STUDIES ON SUPERFLUID TURBULENCE

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STUDIES ON SUPERFLUID TURBULENCE

K. Schwarz

To cite this version:

K. Schwarz. STUDIES ON SUPERFLUID TURBULENCE. Journal de Physique Colloques, 1978, 39

(C6), pp.C6-1322-C6-1326. �10.1051/jphyscol:19786565�. �jpa-00218055�

JOURNAL DE PHYSIQUE Colloque

C6,

supplimenr au no

8,

Tome

39,

aofit

1978,

page

C6-1322

STUD1 ES ON SUPERFLUID TURBULENCE

K.W. Schwarz

IBM Thomas J. Watson R e s e a r c h C e n t e r , P.O. Box 218, Y o r k t a m H e i g h t s , New Y o r k 10598, U.S.A.

Rdsum6.- NOUS envisagerons les ~aractdristi~ues des turbulences superfluides et discuterons quelques rdsultats expdrimentaux concernant l'dtat compl6tement turbulent. Un modzle thCorique expliquant ces propridtds sera dCcrit.

Abstract.- The qualitative nature of superfluid turbulence and some experimentally established pro- perties of the fully turbulent state are reviewed. A theoretical model explaining these properties is described.

INTRODUCTION.- Superfluid 4 ~ e remains a liquid down to arbitrarily low temperatures, and can therefore always be made to move with a non-zero velocity

-+ -+

field

v

(r,t). At temperatures above absolute zero, this fluid is permeated by a dilute gas of elemen- tary excitations. If conditions are such that this excitation gas comes to local thermodynamic equili- brium in some characteristic distance and time,then for distance and time scales much greater than the- se, the gas can be usefully described in terms of

-+ -+

an additional velocity field vn(r,t).

For laminar flows, the hydrodynamical beha- vior of these velocity fields is well described by the Landau two-fluid model / I / , the success and li- mitations of which may be illustrated in terms of

the flow experiments shown in figure 1. At low heat inputs, V and Vs are small and the measured values of AT and AP are in good agreement /2/ with those computed from the two-fluid equations. As the flow is increased to some critical value. however. a new

in terms of the macroscopic two-fluid equations, and the study of this regime has consequently been a matter of continuing interest. It would not be dif- ficult to assemble a list of a hundred papers repor- ting not only on experiments of the type shown in figure 1 , but also on many others in which different geometries and various ingenious methods are used to probe the properties of the turbulent superfluid.

Despite this immense effort, our understanding of the subject remains in a primitive and somewhat con- troversial state, partly because it has proved ex- tremely difficult to devise turbulent flow experi- ments which yield any kind of s i m p l e information, and partly because the subject has not received the amount of theoretical attention it ~erhaps deserves.

JLATING WALL

- -

regime appears in which AT becomes much larger than

expected. The appearance of this new regime is in

I

many ways reminiscent of the onset of turbulence in

a classical fluid, and it is natural to interpret it in terms of a breakdown in the laminar character of

+

-+

v

or vn, or both. The experimental consensus seems to

-+

be that in fact it is V which usually goes unsta- ble, due to the appearance and growth of quantized vortex lines which begin to fill the channel in a more or less random way. Since elementary excita- tions are scattered by these vortices, there is now

-+ +

a new, dissipative interaction between v and v which qualitatively accounts for the anomalously large values of

AT.

Neither the onset nor the hydrodynamical be-

BATH

I

I

HEATER

6

Fig.

