Detection of vortex coherent structures in superfluid turbulence
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Detection of vortex coherent structures in superfluid turbulence
E. Rusaouen , B. Rousset and P.-E. Roche Institut NEEL, CNRS, Universit´e Grenoble Alpes, F-38042 Grenoble, France SBT/INAC CEA, Universit´e Grenoble Alpes, F-38054 Grenoble, France
PACS – Turbulent flows:Coherent structures
PACS – Hydrodynamic aspects of superfluidity; quantum fluids
PACS – Vortices and turbulence
Abstract – Filamentary regions of high vorticity irregularly form and disappear in the turbulentflows of classical fluids. We report an experimental comparative study of these so-called “coherentstructures” in a classical versus quantum fluid, using liquid helium with a superfluid fractionvaried from 0% up to 83%. The low pressure core of the vorticity filaments is detected by pressureprobes located on the sidewall of a 78-cm-diameter Von K´arm´an cell driven up to record turbulentintensity ( R λ ∼ √ Re (cid:39) Introduction. –
Motivation.
Turbulent flows of water, air or otherclassical fluids are populated by so-called “coherent struc-tures”. These structures are localized in space and charac-terized by an organized flow motion. In particular, worm-shaped regions of high vorticity -often referred as “vortexfilaments”- irregularly spring up, and after a life-time sig-nificantly larger than their turn-over time, destabilize andvanishe [1–5].A few numerical studies of superfluid helium have shownthat bundles of quantum vortex lines should be the coun-terparts of classical vortex filaments in quantum fluids.The formation of such bundles in a freely evolving quan-tum fluid have been recently reported in Ref. [6]. This re-sult was preceded by a number of numerical studies wherean external field was promoting the formation of vortexbundles in a superfluid (e.g. see Ref. [7, 8]).The motivation of the present study is to detect exper-imentally coherent structures in quantum turbulence.
Experimental context.
The comparison between clas-sical and quantum (or superfluid) turbulence has focuseda lot of attention over the last years [9]. Regarding ex-perimental studies of turbulent fluctuations, the situationis contrasted [10]. On the one hand, several similaritieshave been reported including on velocity spectra [11, 12] and energy transfer between eddies of different sizes [13].On the other hand, differences between classical and quan-tum turbulences are reported when vorticity (instead thanvelocity) is directly or indirectly probed, by spectral mea-surements of the vortex line density [14, 15] and by visu-alization of reconnections of individual vortices [16, 17].In this context, coherent vortex structures are interest-ing objects to compare classical and quantum turbulence.Indeed, a bundle of quantum vortices is an intermedi-ate structure living between the quantum scales (wherea quantized vortex line can move without dissipation) andmacroscopic scales (where classical turbulent propertiesare expected).
Methodology.
We use liquid helium He, both aboveits superfluid transition (where it is a classical fluid) andbelow it, where it acquires properties of a quantum fluid[18,19]. In the later case, according to the two-fluid modelof Landau and Tisza, it behaves as an intimate mixtureof a ”normal” fluid and a ”superfluid”, which are coupledby a mutual friction force. The normal fluid follows theNavier-Stokes equation, while the superfluid has zero vis-cosity and can be described as a tangle of quantized vor-tex lines. In the zero temperature limit, the normal fluiddensity (volumetric mass) ρ n vanishes and He becomes apure superfluid. Conversely, near the transition tempera-p-1 a r X i v : . [ c ond - m a t . o t h e r] M a y . Rusaouen , B. Rousset and P.-E. Roche ture ( (cid:39) K ), the superfluid density ρ s = ρ − ρ n vanishes.In the present study, the superfluid fraction ρ s /ρ variedfrom 0% to 83% (2 . K ≥ T ≥ . K ).To detect coherent vortex structures, we look for the lowpressure appearing in their core due to centrifugal force.This pressure depletion can be assessed from the Poissonequation for pressure p in an incompressible flow [20], de-rived by taking the divergence of Navier-Stokes equation(a generalization for compressible flow is proposed in [21]):∆ p = ρ (cid:0) ω − σ (cid:1) (1)where ρ are the fluid density, ω , and σ are the flow vorticityand rate of strain defined as ω = 12 (cid:88) i,j ( ∂ i v j − ∂ j v i ) (2) σ = 12 (cid:88) i,j ( ∂ i v j + ∂ j v i ) (3)By analogy with electrostatics, equation 1 shows that alocalized region of high vorticity is a (negative) sourceterm for pressure . The technique of tracking low pressurespikes to detect coherent structures has been widely usedin classical turbulent flows, in particular the Von K´arm´angeometry (eg. see Ref. [23–28]). In practice, a