Coupled dynamics for superfluid 4 He in the channel
NNoname manuscript No. (will be inserted by the editor)
Coupled dynamics for superfluid He in the channel.
D. Khomenko · P. Mishra · A. Pomyalov
Received: date / Accepted: date
Abstract
We study the coupled dynamics of normal and superfluid compo-nents of the superfluid He in the channel considering the counterflow turbu-lence with laminar normal component. In particular, we calculated profiles ofthe normal velocity, the mutual friction, the vortex line density and other flowproperties and compared them to the case when the dynamic of the normalcomponent is ”frozen”. We have found that the coupling between the normaland superfluid components leads to flattening of the normal velocity profile,increasingly more pronounced with temperature, as the mutual friction, andtherefore coupling, becomes stronger. The commonly measured flow propertiesalso change when the coupling between two components is taken into account.
Keywords
Superfluid helium · Coupled Dynamics · Thermal Counterflow
Superfluid He below transition temperature T λ (cid:39) .
17 K may be viewed asa two-fluid system[1,2,3] consisting of normal fluid with the very low kine-matic viscosity ν n ( T ) and inviscid superfluid component, that have their owndensities, ρ n ( T ), ρ s ( T ), and velocity fields, u n ( r , t ), u s ( r , t ). Due to quantummechanical restriction, the circulation around the superfluid vortices is equalto κ = h/m (cid:39) − cm /s, where h is the Plank constant and m denotes themass of He atom. The singly quantized vortices usually arrange themselvesin a tangle, referred to as quantum turbulence , that may be characterized byvortex line density (VLD) L , i.e., total length of the quantized vortex line ina unit volume.The large scale motion of such a system may be described by Hall-Vinen-Bekarevich-Khalatnikov equations(HVBK)[6,7], where both components are D. Khomenko · P. Mishra · A. PomyalovDepartment of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, IsraelE-mail: [email protected] a r X i v : . [ c ond - m a t . o t h e r] A ug D. Khomenko et al. considered as continuous fluids, coupled by a mutual friction force. The mi-croscopic description of the vortex lines dynamics at scales that are still muchlarger than the vortex core size a ≈ − cm, was proposed by Schwarz[13,14] in the framework of Vortex Filament Method (VFM). In his approach,the influence of the normal component the dynamics on the quantum vortexlines is accounted through the mutual friction force with the predefined time-independent normal fluid velocity. It was soon realized that the back-reactionof the vortex tangle on the dynamics of the normal component may be impor-tant and several self-consistent methods, coupling Navier-Stokes equation forthe normal component with Lagrangian description of the vortex lines dynam-ics, were proposed[8,9,10]. These methods are computationally challenging,given the wide range of scales involved in the problem. Such a self-consistentstudies were limited to space-homogeneous flows with diluted vortex tangles.The VFM methods were extended to the wall-bounded flows[15,16] andmore recently included full Biot-Savart description of the tangle [17,18,19,20]. Also here, a time-independent mean normal velocity profile, or a snapshotof the turbulent velocity field were imposed to generate the quantum vortextangle.Depending on the way the quantum turbulence is generated in the channelor pipe, the normal and superfluid components flow in the same direction, orin opposite directions. In the latter case (the thermal counterflow), a relative,counterflow velocity V ns is established in the channel. At sufficiently large val-ues of V ns the quantum vortex tangle is created. As V ns increases, the thermalcounterflow passes several stages. At relatively low values of counterflow veloc-ity, the normal components remains laminar, while a fully-developed quantumvortex tangle is formed, indicating superfluid turbulence. This regime was la-beled at T1 state [21]. At higher values of V ns , the normal component alsobecomes turbulent, forming so-called T2 state.In this paper we address the back-reaction of the quantum vortex tangle onthe normal velocity profile, restricting our study to T1 state of