I

: Schematic of a typical counterflow experi- ment. V and V are the average normal and super- fluid vglocitigs. The measured quantities are AT and AP

With these considerations in mind, and with apologies to the many distinguished scientists whose work will not be mentioned,

I

have chosen not to give a historical survey of the field. Instead, I will review only a few properties of the fully tur- bulent state, and then discuss the extent to which they are presently understood.

havior of the "turbulent regime can be understood

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786565

ASPECTS OF FULLY TURBULENT FLOW.- One r e a s o n f o r con- c e n t r a t i n g on t h e f u l l y t u r b u l e n t s t a t e , r a t h e r t h a n t h e o n s e t r e g i o n , i s t h a t one c a n e x p e c t i t t o be somewhat s i m p l e r . I t i s assumed t h a t t h e c h a n n e l be- comes f i l l e d w i t h a random t a n g l e o f q u a n t i z e d v o r - t e x l i n e s which move a b o u t under t h e combined i n f l u -

* e n c e o f t h e i d e a l f l u i d e q u a t i o n s g o v e r n i n g v and t h e f r i c t i o n a l f o r c e s e x e r t e d on t h e v o r t i c e s by t h e normal f l u i d . As t h e a v e r a g e d r i v i n g v e l o c i t y V -V

n s i s i n c r e a s e d beyond t h e c r i t i c a l v a l u e , t h i s t a n g l e becomes i n c r e a s i n g l y d e n s e , and t h e c h a r a c t e r i s t i c i n t e r l i n e s p a c i n g 6 becomes s m a l l e r . Presumably,when 6 i s much l e s s t h a n t h e c h a n n e l s i z e , t h e t u r b u l e n c e c a n b e d e s c r i b e d by l o c a l e q u a t i o n s , t h a t do n o t de- pend on t h e geometry o f t h e system, e x c e p t t h r o u g h boundary c o n d i t i o n s . I n m a c r o s c o p i c (d 1

o - ~

cm) c h a n n e l s t h i s l i m i t i s e a s i l y a t t a i n e d : f o r V -V o f

n s a few cm s - l , 6 i s a l r e a d y of o r d e r t o cm.

I t i s i n s t r u c t i v e t o n o t e t h a t t h i s c o r r e s p o n d s t o a t o t a l l i n e l e n g t h o f 10 t o 1000 km p e r cm3 of f l u i d 131.

As f a r a s we c a n t e l l a t p r e s e n t , i t does a p p e a r t h a t t h e f u l l y d e v e l o p e d t u r b u l e n t s t a t e h a s some w e l l - d e f i n e d g e n e r a l p r o p e r t i e s . By f a r t h e most w e l l e s t a b l i s h e d o f t h e s e was o b s e r v e d a s e a r l y a s 1940 by Keesom, S a r i s , and Meyer / 4 / i n a n expe- r i m e n t s u c h a s t h a t shown i n f i g u r e l . They found t h a t t h e e x c e s s AT d u e t o t h e t u r b u l e n c e went a s t h e cube o f t h e h e a t c u r r e n t

6.

T h i s o b s e r v a t i o n h a s been c o n f i r m e d a p p r o x i m a t e l y by many s u b s e q u e n t i n v e s t i - g a t o r s 15-111. I t c a n b e u n d e r s t o o d phenomenological- l y by assuming t h a t i n t h e t u r b u l e n t s t a t e a new mutuaZ friction force d e n s i t y 151.

a c t s t o c o u p l e t h e normal and s u p e r f l u i d components o f t h e motion. Here, A(T) i s a n e x p e r i m e n t a l l y d e t e r - mined f u n c t i o n o f t e m p e r a t u r e , ps and pn a r e t h e su-

3

-+

-+

p e r f l u i d and normal f l u i d d e n s i t i e s V = V -V and n s n s ' v i s a s m a l l a d j u s t a b l e p a r a m e t e r . F i g u r e 2 shows

t h e c o e f f i c i e n t A ( T ) a s d e t e r m i n e d i n v a r i o u s expe- r i m e n t s . Although t h e r e i s r e a s o n a b l e q u a l i t a t i v e c o n s i s t e n c y , t h e magnitudes o f A a g r e e o n l y v e r y ap- p r o x i m a t e l y . ~ i f f e r e n c e s o n t h e o r d e r of t h o s e s e e n i n f i g u r e 2 seem t o c r o p up r a t h e r c h a r a c t e r i s t i c a l l y when o n e t r i e s t o compare v a r i o u s f l o w e x p e r i m e n t s ,

f o r r e a s o n s I s h a l l d i s c u s s s h o r t l y . At o u r p r e s e n t l e v e l of u n d e r s t a n d i n g , t h e s e d i f f e r e n c e s a r e of l e s s c o n c e r n t h a n t n e common f e a t u r e s r e p r e s e n t e d by equa- t i o n ( I ) and f i g u r e 2.

