Determination of the effective kinematic viscosity for the decay of quasiclassical turbulence in superfluid 4 He
aa r X i v : . [ c ond - m a t . o t h e r] F e b Determination of the effective kinematic viscosity for the decay of quasiclassicalturbulence in superfluid He J. Gao,
1, 2
W. Guo ∗ ,
1, 2 and W.F. Vinen National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA Mechanical Engineering Department, Florida State University, Tallahassee, FL 32310, USA School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom (Dated: February 4, 2020)The energy dissipation of quasiclassical homogeneous turbulence in superfluid He (He II) iscontrolled by an effective kinematic viscosity ν ′ , which relates the energy decay rate dE/dt to thedensity of quantized vortex lines L as dE/dt = − ν ′ ( κL ) . The precise value of ν ′ is of fundamentalimportance in developing our understanding of the dissipation mechanism in He II, and it is alsoneeded in many high Reynolds number turbulence experiments and model testing that use He II asthe working fluid. However, a reliable determination of ν ′ requires the measurements of both E ( t )and L ( t ), which was never achieved. Here we discuss our study of the quasiclassical turbulence thatemerges in the decay of thermal counterflow in He II at above 1 K. We were able to measure E ( t )using a recently developed flow visualization technique and L ( t ) via second sound attenuation. Wereport the ν ′ values in a wide temperature range determined for the first time from a comparisonof the time evolution of E ( t ) and L ( t ). PACS numbers: 67.25.dk, 29.40.Gx, 47.27.-i
Below about 2.17 K, liquid He transits to the super-fluid phase (He II) in which an inviscid irrotational su-perfluid component (i.e. the condensate) coexists with aviscous normal-fluid component (i.e. the thermal excita-tions) [1]. The fraction of the normal fluid drops drasti-cally with decreasing temperature and only amounts toabout 0.7% of the total density at 1 K [2]. This quantumfluid system exhibits fascinating hydrodynamic proper-ties. For instance, the rotational motion of the superfluidin a simply-connected volume can occur with the forma-tion of topological defects in the form of vortex lines.These vortex lines all have identical cores with a radius a ≃ κ =10 − cm /s [3]. Turbulence in the superfluidtherefore takes the form of an irregular tangle of vortexlines (quantum turbulence). Turbulence in the normalfluid is expected to be more similar to that in a classicalfluid, but a force of mutual friction between the two flu-ids, arising from the scattering of thermal excitations bythe vortex lines, can affect the flows in both fluids [4].At above 1 K, despite being a two-fluid system withmany properties restricted by quantum effects, He II isobserved to behave very similarly to classical fluids whena turbulent flow is generated by methods conventionallyused in classical fluid research, such as by a towed grid [5]or a rotating propeller [6]. Even in a non-classical ther-mal counterflow induced by an applied heat current inHe II, it has been revealed that quasiclassical turbulencecan emerge during the decay of counterflow after the heatcurrent is switched off [7–9]. The quasiclassical behavior ∗ Corresponding: [email protected] of He II is interpreted as the consequence of a strong cou-pling of the two fluids by mutual friction at large scales[10]. It is suggested that the turbulent eddies in the nor-mal fluid are matched by eddies in the superfluid pro-duced by polarized vortices [11, 12], although differentviews regarding the bundling of the vortices exist [13–15]. The coupled fluids behave at large length scales likea single-component viscous fluid at high Reynolds num-ber. At small scales, due to the viscous dissipation inthe normal fluid and the discrete vortex-line structure inthe superfluid, the flows in the two fluids become decou-pled. Mutual friction dissipation sets in at these smallscales. This mechanism of coupling becomes weaker atlower temperatures. Nevertheless, at temperatures be-low 0.5 K where the normal-fluid fraction is essentiallyzero, quasiclassical turbulence in the superfluid was stillobserved [16, 17]. In this case, it is generally believedthat a classical Richardson cascade of the turbulence en-ergy in the superfluid exists at scales greater than themean intervortex distance ℓ = L − / (where L is the linedensity, i.e. vortex length per unit volume). But un-like at higher temperatures, this energy cascade can nolonger be terminated by mutual