Eulerian and Lagrangian second-order statistics of superfluid 4 He grid turbulence
EEulerian and Lagrangian second-order statistics of superfluid He grid turbulence
Y. Tang, W. Guo
National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA andMechanical Engineering Department, Florida State University, Tallahassee, FL 32310, USA
V.S. L’vov, A. Pomyalov
Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel
We use particle tracking velocimetry to study Eulerian and Lagrangian second-order statistics ofsuperfluid He grid turbulence. The Eulerian energy spectra at scales larger than the mean distancebetween quantum vortex lines behave classically with close to Kolmogorov-1941 scaling and arealmost isotropic. The Lagrangian second-order structure functions and frequency power spectra,measured at scales comparable with the intervortex distance, demonstrate a sharp transition fromnearly-classical behavior to a regime dominated by the motion of quantum vortex lines. Employingthe homogeneity of the flow, we verify a set of relations that connect various second-order statisticalobjects that stress different aspects of turbulent behavior, allowing a multifaceted analysis. We usethe two-way bridge relations between Eulerian energy spectra and second-order structure functionsto reconstruct the energy spectrum from the known velocity second-order structure function andvice versa. The Lagrangian frequency spectrum reconstructed from the measured Eulerian spectrumusing the Eulerian-Lagrangian bridge differs from the measured Lagrangian spectrum in the quasi-classical range which calls for further investigation.
Introduction
The statistical description of turbulent flows followstwo distinct approaches. In the Eulerian approach, themain objects of interest are velocity differences for vari-ous spatial separations. In the Lagrangian approach, thefluid particles are followed along their trajectories, andthe main focus is the velocity differences at sequentialtime moments.These two approaches complement each other. TheEulerian approach to turbulence is more convenient forphenomena dominated by large scale motions, such aswall-bounded turbulence. The Lagrangian description ofturbulence has unique physical advantages that are espe-cially important in studies of phenomena dominated bysmall-scale motions or many-point correlation functions,like turbulent mixing and particle dispersion.The Eulerian second-order statistics, namely the veloc-ity structure functions S ( r ) (square of the spatial velocityincrements across a separation r ) and the energy spec-tra of turbulent velocity fluctuations in the wavenumber k -space E ( k ), have been studied in depth over years .The typical experimental studies use one-probe measure-ments (e.g. by hot-wire anemometer), which provide re-searchers with the time dependence of the Eulerian veloc-ity u ( r pr , t ) at the position of the probe r pr . In the pres-ence of large mean velocity (such as wind in atmosphericmeasurements), Taylor hypothesis of frozen turbulenceallows one to transform these data into coordinate depen-dence u ( x, t ) in the streamwise direction (cid:98) x practically atthe same moment of time t .One-probe measurements cannot give informationabout a velocity v ( r | t ) of a Lagrangian tracer (a mass-less particle, swept by turbulent velocity field withoutslippage) positioned at r = r for t = t . The exper-imental study of Lagrangian velocity v ( r | t ) was done by Snyder and Lumley who provided the first system-atic set of particle-tracking velocity measurements fromthe optical tracking of tracer particles in wind-tunnelgrid turbulence. The technique is known as particletracking velocimetry (PTV), when velocities are con-cerned, or Lagrangian particle tracking, when the po-sition and the acceleration are also determined. Par-ticles can be passive tracers that approximate the La-grangian motion of fluid elements, have inertia, or havea size larger than the smallest scales of the flow. Re-cently, flow visualization was extended to the super-fluid He, providing invaluable information about its richand unusual physics.Later experiments and Direct Numerical Simulations(DNS) [see, e.g. Refs. ] gave more detailed infor-mation on the spatial and temporal correlations in theclassical and superfluid turbulence.Recall that liquid He becomes a quantum fluid when it is cooled below a critical temperature T λ =2 .
17 K. At these conditions, fluid vorticity may be di-vided into two parts. A quantized part forms a quantumground state, while thermal excitations represent a nor-mal fluid with continuous vorticity. The vorticity quan-tization results in the creation of thin quantum vortexlines of fixed circulation that interact with the normalfluid via mutual friction force, forming dense tangles.Large-scale hydrodynamics of such a system is usu-ally described by a two-fluid model, interpreting He asa mixture of two coupled fluid components: an invis-cid superfluid and a viscous normal fluid. The tangle ofquantum vortexes mediates the interaction between fluidcomponents via mutual friction force .There are many indications that the mechani-cally driven turbulence in the superfluid He is similar tothe turbulence in the classical flows. In this paper, weapply the particle tracking velocimetry to study the sec- a r X i v : . [ c ond - m a t . o t h e r] D ec ond order statistics in the grid superfluid He turbulence.We confirm the near-Kolmogorov scaling of the Eule-rian energy spectra, which are also almost isotropic. Thesmall-scale statistics studied using Lagrangian structurefunctions and energy spectra deviate from the classicalbehavior, illustrating a clear transition from a randomfluid motion to a motion dominated by the velocity ofindividual vortex lines at scales comparable with the in-tervortex spacing.The paper is organized as follows. In Sec. I we sum-marize some important information about the second-order statistics and the relation between the Eulerianand Lagrangian energy spectra in the classical turbu-lence. In Sec. II we describe the experimental techniques(Sec. II A) and the method to extract the 2D velocities(Sec. II B) and the Eulerian energy spectra from the PTVdata (Sec. II C). Sec. III is devoted to the experimentalresults and their discussion. We start with the Eulerianstatistics in Sec. III A. We discuss various forms of the en-ergy spectra as well as relations between the spectra andthe structure and correlation functions. Then we turnto the Lagrangian statistics (Sec. III B) where we con-sider the second-order structure functions (Sec. III B 1)and the energy spectra (Sec. III B 2). In Conclusions wesummarize our findings. I. ANALYTICAL BACKGROUND
The goal of this Section is to recall well-known defi-nitions of the second-order statistical objects in the Eu-lerian and Lagrangian description of turbulence, to in-troduce their notations, and to remind relations betweenthem.
A. Second-order statistical description ofturbulence
The complete statistical description of spatially andtemporally homogeneous random processes with Gaus-sian statistics can be done on the level of their quadratic(second-order) statistical objects. Statistics of developedhydrodynamic turbulence is not Gaussian. Neverthe-less, the most basic characteristics of turbulence, the en-ergy distributions among spatial and temporal scales, aregiven by their second-order objects, i.e., the energy spec-tra and the structure functions.