TEMPERATURE ( K)

F i g . 2 : Mutual f r i c t i o n c o e f f i c i e n t o b s e r v e d under v a r i o u s c o n d i t i o n s : ( a ) d ²⁾ 0.5 cm, c o u n t e r f l o w 161;

(b) d ^?, 0.01 cm, c o u n t e r f l o w 1 8 1 ; ( c ) d ^?. 0.3 em, c o u n t e r f low / 7 / ; (d) d ^1.0 . 0 3 cm, p u r e normal f l u i d f l o w / l o / ; ( e ) d 'L 0.03 cm, p u r e s u p e r f l o w / l o / . The d o t t e d l i n e shows t h e t h e o r e t i c a l p r e d i c t i o n .

-+

A s mentioned e a r l i e r , Fsn i s j u s t a f r i c t i o - n a l f o r c e which a r i s e s b e c a u s e t h e q u a n t i z e d v o r t e x t a n g l e s c a t t e r s t h e t h e r m a l e x c i t a t i o n s t h a t makeup t h e normal f l u i d . T h i s i n t e r p r e t a t i o n was f i r s t pro- posed b y Vinen 1121, i n a n a l o g y t o t h e f r i c t i o n a l f o r c e t h a t a c t s upon second sound i n a r o t a t i n g b u c k e t / 1 3 / . By m e a s u r i n g t h e a t t e n u a t i o n o f a s m a l l second sound wave by t h e t u r b u l e n c e i n a c h a n n e 1 , h e was a b l e t o t u r n t h e argument around and deduce t h a t

t h e t o t a l l i n e l e n g t h d e n s i t y o b e y s t h e r e l a t i o n

E q u a t i o n ( 2 ) i s c o n s i s t e n t w i t h e q u a t i o n ( I ) , b u t it p r o v i d e s a somewhat d i f f e r e n t k i n d o f i n f o r m a t i o n . I t h a s n o t b e e n s t u d i e d n e a r l y a s e x t e n s i v e l y a s r e - l a t i o n ( I ) , a l t h o u g h i t h a s r e c e i v e d some i n d i r e c t s u p p o r t from e x p e r i m e n t s u s i n g i o n p r o b e s 114,151.

A n improved v e r s i o n of V i n e n ' s e x p e r i m e n t i s c u r - r e n t l y underway, / 1 6 / and we e x p e c t t o h e a r some of t h e r e s u l t s a t t h i s c o n f e r e n c e .

A f i n a l f e a t u r e of v o r t e x t u r b u l e n c e was d i s - covered r e c e n t l y by Ashton and Northby 1171. By f o l - lowing t h e m o t i o n o f i o n s which had become.trapped on t h e q u a n t i z e d v o r t i c e s , t h e y were a b l e t o show t h a t t h e v o r t e x t a n g l e h a s a n e t a v e r a g e d r i f t ve- l o c i t y w i t h r e s p e c t t o t h e s u p e r f l u i d r e s t f r a m e , o f t h e form

T h i s i m p l i e s t h a t t h e v o r t e x t a n g l e i s a n i s o t r o p i c , a r e s u l t which, i n r e t r o s p e c t , i s n o t s u r p r i s i n g .

C6- 1324

JOURNAL DE PHYSIQUE

Only t h e one e x p e r i m e n t h a s b e e n done, and f u r t h e r work t o e s t a b l i s h t h e v a l i d i t y of e q u a t i o n (3) would b e d e s i r a b l e .

R e l a t i o n s ( I ) t o (3) make up a r a t h e r s h o r t l i s t , one which e x c l u d e s many i n t e r e s t i n g phenomena.

