Inhomogeneous distribution of particles in co-flow and counterflow quantum turbulence
IInhomogeneous distribution of particles in co-flow and counterflow quantumturbulence
Juan Ignacio Polanco and Giorgio Krstulovic
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, Nice, France
Particles are today the main tool to study superfluid turbulence and visualize quantum vortices. Inthis work, we study the dynamics and the spatial distribution of particles in co-flow and counterflowsuperfluid helium turbulence in the framework of the two-fluid Hall-Vinen-Bekarevich-Khalatnikov(HVBK) model. We perform three-dimensional numerical simulations of the HVBK equations alongwith the particle dynamics that depends on the motion of both fluid components. We find that,at low temperatures, where the superfluid mass fraction dominates, particles strongly cluster invortex filaments regardless of their physical properties. At higher temperatures, as viscous dragbecomes important and the two components become tightly coupled, the clustering dynamics inthe coflowing case approach those found in classical turbulence, while under strong counterflow, theparticle distribution is dominated by the quasi-two-dimensionalization of the flow.
I. INTRODUCTION
Turbulence has fascinated physicists and mathematicians for centuries, and is one of the oldest yet still unsolvedproblems in physics. In a turbulent fluid, energy injected at large scales is transferred towards small scales in a cascadeprocess [1]. At small scales, a turbulent fluid develops strong velocity gradients resulting in the appearance of vortexfilaments [2]. Such vortices have an important counterpart in turbulent quantum fluids, such as superfluid heliumand Bose-Einstein condensates (BECs) made of dilute alkali gases.At finite temperatures, a quantum fluid consists of two immiscible components: a superfluid with no viscosity, anda normal fluid described by the Navier-Stokes equations. In the case where the mean relative velocity of these twocomponents is non-zero, the two-fluid description leads to a turbulent state with no classical analogous known ascounterflow turbulence [3]. Such out-of-equilibrium state is typically produced by imposing a temperature gradientin a channel [4, 5]. Another defining property of superfluids is that the circulation around vortices is quantized.Such objects, known as quantum vortices, have been the subject of extensive experimental studies since the earlydiscovery of superfluidity. Rectilinear quantum vortices were first photographed at the intersection with helium-freesurfaces in 1979 [6]. There is a renewed interest since 2006, when they were first visualized in superfluid heliumusing hydrogen particles [7]. Further progress on particle tracking methods has enabled the observation of quantumvortex reconnections [8] and Kelvin waves [9], as well as unveiling the differences between classical and quantumturbulence [10, 11].Particles have been also actively used to study vortex dynamics in classical fluids [12]. Particle inertia generallyleads to a non-uniform spatial distribution of particles in turbulent flows [13]. Light particles such as bubbles in waterbecome trapped in vortices allowing their visualization [14], while heavy particles tend to escape from them [15]. Inquantum turbulence, the situation is more complex since particles interact with both components of the superfluid [16].At low temperatures where the normal fluid fraction is negligible, the particle dynamics is dominated by pressuregradients leading to their trapping by quantum vortices [3, 17, 18]. As temperature increases, particles additionallyexperience a viscous Stokes drag from the normal component.There exist different models to describe superfluid turbulence. At very low temperatures, the Gross-Pitaevskiiequation describes well a weakly interacting BEC, and is expected to provide a qualitative description of superfluidhelium. In this model, vortices are by construction topological defects and their circulation is therefore quantized.The Gross-Pitaevskii equation has been generalized to include the dynamics of classical particles [19, 20], and hasbeen used to study particle trapping by quantum vortices [18] and the particle-vortex interaction once particlesare trapped [21]. A second approach is the vortex filament method, where each vortex line advects each otherthrough Biot-Savart integrals [22]. This method has also been adapted to describe the interaction of particles andvortices [17, 23]. Finally, a third kind of model is given by the coarse-grained Hall-Vinen-Bekarevich-Khalatnikov(HVBK) equations [3]. This approach is well adapted to describe the large-scale motion of a turbulent superfluid atfinite temperature, although the quantum nature of vortices is lost. In particular, it has been recently used to studyco-flow and counterflow turbulence [24, 25].Liquid helium experiments commonly use solid hydrogen or deuterium particles with typical diameters of a fewmicrons [26, 27]. Although such particles are much larger than the vortex core size a ≈ a r X i v : . [ c ond - m a t . o t h e r] M a r T (K) − − ν ν s n / ρ ρ s n / α a T (K) S t ok e s nu m b e r a p b FIG. 1. (a) Temperature-dependent properties of superfluid He. Mutual friction coefficient α from [31], density ratio ρ s /ρ n and viscosity ratio ν s /ν n . (b) Dependence of Stokes numbers St = τ p /τ (n) η on temperature. Stokes numbers are estimated forspherical solid hydrogen particles ( ρ p /ρ ≈ . µ m, using τ (n) η = ( ρ/ρ n ) / τ exp η and τ exp η = 0 . on vortices depending on their physical properties. II. GOVERNING EQUATIONSA. Coarse-grained HVBK model
