Coarse-grained Hydrodynamics of turbulent superfluids: HVBK approach and the bundle structure of the vortex tangle
aa r X i v : . [ c ond - m a t . o t h e r] A ug Coarse-grained Hydrodynamics of turbulent superfluids: HVBK approach and thebundle structure of the vortex tangle.
Sergey K. Nemirovskii
Institute of Thermophysics, Lavrentyev ave., 1, 630090, Novosibirsk, Russia
In the paper I develop a critical analysis of the use of the HVBK method for the study ofthree-dimensional turbulent flows of superfluids. The conception of the vortex bundles formingthe structure of quantum turbulence is controversial and does not justify the use of the HVBKmethod. In addition, this conception is counterproductive, because it gives incorrect ideas aboutthe structure of the vortex tangle as a set of bundles containing parallel lines. The only type ofdynamics of vortex filaments inside these bundles is possible, namely, Kelvin waves running alongthe filaments. At the same time, as shown in numerous numerical simulations, a vortex tangleconsists of a set of entangled vortex loops of different sizes and having a random walk structure.These loops are subject to large deformations (due to highly nonlinear dynamics), they reconnectwith each other and with the wall, split and merge, creating a lot of daughter loops. They also bearKelvin waves on them, but the latter have little impact.I also propose and discuss an alternative variant of study of three-dimensional turbulent flows,in which the vortex line density L ( r, t ) is not associated with ∇ × v s , but it is an independentequipollent variable described by a separate equation. I. INTRODUCTION L ( r, t ) with the coarse-garined vorticity ∇ × v s via the favor Feynman rule, was initially elaborated andis only suitable for rotating stationary cases. Meanwhile, at present, the use of this method for three-dimensionalunsteady flows is a widespread practice. Sometime this is done without justification, sometimes they refer to theso-called vortex bundle structure of quantum turbulence. The conception of the vortex bundles forming the structureof quantum turbulence is also critically discussed in the paper. I also propose an alternative variant in which thevortex line density L ( r, t ) is not associated with ∇ × v s , but it is an independent and equipollent variable described bya separate equation. The structure of paper is following. The next, second section is devoted to general formulation ofcoarse-grained hydrodynamics of superfluids in presence of the vortex tangle. In the third Section the problem of theuse of the HVBK methods for rotating superfluids and in the three -dimensional flows are discussed. An alternativevariant of study of three-dimensional turbulent flow is described in fourth Section. II. COARSE-GRAINED HYDRODYNAMICS OF TURBULENT SUPERFLUIDS.
In presence of the vortex filaments the two-fluid hydrodynamics of the superfluid helium should be modernized andbe represented as follows: ρ n ∂ v n ∂t + ρ n ( v n · ∇ ) v n = − ρ n ρ ∇ p n − ρ s S ∇ T + F mf + η ∇ v n , (1) ρ s ∂ v s ∂t + ρ s ( v s · ∇ ) v s = − ρ s ρ ∇ p s + ρ s S ∇ T − F mf (2)We assume that the motion of both components is incompressible, ∇ · v n = 0, ∇ · v s = 0. and where v n and v s are the coarse-grained velocity of the normal and superfluid component (averaged over a small volume V ), p n and p s are the effective pressures acting on the normal and the superfluid component ( ∇ p n = ∇ p + ( ρ s / ∇ v ns and ∇ p s = ∇ p − ( ρ n / ∇ v ns ), p is the total pressure, S is the entropy, T is the absolute temperature, η is the dynamicviscosity of the normal component, and v ns = v n − v s .The effects of the vortices on the two components (normal andsuperfluid) are described by the friction force exerted by the superfluid component on the normal component F mf .When these forces are averaged over all vortices inside the small volume V , then the following expression of F mf isobtained (see e.g. [3]) F mf = L < f MF > = αρ s κ L h s ′ × [ s ′ × ( v ns − v i )] i + α ′ ρ s κ L h s ′ × ( v ns − v i ) i . (3)In this equation s ′ ( ξ ) is the tangent vector along the vortex filaments s ( ξ ), composing the vortex tangle, α, α ′ aretemperature-dependent dimensionless mutual friction parameters, v i .is the self-induced velocity of the vortex filament.Quantity L is the vortex line density, averaging h·i is performed over various configurations of vortex filaments.Equations (1) - (3) are coarse-grained equations, hence the inclusion of the effects of the vortex lines requires a highvortex line density per unit volume. These equations had been written and discussed for a long time, a classic variant,close to the stated above can be found in the book [4].In the written form these equations are common to any type of flow, such as counterflow, flow past obstacles,acoustic waves, etc. The main difference for different types of