CHAPTER V I I
APPLICATION OF A MATHEMATICAL LODEL TO ILLUSTRATE RELATIONS CHARACTERISTIC OF TURBULENCE
50, A number o f r e l a t i o n s c h a r a c t e r i s t i c f o r t h e e n e r g y exchange between t h e components o f t u r b u l e n c e and, when p r e s e n t , botvjeon t h e
main m o t i o n and t h e s e components, can be i l l u s t r a t e d by means o f t h e s o l u t i o n s o f a d i f f e r e n t i a l e q u a t i o n , much s i m p l e r t h a n t h e h y d r o d y n a -mi c a l e q u a t i o n s b u t r e t a i n i n g c e r t a i n i m p o r t a n t f e a t u r e s o f t h e l a t t e r . I t has boon found t h a t t h o s e s o l u t i o n s a l s o l e a d t o s t a t i s t i c a l p r o b lems, i n many ways a n a l o g o u s t o s t a t i s t i c a l problems o c c u r r i n g i n t u r b u -l e n c e . I t i s v / o r t h w h i -l e t h e r e f o r e t o g i v e a t t e n t i o n t o t h i s e q u a t i o n and i t s s o l u t i o n s , w h i c h may be c o n s i d e r e d as g i v i n g an i n t r o d u c t i o n i n -t o some o f -t h e deeper -t u r b u l e n c e p r o b l e m s .
The p a r t i c u l a r e q u a t i o n chosen f o r t h i s purpose has t h e f o r m :
- t * " - ^ = v ^ ( 1 )
The V a r i a b l e v can be t a k e n as t h e analogue o f t h e t u r b u l e n t v e l o c i t y ; i t i s dependent on t h e t i m e and a c o o r d i n a t e y ( s o m e t h i n g l i k e t h e t r a n s -v e r s e d i m e n s i o n s o f t h e f i e l d of f l o w i n 3 t u b e ) . The e q u a t i o n has a n o n - l i n e a r t e r m o f t h e f i r s t o r d e r , and a t e r m o f t h e second o r d e r m u l t i p l i e d by a c o e f f i c i e n t v virhich i s assumed t o be v e r y s m a l l (analogue o f t h e k i n e m a t i c v i s c o s i t y ) . The i n t e r p l a y o f t h e s e t e r m s i s o f p r i m a r y i i i p o r t a n c e i n t h e dynamics o f t u r b u l e n c e , "^ince t h e e q u a t i o n r e f e r s t o a s i n g l e v a r i a b l e and a s i n g l e coorodinaii^ i t does n o t p i c t u r e t h o s e g e o m e t r i c r e l a t i o n s o f hydrodynaraic t u r b u l e n c e , w h i c h e r e de-pendent on t h e t h r e e d i m e n s i o n a l n a t u r e o f t h e f i e l d , t h e p r o p e r t i e s o f v o r t i c e s and t h e phenomena o f s h e a r i n g raotione There i s no p r e s s u r e t e r m i n t h e e q u a t i o n and t h e r e i s no e q u a t i o n o f c o n t i n u i t y ; hence t h e r e i s n o t h i n g v/hich r e f l e c t s t h e c o n d i t i o n o f i n c o m p r e s s i b i l i t y o f o r d i n a r y hydrodynamics0 I n a v/ay t h e e q u a t i o n i s more i l l u s t r a t i v e o f c e r t a i n phenomena p e c u l i a r t o c o m p r e s s i b l e media, i n p a r t i c u l a r t o shock xmvesc I n t h e r e s u l t s v/hich can be o b t a i n e d f r o m e q . ( l ) t h e r e i s no approach
t o t h e s p e c i a l g e o m e t r i c f e a t u r e s ( t e n s o r r e l a t i o n s e t c ) w h i c h a r e c o n s i d e r e d i n t h e t h e o r y o f i s o t r o p i c t u r b u l e n c e ,
The e q u a t i o n can bo com.pletcd on t h e r i g h t hand s i d e b y i n t r o -d u c i n g e i t h e r a t e r m r e p r e s e n t i n g an e x t e r i o r f o r c e , o r a t e r m - des-c r i b i n g a des-c o u p l i n g between v and a n o t h e r parameter, t o be des-c o n s i d e r e d as t h e analogue o f t h e v e l o c i t y o f t h e main m o t i o n . I t i s n e c e s s a r y t o s p e c i f y t h e domain o f t h e v a r i a b l e y t o v/hich t h e e q u a t i o n s h a l l r e f e r . T h i s can be e i t h e r t h e complete y - a x i s , from. - oo , t o + oo ; o r i t can be a l i m i t e d domain, f o r i n s t a n c e 0 y b , i n w h i c h case we s h a l l r e q u i r e t h a t v v a n i s h e s a t b o t h l i m i t s .
E q u a t i o n ( l ) can be reduced t o a l i n e a r e q u a t i o n b y s u b s t i t u t i n g a new v a r i a b l e f o r v. I t i s u s e f u l , however, f i r s t t o c o n s i d e r t h e e q u a t i o n as i t s t a n d s . We l o o k f o r a p a r t i c u l a r s o l u t i o n o f t h e f o r m :
where p and V a r e f u n c t i o n s o f t . When t h i s e x p r e s s i o n i s s u b s t i t u t e d i n t o ( l ) , t h e e q u a t i o n i s s a t i s f i e d i d e n t i c a l l y i f
p = l / ( t - t ^ ) | C*= c o n s t a n t (3)
t ^ b e i n g a constant» Hence a s o l u t i o n r e p r e s e n t e d b y a s t r a i g h t l i n o w i l l t u r n t o t h e r i g h t , w h i l e i t s p o i n t o f i n t e r s e c t i o n w i t h t h e a x i s
( ^ r e m a i n s unchanged. The a n g l e a b e t w e e n t h e segment and t h e v e r t i -c a l d i r e -c t i o n i n -c r e a s e s a -c -c o r d i n g t o t h e r e l a t i o n :
t a n a = t - t , o
The r e s u l t can be extended t o t h e case where t h e course o f v i s r e p r e s e n t e d b y a s e r i e s o f s t r a i g h t segments. To p r e v e n t d i f f i c u l t i e s w i t h q u a n t i t i e s becoming i n f i n i t e , we assume t h a t t h e a n g l e s between c o n s e c u t i v e segments a r e s l i g h t l y rounded o f f . S i n c e v/'Q y i s n o t z e r o i n t h e n e i g h b o r h o o d o f t h e a n g l e s , t h e t e r m v ( ^ ^ v / c>y2) v d l l have i n f l u e n c e on t h e f o r m o f t h e s o l u t i o n . Keeping i n mind t h e e f f e c t o f a s i m i l a r t e r m i n t h e e q u a t i o n o f d i f f u s i o n , we may expect
t h a t t h e r o u n d i n g o f f w i l l g r a d u a l l y spread t o a more e x t e n d e d region» I n t h e f i r s t phases o f t h e m o t i o n t h i s d e t a i l w i l l n o t have much i n -f l u e n c e and we can -f i n d t h e a p p r o x i m a t e h i s t o r y o -f t h e s o l u t i o n s i m p l y by a p p l y i n g t h e r u l e t h a t e v e r y segment t u r n s about i t s " h i n g e p o i n t " , 6, a c c o r d i n g t o f o r m . (2) and ( 3 ) . The p o i n t o f i n t e r s e c t i o n o f t w o c o n s e c u t i v e segments r e t a i n s a c o n s t a n t v a l u e o f v and moves p;ith a v e l o c i t y eoj^ual t o Vo I f t h e p o i n t i s b e l o w t h e a x i s , v i s n e g a t i v e and t h e movement i s d i r e c t e d t o t h e l e f t ,
51. When t h e i n i t i a l s l o p e o f t h e segment was p o s i t i v e , i t w i l l r e m a i n so w i t h t h e s l o p e g r a d u a l l y d e c r e a s i n g t o z e r o . I f t h e i n i t i a l s l o p e i s n e g a t i v e , t h e s l o p e w i l l i n c r e a s e i n a b s o l u t e measure and a f t e r a f i n i t e l a p s e o f t i m e t h e segment w i l l approach t o a v e r t i c a l p o s i t i o n . Our s i m p l e r e s u l t can t h e n no l o n g e r be used.
To s e e what happen i n such a case, we c o n s i d e r a p a r t i c u l a r s o l u t i o n o f ( l ) , o b t a i n e d b y assuming t h a t v i s d e t e r m i n e d b y an ex-p r e s s i o n o f t h e f o r m : - 1/2 V = V ( 7 9 ) ( t - t ^ ) where V {TJ ) i s a f u n c t i o n o f t h e v a r i a b l e = ( y - 6 ) ( t - t ^ ) E q u a t i o n ( l ) i s t h e n t r a n s f o r m e d i n t o : V V " - VV' + I -77 V> + I V = 0 w h i c h can be i n t e g r a t e d ond g i v e s : v V - I + I r ; V = 0,
( t h e i n t e g r a t i o n c o n s t a n t has been a d j u s t e d so t h a t V may v a n i s h a t i n f i n i t y ) , \'!o now p u t :
V = - 2V d ( l n u ) / d 7 ; , and o b t a i n a l i n e a r e q u a t i o n f o r u:
I n t o g r a t i o n o f t h e l a t t e r e q u a t i o n f i n a l l y l e a d s t o t h e f o l l o v a n g ex-p r e s s i o n f o r V! V= 2v h e ^ c p ( ( h ^ - r ? ^ ) A V V hv - h ^ drj^ exp i^ ( h ^ - v l)/hv ' h h b e i n g a n o t h e r i n t e g r a t i o n c o n s t a n t . S i n c e t h i s e x p r e s s i o n l o o k s r a t h e r i n v o l v e d , i t i s u s e f u l t o n o t e t h e f o l l o w i n g a p p r o x i m a t i o n s , v a l i d when h / v i s l a r g e ; ( a ) when ^ = h + Ö v;here 6 « h , I h [ 1 - f a n h (hö/i^v)] ;
(b) when 0 C?; Ox end n o t t o o n e a r t o one o f t h e end p o i n t s :
( c ) when 7? i s near zero o r i s n e g a t i v e , t h e v a l u e o f V becomes p r a c -t i c a l l y z e r o .