There i s o f c o u r s e t h e whole q u e s t i o n o f what hap- p e n s i n t h e o n s e t regime ; b u t one might a l s o men- t i o n t h e e x p e r i m e n t s o f Peshkov and Tkachenko / 1 8 / on t h e p r o p a g a t i o n of t u r b u l e n c e down v e r y l o n g c h a n n e l s , o r t h e v e r y r e c e n t o b s e r v a t i o n s o f random f l u c t u a t i o n s i n t h e v o r t e x l i n e d e n s i t y 119,201.

Phenomena of t h e s e v a r i o u s t y p e s a p p e a r , h o w e v e r , t o b e more c o m p l i c a t e d t h a n t h o s e which I have s t r e s s e d .

As a f i n a l comment on t h e e x p e r i m e n t a l as- p e c t s , I would l i k e t o r e c a l l a well-known r e s u l t from c l a s s i c a l hydrodynamics 1211 : when a v i s c o u s f l u i d moyes down a c h a n n e l o f l a t e r a l d i m e n s i o n d , t h e d i s t a n c e i t must t r a v e l downstream b e f o r e end e f f e c t s become s m a l l i s Z

*

0.1 pvd2/n, where p i s t h e d e n s i t y , V i s t h e c h a r a c t e r i s t i c v e l o c i t y , and

n

i s t h e v i s c o s i t y of t h e f l u i d . L e t u s a p p l y t h i s e q u a t i o n t o t h e normal f l u i d , assuming i t s f l o w t o b e l a m i n a r . For t h e t y p i c a l s u p e r f l o w e x p e r i m e n t s done i n narrow (d 1. lod2 cm) c h a n n e l s , Z is v e r y s m a l l and end e f f e c t s a r e u n i m p o r t a n t . However, s u c h e x p e r i m e n t s have been l i m i t e d t o m e a s u r i n g AT and A P . The second sound and t h e i o n p r o b e e x p e r i - ments h a v e been done i n c h a n n e l s w i t h d

*

1 cm. Here

Z t u r n s o u t t o b e on t h e o r d e r of m e t e r s , and t h e r e i s e v e r y r e a s o n t o e x p e c t end e f f e c t s t o b e v e r y i m p o r t a n t . T h i s , and t h e f a c t t h a t o f c o u r s e t h e f l o w p a t t e r n s a r e always inhomogeneous a c r o s s t h e c h a n n e l , may a c c o u n t f o r t h e c h a r a c t e r i s t i c d i f f e - r e n c e s between v a r i o u s e x p e r i m e n t s shown i n f i g u r e 2 .

THEORETICAL CONSIDERATIONS.- T h i s s e c t i o n w i l l b e c o n c e r n e d 6 t h d e s c r i b i n g a r e c e n t model 122,231 o f t h e v o r t e x t a n g l e dynamics which, d e s p i t e i t s appro- x i m a t e n a t u r e , a p p e a r s t o b e r a t h e r s u c c e s s f u l i n e x p l a i n i n g t h e g r o s s f e a t u r e s d i s c u s s e d above. L e t i t b e s a i d a t t h e o u t s e t t h a t t h e g e n e r a l q u a l i t a - t i v e background and s e v e r a l o f t h e s p e c i f i c i d e a s which e n t e r i n t o t h i s model were i n t r o d u c e d b y Vinen /12/ t w e n t y y e a r s ago. I n a d d i t i o n , Ashton and Northby / 1 7 / have g i v e n a q u a l i t a t i v e i n t e r p r e - t a t i o n o f t h e i r v o r t e x d r i f t e x p e r i m e n t which h a s c e r t a i n e l e m e n t s i n common w i t h t h e model t o b e des- c r i b e d .