1. Bridge equations for Eulerian structure functions andenergy spectra
The Eulerian approach to the statistics of homoge-neous stationary turbulence is applied to the field of theturbulent velocity fluctuations u ( r , t ) with zero mean (cid:104) u ( r , t ) (cid:105) = 0. Here (cid:104) . . . (cid:105) is a “proper” averaging, whichcan be understood as the time averaging in stationary turbulence, the ensemble averaging over realizations inexperiments or numerical simulations, etc .As the basic objects in the second-order statistical de-scription of turbulence in the Eulerian framework, wecan take the simultaneous two-point correlation function(covariance) C αβ ( r ) ≡ (cid:104) u α ( r + r (cid:48) , t ) u β ( r (cid:48) , t ) (cid:105) (1a)and its Fourier transform F αβ ( k ) ≡ (cid:90) d r C αβ ( r ) exp( i k · r ) . (1b)Here the indices α, β = { x, y, z } denote Cartesian co-ordinates. The inverse transform to Eq. (1b) takes theform: C αβ ( r ) ≡ (cid:90) F αβ ( k ) exp( − i k · r ) d k (2 π ) . (1c)Alternatively, the function F αβ ( k ) can be expressed interms of the Fourier transform of the velocity field u ( k ) = (cid:90) d r u ( r ) exp( i k · r ) . (2a)as follows(2 π ) δ ( k − k (cid:48) ) F αβ ( k ) = (cid:10) u α ( k ) u ∗ β ( k (cid:48) ) (cid:11) , (cid:88) α F αα ( k ) ≡ F αα ( k ) ≡ F ( k ) . (2b)Hereafter we adopt Einstein summation over repeatedindices and skip for shortness the repeated vector indicesin the resulting traces of all tensors, e.g. F αα ( k ) ≡ F ( k ), C αα ( r ) ≡ C ( r ), etc.One sees that F ( k ) in Eq. (2b) describes the turbu-lent energy spectrum – the density of twice the tur-bulent kinetic energy per unit mass in the wave-vectorthree-dimensional space k . In particular, it follows fromEqs. (1) that (cid:90) F ( k ) d k (2 π ) = C (0) = (cid:10) | u ( r ) | (cid:11) . (2c)In their turn, the correlations (1a) are simply connectedto the second-order structure functions of the velocitydifferences S ( r ) in the homogeneous turbulence S ( r ) ≡ (cid:104)| u ( r + r (cid:48) ) − u ( r (cid:48) ) | (cid:105) = 2 (cid:2) C (0) − C ( r ) (cid:3) . (3)The equation (1), the one-dimensional (1D) version ofwhich is known in the theory of stochastic processes asWiener-Khinchin-Kolmogorov theorem , can be consid-ered in the theory of turbulence as two-way bridge equa-tions which allows one to find the Eulerian correlationfunction C ( r ) if one knows the energy spectrum F ( k )and vice versa: to find C ( r ) from known F ( k ). In theisotropic turbulence, where F ( k ) and C ( r ) depend onlyon the wavenumber k = | k | and r = | r | , these equationsmay be simplified: C ( r ) = ∞ (cid:90) F ( k ) sin( kr ) r kdk π , (4a) F ( k ) = 4 π ∞ (cid:90) C ( r ) sin( kr ) k rdr . (4b)For 1D case, one should replace (cid:82) d r in Eq. (1b) by (cid:82) ∞−∞ dr and replace (cid:82) d k in Eq. (1c) by (cid:82) ∞−∞ dk . Thisgives the 1D k ⇔ r -bridge equations C ( r ) = 1 π ∞ (cid:90) E ( k ) cos( k r ) dk , (5a) C (0) = 1 π ∞ (cid:90) E ( k ) dk , (5b) S ( r ) = 4 π ∞ (cid:90) E ( k ) sin (cid:16) k r (cid:17) dk , (5c) E ( k ) = 2 ∞ (cid:90) C ( r ) cos( k r ) dr . (5d)In the rest of the paper E ( k ) denotes 1D Eulerian energyspectrum. We should stress here that the assumption ofspace homogeneity and stationarity of the turbulent flowis essential in derivations of Eqs. (1), (4) and (5).
2. Bridge equations for Lagrangian structure functions andfrequency power spectra
Similar to the Eulerian approach to statistics of turbu-lence, important information in the Lagrangian frame-work is contained in the 2 nd -order statistical objects:the Lagrangian correlation C ( τ ) and structure functions S ( τ ): C ( τ ) ≡ (cid:104) v ( r | t + τ ) · v ( r | t ) (cid:105) , S ( τ ) ≡ (cid:104)| v ( r | t + τ ) − v ( r | t ) | (cid:105) , (6a)together with the Lagrangian energy spectrum E ( ω ), de-fined via v ( r | ω ), the Fourier transform of v ( r | t ):(2 π ) δ ( ω − ω (cid:48) ) E ( ω ) ≡ (cid:104) v ( r | ω ) · v ∗ ( r | ω (cid:48) ) (cid:105) , v ( r | ω ) ≡ (cid:90) v ( r | t ) exp( iωt ) dt . (6b)Similar to Eqs. (1b) and (5), E ( ω ) is the Fourier trans-form of C ( τ ): E ( ω ) = 2 ∞ (cid:90) C ( τ ) cos( ωτ ) dt . (7a) This relation can be considered as the τ ⇒ ω -bridge equa-tion, which expresses E ( ω ) in terms of C ( τ ) and S ( τ ) S ( τ ) = 2 (cid:2) C (0) − C ( τ ) (cid:3) . (7b)The inverse ω ⇒ τ -bridge has the form similar to Eq. (5): S ( τ ) = 2 π (cid:90) sin (cid:16) ω τ (cid:17) E ( ω ) dω . (7c)
3. Kolmogorov-1941 scaling
The shapes of the energy spectra E ( k ) and E ( ω ) to-gether with the shapes of the structure functions S ( r ) and S ( τ ) in the classical isotropic fully developed turbulencewere the subject of intensive studies over decades, seee.g. Refs. . Up to relatively small intermittency cor-rections, these objects obey the Kolmogorov-1941 (K41)scaling : E K41 ( k ) (cid:39) ε / k / , E K41 ( ω ) (cid:39) εω , (8a) S K41 ( r ) (cid:39) ( ε r ) / , S K41 ( τ ) (cid:39) ε τ . (8b)Here ε is the rate of dissipation of kinetic energy per unitmass.
