Local measurement of vortex statistics in quantum turbulence
eepl draft
Local measurement of vortex statistics in quantum turbulence
Eric Woillez and
Philippe-E. Roche
Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France
PACS – Vortices and turbulence
PACS – Hydrodynamic aspects of superfluidity: quantum fluids
PACS – Transport, hydrodynamics, and superflow
Abstract – The density fluctuations of quantum vortex lines are measured in a turbulent flowof superfluid He, at temperatures corresponding to superfluid fraction of 16%, 47% and 81%.The probe is a micro-fabricated second sound resonator that allows for local and small-scalemeasurements in the core of the flow, at a 10-mesh-size behind a grid. Remarkably, all the vortexpower spectra collapse on a single master curve, independently from the superfluid fraction andthe mean velocity. By contrast with previous measurements, we report an absence of power lawscaling of the power spectra. The vortex density probability distributions are found to be stronglyskewed, similarly to the vorticity distributions observed in classical turbulence. Implications ofthose results are discussed.
Introduction. –
At zero temperature, quantum flu-ids exhibit two fascinating superfluid properties [1] : theabsence of viscous dissipation and the concentration of itsvorticity along vortex lines, of atomic diameter in the caseof He vortices. Besides, the vortices are quantized in thesense that the circulation of velocity around one vortex hasa fixed value ( κ (cid:39) − m s − in He). At finite temper-ature, the superfluid behaves as if it experienced frictionwith a background viscous fluid, called the “normal fluid”.The relative mass density of the superfluid ρ s /ρ (where ρ is the total mass density) decreases from one at 0 K to zeroat the superfluid transition temperature ( T λ (cid:39) .
18 K in He), suggesting a possible continuous cross-over betweenthe properties of quantum and classical hydrodynamics.These peculiar dissipation and vortical properties offerunique opportunities to revisit open questions in classi-cal turbulence. Thus, the last two decades have seen theemergence a new research area applying the methodol-ogy and tools of classical turbulence to quantum flows [2],and in particular the statistical study of local fluctuationsin highly turbulent canonical flows such as von K´arm´an,wakes and grid flows [3].Local velocity statistics have been successfully probedwith a variety of anemometers in highly turbulent flows,and similarities have been systematically found betweenquantum and classical turbulence (eg. [4–7]). This was at-tributed to a locking of the superfluid and normal fluid atlarge and intermediate flow scales, resulting in an appar-ent single-fluid viscous dynamics [3]. Still, the smallest scales of the flow -where quantum effects prevail- couldnot yet be resolved with existing anemometers ).To circumvent this shortcoming of anemometers, twoalternative types of probe have explored the statistics ofsmall scale features of intense quantum turbulence.First, using parietal pressure probes, the statistics ofvorticity filaments in classical turbulence have been com-pared with their quantum counterparts: superfluid vortexbundles. Here again, a strong similarity between bothtypes of turbulence was found [9].Second, using “second sound tweezers”, the spectrumof local density of superfluid vortex lines L has been mea-sured [10] for ρ s /ρ (cid:39) .