The s i t u a t i o n t o be c o n s i d e r e d i s a homoge- neous v o r t e x t a n g l e s u b j e c t t o c o n s t a n t , c o u n t e r f l o -

wing normal and s u p e r f l u i d v e l o c i t i e s . The f a s c i n a - t i n g t h i n g a b o u t t h i s problem i s t h a t t h e microsco- p i c b e h a v i o r o f t h e t a n g l e i s r e a s o n a b l y w e l l under- s t o o d . To s t a r t w i t h , one c a n always d e s c r i b e t h e i n s t a n t a n e o u s c o n f i g u r a t i o n o f t h e q u a n t i z e d v o r t i - c e s e x a c t l y by s p e c i f y i n g t h e p a r a m e t r i c form

* *

4

s = s ( C , t ) , where s d e r i o t e s a p o i n t on t h e l i n e a n d

5

i s t h e a r c l e n g t h . The t i m e development o f t h e t a n g l e i s t h e n c o n t a i n e d i n t h e s t a t e m e n t t h a t any g i v e n p o i n t on t h e l i n e moves w i t h a n i n s t a n t a n e o u s v e l o c i t y

w i t h r e s p e c t t o t h e s u p e r f l u i d r e s t f r a m e . The f i r s t term on t h e r i g h t , t h e s e l f - i n d u c e d v e l o c i t y o f t h e l i n e , r e p r e s e n t s t h a t p a r t o f t h e m o t i o n which a r i s e s from t h e i d e a l f l u i d e q u a t i o n s . I t i s g i v e n t o a good a p p r o x i m a t i o n by

* *

where s ' = a s l a g i s t h e v e c t o r t a n g e n t o f t h e l i n e -+

a t t h e p o i n t i n q u e s t i o n ,

2'

⁼ a 2 s / a C 2 i s t h e vec- t o r c u r v a t u r e , and 0 i s a n a p p r o x i m a t e c o n s t a n t of o r d e r cm2 s - l / 2 4 / . I t i s e a s y t o show t h a t t h e m o t i o n d e s c r i b e d by e q u a t i o n ( 5 ) d o e s n o t change

t h e l e n g t h All o f a l i n e e l e m e n t and t h u s c a n n o t cau- s e t h e v o r t e x l i n e t a n g l e t o grow o r d e c a y . The se- cond term on t h e r i g h t of e q u a t i o n ( 4 ) g i v e s t h a t p a r t of t h e l i n e motion which a r i s e s from t h e f r i c - t i o n a l f o r c e s e x e r t e d on t h e l i n e by t h e n o r m a l f l u i d . The temperature-dependent f r i c t i o n c o e f f i c i e n t a i s l e s s t h a n 0.1 up t o 1.8 K , b u t i n c r e a s e s r a p i d l y n e a r t h e A-point. T h i s p a r t o f t h e m o t i o n d o e s n o t c o n s e r v e A l l , and c a n r e s u l t i n e i t h e r t h e growth o r decay o f t h e v o r t e x l i n e d e p e n d i n g on t h e d i r e c t i o n

-+ .+

and m a g n i t u d e o f V -U n s I '

One c a n t a k e t h e i n s t a n t a n e o u s c o n f i g u r a t i o n

*

s ( C , t ) , u s e e q u a t i o n s (4) and ( 5 ) t o f i n d G(C) a t

e v e r y p o i n t on t h e l i n e , and t e l l a computer t o t a - k e a t i m e s t e p Z ( 6 , t

+

d t ) = Z ( C , t )

+

Z ( 6 ) d t . The t i m e development o f t h e t a n g l e c a n t h e n b e o b t a i n e d b y i t e r a t i n g t h i s p r o c e d u r e . A n u m e r i c a l s i m u l a t i o n o f t h i s k i n d may e v e n t u a l l y prove u s e f u l i n studying t h e o n s e t of t u r b u l e n c e . F o r o u r p u r p o s e s , however, i t makes more s e n s e t o d e s c r i b e t h e v o r t e x t a n g l e i n s t a t i s t i c a l t e r m s , and t o d i s c u s s its development i n a n a p p r o x i m a t e manner. To t h i s end, we c o n s i d e r w h a t c a n b e s a i d a b o u t t h e d i s t r i b u t i o n A(Vl. t ) -+