4. Eulerian-Lagrangian bridge equation for the energyspectra
In addition to the bridge equations between the Eu-lerian objects and similar relations for the Lagrangianobjects, it is possible to formulate the relationship be-tween the Eulerian E ( k ), and Lagrangian E ( ω ) spectrain the classical isotropic turbulence.This relation E ( ω ) = 2 π (cid:90) exp (cid:104) − ωγ ( k ) (cid:105) E ( k ) dkγ ( k ) , (9)was derived for the turbulence of Newtonian fluidsin the framework of the Navier-Stokes equation in theBelinicher-L’vov sweeping-free representation . InEq. (9), the turnover frequency γ ( k ) of turbulent fluc-tuations of size ∼ π/k (referred below as k -eddies) canbe estimated as usual: γ ( k ) = C γ (cid:112) k E ( k ) . (10)Here C γ is a dimensionless constants of the order of unity.To clarify the physical mechanisms behind the bridgeEq. (9), we consider the developed turbulence as consist-ing of an ensemble of k -eddies of the energy density E ( k ).Each such k -eddy is a random motion with velocities v k ( τ ) in the reference frame comoving with the center (a) (b) (c) -1 -4 -2 (a) 0 2 4 6 8 10-0.500.51 (b) 10 -2 -2 (c) FIG. 1: (a) Energy spectra Eq. (14): E ( q, m ) for m = 0 (orange line) and m = 2 (red line), and E ( q, m ) for m = 2 (blue line)and m = 4 (green line). The black dashed line corresponds to K41 scaling q − / . (b) Normalized correlation functions (cid:98) C ( y, m ),found from the corresponding spectra via bridges Eqs. (5). The color code is same as in (a). The black dashed line denotesthe asymptote 1 − y / . Note the linear scale. (c) The normalized structure functions (cid:98) S n ( y ) = 1 − (cid:98) C ( y, m ). The dashed linesdenote the viscous asymptote (cid:98) S ( y, m ) ∝ y , the dot-dashed lines denote K41 scaling (cid:98) S ( y, m ) ∝ y / , the horizontal thick grayline marks the saturation level (cid:98) S ( y, m ) = 1. Note the logarithmic scale. of the eddy. To describe its statistical behavior, we in-troduce a new object – partial correlation function of k -eddies (cid:101) C ( τ, k ) = (cid:104) v k (0) v ∗ k ( τ ) (cid:105) k (cid:104)| v k (0) | (cid:105) k , (11a)where (cid:104) . . . (cid:105) k denotes averaging of ensemble of k -eddieswith a given value of k . By definition, (cid:101) C (0 , k ) = 1. Us-ing Eq. (11a), we introduce a decorrelation time of k -eddies in the Lagrangian framework τ k , defined such that (cid:101) C ( τ k , k ) (cid:39) . In developed turbulence, τ k is about thelife-time of k -eddies or their turnover time. Below wesometimes use the frequency γ k ≡ /τ k instead of τ k .The Fourier transform of (cid:101) C ( τ, k ) (cid:101) E ( ω, k ) = ∞ (cid:90) −∞ exp( iωτ ) (cid:101) C ( τ, k ) dτ , (11b)according to the bridge Eq. (7a), is nothing but the fre-quency power spectrum of the considered k -eddy. Thecorresponding inverse Fourier transform (cid:101) C ( τ, k ) = ∞ (cid:90) −∞ exp( − iωτ ) (cid:101) E ( ω, k ) dω π (11c)dictates the normalization of (cid:101) E ( ω, k ) ∞ (cid:90) −∞ (cid:101) E ( ω, k ) dω π = (cid:101) C (0 , k ) = 1 . (11d)It can be shown that for ωτ k (cid:29)
1, the contribution of k -eddies to (cid:101) E ( ω, k ) decays much faster than 1 /ω .Assuming for simplicity the exponential decay, it wassuggested that (cid:101) E ( ω, k ) ∝ exp( − ωτ k ) = exp( − ω/γ k ). This assumption, together with the normalization (11d),results in the model expression for the contribution of k -eddies to the Lagrangian frequency energy spectrum: (cid:101) E ( ω, k ) = 2 πγ k exp (cid:16) − ωγ k (cid:17) . (12a)To sum up contributions of all k -eddies to the frequencyspectrum, we have to integrate the k -eddy contribution (cid:101) E ( ω, k ) over k with the weight E ( k ), i.e. the energy dis-tribution of k -eddies: E ( ω ) = (cid:90) (cid:101) E ( ω, k ) E ( k ) dk . (12b)Combining Eq. (12a) and (12b), we finally get thebridge Eq. (9) for E ( ω ). This equation satisfies the exactgeneral requirement: the total energy density per unitmass does not depend on the representation E = (cid:90) E ( k ) dk = (cid:90) E ( ω ) dω π = C (0) . (13)The Eulerian-Lagrangian one-way bridge Eq. (9) allowsus to find the Lagrangian (frequency) kinetic energy spec-trum E ( ω ), for a given Eulerian energy spectrum E ( k ).It is important to note that Eq. (9) is not limited by ei-ther the inertial interval of scales, or by the requirementof large Reynolds numbers. B. Bridge equations for model spectra
To illustrate the bridge Eqs. (5), we suggest a set ofmodel expressions for the Eulerian energy spectra, basedon the Kolmogorov scaling: E ( q, m ) = q m q m +5 / ,E ( q, m ) = 10 + q . q E ( q, m ) . (14)plotted in Fig. 1(a) as functions of the dimensionlesswavenumber q . These spectra have the K41 scaling E ( q, m ) ∝ q − / for q >
1. To have large enough iner-tial interval, we choose for concreteness q max = 1024 andtake E ( q, m ) = E ( q, m ) = 0 for q > q max . In the rangeof small q , the model functions E ( q, m ) demonstrate avariety of possible behaviors: E ( q,
0) (the orange line)is approaching a plateau, while spectra E ( q, E ( q, E ( q,
4) (the red, blue and green lines respectively)decay for q (cid:28)
1, going through a maximum for last twocases.The normalized correlation functions (cid:98) C ( y, m ) ≡ C ( y, m ) /C n (0 , m ) , (cid:98) C ( y, m ) ≡ C ( y, m ) /C n (0 , m ) , (15a)found from the corresponding spectra E ( q, m ) , E ( q, m )with the help of Eqs. (5a) and (5b), are shown in Fig. 1(b)as a function of a dimensionless coordinate y with thesame color code as in Fig. 1(a). Correlation functions (cid:98) C ( y,
0) and (cid:98) C ( y,
2) (originated from the spectra whichfor small q lie below K41 asymptote) have asymptoticbehavior (cid:98) C n ( y ) = 1 − y / , shown in Fig. 1(b) as blackdashed line, over relatively large interval y <
1. On theother hand, (cid:98) C ( y,
2) and (cid:98) C ( y,
4) deviate from it every-where except for a narrow range y < .
1, not visible inthe linear scale. As expected, all correlation functions (cid:98) C ( y, m ) vanish for large y (in our case for y >
10) butin different ways: monotonically [e.g. (cid:98) C ( y, (cid:98) S n ( y ) ≡ S ( y )2 C (0) = 1 − (cid:98) C n ( y ) , (15b)plotted in Fig. 1(c). Indeed, for small y < .
003 theviscous behavior (cid:98) S ( y, m ) ∝ y (shown by black dashedlines) is observed. This scaling is expected whenever theenergy spectrum decays faster than q − , including thesharp cutoff of the energy spectra for q > q max = 1024in our model. For larger y , the structure functions fol-low closely the K41 scaling (cid:98) S ( y, m ) ∝ y / , shown inFig. 1(c) as dot-dashed lines. At large scales, all structurefunctions demonstrate a tendency to approach plateau (cid:98) S ( y, m ) →
1, as expected.Note that substituting (cid:98) C ( y, m ) into Eqs. (5d), we ob-tain again the initial spectra E ( q, m ), shown in Fig. 1(a).We conclude that using the bridges Eqs. (5) with one ofthe functions E ( q ), C ( y ) or S ( y ), we can compute the restof them. This means that all three considered character-istics of turbulence, the energy spectra, the correlationand the structure function, contain the same informationabout the second-order statistics of turbulence. However,they stress different aspects of the turbulence statistics:the small scale characteristics are highlighted by (cid:98) S ( y )[Fig. 1(c)], the large scale behavior is exposed by (cid:98) C ( y )[Fig. 1(b)], while the energy distribution in wavenumberspace is described by E ( q ) [Fig. 1(a)]. The Eulerian-Lagrangian bridge equation (9) was veri-fied in Ref. by DNS of Navier Stokes equations for sta-tionary isotropic developed turbulence with 1024 gridpoints, as is illustrated in Fig. 2(a). In these simulations,Re λ = 240 was reached, which allowed the extend of aninertial interval of about 20 with K41 scaling E DNS ( k ) ∝ k − / and well resolved viscous subrange, see the insetof Fig. 2(a). In this turbulent flow, the Lagrangian tra-jectories of 5 × fluid points were computed duringabout 20 large-eddy turnover times. The instantaneousLagrangian velocity of the fluid point, required to inte-grate its trajectory, is computed using the fourth-orderthree-dimensional Hermite interpolation from the fluidvelocity at the eight Eulerian grid nodes surroundingthe fluid point. Fig. 2(a) shows excellent agreement di-rectly between the Lagrangian spectrum E DNS ( ω ) foundin the DNS (black solid line) and the spectrum E br ( ω )(red dashed line), calculated from the Eulerian spectrum E DNS ( k ) using the bridge.To illustrate the bridge equations Eqs. (9) and (7c), weuse a simple model spectrum that has a form E mod ( k ) = k − / , γ ( k ) = k / , (16)in the inertial interval between k min = 0 .