W i t h t h e a i d o f t h e a p p r o : d m a t i o n s we can o b t a i n a g e n e r a l p i c t u r e o f t h e course o f V, T h i s can be r e a d i l y t r a n s l a t e d i n t o a p i c t u r e f o r V as a f u n c t i o n o f y and t , l e a d i n g t o t h e r e s u l t g i v e n i n t h e d i a g r a m ;
I t w i l l be seen t h a t t h e course o f v i s r e p r e s e n t e d a p p r o x i m a t e l y by a t r i a n g u l a r f i g u r e v d t h c o n s t a n t a r e a ™- h^., The s l o p e o f t h e hjrpotjfienuse i s l / ( t - t ^ ) as b e f o r e . The v e l o c i t y o f advance o f t h e r i g h t hand s i d e ( t h e " f r o n t v e l o c i t y " ) i s - | h ( t - t ) " ^^'^, w h i c h i s e q u a l t o one h a l f t h e h e i g h t o f t h e f r o n t . An a p p r o x i m a t i o n t o t h e course o f v a t t h e f r o n t i s g i v e n b y ; V = h 2 - ^ t ' T t ' ; 1 „ tanh - i ^ i l 4 . v ^ r ~ t X " - i (4) w i t h % h ( t - t ^ ) I t s h o u l d be n o t e d t h a t t h e r e i s a l s o a s o l u t i o n i n w h i c h t h e s i g n s o f V and b o t h a r e changed; t h e t r i a n g l e p o i n t s dovmward and i t s f r o n t moves t o t h e l e f t ,
52, The r e s u l t t h a t t h e a r e a o f t r i a n g l e i s c o n s t a n t expresses t h e " c o n s e r v a t i o n o f momentum" f o r t h e s o l u t i o n . By i n t e g r a t i n g eq.
( l ) w i t h r e s p e c t t o y we o b t a i n
J v dy = 0, ( 5 ) f o r any domain a t t h e l i m i t s o f w h i c h v v a n i s h e s and v ('öv/'fey) i s
e i t h e r r i g o r o u s l y zero o r s u f f i c i e n t l y s m a l l t o be n e g l i g i b l e .
I f we m u l t i p l y e q , ( l ) b y v and i n t e g r a t e , we o b t a i n an " e q u a t i o n o f e n e r g y "
d
d t r d y — V J ( - ^ dy — V I ( - ^ ) dy, ( 6 )
f o r any domain a t the l i m i t s of which v v a n i s h e f . I n t h e case of t h e s o l u t i o n considered above the energy i n t e g r a l amounts t o ;
Making use o f (4) t h e " d i s s i p a t i o n i n t e g r a l " i s found t o be:
. 0 - 3/2
-12 - ^ o )
I t i s e a s i l y v e r i f i e d t h a t t h e s e e x p r e s s i o n s s a t i s f y ( 6 ) .
The r e s u l t t h a t a n e a r l y v e r t i c a l f r o n t i n t h e course o f v can be d e s c r i b e d a p p r o x i m a t e l y by a h y p e r b o l i c t a n g e n t f u n c t i o n and t h a t i t has a c e r t a i n v e l o c i t y o f advance, can be g e n e r a l i z e d . We r e t u r n t o eq, (1) and i n t r o d u c e a c o o r d i n a t e system which s h a l l move \»fith t h e f r o n t . To t h i s end T*e p u t :
y' = y - ^ ; t ' = t ,
fej b e i n g a f u n c t i o n o f t . From t h e s e f o r m u l a s we deduce:
^ 'ö '5 a c •— ^ y ~ -öy'
'
- g i t ^ t ' " ^ y « • vjhere c = d & i / d t o S u b s t i t u t i o n o f t h e s e e x p r e s s i o n s i n t o ( l ) g i v e s : '0 V,
^ ^ v ^ ^v + (v - c) ^ — 7 = V W - -©y» " 3 ,2 • -1 We e x p e c t t h a t t h e d e r i v a t i v e v / "© y' w i l l be o f t h e o r d e r V 2 P i n t h e r e g i o n w i t h a s t e e p g r a d i e n t , and t h a t "c) v/'öy^" V i / i l l be o f o r d e r .V " , I f c i s p r o p e r l y a d j u s t e d , v / ö t * v d l l be o f n o r m a l o r d e r o ^ magnitude b o t h a t t h e s t e e p f r o n t and elsev/here. T h i s t e r m can t h e n be d i s c a r d e d and t h e r e r e m a i n s :I n t e g r a t i o n g i v e s :
(v - c ) ^ -V j : ^ ^ = c o n s t a n t .
S i n c e t h i s e x p r e s s i o n must be v a l i d t h r o u g h t h e r e g i o n o f a r a p i d change o f v and a l s o on b o t h s i d e s o f i t , v;here T> v / ' t ) y ' r e t u r n s t o
t o a n o r m a l o r d e r o f m a g n i t u d e and v ( v / T S y ' ) can be n e g l e c t e d , we f i n d ;
1 2 1 2
~2~ ("^j - c) = {vjj - c ) = c o n s t a n t ,
where t h e s u b s c r i p t s I and I I denote t h e v a l u e s o f v on t h e two s i d e s o f r a p i d change. From t h i s : C = ( V j + V J J ) (7) Hence t h e v e l o c i t y o f advance o f a s t e e p f r o n t i s g i v e n b y h a l f t h e sum.of t h e v a l u e s o f v a t b o t h ends, A f u r t h e r i n t e g r a t i o n g i v e s : 1 ^ 1 ( V j - ) ( y - è,, ) V = - ( V j + V J J ) - - ( v j - V J J ) t a n h » (8)
By making use o f t h i s r e s u l t we can now c o n s t r u c t t h e d e v e l o p -ment o f V f r o m any i n i t i a l s t a t e g i v e n by a c h a i n o f s t r a i g h t seg-ments, a t l e a s t so l o n g as t h e r o u n d i n g o f f o f t h e a n g l e s i n conse-quence o f t h e v i s c o s i t y has n o t proceeded t o o f a r . E v e r y segment t u r n s a b o u t i t s "hinge p o i n t " a c c o r d i n g t o t h e f o r m , ( 2 ) and ( 3 ) ; a n y t i m e vjhon a downward s l o p i n g segment reaches t h e v e r t i c a l p o s i t i o n , i t does n o t t u r n any f u r t h e r b u t r e m a i n s v e r t i c a l , o b t a i n i n g a v e l o c i t y o f a d -vance g i v e n by f o r m , ( 7 ) . T h i s v e l o c i t y o f ad-vance w i l l n o t be c o n s t a n t , s i n c e t h e v a l u e s o f V j and V j j u s u a l l y w i l l change i n c o u r s e o f t i m e .
I t i s p o s s i b l e t h a t c o n s e c u t i v e v e r t i c a l segments o v e r t a k e each o t h e r . When t h i s o c c u r s we combine them t o f o r m a s i n g l e segment, moving f r o m now on w i t h a v e l o c i t y a g a i n g i v e n b y (7) p r o v i d e d V j and V J J r e f e r t o t h e v e l o c i t i e s a t t h e ends o f t h e new segment.
I t i s f o u n d t h a t e v e r y v e r t i c a l segment i s t h e s e a t o f d i s s i p a -t i o n o f energy, -t o -t h e amoun-t ( V j - V j j) V l 2 . ( i t s h o u l d be o b s e r v e d t h a t V j - V J J i s p o s i t i v e f o r e v e r y v e r t i c a l segment, as w i l l be e v i -d e n t f r o m t h e p r o c e s s b y w h i c h t h e s e segments a r e g e n e r a t e -d ) .
A l l t h e s e f e a t u r e s can be checked by making use o f t h e e x a c t s o l u t i o n o f eq, ( l ) , w h i c h can be o b t a i n e d , as xms i n d i c a t e d by J . D, Cole and V, Bargmann, b y making t h e s u b s t i t u t i o n :
w h i c h i s s i m i l a r t o t h e one used t o t r a n s f o r m t h e o r d i n a r y ( n o t p a r t i a l ) d i f f e r e n t i a l e q u a t i o n f o r V.
53, Ono may ask what these c o n s i d e r a t i o n s have t o do w i t h t h e t u r b u l e n c e p r o b l e m .
The f i r s t p o i n t , a l r e a d y n o t e d , i s t h a t e q , ( l ) has t h e two i m -p o r t a n t f e a t u r e s o f t h e hydrodynamic e q u a t i o n s ; a t y -p i c a l n o n - l i n e a r t e r m o f t h e f i r s t o r d e r and a l i n e a r t e r m o f t h e second o r d e r m u l t i -p l i e d b y a s m a l l c o e f f i c i e n t . I t i s -p o s s i b l e t o a -p -p l y t h e s i m i l a r i t y t h e o r y t o eq, ( l ) and i t i s f o u n d t h a t i t s s o l u t i o n s a r e c h a r a c t e r i z e d b y a Reynolds number formed o u t o f t h e p r o d u c t o f a t y p i c a l v e l o c i t y and a t y p i c a l l e n g t h , d i v i d e d by t h e k i n e m a t i c v i s c o s i t y v » Moreover, the s o l u t i o n s o f eq, ( l ) which v a n i s h a t i n f i n i t y s a t i s f y t h e c o n d i t i o n o f c o n s e r v a t i o n o f momentum, and an energy e q u a t i o n can be formed, ex-p r e s s i n g t h e l o s s o f energy t h r o u g h d i s s i ex-p a t i o n , (The n o n - l i n e a r t e r m of e q , ( l ) v a n i s h e s b o t h f r o m t h e momentum and f r o m t h e e n e r g y e q u a t i o n ) .
The presence o f t h e n o n - l i n e a r terra has t h e consequence t h a t f e a t u r e s o f t h e s o l u t i o n s a r e propagated w i t h a v e l o c i t y g i v e n b y t h e m a g n i t u d e o f v i t s e l f , w h i c h has t h e f u r t h e r consequence t h a t s t e e p f r o n t s can bo g e n e r a t e d v f h i c h , once formed, keep t h e i r i n d i v i d u a l i t y so l o n g as t h e y do n o t merge vilth a n e i g h b o r i n g f r o n t . T h i s p r o p e r t y can be compared w i t h t h e tendency found i n f l u i d m o t i o n t h a t masses,
h a v i n g a c q u i r e d a c e r t a i n v e l o c i t y , d i s p l a c e themselves v d t h t h i s v e l o c i t y and push a s i d e e l e m e n t s h a v i n g a s m a l l e r v e l o c i t y , i n which p r o c e s s
u s u a l l y s u r f a c e s o f i n t e n s i v e s h e a r i n g m o t i o n a r e g e n e r a t e d . S h e a r i n g m o t i o n i s n o t r e p r e s e n t e d b y t h e s o l u t i o n s o f eq„ ( l ) , b u t t h e a n a l o g y i s r e t a i n e d when we do n o t l o o k a t t h e shear i t s e l f , b u t a t t h e d i s s i -p a t i o n accom-panying i t ; a s i m i l a r d i s s i -p a t i o n i s t o be found i n each s t e e p f r o n t d e v e l o p i n g i n t h e course o f v .