flows is the determination of the vortex line density L entering into the expression for the macroscopic mutual friction force (3), and the choice of averaging method h·i .Another question concerns the temperature range for applying equations (1) - (3), In general, the quantum turbu-lence is the chaotic dynamics of three strongly interacting nonlinear fields. These are the motion of the normal andsuperfluid component and stochastic evolution of a set of vortex filaments (vortex tangle). The type of interactiondepends of the temperature T (via mutual friction). For large T the coupling is strong, both components are clumpedand move together. As a result, we got almost the one-fluid hydrodynamics. The case of very low temperature is veryinteresting because it is ideal for testing the idea of whether the dynamics of discrete vortices (quantized vortex lines)can imitate classical turbulence. The study of superfluid turbulence for intermediate temperatures is not suitable forthis purpose (due to the presence of a normal component). Quantum turbulence in superfluid helium (for intermedi-ate temperatures) is rather the separate problem, not identical to classical turbulence. And it is precisely this casethat requires the study of two coupled Navier-Stokes Equations. Thus, we can say that we study the intermediatetemperature range.The above equations (1) - (3) are a point of consensus among physicists. Disagreements begin with the question howto fulfil averaging and to treat variable L . Here we discuss two main ways to perform these procedures. They are theHall-Vinen-Bekarevich-Khalatnikov (HVBK) model (see e.g., book by Khalatnikov, [5] 1965) and the Hydrodynamicsof superfluid turbulence model (Nemirovskii & Lebedev, 1983 [6],[7]) III. HVBK APPROACHA. HVBK approach for rotating superfluids
The Hall-Weinen-Bekarevich-Khalatnikov (HVBK) model (see, for example, the book Khalatnikov [5] became thebasis for the mathematical formalism of the hydrodynamics of rotating superfluids. As is well known (see [2] , in avessel rotating with an angular velocity Ω, appears a regular array of vortex filaments with a density n = 2Ω /κ . Sucha distribution of vortices creates an average coarse-grained superfluid velocity h v s i , which satisfies the condition ofsolid body rotation h v s i = Ω × r . The vorticity filed ω is ω = 2 Ω . Therefore, the density of the vortex filaments inthis case can be related to the vorticity field by the following relation: ∇ × v s = κ L (4)Due to the smallness of the quantum of circulation κ , even a relatively weak rotational speed produces a highdensity of vortex lines. Thus, it is possible to construct a coarse-grained hydrodynamics of hydrodynamic equations,which the average contribution of many individual vortex lines and incorporate their contribution to the macroscopicevolutionary equations for superfluid and normal He II velocities.Combining (3) with (4) one get expression for average coarse-garined mutual friction F ( HV BK ) mf F ( HV BK ) mf = ρ s α ˆ ω × [ ω × ( v − ˜ β ∇ × ˆ ω )] + ρ s α ′ ω × ( v − ˜ β ∇ × ˆ ω ) , (5)where ω = ∇ × v s is the averaged superfluid vorticity, ˆ ω = ω/ | ω | . Thus, the question of elimination of the vortex linedensity is resolved with the use of the Feynman rule, it allows to study various problems of coarse-grained dynamicsof rotating supefluids (see, e.g. book be Sonin [8]). B. HVBK approach for three -dimensional flows
The HVBK model is a fruitfull and elegant approach, however it is principally assigned for rotating superfluids .Nevertheless this approach, which uses ansatz ∇ × v s = κ L (4), and the force F ( HV BK ) mf (5) is widely used for nu-merical and analytical studies of coarse-grained hydrodynamic problems of turbulent superfluid in three -dimensionalsituations.This approach seems to be unfounded. In my opinion, there is no way to apply it to three-dimensional hydro-dynamics. Anticipating the objections of readers, I would like to discuss the usual arguments for using the HVBKmethod. Probably the only argument is the conviction that the vortex tangle consists of so-called vortex bundles,unifying many near parallel vortex filaments.There are few papers (see, e.g. [9], [10]), where the authors claim of the existence of the bundles. In fact, they onlydemonstrated how, with the help of statistical analysis, one can get a small polarization (prevailing of one directionover the other) in the vortex tangle. But, firstly, it is just a statistical effect and, secondly, under no circumstancesthis small polarization allows to use the Feynman rule (4), which is crucially needed in the HVBK equations. To usethis ansatz, it is necessary that all filaments in the vortex tangle are involved in the rotation. But if the polarization ispartial, then there is a lot of free (randomly orientated) vortex filaments that contribute to mutual friction and do notcontribute to the