grid and tank walls. The stroke was 12 cmpeak to peak, powered by an 11 kW motor. A typicalgrid frequency was 3 Hz, but was raised to 5 Hz to in-vestigate Reynolds number dependence. Water coolingmaintains the temperature at ± 0.1C during each run.Neutrally buoyant 136 µm diameter polystyrene tracerparticles were added to the flow until approximately 50were seen by each camera. This particle density was cho-sen to maximize data per frame while minimizing track-ing errors. Particle density could greatly increase withthe planned addition of two more cameras26,27. One dif-ficulty of oscillating grid flows is that vibrations from theoscillatory drive can couple to the camera supports anddegrade imaging accuracy. We mounted a custom cam-era support on an optical table to minimize vibrations.Air bubble suppression was an additional concern. Wedeveloped a method to keep all water seals and bearingssufficiently wet to maintain an air tight seal.

B. Detection

These data were acquired using 3D PTV (particletracking velocimetry) measurements using two BasslerA504K video cameras capable of 1280 × 1024 pixel res-olution at 500 frames per second (a data rate of approxi-mately 625 MB per second per camera). Recording sucha high data rate is a significant technological hurdle. Atypical system would store data in 4 GB of video RAM,so that one run would last just 7 seconds before wait-ing approximately 7 minutes for the data to downloadto hard disk. We use an image compression circuit tothreshold images in real-time so that only pixels above

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FIG. 1: Experimental apparatus diagram. Two oscillatinggrids were held 56.2 cm apart in a 1,100 liter octagonal prismPlexiglas tank. Two high speed cameras are used to stereo-scopically image chosen regions of the tank in order to record3D particle positions. Illumination is provided by a Nd:YAGlaser with 50 W average power.

a user defined brightness limit are regarded as particledata and retained while the dark background pixels arediscarded25. This technique produces a dynamic datacompression factor of 100-1000, which enables continu-ous data collection and storage to hard disk.

Our first implementation of the image compression cir-cuits have faced two major challenges. First, the simplethresholding compression reduces particle center accu-racy. However, particle finding accuracy is typically de-graded by only 0.1 pixel, which is typically less than theuncertainty in particle finding from unthresholded im-ages. Secondly, because frame number information wascreated and recorded separately on each computer, anyoperating system delay can lead to frames lost and tim-ing mismatch between the cameras. For the measure-ments in this paper, frame number errors were correctedin post processing. Updated versions of the image com-pression circuit have solved this problem by includingcamera frame number in the data stream the computersrecord.

Particles are illuminated using a 532 nm pulsedNd:YAG laser with 50 W average power. The beamwas expanded to create an illumination volume approx-imately 7cm × 4cm × 5cm. Images were processed tofind the center of each particle as seen by each cameraand then stereomatched to find the 3D position in realspace. Stereomatching was accurate to approximately11 µm (0.08 particle diameters or 0.2 pixels). At this levelof accuracy it is essential to have a very good calibrationof camera position parameters to use for stereomatching.We start with a traditional calibration to obtain initial3D stereomatching 28. We then use known stereomatched

pairs from the two cameras, and run a non-linear opti-mization to minimize the stereomatching error and findoptimal camera position parameters26.

III. RESULTS

A. Characterizing the Flow

We define a characteristic velocity by u = (〈uiui〉/3)1/2and a characteristic length scale by L = u3/ε whereε is the energy dissipation rate per unit mass definedin section III C. For the center region u = 6.0 cm/s,L = 9.0 cm, and for the near grid region u = 8.3 cm/s,L = 4.5 cm. The Taylor Reynolds number, Rλ =(15uL/ν)1/2, (where ν is the kinematic viscosity) rangesfrom 285 for 3 Hz grid frequency to 380 for 5 Hz gridfrequency in the center. Near the grid at 3Hz Rλ = 230.The Kolmogorov length and time scales are η = 140µm,τη = 20 ms in the center region, and η = 94 µm, τη = 8.8ms in the near grid region.

Figure 2(a) shows the mean vertical velocity as a func-tion of the vertical position along the central axis of thetank. The top and bottom grids are separated by 56.2cm, approximately 7 L. In Fig. 2, the dot-dashed line in-dicates the maximum amplitude of the bottom grid, 22.1cm below the center of the tank. Data was collected at5 separate heights in order to measure the complete flowprofile from the center of the tank to the bottom grid.Mapping the bottom half of the tank is sufficient becausethe geometrical symmetry produces a mirror image abovethe midplane. The two volumes which we will focus onthroughout this paper are bounded by the dashed lines,and will be referred to as the center (C), and near grid(NG) observation volumes. At this grid separation dis-tance the mean flow traces four torii, two above and twobelow the center plane of the tank, as shown in the sketchin figure 3 (drawn to scale). In the large central region,the effect of the mean flow is to pump highly energeticfluid from the region near the grid towards the center ofthe tank. In figure 2(a), there are two points where themean vertical velocity reaches zero: one near the center,the other 18 cm from the center just below the near gridobservation volume. The existence of this second stag-nation point and reverse circulation region depicted inFig. 3 is a common feature in mean flows generated byoscillations28. In the following measurements, the smallmean velocity has been subtracted so that we study thefluctuating velocity field.