S u r f a c e s o f shear f l o v j , s e p a r a t i n g masses o f l i q u i d w i t h d i f f e r -e n t v -e l o c i t i -e s , can m-erg-e t o g -e t h -e r , i n a s i m i l a r way as t h -e s t -e -e p
f r o n t s i n t h e curve d e s c r i b i n g o u r f u n c t i o n v; b o t h i n t h e case o f f l u i d m o t i o n and i n t h a t case o f eq. ( l ) t h i s p r o c e s s has t h e i m p o r t a n c e t h a t i t l e a d s t o an i n c r e a s e o f t h e s c a l e o f t h e p a t t e r n ,
A system i n t e r m e d i a t e i n i t s c h a r a c t e r between t h e c o m p l e t e h y d r o -dynamic e q u a t i o n s and e q , ( l ) i s formed b y e q u a t i o n s ( 1 3 ) c o n s i d e r e d
s h o r t l y i n s e c t i o n Z+S o f t h e p r e c e d i n g c h a p t e r . The s o l u t i o n s o f t h e system (13) a c t u a l l y e x h i b i t t h o f o r m a t i o n o f r e g i o n s o f s h e a r f l o w , w h i c h a l r e a d y a r c q u i t e comparable t o t h o s e o f a c t u a l f l u i d f l o w .
I t i s p o s s i b l e t o make a F o u r i e r a n a l y s i s o f t h e s o l u t i o n s o f eq. ( l ) , i n a s i m i l a r way as t h i s can be done i n h y d r o d y n a m i c s . I t i s t h e n f o u n d , i n b o t h cases, t h a t t h e r e i s a c o u p l i n g b e t w e e n t h e
v a r i o u s components, d e p e n d i n g upon t h e n o n - l i n e a r t e r m s o f t h e e q u a t i o n s , of such n a t u r e , t h a t t h e i n t e r a c t i o n b e t v j e o n any two components l e a d s t o t h o appearance, o r t h e i n f l u e n c i n g , o f o t h e r components, h a v i n g t h o sum or t h e d i f f e r e n c e o f t h e f r e q u e n c i e s . Hence, when we d e s c r i b e t h e system b y means o f a spectrum, t h e r e i s a t r a n s f e r o f energy b o t h up-ward and downv/ard a l o n g t h e f r e q u e n c y s c a l e . V i s c o s i t y i n b o t h cases o p e r a t e s most e f f e c t i v e l y on t h e components o f s m a l l w a v e l e n g t h ( h i g h f r e q u e n c y ) ; hence i n b o t h cases t h e r e must bo an o v e r a l l f l o w o f energy t o w a r d s t h e h i g h f r e q u e n c y end o f t h e s p e c t r i r a i . I n a l l t h e s e a s p e c t s and i n a number o f s t a t i s t i c a l f e a t u r e s , t h e s o l u t i o n s o f eq, ( l ) can h e l p t o u n d e r s t a n d r e l a t i o n s a p p e a r i n g i n a c t u a l t u r b u l e n c e . This w i l l be shown i n g r e a t e r d e t a i l i n t h e n e x t s e c t i o n s .
A t t e n t i o n may be drawn t o one f u r t h e r p o i n t : Hov; c o n v e n i e n t t h e r e s o l u t i o n o f a p a t t e r n o f f l u i d m o t i o n i n t o F o u r i e r components may be f r o m many p o i n t s o f viev;, t h e r e i s a l s o s o m e t h i n g u n n a t u r a l i n i t and t h e s t r o n g c o u p l i n g betv/een t h e v a r i o u s components duo t o t h e n o n - l i n e a r t e r m s o f t h e e q u a t i o n s p r i ^ s e n t a m a t h e m a t i c a l p r o b l e m w h i c h t h u s f a r can-n o t be s o l v e d . I t i s d i f f i c u l t , t h e r e f o r e , t o p r e d i c t t h e h i s t o r y o f acan-ny s i n g l e F o u r i e r component. The same a p p l i e s t o any o t h e r method o f r e s o -l u t i o n , based on t h e c h a r a c t e r i s t i c s o -l u t i o n s o f some -l i n e a r p a r t i a -l d i f f e r e n t i a l e q u a t i o n ( f o r i n s t a n c e , t h e e q u a t i o n used i n i n v e s t i g a t i o n s on t h e s t a b i l i t y o f l a m i n a r m o t i o n ) . On t h e o t h e r hand t h e t y p e o f s o l u -t i o n s o f eq, ( l ) c o n s i d e r e d i n -t h e p r e c e d i n g s e c -t i o n , w i -t h -t h e i r s -t e e p f r o n t s , a r e d i r e c t l y c o n n e c t e d w i t h t h e n o n - l i n e a r terra and p r e s e n t , so t o say, an " i n d i v i d u a l i z e d " c h a r a c t e r . S t e e p f r o n t s have a c e r t a i n l i f e t i m e and i t i s p o s s i b l e t o d e s c r i b e a c e r t a i n c l a s s o f s o l u t i o n s o f ( l ) b y g i v i n g a l i s t o f t h e p o s i t i o n s and s t r e n g t h s o f t h e f r o n t . The h i s t o r y o f t h e f i e l d t h e n becomes a d e s c r i p t i o n o f t h e m o t i o n o f t h o s e f r o n t s and t h e i r m e r g i n g t o g e t h e r , l e a d i n g t o a g r a d u a l d e c r e a s e o f t h e i r number
end i n c r o a s G o f t h e i r mean d i s t a n c e . Such a d e s c r i p t i o n b r i n g s i n t o e v i d e n c e t h o s e p r o p e r t i e s o f t h e f i e l d \7hich a r e dependent on t h e p r e s e n c e o f t h e n o n - l i n e a r t e r m i n t h e e q u a t i o n , T/heroas t h e F o u r i e r r e s o l u t i o n i s based p r i m a r i l y on t h o l i n e a r terms and i s o n l y p o o r l y adapted t o t h o t r e a t m e n t o f n o n - l i n e a r e f f e c t s .
I n t h o n e x t s e c t i o n s v/e s h a l l c o n s i d e r v a r i o u s p a r t i c u l a r cases o f f i e l d s d e s c r i b e d by eq. ( l ) .
g n l j j t l o n a Qf Bq « ( l ) R e p r e s e n t i j i g _ . t h e _ ^
o f I m p u l s e s i n One Direc.tion„_alo.M..^>.e-Xr4SlS
54, I n s e c t i o n 51 we have become a c q u a i n t e d w i t h a p a r t i c u l a r s o l u t i o n o f eq. ( l ) c o r r e s p o n d i n g t o an i m p u l s e o f magnitude h /2 i n t r o -duced a t t h e i n s t a n t t a t t h e p o i n t y = ^. The course o f v i s r e p r e ¬ s e n t e d by a t r i a n g l e , h a v i n g t h e a r e a h / 2 , w h i l e t h e v e l o c i t y o f a d -vance o f t h e f r o n t i s g i v e n b y one h a l f t h e h e i g h t o f t h e f r o n t . These two r u l e s c o m p l e t e l y d e t e r m i n e t h e solution»
We now i m a g i n e t h a t a s e r i e s o f i m p u l s e s o f magnitudes ^ J - ^ j t / J ^ j y ...o i s i n t r o d u c e d a t t h e p o i n t y = 0 a t i n s t a n t s t ^ , t^, ty a l l i m p u l s e s b e i n g p o s i t i v e . Each i m p u l s e l e a d s t o t h e appearance o f a t r i a n g l e and t h e v a r i o u s t r i a n g l e s a r e superposed on each o t h e r i n t h e way as i n d i c a t e d i n t h e accompanying f i g u r e . F o r any i n s t a n t t h i s f i g u r e
can be c o n s t r u c t e d b y d r a w i n g a s e t o f s t r a i g h t l i n e s s t a r t i n g f r ^ % y = 0 and h a v i n g s l o p e s g i v e n b y l / ( t - t ^ ) ; between t h e s e l i n e s i/e drav/ v e x - t i c a l segments i n such a way t h a t a t r i a n g l e w i t h t h e a r e a
OJ^ i s foriTied between t h e l i n e s v / i t h s l o p e s l / ( t - t^^^) and l / ( t - 'tj_„-]_)» \7e d e n o t e t h e p o s i t i o n o f t h e c o r r e s p o n d i n g v e r t i c a l segment b y ^ and p r o v i s i o n a l l y assume t h a t a l l s a t i s f y t h e c o n d i t i o n s <
\le t h e n have a r e p r e s e n t a t i o n o f t h e course o f v a t t h e i n s t a n t t ( h e a v y b r o k e n l i n e i n t h e d i a g T a m ) , The development i n t i m e o f t h e
c u r v e i s d e t e r m i n e d b y t w o r u l e s : a l l i n c l i n e d l i n e s t u r n t o t h e r i g h t i n c o n f o r m i t y v d t h t h e e x p r e s s i o n l / ( t - t ^ ) f o r t h e i r s l o p e ; e v e r y t r i a n g l e must r e t a i n a c o n s t a n t a r e a CO ^» One f i n d s :
2 a> . ( t - t . ) ( t •- t . J
^ = l / ~ - - t - ^
Tie assume t h a t new i m p u l s e s a r e i n t r o d u c e d a t y = 0 as t i m e advances, v/hich w i l l g i v e r i s e t o new t r i a n g l e s t o be superposed a t t h e l e f t end o f t h e c u r v e , - The c o r r e c t i o n s connected v / i t h v i s c o s i t y a r e n e g l e c t e d i n t h i s p i c t u r e , b u t we c a n assume t h a t t h e y a r e v e r y s m a l l .