We consider the dynamics of turbulent superfluid helium at finite temperature driven by the HVBK equations,describing the flow at scales larger than the mean distance between vortices. At these scales, the quantum vortexdynamics can be approximated by a coarse-grained superfluid velocity field u s , which interacts with the viscous normalcomponent u n via two coupled Navier-Stokes equations, ∂ u n ∂t + u n · ∇ u n = − ρ n ∇ p n + ν n ∇ u n − ρ s ρ n f ns + Φ n , (1) ∂ u s ∂t + u s · ∇ u s = − ρ s ∇ p s + ν s ∇ u s + f ns + Φ s , (2) ∇ · u n = ∇ · u s = 0 , f ns = α Ω ( u n − u s ) . (3)The total density of the fluid is ρ = ρ n + ρ s . The normal fluid viscosity ν n is related to the helium dynamic viscosity µ by ν n = µ/ρ n . The two fluids are coupled through the mutual friction force f ns that originates from the scattering ofthe excitations constituting the normal fluid component on quantum vortices. To be included in the HVBK dynamics,this microscopic process has to be averaged on the relevant scales (for a detailed discussion see Ref. [25]). A numberof models have been proposed to estimate this characteristic time scale for the HVBK description. In general, it isproportional to the temperature dependent mutual friction coefficient α (see Fig. 1a) and to a characteristic superfluidvorticity Ω . The frequency Ω is in principle proportional to the vortex line density and to the quantum of circulation.As in Ref. [24], we estimate it as Ω = (cid:104)| ω s | (cid:105) /
2, where ω s = ∇ × u s is the superfluid vorticity, and (cid:104)·(cid:105) denotes aspace average. When there is a very strong counterflow, this superfluid vorticity-based estimate may underestimatethe mutual friction frequency. In this case, one can instead take Ω as an external control parameter depending onthe particular flow [28].The two velocity fields are stirred by independent large-scale Gaussian random forces Φ s ( x ) and Φ n ( x ) of unitvariance. In the present formulation, a mean counterflow velocity U ns = (cid:104) u n (cid:105) − (cid:104) u s (cid:105) may be optionally imposedby setting the average forces to (cid:104) Φ s (cid:105) = − α Ω U ns and (cid:104) Φ n (cid:105) = ( ρ s /ρ n ) α Ω U ns . In Eq. (2), the effective superfluidviscosity ν s models the small-scale physics not resolved by the HVBK equations, including energy dissipation due toquantum vortex reconnections and Kelvin wave excitation. The values of the effective viscosity are taken from themodel described in Refs. [29, 30]. The viscosity ratio ν s /ν n resulting from this model is shown in Fig. 1(a). B. Inertial particles in the HVBK model
Particles in superfluid helium experience a Stokes drag associated to the viscosity of the normal fluid, while alsofeeling the pressure gradient force from both fluid components. Particles are considered to be much smaller than theKolmogorov scales of the flow. Hence finite-size effects can be neglected, as well as the action of particles on the flow,since any disturbance of the flow is immediately damped. The Basset history term is also neglected. The equationsgoverning the particle dynamics then read [16, 33]d v p d t = 1 τ p ( u n ( x p ) − v p ) + β (cid:18) ρ n ρ D u n D t + ρ s ρ D u s D t (cid:19) (4) τ p = a βν , β = 3 ρ ρ p + ρ , (5)where ρ p is the particle density and a p its radius, and D / D t are the corresponding material derivatives. The densityparameter β accounts for added mass effects, while the Stokes time τ p represents the particle response time to normalfluid fluctuations. Note that, even though there is a viscous term in Eq. (2), there is no Stokes drag resulting fromthe superfluid component. Particles moving at velocities close to the speed of sound could in principle trigger vortexnucleations, which would result in an additional effective drag. Here we neglect such small-scale effects.The superfluid pressure gradient term in Eq. (4), proportional to D u s / D t , is responsible for particle attractiontowards superfluid vortices. Note that the present model does not explicitly account for particles that become trappedby quantum vortices, whose behavior is expected to be different from that of untrapped particles. For instance, inthermal counterflow experiments, trapped particles move towards the heat source along with the superfluid flow, whileuntrapped ones are transported away from it by the normal component [34]. With regard to the spatial distributionof particles, one can expect that accounting for trapping would further increase the concentration of particles insuperfluid vortices compared to the present model.The Stokes number St = τ p /τ (n) η quantifies the particle inertia. Here the Kolmogorov time scale associated to thenormal component is τ (n) η = ( ν n /ε n ) / , where ε n is the mean energy dissipation rate of the normal fluid. In thelimit St →