friction dissipation. In-stead, the turbulence energy is further transferred downto smaller scales via a cascade of Kelvin wave excitationson the vortices, which eventually leads to phonon emis-sion [18, 19].The classical behavior of He II, especially in the two-fluid regime at above 1 K, has brought up the feasibil-ity of using He II in classical turbulence research andfor practical model testing. He II has very small kine-matic viscosity which allows the generation of flows withextremely high Reynolds numbers that can hardly beachieved with other conventional fluid materials [20].Various projects have been launched for this purpose[21–23]. However, the viscosity that controls the en-ergy dissipation of the quasiclassical turbulence above1 K is not the normal-fluid viscosity but instead an ef-fective kinematic viscosity ν ′ that accounts for both theviscous dissipation in the normal fluid and the mutualfriction dissipation at small scales. The precise value of ν ′ is needed in the design of these He II based quasi-classical turbulence experiments. Furthermore, makingreliable measurements of the ν ′ values will be indispens-able in rigorously testing the various theories about thedissipation mechanism in He II [4, 5, 24], which will befundamentally important in advancing our knowledge ofquantum turbulence.Stalp et al. first introduced ν ′ in a theoretical modelfor interpreting their measured vortex density L ( t ) dur-ing the decay of a towed-grid generated turbulence inHe II above 1 K [5]. By analogy with the energy decayequation for classical turbulence [25], Stalp et al. pro-posed that the total turbulence kinetic energy per unitHe II mass, E ( t ), decays as dE/dt = − ν ′ ( κL ) [5]. Ap-proximately, E can be evaluated as E = E + E , where E comes from the flows in the superfluid on scales ator below ℓ associated with individual vortex lines and E represents the kinetic energy density associated withlarge-scale flows in the coupled turbulence. E can be es-timated as E = B ( ρ s /ρ ) κ L [3], where the dimensionlessfactor B ≃ π ln ( l/a ) is typically about unity, and ρ s /ρ denotes the ratio of the superfluid density to the totaldensity of He II. E can be evaluated as E = (∆ U ) ,where ∆ U = h ( U − U ) i / denotes the root mean squarevelocity fluctuation of the coupled flows. For quasiclassi-cal turbulence in He II, E is normally much greater than E [10]. The energy decay rate equation can therefore beformally written as dEdt = Bκ ρ s ρ dLdt + dE dt = − ν ′ ( κL ) (1)Based on Eq. (1), Stalp et al. neglected the E contribu-tion and derived an explicit expression for L ( t ) at largedecay times as L ( t ) ≃ D (3 C ) / πκ √ ν ′ · t − / (2)where D is the width of the flow channel and C =1 . D ; and 2) the coupled turbulence has a classicalKolmogorov energy spectrum that extends to all scales(i.e. E = R e E ( k ) dk , where e E ( k ) depends on the wavenumber k as e E ( k ) ∝ k − / ) [5]. These hypotheses are,to some extend, supported by the observed L ( t ) ∝ t − / behavior at large decay times. The value of ν ′ was then determined by fitting the measured L ( t ) using Eq. (2)[26, 27]. This method was later used by other groupsfor estimating ν ′ in decaying counterflow and decayingco-flow in a channel [8, 28]. A similar idea was also ap-plied to the study of quasiclassical turbulence in puresuperfluid at very low temperatures where the effectiveviscosity ν ′ arises from completely different dissipationmechanism [29–31]. Nevertheless, in all these studies the ν ′ values obtained using Eq. (2) are indeed dubious asdiscussed by Zmeev et al. [32]. There is no evidenceshowing that the energy-containing eddy size must bethe same as the channel width D . Although intuitivelythey should not be too different, any possible differencecan result in significant change in the fitted ν ′ valuessince ν ′ ∝ D according to Eq. (2). Furthermore, ifthere is indeed a Kolmogorov spectrum, this spectrummust break down near the cut-off scale D . We also notethat Skrbek’s group managed to evaluate ν ′ in steady-state co-flows [33]. But their analysis requires additionalassumptions which further limit the result accuracy. Areliable determination of ν ′ for quasiclassical turbulencein He II can be made only if one can measure directlyboth L ( t ) and E ( t ).In He II, the vortex density L ( t ) can be readily mea-sured using either second sound attenuation [5, 34] ortrapping of negative ions [30, 35]. However, a direct mea-surement of the turbulence energy is challenging. Typicalmeasurement