Feynman rule (4). In addition, these chaotic lines interact with polarized lines, thereby destroyingthe polarization and, correspondingly, the quasi-bundle structure.There are other examples of observation of vortex bundles (in numerical works), when they are artificially preparedstructure, or are initiated by eddies of normal component (see, e.g. [11],[11]). However, there are works in theliterature, in which it is stated that even if the vortex bundles are artificially created, they can be destroyed rathersoon. For instance, G. Volovik [12]) have shown that at low temperatures, where the mutual friction is small, theexistence of the bundles is impossible. They should melt, changing into a highly irregular structure. Other exampleis a series of numerical simulations by Kivotides [13], [14],[15],[16]), who studied the exact (not HVBK) dynamicsof quantum vortices in the turbulent flows (at finite temperature) and concluded ”that the results do not show thata turbulent normal-fluid with a Kolmogorov energy spectrum induces superfluid vortex bundles in the superfluid”.In paper [15] Kivotides reported about observation of clusters with weakly polarized vortex lines and associatedKolmogorov - type spectra E ( k ) ∝ k / . To some measure this is expected result since the appearance of theKolmogorov spectrum requires formation of coherent structures. However, the partial polarization also prevents theuse of closure ∇ × v s = κL in the mutual friction, since ALL the lines contribute into friction, and partial polarization(if any) includes A SMALL FRACTION of the total vortex line density.In this regard, it seems appropriate to discuss question of relation of the vortex bundle arrangement and theKolmogorov-type spectrum. For the uniform array of vortex filaments the coarse grained velocity field is v ( r ) = Ω × r .Accordingly the Fourier transform v ( k ) scales as 1 /k − . This implies that two-dimensional spectrum E ( k ) = dE/d k should behave as 1 /k − and the isotropic spectrum E ( k ) depends on absolute value of the wave vector k as E ( k ) ∝ /k . Thus, we state that the uniform vortex bundles do not generate the Kolmogorov type spectra. It can be shownthat the nonuniform vortex array with the distribution of density of vortex filaments n ( r ) = ∆ N/ ∆ r ∝ /r − / doesgenerate Kolmogorov type spectra E ( k ) ∝ /k / (see [17], [18]).Of course, for very strong mutual friction vortex filaments can be completely trapped by the eddies of normalcomponent and follow the dynamics of normal fluid. In fact, in this situation the coarse- grained hydrodynamicsbecome the one-fluid dynamics, when both components move together.There are many physical mechanisms that result in the destruction of the regular vortex bundle structure. Theapparent source of the destruction of the bundle structures is the various reconnections. Thus, as demonstratedin the paper by Kursa, Bajer & Lipniacki, [19], and in work by Kerr [20] even a single reconnection results ina cascade of vortex loops of various sizes being chaotically radiated from the reconnection point. Clearly thesepropagating loops collide with the lines composing bundles, triggering new reconnections, and developing an avalanche-like randomization.Some authors(see e.g. [21],[22]), based on their numerical results, claim that the vortex bundles are the robuststructures with respect to reconnection of two adjacent bundles. This conclusion, however, concerns the situationwhen the different bundles have the same structure (N strands in each bundle) . In quantum fluids full reconnectionbetween bundles carrying different numbers of threads is not possible for topological reasons, and the residual structure,analogous to the ”bridging” in classical hydrodynamics, should accompany the collision of such bundles (see e.g. papers[23], [24], [25], [26]). The analog of classical ”bridging” leads to the randomization and violation of the structure ofthe bundles and to the creation of vortex loops. Also, the long-range interaction between vortex filaments in thebundles and the ”external” vortices also destroy regular array due to the action of tidal forces.As for the filaments inside bundle, the direct reconnection event for them is impossible, since for the reconnectionthe approaching vortices must be antiparallel. Reorganization of lines destroys the parallel array of vortex filaments.Similarly, the processes of emission and re-absorption of the ring by the vortices (the ”anti-bottleneck”, proposedby Svistunov [17, 27]) should also lead to the fragmentation of the regular arrays and the appearance of chaotic loops.There are also experimental results which do questionable the idea of bundles. Thus, in experiments by Roche etal. [28], and by Bradley et al.[29], it was observed that the spectrum of the fluctuation of the VLD L is compatiblewith a − / L (via Eq. (4))) should scale as 1 / C. Where is it from?