0, particles behave as perfect tracers of the normal component. In the opposite limit St → ∞ the particlemotion is ballistic and not modified by turbulent fluctuations. The Stokes numbers of micrometer-sized hydrogenparticles based on dissipation measurements in the SHREK experiment [32] are estimated in Fig. 1(b). Remarkably,the temperature dependence of St for fixed particle parameters ( a p , ρ p ) is non-monotonic due to the variation ofhelium properties with temperature, and presents a maximum value at T ≈ .
04 K.As noted above, the present model is valid in the limit of small particle size compared to the Kolmogorov scale ofthe normal fluid. In addition, particles should be in principle smaller than the mean inter-vortex distance, so thatthey do not interact strongly with quantized vortices, and do not get often trapped by them [33]. Consistently withthe HVBK approach, which does not explicitly account for quantized vortex dynamics, Eq. (4) should be interpretedas describing the coarse-grained particle dynamics, neglecting the physics at smaller scales. Whether such small-scale phenomena have an impact on the coarse-grained particle dynamics is a challenging question that can only beanswered by confronting this model to new experimental results. Note that, in recent superfluid He experiments, theinter-vortex distance is of order 10 µ m [5, 32, 35], comparable both to the Kolmogorov scales and to the typical sizeof hydrogen particles in experiments. C. Numerical procedure
We investigate the spatial distribution of inertial particles in superfluid He by numerically solving the HVBKequations (1–3) in a triply periodic box using a parallel pseudo-spectral code (see [36] for details). Point particles arerandomly initialized in the domain, and their trajectories are evolved in time until the system reaches a statisticallysteady state. The time advancement of both particles and fields is performed using a third-order Runge-Kutta scheme.Fluid fields are interpolated at particle positions using fourth-order B-splines [37].Simulations are performed at temperatures T = 1.3, 1.9 and 2.1 K. Navier-Stokes simulations are also performedfor comparison with the classical turbulence case. The number of collocation points in each direction is either N =256 or 512. Both resolutions only differ on the numerical value of the viscosities ν n and ν s and on the resultingReynolds numbers, but the ν s /ν n ratio is kept the same. The Reynolds numbers associated to the normal andsuperfluid components are defined as Re α = u ( α )rms / ( ν α k ), where α = { n , s } , u ( α )rms is the root-mean-square of thevelocity fluctuations, and 1 /k = 1 is the scale of the external forcing. Reynolds numbers are fixed by the resolution,as the smallest scales of the most turbulent component have to be well resolved. For each run, N p particles of a given a b c d FIG. 2. Quasi-two-dimensional slices of the instantaneous particle distribution for St = 1 and ρ p /ρ = 0 . β = 1 . T = 1 . T = 1 . T = 1 . class are tracked, each class being defined by a set of parameters ( a p , ρ p ). Simulation parameters are summarized inTable I.Two counterflow simulations (runs IV and V in Table I) are performed at the temperature T = 1 . = (cid:104)| ω s | (cid:105) / as an externalcontrol parameter with a value 4 times larger than steady value of the first run. The values of Ω , normalized by k u (n)rms are also displayed in Table I. Note that effectively, the coupling between the two fluid components is strongerfor run V than run IV. As discussed in Sec. II A, this is to account for a likely underestimation of the mutual frictionintensity by the superfluid vorticity-based estimate. This also allows to clarify the effect of the mutual friction onparticle concentration statistics. In both cases, the imposed mean counterflow velocity is U ns /u (n)rms ≈ TABLE I. Simulation parameters. NS denotes Navier-Stokes simulations. (See text for definitions.)Run T (K) U ns /u (n)rms N α Ω / ( k u (n)rms ) ρ s /ρ ρ n /ρ ν s /ν n Re n Re s N p / I 1.3 0.0 256 0.034 8.7 0.952 0.048 0.043 28 707 2.0II 0.0 512 14.2 59 1479 3.2III 1.9 0.0 256 0.206 7.9 0.574 0.426 1.25 632 516 2.0IV 4.3 256 2.3 592 426 0.4V 5.6 256 11.2 447 386 0.4VI 0.0 512 11.2 1299 1053 3.2VII 2.1 0.0 256 0.481 7.4 0.259 0.741 2.5 695 268 0.4VIII 0.0 512 9.8 1332 515 3.2IX NS 0.0 256 0.0 – 0.0 1.0 – 780 – 0.4X 0.0 512 – 1639 – 3.2
III. SPATIAL DISTRIBUTION OF PARTICLES
To illustrate the effect of temperature on particle clustering, we show in Fig. 2 the instantaneous particle distributionobtained from different simulations. Particle parameters are St = 1 and ρ p /ρ = 0 .