The c u r v e o b t a i n e d g i v e s a crude a n a l o g y t o t u r b u l e n c e produced b y a s c r e e n i n a w i n d t u n n e l and c a r r i e d a l o n g b y t h e g e n e r a l f l o w . I n t h e p r e s e n t case t h e r e i s no s p e c i a l " c a r r y i n g f l o w " : t h e p r o p a g a t i o n a l o n g t h e y - a x i s i s d e t e r m i n e d by t h e m a g n i t u d e o f t h e i m p u l s e s them-s e l v e them-s , ITe them-suppothem-se t h a t a t y = 0 i m p u l them-s e them-s a r e c o n t i n u o u them-s l y i n t r o d u c e d i n a random manner, b u t so t h a t i t i s p o s s i b l e t o d e f i n e a mean v a l u e
o f t h e 6 0 and a mean v a l u e 0 o f t h e t i m e i n t e r v a l s T. = t . - t . -.,
1 1 1 1—X
and a l s o a mean v a l u e o f t h e r a t i o (a.^ -^^^^^ p i c t u r e t h e n o b t a i n e d has a s t a t i o n a r y s t a t i s t i c a l c h a r a c t e r f o r e v e r y g i v e n p o i n t o f t h e y - a x i s ; i t changes g r a d u a l l y as we go a l o n g t h e y - a x i s .
The s u p p o s i t i o n t h a t a l l s a t i s f y < ^^j^ c a n n o t be u p h e l d i n g e n e r a l . The v e l o c i t i e s o f advance o f t h e v a r i o u s v e r t i c a l f r o n t s w i l l be u n e q u a l and whenever we f i n d d ^ ^ / d t ^ d fep ^_j/'^'^> t h e r e w i l l bo a d e c r e a s e o f t h e d i s t a n c e è-i, . ^ - ï . . '^hen t h i s has
" ^ x - 1 ^ 1 become z o r o , t h e v e r t i c a l f r o n t ^ has o v e r t a k e n t h e segment
f r o m t h e n onv/ard we must c o u n t t h e s e f r o n t s a s a s i n g l e one. T h i s r e q u i r e s t h a t f r o m nov; onvrard we l e a v e o u t t h e i n c l i n e d l i n e v.dth t h e s l o p e l / ( t - t_^ j ) and vjork as i f t h e combined i m p u l s e OJ ^ + cO j_ -j_ had been i n t r o d u c e d a t t h e i n s t a n t t ^ , so t h a t f r o m now on i t f o l l o v ^ s t h e i m p u l s e . ^ i n t r o d u c e d a t t . „,
T h i s p r o c e s s l e a d s t o a g r a d u a l d e c r e a s e o f t h e number o f f r o n t s . S i n c e new f r o n t s a r e c o n t i n u a l l y i n t r o d u c e d a t y = 0, t h e s t a t i s t i c a l s t a t e o f t h e system can b e s t a t i o n a r y .
ÏÏe now come b a c k t o t h e r u l e f o r c o n s t r u c t i n g t h e s t a t e o f t h e system a t a g i v e n i n s t a n t t o V,Tienever i n c a r r y i n g o u t t h e r u l e s g i v e n b e f o r e , we f i n d a case where S> ^ > ^ '^-'^^^^ '^^'^ i n s t a n t
t ^ J and combine 6 0 ^ and <^ ^^j^ i n t o a s i n g l e i m p u l s e c o r r e s p o n d i n g
t o t h e i n s t a n t t . o I f we s h o u l d f i n d t h a t t h o new v a l u e o f would s u r p a s s ^ j_ 2' ° ^ v/ould be s m a l l e r t h a n ^ i + i ' p r o c e s s has t o bo r e p e a t e d , u n t i l a l l cases w h i c h d i d n o t s a t i s f y t h e c o n d i t i o n
^ . < ^ . -, have been eliminated» The r e s u l t o f t h i s e l i m i n a t i o n i s n o t dependent on t h o o r d e r i n I'fhich i t i s c a r r i e d o u t ,
A p a r t i c u l a r case i s o b t a i n e d when a l l i m p u l s e s a r e e x a c t l y o f t h e same m a g n i t u d e '^O and a r e spaced a t e x a c t l y e q u a l i n t e r v a l s o f t i m e 0, I n t h a t case m^erging o f f r o n t s w i l l o c c u r o n l y a t t h e f o r e m o s t end o f t h e sequence. I f t h e pjrocess has boon g o i n g on f o r a l o n g t i m e , t h i s end w i l l have moved f a r o u t t o t h e r i g h t , and i n c e r t a i n cases can be c o n s i d e r e d as h a v i n g d i s a p p e a r e d f r o m t h o f i e l d under o b s e r v a t i o n . The v e l o c i t y o f advance o f a l l o t h e r v o r t i c a l f r o n t s w i l l t h e n be v e r y n e a r l y e q u a l t o /0. - I t s h o u l d be observed t h a t v/hen t h e i n i t i a l i m -p u l s e s v/ore n o t e x a c t l y e q u a l , o r when t h e r e s h o u l d have been s l i g h t i n e q u a l i t i e s o f t h e i r s p a c i n g i n t i m e , t h e s e i n e q u a l i t i e s v/ould n o t d i s a p p e a r i n t h e p r o c e s s , b u t v/ i l l l e a d t o t h e merging o f f r o n t s some-where i n s i d e t h e s e r i e s . T h i s i s a t y p i c a l f e a t u r e c h a r a c t e r i s t i c f o r systems v/hich o r e g o v e r n e d by a n o n - l i n e a r e q u a t i o n , b e l o n g i n g t o t h e h y p e r b o l i c t y p e ( h a v i n g r e a l c h a r a c t e r i s t i c s ) , where t h e r e i s no smoothing o u t o f i r r e g u l a r i t i e s i n t r o d u c e d by t h e b o u n d a r y c o n d i t i o n s o r by t h e i n i t i a l c o n d i t i o n s .
I n t h o b e g i n n i n g o f t h i s c h a p t e r i t had been remarked t h a t eq, (1) i s a l s o i l l u s t r a t i v e o f phenomena p e c u l i a r t o c o m p r e s s i b l e media, Tho s o l u t i o n we have been c o n s i d e r i n g can bo t a k e n as a s i m p l i f i e d p i c t u r e o f t h o b e h a v i o r o f a s e r i e s o f p l a n e shock waves, i n t r o d u c e d one a f t e r t h e o t h e r i n t o a gas. R e a l shock waves v/ i l l show t h e same f e a t u r e o f o v e r t a k i n g one a n o t h e r and o f m e r g i n g t o g e t h e r , as do t h e s t e e p f r o n t s f o u n d i n o u r s o l u t i o n o f e q . ( l ) .
55. S t a t i s t i c a l P r o b l e m s Connected w i t h t h e P r o p a g a t i o n o f ... C o n s e c u t i v e I m p u l s e s . I f t i s chosen so t h a t f o r a group o f consecut i v e i m p u l s e s consecut consecut ^ i s l a r g e compared w i consecut h consecut h e consecut i m e i n consecut e r v a l s m consecut h -i n t h -i s group, we can r e p l a c e ( 9 ) by t h e a p p r o x -i m a t -i o n :
^ i ^ ^ • \ ^ i ^ i ) (10)
where u. = "\l2 OJ Jl. ,
I n t h i s case t h e c o n d i t i o n | y ^ < ^^^-|_ i s n o t c o m p a t o b l e m t h
a l a r g e p o s i t i v e d i f f e r e n c e u^ - •u^__-[_<. fJhenever ' ^ i <^an f i n d a v a l u e o f t f o r ?jhich t h e c o n d i t i o n • < • w i l l 'be v i o l a t e d
so t h a t we must combine i m p u l s e s i n t h e way as descrj.bed b e f o r e . S i n c e we have assumed t h a t t h e system has a s t a t i s t i c a l l y s t a t v ^ o ^ y c h a r a c t e r we can expect t h a t i n t h e course o f t i m e a l l a p p r e c i a l & ^ ^ d i f f e r e n c e s between c o n s e c u t i v e u^ w i l l be e l i m i n a t e d i n t h i s way. There m i g h t r e m a i n l a r g e n e g a t i v e d i f f e r e n c e s u^ - ^x-X °^ s e r i e s w i t h c o n s e c u t i v e n e g a t i v e d i f f e r e n c e s . B u t i n v i e w o f t h e randomness o f t h e v a l u e s o f t h e u ^ , a l a r g e v a l u e o f u ^ _ j ^ would have g r e a t chance t o be f o l l o w e d by a s m a l l e r one, and t h e n e l i m i n a t i o n a g a i n vrould bo n e c e s s a r y .
To make t h e s e c o n s i d e r a t i o n s more p r e c i s e , we observe t h a t when two v e r t i c a l f r o n t s merge t o g e t h e r i n t o a s i n g l e one, t h e v a l u e s o f t h e
OJ ^ a r e added, so as a l s o a r e t h e v a l u e s o f t h e i n t e r v a l s T. , The new v a l u e o f u^ r e s u l t i n g f r o m t h e p r o c e s s i s g i v e n b y :
V
\ ' ^ i - V l . ' \ ' ^ i - 1
I.-: now we d e f i n e a mean s q u a r e v a l u e o f t h e u^ b y means o f t h o f o r m u l a : 2 ^ ^ i
U ^ » ^ ^ i
where t h e summation i s extended o v e r a s e r i e s o f c o n s e c u t i v e i - v a l u e s , i t f o l l o w s t h a t t h i s mean square v a l u e i s n o t a f f e c t e d b y t h o p r o c e s s o f m e r g i n g t o g e t h e r o f f r o n t s and t h a t i t i s a c o n s t a n t . I t i s e a s i l y seen t h a t t h i s c o n s t a n t has t h e v a l u e 2 / 9 ,
On t h e o t h e r hand, s i n c e :
u.
1 1 ( V i \ - l ^ i - l ) ' - ^ " \ - . l ) ' ' (12) i t i s seen t h a t 5^ " ' i ' ^ i i i ^ ^ r e a s e s each t i m e two f r o n t s merge. Hence t h e ( l i n e a r ) mean v a l u e u o f t h e u^ i n c r e a s e s . I t t h u s f o l l o w s t h a t t h e f l u c t u a t i o n s o f t h e u. t e n d t o become s m a l l e r and s m a l l e r as t i m e goes o n .
T h i s can be p u t i n t e a s l i g h t l y d i f f e r e n t f o r m i f v/e w r i t e ;
u. = U ( l + 6 ^ ) . (13)
Wo t h e n have f o r t h e mean v a l u e o f t h e square o f 1 + 6 ^ 5
( 1 + ö.)2 = 1 ,
f r o m w h i c h ;
25 = - d^.