tools for ∆ U , such as pitot pressure tubes[6], normally have limited spatial resolution, and theirapplication requires a large mean flow velocity whichis not always present in decaying turbulence. Anotherroute to probe the turbulence energy is to measure theresulting heat input to the fluid as the turbulence decays[36]. Along this line, Bradley et al developed a uniqueAndreev scattering technique and made the first directmeasurement of the energy decay in superfluid He-B atzero temperature limit [37]. However, they could not de-termine the vortex density in the same experiment andthus cannot deduce the values for ν ′ . In this paper, wereport the measurements of both L ( t ) and E ( t ) in de-caying counterflow turbulence in He II, by combining thesecond sound attenuation technique and a recently de-veloped tracer-line tracking flow visualization technique[38, 39]. Our method is applicable to the two-fluid regimeat above 1 K. We show that a reliable determination ofthe ν ′ values can be made.Our experimental setup is shown in Fig. 1 (a). Astainless steel channel (square cross-section: (9.5 mm) ;length 300 mm) is attached to a pumped helium bathwhose temperature can be controlled within 0.1 mK. Aplanar heater at the lower end of the channel can be usedto drive a thermal counterflow, i.e. the superfluid flow-ing towards the heater and the normal fluid away from it[40]. The mean velocity U of the normal fluid is relatedto the heat flux q by U = q/ρsT , where s is the specificentropy of the helium. When the heat flux is greaterthan a small critical value, it is known that the super-fluid can become turbulent and a self-sustained vortextangle is generated by the mutual friction between thetwo fluids [4]. We have reported that the normal fluidcan also become turbulent above a threshold heat flux q c (e.g. q c ∼
60 mW/cm for 1.65 K), exhibiting a novel k − energy spectrum [39]. This energy spectrum is likelycaused by the mutual friction dissipation that occurs ina wide range of scales in the normal fluid, since the twofluids have opposite mean velocities and therefore cannotget completely coupled. As the heater is turned off, theheat current decays to zero with a thermal time constant τ [34, 45]. In the absence of the heat current, the twofluids can then get coupled at large scales by mutual fric-tion. The time it takes to establish the coupling can beestimated using the formula derived by Vinen [10] and istypically in the range of 1-10 ms in our experiment. Ouranalysis on ν ′ will relate to times that are greater thanboth the thermal time constant and the time required forcomplete coupling. FIG. 1: (color online). (a) Schematic diagram of the experi-mental setup. (b) Typical images showing the deformation ofthe He ∗ molecular tracer lines in steady-state thermal coun-terflow. The white dashed lines indicate the initial locationsof the tracer lines. In order to extract quantitative flow field information,we have adopted a recently developed flow visualizationtechnique by tracking thin lines of He ∗ molecular trac-ers. These tracers are created via ionizing ground statehelium atoms using a focused femtosecond laser pulse[38]. Above 1 K, He ∗ tracers are completely entrainedby the normal fluid and can be imaged via laser-inducedfluorescence [41–44]. Fig. 1 (b) shows typical images ofthe He ∗ tracer lines in steady-state counterflow at 1.65K. The streamwise velocity field of the normal fluid canbe determined from the vertical displacements of the linesegments [38]. Using this tracer-line tracking technique,we can probe the normal fluid motion (and hence the coupled-fluid motion) at scales from the channel width( ∼ ∼ µ m). (a)(b) () z P U ( s / mm ) z U (mm/s) 8 ms53 ms513 msDecay time
426 mW/cm q = T =
426 mW/cm
150 mW/cm ~ t − z U ∆ FIG. 2: (color online). (a) Velocity probability density func-tions (PDFs) in decaying counterflow turbulence at 1.65 Kwith an initial heat flux of 426 mW/cm . The solid curvesrepresent Gaussian fits to the data. (b) Streamwise veloc-ity fluctuation ∆ U z determined from the Gaussian fits of thevelocity PDFs. For a given heat flux q , we normally maintain a steady-state counterflow for over 20 s and then switch off theheat current. We repeat the experiment 200 times andanalyze the 200 images acquired at every decay time toproduce velocity probability density functions (PDFs).Typical results for an initial heat flux of q =426 mW/cm at 1.65 K are shown in Fig. 2 (a). These velocity PDFscan be well fitted with Gaussian functions, which allowus to