Robust inverse energy cascade and turbulence structure in three-dimensional layers of fluid
RRobust inverse energy cascade and turbulence structure in three-dimensional layersof fluid
D. Byrne, H. Xia, a) and M. Shats Research School of Physics and Engineering,The Australian National University, Canberra ACT 0200,Australia (Dated: 21 November 2018)
Here we report the first evidence of the inverse energy cascade in a flow dominatedby 3D motions. Experiments are performed in thick fluid layers where turbulenceis driven electromagnetically. It is shown that if the free surface of the layer is notperturbed, the top part of the layer behaves as quasi-2D and supports the inverseenergy cascade regardless of the layer thickness. Well below the surface the cascadesurvives even in the presence of strong 3D eddies developing when the layer depthexceeds half the forcing scale. In a bounded flow at low bottom dissipation, theinverse energy cascade leads to the generation of a spectral condensate below thefree surface. Such coherent flow can destroy 3D eddies in the bulk of the layer andenforce the flow planarity over the entire layer thickness.
I. INTRODUCTION
There has been remarkable progress in theunderstanding of turbulence in fluid layers.Such layers, characterized by large aspect ra-tios, are ubiquitous in nature. 2D turbulencetheory by Kraichnan , in particular the in-verse energy cascade, has been confirmed inexperiments in thin fluid layers . More re-cent studies showed that theoretical resultsderived for idealized 2D turbulence are validin a variety of conditions, even when the the-ory assumptions are violated .In bounded turbulence, the inverse en-ergy cascade may lead to the accumulation ofspectral energy at the box size scale and thegeneration of a spectral condensate, a largevortex coherent over the flow domain . Goodagreement with the Kolmogorov-Kraichnantheory was found in the double layers of fluidsin spectrally condensed turbulence .It should be noted that though spectralcondensation was observed in the double-layer configurations , in single layers ofelectrolytes not only spectral condensation,but also the very existence of the inverse en- a) Electronic mail: [email protected] ergy cascade has been questioned. For ex-ample, in Ref. the flow generated in a sin-gle layer was referred to as “spatio-temporalchaos” to stress the absence of the turbu-lent energy cascades. It is thus important tobetter understand spatial structure of turbu-lence in such layers as well as differences be-tween turbulence in double and single layers.(Here we do not discuss MHD flows whichalso show spectral condensation , but wheretwo-dimensionality is imposed by homoge-neous magnetic field and bottom dissipationis restricted to a thin Hartmann layer.)A comparison of turbulent flows in a singleand double-layer configuration is also impor-tant to improve our understanding of turbu-lence in atmospheric boundary layers. Theselayers are very different over terrain and theoceans, with the former being substantiallythicker than the latter ones. Stable immis-cible layers of fluids have been generatedin the laboratory by placing a heavier non-conducting fluid at the bottom of the cell anda lighter layer of electrolyte resting on top ofit . In this case the electromagnetic forc-ing is detached from the solid bottom and itis maximal just above the interface betweenthe two fluids. The structure of the flow in1 a r X i v : . [ phy s i c s . f l u - dyn ] S e p he top layer close to the interface with thebottom layer may be similar to that of theatmospheric boundary layer over the ocean.In this paper we present new results onthe spatial structure of turbulence in a single-and a double-layer turbulent flow. In con-trast to the previous studies, we focus hereon thick layers. Recent 3D direct numericalsimulations of the Navier-Stokes equations have shown that the finite layer depth leadsto splitting of the energy flux. In thin lay-ers the energy flux injected by forcing is in-verse. As the layer thickness h is increased, alarger fraction of the flux is redirected downscale, towards the wave numbers larger thanthe forcing wave number k f . At h/l f > . . It hasbeen shown that at h/l f > . k > k f no k − / spectrum wasobserved as should be expected in the pres-ence of the direct energy cascade. Probably3D eddies over the solid bottom introducehigher damping for small scales ( k > k f ) anddo not allow the direct cascade to develop.In the double layers the effects of the solidbottom on the top layer turbulence are iso-lated by a layer of non-conducting, low vis-cosity heavier fluid at the bottom. It wasrecently found that the range of depths inwhich a top layer flow remains planar andsupports spectral condensation, is greatly ex-tended, h/l f > .