pressuretransducer is imbedded in the sidewall of the cell; when avortex filament passes by the probe, the resulting negativespike greatly exceeds in magnitude the standard deviationof the pressure fluctuations generated by the“background”turbulence. Thus, the vortex filament can be detected.Generalization of this equation in a quantum fluid atfinite temperature is straightforward in the framework ofHVBK equations, discussed in [29]. In this approach, thesuperfluid tangle is coarse-grained into continuous velocity (cid:126)v s and vorticity (cid:126)ω s fields. The detail of individual vorticesis lost but the resulting equation for the superfluid canaccount for fluid motion at scales much larger than thetypical inter-vortex distance. The HVBK equations arean Euler equation for the superfluid (underscript s ) anda Navier-Stokes equation for normal fluid (underscript n ),both coupled together : ρ s (cid:2) ( ∂(cid:126)v s /∂t ) + ( (cid:126)v s · ∇ ) (cid:126)v s (cid:3) = − ρ s ρ ∇ p + ρ s S ∇ T − (cid:126)F (4) ρ n (cid:2) ( ∂(cid:126)v n /∂t )+( (cid:126)v n ·∇ ) (cid:126)v n (cid:3) = − ρ n ρ ∇ p − ρ s S ∇ T + (cid:126)F + µ ∇ (cid:126)v n (5)where µ is the dynamic viscosity, S is the entropy, andwhere the coupling term (cid:126)F accounts mutual coupling.Assuming incompressibility, and taking the divergenceof the sum of Eq. 4 and 5, one gets a generalized Poissonequation in the two-fluid model : contrary to a frequent assumption, ω and σ don’t balance eachother on average in closed flows [22]. ∆ p = ρ s (cid:0) ω s − σ s (cid:1) + ρ n (cid:0) ω n − σ n (cid:1) (6)The above equations shows that negative pressure spikesin a quantum fluid remain markers of high vorticity re-gions. Superfluid and normal fluid vorticities are probedsimultaneouly, and weighted in proportion of the densityof each fluid. Note that the low pressure on individualquantum vortices has been invoked to explain the trap-ping of light particles along vortices (see [16, 30, 31] andreferences within). Experimental set-up. –
The Von K´arm´an flow.
The Von K´arm´an flow usedfor this experiment has been extensively described in adedicated paper [32]. We only recall below its main spec-ifications, see figure 1.
Pumped He bathPressurized HeI / He II (Ø 780 mm cell)Bottom propellerHeat exchanger mm Parietal pressure probes ( Ø 1 mm tap holes 34 mm below equator)Transmission shaftTop propellerMixing layer
Fig. 1: Schematic of the experiment.
The liquid helium He used in this experiment was se-quentially set to temperatures of 2.4 K, 2.1 K and 1.6K, that is both above and below the superfluid transitiontemperature ( T λ (cid:39) . K at 3 bars). These three tem-peratures correspond respectively to superfluid fractionsof 0%, 19% and 80% at the pressures of interest (see Ta-ble 1). The pressurization of the flow prevents occurrenceof cavitation for all flow conditions.The flow is enclosed in a 780-mm-diameter cylindri-cal vessel and it is mechanically stirred by two co-axialbladed-disks of radius R = 360 mm , located 702 mm away,counter-rotating in this work. The 8 blades on each diskare curved, and the direction of rotation is such that theconvex side of the blades moves into the fluid. This spe-cific direction is chosen because it results in a stable largescale circulation between the disks [32].p-2etection of vortex coherent structures in superfluid turbulenceSuch a stirring gives rise to two counter-rotating sub-flows separated by a mixing layer, as depicted in Fig. 1.The (mean) position of this mixing layer is determined bythe relative angular velocities Ω b and Ω t of the bottomand top disks. For exact counter-rotation (Ω b = Ω t ), themixing layer is located at mid-height. In this study, we setΩ b > Ω t , to position the mixing layer above the mid-planeaway from the probes which are located 34 mm below thismid-plane. The relative angular velocity of the disks ischaracterized by : θ = Ω b − Ω t Ω b + Ω t (7)The parameter θ was set to 11-12% and 20% to probe theflow at two distances from the mixing layer. In classicalVon K´arm´an flow, the Reynolds number is often definedas : Re = ρR (Ω b + Ω t )2 µ = ρR Ω µ (8)where ρ is the density of the fluid and Ω is the mean an-gular velocity. For our purposes, this definition remainsa convenient control parameter below the superfluid tran-sition. Indeed, at large scales, the superfluid and nor-mal fluid are strongly locked by the mutual coupling forcewhich make them behave as a single fluid of viscosity µ [8, 33].The flow parameters θ and Re used in the present studyare given in Table 1. We stress that this study is performedat ultra large Reynolds number, of order Re (cid:39) rarelyreached in laboratory conditions. Following [34], the typ-ical Taylor microscale Reynolds number can be assessedfrom Re as R λ (cid:39) (cid:112) ( Re ) (cid:39) Instrumentation. –
The parietal pressure probes.