Hence ö 0 and u U. Since u i n c r e a s e s , ö i n c r e a s e s t o w a r d s 2 zero and ö d e c r e a s e s . T h i s i m p l i e s t h a t t h e a b s o l u t e v a l u e s o f t h e 0^ g r a d u a l l y become s m a l l e r , A consequence o f t h e s e c o n s i d e r a t i o n s i s t h a t , i f we have s t a r t e d w i t h t w o i n d e p e n d e n t d i s t r i b u t i o n s f o r t h e v a l u e s o f t h e U) . and t h o T. t h e s e v/ i l l n o t r e m a i n i n d e p e n d e n t o f each o t h e r . I f t h e y a r e r e p r e s e n t e d s i m u l t a n e o u s l y by moans o f p o i n t s i n an CJ ., T. - diagram, t h e combi¬ n a t i o n s o c c u r r i n g e v e r y t i m e v/hen two f r o n t s merge, w i l l l e a d t o t h e appearance o f b o t h l a r g e r Oü ^ - v a l u e s and l a r g e r T.- v
such a way t h a t t h e r a t i o s OJ ^/T. w i l l c l u s t e r more and more c l o s e l y r o u n d t h e mean v a l u e ^ / 9 ,
56. A p p l d c a t i o n o f S i m i l a r i t y Considerations» We v/ i l l f o l l o w t h e h i s t o r y o f a group, o r i g i n a l l y formed f r o m i m p u l s e s o f t o t a l amount S~l , i n t r o d u c e d a t y = 0 i n a p e r i o d o f d u r a t i o n ® » When v/e
d e c r c a s e d t o say and t h e mean v a l u e s U) and 9 a r e g i v e n b y ; 'ÖJ = J l / u ; e = (B> /N.
When t h o m e r g i n g p r o c e s s has a l r e a d y gone so f a r t h a t t h e 0^
a r e s m a l l compared w i t h u n i t y , i t f o l l o w s f r o m (12) t h a t when t?jo f r o n t s merge ( w h i c h makes E d e c r e a s e v d t h l ) t h o sum 2. u^ ^ c r e a s e s b y a p p r o x i m a t e l y ;
2(u. T. . u.„j T . _ j )
I t seems f a i r l y s a f e t o assume t h a t t h e mean v a l u e o f t h i s q u a n t i t y i s g i v e n hyi 1 2 U 9 0"^ 1 ^2 A w h i c h t h e r e f u r t h e r f o l l o w s ; Honce t h e mean v a l u e u i n c r e a s e s on t h e a v e r n g e b y 2 ^ ^^'-''^ average i n c r e a s e o f S = ^ ö^/N and: average d e c r e a s e o f Ö = ^ö./ M I f v;e assume t h a t a c e r t a i n s t a t i s t i c a l s i m i l a r i t j ' " i s conserved i n t h e p r o c e s s , we must w r i t e t h i s i n t h e fox-m; ~2 decrease o f Ö _ decrease o f N H N and f i n d : 0^ ^ N . (14) iaètant'^b v i r i s g i v e n b y ;
Wow t h e i n S t a n t ^ c t w h i c h '£> ^ and '£3, j_ j merge, ( t h i s nas t h e i - \ - r
,* , , 1 rp. ^ 1 - 1 ' 1 1-1 _ .JL ..1,1 1 - 1 * " 1 2 " 2 ( u . - u. J " T T ö . - Ö. J
1 1 - 1 1 1 - 1
v/e can t a k e t h i s e x p r e s s i o n as an e s t i m a t e o f t h o "age" o f t h e group under consideration„ S i n c e t h e average v a l u e o f «5. - 6. T
1 1 - 1 ^ 25'~ , we o b t a i n : average age o / "\ I n t h i s way ve a r r i v e a t t h o r e s u l t s : - 2/3 N ''-•^ ( a g e ) ~ — - 3/2 20^ CO (E) 11 I S (15) Ö'^to ( a g e ) 2/3 I t a l s o f o l l o v j s t h a t ! (a) and T N" ( a g e ) 2/3 (16) (17)
The dependence on t h e age can be t r a n s f o r m e d i n t o a dependence on t h e moan p o s i t i o n o f t h e group a l o n g t h o y - a x i s , i f ve o b s e r v e t h a t a c c o r d i n g t o ( l O ) : r- , '-2 U. ( a g e ) , (18) 57» An o b s e r v e r l o c a t e d a t a f i x e d v a l u e o f y w i l l see t h e b r o k e n c u r v e r e p r e s e n t i n g t h e c o u r s e o f v pass o v e r h i m w i t h a v e l o c i t y a p p r o x i -m a t e l y e q u a l t o ü» H i s o b s e r v a t i o n s w i l l show a slow i n c r e a s e o f v d u r i n g p e r i o d s T^, a l t e r n a t i n g w i t h a sudden d e c r o a s e . The m a g n i t u d e o f t h e decrease, observed a t t h e i n s t a n t v;hen t h e f r o n t ^ passes t h e o b s e r v e r , i s g i v e n b y : L ^ i _ 2 ^ T^ C O U ( t - 1 . ) ( t : t . _ ^ 2U> .T. 1 1 t . + 1 T. 1 ~ 1 (19)
From t h i s e x p r e s s i o n i t a p p e a r s t h a t so l o n g as m e r g i n g o f f r o n t s does c o n s i d e r e d i n s e c t i o n 52) d e c r e a s e i n v e r s e l y p r o p o r t i o n a l l y w i t h t h e age, and c o n s e q u e n t l y a l s o i n v e r s e l y p r o p o r t i o n a l l y w i t h t h e d i s t a n c e o f t h e o b s e r v e r f r o m t h e o r i g i n on t h e y-axis» However, e v e r y m e r g i n g o f two s u c c e s s i v e f r o n t s i n t r o d u c e s an i n c r e a s e o f -CJ ^ and t h e d e c r e a s e o f t h e average v a l u e o f t h e v ^ c o n s e q u e n t l y w i l l be slov/ero From (17) ¥/e see t h a t t h e n u m e r a t o r i n c r e a s e s p r o p o r t i o n a l l y w i t h ( a g e ) ^ hence we f i n d :
C e r t a i n o t h e r p r o b l e m s can be t a k e n i n v i e w connected w i t h t h e v -c u r v e p a s s i n g o v e r a s t a t i o n a r y o b s e r v e r . S i n -c e t h e p r o -c e s s s h o u l d be s t a t i o n a r y i n t h e s t a t i s t i c a l sense, we can c o n s i d e r t h e E u l e r i a n t i m e -c o r r e l a t i o n v ( t ) v ( t + ' i : ) f o r a f i x e d v a l u e o f y. I t -can be e s t i m a t e d t h a t t h i s c o r r e l a t i o n w i l l becomie s e r o v/hen T exceeds a fev/ t i m e s Q.
We can a l s o f i x a t t e n t i o n t o a p a r t i c u l a r p o i n t o f t h e c u r v e f o r v . Such a p o i n t d i s p l a c e s i t s e l f v/ i t h o u t change o f h e i g h t w i t h i t s v e l o c i t y V, u n t i l t h e i n s t a n t where t h e tv/o f r o n t s betv/een w h i c h i t i s s i t u a t e d , happen t o merge. The i n t e r v a l o f t i m e f r o m t h e i n s t a n t a t v/hich t h e
c o r r e s p o n d i n g i m p u l s e s v/ere i n t r o d u c e d u n t i l t h e i n s t a n t a t v/hich t h e f r o n t s merge, i s o f t h e o r d e r o f t h e age o f t h e group t o w h i c h t h e f r o n t s b e l o n g . S i n c e t h i s age i s much l a r g e r t h a n Q, an i n d i v i d u a l p o i n t o f t h e c u r v e , f o l l o v / e d i n i t s m o t i o n , w i l l keep i t s v e l o c i t y u s u a l l y o v e r a l o n g p e r i o d . Hence a L a g r a n g i a n c o r r e l a t i o n c a l c u l a t e d f o r t h e m o t i o n o f i n d i v i d u a l p o i n t s w i l l be d i f f e r e n t f r o m z e r o over a much l o n g e r i n t e r v a l t h a n t h e E u l e r i a n t i m e c o r r e l a t i o n .
When t h e age o f a group i s l a r g e , and t h e group s t i l l c o n t a i n s many members, i t i s a l s o p o s s i b l e t o c a l c u l a t e an E u l e r i a n space c o r r e l a t i o n v ( y ) v ( y + 'f^ ) f o r t h e group, w h i c h q u a n t i t y w i l l be a f u n c -t i o n o f -t i m e . These E u l e r i a n space c o r r e l a -t i o n s , however, can be b e -t -t e r i n v e s t i g a t e d i f we t u r n t o a d i f f e r e n t t y p e o f s o l u t i o n o f e q , ( l ) , i n w h i c h t h e b r o k e n curve g i v i n g t h e c o u r s e o f v i s formed o f p a r a l l e l up-ward s l o p i n g segments ( i n s t e a d o f segments a l l s t a r t i n g f r o m t h e p o i n t
I I
y = o ) . Such a s o l u t i o n can bo o b t a i n o d f r o m t h o one c o n s i d e r e d h e r e b y means o f a l i m i t i n s p r o c e s s , b u t i t i s more c o n v e n i e n t t o d e f i n e i t i n an i n d e p e n d e n t viaVo 1 2 I t i s u s e f u l t c n o t i c e t h e q t t a n i t y ^ ^ •• U f o r t h e s o l u t i o n t h u s f a r c o n s i d e r e d . T h i s q u a n t i t y r e p r e s e n t s t h e d i f f e r e n c e 4 ^
i n a r e a between t h e t w o t r i a n g l e s OAB and OCD connected w i t h t h e i n t e r -v a l • ' ^ i ~ \ X» 1'^*^ -v a l u e can be p o s i t i -v e o r n e g a t i -v e and t h e mean V a l u e i s z e r o . Since
W ^ - I U^T. = T^ ( Ö. + I 5.2) ^ U^T. 0^ ,
we must e x p e c t t h a t t h e a v e r a g e a b s o l u t e V a l u e w i l l i n c r e a s e p r o p o r t i o n a w i t h ( a g e ) ^ .