01 , k max = 100and is zero elsewhere. The resulting Lagrangian spec-trum E cl ( ω ) is shown in Fig. 2(b). As expected, it hasthe K41 scaling (8a), E ( ω ) ∝ /ω in the inertial intervalof frequencies (shown by the red dashed line), in our casebetween ω min = 0 . ω max = 20. It approaches someconstant value in the limit of k → E ( ω ) = 2 π E ω min (cid:104) ω + ω min ) − ω max + ω min ) (cid:105) (17)for all frequencies ω < ω max . The only difference is asharp cutoff at ω = ω min except of a smooth exponentialdecay. For ω max (cid:29) ω min , Eq. (12) satisfies the same fre-quency sum-rule (13) as the numerical spectrum E cl ( ω ).The corresponding Lagrangian structure function S cl ( τ ) calculated using Eq. (7c), is shown in Fig. 2(c). Asexpected, it has the K41 scaling S ( τ ) ∝ τ in the inertialinterval of scales, shown by red dashed line, and the vis-cous behavior S ( τ ) ∝ τ , shown by blue dot-dashed line.At very large times, S ( τ ) approaches a constant value. II. EXPERIMENT AND DATA ANALYSISA. Description of the experiment and mainparameters of the flow
The experimental apparatus is described in detail inRef. . In particular, a transparent cast acrylic flowchannel with a cross-section area of 1.6 × and a -2 -5 k -4 -2 E DN S / E (a) 10 -10 -5 (b) 10 -5 -5 (c) FIG. 2: Eulerian and Lagrangian statistics in classical turbulence. (a) Energy spectra in DNS with N = 1024 and Re λ = 240.The Lagrangian energy spectrum E ( ω ) (solid line) in comparison with the spectrum E br (dashed line) calculated from the bridgeEq. (9) by integrating E DNS ( k ), shown in the inset. The data of the dashed line (0 . ≤ ω/ω η ≤ .
6) are limited by the availableDNS data. (b) The Lagrangian energy spectrum E cl ( ω ), reconstructed using the bridge (9) from K41 Eulerian spectrum Eq. (16)in the interval 0 . < k <
100 and zero elsewhere. Blue dot-dashed line shows the asymptote lim ω → E ( ω ), red dashed line denotesK41 scaling in the inertial range E ( ω ) ∝ ω − . (c) The Lagrangian structure function Eq. (7c) calculated using the spectrum E ( ω ), shown in (b). Blue dot-dashed line shows viscous asymptotic behavior S ( τ ) ∝ τ , red dashed line denotes K41 inertialrange scaling S ( τ ) ∝ τ and gray horizontal line marks large τ limit S ( τ ) =const.TABLE I: Key parameters of the grid turbulence in He II.1.65 K 1.95 K 2.12 KVortex line density L , (cm − ) 2.1 × . × . × Intervortex distance (cid:96) = 1 / √L , (mm) 0.07 0.05 0.07The crossover wavenumber k cr = 2 π/(cid:96) , (mm − ) 89.7 125.6 89.7The outer scale ofturbulence L out , (mm) 3 2 3 k out (cid:39) π/L out , (mm − ) 2.1 3.1 2.1Mean density of thekinetic energy perunit mass E ≡ (cid:104)| u ( r , t ) | (cid:105) r , (mm /s ) 13.2 16.6 9.9RMS of the turbulentvelocity v T = √ E , (mm/s) 3.6 4.1 3.1Outer-scale turnoverfrequency ω out (cid:39) πv T /L out , (s − ) 7.5 12.9 6.5Turnover frequency ofthe smallest eddies ofscale (cid:96) ω (cid:96) (cid:39) ω out ( L out /(cid:96) ) / , (s − ) 92.3 150.6 79.5 * The factor 1 / length of 33 cm is immersed vertically in a He II bathwhose temperature can be controlled by regulating thevapor pressure. A brass mesh grid with a spacing of 3mm and 40% solidity is suspended in the flow chan-nel by four stainless-steel thin wires at the four corners.These wires are connected to the drive shaft of a linearmotor whose speed can be tuned in the range of 0.1 and60 cm/s. In the current work, we used a fixed grid speedat 30 cm/s.To probe the flow, we adopt the PTV method usingsolidified D tracer particles with a mean diameter ofabout 5 µ m . Due to their small sizes and hence small Stokes number in the normal fluid , these particles areentrained by the viscous normal-fluid flow. But whenthey are close to the quantized vortices in the super-fluid, a Bernoulli pressure owing to the induced super-fluid flow can push the particles toward the vortex cores,resulting in the trapping of the particles on the quantizedvortices .A continuous-wave laser sheet (thickness: 200 µ m,height: 9 mm) passes through the center of the chan-nel to illuminate the particles. We then pull the grid anduse a high-speed camera (120 frames per second) to filmthe motion of the particles. Following the passage of thegrid, we record the particle positions for a period of 0.28s (i.e., 34 images) for every 2 s. A modified feature-pointtracking routine is adopted to extract the trajectoriesof the tracer particles from the sequence of images. In thecurrent work, we focus on analyzing the data obtained at4 s ( T = 1 .
95 K) and 6 s ( T = 1 .
65 K, T = 2 .
12 K) fol-lowing the passage of the grid. As discussed in Ref. ,the turbulence at these decay times appears to be rea-sonably homogeneous and isotropic, and its turbulencekinetic energy density is relatively high such that an in-ertial interval exists. We have also installed a pair ofsecond-sound transducers for measuring the mean vortexline density L ( t ) using a standard second-sound attenu-ation method .At a given temperature T , we normally repeat our mea-surement up to 10 times so that an ensemble statisti-cal analysis of the particle trajectories can be performed.These 10 acquisitions, denoted as A = 1 , . . .
10, eachcontains 34 consecutive images. The velocity of a parti-cle can be calculated by dividing its displacement fromone image to the next by the frame separation time. Weonly select the middle 24 images ( I = 1 , . . .
