Numerical Studies of Counterflow Turbulence

Transcription

1 Nonae anusript No. will be inserted by the editor Nuerial Studies of Counterflow Turbulene Veloity Distribution of Vorties Hiroyuki Adahi Makoto Tsubota Reeived: date Aepted: date Abstrat We perfored the nuerial siulation of quantu turbulene produed by theral ounterflow in superfluid 4 He by using the vortex filaent odel. The pioneering work was ade by Shwarz, whih has two defets. One is negleting non-loal ters of the Biot-Savart integral loalized indution approxiation, IA, and the other is the unphysial ixing proedure in order to sustain the statistially steady state of turbulene. We sueeded in aking the statistially steady state without the IA and the ixing. This state shows the harateristi relation = γ 2 v 2 ns between the line-length-density and the ounterflow relative veloity v ns with the quantitative agreeent of the oeffiient γ with soe typial observations. We opare our nuerial results to the observation of experient by Paoletti et al, where theral outerflow was visualized by solid hydrogen partiles. Keywords superfluid 4 He quantized vorties ounterflow Introdution Quantu turbuleneqt, whih onsists of a tangle of quantized vorties, has been investigated sine the theral ounterflow experients of Vinen [-4] half a entury ago, while the underlying physis is far fro being fully understood [5,6]. The nuerial siulations are the useful soure of knowledge about QT, beause the whole dynais of this syste is too opliated to be desribed analytially. One of the powerful shees of the siulation is the vortex filaent odel based on pioneering works by Shwarz [7,8]. Shwarz perfored the nuerial siulation of ounterflow turbulene under the periodi boundary ondition by using the loalized indution approxiation IA whih neglets a non-loal ter of the Biot-Savart integral [8]. However he ould not obtain the statistially steady state SSS beause the vorties lie in planes noral to v ns = v n v s to prevent the fro reonneting. Therefore the unphysial ixing H.Adahi, M.Tsubota Departent of Physis, Osaka City University, Suiyoshi-Ku, Osaka Tel.: Fax: E-ail: adahi@si.osaka-u.a.jp

2 2 proedure, in whih half of the vorties are randoly seleted to be rotated by 9 around the axis defined by the flow veloity, was used, and only this ethod enabled hi to obtain the SSS. This failure reinds us that the IA is unsuitable through the absene of the interation between vorties. Reently, theral ounterflow in superfluid 4 He was visualized by using solid hydrogen partiles, and the veloity distribution of partiles was observed [9]. In this experient, two types of partiles appeared. Soe partiles ove straight along the noralflow with the approxiately sae veloity as noralflow. Other partiles ove zigzag along the diretion of superfluid and has different veloity fro superfluid. In the latter ase, partiles see to be trapped in the ore of the vorties. We try to apply our nuerial results for understanding these observations. Setion 2 desribes the equation of otion of vorties and the ethod of nuerial alulation. In Se.3 we show the typial nuerial results of vortex tangle. Setion 4 studies the validity of the IA by oparing the IA alulation with the full Biot- Savart alulation. In the setion 5 we present the veloity distribution of vorties in the SSS of vortex tangle and opare it with that of partiles in the experiental observation. 2 Equations of Motion and Nuerial Siulation At K the veloity ṡ of the filaent at the point sξ,t is given by ṡ = κ 4π s s 2l+ l 2 ln e 4 a + κ s s ds 4π s s 3 + v s,a, where the prie denotes the derivatives with respet to the ar length ξ, and κ is the quantized irulation, a is a utoff paraeter orresponding to a vortex ore radius, l + and l are the length of the two adjaent line eleents that hold the points between. The first ter shows the loalized indution field arising fro a urved line eleent ating on itself. The seond ter represents the non-loal field obtained by arrying out the integral of the Biot-Savart integral along the rest of the filaent. The third ter v s,a is an applied field. The IA used in soe works e.g.,[7,8,,] eans negleting the seond non-loal ter. In ontrast the alulation without the IA is alled the full Biot-Savart alulation. At finite teperatures, the veloity ṡ is given by ṡ = ṡ + αs v n ṡ α s [s v n ṡ ], 2 where α and α are the teperature-dependent frition oeffiients, v n is the noralfluid veloity, and ṡ is alulated fro Eq.. In the work [8] the author negleted the third ter of Eq.2. We perfored the full Biot-Savart alulation in this work. The onrete reonnetion proedure used in this work is the following. Every vortex is represented by a string of points at intervals of alost δξ. When a point on a vortex approahes another point on another vortex ore losely than the fixed spae resolution ξ, we join these two points and reonnet the vorties [2]. For the integration of the otion of Eq.2 in tie we used the lassial 4th order Runnge-Kutta ethod. We usually start with an initial vortex onfiguration of six vortex rings as shown in Fig.upper left.