S o l u t i o n s o f Ego ( l ) I l l u s t r a t i n g , SpatiaJjLY Homogeneous T u r b u l e n c e D e c a y i n g w i t h Time
58o ITe c o n s i d e r a s o l u t i o n o f eq<, ( l ) i n which t h e i n i t i a l f o r m of t h e c u r v e f o r v i s g i v e n b y a s e r i e s o f s t r a i g h t p a r a l l e l segments, i n consequence o f f o r m u l a s ( 2 ) and ( 3 ) o f s e c t i o n 50, t h e p a r a l l e l i s m w i l l be r e t a i n e d d u r i n g t h e whole h i s t o r y , though t h e m a g n i t u d e o f t h e s l o p e \ 7 i l l decrease. I t i s c o n v e n i e n t t o t a k e t h e s l o p e e q u a l t o l / t , o m d t t i n g t h e u n i m p o r t a n t c o n s t a n t t ^ . The upward s l o p i n g segments a r e a g a i n s e p a r a t e d b y v e r t i c a l f r o n t s . I f a c c o u n t must be t a k e n o f t h e i n f l u e n c e o f v i s c o s i t y , a m.ore c o r r e c t e x p r e s s i o n f o r t h e c o u r s e o f v i n t h e s e f r o n t s can be g i v e n hy making use o f eq„ (ö) o f s e c t i o n 52. I f we i n t r o d u c e t h e n o t a t i o n which i s i l l u s t r a t e d i n t h e accompanying d i a g r a m , i t v d l l be seen t h a t t h e h e i g h t o f a v e r t i c a l f r o n t i s measured •by y _ v-^.^ ^ T: ./t, We change t h e f i r s t t e r m o f ( 8 ) i n s u c h a v/ay
t h a t a c c o u n t i s t a k e n o f t h e s l o p e o f t h e c u r v e t o t h e l e f t and t o t h e r i g h t o f t h e f r o n t and o b t a i n s y iS V t V = 2 t (21)
V
A t t h e sarne t i m e :
d t t * ^ ^ The d i s t r i b u t i o n o f t h e l e n g t h s T ^ and A ^ t o a l a r g e e x t e n t can be chosen a r b i t r a r i l y a t t h e i n i t i a l i n s t a n t , p r o v i d e d t h e d i s t r i b u ¬ t i o n i s s u f f i c i e n t l y homogeneous i n o r d e r t h a t mean V a l u e s f and X s h a l l e x i s t . These mean v a l u e s a r e o b t a i n e d b y t a k i n g a c e r t a i n number M o f c o n s e c u t i v e nr ^ o r X ^ and d i v i d i n g t h e i r sum b y N; t h e r e -s u l t mu-st approach t o a d e f i n i t e l i m i t when N i n c r e a -s e -s more and more, w h i c h l i m i t s h o u l d be i n d e p e n d e n t o f t h e chodce o f t h e s t a r t i n g p o i n t . The mean v a l u e s % and X moreover s h a l l be e q u a l . I n consequence o f t h e c i r c u m s t a n c e t h a t t h e development o f t h e system i n t h e c o u r s e o f t i m e l e a d s t o t h e m e r g i n g o f f r o n t s , t h e r e can a r i s e a c e r t a i n i n t e r -dependence between t h e d i s t r i b u t i o n s o f t h e T ^ and X ^, and we s h a l l see l a t e r ( s e c t i o n 65.) t h a t such an i n t e r d e p e n d e n c e may be o f i m p o r t a n c e ,
I n t h e c a l c u l a t i o n s we s h e l l a l s o have t o do w i t h mean v a l u e s o b -t a i n e d b y i n -t e g r a -t i n g a q u a n -t i -t y w i -t h r e s p e c -t -t o y over a c e r -t a i n
l e n g t h S o f t h e y - a x i s and d i v i d i n g b y t h i s l e n g t h ; such mean v a l u e s m i l be d e n o t e d by means o f a s i m p l e b a r , as f o r i n s t a n c e y. I t i s o f t e n
c o n v e n i e n t t o make S = N o
There i s s t i l l a f r e e d o m i n t h e s y s t e m : t h e r e l a t i v e s i t u a t i o n of t h e t¥Jo s e r i e s o f p o i n t s ör^ and , YJe assume t h a t t h i s i s d e t e r m i n e d i n such a way t h a t t h e s t a t i s t i c a l p r o p e r t i e s o f t h e system do n o t change when t h e d i r e c t i o n o f y and t h e s i g n o f v a r e s i m u l t a n e o u s l y changed. The mean v a l u e v o f v v d t h r e s p e c t t o y v d l l t h e n b e a e r o .
The p r o p e r t i e s o f h o m o g e n e i t y and o f s t a t i s t i c a l i n v a r i a n c e v d t h r e s p e c t t o s i m u l t a n e o u s change o f o r d e r and o f t h e s i g n o f v j u s t men-t i o n e d , r e m a i n v a l i d men-t h r o u g h o u men-t men-t h e d e v e l o p m e n men-t o f men-t h o sysmen-tem i f men-t h e y have e x i s t e d a t some i n i t i a l i n s t a n t . T h i s i s a consequence o f t h o i n -v a r i a n c e o f t h e d i f f e r e n t i a l e q u a t i o n , b o t h -v d t h r e s p e c t t o a s h i f t a l o n g t h e y - a x i s and w i t h r e s p e c t t o a s i m u l t a n e o u s change o f s i g n s o f V and y ,
C o n c e r n i n g t h o development o f t h e system i n t h e c o u r s e o f t i m e
move a c c o r d i n f t o eq„ (22); i t f o l l o w s t h a t t h e l e n g t h s X ^ a r e f u n c t i o n s o f t i m e and d t t I t i s c o n v e n i e n t t o i n t r o d u c e q u a n t i t i e s ^ ^ d e f i n e d b y : t h e n : d t d t t ^ d t t
The ^ ^ can be p o s i t i v e as w e l l as n e g a t i v e , and t h e mean v a l u e ^ must be z e r o .
The p r o p e r t y o f s t a t i s t i c a l i n v a r i a n c e w i t h r e s p e c t t o a s i m u l t a n e o u s change o f s i g n o f v and y, r e f e r r e d t o b e f o r e , can a l s o be e x p r e s s e d b y t h e r u l e t h a t any s t a t i s t i c a l q u a n t i t y f o r m e d o u t o f t h e j _ ' j_ ^'^^
^ . w i l l r e m a i n unchanged when t h e s i g n s o f a l l t h e ^ . a r e changed s i m u l t a n e o u s l y w i t h a change o f t h e d i r e c t i o n i n which i i s c o u n t e d , no change o f s i g n b e i n g made i n T . and X . . F o r i n s t a n c e :
1 1 1 1-1' 1 ^ 1 i ^ 1 - 1 ' 1 > i
IThen two c o n s e c u t i v e ( s a y andC,^) become e q u a l t o each o t h e r , t h e p o i n t ^ ^ _ j and t h e segment X^ d i s a p p e a r f r o m t h e a r r a n g e m e n t . The v a l u e s o f TT _ ^ _ j and ^ a r e added and t h e lavi o f m o t i o n o f t h e r e s u l t i n g segment i s a g a i n d e t e r m i n e d b y ( 2 2 ) , p r o v i d e d we r e p l a c e x . b y T . + t . „ By t h i s p r o c e s s b o t h t h e number o f segments T . and t h a t o f segments X . a r e g r a d u a l l y reduced i n t h e course o f t i m e , 59. S i m p l e Mean V a l u e s . We v / r i t e : r — 1 r—-1 • t = A . ,£ (243.) and f u r t h e r : T:'" = - 1 ^ ( 1 + C . J ) - •^li\i,J);rt^ ^ = (24b) To c a l c u l a t e t h e mean v a l u e o f v ( w h i c h i s zero, as m e n t i o n e d b e f o r e ) ,
we o b s e r v e t h a t v changes l i n e a r l y w i t h y o v e r a segment X^| hence f o r a s i n g l e segment:
f^e suiia t h i s e x p r e s s i o n w i t h r e s p e c t t o i o v e r a l l segments c o n t a i n e d i n a p a r t o f t h e y - a x i s o f g r e a t l e n g t h S| t h e sum must be d i v i d e d b y S = N X , where K i s t h e number o f segments X ^ c o n t a i n e d i n S (and a l s o t h e number o f segments t . c o n t a i n e d i n S ) , H a v i n g r e g a r d t o t h e f 1 r e l a t i o n T . > , = 0, i t i m m e d i a t e l y f o l l o w s t h a t v = 0. 2 To f i n d t h e mean v a l u e o f v a s i m i l a r p r o c e d u r e i s a p p l i e d . I n t e g r a t i o n o v e r t h e l e n g t h o f t h e segment X ^ g i v e s :
Hence t h e mean v a l u e becomes:
^ 2
~ ' ~- j 7 ( T i 7 * ï i - ^ = - f - { ' ^ ^ i r ) } •
I t has been m e n t i o n e d i n s e c t i o n 52 t h a t t h e d i s s i p a t i o n i s t o be f o u n d o n l y i n t h e v e r t i c a l segments, each segment g i v i n g a c o n t r i b u t i o n
1 3 w h i c h i n t h e p r e s e n t n o t a t i o n i s e q u a l t o z-^r • 1^" • I't f o l l o w s ^ t h a t -L/C 1 t h e mean d i s s i p a t i o n o f e n e r g y i n u n i t t i m e p e r u n i t l e n g t h o f t h e y - a x i s i s g i v e n b y : € = — - ^ - ^ 2 = ( 1 + W ( 2 6 ) 12 X t - ^ 12 t ^
I n consequence o f t h e m e r g i n g o f f r o n t s t h e number N o f segments c o n t a i n e d i n a c o n s t a n t g r e a t l e n g t h S d e c r e a s e s and i n c r e a s e s v d t h t i m e . I t i s a l s o p o s s i b l e t h a t t h e v a l u e s o f t h e dimensionlcs-s
p a r a m e t e r s OJ , {jtj'^\ ^UJ ^ v d l l be f u n c t i o n s o f t . I f t h e arrangement r e t a i n s a s t a t i s t i c a l s i m i l a r i t y , t h e s e p a r a m e t e r s w i l l be c o n s t a n t s .