24) for ourvelocity-field analysis. The velocities (cid:8) u ( X ) (cid:9) I,A at theparticle locations X = ( x, z ) are determined, where x and z denote the horizontal (wall-normal) and the verti-cal (streamwise) coordinates, respectively.In Table. I, we list some key parameters relevant to theflows that will be used in our result analysis. B. 2D velocity on a periodic lattice from PTV data (cid:2)
FIG. 3: An illustration to the division of the observation areainto cells to calculate the averaged velocity field.
In order to analyze the Eulerian turbulence energyspectra, it is highly desired to generate a two-dimensionalvelocity field on a periodic lattice R n,m = (cid:8) X n = n ∆ , Z m = m ∆ (cid:9) . For this purpose, we first combine thevelocity data u ( X ) obtained from all 24 images in eachacquisition into a single image. Then, we divide the com-bined image into square cells with ∆ = 0 .
02 mm which islarge enough to contain at least 1-2 data points in mostof the cells, as shown in Fig. 3. The velocity assignedto the center of each cell is calculated as the Gaussian-averaged velocity of particles inside the cell, u ( X ), withthe variance σ ≈ ∆ / (cid:8) u ( R n,m ) (cid:9) is thenFourier transformed and the edge-related artifacts canbe removed using known algorithms . The ensemble-averaged Eulerian energy spectra can be derived basedon these Fourier-transformed velocity fields. C. 2D-energy spectra and its 1D reductions
Having obtained the velocity field (cid:8) u ( R n,m ) (cid:9) on theperiodic lattice R n,m , we then perform the Fourier trans-form u ( k x , k z ) = 1 N M N − (cid:88) n =0 M − (cid:88) m =0 u ( R n,m ) × exp[ − i ( k x n + k z m )∆] , (18a)where N and M are the numbers of cells in x and z directions, respectively. The 2D energy spectrum can beobtained as F α ( k x , k z ) = (cid:104)| u α ( k x , k z ) | (cid:105) , (18b)where α = x, z and (cid:104) . . . (cid:105) denotes an ensemble averageover the the acquisitions A . Notice that Eqs. (18) are thediscrete 2D version of Eqs. (2).In addition, we introduce the 1D Fourier transforms u ( k x , z ) = 1 N N − (cid:88) n =0 u ( R n,m ) exp[ − ik x n )∆] , u ( x, k z ) = 1 M M − (cid:88) m =0 u ( R n,m ) exp[ − ik z m )∆] , (19)and 1D “linear” energy spectra E (cid:104) z (cid:105) α ( k x ) = (cid:104)| u α ( k x , z ) | (cid:105) z ,E (cid:104) x (cid:105) α ( k z ) = (cid:104)| u α ( x, k z ) | (cid:105) x , (20)where (cid:104) . . . (cid:105) x (and (cid:104) . . . (cid:105) z ) denotes averaging over x (andthe z ) axis in addition to ensemble averaging over differ-ent acquisitions. Linear 1D spectra (20) are related tothe 2D-spectra (2) as follows: E (cid:104) z (cid:105) α ( k x ) = M − (cid:88) m =0 F α (cid:16) k x , πmM ∆ (cid:17) ,E (cid:104) x (cid:105) α ( k z ) = N − (cid:88) n =0 F α (cid:16) πnN ∆ , k z (cid:17) . (21)In order to examine possible anisotropy of the turbu-lence, we perform SO(2) decomposition to get 1D energyspectra. The decomposition is done by projecting 2Dspectra (2) defined on the ( x, z )-plane, on 0 th and p th components of an orthonormal basis, which is propor-tional to exp( − i p ϕ ): E (0) α ( k ) = k π (cid:90) F α ( k cos ϕ, k cos ϕ ) dϕ ,E ( p ) α ( k ) = p √ k π (cid:90) F α ( k cos ϕ, k sin ϕ ) cos( pϕ ) dϕ . (22)If we keep the lowest two components, the original 2Dspectra (2) can be approximately expressed as: F α ( k x , k z ) = E (0) α ( k ) + 2 √ ϕ ) E (2) α ( k ) . (23)For an isotropic turbulence, E (2) α ( k ) = 0. For flows withweak anisotropy, the strength of the anisotropy can beevaluated by the ratio E (2) α ( k ) (cid:14) E (0) α ( k ). III. RESULTS AND THEIR DISCUSSIONA. Eulerian statistics
1. Eulerian energy spectra
Experimental results for 2D energy spectra F ( k x , k z )for T=1 .
65 K, 1 .
95 K and 2 .
12 K, are shown in Fig. 4(a),(b) and (c), respectively. At all temperatures thesespectra are nearly isotropic with small corrections.In Fig. 4(d), (e) and (f), the ratios E (2) ( k ) /E (0) ( k ), E (4) ( k ) /E (0) ( k ) and E (8) ( k ) /E (0) ( k ) of the SO(2) de-composition components are shown for all temperatures.We see that except in regions of k <
10 mm − , allanisotropic corrections are very small (below the levelof 5%) with respect of the isotropic contribution E (0) ( k ).We attribute the residual star-like structures in the 2Dspectra to regions of the velocity field originating fromthe cells with a small number of particles.In Figs. 5(a), (b) and (c) we compare angular-averagedenergy spectra E (0) ( k ) with two linear spectra E (cid:104) x (cid:105) ( k z )and E (cid:104) z (cid:105) ( k x ). All spectra are normalized by the en-ergy density E and compensated by κ / with κ ≡ k/ ∆ k x . As expected, all spectra in the inertial inter-val (for k >
20 mm − in our case) have K41 scaling ∝ k / . We see that two linear spectra almost coin-cide: E (cid:104) x (cid:105) ( k ) ≈ E (cid:104) z (cid:105) ( k ) confirming again the isotropyof the spectra. Note that the angular-averaged spectra E (0) ( k ), shown by green line, settle at larger values. Thereason is that the K41 scaling is proportional to the en-ergy dissipation rate: E (cid:104) x (cid:105) ( k z ) ∝ ε / z , E (cid:104) z (cid:105) ( k x ) ∝ ε / x and E (0) ( k ) ∝ ε / k . Here ε x ≈ ε z are the energy fluxesin the ˆ x and ˆ z -directions, while ε k = (cid:112) ε x + ε z ≈ √ ε x . Therefore, we expect that E (0) ( k ) ≈ / E (cid:104) z (cid:105) ( k ). Indeed, E (0) ( k ) / / shown by cyan lines in Figs. 5(a)-(c), prac-tically coincide with the plots of E (cid:104) x (cid:105) ( k z ) and E (cid:104) z (cid:105) ( k x )for all temperatures.Therefore, even in fully isotropic turbulence, 1D spec-tra obtained by different methods – e.g. by one probemeasurements with Taylor hypothesis of frozen turbu-lence, by axial averaging of two-dimensional spectra ob-tained via PIV or PTV method and by full averaging ofthree-dimensional spectra of turbulence – all have differ-ent pre-factors, which need to be accounted to compareexperimental data with theoretical or numerical results.Lastly, in Figs. 5(d), (e) and (f) we compare linearspectra E (cid:104) x (cid:105) α ( k ) and E (cid:104) z (cid:105) α ( k ) of the streamwise and wall-normal components ( α = x, z ). We see that for T =1 .