7, comparable to those typicallyfound in experiments (see Fig. 1b). In turbulent coflow at T = 1 . T = 1 . D [39, 40], estimated as the small-scalepower law scaling of the probability P ( r ) of finding two particles at a distance smaller than r (i.e. P ( r ) ∼ r D for r small). In three dimensions, D = 3 indicates that particles are uniformly distributed in space, while smaller values − − − − P r () ≈ ≈ a St − − St b T (K) N r η / (n) FIG. 3. Particle concentration at constant density ρ p /ρ = 0 . β = 1 . P ( r ) for T = 1 . η (n) = ( ν /ε n ) / . (b) Correlation dimension D as a function of Stokes number for all runs. Different markers representdifferent cases. CF: counterflow turbulence (run IV). CF*: counterflow turbulence with strong mutual friction (run V). NS:classical turbulence (runs IX, X). are evidence of fractal clustering.We first consider particles of relative density ρ p /ρ = 0 . a p . The separation probability P ( r ) for different Stokes numbers is shown in Fig. 3(a) for the 1.3 K cases. At small scales, the curves present a clearpower law scaling, with an exponent D that varies significantly with St. At this temperature, particle clustering ismaximal for St ≈ .
4, which would roughly correspond to 6 µ m-diameter hydrogen particles in SHREK (Fig. 1b) orin the Prague oscillating grid experiments [41].We plot in Fig. 3(b) the correlation dimension D from all runs. As in classical turbulence [15], for all temperaturesparticle clustering is maximal at Stokes numbers of order unity. At temperatures close to T λ , the minimum value of D is close to 2.3, comparable to the case of heavy particles in turbulence [42]. In particular, both counterflow casesat T = 1 . ≈
1, similarly to the coflow runs at the same temperature. The twocounterflow curves nearly collapse, suggesting that there is no significant effect of the mutual friction intensity on D .As anticipated from Fig. 2, particle clustering changes dramatically in turbulent coflow at lower temperatures. At T = 1 . D decreases to 0.75, indicating that particles become concentrated in worm-likestructures such as those seen in Fig. 2(a).To understand the above observations, we consider the particle equation of motion in the small Stokes number limit( τ p (cid:28) τ (n) η ). In this case, particles follow an effective compressible velocity field v eff ( x , t ) [13, 43]. From Eq. (4), thisfield writes v eff ≈ u n + τ p (cid:16) β ρ n ρ − (cid:17) D u n D t + τ p β ρ s ρ D u s D t . Taking its divergence, one finds1 τ p ∇ · v eff ≈ (cid:18) β ρ n ρ − (cid:19) ( S − Ω ) + β ρ s ρ ( S − Ω ) , (6)where S s , S n , Ω s and Ω n are the norms of the strain-rate and rotation-rate tensors of the two fluids.In the classical limit where ρ s = 0, Eq. (6) indicates that light particles ( β >
1) tend to concentrate in vorticity-dominated regions (where Ω n > S n ), while heavy particles ( β <
1) accumulate in strain-dominated regions [44]. Forneutral particles ( β = 1), the effective velocity field is incompressible and no preferential concentration is expected.The classical picture changes in low-temperature He when ρ s (cid:29) ρ n . In this case, Eq. (6) becomes τ p − ∇ · v eff ≈− ( S − Ω ) + β ( S − Ω ), implying that the remaining normal component acts on the particle dynamics only throughthe Stokes drag. Due to its higher viscosity, the normal velocity field is smoother (has weaker gradients), hence ingeneral | S − Ω | (cid:29) | S − Ω | . As a consequence, for β of order unity, the superfluid term dominates, and thusparticles cluster in regions of high superfluid vorticity. We stress that this behavior is unique to quantum turbulence,since the absence of superfluid drag on the particles implies that there is no force counteracting the dominant effectof the superfluid pressure gradient.In the opposite limit T → T λ , the superfluid fraction vanishes and the classical behavior discussed above is recovered.More interesting is the intermediate case where the two fluid