So l o n g as we do n o t know hovv X changes v d t h t i m e , v e r y l i t t l e can be s a i d a b o u t t h e b e h a v i o r o f v ^ a n d Ê. , 60. Q u a n t i t i t e s Connected v d t h t h e Momentum I n t e g r a l , We w r i t e : = . / t ( 2 7 ) E v e r y q u a n t i t y c o r r e s p o n d s t o a segment ^ and t o a v e r t i c a l f r o n t w i t h p o s i t i o n W i t h t h e a i d o f t h e r e l a t i o n s i n d i c a t e d i n t h e d i a g r a m o f s e c t i o n 58, i t can e a s i l y be p r o v e d t h a t i s t h e a l g e b r a i c
v a l u e o f t h e a r e a ABODE
c o n t a i n e d between t h e two s l o p i n g l i n e s c o n n e c t e d w i t h t h e segment T . ( c o u n t i n g iiBG p o s i t i v e and ODE n e g a t i v e ) . For t h e segment i . , i n t h e d i a g r a m g i v e n above, where t h e s i t u a t i o n i s somevYhat d i f f e r e n t , t h e c o r r e s p o n d i n g Oj.uantity vjould be t h e a r e a A'B'G'A, counted p o s i t i v e , The q u a n t i t i e s M., w h i c h t h u s can be p o s i t i v e as w e l l as n e g a t i v e , a r e 1 2 t h e a n a l o g u e s o f t h e q u a n t i t i e s . - - r - U Ï. c o n s i d e r e d a t t h e end o f s e c t i o n 57. I t V f i l l be seen t h a t t h e i n t e g r a l y+S V d y e x t e n d e d o v e r a c e r t a i n l e n g t h S o f t h e y - a x i s w i l l be e q u a l t o N-1 k=0
vjhore t h e sum r e f e r s t o t h e v a l u e s o f f o r a l l segments nr ^ c o n t a i n e d i n S, virhile A s t a n d s f o r t h e a d d i t i o n a l p a r t s a p p e a r i n g a t t h e ends o f t h e i n t e g r a t i o n i n t e r v a l *
S i n c e d ^ ^ / d t - ^^/t, f r o m which i t f o l l o w s t h a t t h e ^ ^ a r e p r o p o r t i o n a l v d t h t , t h e a r e i n d e p e n d e n t o f t h e t i m e , Wïienever two
^ ^ come t o c o i n c i d e n c e and t h e c o r r e s p o n d i n g v e r t i c a l f r o n t s merge i n t o a s i n g l e f r o n t , t h e jU^ c o r r e s p o n d i n g t o t h e s e f r o n t s a r e added. Hence ^ iu.^ does n o t change^ T h i s r e s u l t i s e v i d e n t l y .connected v d t h t h e p r o p e r t y o f t h e momentum i n t e g r a l , m e n t i o n e d i n s e c t i o n 5 2 .
17e nov.r c o n s i d e r t h e moan v a l u e M o f t h e q u a n t i t y ;
- i - ( s , , ) k=0
w h i c h e v i d e n t l y i s i n d e p e n d e n t o f t h e t i m e . I f t h e would be com-p l e t e l y i n d e com-p e n d e n t o f each o t h e r , t h e mean v a l u e would r e d u c e t o :
i———)
I t i s p o s s i b l e , however, t h a t r e l a t i o n s e x i s t between s u c c e s s i v e [l., so t h a t t h e mean v a l u e s may be d i f f e r e n t f r o m z e r o , lïe can
e x p e c t n e v e r t h e l e s s t h a t t h e s e mean v a l u e s r a p i d l y v d l l a p p r o a c h zero when k i n c r o a s o s . Hence, v/hon N i s l a r g e , we may w r i t e :
1 '
^'^ = - £ \ + 2 -Z M. M-^+^kV ( 2 8 )
w h i c h f o r m u l a c o n t a i n s t h e p r e c e d i n g one as a s p e c i a l case.
I f t h e a r r a n g e m e n t s h o u l d r e t a i n a s t a t i s t i c a l s i m i l a r i t y ( w h i c h i n i t s e l f i s n o t q u i t e c e r t a i n , we s h a l l see a f t e r w a r d s ) , an iïüportant argument c o u l d be deduced f r o m t h e f a c t t h a t M i s i n d e p e n d e n t o f t h o t i m e , ^ J f t h e r e would be s t a t i s t i c a l s i m i -l a r i t y , mean v a -l u e s -l i k e ijF and 7 ^ 1 7 7 iTOij-ld be p r o p o r t i o n a -l t o t h o f o u r t h power o f t h e mean l e n g t f e ^ j f t h e segments T . , d i v i d e d by t h e s q u a r e o f t h e t i m e t . Hence M would become ^ p r o -p o r t i o n a l t o t h e t h i r d -power o f £ d i v i d e d b y t ^ , and s i n c e t h i s must be a c o n s t a n t , i t f o l l o w s t h a t must i n c r e a s e a c c o r d i n g t o t h e f o r m u l a :
As UJ , OJ , <^~> vrould be c o n s t a n t s , i t f u r t h e r f o l l o w s f r o m eqs. ( 2 5 ) and ( 2 6 ) t h a t :
Theso r e s u l t s vrould be i n a c c o r d a n c e v/ i t h t h o s o o b t a i n e d i n s e c t i o n s 56 and 57 f o r t h e p r e c e d i n g t y p o o f s o l u t i o n s (see oqso 17 and 2 0 ) ,
Even i f t h e r e s h o u l d be s i m i l a r i t y , t h e argument w o u l d brea.k down i f would b e z e r o . T h i s can occur v/here t h e r e
^ e x i s t c e r t a i n r e l a t i o n s betv;een t h o mean v a l u e s /Li.M. and o f such n a t u r e t h a t t h e mean v a l u e o f ( ^ ' ^ i + k ^ ^ vrould n o t i n c r e a s e p r o p o r t i o n a l l y v / i t h N, b u t a t a s l o w e r r a t e o r w o u l d approach t o a c o n s t a n t V a l u e .
W i t h r e f e r e n c e t o t h i s p o i n t i t i s o f i m p o r t a n c e t o men-t i o n men-t h a men-t men-t h e s o l u men-t i o n s v/e have been c o n s i d e r i n g can be o b t a i n e d f r o m a n i n i t i a l s t a t e i n which a s e r i e s o f c o n c e n t r a t e d i m p u l s e s , each o f a f i n i t e i n t e g r a t e d magnitude A , i s i n t r o d u c e d a t an
i n f i n i t e s e r i e s o f p o i n t s o f t h e y - a x i s , a r b i t r a r i l y spaced, b u t so t h a t t h e d i s t r i b u t i o n i s s t a t i s t i c a l l y homogeneous. The A can be p o s i t i v e o r n e g a t i v e and f o l l o w each o t h e r a t random,
v/ h i l e t h e mean v a l u e o f t h e A o v e r a n y l a r g e domain o f t h e y
-a x i s s h -a l l be z e r o . I t c-an tSen be p r o v e d t h -a t t h e IJ.^ -a r e e i t h e r e q u a l t o c e r t a i n A^ o r a r e e q u a l t o sums o f c o n s e c u t i v e A ( i n
conseciuenco o f t h e m e r g i n g t o g e t h e r o f c o n s e c u t i v e i m p u l s e s ) , so t h a t 21, /^.ij-v 0^0^ P^J^y l e n g t h S o f t h e y - a x i s i s e q u a l t o y\ A
X "^it —^ m
f o r t h e same l e n g t h . Hence i f t h o random d i s t r i b u t i o n o f t h e A^ i s s u b j e c t e d t o t h e c o n d i t i o n t h a t t h e moan v a l u e o f ( "E- A ) w i l l n o t I n c r e a s e p r o p o r t i o n a l l y v / i t h t h e l e n g t h S, b u t i n c r o a s o s a t a s l o w e r r a t e o r approaches t o a c o n s t a n t v a l u e , t h e same p r o p e r t y w i l l a p p l y t o ( X ^^^+•^)^' Such arrangements can be o b t a i n e d b y c h o o s i n g a p a r t i c u l a r r u l e f o r d e t e r m i n i n g t h e magni-t u d e s o f magni-t h e A ,
m
I n t h o n e x t s e c t i o n v/e s h a l l see t h a t M i s connected w i t h an i m -p o r t a n t i n v a r i a n t r e f e r r i n g t o c o r r e l a t i o n f u n c t i o n s *
6 l . C o r r e l a t i o n s . F o r t h e t y p e o f s o l u t i o n s we a r e c o n s i d e r i n g nov/, t h e E u l e r i a n space c o r r e l a t i o n v ( y 7 v ( y + 7j) can be o b t a i n e d , f o r w h i c h wo s h a l l w r i t e v^^v^. A n o t h e r i m p o r t a n t E u l e r i a n c o r r e l a t i o n f u n c t i o n i s [ v ( y ) ] t o be w r i t t e n v ^ ^ . I t v/ i l l be e v i -d e n t t h a t Vj^Vg i s a s y m m e t r i c f u n c t i o n o f ^ , h a v i n g i t s maximum a t V = 0, v/horoas '^-[_''^2 "^^^ f u n c t i o n o f ^ , \.'hxch i s z e r o ' f o r . = 0. B o t h f u n c t i o n s d e c r e a s e t o z e r o when 7^ becomes l a r g e , - I t i s u n d e r s t o o d t h a t t h e mean v a l u e s r e f e r t o a p a r t i c u l a r i n s t a n t . B o t h q u a n t i t i e s t h e r e f o r e a r e f u n c t i o n s o f t h e t i m e . S t a r t i n g f r o m e q , ( l ) t h e f o l l o w i n g e q u a t i o n can be f o r m e d
wher L-e V j = v ( y ) | = v ( y + 7 ) : 2 , ^ 2 ^ ^2 r J t ' -1-2'- •1-2 \ - Ö y ' y / ' A "2 ^2 ^1 ^ ^2 Ylhen meci,n v a l u e s a r e t a k e n f r o m b o t h s i d e s , we o b t a i n ( a c c o r d i n g t o a v r e l l k n o ™ p r o c e d u r e ) : (v-,v^) = ( v / v j + 2 v — j- ( v , v j . ( 3 l ) t ^ t -1^2' - ^ r ? ^^1 '2' ' .,_2 ^^1^2 T h i s e q u a t i o n i s t h e analogue o f t h e I m p o r t a n t e q u a t i o n o f von