84 . Unexpectedly, the spectrumwas consistent with a ∼ f − / power law at intermediatescales, at odds with the spectrum of the absolute valueof vorticity in classical turbulence. This spectral observa-tion, studied numerically [11,12] and theoretically [13–15],has been up to now the only existing experimental con-strain for the small scale closure of the models of quantumhydrodynamics.The present study reports the first statistical character-isation of local vortex line density (VLD) statistics in well-controlled highly turbulent flow, and over a wide range of for a recent attempt, see eg. [8] Literature also reports experimental [16] and numerical [17–21]spectra of the vortex line density spatially integrated across thewhole flow. Still, spectra of such “integral” quantities differ in naturefrom the spectra of local quantities, due to strong filtering effects ofspatial fluctuations. p-1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b ric Woillez Philippe-E. Roche Turbulence generating gridHoney combRing withprobesPitot tubePump
Hebath Fig. 1: Sketch of the flow and the experimental setup withprobes. ρ s /ρ spanning from 0.16 to 0.81. Experimental setup. –
The experimental setup hasbeen described in details in a previous publication [7], andwe only review in this section the major modifications.The setup consists in a wind tunnel inside a cylindricalcryostat (see Fig. 1) filled with He-II. The flow is contin-uously powered by a centrifugal pump located at the topof the tunnel. At the bottom, an optimized 3D-printedconditioner ensures a smooth entry of the fluid, withoutboundary layer detachment, inside a pipe of Φ = 76 mminner diameter. Spin motion is broken by radial screensbuilt in the conditioner. The fluid is then “cleaned” againby a 5-cm-long and 3-mm-cell honeycomb. The mean flowvelocity U is measured with a Pitot tube located 130 mmupstream the pipe outlet. We allow a maximal mean ve-locity U = 1 . M = 17 mm meshwith square bars of thickness b = 4 mm, which gives aporosity of β = (1 − b/M ) ≈ . ∼ M downstream the grid is the result of a compromise betweenthe desire to have a “large” turbulence intensity, and thenecessity to leave enough space for turbulence to developbetween the grid and the probes. In-situ measurements ofthe mean vortex line density can be used to indirectly (via Eq. 6) give an estimation of the the turbulence intensity τ = u rms /U (cid:39) −
13% (where u rms is the standard de-viation of longitudinal velocity component). We presentthe results later in Fig. 5). For comparison, Vita and co.[22] report a turbulence intensity around τ = 9% percentsat 10 M in a classical grid flow of similar porosity. Thedifference between both values of τ could originate from aprefactor uncertainty in Eq. (6) or from differences in flowdesign (e.g. the absence of a contraction behind the hon-eycomb). This difference has no important consequencesfor the present study devoted to the measure of quantumvortex statistics, and was not further examined.The longitudinal integral length scale of the flow H (cid:39) . . Re defined with u rms H and thekinematic viscosity 1 . × − m s − of liquid He justabove T λ , is Re = 2 . × for U = 1 m/s. Using standardhomogeneous isotropic turbulence formula [23], this Re corresponds to a Taylor scale Reynolds number R λ = 690(for τ = 9% and H = 5 mm). This gives an indication ofturbulence intensity of the flow below T λ .Temperature of the helium bath is set via pressure regu-lation gates. The exceptional thermal conductivity of He-II ensures an homogeneous temperature inside the bathfor T < T λ . Two Cernox thermometers, one located justabove the pump, the other one on the side of the pipe closeto the probes, allow for direct monitoring of T . Probes. –
We have four probes to measure quantumturbulence characteristics. The first one is a miniaturePitot tube that allows for in situ measurements of velocityfluctuations for monitoring purposes. It is composed of acapillary tube of 0 . . . Fig. 2: Ring with probes. The inset is a zoom on the heatingand the thermometer plates of a second sound tweezers. quantum vortex lines inside the cavity causes an attenu-ation of the wave [24, 25] with a very minor phase shift[26]. This attenuation can be very accurately modelizedby a bulk dissipation coefficient inside the cavity denoted ξ L . The second effect is a ballistic advection of the waveout of the cavity. It is related to both an attenuation ofthe temperature oscillation and an important phase shift.Depending on the flow mean velocity U , the size of thetweezers, and the frequency of the wave, one of these twoeffects can overwhelm the other. We have thus designedtwo models of tweezers: one model to take advantage ofthe first effect to measure the vortex lines density (VLD),and the other one to take advantage of the second effectto measure the velocity.The two largest tweezers displayed in Fig. 2 are de-signed to measure the quantum vortex lines density. Theplates size is l = 1 mm and the gaps between the plates are D = 1 .
32 mm and D = 0 .
83 mm respectively. The platesface each other with positioning accuracy of a few microm-eters. The tweezers are oriented parallel to the flow (seeFig. 2) to minimize the effect of ballistic advection of thewave.The smallest tweezers displayed in Fig. 2 are designed tomeasure the velocity fluctuations parallel to the mean flow.The two plates have a size l = 250 µ m, and are separatedby a gap of D = 0 .
431 mm. The tweezers are orientedperpendicular to the mean flow (see Fig. 2) with an inten-tional lateral shift of the heater and the thermometer ofabout l/
2. This configuration is expected to maximize thesensitivity to ballistic advection, and thus to velocity fluc-tuations. However, due to excessive heating of this probe,we were not able to calibrate it reliably. Consequently wedo not use it to estimate the turbulence intensity. Thevelocity spectrum (in arbitrary units) of this probe is dis-played in the bottom panel of Fig 6. This anemometeris mostly used in the present study to assess the integralscale and qualitatively check the stability of the flow.