Fluctuations of parietalpressure are monitored at two locations, both 34 mm be-low the mid-plane and at 80 mm from each other (mea-sured along the sidewall circumference). At each location,a differential transducer senses the pressure difference be-tween an orifice in the sidewall and a pressure reference.The pressure reference is low-pass-filtered by an hy-draulic impedance so that its mirrors the static pressureinside the flow, and follows its possible slow drift. Fromthe spectral analysis of the measured pressure fluctuations,this lower cut-off frequency of the probe is estimated tobe significantly lower than 100 mHz.The orifice in the flow sidewall is a square-edge 1-mm-diameter hole, perpendicular to the wall, with an effectivedepth of around 20 mm. The membrane of the pressuretransducer is mounted at the end of this connecting pipe.The Helmholtz resonance is close to 1 kHz and mechanicalvibrations of the transducers are damped using a mechan-ical filter.In practice, the largest useful frequencies of the mea-sured signal was not limited by the probe itself but by broadband pressure oscillations in the flow, in the hun-dreds of Hertz range. Those oscillations were probablyoriginating from the cryogenic system maintaining the ex-periment cold.
Electronics and acquisition.
Each piezo-resistive pres-sure transducer consists in a Wheatstone bridge layingover a deflecting membrane. Each bridge is polarized bya battery-based ∼
350 mA current source. The bridgeoutput voltage is amplified using a low noise instrumenta-tion preamplifier (0.6 nV / √ Hz , model EPC1-B). A 8th-order linear-phase anti-alias filter at frequency f c (Kemo1208/20/41LP) is inserted before an 18-bits acquisitionboard (National Instrument 6289). Acquisitions are per-formed at sampling frequency 20 kHz (with f c = 6 kHz)and last between 25 and 45 min, except for a few sampledat 1 kHz (with f c = 200 Hz) for practical reasons. Alltimes series are post-processed by a numerical low-passfilter at 160 Hz to avoid possible post-processing artifactcaused by the Helmholtz resonance. Results. –
Detection of coherent structures in turbulent superfluid.
We first discuss the classical flow regime ( ρ s /ρ = 0).The red time series plotted on Figure 2 illustrated therecording of several sharp depressions during 100 rotationsof the disks. The time axis is scaled by 2 π/ Ω so that itcorresponds to a number of turns of the disks.
Time [number of turns] P [ a r b i t r . un i t s and o ff s e t] ρ s / ρ = 0 % , Re = 6.6e7 [ θ =0.12] ρ s / ρ = 19 %, Re = 8.6e7 [ θ =0.12] ρ s / ρ = 83 %, Re = 8.9e7 [ θ =0.11] Fig. 2: Pressure time series at 3 temperatures for roughly sim-ilar forcing. The superfluid fraction ranges from 0% to 84%.Time on the x axis is rescaled by the mean rotation time 2 π/ Ωof the disks. The sharp depressions are interpreted as the sig-nature of vortical coherent structures passing over the pressuretap.
Two possible artifacts of the measurements are acousticnoise within the fluid and mechanical noise propagatingalong the mechanical structure of the experiment. Pres-sure fluctuations were simultaneously recorded from twonearby sensors (as previously done in [24], for example),p-3. Rusaouen , B. Rousset and P.-E. Roche Table 1: Characteristics of the times series.
Superfluid Temperature Pressure Reynolds Rotation Mean Azimutal Skewness Flatnessfraction number dissym. rotation velocity ρ s /ρ [K] [Bar] Re θ Ω [rad/s] V (cid:63) [m/s]0 % 2 .
42 ( > T λ ) 3.4 5 . × .
41 ( > T λ ) 3.4 5 . × .
46 ( > T λ ) 3.6 6 . × .
10 ( < T λ ) 2.7 5 . × .
10 ( < T λ ) 2.7 8 . × .
10 ( < T λ ) 2.8 1 . × .
64 ( < T λ ) 3.0 1 . × .
64 ( < T λ ) 3.0 1 . × .