This is a somewhat mysterious question: how did at all the HVBK method, designed to a rotational or two-dimensional case, become used for three-dimensional turbulent flows? Trying to find the origin I analyzed a largemass of literary sources. The most frequent references are to the papers by Sonin [30] and [31]. But these linksare absolutely irrelevant, since the authors definitely wrote that they work with rotating helium. Probably, the oneof the first papers in which the use of this method for three-dimensional turbulent flows is discussed is the work ofHolm [32]. It is interesting, however, that he started paper with the text ”Recent experiments establish the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations as a leading model for describing superfluid Helium turbulence.See Nemirovskii and Fiszdon [1995] and Donnelly [1999] for authoritative reviews.”But R. Donnelly [33] discussed HVBK approach namely for rotating helium. As for my (with W. Fiszdon) paper[7] on superfluid turbulence, then, firstly, there was no mention on HVBK theory at all, and, secondly, I generallyopposed to this method to treat three-dimensional quantum turbulence.Thus, the origin of the idea of the using a pure rotational HVBK approximation for a three-dimensional turbulentflows is rather vague.Resuming this Section, I can state the HVBK ansatz ∇ × v s = κ L is in general unfounded in the three-dimensionalcase, therefore, works using this approach are questionable, and the corresponding results are not reliable. IV. OTHER WAYS TO TREAT THE VORTEX LINE DENSITY
The main attractive advantage in HVBK approach was to rid out of vortex line density L in equations of motions(1) - (3). It seems,however, that the question of ”elimination” the vortex line density L ( r , t ) should be solved in afundamentally different way. We have not to ”eliminate” the quantity L ( r , t ), but on the contrary, to include it intoconsideration as an independent and equipollent variable. Correspondingly we have to consider the problem, in whichthere are three independent variables - velocities v n ( r , t ) and v s ( r , t ), and vortex line density L ( r , t ). We also canadd the density field ρ ( r , t ) , as well as the entropy field S ( r , t ) if it is pertinent. In this way, however, we need anadditional independent equation for the temporal and spacial evolution of quantity L ( r , t ).This is a particular task that requires a lot of efforts. Derivation of such an equation, certainly, depends on thetype of flow, such as counterflow, co-flow, flow past objects, unsteady rotation, etc. In fact, so far the correspondingequation exists only for the case of counterflow, this is the famous Vinen equation (or some modernized versions ofthis equation). Although, there are some problems with this equation (see e.g. Sec. IV in [34]), it works well forhydrodynamic e.g. for acoustic or engineering problems. It is important to stress that the construction of a theoryof the evolution of three fields ( v n , v s and L ( r , t )) is not an automatic addition of Vinen equation to the Landautwo-fluid hydrodynamics. It is a more involved procedure, since all variables (energy, entropy, etc.) change in thepresence of the vortex tangle. This self-consistent procedure was implemented in [6], it is called Hydrodynamics ofSuperfluid Turbulence (HST).This theory was successful, it explained a lot of experimental results on the nonlinear acoustics of the first andsecond sounds, the evolution of strong heat pulses, the formation of shock waves etc. It was also successfully appliedto the problems of unsteady heat transfer and boiling of He II ([7],[35],[36]). These examples demonstrate that theway to treat the vortex line density L ( r , t ) as an additional and equipollent variable seems to be productive andfruitful.Unfortunately, for other types of flows there is no theory describing temporal - spatial evolution of the vortex linedensity, and there is no ideas (similar to the Feynman qualitative scenario) how to obtain the according equation.Vinen’s equation reflects the fact that the vortex line density L ( r , t ) grows due to the relative velocity v n − v s andattenuates, probably, due to the cascade like breaking down of vortex loops, described by Feynman [2]. That is goodguideline how to develop an