,

where

A ( z l , t ) d % l i s d e f i n e d a s t h e t o t a l l i n e l e n g t h p e r u n i t volume f o r which t h e s e l f - i n d u c e d v e l o c i t y l i e s between W l and v , + d v l . Because of t h e key r o l e t h a t

3 v p l a y s i n e q u a t i o n ( 4 ) , t h i s t u r n s o u t t o be a ma- 1

t h e m a t i c a l l y u s e f u l d e s c r i p t i o n . I n a d d i t i o n , s i n c e A(vl, t ) t e l l s how much l i n e t h e r e i s and how t h e

+

t a n g l e i s moving, it p r o v i d e s j u s t t h e kind of in- f o r m a t i o n we a r e looking f o r .

How does A changes a s t h e t a n g l e develops ac- c o r d i n g t o e q u a t i o n s (4) and (5) ? To determine t h i s i t i s n e c e s s a r y t o know t h e i n s t a n t a n e o u s r a t e of change of t h e l e n g t h All of every l i n e elementand of t h e v l a s s o c i a t e d w i t h it. An e n t i r e l y s t r a i g h t - 3

forward c a l c u l a t i o n shows t h a t t h e normal f l u i d term i n e q u a t i o n (4) g i v e s r i s e t o

-t + + -t a + -t 3

v

1

2

-

%

^{V l}

^vie

^(Vns

^-

^{v l )}

^-

v l x (vl x Vns), (6b)

w h i l e t h e i d e a l f l u i d p a r t o f t h e motion makes a c o n t r i b u t i o n

Our only purpose i s resenting t h e s e e q u a t i o n s ex- p l i c i t l y i s t o i l l u s t r a t e t h e d i f f e r i n g n a t u r e s of t h e two c o n t r i b u t i o n s t o t h e e v o l u t i o n of t h e tan- g l e . The e f f e c t of t h e normal f l u i d can b e determi- ned i m e d i a t e l y , s i n c e e q u a t i o n s (6a) and (6b) in- volve only v +

.

I f one keeps t r a c k of what h a p p e n s t o

1

t h e l i n e elements i n i t i a l l y contained i n some e l e - ment d3vl of v s p a c e , one o b t a i n s +

1

A(;

+ ;

d t , t + d t ) d 3 v ( t + d t ) = ( l + ~ i d t / A ~ ) ~ ( $ ~ , t ) d ~ v ~ ( t )

1 1 1

This simple bookkeeping formula, when expanded t o f i r s t o r d e r i n d t and combined w i t h e q u a t i o n s (6a) and ( 6 b ) , y i e l d s t h e normal f l u i d c o n t r i b u t i o n t o

a A / a t .

This c o n t r i b u t i o n c o n t a i n s c o n v e c t i v e terms which move t h e t a n g l e around i n v space, and sour- +

1

c e terms which i n c r e a s e and d e c r e a s e t h e l i n e length i n v a r i o u s p a r t s of v space, b u t

+

i t s a y s n o t h i n g

1

about the random n a t u r e of t h e t a n g l e . By i t s e l f , i t does not l e a d t o a s t e a d y s t a t e .

The i d e a l f l u i d p a r t of t h e motion c ~ ~ u a t i o n

$ (7b)l i s e n t i r e l y d i f f e r e n t i n n a t u r e , s i n c e V de-

1 pends on l o c a l p r o p e r t i e s

zl", zl:""

of t h e l i n e e l e - ment about which A ( v l , t ) provides no information. -f

Thus, e x t r a p h y s i c a l arguments must be i n t r o d u c e d t o d e a l witE t h e e f f e c t of t h i s term on A . Now, t h e l i n e i n t h e v o r t e x t a n g l e w i l l have some c h a r a c t e - r i s t i c average c u r v a t u r e <s"> and hence a characte- r i s t i c self-induced v e l o c i t y <v > 2 B<s">. Various

