4 He Counterflow Differs Strongly from Classical Flows: Anisotropy on Small Scales
L. Biferale, D. Khomenko, V. L'vov, A. Pomyalov, I. Procaccia, G. Sahoo
44 He Counterflow Differs Strongly from Classical Flows: Anisotropy on Small Scales
L. Biferale , D. Khomenko , V. L’vov , A. Pomyalov , I. Procaccia and G. Sahoo Dept. of Physics, University of Rome, Tor Vergata, Roma, Italy Laboratoire de physique th´eorique, D´epartement de physique de l’ENS,´Ecole normale sup´erieure, PSL Research University,Sorbonne Universit´es, CNRS, 75005 Paris, France. Dept. of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel Dept. of Mathematics and Statistics and Dept. of Physics, University of Helsinki, Finland
Three-dimensional anisotropic turbulence in classical fluids tends towards isotropy and homo-geneity with decreasing scales, allowing –eventually– the abstract model of “isotropic homogeneousturbulence” to be relevant. We show here that the opposite is true for superfluid He turbulencein 3-dimensional counterflow channel geometry. This flow becomes less isotropic upon decreasingscales, becoming eventually quasi 2-dimensional. The physical reason for this unusual phenomenonis elucidated and supported by theory and simulations.
All turbulent flows in nature and in laboratory exper-iments are anisotropic on the energy injection scales [1].Nevertheless the model of “isotropic homogeneous turbu-lence” had been shown to be highly relevant and success-ful in predicting the statistical properties of turbulentflows on scales much smaller than the energy injectionscales (but still larger than the dissipative scales). Thereason for this lies in the nature of the nonlinear termsof the equations of fluid mechanics; these terms tendto isotropize the flow upon cascading energy to smallerscales, redistributing the anisotropic velocity fluctuationsamong smaller scales with a higher degree of isotropy.Eventually, at small enough scales, the flow becomes suf-ficiently isotropic to allow the application of the idealmodel of isotropic homogeneous turbulence [2]. In thepresent Letter, we show that in turbulent superfluid Hein a channel geometry with a temperature gradient alongthe channel, the opposite phenomenon takes place: theflow becomes less and less isotropic upon decreasing thescales. Eventually, the flow becomes quasi 2-dimensionalwith interesting and unusual properties as detailed be-low.An easy way to account for this difference in tendencytowards isotropy is furnished by the two-fluid model ofturbulence in superfluid He [3–5]. Denote by u s and u n the superfluid and normal-fluid turbulent velocities, re-spectively. In counterflow geometry, with a temperaturegradient directed along the channel, the mean superfluidvelocity U s is directed towards the heater, and the meannormal velocity U n away from the heater. Importantly,one finds that there exists a mutual friction force f ns be-tween these two components [4–9], proportional to thedifference in velocities, i.e f ns ∝ ( u n − u s ). As long asthe fluctuations between these two velocities are corre-lated, this force remains small. Upon loss of correlationthis force becomes large and will lead to a suppression ofthe corresponding fluctuations. Consider then two typesof velocity fluctuations, one elongated along the channeland the counterflow and the other orthogonal to them,see Fig. 1. Due to the mean flow in opposite directions, 𝑈 ns 𝑢 n 𝑢 s 𝑈 n 𝑈 s xy z FIG. 1: Schematics of the superfluid He channel counterflow.The normal-fluid eddies (solid red lines) and the superfluid ed-dies (the blue dashed lines) are swept by the correspondingmean velocities U n and U s away and towards the heater, re-spectively. The resulting counterflow velocity U ns is orientedalong the positive x -direction. The streamwise-elongated ed-dies have longer overlap time than the cross-stream-elongatededdies. the velocity fluctuations oriented orthogonally will have ashort overlap time and will decorrelate quickly, whereasthe velocity fluctuations along the counterflow will re-main correlated for a longer time. The result will be astrong suppression of the former type of velocity fluctu-ations with respect to the latter. This will eventuallylead to a turbulent flow in which the fluctuations consistmostly of the stream-wise component, while the energyis concentrated in the plane orthogonal to the counter-flow direction. The rest of this Letter will elaborate thispicture by using an analytical approach and will supportit