Note on the superfluid Reynolds number for turbulent flow of superfluid 4 He around an oscillating sphere
NNote on the superfluid Reynolds number for turbulent flow of superfluid He aroundan oscillating sphere
W. Schoepe
Fakult¨at f¨ur Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
The superfluid Reynolds number Re s = ( v − v c ) D/κ can be expressed simply by the number ofvortex rings that are shed during a half-period of the oscillation.
PACS numbers: 67.25.dk, 67.25.dg, 47.27.Cn
In a recent work by Reeves et al. [1] the two-dimensional Gross-Pitaevskii equation was investigatednumerically in the vicinity of the critical velocity v c forthe onset of quantum turbulence. The central result ofthat work was the observation of a dynamic similarity inthe wake of a cylindrical object and its breakdown dueto vortex shedding at a superfluid Reynolds number isgiven by Re s = ( v − v c ) Dκ , (1)where v c is the critical velocity for the onset of turbu-lence, D is a characteristic length scale, and κ is thecirculation quantum. Applying this result to our experi-ments with a sphere oscillating in He below 0.5 K (for arecent review, see [2]), we choose D = 2 R as the charac-teristic length scale, where R = 0.12 mm is the radius ofthe sphere. Defining ∆ v ≡ ( v − v c ) and v ≡ κ/ R wewrite Re s = ∆ vv , (2)where in our case v = 0 .
40 mm/s. This result is validfor a sphere, but no assumptions have been made con-cerning the dimension of the flow (2D or 3D) nor of itstype (steady or oscillatory). We note, that the ratio κ/R determines the self-induced velocity of a vortex ring ofradius R .In the following we show from the data analysis of ourexperiments, how Eq.(2) can be interpreted in very sim-ple way, namely that Re s is given by the number n ofvortex rings that are shed from the sphere during one-half period of the oscillation 1 / (2 f ), where f = 119 or160 Hz is the frequency of the oscillating sphere.In a small interval of ∆ v above v c ≈
20 mm/s with∆ v/v c ≤ F , we analyze the distribution of the lifetimes t of theturbulent phases and find an exponential distributionexp( − t/τ ), and mean lifetimes τ increasing very fast with the driving force amplitude, namely as τ = τ exp[ ( F/F ) ] , (3)where the fitting parameters τ = 0.5 s at 119 Hz and 0.25s at 160 Hz, and F = 18 pN and 20 pN, respectively. Theforce F can interpreted as being caused by the loss ofkinetic energy of the sphere due to the shedding of one vortex ring of radius R during one half-period.[2, 3] Froma fit to the data we find F = 1 . ρκR √ κω, (4)where ρ is the density of the liquid and ω = 2 πf . Thedriving force is obtained from the data v ( F ) and is givenby F ( v ) = (8 / π ) γ ( v − v c ) . (5) γ is identical to the expression for classical turbulent flowaround a sphere, namely γ = c D ρπR / c D ≈ / π = 0.85 takes into account the energy balance for anequilibrium oscillation amplitude: energy gain from thedrive and loss from a quadratic damping must cancel.While Eq.(5) is deduced from the experiment up to ve-locities of ca. 100 mm/s, which is 5 times larger than v c ,Eq.(4) is proven valid only in the small interval ∆ v/v c ≤ τ was measurable. In this regime we mayapproximate Eq.(5) by F ( v ) = (8 / π ) 2 γ v c ∆ v. (6)We assume that the number n = F/F is the averagenumber of vortex rings emitted per half-period. InsertingEq.(4) and Eq.(6), and using our results v c = 2 . √ κω ,we find n = FF = (8 / π ) 2 γv c ∆ v . ρ κ R √ κ ω = ∆ vv , (7)where v = 0 . κ/R = 0.39 mm/s.In Fig.1 we plot the normalized mean lifetime τ ∗ (∆ v ) ≡ τ /τ = exp [(∆ v/v ) ] . (8)The salient feature is that τ ∗ is independent of the oscil-lation frequency, of the temperature, and is not affected a r X i v : . [ c ond - m a t . o t h e r] J a n τ * Δ v (mm/s) superfluid Reynolds number FIG. 1: (Color online, from [2]) The normalized mean life-times τ ∗ = τ /τ as a function of ∆ v = v − v c for the 119 Hzoscillator at 301 mK (blue squares) and the 160 Hz oscillatorat 30 mK with 0.05% He (black dots). Note the rapid in-crease of τ ∗ by 3 orders of magnitude over the small velocityinterval of ca. 0.7 mm/s . The frequency, the temperature,and the He concentration have no effect on the data. Thedashed curve is calculated from Eq.(8). by He impurities. The only frequency dependence is in τ .Moreover, we see that within our estimated experimentalresolution of about 10% (from the accuracy of the numer-ical factors of v c and F in Eq.(7)), the velocities v and v are identical. Hence, we have our main result: Re s = n. (9) That means, in our experiments (where Re s <
3, seeFig.1) the superfluid Reynolds number is given by thenumber of vortex rings that are shed from the sphereduring one-half period of the oscillation. This is a sur-prisingly simple result. Because Eq.(2) is a rather generalexpression it may be possible that Eq.(9) remains validfor larger values of Re s as well. But this remains to beproven.Finally, it should be mentioned that in a completelydifferent context an equally simple superfluid Reynoldsnumber has recently been calculated for 2D superfluidturbulence in the limit of Re s (cid:29) [1] M.T. Reeves, T.P. Billam, B.P. Anderson, and A.S.Bradley, Phys. Rev. Lett. , 155302 (2015).[2] M. Niemetz R. H¨anninen, and W. Schoepe, J. Low Temp.Phys. , 195 (2017), and references therein. A newversion with an update of Section 4.4 can be found inarXiv:1701.05733v2 [cond-mat.other].[3] W. Schoepe J. Low Temp. Phys. , 170 (2013).[4] L.D. Landau, E.M. Lifshitz, Fluid Mechanics , 2nd edn.(Butterworth, Stoneham, 1987).[5] M.T. Reeves, T.P. Billam, X. Yu, B.P. Anderson, A.S.Bradley, Phys. Rev. Lett.119