appropriate theory for any flow, involving, of course, some auxiliary speculations.Beside of introduction of the vortex line density into the coarse-grained hydrodynamics of superfluid turbulence,there was one more, crude and simplified way, which had been applied on the early stages of research on the superfluidturbulence. It was the use the Gorter - Mellink formula for mutual friction which immediately follows, from the Vinentheory F mf ∝ A ( T )( v n − v s ) ( v n − v s ) . (6)Here A ( T ) is the Gorter - Mellink. In fact, in this formula it was used the ansatz L ∝ ( v n − v s ) , well known inthe theory of quantum turbulence The use of equation (6) also ”resolves” the problem of elimination the vortex linedensity L ( r, t ). V. CONCLUSION
The two approaches how to investigate turbulent flow in superfluids - HVBK and HST have been described in thepaper. In the first one, the vortex line density L ( r , t ), crucial for the whole dynamics is straightforwardly excludedfrom equations of motion (1) and (2) with the use of Feynman rule (4). In the second approach, the variable L ( r, t )is considered as an additional independent variable obeying the according evolution equation.In the present paper it had been argued that the HVBK ansatz ∇ × v s = κ L is suitable only for rotating cases andfails in three-dimensional situations. The attempts to justify this procedure give rise to a whole scientific direction(trend), which asserts that the vortex tangle in quantum turbulence is composed of the so called vortex bundlescontaining a set of near parallel lines. In the paper, I put a number of arguments criticizing the conception of thevortex bundle structure. References to the fact that in some numerical works a partial polarization of vortex filamentshad been observed cannot be considered as justification for using the ansatz ∇ × v s = κ L , since ALL the linescontribute into friction, and polarization (if any) includes a small fraction of the total vortex line density.Furthermore, it is of great concern that the concept of vortex bundles has gone beyond the coarse-grained hydrody-namics of superfluid turbulence and often serves as the basis for other (more subtle) aspects of the theory of quantumturbulence. This seems counterproductive, since after the pioneering works of Feynman, Vinen, Donnelly, Schwartzand others, it was customary to present a vortex tangle as a set of stochastic loops with rich and diverse dynamics.These loops are subject to large deformations (due to highly nonlinear dynamics), they reconnect with each otherand with the wall, split and merge, creating a lot of daughter loops. This vision was observed in numerous numericalsimulations (see e.g. [3],[37],[38],[39], [40],[15]). This, let’s say, Feynman-Vinen model is very different. from thevortex bundle model, where almost the only possible dynamics of the vortex filaments is the Kelvin waves evolutionalong the lines composing the bundles.In summary, the use of ansatz ∇ × v s = κ L , for the closure procedure for coupled Navier-Stokes equations (1)- (3) in the 3D turbulent flow is not motivated and would lead to unreliable results. And the commonly used thevortex bundle model, which justifies the use of this method, is questionable and unfounded. Moreover, the vortexbundle concept disavows the real structure of the vortex tangle as a set of vortex loops, and prevents developing of anadequate theory. We assert that the introduction of an additional independent field L ( r, t ) to the classical two-fluidhydrodynamics of supefluids is the only correct way to construct the coarse-garined hydrodynamics of turbulent flows. VI. ACKNOWLEDGEMENTS.
The study on the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) approach was carried out under state contract withIT SB RAS (No. 17-117022850027-5), the study on the Hydrodynamics of Superfluid Turbulence (HST) method wasfinancially supported by RFBR Russian Science Foundation (Project No. 18-08-00576). [1] C. J. Gorter and J. H. Mellink, Physica , 285 (1949).[2] R. P. Feynman, Progress in Low Temperature Physics, Vol. 1 (North-Holland, Amsterdam, 1955), p. 17.[3] K. W. Schwarz, Phys. Rev. B , 2398 (1988).[4] R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, UK, 1991).[5] I. M. Khalatnikov,
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