using direct numerical simulations (DNS). The basic equations . The two-fluid model describes su-perfluid He of density ρ as a mixture of two interpen-etrating fluid components: an inviscid superfluid and aviscous normal-fluid. The densities of the components ρ s , ρ n : ρ s + ρ n = ρ define their contributions to themixture. The fluid components are coupled by a mu-tual friction force, mediated by the tangle of quantumvortices [4–8] of a core radius a ≈ − cm and a fixedcirculation κ = h/M ≈ − cm /s, where h is Planck’sconstant and M is the mass of the He atom [10]. A a r X i v : . [ c ond - m a t . o t h e r] J a n complex tangle of these vortex lines with a typical inter-vortex distance [5] (cid:96) ∼ − − − cm is a manifestationof superfluid turbulence.To proceed it is sufficient to employ coarse-grained dy-namics, following the gradually-damped version[11] ofthe Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equa-tions for counterflow turbulence [11–15]. It has a formof two Navier-Stokes equations for the turbulent velocityfluctuations u j ( r , t ) of the normal-fluid ( j = n) and thesuperfluid ( j = s): (cid:104) ∂∂t + ( u j + U j ) · ∇ (cid:105) u j − ∇ p j ρ j = ν j ∆ u j + f j + ϕ j , (1)coupled by the mutual friction forces f j in the mini-mal form [16]: f s (cid:39) Ω s ( u n − u s ), f n (cid:39) Ω n ( u s − u n ),Ω s = ακ L , and Ω n = ρ s Ω s /ρ n . The mutual fric-tion frequency Ω s depends on the temperature-dependentdimensionless mutual friction parameter α ( T ) and onthe vortex line density L . In Eqs. (1) p j are the pres-sures of the normal-fluid and the superfluid components.The kinematic viscosity of the normal-fluid componentis ν n = η/ρ n with η being the dynamical viscosity of He [17]. The energy sink in the equation for the super-fluid component, proportional to the effective superfluidviscosity, ν s , accounts for the energy dissipation at theintervortex scale (cid:96) , due to vortex reconnections and en-ergy transfer to Kelvin waves [5, 11]. The contributions,involving the reactive (dimensionless) mutual friction pa-rameter α (cid:48) , that renormalizes the nonlinear terms, wereomitted due to its numerical smallness [17].The large-scale motion in the thermal counterflow issustained by the temperature gradient, created along thechannel. Here we use the fact that the center of the chan-nel flow at large enough Reynolds numbers can be con-sidered as almost space-homogeneous [18]. To simplifythe analysis we consider homogeneous turbulence underperiodic boundary conditions and mimic the steering ofturbulence at large scales by random forces ϕ j . Equa-tions (1) describe the motion of two fluid componentsin the range of scales between the forcing scale and theintervortex distance. Statistics of anisotropic turbulence . The most generaldescription of homogeneous superfluid He turbulence atthe level of second-order statistics can be done in termsof the three-dimensional (3D) Fourier-spectrum of eachcomponent and the cross-correlation functions:(2 π ) δ ( k − k (cid:48) ) F αβij ( k ) = (cid:68) v αi ( k ) v ∗ βj ( k (cid:48) ) (cid:69) , (2)where v j ( k ) is the Fourier transform of u j ( r ); the indices i and j refer to the fluid components; the vector indices α, β = { x, y, z } denote the Cartesian coordinates and ∗ stands for complex conjugation. In the following, wechoose the counterflow velocity, U ns = U n − U s along theˆ x -direction as depicted in Fig.(1). Next denote the traceof any tensor according to F jj ( k ) ≡ (cid:80) α F ααjj ( k ). With this notation, the kinetic energy density per unit mass E j reads E j ≡ (cid:10) | u j ( r ) | (cid:11) = 12 (cid:90) F jj ( k ) d k (cid:14) (2 π ) . (3)Due to the presence of the preferred direction, definedby the counterflow velocity, the counterflow turbulencehas an axial symmetry around the ˆ x axis. Then F ij ( k )depends only on the two projections k (cid:107) = k x and k ⊥ = (cid:113) k y + k z of the wave-vector k , being independent of theangle φ in the ⊥ -plane, orthogonal to U ns . This allows usto define a set of two-dimensional (2D) objects that stillcontain all the information about 2 nd -order statistics ofthe counterflow turbulence F ij ( k (cid:107) , k ⊥ ) ≡ k ⊥ π F ij ( k (cid:107) , k ⊥ ) . (4a)Another way to represent the same information is to in-troduce a polar angle cos ( θ ) = ( k , U ns ) / | k || U ns | , and touse spherical coordinates:˜ F ij ( k, θ ) ≡ k π F ij ( k cos θ, k sin θ ) . (4b) Physical origin of the strong