Figure 2(b) shows the vertical velocity variance alongthe central axis as a function of vertical position. Thevelocity variance is large near the grid and quickly fallsoff towards the center where it is nearly homogeneous.The center and near grid observation volumes were cho-sen to provide a contrast between the large homogeneousregion in the center and the much more inhomogeneousregion near the grid. In the center, the variance of thevelocity is homogeneous for several L in either direction.

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FIG. 2: Mean and variance of vertical velocity along thecentral vertical axis of the tank. Grid frequency 3 Hz, gridseparation distance 56.2 cm. The dot-dash line represents thegrid height at maximum amplitude. The remainder of thepaper will focus on measurements in two regions designatedby the vertical dashed lines: one at the center of the tank andone near the grid.

The velocity variance ranges moderately in the near gridobservation volume, and enormously within one L belowthis region. In Fig. 2, deviations from a smooth curveare not due to statistical uncertainty, but are a result ofpatching 5 calibrated regions together with the majorityof error coming from measuring absolute position in thetank.

It is interesting to note that we made measurements ina flow with a smaller grid separation of 35 cm and foundthat the Reynolds number in the center was lower. Thecharacteristic velocity in the center did increase due tothe closer proximity of the grids, but L was reduced by alarger amount resulting in approximately 8% decrease inRλ. The reason for the unexpected decrease in Reynoldsnumber is a reversal of the mean velocity compared withlarger grid separations. For larger grid separation dis-tances, energetic fluid from near the grids is carried tothe center by the mean flow. However, at 35 cm gridseparation the mean velocity reverses which results in alower Reynolds number in the center.

B. Structure Functions

To measure Eulerian structure functions we first findthe instantaneous longitudinal velocity difference be-tween two particles a distance r apart ∆ur = [u(x) −

FIG. 3: Scale diagram of 56cm × 100cm area between gridsshowing the mean circulation torii which are nearly rotation-ally symmetric about the central vertical axis. Center andNear Grid observation volumes are drawn in dashed lines,which shows the relative size and position of the observationvolumes in figure 2. Horizontal dot-dashed lines representrange of motion of the top and bottom grids.

u(x + r)]L, where the L subscript denotes the longitu-dinal component, found by projecting the 3D velocitydifference vector onto the vector connecting the two par-ticles. The longitudinal structure functions are definedas Dp = 〈(∆ur)p〉 where p represents the order of thestructure function and the brackets represent the ensem-ble average. In the inertial range, Kolmogorov (1941)gives

〈∆upr〉 = C(E)p (εr)p/3, (1)

where the C(E)p are Eulerian Kolmogorov constants and

ε is the energy dissipation rate.Figures 4 and 5 show the measured second and third

order longitudinal velocity structure functions with thestraight thin lines representing Kolmogorov’s predictionfrom Eq. 1. The insets show the structure functions com-pensated by Eq. 1. At Rλ = 285, any scaling range isvery limited, but the plateaus can be used to estimatethe inertial range.

Lagrangian structure functions were measured fromtemporal velocity differences along a particle trajectory.The velocity difference now becomes ∆uτ = u(t)− u(t+τ), where τ is the time interval between measurements.We use the vertical component of the velocity for La-grangian velocity differences throughout this paper, al-though results for the other components are similar. ForLagrangian structure functions Kolmogorov (1941) pre-dicts

〈∆upτ 〉 = C(L)p (ετ)p/2. (2)

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FIG. 4: Eulerian second order longitudinal velocity structurefunction shown as a function of pair separation r scaled by theKolmogorov length η. The inset shows this data compensatedby Eq. 1 for p=2.

C. Energy Dissipation Rate Measurement

The energy dissipation rate ε is an important valuethroughout this analysis, it is worth a moment to discusshow it is determined. Limitations in particle density pre-clude direct measurement via the definition

ε = 2ν〈sijsij〉

with

sij =12

(∂ui∂xj

+∂uj∂xi

).

Instead we utilize Kolmogorov’s 4/5 law: Eq. 1 withp = 3 where the coefficient C(E)

3 = −4/5 can be de-rived from the Navier-Stokes equations. We identify theinertial range with the plateau in the compensated thirdorder structure function (Fig. 5 inset). The inertial rangeis chosen to be 25 to 91 r/η (.35 to 1.3 cm). If the sameinertial range is used in the second order structure func-tion, the energy dissipation rate determined from it (us-ing the empirical coefficient C(E)

2 = 2.0)29 is within 3%of the value calculated from the third order.

FIG. 5: Eulerian third order longitudinal velocity structurefunction. The inset shows this data compensated by Eq. 1 forp=3.