densities and viscosities are similar, at T ≈ . β a T (K) β p αα / b n s FIG. 4. Clustering as a function of particle density for St = 1. (a) Correlation dimension D . (b) Relative enstrophy sampledby the particles, W p α / W α , for α = n , s. Solid lines, normal fluid enstrophy; dotted lines, superfluid enstrophy. scales [24]. Hence, S n ≈ S s and Ω n ≈ Ω s , and the clustering behavior predicted by Eq. (6) falls back to the classicalcase. This does not apply in the counterflow case, where the normal and superfluid motions are decorrelated at thesmall scales along the counterflow direction [25].To support the above predictions and to extend our results to different particle densities, Fig. 4(a) shows D as afunction of the density parameter β , for St = 1. In the classical case, heavy particles concentrate in planar structures( D (cid:38)
2) while light particles form localized linear clusters ( D (cid:46) D strongly decreases at the lowest temperature, suggesting the formation of linear clusters. Theabove discussion suggests that these clusters form in high superfluid vorticity regions. This is verified in Fig. 4(b),where the relative enstrophy sampled by the particles, W p α / W α (with α = n , s), is plotted. Here, W α = (cid:104)| ω α | (cid:105) is theenstrophy of a given fluid component (Eulerian average), and W p α = (cid:104)| ω α ( x p ) | (cid:105) where the average is over particlepositions. For T = 1 . β , here particles formclusters of dimension D ≈ D , consistently withthe observations from Fig. 3(b). This is however not the case for the relative enstrophy sampled by the particles(Fig. 4b), which displays a striking variation with the mutual friction frequency Ω . In the low Ω case, light particlestend to cluster in regions of very high normal fluid vorticity, while this is not the case when Ω is increased. Thisis a consequence of the change of Eulerian fields with the mutual friction intensity. A strong mutual friction resultsin weaker enstrophy fluctuations in the flow (data not shown). Furthermore, mutual friction suppresses the velocityfluctuations in the counterflow direction [28], enhancing the two-dimensionalization of the flow and thus the formationof vortex sheets that drive particle clustering. This finally explains why the correlation dimension D remains closeto two when the mutual friction frequency is increased. IV. SUMMARY
We have studied the spatial organization of inertial particles in the HVBK framework for superfluid helium. Inthe absence of a mean counterflow, the most striking difference with classical fluids is observed at low temperatureswhen the superfluid mass fraction is dominant. In this case, particles are attracted towards high superfluid vorticityregions regardless of their density relative to the fluid, thus forming quasi-one-dimensional clusters. This attractionis explained by the dominant effect of the superfluid pressure gradient on the particles. At higher temperatures, asthe two fluid components become strongly coupled by mutual friction, the classical turbulence behavior is recovered.Namely, light particles concentrate in vortex filaments, while heavy particles are expelled from them. Finally, in thepresence of a strong counterflow, the clustering dynamics is governed by the two-dimensionalization of the velocityfields and the formation of large-scale vortex columns or sheets, which either attract or repel particles as a functionof the particle density and/or inertia. In this case, particles cluster in quasi-2D structures almost regardless of theirdensity and of the imposed mutual friction intensity.
ACKNOWLEDGMENTS
This work was supported by the Agence Nationale de la Recherche through the project GIANTE ANR-18-CE30-0020-01. Computations were carried out on the M´esocentre SIGAMM hosted at the Observatoire de la Cˆote d’Azur,and on the Occigen cluster hosted at CINES through the GENCI allocation A0072A11003. [1] U. Frisch,
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