3 3 3 Siulation of Counterflow Turbulene In this setion we present the nuerial siulations of ounterflow turbulene at the teperature T =.9K, in the oputing box... 3, at applied onterflow veloity v ns=.286,.38,.572 s. A typial result is shown in Fig.. Here the Fig. Developent of a vortex tangle by the full Biot-Savart alulation in a periodial box with the size.. Here teperature T =.9K, and onterflow veloity v ns=.572s is along the vertial axis. Upper left t= s, upper right t=.5 s, lower left t=.2 s, lower right t=2.5 s initial onfiguration of vortex loops evolves in the periodial box to a highly haoti vortex tangle. The vortex line density t is defined as the vortex line length per unit volue. In Fig.2 we depit the tie evolution of quantity t. It is seen that the vortex tangle goes to the SSS after growth period. In Fig.2b we found out that line length density satisfied the harateristi relation = γ 2 v 2 ns whih have been obtained in previous experients [3]. This relation is derived fro the Vinen s equation [3], and also obtained by Shwarz using the IA and the dynaial saling [8]. We obtain γ 39s 2 whih quantitatively agrees with the experiental observation γ 3s 2 [4]. 4 Validity of the IA In the siilar works [8,] the authors did not obtain the SSS of turbulene the work [8] reated the SSS only with the ixing. The SSS was realized in the work [] by

4 4 a 8 7 Vns=.286s b Vns=.38s Vns=.572s t s Vns s Fig. 2 Vortex line density as a funtion of tie for different driving noralfruid v n equal to.286,.38,.572 s using the IA, in whih the authors entioned that failures of previous works were due to the unsuitable reonnetion proedure. We will disuss the ain reason why the SSS was not realized in previous works. In order to onsider the validity of the IA, we opare two alulations. One uses the IA [Fig.3 left], and the other uses the full Biot-Savart law [Fig.3 right]. We run both alulations at the teperature Fig. 3 Side view of vortex onfiguration by the IA alulation left and by the full Biot- Savart law right at t=34.5s. The syste is a.2 3 ube. Applied noralfluid veloity v ns =.367s. T =.6K, in the oputing ubi box , and applied ounterflow veloity v ns =.367s. To explain the differene between the results of the IA and full Biot-Savart law we introdue the diensionless anisotropy paraeter [8] I = Ω [ s ˆr 2 ]dξ. 3 Here ˆr stands for a unit vetor parallel to the v ns diretion, and Ω is the saple volue. An isotropi tangle yields I = 23. At the other extree, if the tangle onsists entirely of urves lying in planes noral to v ns, I =. The tie evolution of t and I t is shown in Fig 4. Figure 3 left and Fig.4 b show that the any vorties lie in the planes noral to v ns, the vortex tangle being degenerate. Sine the dense part of vorties ath other vorties oving freely, vorties beoe to huddle in periodial