14 15 16 17 f (kHz) T ( m K ) -0.15 -0.1 -0.05 0 0.05 X (mK) -0.15-0.1-0.0500.05 Y ( m K ) U ( m / s ) Fig. 3:
Top: second sound resonance of the tweezers around7 . U increases from top curve to bottomcurve. The vertical axis gives the amplitude of the thermalwave in K. Bottom: representation of the same resonance inphase and quadrature.
Method. –
Figure 3 displays a resonance of a largetweezers at frequency f = 15 . T measured by the thermometer is demodulated by a Lock-in amplifier NF LI5640. T can be accurately fitted by aclassical Fabry-Perot formula T = A sinh (cid:16) i π ( f − f ) Dc + ξD (cid:17) (1)where f is the resonant frequency for which the wave lo-cally reaches its maximal amplitude, c is the second soundvelocity, A is a parameter to be fitted, and ξ is related tothe energy loss of the wave in the cavity. The top panelof Fig. 3 displays the amplitude of the thermal wave (inmK) as a function of the frequency, and the bottom panelshows the same signal in phase and quadrature. Whenthe frequency is swept, the signal follows a curve closeto a circle crossing the point of coordinates (0 , U increases, which is interpreted as attenua-tion of the wave inside the cavity. The red points displaythe attenuation of the signal at constant value of f . Itcan be seen on the bottom panel that the variation of thesignal is close to a pure attenuation, that is, without phaseshift. ξ can be decomposed as ξ = ξ + ξ L (2)p-3ric Woillez Philippe-E. Rochewhere ξ is the attenuation factor when U = 0 m/s and ξ L is the additional attenuation created by the presenceof quantum vortex lines inside the cavity. ξ L is the signalof interest as it can be directly related to the vortex linesdensity (VLD) using the relation ξ L = BκL ⊥ c , (3) L ⊥ = 1 V (cid:90) sin θ ( l )d l (4)where B is the first Vinen coefficient, κ ≈ . × − m /s is the quantum of circulation, V is the cavity volume, l is the curvilinear absciss along the vortex line, θ ( l ) isthe angle between the vector tangent to the line and thedirection perpendicular to the plates. We note that thesummation is weighted by the distribution of the secondsound nodes and antinodes inside the cavity and does notexactly corresponds to a uniform average but we neglectthis effect in the following. Our aim is to measure boththe average value and the fluctuations of L ⊥ , as a functionof U and the superfluid fraction.The method goes as follows: first, we choose a reso-nant frequency f where the amplitude of the signal hasa local maximum and we fix the frequency of the heat-ing to this value f . Then we vary the mean velocity U and we record the response of the thermometer platein phase and quadrature. The measurements show thatthe velocity-induced displacement in the complex planefollows a straight line in a direction −→ e approximately or-thogonal to the resonant curve. Expressions (1-2) give ξ L from the measured amplitude T by [10] ξ L = 1 D asinh (cid:18) AT (cid:19) − ξ . (5)The colored dots of Fig. 4 illustrate the fluctuations ofthe signal in phase and quadrature, for different values of U . The average signal moves in the direction of the atten-uation axis. The figure also shows a part of the resonantcurve for U = 0. The fluctuations have two components inthe plane, both associated with different physical phenom-ena. Fluctuations in the direction tangent to the resonantcurve can be interpreted as a variation of the acousticpath π ( f − f ) Dc without attenuation of the wave. Thosefluctuations can occur for example because the two armsof the tweezers vibrate with submicron amplitude, or be-cause the temperature variations modify the second soundvelocity c . To isolate only the fluctuations associated toattenuation by the quantum vortices, we split the signalinto a component along the attenuation axis, and anotherone along the acoustic path axis. We then convert thedisplacement along the attenuation axis into vortex linedensity (VLD) using expressions (3-5). Results. –
As a check of the validity of our ap-proach, we measured the average response of the secondsound tweezers as a function of the mean velocity U . Ac-cording to literature [27], we were expecting the scaling -2 -1.8 -1.6 -1.4 -1.2 X (V) -6 -12-10-8 Y ( V ) -7 Attenuationaxis0.90 m/s1.20 m/s0 m/s0.26 m/s0.48 m/sAcoustic pathaxis
Fig. 4: Fluctuations of the thermal wave in phase and quadra-ture. The colored clouds show the fluctuations of the signal,for different values of U . The blue curve shows the resonancefor U = 0 m/s. The fluctuations tangent to the resonant curveare created by a variation of the acoustic path. The quantumvortices are associated to attenuation of the wave and create adisplacement along the attenuation axis. (cid:104) L ⊥ (cid:105) ∝ U , with a prefactor related to the flow maincharacteristics. The function (cid:104) L ⊥ (cid:105) was thus measured fora range 0 . < U < .
