SELF-PRESERVING STRUCTURES IN HOMOGENEOUS SUPERFLUID TURBULENCE

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SELF-PRESERVING STRUCTURES IN

HOMOGENEOUS SUPERFLUID TURBULENCE

J. Tough, D. Ladner, C. Piotrowski

To cite this version:

J. Tough, D. Ladner, C. Piotrowski. SELF-PRESERVING STRUCTURES IN HOMOGENEOUS SUPERFLUID TURBULENCE. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-158-C6-159.

�10.1051/jphyscol:1978670�. �jpa-00218350�

JOURNAL DE PHYSIQUE

Colloque C6, supplhent au

no

8, Tome 39, aoQt 1978, page C6-158

SELF-PRESERVING STRUCTURES IN HOMOGENEOUS SUPERFLUID TURBULENCE J.T. Tough, D.R. Ladner and C. Piotrowski

Physics Department, The Ohio State University, CoZwnbus, Ohio, U.S.A. 43210

R6sumd.- La force de frottement mutuel dans le contrecourant thermique d'He I1 a 6td expliqude par Vinen dans un modble de la turbulence superfluide. Ce modSle

a

6t6 applique avec succZs

1

plusieurs phdnomSnes de contrecourant. Nous avons essay6 d'examiner l'homo- gdneitd de la turbulence en faisant des comparaisons entre ces gradients de tempdrature et les vitesses critiques dans des canaux de contrecourant circulaires et rectangulaires.

Les rdsultats suggbrent que la turbulence est homogsne mais que chaque configuration g60- mStrique possbde sa propre structure self-entrafnante.

Abstract.- The mutual friction force in He I1 thermal counterflow has been accounted for by Vinen in a model for superfluid turbulence. The model has been successfully applied to

a

number of counterflow phenomena. We have attempted to examine the homogeneity of the turbulence by comparing temperature gradients and critical velocities from circular and rectangular counterflow channels. The data suggest the turbulence is homogeneous, but that each geometry possesses a different self-sustaining structure.

The non-linear thermal resistance obser- ved in He I1 flowing at an average counterflow velo- city V was interpreted by Gorter and Mellink

/ 1 /

in terms of a mutual friction force F

+

between the

sn

normal and superfluid components. Vinen's experiments /2/ extended to much lower values of

5

and demons- trated that F

+

vanished at a critical velocity

t

sn c'

below which the He I1 became super-thermal conduc- ting. Vinen /3/ proposed that the mutual friction force resulted from turbulence in the superfluid in the form of a "tangled mass" of vortex lines. If lines are present in the flow at a density Lo(V), then the mutual friction force is

-+ F

( 5 =

?L~(V)

s n (1)

where f is the force on unit length of vortex line,

+

and the factor 2/3 arises from spatial averaging.

The line density Lo(V) is maintained by a dynamic balance of production and annihilation processes with amplitudes proportional to parameters

XI

andx2

respectively. At a given counterflow velocity Fsn + is then determined by the magnitude of x1/x2. The critical velocity is also determined by X1/x2.

Many features of the Vinen model have been experi- mentally confirmed. In a recent theoretical paper

Schwarz

/6/

has also given a detailed microscopic equation for the line dynamics which reduces to the Vinen dynamic balance equation when integrated over the line distribution fuxtion . Central to any further understanding of superfluid turbulence, however, is the question of homogeneity. Gorter and

Mellink, Vinen, and Schwarz all assume that Fsn is

+

homogeneous -- that is, independent of position in the flow channel. The reason for this assumption becomes obvious when alternative ones are examined.

Consider flow in a circular channel

:

axial coordi- nate Z, radial coordinate r. It is straightforward to demonstrate from the two-fluid equations of mo- tion that even-the simple inhomogeneous case where

+

Fsn =

ZF

(r) is not allowed. In fact, the only sen- sible inhomogeneous force allowed by the equations is one for which F has both Z and r components,

sn

each of which depends on Z and r. It seems that if the simple homogeneous distribution of lines con- sidered by Vinen is not correct, the situation is .incredibly complex.

There is nevertheless at least one reason to suppose that L (F) is not homogeneous. In the Vinen and Schwarz models it is the local value of V which determines Lo. Since ; is a function of r,

n

5

must be the same function of r if F

=

V

_s

- _n ^is

to be uniform and L

( 0 )

homogeneous. While the assumption of a uniform 5 and corresponding homo- geneous Lo(V) is convenient (for example, in des- cribing the eddy viscosity of the turbulent super- fluid /7/, it is not proven.