5 5 a 25 b.95 2 IA.9 IA Full Biot-Savart t I Full Biot-Savart t s t s Fig. 4 Coparison of a vortex line density t a and an anisotropy paraeter I t b. planes as shown in Fig.3 left. This ill behavior oes fro the utual frition, whih tends to expand vorties perpendiular to the v ns, so that vorties gradually lie in planes noral to v ns. However, a non-loal ter of the Biot-Savart integral ould yield the veloity in a diretion parallel to v ns even when vorties align in a plane perpendiular to v ns, thus destroying the ill struture. Hene, the alulation with the full Biot-Savart law an sustain the SSS in ontrast to that with the IA. 5 Veloity Distribution of Vorties For oparison with experiental results [9], we present the veloity distributions of vorties in the SSS. We obtained the distributions as shown in [Fig.5 left] by easuring the z-oponent of the veloity ṡ at eah points on the vortex filaents fro the results of ounterflow turbulene in whih applied veloity v ns is direted to z-axis. The veloity ṡ does not neessarily express the partile otion on the vortex filaent whih probably our in the experients. Therefore we brought this effet in our alulation. For the sake of sipliity, suppose the trapped partiles do not have a signifiant influene on the loal otion of the lines, and that visous interation of a partile with the noral fluid auses the partiles to ove along the filaent at a rate given by the Stokes law with the fore equal to the oponent of the visous fore along the line. With these assuption, the z-oponent of the veloity ṡ p inluding the effet of otion is derived fro the siilar fashion of the work [5] like ṡ pz = ṡ n ẑ + v n ṡ n ẑ os 2 θ, 4 where ṡ n is the noral oponent of the vortex veloity to the vortex filaent, θ is a polar angle of the vortex relative to v ns. The veloity distributions of both ṡ z and ṡ pz are shown in [Fig.5 right]. The vorties in our siulations have lower vertial veloities than the superfluid veloity just like the experiental observation. For the quantitative agreeent with observation, we need to introdue other effets suh as a distortion of vorties by partiles. 6 Conlusions The obtained nuerial results with the full Biot-Savart law deonstrate that the initially sooth vortex rings develop to a vortex tangle of the statistially steady state.

6 6 a. b s z -.5 z V r P.5 s pz s z V s z s pz Vz s Vs s Fig. 5 a The vertial veloity v z distribution at teperature T =.9K, applied veloity v ns =.572s. The superfluid veloity v s is shown by vertial line. b Vertial veloity of vortex tangle v z, whih is a peak veloity of the distribution, as a funtion of v s. The dashed line orresponds to v z = v s. We opared the nuerial results by the full Biot-Savart alulation to that by the IA alulation. The IA alulation ould not sustain the statistially steady state in ontrast to the full Biot-Savart alulation. We opared the veloity distributions of vorties obtained by our nuerial siulations to the observations by the visualization experient of outerflow by using solid hydrogen partiles. Our results agree with the experiental observation qualitatively in that ost vorties have lower veloities than the superfluid veloity. Referenes. W.F.Vinen, Pro. Roy. So. ondon A 24, W.F.Vinen, Pro. Roy. So. ondon A 24, W.F.Vinen, Pro. Roy. So. ondon A 242, W.F.Vinen, Pro. Roy. So. ondon A 243, M.Tsubota, J. Phys. So. Jpn. 77, Progres in ow Teperature Physis volue XVI, edited by W.P.Halperin and M.Tsubota, ESEVIER, Asterda K.W.Shwarz, Phys. Rev. B 3, K.W.Shwarz, Phys. Rev. B 38, M.S.Paoletti, R.B.Fiorito, K.R.Sreenivasan and D.P.athrop, J. Phys. So. Jpn. 77, R.G.M.Aarts, A nuerial study of quantized vorties in HeII, Tehnishe Universiteit Eindhoven, Eindhoven P.Kondaurova, V.A.Andryushenko and S.K.Neirovskii, J. ow Tep. Phys. 5, M.Tsubota, T.Araki and S.K.Neirovskii, Phys. Rev. B 62, J.T.Tough, Progres in ow Teperature Physis volue VIII, edited by D.F.Brewer, E- SEVIER, Asterda R.K.Childers and J.T.Tough, Phys. Rev. B 3, D.R.Poole, C.F.Barenghi, Y.A.Sergeev and W.F.Vinen, Phys. Rev. B 7,