HYDRODYNAMIC INSTABILITIES IN THE ROTATING COUETTE FLOW OF SUPERFLUID HELIUM

(1)

HAL Id: jpa-00218352

https://hal.archives-ouvertes.fr/jpa-00218352

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

HYDRODYNAMIC INSTABILITIES IN THE ROTATING COUETTE FLOW OF SUPERFLUID

HELIUM

P. Elleaume, J. Hulin, Bernard Perrin

To cite this version:

P. Elleaume, J. Hulin, Bernard Perrin. HYDRODYNAMIC INSTABILITIES IN THE ROTATING

COUETTE FLOW OF SUPERFLUID HELIUM. Journal de Physique Colloques, 1978, 39 (C6),

pp.C6-163-C6-164. �10.1051/jphyscol:1978672�. �jpa-00218352�

(2)

JOURNAL DE PHYSIQUE Colloque C6, suppl&ment au no 8, Tome 39, aozit 1978, page C6-163

HYDRODYNAMIC INSTABILITIES IN THE ROTATING COUETTE FLOW OF SUPERFLUID HELIUM

P. Elleaume, J.P. Hulin and B. Perrin

Groupe de Physique des Solides de Z'EeoZe NomaZe Sup&ieure, 24 rue Lhomond, 75231 Paris Se, France

Rdsum6.- Nous prdsentons une etude experimentale des diffdrents regimes d'dcoulement de l'hdlium su- perfluide en geometric de Couette. Nous avons mesure l'attenuation de rdsonances de 2Sme son pour differentes valeurs de temperatures et d'espacements entre cylindres en fonction de la vitesse de rotation. Nous avons ainsi successivement identifid trois diffdrents rdgimes conduisant B l'bcoule- ment turbulent.

Abstract.- We present an experimental study of the different flow regimes of superfluid helium in the Couette geometry. We have measured the attenuation of second sound resonances with the rotation velocity for different temperature values and spacings between cylinders. Three flow regimes leading to turbulent flow have been successively identified.

INTRODUCTION.- We present an experimental study of the Couette flow of He I1 between two concentric cylinders of radii R1, R2. Using the two-fluids model, this problem appears as the search for the flow instabilities of a two-fluids system : a super- f l k d with zero viscosity and entropy and a normal fluid of excitations (with a viscosity 11 ) moving

n

separately but coupled by'a mutual friction force due to vortex lines present in the bath. Using the classical Rayleigh and Taylor theories /I/, very different thresholds of hydrodynamic instabilities are predicted if the two fluids were supposed to move independently ; in our experimental case(insi- de cylinder only rotating at velocity a ) , the laminar flow of the superfluid should be instable as

soon as $I # 0, but the normal flow would remain laminar up to a critical Taylor number T = 3390. The questions are : how are the instability thresholds modified by the mutual friction force between the two fluids and how does the peculiar microscopic structure of superfluid turbulence influence the transition towards a turbulent state.

We have studied the Couette flow using a similar second sound method as Bendt /2/ and Snyder /3/ in a large range of rotations velocities (up to 13 turns per second) and with spacings d between cylinders always smaller than 20 % of the mean ra- dius and as low as 3 %. Working at low d values permits us to modify the role of viscosity forces relative to vorticity. We can expect to reach at low d values a regime of dominant viscosity forces.

EXPERIMENTAL PROCEDURE.- Experimentally, we measure the Q's of second sound resonances in the space between cylinders. In contradiction to references /2/

and / 3 / , to study the case d << R2, we chose to use resonances corresponding to an orthoradial propa- gation along the perimeter of the cylinder. For these modes, if d << Rg, the resonance condition is II(R +R ) = nX (n integer) ; hence their frequencies

1 2

do not depend on d and the noise level is minimized even on rotation. The second sound is emitted and detected by 4 aluminized mylar foil strips parallel to the axis of rotation. These strips produce a very uniform second sound field : hence the pure orthoradial modes are the only ones observed up to the frequency of the first radial mode ; the Q values are of the order of 2000.

DOPPLER SPLITTING OF ORTHORADIAL RESONANCES.- Figu- re 1 shows the output of a spectrum analyser con- nected to the receiving strips. The frequency is swept across an orthoradial resonance for different rotation rates. We see that the resonance splits into two components more and more separated and at- tenuated as the rotation velocity increases. This splitting effect is due to the difference in velocity between sound propagating in the direction of helium flow and in the opposite one ; it is propor- tional to the flow velocity. Contrary to reference

/ 4 / , the velocity is not uniform and the eigenmo-

des are not plane waves. We solved the problem by numerical integration of the equations of motion.

The agreement between the theoretical and experi-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978672

(3)

mental results is better than 1 % if a laminar velocity profile with v = vn is assumed for the mean flow. We cannot compute directly the attenuation form amplitude measurements since the two resonances components partially overlap. Therefore, we used a Kennelly circle method /5/ implying a compu- ter analysis of the datas.

Fig. 1 : Splitting and attenuation of an orthoradi- a1 resonance mode at various rotations rates.

ATTENUATION MEASUREMENTS RESULTS.- Figure 2 shows a typical variation with the angular velocity of the extra attenuation l/Q(D)

-

l/Q(O) due to rotation. Below a velocity Dl there is no extra attenuation. Hence D marks the onset of a vortex penetra-

1

tion into the space between cylinders.

Pig. 2 : Variation with rotation of the extra-atte- 1 1

nuation (-

-

^-) of second sound.

Q(a) Q(0)

1 1

slopes n = d (-

-

^-) / d52 to the theoretical slope B R ~ Q Qo

(aO excitation frequency, B mu-

%"=

2woiR;

-

^R;)

tual friction coefficient) which corresponds to an arrangement of straight filaments parallel to the rotation a;xis. The remarkable feature is that the r a t i o s 5 are independent of both d and T and are

- nth

approximatively equal to 1,2 and 3 in the three flow regimes above Dl. So, the first flow regimes between ill and D2 (figure 2)corresppnds to an arrangement of parallel vortices. We believe that in the second regime appears an hydrodynamic instability (similar to Taylor cells) distorting the vortices and increasing their length. With a model using the Landau transition theory, we showed that the extra

52-52

attenuation increases linearly as

2

above n2, 522

with coefficient independent of d and T. The 0 va- 2 lues are lower than the classical Taylor value for the normal fluid but the disagreement is smaller as d decreases and viscosity forces become more impor- tant. Above Ci3, appears a more disordered turbulence initiated by the normal fluid in which vortices have random orientations ; this is supported by the value of the critical Reynolds number which is about 5000 for all d and T values. Of course, it will be necessary to precise these results and es- pecially to obtain further proofs of the existence of a secondary cell flow above D2.

References

/I/ Taylor, G.I. Phil. Trans. A

223

(1923) 289 /2/ Bendt, P.J. Phys. Rev.

153

(1967) 280

/3/ Snyder, H.A., proceeding of the 13th Internati- nnal Conference of Low Temperature

1,

(Plenum Press -New York -London 1974) 283

/ 4 / Kojima, H., Veith, W., Guyon, E. and Rudnick, I.

3. Low. Temp. Phys.

8

(1972) 187

/5/ Mathieu, P., Serra, A., and Simon, Y. Phys. Rev.

14 (1976)3753

-

At higher velocities, a sharp slope change occurs at a critical velocity Q2 and a smoother one around a third value Q3.. In order to identify these different flow regimes, we have compared the experimental