D. Phase Dependence

A simple energy cascade has constant energy input atthe largest length scales. An obvious departure from con-stant energy input is the oscillating grid driving mecha-nism. The sinusoidal motion of the grid directly corre-sponds to energy input with periodic time dependence.It seems likely that such a strongly periodic energy in-put would have a signature throughout the whole energycascade.

The method we employ throughout this work to detectsignatures of the large scales is to condition various statis-tics on some measurement of the state of the large scales.In this case, we condition on the phase of the grid mo-tion, φ. Conditioning instantaneous single particle statis-tics such as the mean and variance of the velocity showssome sinusoidal dependence on grid phase. For exam-ple, in the center the conditional variance, 〈(u−〈u〉)2|φ〉,varies by 1% over the cycle of the grid. In the near gridregion, the conditional variance varies by 10%. The meanvertical velocity in the center, 〈u|φ〉, varies by 0.8 cm/sover a cycle of the grid which is 10% of the standarddeviation. Near the grid, the conditional mean velocityvaries by 2 cm/s, which is 20% of the standard deviationat that location.

Figure 6 shows the compensated second order lon-

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gitudinal structure functions conditioned on the phase,〈∆u2

r|φ〉. In the center of the flow (Fig. 6(a)) the struc-ture functions have essentially no change with phase.Near the grid (Fig. 6(b)) there is a slight dependenceon grid phase. To emphasize the differences betweenstructure functions at different phases, we compensatedthe structure functions by a single energy dissipationrate in each figure, ε3 = 24.6 cm2/s3 in the center andε3 = 131 cm2/s3 near the grid. These values were de-termined when the grid is in mid amplitude (the thirdbin). The good collapse of the structure functions at allphases across the entire range of r shows the minimaldependence of the small scales on the large scale peri-odicity of the flow created by the oscillating grids. Onepossible source of dependence of small scale statistics onlarge scales has been shown to be minimal.

FIG. 6: (Color Online) Second order compensated veloc-ity structure functions conditioned on grid phase. The col-lapse shows the very weak phase dependence. a) center of thetank (C). b) near the grid (NG). Zero and 2π phase representsgrid at lowest possible amplitude. φ: + = 0 - 2π/5, ∗ = 2π/5- 4π/5, � = 4π/5 - 6π/5, 4 = 6π/5 - 8π/5 , � = 8π/5 - 2π.

E. Dependence on Large Scale Velocity

1. Eulerian structure functions conditioned on the largescale velocity: center region

A more revealing dependence on the large scales of theflow is found by conditioning the velocity structure func-tions directly on the large scale velocity. A convenientmeasurable quantity that reflects the local instantaneousstate of the large scales is the average velocity of theparticle pair used for the structure function, defined asΣuz = (uz(x) +uz(x + r))/2. Alternatively, conditioningon the average velocity of many particles, not just onepair, was studied and found to have very similar results,but we choose to focus on Σuz because it can be moreeasily measured and does not depend on the observa-tion volume and seeding density. Additional conditioningquantities will be discussed in section III G.

Figure 7(a) shows the second order Eulerian velocitystructure function conditioned on Σuz. The smallest val-ues of the structure function correspond to pair velocitiesnear zero, represented by �, while large |Σuz| results inlarger values of the structure functions. For the bins wechose, the structure function conditioned on large valuesof Σuz is nearly twice the value when conditioned on Σuznear zero.

Figure 7(b) shows the data in Figure 7(a) compen-sated by Kolmogorov inertial range scaling. The func-tional forms are quite similar, confirming the impressionfrom Fig. 7(a) that all length scales are affected simi-larly by the instantaneous state of the large scales. InFig. 7(b) we used a different energy dissipation rate, εuz

to compensate each of the 5 individual large scale verti-cal velocity bins. This insures all conditions plateau atapproximately the same value, and allows for direct com-parison of the functional forms of the conditional struc-ture functions.

The strong dependence of the conditional structurefunctions on the large scale velocity at all scales revealsthat the small scales are not statistically independent ofthe large scales in this flow. There is not any detectabletrend toward the smaller scales becoming less dependenton the large scale velocity than somewhat larger scales.

2. Lagrangian structure functions conditioned on the largescale velocity: center region

In much the same way we can evaluate the conditionalLagrangian structure functions. Figure 8(a) shows thesecond order Lagrangian structure function conditionedon the vertical component of the large scale velocity, Σuz.Here we find Σuz by averaging the velocity of the particleat the two times used to determine ∆uτ . The conditionalstructure functions for different large scale velocity aredifferent by a factor of about 2.5, and they remain nearlyparallel throughout the entire time range.

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FIG. 7: (Color online) Second order velocity structure func-tion conditioned on particle pair velocity (vertical component)in the center of the tank. a) Uncompensated structure func-tions b) Individually compensated by the energy dissipationrate for each conditional data set. Symbols represent the fol-lowing dimensionless vertical velocities, Σuz/

p〈u2z〉: + = 4.2

to 2.5, ∗ = 2.5 to 0.84, � = 0.84 to -0.84, 4 = -0.84 to -2.5,� = -2.5 to -4.2.