anisotropy . The physical ori-gin of the strong anisotropy in the counterflow turbu-lence is best exposed by considering the balance equationfor the 2D energy spectra ˜ F nn ( k, θ ) , ˜ F ss ( k, θ ). For thatwe start with Eqs. (1), follow the procedure described inRef. [15] and average the resulting equations for the 3Dspectra over the azimuthal angle ϕ . Finally, for the nor-mal component we get: ∂ ˜ F nn ( k, θ, t ) ∂t +div k [ ε n ( k )] = −D mfn ( k, θ ) −D kvn ( k, θ ) , D mfn ( k, θ ) = Ω n [ ˜ F nn ( k, θ ) − ˜ F ns ( k, θ ) (cid:3) , (5) D kvn ( k, θ ) = 2 ν n k ˜ F nn ( k, θ ) , where div k [ ε j ( k )] is the transfer term due to inertial non-linear effects, D mfn ( k, θ ) describes the rate of energy dissi-pation by the mutual friction, while D kvn ( k, θ ) stands forthe rate of dissipation by the kinematic viscosity. A sim-ilar equation is obtained for the superfluid component byreplacing n with s everywhere. For a qualitative analysisof the origin of the anisotropy in our system it is impor-tant to develop a closure of the cross-correlation function˜ F ns ( k, θ ) in D mf j ( k, θ ) in terms of the spectral propertiesof each fluid component and of the counterflow velocity.According to Ref. [12]:˜ F ns ( k, θ ) = AB/ [ B + ( k · U ns ) ] . (6)Here A = Ω s ˜ F nn ( k, θ )+Ω n ˜ F ss ( k, θ ) and B can be approx-imated as B = Ω n +Ω s , as shown in [15]. We further sim-plify ˜ F ns ( k, θ ) in Eqs. (6) by noting [15] that when twocomponents are highly correlated, the cross-correlation k -3 -2 -1 CHTCounterflow Coflow(a) k Counterflow Coflow(b) -3 -2 -1 Counterflow Coflow (c) k k k FIG. 2: (a) The spherical energy spectra E jj ( k ) of the normal-fluid (circles) and the superfluid (squares), (b) the cross-correlation function R ( k ) and (c) the angular dependence of the cross correlation function ˜ R ( k, θ ) for the coflow and thecounterflow. In panel (c), the data for the coflow all coincide with the isotropic result. For the counterflow, red lines correspondto the ˜ R ( k, θ ) averaged over the wavenumber range 10 ≤ k <
20, green lines – to averaging over 20 ≤ k <
60 and blue lines –to the averaging over 60 ≤ k ≤
80 (labeled as k , k , and k , respectively). Note the log-linear scale. may be accurately represented by the corresponding en-ergy spectra. For wavenumbers where the componentsare not correlated, as is quantified by the decorrelationfunction D ( k, θ ) [12], ˜ F ns ( k, θ ) is small and the accuracyof its representation is less important. We therefore geta decoupled form of the cross-correlation:˜ F ns ( k, θ ) = ˜ F jj ( k, θ ) D ( k, θ ) , (7a) D ( k, θ ) = (cid:104) (cid:16) kU ns cos θ Ω n + Ω s (cid:17) (cid:105) − , (7b)and finally determine the rate of energy dissipation dueto mutual friction: D mf j ( k, θ ) = Ω j ˜ F jj ( k, θ ) (cid:2) − D ( k, θ ) (cid:3) . (7c)Equations (7) are the central analytical result of this pa-per.The impact of U ns on the anisotropy follows from theclosure (7c). Indeed, for small k or even for large k with k almost perpendicular to U ns (i.e cos θ (cid:28) D ( k, θ ) (cid:39)
1, the normal-fluid and superfluid velocitiesare almost fully coupled and the dissipation rate is small: D mf j ( k, θ ) (cid:28) Ω j . In this case, the mutual friction doesnot significantly affect the energy balance and we ex-pect the energy spectrum ˜ F jj ( k, θ ) to be close to theKolmogorov-1941 (K41) prediction E K41 ( k ) ∝ k − / forboth components. For large k and with cos θ ∼
1, thevelocity components are almost decoupled D ( k, θ ) (cid:28) D mf j ( k, θ ) ≈ Ω j (cid:101) F jj ( k, θ ). This situation is similar to thatin He with the normal-fluid component at rest [13]. Insuch a case, we can expect that the energy dissipation bymutual friction strongly suppresses the energy spectra,much below the K41 expectation E K41 ( k ). Combiningall these considerations, we expect the energy spectra˜ F jj ( k, cos θ ) to become more anisotropic with increasing k , with most of the energy concentrated in the range ofsmall cos θ , i.e. in the orthogonal plane. Numerical results . Direct numerical simulations of thecoupled HVBK Eqs. (1) were carried out using a fully de-aliased pseudospectral code with a resolution of 256 col-location points in a triply periodic domain of size L = 2 π .To reach a steady state flow, velocity fields of the normaland superfluid components are stirred by two indepen-dent random Gaussian forces ϕ s and ϕ n with the forceamplitudes | ϕ | = 0 . k ϕ ∈ [0 . , . T = 1 .