25 m/s with a time averaging over300 ms, at the three different temperatures 1 .
65 K, 1 .
99K and 2 .
14 K.An effective superfluid viscosity ν eff is customarily de-fined in quantum turbulence by (cid:15) = ν eff ( κ L ) where (cid:15) isthe dissipation and L = 3 (cid:104) L ⊥ (cid:105) / R λ ho-mogeneous isotropic flows, we also have (cid:15) (cid:39) . U τ /H (eg see [23] p.245), which entails τ (cid:39) . ν eff Hκ (cid:104) L ⊥ (cid:105) U (6)Using Eq. (6), we compute the turbulence intensity asa function of U , for the three considered temperatures.The result is displayed in Fig. 5. The figure shows thatthe turbulence intensity reaches a plateau of about 12%above 0 . (cid:104) L ⊥ (cid:105) ∝ U is reached in our experimentfor the range of velocities U > . ν eff in Eq. (6)has been measured in a number of experiments (eg seecompilations in [15, 27, 29]). Still, the uncertainty on itsvalue exceeds a factor 2. For the temperatures 1 .
65 K and1 .
99 K, we used the average values 0 . κ and 0 . κ . Bylack of reference experimental value of ν eff above 2 . τ ( U ) datasets obtained at2 .
14 K with the two others. We found the value ν eff ≈ . κ p-4ocal measurement of vortex statistics in quantum turbulence U (m/s) u r m s / U ( % ) Fig. 5: Indirect measurement of the turbulence intensity τ = u rms /U as a function of U using Eq. (6). The three differentsymbols correspond to three values of the mean temperature. at 2 .
14 K.Assuming isotropy of the vortex tangle, the value of L gives a direct order of magnitude of the inter-vortex spac-ing δ = 1 / √L . We find δ ≈ µm at 1.65 K and a meanvelocity of 1 m/s. This shows the large scale separation be-tween the inter-vortex spacing and the flow integral scale H , a confirmation of an intense turbulent regime.Fig. 6 presents the main result of this letter. We displayon the top panel the VLD power spectral density P L ( f )of L ⊥ / (cid:104) L ⊥ (cid:105) . With this definition, the VLD turbulenceintensity L rms ⊥ / (cid:104) L ⊥ (cid:105) is directly given by the integral of P L ( f ). We have measured the VLD fluctuations at thetemperatures T = 1 .
65 K and superfluid fraction ρ S /ρ =81%, T = 1 .
99 K and ρ S /ρ = 47%, T = 2 .
14 K and ρ S /ρ = 16%. At each temperature, the measurement wasdone for at least two different mean velocities.The first striking result is the collapse of all the spectraindependently of the temperature, when properly rescaledusing f /U as coordinate (and P L ( f ) × U as power spectraldensity to keep the integral constant). The VLD spectrumdoes not depend on the superfluid fraction even for van-ishing superfluid fractions, when T comes very close to T λ .Only one measurement with one of the large tweezers at T = 1 .