Figure 8(b) shows the second order conditional La-grangian structure function compensated by Eq. 2 whereε is individually chosen so the maxima of all of the con-ditioned structure functions coincide. This aids compar-ison of the functional forms of the conditioned structurefunctions. Again, the functional form is nearly identi-cal for different large scale velocities, indicating that thelarge scales affect all time scales in the same way. Theremay be a small trend towards larger values of the com-pensated Lagrangian structure functions at small timeswhen the magnitude of the large scale velocity is large.

It should be noted that there is a bias present in La-grangian measurements that is not present in Eulerianmeasurements. A sample of measured Lagrangian tra-

FIG. 8: (Color online) Second order Lagrangian velocitystructure function conditioned on instantaneous velocity (ver-tical component) in the center of the tank. a) Uncompensatedstructure functions. b) Individually compensated to have thepeak values match. Symbols represent the following dimen-sionless vertical velocities, Σuz/

p〈u2z〉: + = 3.1 to 1.9, ∗ =

1.9 to 0.62, � = 0.62 to -0.62, 4 = -0.62 to -1.9 , � = -1.9 to-3.1.

jectories is biased towards low velocity particles sincethe high velocity particles are more likely to have leftthe measurement volume. This bias becomes larger forlarger τ . Berg et al..30 have studied this bias and findthat it can be quite large for typical experimental con-ditions. We quantified this bias in our data by measur-ing the Lagrangian structure functions using trajectoriesthat remained inside artificially restricted measurementvolumes. From a simple extrapolation of the dependenceon the size of the artificial detection volume, we esti-mate that our experimental Lagrangian structure func-tions underestimate the true value by 17% for τ = 8τη.This is roughly consistent with the size of the error weexpect based on the critical time lag defined in Ref.30.Note that we have not performed the compensation theyrecommend and we are roughly translating their uncom-pensated results. Because of this bias, we will focus at-tention on τ < 10τη. As we’ll discuss in section III F 3,the dependence of the conditioned Lagrangian structurefunctions on the large scale velocity does not seem to besignificantly influenced by this bias.

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3. Eulerian structure functions conditioned on the largescale velocity: near grid region

By comparing separate regions of the tank we are ableto explore the effects of inhomogeneity on this condi-tional dependence. Figure 9 shows the Eulerian struc-ture functions, similar to Fig. 7, but with data collectedin the inhomogeneous region near the bottom grid (NG).The separation between Eulerian structure function con-ditions has doubled to approximately a factor of four.Note the different ordering of the structure functions.The up-down symmetry is now broken. Fluid travel-ing upwards (∗ symbols) has a large structure functionwhile fluid traveling downward with the same magnitudeof vertical velocity (4 symbols) has the lowest value ofthe structure function. We interpret this as highly en-ergetic fluid originating near the bottom grid and beingturbulently advected into the observation volume. Simi-larly, fluid carried down from the more quiescent regionabove the detection volume has low energy and a smallerstructure function.

Figure 9(b) shows the compensated Eulerian struc-ture functions, similar to figure 7(b), but reveals a novelinsight. Stepping through the vertical velocity bins isequivalent to stepping through the energy cascade. Fluidcoming directly upward from the bottom grid (symbol +)carries energy that was recently injected into the largescales. As a result, the compensated structure functionfor upward moving fluid is biased towards the large scales.Fluid that has downward vertical velocity (symbol 4)comes from the center region far away from the grid. Ithas had more time to mature, and in this process the en-ergy is transported to smaller length scales. Conditionalstructure functions appear to be an effective tool to eval-uate whether or not a turbulent flow is fully developedand has established a stable cascade.

4. Third order Eulerian structure functions conditioned onthe large scale velocity: center region

Figure 10 shows the third order structure function in-dividually compensated and conditioned on Σuz in thecenter of the tank. Convergence of third order statis-tics was more difficult, so elimination of the two extremeconditions was required. The third order structure func-tion proves to be similar to the second order in separa-tion, symmetry, and collapse to a single functional form.The energy dissipation rates found for the three con-ditions are: ε∗ = 25.2cm2/s3, ε� = 21.7cm2/s3,ε4 =28.7cm2/s3.

FIG. 9: (Color online) Second order velocity structure func-tion conditioned on particle pair vertical velocity (Z direction)in the region near the bottom grid. The condition with thelargest downward velocity has been eliminated due to lack ofstatistical convergence. Symbols represent the following par-ticle pair vertical velocities Σuz/

p〈u2z〉: + = 3.8 to 2.3, ∗ =

2.3 to 0.75, � = 0.75 to -0.75, 4 = -0.75 to -2.3, a) Uncom-pensated structure functions b) Individually compensated bythe energy dissipation rate for each conditional data set.