85 K, at which the densities and viscosities ofthe normal-fluid and superfluid components are close: ρ s /ρ n = 1 .
75 and ν s /ν n = 1 .
07. The mutual frictionparameter for this temperature is α = 0 .
18. The simu-lations were carried out with both the normal-fluid andsuperfluid viscosity ν n = ν s = 0 . wRe j = ( u j T )( /ν j k ) , w = U ns /u n T . (8)Here u j T = (cid:112) (cid:104) u j (cid:105) is the root mean square (rms) of theturbulent velocity fluctuations, k = 1 is the outer scaleof turbulence. To emphasize the importance of the coun-terflow, we compare the results with the simulations forthe so-called coflow with the rest of the parameters be-ing the same. In the coflow, the two components of themechanically driven He, being coupled by the mutualfriction force, move in the same direction with the samemean velocities, U ns = 0. The statistics in the coflowconfiguration is known to be similar to that of classi-cal isotropic turbulence [14, 19–21]. In our simulations,the values of the Reynolds numbers in the counterfloware Re n = 1051 and Re s = 1056, while in the coflow, Re n = 1179 and Re s = 1181. The rms velocities of bothcomponents in both flows are u s T = u n T = 3 .
5. The dimen-sionless values of the mutual friction frequency Ω s = 20 -20 -15 -10 -5 (a) k Coflow Counterflow (b)
FIG. 3: (a)The superfluid component energy spectrum F ss ( k (cid:107) , k ⊥ ) in the counterflow. (b) The tensor decomposition of thenormalized spherical energy spectra K αj ( k ) for the normal-fluid (circles) and the superfluid (squares). (c) The superfluid velocitycomponents u x s ( r )(top) and u y s ( r )(bottom). The u z s ( r ) (not shown) is similar to u y s ( r ). The velocity magnitude is color-codedwith red denoting positive and blue denoting negative values. and the counterflow velocity U ns = 15 . s dependence of the energy spectra will be reportedelsewhere. The flow conditions were controlled by thesimulations of the uncoupled equations without counter-flow ( U j = Ω j = 0), which represent here the classicalhydrodynamic isotropic turbulence (CHT).The energy spectra are influenced by the viscous dis-sipation, by the dissipation due to mutual friction andby the counterflow-induced decoupling. To clarify therole of each of these factors, we first ignore the expectedanisotropy and compare in Fig. 2(a) the normal-fluid andsuperfluid energy spectra E nn ( k ) and E ss ( k ) and thecross-correlation E ns ( k ), integrated over a spherical sur-face of radius k , i.e. over all directions of vector k : E ij ( k ) = (cid:90) F ij ( k ) dφ d cos θ (2 π ) . (9)The corresponding normalized cross-correlation functions R ( k ) = 2 E ns ( k ) / [ E nn ( k ) + E ss ( k )] (10)are shown in Fig. 2(b). The effect of viscous dissipationis clearly seen in the spectra of the uncoupled compo-nents, corresponding to classical hydrodynamic turbu-lence (marked “CHT”, black lines). The spectra almostcoincide, since at T = 1 .