58 ( < T λ ) 3.1 8 . × V (cid:63) using the 8 cm probe separation. It is found in the m.s − range, as given in Table 1. With V (cid:63) = 1 . m.s − and taking 160 Hz as the effective noise-free probe dy-namics, we find a noise-free effective probe resolution of1 cm but wavelet analysis of the raw time series (withoutthe 160 Hz low-pass filter) allows to track the signature ofthe depression nearly up to the (cid:39) kHz probe resonancefrequency, showing that the coherent structures can be atleast as thin as 1 . m.s − / kHz (cid:39) mm , to be comparedwith the large scale L of such Von K´arm´an flows ( [28]). L (cid:39) R/ (cid:39) mm, (9)and to rough estimates of the Taylor and Kolmogorov dis-sipative scales λ and η based on the homogeneous isotropicturbulence equations. λ ∼ L · (cid:112) /Re (cid:63) (cid:39) . mm (10) η ∼ L/Re (cid:63) / (cid:39) − mm (11)where we took Re (cid:63) = LV (cid:63) ρ/µ (cid:39) . · . Surely, the flowis neither homogeneous nor isotropic, but these equationscan still provide useful orders of magnitude, and show thatthe present probe is partly resolving the inertial range ofthe turbulent cascade, which extends from ∼ L down to ∼ η .We now address the superfluid regime. Figure 2 illus-trates two typical times series with superfluid fractions of ρ s /ρ = 19% and 83% acquired at Reynolds numbers sim-ilar to the classical regime ( Re = 7 . ± -10 -5 010 -4 -3 -2 -1 P [standard deviation unit] P r obab ili t y den s i t y ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.12] ρ s/ ρ = 0 %, Re=6.6e7 [ θ =0.12] ρ s/ ρ = 19 % Re=5.9e7 [ θ =0.20] ρ s/ ρ = 19 % Re=8.6e7 [ θ =0.12] ρ s/ ρ = 19 % Re=1.1e8 [ θ =0.12] ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.20] ρ s/ ρ = 79 % Re=1.3e8 [ θ =0.11] ρ s/ ρ = 83 % Re=8.9e7 [ θ =0.11]gaussian (standard deviation=1) ρ s/ ρ = 0 %, Re=5.5e7 [ θ =0.20] Fig. 3: Probability density function (pdf) of the pressurefluctuations normalized to unity standard deviation.
Histogram of pressure : density and strength of coher-ent structures.
Figure 3 shows the probability densityfunctions (pdf) of pressure time-series normalized by thestandard deviation of their positive pressure fluctuations.The pdf shape is compatible with the description givenin classical turbulence literature for Von K´arm´an flows[23, 24, 26–28]. It can be approximated as gaussian com-plemented with a long exponential tail associated to therare but intense negative pressures spikes associated withthe coherent structures. Such skewed pressure pdf havebeen reported in a number of classical turbulent flows,for instance in homogeneous isotropic turbulence [35, 36],p-4etection of vortex coherent structures in superfluid turbulencealong the centerline of pipes [37] and in jets [38] . Oneadvantage of the Von K´arm´an geometry over these otherflows is the efficient generation of vortex filaments in itsmixing layer, and the resulting significant enhancement ofthe pressure skewness compared to the background skew-ness resulting from the quadratic velocity dependence ofpressure [40].Whatever the superfluid fraction and Reynolds num-ber, all the pdf corresponding to a given θ are found tocollapse, up to our statistical uncertainty. In other words,the density and strength of coherent structures are foundindependent of the superfluid fraction from 0% up to 83%of superfluid. This is the second important result of thisstudy. When θ is lowered, the mixing layer gets closerto the probes and the density of coherent structures in-creases. It suggests that the mixing layer is an intensesource of coherent structures, both in classical and super-fluid turbulence. The dependence with the distance to themixing layer can then be understood as the result of the fi-nite life time [27] of the vortical coherent structures. Thisprovides an indirect indications that the lifetime of the co-herent structures is similar in the classical and superfluidcases.The asymmetry and flatness of the pdf can be assessedquantitatively from two statistical quantities : the skew-ness and kurtosis of the pressure fluctuations. They arerespectively defined as the centered third and fourth mo-ment of the fluctuations normalized by their standard de-viation. Re sk e w ne ss Re f l a t ne ss Fig. 4: Upper (lower) plot : skewness (flatness) of pressurefluctuations. The open (full) symbols correspond to measure-ments in superfluid (in classical liquid helium). The square-shaped (circle-shape) symbols are for a differential rotation pa-rameter of θ = 0 . θ = 0 . − . Figure 4 shows the measured skewness and flatness (kur-tosis) below and above the superfluid transition tempera-ture. The numerical values are given in Table 1. Appli- in boundary layers more symmetrical pdf can be found, see e.g.[37, 39] cation of an additional 160 Hz/ (cid:39) Hz low-pass filteron the time series don’t alter significantly those quanti-ties suggesting that we don’t have time resolution issues.For a given value of θ , no Reynolds number dependenceemerges from our measurements when Re ∼ is variedby a factor 2.3, justifying a-posteriori that the definitionof a Reynolds number below the superfluid transition isnot critical in the present study. On the contrary, the de-pendence of both parameters versus θ is around a factor3. δ T [in numbers of turns] c oun t s Classical fluid ( ρ s/ ρ =0%), Re = 5.5e7 [ θ =0.12]Superfluid ( ρ s/ ρ = 83 %), Re = 8.9e7 [ θ =0.11]independent events stat. (1 mean occurence / 11.5 turns) Fig. 5: Histogram of the intervals between successive coherentstructures which are larger that δT . To improve statisticalconvergence, the statistics from two pressure taps (thin lines)have been averaged (thicker line). The dash line correspondsto the expected dependence of independent events with a meanseparation of 11.5 mean rotations (see text). Spatial distribution of superfluid coherent structures.