F. A Powerful Method for Plotting ConditionalStructure Functions

1. Eulerian structure functions conditioned on the largescale velocity: center region

An alternative, and in many ways a more powerful,method of visualizing the same data is presented inFig. 11. Here we show the second order Eulerian struc-ture function conditioned on the vertical component ofthe large scale velocity (the same data as Fig. 7). How-ever, the scaled vertical pair velocity is plotted on thehorizontal axis with conditioned structure functions on

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FIG. 10: (Color online) Third order velocity structure func-tion plots conditioned on particle pair vertical velocity andindividually compensated for each conditional data set. Datais taken in the center region of the tank, and the extremevertical velocity plots have been eliminated due to lack ofstatistical convergence. Symbols represent the following ver-tical velocities Σuz/

p〈u2z〉: ∗ = 2.5 to 0.84, � = 0.84 to to

-0.84, 4 = -0.84 to -2.5.

the vertical axis. When the structure functions are scaledby their value at Σuz = 0, we find very good collapse ofthe data. The fact that these curves for different r/η col-lapse so well is a striking demonstration that the largescales affect all length scales in the same way. The factthat the conditional structure functions vary by a factorof 2.5 demonstrates the strong dependence on the largescales. Note that for Gaussian random fields, the plot inFig. 11 would be flat, and a nearly flat result is observedin DNS and grid turbulence19.

In Fig. 11, it may be expected that the structure func-tion at the largest length scales (×) are a function of thelarge scale velocity. We see here that the dependence isa steep parabola. What is now more clear with this plot-ting method is the extent to which all the smaller scalesare also affected by the large scale velocity; in fact, alllength scales collapse nearly perfectly onto one parabola.The large scale velocity affects all length scales in nearlythe exact same way, all the way down to the dissipativerange.

FIG. 11: (Color Online) Eulerian second order conditionalstructure function versus large scale velocity. Data taken inthe center region. Each curve represent the following separa-tion distances r/η: + = 0 to 40, ∗ = 40 to 70, � = 70 to 110,4 = 110 to 140, � = 300 to 370, × = 370 to 440.

2. Eulerian structure functions conditioned on the largescale velocity: higher Reynolds number

Figure 12 shows the effect of increasing the Reynoldsnumber. This data is at the center of the tank with thegrids oscillating at 5 Hz which increases Rλ to 380. Thecollapse of the structure function remains. The curvaturein this figure is not significantly different from the lowerReynolds number data in Fig. 11 indicating that if thereis a Reynolds number dependence it is weak.

Figure 13 shows a comparison of our data with datataken in the atmospheric boundary layer19 with Rλ >104. Atmospheric boundary layer turbulence shows asimilar collapse of conditional structure functions at alllength scales. The curvature is also similar in both datasets, indicating that the dependence on the large scalesis similar even at these very large Reynolds numbers.

3. Lagrangian structure functions conditioned on the largescale velocity: center region

Figure 14 shows the Lagrangian structure functionsplotted versus the large scale velocity, comparable to theEulerian data shown in Fig. 11. The parabolic shaperemains, but the curvature is greater for all Lagrangiantime scales than it is in the Eulerian data. All time scalesare affected by the large scale velocity. To determine theeffect of measurement volume bias, we have done this

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FIG. 12: (Color Online) Eulerian second order conditionalstructure function versus large scale velocity. Data taken inthe center region at higher grid frequency, 5Hz, resulting inhigher Taylor Reynolds number 380. Symbols represent thefollowing separation distances r/η: + = 0 to 50, ∗ = 50 to100, � = 100 to 150, 4 = 150 to 200, � = 310 to 420, × =420 to 520.

analysis for artificially restricted measurement volumes.By decreasing the volume by a factor of 2, we observethe large τ curves to shift by approximately the devia-tions between the curves. We conclude that the bias doesnot have a significant effect on the conditional depen-dence shown in Fig. 14 for the time differences presented(τ ≤ 10τη).

4. Eulerian structure functions conditioned on the largescale velocity: near grid region

Figure 15 shows conditioned Eulerian structure func-tion similar to figure 11, but measured in the inhomoge-neous region near the grid (NG). The structure functionshere are strikingly different than in the center. The min-imum is shifted by more than one standard deviation tothe left. The inhomogeneity breaks the up-down sym-metry so that fluid coming directly up from the bottomgrid is markedly different then fluid coming down fromthe more quiescent region above (analogous to the ∗ and4 separation in Fig. 9). It follows that fluid with an up-ward velocity has higher energy than fluid with the samevelocity magnitude in the downward direction. The at-mospheric boundary layer data in Fig. 13 also shows thiseffect with a minimum at Σuz/

√〈u2z〉 = −0.5, presum-

ably as a result of weaker inhomogeneity. Also notable

FIG. 13: (Color Online) Eulerian second order conditionalstructure function versus large scale velocity. The thin plotsare from atmospheric boundary layer data 19 r/η: ∗ ∼ 100,4 ∼ 400, � ∼ 1000, × ∼ 1250. The thick line is from fig. 11,which has been overlaid for comparison, r/η: � = 70 to 110.

is that the collapse of plots for various r values is notas complete as in the central region. This is consistentwith Fig. 9(b) which shows that the conditional structurefunctions have somewhat different r dependence.