determine the time evolution of both the mean flowvelocity U and the streamwise root mean square velocityfluctuation ∆ U z . The time taken for U to decay to nearlyzero is about 100 ms for q =426 mW/cm and shorter atlower heat fluxes, in agreement with the expected thermaltime constant. The measured decay of ∆ U z for typicalinitial heat fluxes is shown in Fig. 2 (b). We observethat after the two fluids get coupled, the decay of ∆ U z is very slow and nearly flattens off at relatively smalldecay times. At large decay times, ∆ U z ∝ t − and hencethe energy E ( t ) ∝ t − . The late decay behavior is inaccordance with the decay of a quasiclassical turbulencewith a Kolmogorov spectrum [5], but the initial flatteningis more severe than the expected ( t + t ) − behavior. FIG. 3: (color online). The calculated 2nd order transversestructure function at different decay times in decaying coun-terflow with an initial heat flux of 426 mW/cm . We note in passing that the energy spectrum of thecoupled turbulence can be directly probed in our experi-ment by calculating the second-order transverse structurefunction S ⊥ ( r ) = h ( U − U ) i [39], where r is the sep-aration of two line segments (see Fig. 1 (b)). The timeevolution of the calculated S ⊥ ( r ) is shown in Fig. 3. Weobserve that S ⊥ ( r ) ∝ r n below a few millimeters. Thisexponent n leads to an energy spectrum e E ( k ) ∼ k − ( n +1) [46, 47]. The observed variation of n reveals that the cou-pled turbulence evolves from a non-classical form at smalldecay times with a spectrum close to that in steady state(i.e. n =1) to a quasiclassical turbulence at large decaytimes with a Kolmogorov spectrum (i.e. n =2/3). Thisspectrum transition is found to be responsible for theinitial slow decay of ∆ U z and E ( t ) [9].We also measured the vortex-line density L ( t ) in decay-ing counterflow using the standard second sound attenu-ation method [34]. The typical decay behavior of L ( t ) at1.65 K is shown Fig. 4. We observe that when the normalfluid is turbulent in the steady state, the decay of L ( t )always exhibits three distinct regimes. The first regimeoccurs at very short decay times where L ( t ) decays fastand in accordance with Vinen’s phenomenological model[34]. Subsequently, L ( t ) can grow with time and show a“bump” structure. At large decay times, L ( t ) ∝ t − / .This L ( t ) decay behavior was reported in the past [7, 34].Skrbek et al first realized that the t − / behavior at largedecay times indicated the decay of a quasiclassical turbu-lence in the coupled two fluids, similar to those generatedby a towed grid [7]. However, the underlying mechanismfor the appearance of the bump and the switching to the ~ t −
426 mW/cm150 mW/cm T=1.65 K
FIG. 4: (color online). The decay of the vortex line density L ( t ) measured at different initial heat fluxes at 1.65K t − / decay was unclear for many years despite varioustheoretical efforts [48–50]. With the aid of our flow vi-sualization, we have recently elucidated that the energyspectrum transition in the coupled turbulence is respon-sible for the observed complex L ( t ) behavior [9].In order to determine the effective kinematic viscosity ν ′ , we integrate Eq. (1) from t to infinity on both sidesand write the total energy density E ( t ) as Bκ ρ s ρ L ( t ) + E ( t ) = ν ′ · Z ∞ t κ L ( t ′ ) dt ′ (3)Here E ( t ) can be evaluated as E ( t ) = (∆ U z ) , assum-ing that the large-scale turbulence in decaying counter-flow is isotropic. This assumption should hold reason-ably well at least at large decay times where the coupledflow shows a Kolmogorov spectrum for isotropic turbu-lence. The total turbulence energy density E can be cal-culated based on our measured L ( t ) and ∆ U z using theexpression on left-hand side of Eq. (3). The results for q =426 mW/cm and 150 mW/cm at 1.65 K are shown inFig. 5 as circles and triangles, respectively. For both heatfluxes, the contribution from E dominates. The solidcurve and the dashed curve shown in Fig. 5 are calculatedbased on the integral on the right-hand side of Eq. (3). Toevaluate this integral, we assume that the t − / behaviorof L ( t ) continues for decay times beyond the maximummeasurement time in our experiment (about 40 s). Dueto the fast decay of L ( t ), the contribution to the inte-gral at very long decay times is negligible. We then vary ν ′ and determine its value by requiring that the energydensities calculated with the expressions on either sideof Eq. (3) give the best agreement at large decay times.At 1.65 K, ν ′ /κ = 0 .