1

p a r t s of t h e l i n e a r e moving about randomly w i t h

v e l o c i t i e s of t h i s o r d e r , producing a complicated i n t e r n a l motion. T h i s motion w i l l l e a d t o f r e q u e n t l i n e - l i n e c r o s s i n g s , which i n t u r n g e n e r a t e l a r g e random d i s t o r t i o n s a t l o c a l i z e d r e g i o n s on t h e l i n e . These d i s t o r t i o n s t h e n propagate along t h e l i n e , s o t h a t any given p o i n t on t h e l i n e w i l l r e c e i v e i n t e r - m i t t e n t randomizing " s i g n a l s " . More d e t a i l e d c o n s i d e -

r a t i o n s show t h a t i n f a c t t h e t y p i c a l randomizing s i g n a l o c c u r s i n an average time of o r d e r 6 / < v l >

and t h a t i t a r i s e s from a c r o s s i n g which occured a d i s t a n c e 6 away, where 6 again i s t h e average i n t e r -

l i n e d i s t a n c e .

The c o n c l u s i o n t h a t t h e l i n e i s r e p e a t e d l y chopping i t s e l f i n t o s e c t i o n s of l e n g t h 6 makes i t n a t u r a l t o assume t h a t t h e t a n g l e remains "kinky"

on t h e s c a l e 6. Given <sf'>, t h i s i m p l i e s t h e order- of-magnitude c o n d i t i o n s

13f1 1

^% <s1'>/6, (9)

Furthermore,

31'

^and

2"

randomly t a k e on new v a l u e s i n t h e time Ci/<v:.Equation (7b) t h e n i m p l i e s , roughly

speaking, t h a t t h e v a s s o c i a t e d w i t h any l i n e seg- -t

1

ment changes by a random amount of o r d e r < V l > i n the c h a r a c t e r i s t i c time i n t e r v a l 6/<v >. Thus, any

+ ¹

s t r u c t u r e i n A(v , t ) w i l l spread o u t a d i s t a n c e < v >

+ ^{1 1}

i n v1 space i n t h e time 6 / < v >. T h i s e f f e c t may b e 1

c r e d e l y modelled a s a d i f f u s i v e r e l a x a t i o n p r o c e s s , a c t i n g on t h e d i s t r i b u t i o n A($ , t ) and c h a r a c t e r i z e d

1

by a d i f f u s i o n c o n s t a n t of o r d e r <V > 3 / 6 . 1

The combination of t h e convective and s o u r c e terms due t o t h e normal f l u i d , and t h e d i f f u s i v e term a r i s i n g from t h e self-induced motion of t h e t a n g l e seems t o p r o v i d e a l l t h e n e c e s s a r y ingredients f o r a r e a s o n a b l e , a l b e i t v e r y ~ r i m i t i v e , d e s c r i p - t i o n of t h e v o r t e x t a n g l e dynamics. Indeed, it a l - ready appears t h a t a model of t h i s type can e x p l a i n the e x p e r i m e n t a l l y observed f e a t u r e s d i s c u s s e d e a r - l i e r . A s p e c i f i c e q u a t i o n f o r

a A / a t ,

based on t h e s e arguments, h a s been i n t e g r a t e d numerically t o f i n d t h e s t e a d y s t a t e A(vl,m) f o r v a r i o u s v a l u e s of V

+

ns and T. This e q u a t i o n i s extremely complicated, and i t may b e p o s s i b l e t o f i n d simpler v e r s i o n s . Never- t h e l e s s , t h e numerical work h a s a l r e a d y produced s e v e r a l i n t e r e s t i n g r e s u l t s . F i r s t , k(Sl ,a) i s found t o be s t r o n g l y peaked a t v a l u e s of V. on t h e o r d e r