the counterflowin the channel. Here we describe the laminar normal velocity by its mean profile(neglecting normal velocity fluctuations), but allow its evolution, driven by themutual friction force that couples dynamics of the mean normal velocity withthe evolution of the vortex tangle. To be consistent with the nature of the meanprofile, the mutual friction force, generated by the vortex tangle, is integratedover space and short intervals of time. Such a multi-scale, multi-time approachallows us to follow the coupled dynamics of two components and account forthe influence of the vortex tangle on the normal component.We have found that initially parabolic mean normal velocity profile evolvesto a flatter shape, with effect stronger for higher temperature and counterflowvelocities. As a consequence, the VLD profile becomes more uniform in thechannel core with the peaks pushed towards the wall. Both the global and mi-croscopic properties of the flow change compared to the uncoupled dynamics,although this effect is significant only at high temperatures. oupled dynamics for superfluid He in the channel. 3
In the framework of two-fluid description of the superfluid He, dynamics ofthe normal component is given by HVBK equation[6,7,8]: ∂ u n ∂t + ( u n · ∇ ) u n = ∇ Pρ n + F ns ρ n + ν n ∆ u n , (1)where mutual friction force F ns couples the two components and the effectivepressure P is defined by: ∇ P = − ρ n /ρ ∇ p + ρ s S ∇ T ( S is entropy and ρ = ρ n + ρ s is density of Helium II). For the laminar flow in the channel, Eq. (1)simplifies to an equation for the mean normal velocity: ∂V n ( y ) ∂t = dPdx + F ns ( y ) ρ n + ν n ∂ V n ( y ) ∂y , (2)Here we took into account that in the planar channel geometry the normalvelocity field V n ( y ) depends on the wall-normal direction only and has onlyone non-zero (streamwise) component along channel. The mutual friction term F ns ( y ) is a time average of a streamwise projection of the mutual friction force, F ns = ρ s Ω (cid:90) C (cid:48) ( α s (cid:48) × [ s (cid:48) × V ns ] + α (cid:48) s (cid:48) × V ns ) dξ, (3)defined by dynamics of the quantum vortex tangle[14]. Here s is a radius vectorto the points on the vortex line, (cid:48) denotes derivative along the vortex line and α, α (cid:48) are the mutual friction coefficients. The integral in Eq. (3) is taken alongvortex lines C (cid:48) , residing inside suitably defined coarse-grained volume Ω .The instantaneous counterflow velocity V ns = V n − V s is defined by ve-locities of the normal and superfluid components. The superfluid velocity V s = V + V BS ( s ) contains contributions of the vortex tangle velocity V BS ( s )in the Biot-Savart representation, integrated over entire vortex configuration C and the applied superfluid velocity V , defined by the counterflow condition ρ n (cid:104) V n (cid:105) v + ρ s (cid:104) V s (cid:105) v = 0. Here (cid:104) ... (cid:105) v stands for global volume averaging. Implementation Details
The simulations were set up in a planar channel ofthe size 4 h × h × h, h = 0 .
05 cm. The vortex-lines dynamics was solved usingVFM[14,25,20] with 4th-order difference scheme for the derivatives s (cid:48) and s (cid:48)(cid:48) [23,24]. We used the periodic boundary condition in the streamwise( x ) andthe spanwise( z ) directions, with slip conditions in the wall-normal y direction,the line-resolution ∆ξ = 0 . δt s , defined by the Table 1
Parameters of Heluim-II,used in simulations[26].T(K) α α (cid:48) ρ n (g/cm ) ρ n /ρ s ν = η/ρ n (cm /sec)1.45 6 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − D. Khomenko et al. -1 0 100.20.40.60.81 y † V n / ( h V n i) V n V n - model P x =9Parabolic profile T=1.9 KT=1.6 K T=1.45 K -1 0 100.20.40.60.81 y † V n / ( h V n i) T=1.9 KP x =7Parabolic profile P x =9P x =5 Fig. 1
Comparison of normalized velocity profiles V n ( y ) / (3 / (cid:104) V n (cid:105) ) vs y † = y/ ( h ) , y † ∈ [ − ,
1] for effective pressure gradient P x = 9 and different temperatures (left panel) and for T = 1 .
9K and different values P x (right panel). The normalized parabolic profile is shownby black solid line. In the left panel we also show the model profiles Eq. (8)(dashed lines) for T = 1 . (cid:104) V n (cid:105) = 1 .
22 cm/s], 1 . (cid:104) V n (cid:105) = 1 .