5. It was hypothesized, thata residual inverse energy flux condenses tur-bulent energy into large scale flows even inthick layers. These coherent flows enforce theplanarity by shearing off vertical eddies andthus secure the upscale energy transfer . Itis not clear however how the large scale flow is generated in a thick layer in the first place.In this paper we present new results whichshed light on this. II. EXPERIMENTAL SETUP ANDFLOW CHARACTERIZATION
In these experiments turbulence is gener-ated through the interaction of a large num-ber of electromagnetically driven vortices .An electric current flowing across the con-ducting fluid layer interacts with the spatiallyvarying vertical magnetic field produced bya 24 ×
24 or 30 ×
30 array of magneticdipoles (10 mm and 8 mm separation respec-tively), Figure 1. The magnet arrays areplaced under the bottom of the 0.3 × fluid cell. To ensure that turbulence is forcedmonochromatically at k = k f , vertical mag-netic field produced by the array has beenmeasured using a Hall probe scanned in hori-zontal planes at several heights above the ar-ray. The measured magnetic field (Fig. 1(b))has then been Fourier transformed in 2D,Fig. 1(c). The spectrum shows that J × B forcing is localized in k -space in a narrowspectral range (in this example, for a 10 mmmagnet separation, the spectrum peaks at k ≈
630 rad · m − ). The forcing strengthis controlled by adjusting electric currentthrough the layer in the range 0.5-5 A.Turbulence is generated either in a singlelayer of N a SO water solution, or it is drivenin the top layer of the electrolyte which restsupon a layer of heavier (1820 kg/m ) non-conducting liquid (FC-3283 by 3M) which isnot soluble in water. In the latter case, forc-ing is the strongest just above the interfacebetween the layers.The flow is visualized using neutrallybuoyant seeding particles 50 µ m in diameterilluminated by a horizontal laser sheet. Thethickness of the laser sheet and its height rel-ative to the free surface are adjusted to vi-sualize different regions of the layers. Per-turbations to the fluid surface are monitoredby reflecting a thin laser beam off the free2 erticalmagneticfieldPlatinum electrodes j B MagnetsLight fluid Laser sheet Laser sheet optics LaserHeavy fluid (a)(b) (c) B k ( a r b . un i t s ) FIG. 1. Schematic of experimental setup. (a) Neutrally buoyant seeding particles in the top(conducting) layer are illuminated and their motion is filmed from above. (b) Measured verticalmagnetic field produced by the magnet array: blue and red dots indicate upward and downward B direction. (c) Wave number spectrum of the measured magnetic field. surface onto a distant screen (the detectionsensitivity is ∼ × − mm). In all reportedexperiments no perturbation to the free sur-face is detected.To characterize the vertical structure ofthe flow, defocusing PIV is used . Thistechnique uses a single CCD camera witha multiple pinhole mask to measure three-dimensional velocity components of individ-ual seeding particles in the flow . The hor-izontal turbulent velocity fields are derivedusing particle image velocimetry (PIV). Par-ticle pairs are matched from frame to framethroughout the illuminated volume using aPIV/PTV hybrid algorithm. Derived ve-locities are then averaged over hundreds offrame pairs to generate converged statisticsof the root-mean-square velocities (cid:104) V x,y,z (cid:105) rms throughout the layer. The technique allowsone to resolve vertical velocities (cid:104) V z (cid:105) rms ≥ (cid:104) V xy (cid:105) rms ≥ × − mm/s. Thedamping rate of the flow is studied from the decay of the horizontal energy density afterswitching off the forcing .We also study vertical motion of seedingparticles by illuminating the layer using athick vertical laser sheet. Streaks of the par-ticles in z − x plane are filmed with the expo-sure time of 1-2 s through a transparent sidewall of the fluid cell. III. TURBULENT FLOW IN ATHICK SINGLE LAYER
We first describe turbulent flows in a sin-gle layer. The flow is forced at k f ≈ · m − ( l f ≈ . h l = 10 mm. According to numericalsimulations, turbulence in such a layer shouldshow substantial three-dimensionality , evenwhen the forcing is 2D. Figure 2 shows verti-cal profiles of vertical (a) and horizontal (b)velocities along with the snapshot of the par-ticle streaks in the vertical ( z − x ) plane.3 Vxy (mm/s)
Vz ( mm/s)(a) h (mm) (b) h (mm) h h h (c)xz FIG. 2. Vertical profiles of (a) vertical, V z , and(b) horizontal, V x,y , velocities. A grey box in (a)indicates the sensitivity of the defocusing PIVtechnique. (c) A snapshot of the particle streakstaken at the exposure time of 2 s. RMS vertical velocities < V z > are low in thetop sublayer, 2 mm below the free surface, aswell as in the bottom boundary sublayer. Inthe bulk of the flow (2 - 8 mm) < V z > ishigh, being only a factor of two lower thanhorizontal velocities < V x,y > . < V x,y > shows a maximum at h = (3- 4) mm, which is indicative of the com-petition between the forcing and the bot-tom drag. The forcing f ∼ ( J × B ) is thestrongest near the bottom (magnets under-neath the cell, the current density J is con-stant through the layer) and it decreases in-versely proportional to the distance from thebottom, f ∝ /h . The bottom drag is alsothe strongest near the bottom; in a quasi-2Dflow it should scale faster , α ∝ /h , result-ing in the maximum of < V x,y > at h = (3 − h = (8 −