650 K has given a slight deviation from the mastercurve of the VLD spectra: it is displayed as the thin greycurve in Fig. 6. We have no explanation for this devia-tion but did not observe this particular spectrum with thesecond tweezers, and neither at any other temperature.Second, the VLD spectrum has no characteristic power-law decay. We only observe that the spectrum follows anexponential decay approximately above f /U >
100 m − .This strongly contrasts with the velocity spectrum ob-tained with the small second sound tweezers anemometer(see bottom panel), which displays all the major featuresexpected for a velocity spectrum in classical turbulence:it has a sharp transition from a plateau at large scale toa power law scaling close to − / H = 5mm for the longitudinal integral scale. As a side remark,the apparent cut-off above 10 m − is an instrumental fre-quency cut-off of the tweezers.We find a value of the VLD turbulent intensity close to20%, which is significantly higher than the velocity turbu-lence intensity. We also checked that we obtain the sameVLD spectrum using different resonant frequencies f .Our measurements are limited by two characteristic fre-quencies. First, the tweezers average the VLD over a cubeof side l , which means that our resolution cannot exceed f /U > /l . For the large tweezers, this sets a cut-off scaleof 10 m − , much larger than the range of inertial scalespresented in top panel of Fig. 6. Second, the frequencybandwidth of the resonator decreases when the quality fac-tor of the second sound resonance increases. This againsets a cut-off scale given by f /U = ξ c / (2 U ). The worstconfiguration corresponds to the data obtained at 2.14 Kand U = 1 . m − .For this reason, the VLD spectra of Fig. 6 are conserva-tively restricted to f /U <
300 m − which allows to resolveabout one and a half decade of inertial scales.Figure 7 displays some typical PDF of the rescaled VLDfluctuations L ⊥ / (cid:104) L ⊥ (cid:105) in semilogarithmic scale, for thethree considered temperatures. The PDF have been verti-cally shifted by one decade from each other for readability.The figure shows a strong asymmetry at all temperatures,with a nearly Gaussian left wing, and an exponential rightwing. Contrarily to the VLD spectra, the PDF do not ac-curately collapse at different velocities and temperatures:only the global asymmetric shape seems to be a robustfeature. We do not see a clear trend when increasing thetemperature and/or the mean velocity. By contrast, thedotted curve in Fig. 7 displays one PDF of the smalltweezers anemometer at 1 .
65 K, for which the mean hasbeen shifted and the variance rescaled. It can be seen thatthe general shape of this latter PDF is much more sym-metric and closer to a Gaussian as expected for a PDF ofvelocity fluctuations.
Discussion and conclusion. –
In the present pa-per, we have investigated the temperature dependence ofthe statistics of the local density of vortex lines (VLD)in quantum turbulence. About one and a half decadeof inertial scales of the turbulent cascade was resolved.We measure the VLD mean value and deduce from Eq.(6) the turbulence intensity (Fig. 5), we report the VLDpower spectrum (Fig. 6), and the VLD probability distri-bution (Fig. 7). Whereas the VLD mean value at differ-ent temperatures confirms previous numerical [11, 27] andexperimental studies [27], the spectral and PDF studiesare completely new. Only one measurement of the VLDfluctuations had been done previously [10] but in an ill-defined flow around 1.6K.In the present work, we haveused a grid turbulence, which is recognized as a referenceflow for isotropic turbulence.p-5ric Woillez Philippe-E. Roche -3 P L ( f ) . U ( m ) -4 -5 ρ s/ ρ =81% (1.65 K) ρ s/ ρ =47% (1.99 K) ρ s/ ρ =16% (2.14 K) f/U (m -1 ) -14 -16 -18 P U ( f ) / U ( m ) f/U (m -1 ) -5/30.26 m/s0.48 m/s fi t H -5/3 Fig. 6:
Top:
Power spectral density of the projected vortexline density (VLD) L ⊥ , obtained with the large second soundtweezers, for different values of U and temperatures. All mea-sured spectra collapse using the scaling f/U and P L ( f ) × U .The fluctuations have been rescaled by the mean value of theVLD such that the integral of the above curves directly givethe VLD turbulence intensity. Bottom:
Power spectral den-sity of the uncalibrated velocity signal obtained from the sec-ond sound tweezers anemometer, for two values of U at 1.65 K.The spectra collapse using the scaling f/U for the frequencyand P U ( f ) /U for the spectral density. The straight line dis-plays the − / H . -4 -2 Fig. 7: Normalized probability distributions of the VLD fluc-tuations obtained at three temperatures. The PDF have beenshifted by one decade from each other for readability. By com-parison, the dotted black curve displays a rescaled PDF ob-tained with the small tweezers measuring velocity.