01 cm/s]and 1 . (cid:104) V n (cid:105) = 0 . A GM , calculated from mutual friction force in the coupled dynamics are: A GM (1 .
45) =20 . · s/g, A GM (1 .
6) = 35 . · s/g, A GM (1 .
9) = 47 . · s/g and n = 14.(Color Online) stability condition of the 4th-order Runge-Kutta scheme. The dynamics of themean normal velocity profile V n ( y, t ), Eq. (2), was solved using 2nd-order finite-difference scheme for the viscous term, 2nd-order Adams-Bashforth methodfor the time marching and no-slip conditions on the solid walls. The effectivepressure gradient P x = dP/dx was used as a free parameter of the system.To calculate the mutual friction term, we used two-stage approach. Theterm F ns in Eq. (2) implies averaging over space and time, while the term F ns is instantaneous and is averaged only over space. The parabolic normalvelocity profile was initiated on a grid of a mesh size δy = 0 . ∆t = 10 δt s . At each time step, the mutual friction force F ns was integratedover thin slices of volume 4 h × δy × h and then averaged during ∆t . Resultingvalues of F ns were assigned to the middle points of the slices in y -directionand linearly interpolated to the normal velocity grid points. With this meanmutual friction force, V n ( y ) was then propagated to ∆t using the same δt s .The newly obtained values of V n ( y ) were interpolated to the positions of the -1 0 10.4 y † V n s / h V n s i P x =7, T=1.9 K V n (0)=1, T=1.9 KV n (0)=1.5, T=1.6 KP x =7, T=1.6 K -1 0 10 y † L V n (0)=1.5, T=1.6 KP x =7, T=1.9 KP x =7, T=1.6 KV n (0)=1, T=1.9 K Fig. 2
Comparison of normalized V ns (left) and VLD profiles(right) for coupled dynam-ics (solid lines) and for time-independent parabolic V n (dashed lines) for T = 1 .
6K and1 . He in the channel. 5 line points for the next cycle of the superfluid dynamics, during which V n wasconsidered constant.The resulting profiles of L ( y ), V s,n ( y ) and other related properties wereobtained by integrating over the same slices as the mutual friction, assignedto the middle points of the slices and then averaged over more than 50 steady-state configurations. The simulations were carried out for T = 1 . , . .
9K and a number of P x values. The material properties of the fluid components[26] are given in the Table 1. Notice that we didn’t use the modified mutualfriction coefficients[9], because we only consider the mean normal velocity field. The steady-state profiles of V n ( y ) for different T and values of effective pressureare shown in Fig. 1. To compare the shapes of profiles we normalized themby 3 / (cid:104) V n (cid:105) . For the original parabolic profile this value corresponds to thecenterline velocity. For modified profiles this relation no longer valid. Theprofiles of V n ( y ) become flatter in the center of the channel, increasingly sowith increasing temperature and the applied pressure gradient. The resultingprofile of counterflow velocity V ns ( y ) is shown by solid lines in Fig. 2, left panel,for T = 1 .
6K and 1 .
9K together with profiles, obtained using time-independentparabolic profile with similar (cid:104) V n (cid:105) (dashed lines). The flatter profiles of V n leadto a change in the profiles of V ns , especially in the channel core. The vortexlines tend to concentrate in the regions with smaller V n [20], such that thepeaks in the VLD profiles are pushed further toward the walls.The statistical properties of the vortex tangle are commonly characterizedby relation between mean VLD in the channel and (cid:104) V ns (cid:105) : (cid:104)L(cid:105) / = γ ( T )( (cid:104) V ns (cid:105)− v ), where γ ( T ) is a temperature-dependent coefficient and v is virtual origin.This relation is valid only globally[27]. We compare in Table 2 the values of γ , obtained with different profiles. Notice that results for different profiles attemperatures T = 1 .
45K and 1 .