10) mm,turbulence is expected to behave as quasi-2Ddue to the lower vertical velocities and theabsence of vertical gradients of the horizon-tal velocities. The planarity of the top sub-layer can be seen qualitatively in the particlestreaks of Fig. 2(c). To test if the nature of the turbulent en-ergy transfer changes between the top sub-layer, the bulk flow and the bottom layer, weperform PIV measurements of the horizon-tal velocities by illuminating three differentranges of heights: h = (8 −
10) mm, h =(4 −
7) mm, and h = (1 −
4) mm. The ve-locity fields are analyzed as described, for ex-ample, in Ref. . We compute the wave num-ber energy spectra E k ( k ) and the third-orderstructure function S L = (cid:104) ( δV L ) (cid:105) , where δV L is the increment across the distance r of thevelocity component parallel to r . The third-order structure function is related to the en-ergy flux in k -space as (cid:15) = − (2 / S L /r . Pos-itive S L corresponds to negative (cid:15) and theinverse energy cascade.Figures (3)(a,b) show the kinetic energyspectrum and the third-order structure func-tion as a function of the separation distancemeasured in the top sublayer h . At k < k f the spectrum scales close to k − / , while S L is positive at l > l f and is a linear functionof r . This is in agreement with the expec-tation of the quasi-2D turbulence in the topsublayer.In the bulk flow h , which is dominatedby 3D motions, the spectrum is still close to k − / , though it flattens at low wave num-bers as seen in Fig.3(c). Consistently withthis, the range of scales for which S L is pos-itive and linear is reduced to about r ≈ S L ( r ) isalso typical for thin single layers. The re-duction in the inverse energy cascade rangeis correlated with the increased damping. 3Dmotions present in the bulk flow (Fig.3(c)) in-crease damping rate due to the increased fluxof the horizontal momentum to the bottom ofthe cell . The increased damping arrests theinverse energy cascade at some scale smallerthat the box size.In the bottom sublayer h , the flow is sub-ject to even stronger damping. As a result,the spectrum is much flatter than k − / . S L is positive for a narrow range of scales, givinga hint of same trend as in the bulk and the4
00 1000100 100010-910-810-1010-11 k (rad m ) -1 r (m)(b)0 0.02 0.04 0.0600.40.81.2 S (10 m s ) -8 3 -3 r (m)(d) k ( m ) -1 rad k -5/3 E k ( )m s r (m)(f) k ( m ) -1 rad E k ( )m s k -5/3 k -5/3 E k ( )m s S (10 m s ) -8 3 -3 S (10 m s ) -8 3 -3 FIG. 3. (a,c,e) Wave-number spectra and (b,d,f) the third order structure functions S L measuredin (a,b) the surface sublayer ( h = (8 −
10) mm), (c,d) in the bulk flow ( h = (4 −
7) mm), and(e,f) in the bottom sublayer regions ( h = (1 −
4) mm). top sublayers.The above results suggest that despite thepresence of substantial 3D motion in a thick( h/l f = 1 .
28) single layer, statistics of thehorizontal velocity fluctuations remain con-sistent with that of quasi-2D turbulence andsupports the inverse energy cascade. As seenfrom the energy spectra of Fig.3(a,c,e), thereis no evidence of the direct energy cascade at k > k f . The spectrum shows that E k scalesmuch steeper than the 3D Kolmogorov scal-ing of k − / . A possible reason for the ab-sence of the direct energy cascade in a 3D flowat k > k f is the fact that the Reynolds num-ber is low in these experiments ( Re < h/l f ≈ .
3. If the layer thicknessis further increased in a bounded flow, a spec-tral condensate forms in the top sublayer h .In a layer of a total thickness of 20 mm thespectral condensate penetrates down to 4-5mm below the free surface. The formation ofthe condensate in thick layers is due to thereduction in the bottom damping, since evenin the presence of 3D motions the dampingrate is reduced with the increase in thicknessas α ∼ /h .5
404 0 1 2 -404 0 2 4 6 8
Vz ( mm/s)
Vxy (mm/s) s e n s iti v it y nonconductinglayer nonconductinglayer h (mm) h (mm) (a) (b) start start steadysteady FIG. 4. Vertical profiles of (a) vertical, and (b) horizontal RMS velocities measured at t = 5 safter forcing start, solid diamonds, and at t = 20 s, open circles. IV. TURBULENCE STRUCTUREIN A DOUBLE LAYER FLOW
As shown in the previous section the in-verse energy cascade is sustained in the pres-ence of 3D motions. In bounded turbulenceat low damping such a cascade leads to spec-tral condensation and to the generation of alarge-scale coherent structure. Such a struc-ture could then impose two-dimensionality onthe flow in the layer, as has been found in re-cent experiments . In this section we studyspatial structure of the flow during spectralcondensation in thick layers subject to evenlower bottom drag.The bottom drag is reduced by generat-ing two immiscible layers in which the bot-tom layer (heavier, non-conducting liquid)isolates the conduction layer from the bot-tom. We keep the bottom layer relativelythin, h b /l f < .