To go one step further in the comparison of coherent struc-tures, we now address their relative spatial distribution inthe classical and superfluid regimes. To this end, we focuson the statistics of time interval δT between two consecu-tive coherent structures passing by one probe. We need tochose an arbitrary criterion for identification of coherentstructures. Several criteria have been proposed and stud-ied in the classical turbulence literature, with little inci-dence in the respective conclusions (eg. see [24–26, 41]).Following [26], we choose a pressure threshold at -3 instandard deviation units. Larger thresholds of 4 and 5were also tested and gave compatible results but with aworst statistical convergence. In Figure 5, the y-axis rep-resents the number of intervals between successive coher-ent structures which are larger than δT (x-axis). For bestconvergence, the longest time series at temperatures cor-responding to 0 % and 83 % superfluid have been chosenand the times series from the two probes (thin lines) wereaveraged together (thick lines).If coherent structures were fully independent from eachother, we expect a Poisson statistics for the intervals p ( δT ) ∼ e − δT/τ . By integration, the probability of in-p-5. Rusaouen , B. Rousset and P.-E. Roche terval larger than δT is proportional to τ e − δT/τ . Thisexponential law accounts reasonably well for the resultsfor intervals δT longer than a characteristics correlationtime of ∼
10 mean rotation periods, in good agreementwith classical turbulence literature [26, 41]. A fit gives amean separation time τ = 11 . ± . Concluding remarks. –
If the pressure probes wereable to resolve individual quantum vortices, dissipativescales or the genuine pressure profile of a vortex bundle,some differences between measurements in a classical andin a quantum flows would be apparent. Obviously, theresolution of present probes is not such, but we showedthat it is sufficient to clearly detect the individual co-herent structures, from their measured (low-pass-filtered)pressure profile. Thus, the statistics of occurrence andstrength of coherent structures could be characterized andwe found that they are statistically indistinguishable whenmeasured in a classical flow and with a superfluid fractionof 19% and 79% to 83%. In other words, the microscopicdifferences in internal structures of classical vorticity fila-ment and superfluid vortex bundles do not prevent bothtypes of coherent structures to recover similar macroscopicproperties.Among the perspectives, it would be interesting to re-late this findings to the unexpected f − / vortex line spec-tra [14], which have been interpreted as passive scalarspectra postulating that a large amount for vorticity waslocalized at small scales and carried by the flow [42, 43].The presence of vortex bundles could support very muchthis interpretation (for an alternative interpretation, see[44]). Another interesting perspective is to explore tem-peratures around 1.9K where a singular behavior has beennumerically predicted for intermittency [45, 46], but notyet evidenced experimentally [11, 47]. A third perspectivewould be understand the dissipative interaction betweenthe bundles of superfluid vortices and the (possibly over-lapping) filaments of normal fluid. Acknowledgements. –
Financial support from ECEuhit project (WP21) is acknowledged, and special thanksits coordinator E. Bodenschatz for his initiative. We alsothank members of the SHREK collaboration, with whomthe facility was designed [32], to M. Bon Mardion for facil-ity operation, to A. Girard for Euhit aspects, to P. Dirib-arne and M. Gibert for support in data-logging flow pa-rameters and to B. H´ebral for discussions and proof read-ing. We warmly acknowledge the help from O. Cadot inunderstanding better the origins of the skewness of pres-sure, and the feed-back from Y. Tsuji.
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