To conclude, we discuss below three main findings:1. A master curve of the VLD spectra, independent oftemperature and mean velocity.2. An absence of power law scaling of the VLD spectra.3. A global invariant shape of the strongly skewed PDF.The mean VLD gives the inter-vortex spacing, and thustells how much quantum vortices are created in the flow,whereas the PDF and spectra tell how those vortices areorganized in the flow. From 2.14K to 1.65K, our re-sults confirm that the inter-vortex spacing only weaklydecreases, by less than 23% for a 5-times increase of thesuperfluid fraction. In other words, the superfluid fractionhas a limited effect on the creation of quantum vortices.The current understanding of the homogeneous isotropicturbulence in He-II is that the superfluid and normal fluidare locked together at large and intermediate scales wherethey undergo a classical Kolmogorov cascade [3]. The ex-perimental evidences are based on the observation of clas-sical velocity statistics using anemometers measuring thebarycentric velocity of the normal and superfluid compo-nents. Here, the temperature-independence of (normal-ized) VLD spectra supports this general picture, by remi-niscence of a similar property of He-II velocity spectra.However, in contrast with velocity, this also revealssome temperature-independence of scales smaller than theprobe spatial resolution. For instance, positive and neg-ative velocity fluctuations (e.g like those around a vor-tex core) smaller than the probe resolution are stronglydamped by averaging , while our VLD probe returns thesum of a positive quantity and keeps track of the smallscales fluctuations. Besides, the intermediate scales of 1Dvorticity spectra in classical turbulence are related to thevelocity spectrum at small scales (eg. see [30]) and thesame property is expected to hold in quantum turbulence,in a form yet to be detailed.The observed temperature-independence of spectra isthus constraining for the delicate modeling of the smallscales of quantum turbulence, in particular to developmathematical closures for the continuous description ofHe-II (eg. see [31]).Second, the absence of power law scaling apparentlycontrasts with the spectra reported as “compatible with”a f − / scaling in [10]. We have no definite explanation forthis difference. Still, we note that the averaged spectraldecrease reported in the present study over one and a halfdecade of inertial scales is close to the decrease reportedin [10] over a similar range. Speculatively, the absence ofcomplete development of turbulence at 10 M from the gridcould result in residual imbalance between flow scales, orfrom some memory effect associated with flow forcing atthe grid.As a discussion of the third statement, we compare ourresults with those of numerical simulations done in classi-cal turbulence. The absolute value of vorticity can be seenp-6ocal measurement of vortex statistics in quantum turbulenceas a classical counterpart to the VLD. The work of Iyerand co. [32] for example, displays some enstrophy PDFfrom high resolution DNS, that can be compared to thePDF of Fig. 7. It can be seen in [32] that such PDF arenot universal and depend on the Re number and the aver-aging scale: this probably justifies why the distributionsof Fig. 7 do not collapse. At small scale, the enstrophyPDF are strongly asymmetric and will ultimately convergeto a Gaussian distribution when averaged over larger andlarger scales. Although our tweezers average the VLD overa size much larger than the inter-vortex spacing, they aresmall enough to sense short-life intense vortical events,typical of small scale phenomenology in classical turbu-lence. Thus, the strong asymmetry of the PDF supportsthe analogy between VLD and enstrophy (or its squareroot) and shows the relevance of VLD statistics to explorethe small scales of quantum turbulence.A side result of the present work is to obtain the rela-tive values of the empirical coefficient ν eff = (cid:15) ( κ L ) − atthe three considered temperatures. Models and simula-tions predict that ν eff should steeply increase close to T λ (see [15,27,29] and ref. within), in apparent contradictionwith the only systematic experimental exploration [33].We found in Fig. 5 that the effective viscosity ν eff is twicelarger at 2.14K than at 1.99K. To the best of our knowl-edge, our estimate ν eff (2 . K ) (cid:39) ± . × ν eff (1 . K )is the first experimental hint of such an effective viscosityincrease. ∗ ∗ ∗ We warmly thank J´erˆome Valentin for the process de-velopment and micro-fabrication of the components of thesensors, B. Chabaud for support in upgrading the wind-tunnel and P. Diribarne, E. L´evˆeque and B. H´ebral fortheir comments. We thank K. Iyer with his co-authorsfor sharing data on the statistics of spatially averaged en-strophy analyzed in [34]. Financial support from grantsANR-16-CE30-0016 (Ecouturb) and ANR-18-CE46-0013(QUTE-HPC).
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