6K are close, while for higher T = 1 . γ c issignificantly larger than γ p . The values of γ p are close to those of uniform V n for all temperatures. Notably, our results are higher than those, obtained inRefs.[18,19] for the channel counterflow with parabolic and Hagen-Poiseuilleprofiles. This discrepancy may stem from the fact that in both [18,19] thecounterflow condition did not include contribution from the vortex tangle,which is not negligible for dense tangles. Model profile of V n To rationalize the observed modifications of the normalvelocity profile, we notice that in steady state the mutual friction force isalmost constant across the channel, except for the near-wall region, where itquickly falls to zero. Therefore, qualitatively, the mutual friction redefines theeffective pressure gradient in the middle of the channel P x + (cid:104)F ns (cid:105) /ρ n , whilenear the wall it remains P x . Such a change leads to flattening of the normalvelocity profile (as compared to classic parabolic profile). D. Khomenko et al.
Table 2
The values of γ ( T ) for different profiles: γ c for coupled dynamics, γ p for frozenparabolic profile. For comparison we give also values for the parabolic profile, Ref.[18], theHagen-Poiseuille profile γ hp [19] and for the uniform normal velocity γ uni [25].T(K) γ c (s/cm ) γ p (s/cm ) γ p (s/cm ) [18] γ hp (s/cm )[19] γ uni (s/cm )[25]1.3 - - 67 . . − − − . . . To find new, flattened, V n we first model the shape of mutual friction forceprofile by a function: F ns = − (1 − y † n ) F ns (0) , (4)where n is an even integer and F ns (0) is the mutual friction force in the middleof the channel. To account for the sharp transition from almost a constant valuein the middle of the channel to zero at the wall, large values of n are required.In this work we use n = 14. Substituting (4) into steady state equation (2) for V n we get: νh d V n dy † = − P x + F ns (0) ρ n (1 − y † n ) . (5)This equation is easily solved, giving: V n ( y † ) = A (cid:16) − y † (cid:17) (cid:34) C ( n + 2)( n + 1) 1 − y † n +2 − y † (cid:35) , (6) A = h P x,m / (2 ν ) , C = F ns (0) / ( ρ n P x,m ) , P x,m = P x − F ns (0) /ρ n . Notice, that when mutual force is not taken into account F ns (0) = 0, theprofile (6) reduces to the classical parabolic profile. To proceed, we noticethat (cid:104) V n (cid:105) = 2 A/ AC/ [( n + 1)( n + 3)] and (cid:104)F ns (cid:105) = n F ns (0) / ( n + 1).Using counterflow condition and Gorter-Mellink relation between mean mutualfriction force and (cid:104) V ns (cid:105) , we find (cid:104)F ns (cid:105) = A GM ρ s ρ n (1 + ρ n /ρ s ) (cid:104) V n (cid:105) ≡ ρ n νh (cid:104) V n (cid:105) V . (7)The characteristic velocity V = (cid:0) A GM ρ s h (1 + ρ n /ρ s ) /ν (cid:1) − / corresponds toa balance between mutual friction and viscous terms and A GM is the Gorter-Mellink constant. Combining all above we finally get: V n ( y † ) = 32 (cid:104) V n (cid:105) (cid:104) V (1 − y † ) + V (cid:16) − y † n +2 (cid:17)(cid:105) , (8)where V = (cid:16) − n ( n +3) (cid:104) (cid:104) V n (cid:105) V (cid:105) (cid:17) and V = n +2) n (cid:104) (cid:104) V n (cid:105) V (cid:105) are T -dependentexpansion coefficients. The model profiles are compared with numerical resultsin Fig1, left panel. oupled dynamics for superfluid He in the channel. 7
We have studied back-reaction of the quantum vortex tangle on the mean lam-inar normal velocity profile in the channel counterflow. The initially parabolic V n evolves to a flatter shape. The vortex tangle influences the mean normalvelocity profile via mutual friction force. As a result, both the global prop-erties of flow and the profiles of microscopic properties of the tangle, suchas rms curvature, change compared to the uncoupled dynamics, with effectbeing stronger for high temperatures and counterflow velocities. At low andmoderate temperatures the flow and tangle properties are close to those withthe time-independent normal velocity profile. We propose a model of V n ( y ),expressed via (cid:104) V n (cid:105) , that accounts for the effect of mutual friction. Acknowledgements
We acknowledge useful and encouraging discussions with V.L’vovand I. Procaccia.
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