95 K and T = 2 .
12 K all spectra practically coincide.Only for T = 1 .
65 K the spectrum of E (cid:104) x (cid:105) ( k z ) is slightlymore intense than other contributions.We thus conclude that Eulerian energy spectra ofdeveloped superfluid turbulence behind grid are nearlyisotropic with respect of direction of the wave vector k and with respect of the x - and z - vector projections ofthe velocity field in the available part of the inertial in-terval from k (cid:39)
25 mm − to k (cid:39)
100 mm − . The finitespatial resolution of the Eulerian approach does not allowus to determine the upper edge k max of K41 scaling. InSec. III B we will try solving this issue using Lagrangiananalysis.
2. Experimental Eulerian-Eulerian bridges
In Sec. I A 1, we formulated the Eulerian-Eulerianbridges (5) and analyzed them in Fig. 1 using model en-ergy spectra (14) with large inertial interval. Here wedemonstrate how bridges (5) actually work for real ex-perimental data with a modest inertial interval.First of all, for each realization of the velocity fieldwe calculate the structure and correlation functions us-ing the same ¯ u ( R n,m ) that was used to calculate thespectra, taking displacements R x along the x -directionand R z along the z -direction and averaging the resultingstructure and correlation functions along the other di-rection. Next, we normalize them similar to Eqs. (15)and ensemble-average to get four objects: (cid:98) S (cid:104) z (cid:105) exp ( R x ), (cid:98) S (cid:104) x (cid:105) exp ( R z ), (cid:98) C (cid:104) z (cid:105) exp ( R x ), (cid:98) C (cid:104) x (cid:105) exp ( R z ). Finally, we use the nor-malized directly measured “experimental” linear spectraEq. (20) (cid:98) E (cid:104) z (cid:105) exp ( k x ) ≡ E (cid:104) z (cid:105) exp ( k x )¯ E , (cid:98) E (cid:104) x (cid:105) exp ( k z ) ≡ E (cid:104) x (cid:105) exp ( k z )¯ E (24)to reconstruct the structure functions (cid:98) S (cid:104) z (cid:105) br ( R x ) and (cid:98) S (cid:104) x (cid:105) br ( R z ) according to the bridge Eq. (5c). Similarly, weuse (cid:98) C (cid:104) x (cid:105) exp ( R z ), (cid:98) C (cid:104) z (cid:105) exp ( R x ) to reconstruct the linear spectra E (cid:104) x (cid:105) br ( k z ), E (cid:104) z (cid:105) br ( k x ), according to the bridge Eq. (5d). Inthe latter calculations, to reduce the noise we use the data T = 1 .
65 K T = 1 .
95 K T = 2 .
12 K
T=1.65 KT=1.95 KT=2.12 K (d)
T=1.65 KT=1.95 KT=2.12 K (e)
T=1.65 KT=1.95 KT=2.12 K (f)
FIG. 4: (a), (b) and (c): 2D energy spectra F ( k x , k z ) for T = 1 .
65 K, T = 1 .
95 K and T = 2 .
12 K. Note logarithmic scale ofthe color code. (d), (e) and (f): The corresponding ratios of SO(2), SO(4) and SO(8) decompositions of 2D energy spectra toits isotropic component, E (2) ( k ) /E (0) ( k ), E (4) ( k ) /E (0) ( k ) and E (8) ( k ) /E (0) ( k ). T=1.65 K(a)
T=1.95 K(b)
T=2.12 K(c)
T=1.65 K(d)
20 40 60 80 100 12000.20.40.60.8
T=1.95 K(e) (f) T=2.12 K
FIG. 5: (a), (b) and (c): Comparison of two linear energy spectra E (cid:104) x (cid:105) ( k z ) and E (cid:104) z (cid:105) ( k x ) with angular-averaged spectra E (0) ( k )for T = 1 .
65 K, T = 1 .
95 K and T = 2 .
12 K. All spectra are normalized by total energy density per unit mass ¯ E for the giventemperature and compensated by K41 scaling κ / with κ ≡ k/ ∆ k x with ∆ k x ≈ .
45 mm − . The horizontal gray lines mark(almost) temperature-independent asymptotic level of all normalized and compensated spectra A as ≈ . / ∆ k x . (d), (e) and(f): the individual components of the linear energy spectra. of the correlation functions between R = 0 and the first minimum R min of (cid:98) C exp (not necessarily equal to zero),0 -1 -2 -1 (a) -4 -3 -2 -1 (b) FIG. 6: Experimental Eulerian bridges for T = 1 .
95 K. (a) The measured (cid:98) S (cid:104) z (cid:105) exp and (cid:98) S (cid:104) x (cid:105) exp compared with the (cid:98) S (cid:104) z (cid:105) br and (cid:98) S (cid:104) x (cid:105) br ,reconstructed from the measured energy spectra using the bridge Eq. (5). (b) The measured energy spectra (cid:98) E exp comparedwith E br reconstructed from the measured (cid:98) C exp . supplemented them with the same data, mirror-reflectedaround R min and used Fast Fourier Transform to calcu-late the spectra. The standard 2 / x - and z -directions,we accounted for the respective increments in the calcu-lation of the integrals and for the normalization factors,as in Fig. 5. We also limited the presented data by the R and k ranges available for the spectra, which are smallerthan those accessible by the structure and correlationfunctions.In Fig. 6(a) we compare the “experimental” (cid:98) S (cid:104) x (cid:105) exp and (cid:98) S (cid:104) z (cid:105) exp (cyan and magenta lines, respectively) with the cor-responding (cid:98) S (cid:104) x (cid:105) br and (cid:98) S (cid:104) z (cid:105) br (blue and red lines, respec-tively), reconstructed from the spectra Eq. (24). Thesespectra are shown in Fig. 6(b) by magenta and cyan linesand compared with their counterparts E (cid:104) x (cid:105) br and E (cid:104) z (cid:105) br (redand blue lines, respectively).The overall agreement between the measured and thebridge-reconstructed objects, demonstrated in Fig. 6, isvery encouraging. This allows one to use bridge equations(5) as an efficient tool for the analysis of experimental andnumerical results in studies of hydrodynamic turbulenceat large as well as at modest Reynolds numbers.In particular, we see that the structure functions inFig. 6(a) at small scales do not depend on the orientationbut at large scale they differ. Both the measured (cid:98) S exp and the reconstructed (cid:98) S br show a very narrow range ofscales at which the scaling may be considered close to R / . Over most of the available range, the scaling of thestructure functions gradually changes from R / to R .On the other hand, the reconstructed energy spectra E br exhibit a clear scaling close to k − / behavior over mostof the wavenumbers range, similar to the E exp spectra,see Fig. 6(b).Therefore, as we mentioned in Sec. I A 1, the bridgerelations between the velocity structure and correlationfunctions on one hand, and the energy spectra, on the other hand, do not require large inertial interval. Theonly requirement is the homogeneity of the velocity fields. B. Lagrangian statistics
1. Lagrangian 2 nd -order structure functions Now we consider second-order structure functions ofthe Lagrangian velocity projections S α ( τ ) = (cid:104)| u αj ( t + τ ) − u αj ( t ) | (cid:105) t,j , α = { x, z } , (25)averaged over all traces j during all observation time t . The structure functions are shown in Fig. 7 (a), (b)and (c). We see that their behavior is quite differentfrom what is expected for S cl ( τ ) in classical hydrody-namic turbulence, shown in Fig. 2(b). The only caseswhere the scaling of Lagrangian structure function in su-perfluid turbulence coincides with that in classical K41turbulence E ( τ ) ∝ τ are S z ( τ ) for T = 2 .