46 is obtained. We note that thecalculated energy density curves indeed also show goodagreement at relatively small decay times when the en-ergy spectrum of the coupled turbulence still undergoesthe transition.
426 mW/cm
150 mW/cm T = FIG. 5: (color online). The decay of the total turbulence en-ergy density in decaying counterflow turbulence. The bluecircles and black triangles are calculated based on the ex-pression on the left-hand side of Eq. (3). The black solidcurve and the red dashed curve represent the results calcu-lated using the integral on the right-hand side of Eq. (3). Thebest agreement of the calculated energy density at large decaytimes is achieved with ν ′ /κ = 0 . We have made similar measurements in decaying coun-terflow at other temperatures above 1 K. The overall de-cay behaviors of the vortex density L ( t ) and the rootmean square velocity fluctuation ∆ U z are similar to thoseat 1.65 K. In Fig. 6, we show the effective kinematic vis-cosity ν ′ obtained at different temperatures (blue trian-gles). To aid our discussions, we have also included inFig. 6 the kinematic viscosity ν ′ calculated with our vor-tex density data using Eq. (2) (black squares), the ν ′ values obtained by Stalp et al. in the towed-grid experi-ment [26] (red solid circles), and the kinematic viscosity ν n = µ n /ρ calculated based on the tabulated normal-fluid viscosity µ n [2] (black solid curve). It is clear that ν n is smaller than ν ′ , which reflects the fact that thedissipation processes in quasiclassical turbulence in HeII include not only the normal-fluid viscosity but alsomutual friction. We note that the ν ′ values determinedusing our new methods appear to be greater than boththe values calculated using the traditional method viaEq. (2) and those from Stalp et al. . This difference mayreflect the inherent limitations associated with the hy-potheses involved in deriving the Eq. (2). Indeed, onecan see clearly in Fig. 3 that the structure function atlarge decay times exhibits a peak at a scale smaller thanthe channel width, indicating the energy-containing eddysize being smaller than D . The black squares in Fig. 6would appear much lower if a smaller energy-containingeddy size is used in the calculation. It is worthwhilenoting that the ν ′ values have been determined in sim-ilar temperature range by Skrbek’s group using Eq. (2)in the study of decaying counterflow [28] and decaying bellow-induced co-flow turbulence [51]. Despite the largeerror bars, their data appear to be also greater than thosefrom Stalp et al. Nevertheless, without any informationabout the actual energy spectrum and energy-containingeddy size in these experiments, it is hard to comment onthe reliability of these results.
FIG. 6: (color online). Effective kinematic viscosity in unitsof κ . The blue triangles represent ν ′ values calculated usingour new method via Eq. (3). The black squares are calculatedusing our vortex density data via Eq. (2). The red solid circlesrepresent ν ′ values obtained by Stalp et al. in the towed-gridturbulence experiment [26] that are corrected by Chagovets et al. [8]. The black solid curve is the kinematic viscosity ofHe II calculated based on the normal-fluid viscosity alone [2]. We acknowledge the support from the US Departmentof Energy under Grant DE-FG02 96ER40952 and theNational Science Foundation under Grant No. DMR-1507386. We would also like to thank D.N. McKinseyand S.W. Van Sciver for providing laser and cryogenicsequipment. [1] D.R. Tilley and J. Tilley,
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