5. Lagrangian structure functions conditioned on the largescale velocity: near grid region

Figure 16 shows a Lagrangian structure function takenin the near grid region, similar to the Eulerian data inFig. 15. The minimum is shifted to the left here also asa result of the inhomogeneity in this region of the flow.The conditional dependence on the large scale velocity isagain somewhat larger than in the Eulerian case, and thecollapse at different time scales is not as complete.

6. Third order Eulerian structure functions conditioned onthe large scale velocity: center region

The third order Eulerian velocity structure functionplotted versus the large scale vertical velocity is shown inFig. 17 using data from the center of the tank. Statisticalconvergence is weaker than the second order which limitsthe large scale velocity range available for analysis. Thecollapse seems similar to the second order case shownin Fig. 11, although the measurement uncertainties are

11

FIG. 14: (Color Online) Lagrangian second order conditionalstructure function versus large scale vertical velocity. Datataken in the center region. Symbols represent the followingτ/τη: + = 0.42 , ∗ = 1.3, � = 3.5, 4 = 10.

larger here. The curvature seems to be slightly larger forthe third order than for the second order case.

7. Second order Eulerian structure functions conditionedon the velocity magnitude: center region

The second order Eulerian structure function plottedversus the magnitude of the pair velocity is shown inFig. 18 using data from the center observation volume.The magnitude of the pair velocity is also a useful indi-cator of large scale activity. It has no preferred direction,and it is a more direct indicator of the instantaneous localenergy. A similar dependence remains as in Fig. 11, thecollapse seems similar, and the curvature is significantlylarger.

G. Discussion

We have provided a comprehensive set of measure-ments that shows signatures of the current state of thelarge scales on the inertial range and small scales in tur-bulence. Here we wish to discuss factors that might beresponsible for the dependence of structure functions onthe instantaneous large scale velocity. First we will ad-dress a possible concern that the conditional dependencemay be a kinematic correlation. Then we will discuss pos-sible properties of the large scales that could be impor-tant including Reynolds number, anisotropy, mean shear,

FIG. 15: (Color Online) Eulerian second order conditionalstructure function versus large scale velocity. Data taken inthe near grid region of the tank. The structure function isheavily influenced by the bottom grid which has skewed thesymmetry of the plot minima in the negative direction. Sym-bols represent the following non-dimensional separation dis-tances r/η: + = 0 to 50, ∗ = 50 to 110, � = 110 to 160, 4 =270 to 320, � = 330 to 450, × = 450 to 560.

inhomogeneity, and large scale intermittency. We do notclaim that this list is exhaustive, but it seems that iden-tifying the major contributors will be valuable.

A reasonable suspicion might be that the observed de-pendence is a kinematic correlation, meaning that par-ticle pairs with large velocity may also have a large ve-locity difference simply because the same measurementsare used in both cases. Hosokawa31 identified that Kol-mogorov’s 4/5ths law requires that velocity sums anddifferences be correlated so that

〈u2+∆u−〉 =

εr

30(3)

where u− is half the longitudinal velocity difference andu+ is half the sum. (For comparison, we have used∆ur = 2u− and below Σu‖ = u+.) Khomyansky32 etal provide an experimental confirmation of this and ina more recent paper33 present a list of kinematic rela-tions. However, several lines of evidence indicate thatkinematic correlation does not account for the majorityof the dependence we observe.

First, note that two independent random samples withidentical Gaussian distributions have a difference that isuncorrelated with the sum, so that the conditional de-pendence seen in Fig 11 would be flat. This remains truefor velocity differences and sums from Gaussian randomfields. Both of these results can be obtained by consid-

12

FIG. 16: (Color Online) Lagrangian second order conditionalstructure function versus large scale vertical velocity. Datataken in the near grid region of the tank. Symbols representthe following τ/τη: + = 0.94, ∗ = 2.8, � = 8.0.

ering the joint pdf of the two samples, and then rotating45 degrees to the coordinate system of sums and differ-ences. Because the samples are interchangeable, the sumand difference axes have to be principle axes of the jointgaussian pdf, and the conditional variance of the differ-ence is independent of the sum.

Of course, turbulent velocities are not joint Gaussian.However, from the kinematic relations in the literaturewe have not been able to derive predictions for the con-ditional structure functions that we consider or for thecorrelation 〈(Σu)2(∆u)2〉 that would capture the mainconditional dependence we see.