MESSS ö-ï^d Howarth f o r hydrodynamic t u r b u l e n c e . I t has a c e n t r a l p l a c e i n i n v e s t i g a t i o n s on t h e s t a t i s t i c a l b e h a v i o r o f t h o s o l u t i o n s we a r e c o n s i d e r i n g . I f we c o n s i d e r n o t t o o s m a l l v a l u e s o f •>^ , t h e t e r m w i t h V can be d i s c a r d e d . I n t e g r a t i n g ( 3 l ) w i t h r o s p e c t t o f r o m 0 t o we f i n d s d d / d t = 0 (32) where 00 J =
1
V 2 d 7 (32a) o The q u a n t i t y J i s t h e a n a l o g u e o f Loitsio,nsl<y's i n v a r i a n t i n h y d r o -dynamic t u i ' b u l e n c e . To o b t a i n t h e c o n n e c t i o n botiveen t h i s i n v a r i a n t and t h e q u a n t i t y M o f t h o p r e c e d i n g s e c t i o n , we w r i t e : j+S ^ • 2 yThe e r r o r comjiiitted b y n e g l e c t i n g t h o q u a n t i t y denoted b y A decreases t o z e r o when S i s made l a r g e r and l a r g e r . The mean v a l u e i n d i c a t e d b y t h e b a r must be u n d e r s t o o d i n t h o f o l l o w i n g way: v/e c a l c u l a t e t h e i n t e g r a l w i t h v a r i o u s s t a r t i n g p o i n t s y, k e e p i n g S c o n s t a n t ; t h e n v/e
d e t o r r a i n c t h o mean v a l u o m t h r e s p e c t t o t h e s t a r t i n g point» Tho e x p r e s s i o n f o r M can a l s o be w r i t t e n :
S S
M = -|- \ dy^ \ dy^ v ( y + y^) vCy-py^)
I t w i l l be seen t h a t v ( y + y ^ ) v ( y + y ^ ) r e d u c e s t o t h e f u n c t i o n v ^ v ^ w i t h ^ y ^ - y^. I t i s t h u s e a s i l y f o u n d t h a t t h e d o u b l e i n t e g r a t i o n g i v e s : 1 M = - g - , 23 ^ V j V ^ d r ; = 2 J , o
62o S c a l e s Connected w i t h t h e _ S o l u t i o n s under C o n s i d e r a t i o n o I n a l l i n v e s t i g a t i o n s c o n c e r n i n g t u r b u l e n c e t h e c o n c e p t o f c e r t a i n s c a l e s , r e f e r r i n g e i t h e r t o t h e jna.croscopic a s p e c t o r t o f e a t u r e s o f d e t a i l , p l a y s an i m p o r t a n t p a r t . S i m i l a r q u a n t i t i e s can be formed f o r o u r s o l u t i o n s o f e q , ( l ) , A macroscopic s c a l e o f l e n g t h i s g i v e n b y ^ , r e p r e s e n t i n g t h e r i » ^ mean v a l u e s T = A , An average a m p l i t u d e f o r v i s g i v e n b y x ^ / t . M a k i n g use o f b o t h q u a n t i t i e s VG can d e f i n e a Reyholds number:
Re = I}/ V t , ( 3 3 )
T h i s s h o u l d be a l a r g e number| o t h e i n / i s e t h o a p p r o x i m a t i o n s , i n v o l v i n g e i t h e r t h o n e g l e c t o f t h e v i s c o s i t y o r , i f g r e a t e r a c c u r a c y i s needed, t h e use o f t h e e x p r e s s i o n ( 2 l ) , v;ould n o t bo v a l i d . I f t h e a r r a n g e -ment i n t h o system ?rould r e m a i n s t a t i s t i c a l l y s i m i l a r and t h e r e s u l t s
o b t a i n e d i n s e c t i o n 60 c o u l d be a p p l i e d , i t i s found t h a t Ro would i n -1/3
crease p r o p o r t i o n a l l y w i t h t ,
A raicroscale m can be d e f i n e d by means o f t h o d e v e l o p m e n t o f
f o r r a ; ) * 2
v ~ r = (1 - • - — " — + . . . . ) (34)
A c c o r d i n g t o a r ? e l l known f o r m u l a we t h e n haves
1 2
ïïe i n t r o d u c e t h o moan k i n e t i c energy E = - v and t h e mean d i s s i p a ¬ t i o n é. = V i'^v/ 'öj) I we t h e n have t h e e q u a t i o n s
dE , dv"^ 2 V v ^
- fc. , o r rr- - - ^ (^3^;
d t ^ ' ^ d t 2
m
M a k i n g use o f e q s , (25) ond (26) v/e o b t a i n s F n r — TT
3 2
^ - V t , (36)
T
I t f o l l o w s t h a t f o r a system f o r w h i c h t h e arrangement i n t h o l a r g e would r e m a i n s t a t i s t i c a l l y s i m i l a r , YJO s h o u l d have:
m "\ v t
T h i s vrould mean t h a t t h e r e c o u l d be no s i m i l a r i t y o v e r t h e w h o l e range, s i n c e J:!. /m vrould i n c r e a s e v j i t h t i r a o .
One can d e f i n e a R e y n o l d s number c o n n e c t e d v d t h t h e m i c r o s c a l o and o b t a i n s Re ^ Re-^/2, m _______ 2 63, r.1ien V7e d e v e l o p t h e c o r r e l a t i o n f u n c t i o n v ^ v ^ v d t h r e s p e c t t o 77 , i t i s f o u n d t h a t t h e t e r m o f t h e f i r s t degree d i s a p p e a r s , i n consequence o f t h e c i r c u m s t a n c e t h a t t h e mean v a l u e o f v . ( " ^ i v / 'D y )
"•^In t h o e x i s t i n g l i t e r a t u r e t h i s m i c r o s c e d o i s denoted b y , l?e used rn t o a v o i d c o n f u s i o n v.dth t h e segments \
2
i s zoroo Since v i s an odd f u n c t i o n o f r> , i t s d e v e l o p m e n t b e ¬ g i n s w i t h a t e r m i n "9 ° .
__ 2
Tho s e r i e s f o r t h e two f u n c t i o n s v^v and v ^ v ^ can be v / r i t t e n s 2
2 7 , ' 0 V V ^ ( ^ 2 V ^2
^1^2 - ^ - — ( " i t " ^ " 7 7 " ^ 3
" l "2 " 6 ^ -Dy ^
A p p r o x i m a t e e x p r e s s i o n s f o r t h e moan v a l u e s i n t h e r i g h t hand members can be o b t a i n e d by m a k i n g use o f e q , ( 2 l ) „ The c o n t r i b u t i o n s f r o m t h e upv/ard s l o p i n g segments and f r o m t h e n e a r l y v e r t i c a l f r o n t s can be c a l c u l a t e d s e p a r a t e l y ; t h e v a l u e s o b t a i n e d a r e :
Upv/ard s l o p i n g segments v e r t i c a l f r o n t s
~ ~ v 1 1 "Ö7 t t ^ -öj ' t ^ 12V £t^ ^ ' ^ y t ^ 1 2 0 v 2 i t 5
a y 2
2. 2 2i|0 V ^£t^I t must be observed, hov/over, t h a t t h e s e f o r m u l a s g i v e t h e most i m -p o r t a n t -p a r t s o n l y , and a more r e f i n e d c a l c u l a t i o n v/ould b r i n g terms \ / i t h s m a l l e r powers o f V i n t h e d e n o m i n a t o r s .
I t v / i l l be seen t h a t t h e o v e r a l l mean V a l u e o f v / y i s zero, as i s n e c e s s a r y . The mean v a l u e s o f t h e o t h e r q u a n t i t i e s a r e p r a i i c a l l y g i v e n b y t h o c o n t r i b u t i o n s f r o m t h e f r o n t s , Tho f a c t t h a t t h e mean v a l u o of i '0 v/-Q y i s d i f f e r e n t f r o m z e r o , i s an i n d i c a t i o n o f a
c e r t a i n skowness i n t h o d i s t r i b u t i o n o f v. A s i m i l a r skciimess appears i n hydrodynamic t u r b u l e n c e . A "skevmess f a c t o r " can be d e f i n e d b y :
'4 12 \
. 5
1/2
w h i c h i s p r o p o r t i o n a l t o Re ' . The l a t t e r r e s u l t i s d i f f e r e n t f r o m what i s o b t a i n e d i n hydrodynamic t u r b u l e n c e , where v a l u e s o f o r d e r u n i t y a r e f o u n d ( B a t c h o l o r g i v e s : - 0.395 compare C h a p t e r I V , s e c t i o n 2 l ) . The c o n c e n t r o , t i o n o f t h e g r a d i e n t o f v i n t h e model system, v;hich i s n o t s u b j e c t e d t o any e q u a t i o n o f c o n t i n u i t y , i s much l a r g e r t h a n t h e c o n c e n t r a t i o n o f v o r t i c i t y i n t h e v o r t e x s h e e t s between e d d i e s i n a c t u a l t u r b u l e n c e . Tho s e r i e s f o r t h e c o r r e l a t i o n f u n c t i o n s now t a k e t h e f o r m : V. V„ = V -2 V V , V - - — ^ —' ^1 3 ^ 5
1 2 6 i 2 0 v 2 X t 5
When t h e s o r e s u l t s , w h i c h a r e a p p l i c a b l e o n l y f o r ; v e r y s m a l l v a l u e s o f ''7 > a r e s u b s t i t u t e d i n t o ( 3 l ) and t e r m s i n d e p e n d e n t o f ^ a r c com-p a r e d , we o b t a i n : d* 6lt^ 'w h i c h i s t h e same as e q . (34-) (compare oq. ( 2 6 ) f o r é, ) „ - Comparison ^ 2
o f t h e terms i n does n o t l e a d t o a u s e f u l r e s u l t , s i n c e t h e t e r m s w i t h T s h o u l d bo completed v / i t h o t h e r torras, f o r v;hich no ex-p r e s s i o n has boon o b t a i n e d . I I n c u r r e n t t u r b u l e n c e t h e o r y t h e com¬
2 ^
p a r i s o n o f t h e t e r m s i n 7^ i g usod f o r d i s c u s s i n g t h e decay o f v o r t i c i t y , b u t a l s o h e r o i n a q u a l i t a t i v e way o n l y , s i n c e d e t a i l e d ex-p r e s s i o n s f o r t h e q u a n t i t i e s a ex-p ex-p e a r i n g i n t h e e q u a t i o n s a r e unknovm.