12 K and S x ( τ )for T = 1 .
95 K, see red dashed lines in Fig. 7(b) and (c).In all other cases the S ( τ ) behavior, shown by blue dot-dashed line in Fig. 7, is closer to the scaling S ( τ ) ∝ τ / ,typical for classical K41 scaling of the Eulerian structurefunction, S ( r ) ∝ r / .To rationalize such an observation, we note that someparticle trajectories, shown in Fig. 7 (d), (e) and (f), arequite close to straight lines with more or less equidis-tantly spaced points. Therefore, in these cases the veloc-ities are measured similar to the Eulerian approach, i.e.the Eulerian scaling S ( r ) ∝ r / transforms to the ob-served scaling S ( τ ) ∝ τ / . In some sense, this situationis similar to a one-point measurement of air turbulencein the presence of strong wind, where time-dependence ofthe turbulent velocity is transformed into r -dependencewith the help of the Taylor hypothesis of frozen turbu-lence. The role of strong wind in our case is played by theenergy-containing eddies with a random velocity which ismuch larger than the velocity of small-scale eddies in the1 -2 -1 -2 -1 -2 -1 -0.1 -0.05 0 0.05 0.1 0.15-0.2-0.100.10.2 (d) -0.2 -0.1 0 0.1-0.3-0.2-0.100.10.2 (e) -0.2 0 0.2 0.4-0.3-0.2-0.100.10.20.3 (f) FIG. 7: (a)-(c) Second order Lagrangian structure functions S α ( τ ) for different temperatures, calculated using all trajectorieswith N ≥
5. Vertical lines denote t cr (cid:39) .
04 s. (d)-f) Typical trajectories of particles with 5, 8 and 12 points, respectively. Thetrajectories start at the red point and end at the blue points. inertial interval. The randomness of the sweeping veloc-ity direction is clearly seen in Fig. 7 (d), (e) and (f) as arandom direction of the trajectories, that start at a redpoint and end at the blue points.Another striking observation is that instead of S ( τ ) ∝ τ behavior, originated from the smooth, differential ve-locity in the viscous range of turbulence in classical flu-ids, we observe a saturation of S ( τ ) ≈ const. for small τ < τ cr (cid:39) .
04 s. The independence of S ( τ ) from τ meansthat velocities v ( t + τ ) and v ( t ) are statistically indepen-dent. In this case Eq. (2a) gives S ( τ ) = 2 (cid:104)| v ( t ) | (cid:105) t = const. for τ (cid:46) τ cr . (26)The simple physical picture of statistical independenceof v ( t + τ ) and v ( t ) is based on the assumption that for t (cid:46) τ cr but still larger than the smallest time difference∆ t = 8 · − s available in our experiments, the maincontribution to the tracer velocity consists of sharp peaksuncorrelated during the time interval ∆ t < t (cid:46) τ cr .A possible explanation is that for r (cid:46) (cid:96) the normaland superfluid velocity components become practicallydecoupled. In this regime the normal fluid turbulence isalready damped by viscosity, while the superfluid turbu-lence is supported by the random motion of the quantizedvortex lines. Therefore, the motion of micron-sized par-ticles in the range of scales r (cid:46) (cid:96) is mainly controlled bythe dynamics of the quantum vortex tangle. This allowsus to estimate the decorrelation time of their motion τ cr . The first step is to consider particles trapped on thevortex line. Their velocity can be estimated as the root-mean-square velocity of the vortices v (cid:96) in the Local In-duction Approximation v (cid:96) (cid:39) κ Λ4 π(cid:96) (cid:39) κ(cid:96) , Λ = ln( (cid:96)/a ) . (27a)Here κ (cid:39) − cm /s is the quantum of circulation, a (cid:39) − cm is the vortex core radius. In Eq. (27a)we have accounted for that at (cid:96) (cid:39) .
05 mm the ratioΛ / (4 π ) ≈ . (cid:39)
1. Then the decorrelation time τ cr inthis scenario can be estimated as the time during whichthe configuration of the vortex tangle changes signifi-cantly: τ cr ∼ (cid:96)v (cid:96) (cid:39) (cid:96) κ = 1 κ L . (27b)To consider untrapped particles, we have to take intoaccount that they directly interact with the superfluidcomponent through the inertial and added mass forces .Moreover, in the vicinity of the vortex core, the mutualfriction induces, in the normal fluid, the vortex dipolewhose typical lengthscale is expected to be about0.1 mm, i.e. larger than (cid:96) . It means that untrapped par-ticles will be dragged by induced normal fluid componentwith velocity about κ/r , where r (cid:46) (cid:96) is the distance ofthe particle to the vortex core. For most of the untrappedparticles r ∼ (cid:96) . In such a way, we are coming to the same2 -4 -3 -2 -1 T=1.65 K(a) -4 -3 -2 -1 T=1.95 K(b) -4 -3 -2 -1 T=2.12 K(c) -4 -3 -2 (d) -4 -3 -2 (e) -4 -3 -2 -1 T=1.95K(f)
FIG. 8: (a)-(c) Lagrangian energy spectra E α ( ω ) at different temperatures, calculated using all trajectories with N > E x ( ω )is marked by red squares and E z ( ω ) by the black circles. The vertical line denotes the frequency corresponding to the cross-overtime difference τ cr (cid:39) .
04 s − . (d)-(e) Normalized Lagrangian energy spectra E ( ω ) /E for three temperatures. In (d) the spectrawere calculated using only short ( N ∈ [5 − N > E ( ω )= for T = 1 .
95 K calculated using short (brown dots)and long (green triangles) trajectories. The thin vertical lines denote ω cr = 160 s − , ω longcr = 80 s − . The red line denote theLagrangian spectrum reconstructed from the Eulerian spectrum E ( k ) using the bridge Eq. (9). estimate (27b) τ cr ∼ (cid:96) /κ for the untrapped particles asfor the entrapped ones. With L = 4 · cm − , Eqs. (27a)gives the estimate v (cid:96) as 2 mm/s and the decorrelationtime τ cr (cid:39) .
025 s. This time is close to the critical time τ cr (cid:39) .
04 s below which S ( τ ) saturates, which therebyexplains the saturation (26) of S ( τ ) for τ (cid:46) τ cr .