1

of B/6 and t o be s t r o n g l y a n i s o t r o p i c w i t h

S1

poin-

-t

t i n g p r e f e r e n t i a l l y i n t h e d i r e c t i o n of V T h i s i s n'

i n accord w i t h t h e s p e c u l a t i o n s of Ashton and

C6- 1326

^{JOURNAL D E PHYSIQUE}

and <vlZ> deduced from the computed distributions are in surprinsingly good agreement with experimen- tal observations. Figures 2 and 3 show the calcula- ted values of F as a function of V and T. The

sn ns

predicted results are certainly within the range of experimental variation, and similar agreement is found for L and <v z>. Finally, A(v~,~) + is found nu-

1

merically to obey a certain scaling behavior. By integrating over the equation for ah/at and making use of this scaling property, it is possible to de- rive an equation identical in form to the well- known Vinen equation /12/.

Fig. 3

:

Predicted behavior of the mutual friction F . ^{The d-+.s} are the values actually computed, the l%es are straightline fits. The experimental re- sult is F

a

(Vns - ^{2 v}

^),

to an accuracy of

3

0

In summary, it appears that significant pro- gress can be made in understanding the gross pro- perties of the fully turbulent state. It must be admitted that the theoretical model, as it now stands, raises perhaps as many questions as it ans- wers. However, its initial success is very encouraging and should stimulate further theoretical a d experimen- tal work on this interesting problem.

References

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!J

(1941) 592;

J. Phys. USSR 5 (1941) 71

;

Landau, L.D. and Lifshitz, E.M.,

Fluid Mechanics,

(Pergamon, London) 1959, Chapter 16

/2/ Brewer, D.F., and Edwards, D.O., Proc. Roy. Soc.

A251 (1959) 247

-

/3/ The interline spacing is related to the total line length per unit volume by

6 a

L - ~ / ~ /4/ Keesom, W.H., Saris, B.F., and Meyer, L., Physi-

ca 1 (1940) 817

/5/ Gorter, C.J., and Mellink, J.H., Physica 11

(1949) 285

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/8/ Brewer, D.F., and Edwards, D.O., Phil. Mag. 1

(1962) 721

/9/ Chase, C.E., Phys. Rev. 127 (1962) 361 /I01 Van der Heijden,

G.,

De Voogt, W.J.P., and

Kramers, H.C., Physica 2 (1972) 473

;

Van der Heijden, G., Giezen, J.J., and Kramers, H.C., Physica 61 (1972) 566

;

Van der Heijden, G., Van der Boog, A.G.M., and Kramers, H.C., Physica (1974) 487

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/12/ Vinen, W.F., Proc. Roy. Soc. 242 (1957) 493 /13/ Hall, H.E., and Vinen, W.F., Proc. Roy. Soc. A

238 (1956) 204

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/14/ Sitton, D.M., and Moss, F., Phys. Rev. Lett. a

(1972) 542

/I51 Ashton, R.A., and Northby, J.A., Phys. Rev.

Lett. 2 (1973) 1119

/I61 Kittel, P., Cromar, M.W., Ostermeier, R.M.,and Donnelly, R.J., Bull. Am. Phys. Soc. 23 (1978) 5 14

/I71 Ashton, R.A., and Northby, J.A., Phys. Rev.

Lett. 3 (1975) 1714

/18/ Peshkov, V.P., and Tkachenko, V.K., Zh. Eksp.

Teor. Fiz. dl (1961) 1427

;

Soviet Phys. JETP 14 (1962) 1019

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/19/ Hoch, H., Busse, L., and Moss, F., Phys. Rev.

Lett.

2_4

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/20/ Mantese, J., Bischoff, G., and Moss, F., Phys.

Rev. Lett. 39 (1977) 565

/21/ Schlichting, H.,

Boundary Layer Theory

(McGraw-Hill, New York) 1960

/22/ Schwarz, K.W., Phys. Rev. Lett. 2 (1977) 551 /23/ Schwarz, K.W., Phys. Rev. (to be published) /24/ z' ^, 31, make up a right-handzd, mutually

orthogonaI system of vectors, with respective

magnitudes 1, R-l, B R - ~ , where R is the local

radius of curvature of the line