To make an experimental estimate of the effect ofkinematic correlations, we conditioned the velocity dif-ferences on several other quantities. For each particlepair, we calculated the longitudinal and transverse com-ponents of the average velocity of the particles, denotedΣu‖ and Σu⊥ respectively. We then conditioned thelongitudinal structure functions on the longitudinal andtransverse pair velocities instead of conditioning on thepair vertical velocity. The idea here is that while condi-tioning the longitudinal structure functions on the longi-tudinal component (Σu‖) could have a kinematic corre-lation, conditioning on the transverse component (Σu⊥)should have no kinematic correlation. We found thatconditioning on (Σu‖) had a roughly 30% larger effecton the structure functions than conditioning on (Σu⊥).Conditioning on Σuz should have less kinematic correla-tion than conditioning on Σu‖. So more than 70% of theeffect remains unexplained by kinematic correlation. We

FIG. 17: (Color Online) Eulerian third order conditionalstructure function versus large scale vertical velocity inthe center region. Symbols represent the following non-dimensional separation distances r/η: + = 0 to 40, ∗ = 40 to70, � = 70 to 110, 4 = 110 to 140, � = 220 to 300, × = 300to 370.

conclude that while kinematic correlation may possiblymake a significant contribution to the conditional depen-dence, the majority of the effect comes from the largescales.

An immediate concern when discussing large scale ef-fects is if the oscillating grid flow has a Reynolds num-ber insufficient for adequate scale separation, and it isthis which leads to contamination of the small scale stat-ics by the large scales. Evidence points to large scaledependence not being caused by limited Reynolds num-ber. The comparison in Fig. 13 shows that atmosphericboundary layer data19 with very large Reynolds number(Rλ > 104) has nearly the same dependence on the largescales as our flow. Increasing the Reynolds number in ourflow makes very little difference. Additionally, all lengthscales collapse to nearly the same functional form indi-cating that limited separation of scales is not the primaryfactor. Taken together, these lead us to the conclusionthat merely high Reynolds number alone is not enoughto create small scales that are statistically independentof the large scales.

Our flow is somewhat anisotropic. The ratio of verti-cal to horizontal velocity standard deviations is 1.5:1 inthe center. The effects of large scale anisotropy on thesmall scales has been studied extensively24, but it is not amajor factor in conditional dependence studied here. Wehave analyzed our data by averaging over particle pairswith all orientations, so when the structure functions are

13

FIG. 18: (Color Online) Eulerian second order conditionalstructure function versus magnitude of the velocity pair inthe center region. Symbols represent the following non-dimensional separation distances r/η: + = 0 to 40, ∗ = 40 to70, � = 70 to 110, 4 = 110 to 140, � = 300 to 370, × = 370to 440.

conditioned on a quantity with no preferred direction likethe velocity magnitude (Figure 18) there should be verylittle contribution from anisotropy. In fact, we find thatthe conditional dependence on velocity magnitude is evenstronger than the dependence on the vertical velocitycomponent. We also observe the conditional dependenceremains when conditioned on other quantities withoutpreferred directions like Σu‖, and Σu⊥. We concludethat anisotropy of the large scales is not a significantcause of the conditional dependence we observe.

Sreenivasan and Dhruva19 attribute the strong con-ditional dependence of the Eulerian structure functionson the large scale velocity to shear in the atmosphericboundary layer. In making this argument, they show animportant piece of information in their figure 6 whichshows conditioned structure functions in homogeneousturbulence from both DNS and wind tunnel grid turbu-lence. The conditional statistics in these homogeneousand isotropic flows show no apparent dependence on thelarge scale velocity. However, we conclude that shearis not the fundamental property responsible in our flowsince the oscillating grid flow has a much lower shear butproduces much the same dependence on the large scalevelocity. The mean velocity gradient normalized with theeddy turnover time is 1.2 in the center of our flow andwe estimate it is in the range of 5 or greater for theiratmospheric boundary layer data. There must be someother properties that exist in the shear flow, but also are

important in our flow with small shear.Our data clearly shows the role that inhomogeneity

plays in the observed large scale dependence. Our Eu-lerian and Lagrangian data near the grid in Figs. 15and 16 show that the structure functions depend greatlyon the origin of the fluid being swept into the obser-vation volume. Fluid coming from energetic regions ofthe tank have larger structure functions than fluid com-ing from more quiescent regions. Inhomogeneity is di-rectly responsible for the shift of the minimum in Fig. 15away from zero vertical velocity. In the center of thetank (Fig. 11), the inhomogeneity is much smaller, but itcould be responsible for part of the curvature since bothfluid coming downward and fluid coming upward wouldbe coming from more energetic regions symmetrically.

However, inhomogeneity alone does not account for allof the large scale dependence observed. There is alsoa significant contribution from large scale intermittency,and it is possible that this is the dominant contribution inthe center of the tank. Large scale intermittency has beendifficult to quantify. It can be defined as any temporalfluctuations in the large scales that occur on timescaleslonger than the eddy turnover time, L/u.