85 K the viscosities are close. Inthe coflow, the strongly coupled components are well cor-related at all scales and move almost as one fluid. Notethe additional dissipation due to mutual friction, leadingto further suppression of the spectra compared to theuncoupled case. The presence of the counterflow velocityleads to a sweeping [12] of the two component’s eddies inopposite directions by the corresponding mean velocities.The result is the decorrelation of the components turbu-lence velocities, especially at small scales, for which theoverlapping time is very short, see Fig. 2(b). The dissi-pation by mutual friction is very strong in this case, withboth Ω and the velocity difference being large, leading to very strongly suppressed spectra, with E nn ( k ) ≈ E ss ( k ).This behavior was predicted by the theory [15], based onthe assumption of spectral isotropy. However the spheri-cally integrated spectra and cross-correlations cannot re-veal any properties connected to the anisotropic actionof the mutual friction force. To account for the spectralanisotropy we plot in Fig. 2(c) the normalized 2D cross-correlations˜ R ( k, θ ) = 2 ˜ F ns ( k, θ ) / [ ˜ F nn ( k, θ ) + ˜ F ss ( k, θ )] . (11)Given the discrete nature of the k -space in DNS, we av-erage them over 3 bands of wavenumbers. Leaving aside k ≈ k , influenced by the forcing, we average ˜ R ( k, θ ) overthe k -ranges 10 ≤ k <
20, 20 ≤ k <
60 and 60 ≤ k ≤ θ ≈ k (red lines, labelled k ) and faster as k become larger (green, k , and blue lines, k , respec-tively). Such a strong decorrelation of the componentsvelocities leads to an enhanced dissipation by mutual fric-tion in the counterflow direction, such that most of theenergy is contained in the narrow range cos θ (cid:46) .
1, nearthe plane orthogonal to U ns .Indeed, the superfluid energy spectrum F ss ( k (cid:107) , k ⊥ ),shown in Fig. 3(a), is strongly suppressed in the k || di-rection, while it decays slowly in the orthogonal plane.A similar phenomenon of the creation of quasi-2D tur-bulence is observed in a strongly stratified atmosphere[22–24] and in rotating turbulence [25–27], in which thereexists a preferred direction defined by gravity or by a ro-tation axis. The difference between these examples andthe present counterflow lies in the nature of the velocityfield. The leading velocity components in the classicalflows are in a plane orthogonal to the preferred direc-tion. Moreover, at small scales the isotropy is restored[23, 24]. On the contrary, in He counterflow, the domi-nant velocity component is oriented along the counterflowdirection, with the anisotropy becoming stronger with de-creasing scales, as we show in Fig. 3(b). Here we plot thetensor components of the spherical spectra as the ratios K αj ( k ) ≡ E ααjj ( k ) / E jj ( k ) . (12)The factor 3 was introduced to ensure that for isotropicturbulence K αj ( k ) = 1. Expectedly, the coflow (thealmost horizontal lines) is isotropic at all scales, ex-cept for the smallest wavenumbers. On the other hand,for the counterflow turbulence, the contribution of the K xj ( k ) component (shown by red lines) is dominant andmonotonically increases with k from the isotropic level K xj ( k ) ≈ K xj ( k ) ≈ v xj ( k ) velocity fluctuations. The contribution of v yj and v zj fluctuations for k (cid:38)
10 is negligible. Summa-rizing Fig. 3, the leading contribution to the spectra ofsmall scale counterflow turbulence comes from the tur-bulent velocity fluctuations with only one stream-wiseprojection that depends on the two cross-stream coordi-nates { y, z } : u x ( y, z ). Such type of turbulence can bevisualized as narrow jets or thin sheets with velocity, ori-ented along the counterflow and randomly distributed inthe ⊥ -plane. Indeed the velocity components u y s , shownin Fig. 3c, and u z s have only large scale structures, while u x s has elongated structures at various scales. The energyspectra, corresponding to u x n ( y, t ) were recently measuredexperimentally [28, 29] and were found to agree with pre-dictions [15] in the range of scales where the fluid com-ponents are well correlated, while decaying faster thanpredicted at smaller scales. Summary . The energy spectra of the superfluid Hecounterflow turbulence become more anisotropic upongoing from large scales toward scales about the intervor-tex distance. This strong anisotropy distinguish it fromthe classical turbulent flows that become more isotropicas the scale decreases. Most of the turbulent energybecome concentrated in the plane, orthogonal to thecounterflow direction. Furthermore, contrary to classi-cal quasi-2D turbulent flows in rotation or in stratifiedconfigurations, where dominant velocity components liein the same plane, the only surviving velocity componentat small scales is preferentially oriented along the coun-terflow direction. The selective suppression of the or-thogonal velocity fluctuations has its origin in the stronganisotropy of the energy dissipation by mutual friction,resulting from the angular dependence of the compo-nents’ cross-correlation.
Acknowledgments
LB acknowledges funding fromthe European Unions Seventh Framework Programme(FP7/20072013) under Grant Agreement No. 339032.GS thanks AtMath collaboration at University ofHelsinki. DK acknowledges funding from the SimonsFoundation under grant No. 454955 (Francesco Zam-poni). [1] L. Biferale and I. Procaccia, Phys. Rep.
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