2. Lagrangian energy spectra of turbulence
In order to get information about the turbulence statis-tics at length scales below the cell resolution ∆, we com-pute the Lagrangian energy spectra E ( ω ) by analyzingthe trajectories of individual particles. We first find theLagrangian positions X n and velocities u P ( τ n ) of a par-ticle P in the ( x, z )-plane at consecutive moments oftime τ n = nτ . A Fourier transform in time of the ve-locity u P ( τ n ), as described in Sec. II B, then gives theFourier component v P ( ω ), which allows us to calculatethe ensemble-averaged Lagrangian energy spectra E ( ω ): E ( ω ) = (cid:104)| v ( ω ) | (cid:105) P , (28)where the angled brackets now denote an average over anensemble of particle trajectories P . Notice that Eq. (28)is a discrete version of Eq. (6b) for E ( ω ). The Lagrangian turbulent frequency power spectra E ( ω ) are shown in Fig. 8. The panels (a)-(c) show theenergy spectra components for T = 1 .
65 K, T = 1 .
95 Kand T = 2 .
12 K, respectively. The most prominent fea-ture of all spectra is a sharp fall by about two ordersof magnitude at some ω cr (cid:39)
160 s − equal to 2 π/τ cr ,where the critical time τ cr (cid:39) .
04 s separates the semi-classical regime (for t > τ cr ) from pure quantum regime(for t < τ cr ) in the behavior of S α ( t ). Accordingly, theregion ω > ω cr is expected to be purely quantum, wherewe assume that the main contribution to the tracers’ ve-locity consists of sharp peaks uncorrelated in time. If so, E ( ω ) should be ω -independent for ω > ω cr , as observed.Additional support for our scenario for the quantum-classical crossover frequency ω cr is the estimate ofturnover frequency of (cid:96) -eddies of the intervortex separa-tion scale (cid:96) , ω (cid:96) (cid:39) . − . − for different T , whichis quite close to the measured value ω cr (cid:39)
160 s − . It iscommonly accepted that mechanically driven superfluidturbulence should behave almost classically for ω (cid:28) ω (cid:96) and in a purely quantum manner for ω (cid:29) ω (cid:96) .Notably, this transition does not happen at a singlefrequency. In Fig. 8(c) an overlap region is clearly seen,where both the classical and the quantum behavior coex-ists. It turns out that the velocity field calculated fromshorter trajectories exhibit a longer classical frequency3range, while for longer trajectories the transition is moregradual and starts at lower frequencies ω longcr . This be-havior is temperature independent, see Fig. 8(d,e). Sincethe number of shorter trajectories is typically larger,the spectra that are ensemble-averaged over whole setof available trajectories are characterized by longer clas-sical frequency range. We, therefore, expect that the ac-tual transition occur in a range of frequencies and is notsharp. In Fig. 8(f) we compare for T = 1 .
95 K the spec-tra calculated using short and long trajectories (shownby symbols), and the Lagrangian spectra reconstructedfrom the Eulerian spectra using the bridge Eq. (9) (thickline). The relation (9) is purely classical and does notdescribe the quantum plateau in the spectra. Being nor-malized by the energy contained in the same frequencyrange ω (cid:38)
30 s − as the experimental Lagrangian spec-tra, the reconstructed spectrum partially overlaps withthe measured one in the ”classical” range of frequencies,however with different scaling. The classical bridge hasthe expected ω − scaling, while the measured spectrascale as ω − / , as is shown in Fig. 8(d,e). This scal-ing matches the scaling of the structure functions, seeFig. 7, and originates, as we suggested above, from al-most equidistant position of the tracers and consequentcorrespondence of r and τ dependencies E ( r ) ⇔ E ( τ )according to the Taylor hypothesis of frozen turbulence. Conclusion
In this paper we report a detailed analysis of the Eule-rian and Lagrangian second-order statistics – the veloc-ity structure and correlation functions together with theenergy spectra – measured by the particle tracking ve-locimetry in the superfluid He grid turbulence in a widetemperature range.We measured two-dimensional Eulerian spectra F ( k x , k z ) in the ( x , z )-plane, oriented along the stream-wise z -direction. Using SO(2) decomposition, we demon-strate that the plane anisotropy of the studied grid tur-bulence is very small and can be peacefully neglected.This allows us to further analyze only one-dimensionalenergy spectra. We use three of them: the angular aver-aged spectrum E (0) ( k ) and two “linear” energy spectra E (cid:104) x (cid:105) ( k z ) and E (cid:104) z (cid:105) ( k x ), averaged over the correspondingdirection. We show that with a simple and physically mo-tivated renormalization of the energy fluxes, these threespectra practically coincide. Independent of the way ofaveraging, the Eulerian energy spectra have extended in-ertial scaling range with close to k − / behavior. Theavailable range of the Eulerian spectra, however, does notallow to probe the transition to the viscous or quantumregime. This was achieved by analysis of the Lagrangianstructure functions and spectra. These demonstrate thesharp transition from a near-classical behavior to a time-and frequency independent plateau, respectively. Thetransition occurs at a range of time increments and fre-quencies that are consistent with the intervortex scale, defined by the measured vortex line density. The ap-pearance of such a plateau corresponds to the statisticallyindependent velocities of the tracer particles associatedwith the velocity field dominated by the velocities of thequantum vortex lines. The scaling behavior in the La-grangian spectra in the quasi-classical regime ω < ω cr de-viates from the expected K41 ω − behavior and is closerto ω − / . We suggest that the possible origin of this dis-crepancy is the shape of the particles trajectories. Manyof them are almost straight and particles positions are al-most equidistant at the subsequent measurement. Sucha situation is well described by a Taylor hypothesis offrozen turbulence, in which the time dependence of themeasured velocity effectively corresponds to a one-time r -dependence. Hence the scaling typical to the Eulerianspectra. However, this unexpected scaling behavior nearthe classical-quantum transition requires further investi-gation.The unique feature of PTV measurements, allowingto simultaneously extract from the same set of trac-ers’ velocities the Eulerian and Lagrangian statistical in-formation, allowed us to verify the set of bridge rela-tions that connect various statistical objects, both withinthe same framework (Eulerian-Eulerian and Lagrangian-Lagrangian) and connecting two ways of the statisticalrepresentation (Eulerian-Lagrangian).In particular, we demonstrate using the experimentaldata that two-way bridge Eqs. (5) between the Eulerianenergy spectrum E ( k ), the velocity structure S ( r ) andthe correlation functions C ( r ) allows us to reconstructwith high accuracy any two of these objects using re-maining one of them for any extend of the inertial intervalincluding very modest one. Similar equations (7) connectthe Lagrangian structure functions and the power spec-tra. These bridges may be used as the efficient tool forthe analysis of experimental and numerical data in stud-ies of hydrodynamic turbulence at any Reynolds numbersopening multi-sided view on the statistics of turbulence.For example, if for modest Reynolds numbers S ( r ) doesnot have a visible scaling regime, E ( k ) may reveal itsexistence. The correlation function C ( r ) stresses large-scale properties of turbulence including possible coherentstructures, while S ( r ) highlights the small and moderatescale behavior of turbulence.We also demonstrate how the combination Eulerian-Lagrangian bridge Eqs. (7) and (9) allows one to recon-struct the Lagrangian second-order statistical objects –the energy spectrum E ( ω ), the structure and the correla-tion function from the Eulerian spectrum E ( k ) in classi-cal turbulence. The bridge Eq. (9) does not describe thetransition to the quantum regime. 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