Fernando and DeSilva11 show large scale intermittencycan exist in an oscillating grid flow depending on bound-ary conditions. We have observed clear signatures oflarge scale intermittency in our flow. Although we usetheir recommended boundary conditions, the velocitydistribution in the center of the flow is slightly bimodalindicative of switching between two flow states. This ef-fect is more prominent in preliminary data we took forgrid spacings of 66 cm and 100 cm than it is in the datafor 56.2cm presented in this paper.

Our measurements show a dependence of the condi-tional structure functions on the large scale velocity thatcan not be fully attributed to inhomogeneity, and largescale intermittency appears to be the most likely cause.The clearest evidence for this comes from conditioningthe structure functions on the horizontal components ofthe large scale velocity, Σux and Σuy instead of on thevertical component, Σuz. The horizontal midplane (xand y directions) is much more homogeneous than thevertical axis (z direction). Yet, the structure functionsconditioned on Σuy or Σux show a large scale depen-dence that is only moderately smaller than for the Σuzcondition (85% and 72% of the dependence seen in Σuz).If the inhomogeneous direction shows similar conditionaldependence on the large scales as the homogeneous di-rections show, then it seems that a large part of theconditional dependence must come from fluctuations inthe large scales, and not directly from inhomogeneity.Praskovsky et al.18 attribute large scale intermittency asa crucial component of the large scale dependence theyobserve. More work is needed to isolate the effects oflarge scale intermittency on small scale statistics in tur-bulence.

We have largely ignored considerations of power lawscaling which has been a focus of much of the previ-

14

ous work on this subject. Because of the relatively lowReynolds number of our experiment, we can not makesensitive tests of scaling. However, our data provides aplausible picture about how the large scales should af-fect power law scaling. If the data in Fig. 11 collapsesto a single curve, then the dependence of the conditionalstructure functions on r and uz are separable and thelarge scale dependence will have no effect on the scalingexponents of unconditional structure functions. Whenthis type of plot does not collapse as in Figs. 15 and 16,then the power law scaling will be affected by the largescales.

IV. CONCLUSIONS

We study a flow between oscillating grids with 3D par-ticle tracking and a novel real-time image compressionsystem in order to quantify the effects of various proper-ties of the non-universal large scales on the inertial rangeand small scales.

We measured the mean and variance of the velocityas a function of distance from the grids. The oscillatinggrid motion has produced a weak mean flow as well asa region near the grid with high velocity variance thatfalls off quickly to a very homogeneous, lower velocityvariance, region in the center. This profile has been keyin the determination of the role of inhomogeneity.

Conditional statistics were employed in order to mea-sure the large scale effects. Second order Eulerian veloc-ity structure functions were conditioned on the phase ofthe grid, an obvious source for periodic large scale energyinput. Results show little dependence of the structurefunctions in the center region and surprisingly little evennear the grid.

Eulerian and Lagrangian structure functions were alsoconditioned on the instantaneous large scale velocity. Alarge dependence was found in the center, with the Eu-lerian structure function increasing by a factor of 2 ormore when the large scale velocity is large. The depen-dence of the Lagrangian structure functions is somewhatlarger. Conditioned structure functions show that in the

center of the tank, all length scales are affected in ap-proximately the same way. The region near the grid wasalso analyzed and compared with the region in the cen-ter. Near the grid, we found a much stronger dependenceon the instantaneous large scale velocity for both the Eu-lerian and Lagrangian structure functions than we foundin the center. Near the grid, there are clear signaturesof the effects of large scale inhomogeneity on the smallscales. Fluid coming up from the energetic region nearerthe grid has large structure functions, while fluid comingdown from the quiescent region in the center has muchsmaller structure functions. The functional form of theconditional structure functions are also different indicat-ing the different histories of the different fluid. Thesemeasurements provide a clear picture of the way inho-mogeneity affects the small scales of turbulence.

Plotting the conditional structure functions versus thelarge scale velocity provides a powerful method for visu-alizing the effects of the large scales on all length scales inturbulent flows. We recommend these plots as an effec-tive way to compare the effects of the large scales in differ-ent experiments. This has been done for grid turbulenceand homogeneous, isotropic DNS19 which show almost nodependence of the structure functions on large scale ve-locity. Our oscillating grid flow and high Reynolds num-ber atmospheric boundary layer turbulence19 show verysimilar dependence. Comparison of conditional structurefunctions in other flows has the potential to clarify theeffects of the large scales on small scale turbulence andto guide the search for universal properties of turbulentflows.

V. ACKNOWLEDGEMENTS

This work was supported by Wesleyan University,the Alfred P. Sloan foundation, and NSF grant DMR-0547712. We thank Rachel Brown, Emmalee Riegler andTom Glomann for assistance with the experiment. Webenefitted from discussions with Nick Ouellette, HaitaoXu, Eberhard Bodenschatz, Zellman Warhaft, LaurentMydlarski, and Mark Nelkin.

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