If the mutual friction force were strongly inhomogeneous, similar experiments in different geometries should give conflicting results. To examine this possibility we have measured the tem- perature gradients and critical velocities in long

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978670

(10 cm) g l a s s channels of c i r c u l a r (diameter d ) and assumed, i n analogy t o c l a s s i c a l t u r b u l e n c e , t h a t r e c t a n g u l a r (d x 10d) c r o s s s e c t i o n . The d a t a were / 4 / " a t any i n s t a n t and f o r any v a l u e of L t h e reduced u s i n g t h e (homogeneous) Vinen model t o g i v e geometrical s t r u c t u r e of t h e t u r b u l e n c e i s s i m i l a r ; v a l u e s of

x1/xZ

a t each temperature. Values d e t e r - i . e . t h a t a p a r t from a s c a l e f a c t o r t h e v o r t e x li- mined from t h e c r i t i c a l v e l o c i t y and from t h e tem- nes a r e always arranged i n t h e same geometrical p e r a t u r e g r a d i e n t were g e n e r a l l y w i t h i n experimental way". It i s t h e s e l f - p r e s e r v a t i o n assumption which e r r o r . The r e s u l t s f o r f i v e channels, two c i r c u l a r allows t h e production a n a n n i h i l a t i o n r a t e s t o be (d : 1261.1, and 129y) and t h r e e r e c t a n g u l a r (d = 98p w r i t t e n i n terms of t h e i n s t a n t a n e o u s v a l u e of the 471.1, and 3 2 ~ ) a r e given i n f i g u r e I . l i n e d e n s i t y . The important p o i n t i s t h a t t h e va-

l u e of t h e production parameter ~1 w i l l depend on '.Or--

- ..- . - - - - .- . T I

t h e e x a c t

form

of the geometrical arrangement of v o r t e x l i n e s , and t h i s form mag very w e l l be d e t e r -

10.4 mined by t h e shape of t h e channel boundaries. Dif- f e r e n t geometrical arrangements can be b o t h random ( a "tangled hassl1> and homogeneous. We s u g g e s t t h a t our d a t a i n r e c t a n g u l a r and c i r c u l a r geometries a r e c o n s i s t e n t w i t h s u p e r f l u i d t u r b u l e n c e t h a t i s 0.1 homogeneous and s e l f - p r e s e r v i n g , and t h a t t h e geo- m e t r i c a l s t r u c t u r e of t h e t u r b u l e n c e i s d i f f e r e n t

---

⁰

1.2 1.4 1.6 1.8

TEMP ( K ) i n t h e two c a s e s .

F i g . 1 : Values of

xl/&

obtained from temperature References

g r a d i e n t c r i t i c a l v e l o c i t y d a t a i n c i r c u -

l a r (0) and r e c t a n g u l a r

( D )

^channels. / I / G o r t e r , C . J . and Mellink, J . H . Physica

IS

(1949) 127.

Although t h e v a l u e s f o r

x1/x2

o b t a i n e d /2/ Vinen, W.F., Proc. Roy. Soc. A x (1957) 114.

i n t h e d i f f e r e n t geometries appear t o have t h e same /3/ Vinen, W.F., Proc. Roy. Soc. A242 (1957) 493.

temperature dependence, t h e v a l u e s from t h e r e c t a n - /4/ H a l l , H.E. and Vinen, W.F., Proc. Roy. Soc.

g u l a r channels a r e about 1.8 times l a r g e r i n magni- A= (1956) 215.

tude (note the different scales in figure ,). One 151 Schwarz, K.W., Phys. Rev. L e t t .

36

(1975) 384, and t o be p u b l i s h e d .

p o s s i b i l i t y f o r t h i s d i s p a r i t y might be t h e s p a t i a l

/ 6 / C h i l d e r s , R.K. and Tough, J . T . , Phys. Rev.

averaging f a c t o r (213) i n e q u a t i o n 1 . It i s d i f f i - L e t t .

35

(1975) 527.

c u l t t o s e e how t h i s f a c t o r can vary w i t h channel geometry, and even more d i f f i c u l t t o s e e how i t can be 3.2 times l a r g e r i n t h e r e c t a n g u l a r channels a s

t h e temperature g r a d i e n t d a t a r e q u i r e . F u r t h e r , a v a r i a t i o n i n t h i s f a c t o r would n o t i n f l u e n c e t h e

x1/x2

v a l u e s d e r i v e d from t h e c r i t i c a l h e a t cur- r e n t r e s u l t s a t a l l .

Another p o s s i b i l i t y f o r t h e d i f f e r e n c e i n x 1 / x 2 v a l u e s o b t a i n e d from t h e two geometries i s a n inhomogeneous f o r c e F + o r l i n e d e n s i t y Lo(?).

s n

Since t h e only p o s s i b i l i t y f o r an inhomogeneous allowed by t h e e q u a t i o n s of motion i s extremely s n

g e n e r a l , i t i s of v i r t u a l l y no v a l u e f o r the ana- l y s i s of d a t a . We have t h e r e f o r e attempted t o un- d e r s t a n d the observed v a r i a t i o n with channel geo- metry i n terms'of homogeneous s u p e r f l u i d t u r b u l e n c e .

A c e n t r a l f e a t u r e of Vinen's c a l c u l a - t i o n i s the concept of s e l f - p r e s e r v a t i o n . It i s