5. The top layer, on the otherhand, is thick, h t /l f > .
5, to allow three-dimensionality to develop, even with 2D forc-ing. Here we study the layer configurationdescribed in Ref. , namely h t = 7 mm and h b = 4 mm, which correspond to h b /l f = 0 . h t /l f = 0 .
78 respectively. It has been re-ported that the flow in the top layer showssubstantial 3D motions shortly after turbu-lence is forced. However in the steady state,the development of spectral condensate leadsto a substantial reduction of 3D eddies andto a planarization of the flow. Figure 4 shows vertical profiles of (a)vertical, and (b) horizontal RMS velocitiesmeasured using defocusing PIV in the toplayer. The grey box in Fig.4(a) indicates themethod sensitivity. Shortly after the flow isforced ( t =5 s), vertical velocities peak in themiddle of the layer at (cid:104) V z (cid:105) ≈ (cid:104) V z (cid:105) ≤ (cid:104) V x,y (cid:105) ≈ h = 2 − (cid:104) V x,y (cid:105) near the interface in the steady stateis probably related to the effect of sweepingof the forcing scale vortices by the developingcondensate . Thus, a substantial fraction ofthe top layer is quasi-2D, i.e. (cid:104) V z (cid:105) ≈ ∂ (cid:104) V x,y (cid:105) /∂z ≈ . Indeed,in the steady state, the statistics of horizon-tal velocities is in a good agreement with theKraichnan theory. The spectra and the third-order structure functions in the presence ofspectral condensate are computed after sub-6 -7 -8 -9
100 1000 E k ( ) m s k f k c (g) k -5/3 k ( ) m -1 rad
210 0 0.05 0.1 l f l c (h) r (m) S l (10 m s ) -7 FIG. 5. Statistics of horizontal velocities in adouble layer configuration: top layer thickness h t = 7 mm, bottom layer thickness h t = 4 mm.The forcing scale is l f = 7 . E k , and(b) the third-order structure function S L , com-puted after subtracting time-averaged mean ve-locity field. The entire top layer is illuminated. tracting the time-average velocity field fromthe instantaneous velocity field, as discussedin . After the mean subtraction, the spec-trum shows E k ∼ k − / , as seen in Fig.5(a).The third-order structure function is positiveand is a linear function of the separation dis-tance r up to r ≈
70 mm, Fig.5(b). Thus thespectra and the structure functions in sucha flow agree with quasi-2D expectations andare consistent with the vertical structure ofthe flow of Fig. 4.
V. SUMMARY
We have studied spatial structure of tur-bulent flows in thick layers at low Reynoldsnumbers ( Re ∼ h/l f > .
5, in-troduce additional bottom drag due to theeddy viscosity , but they do not qualita- h h FIG. 6. Schematics of the structure of turbu-lence in thick layers. tively change the statistics of the horizontalvelocity fluctuations, which remains quasi-2Deven in the presence of 3D motions. If thebottom drag is reduced by introducing an im-miscible thin bottom layer, the inverse energycascade leads to spectral condensation andto the formation of the large scale coherentstructures. Such flows, as has recently beenshown , shear off eddies in the vertical planeand reinforce quasi-two-dimensionality of theflow. Measurements presented here, in par-ticular Fig.4, confirm this.Summarizing, flows in thick layers of flu-ids with an unperturbed free surface can beviewed as two interacting sublayers, as illus-trated in Fig.6. The top layer is quasi-2D;it supports the inverse energy cascade. Ina bounded domain at low damping, the in-verse cascade leads to spectral condensationof turbulence. The bottom sublayer is dom-inated by 3D motions which are responsiblefor the onset of the eddy viscosity. A planarcoherent flow (spectral condensate) develop-ing in the top layer can reduce the bottomlayer thickness h through shearing of the 3Deddies. The thickness of the two sublayersthus depends on the competition between thevertical shear and the 3D motions due to theforcing. In the two layer configuration, thespectral condensate formed in the top sub-layer can take over almost the entire layerthickness. ACKNOWLEDGMENTS
The authors are grateful to G. Falkovich,H. Punzmann and A. Babanin for usefuldiscussions. This work was supported bythe Australian Research Council’s Discovery7rojects funding scheme (DP0881544). Thisresearch was supported in part by the Na-tional Science Foundation under Grant No.NSF PHY05-51164.
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