WATERLOOPKUNDIG LABORATORIUM 7 september 1977 O p l e i d i n g e n B e t r e f t : K u r s u s W e t e n s c h a p p e l i j k e S t a f Aan de deelnemers v a n de k u r s u s Het v i e r d e b l o k v a n de k u r s u s W e t e n s c h a p p e l i j k e S t a f " T u r b u l e n t i e " s t a r t i n LD op d i n s d a g 15 november en i n LV d i n s d a g 22 november 1977. Het programma i s a l s v o l g t :
no. onderwerp datum LD datum LV i n l e i d e r
1 I n l e i d i n g 15 november •77 22 november '77 Breusers
2 V e r g e l i j k i n g e n 29 november '77 6_ december '77 V r e u g d e n h i l 3 S t a t i s t i s c h e b e s c h r i j v i n g 13 december '77 20 december '77 V r e u g d e n h i l 4 T r a n s p o r t p r o c e s s e n 17 j a n u a r i '78 24 j a n u a r i '78 V r e u g d e n h i l 5 Wand t u r b u l e n t i e 31 j a n u a r i '78 7 f e b r u a r i •78 Breusers 6 T i j d s a f h a n k e l i j k e g r e n s l a g e n 14 f e b r u a r i •78 21 f e b r u a r i '78 Breusers 7 V r i j e t u r b u l e n t i e 28 f e b r u a r i '78 7 maart '78 Breusers 8 Nieuwe t u r b u l e n t i e theorieën 14 maart •78 21 maart '78 V r e u g d e n h i l
De k u r s u s p l a a t s e n en a a n v a n g s t i j d e n z i j n a l s v o l g t :
C o l l o q u i u m e a a l h o o f d k a n t o o r LD aanvang 16.15 u u r .
P r o f . t e r Veen Lyceum, P r o f . t e r V e e n s t r a a t t e Emmeloord aanvang 19.00
1. INLEIDING
WATERLOOPKUNDIG LABORATORIUM Kursus 1977/1978
KURSUS TURBULENTE STROMINGEN D o e l e n i n h o u d Doel v a n de k u r s u s i s h e t geven v a n : - i n z i c h t i n de f y s i s c h e p r o c e s s e n d i e v a n b e l a n g z i j n i n t u r b u l e n t e s t r o m i n g e n ; - een z o d a n i g e i n t r o d u k t i e i n de t h e o r e t i s c h e a s p e k t e n d a t de l i t e r a t u u r kan worden g e v o l g d en g e b r u i k t ; i n f o r m a t i e u i t m e t i n g e n e.d., d i e g e b r u i k t k a n worden b i j de b e a n t -w o o r d i n g v a n p r a k t i s c h e v r a g e n . De h a n d l e i d i n g h e e f t een i n l e i d e n d k a r a k t e r en d i e n t n i e t t e r v e r v a n g i n g v a n handboeken, waarvan o v e r i g e n s w e l u i t g e b r e i d g e b r u i k i s gemaakt.
De k u r s u s i s verdeeïd i n 8 d e l e n : 1. I n l e i d i n g : d e f i n i t i e s , b e g r i p p e n , k o r r e l a t i e f u n k t i e s , s p e k t r a , t y p e r i n g ; 2 S t a t i s t i s c h e b e s c h r i j v i n g : k o r r e l a t i e f u n k t i e s , s p e k t r a , d i s s i p a t i e -s p e k t r u m , " i n e r t i a l r a n g e " , r e p r o d u k t i e t u r b u l e n t i e i n m o d e l l e n . 3. V e r g e l i j k i n g e n : k o n t i n u i t e i t , b e w e g i n g , e n e r g i e , k o n c e n t r a t i e ; d i s k u s s i e o v e r termen v o o r i m p u l s en m a s s a t r a n s p o r t , " c l o s u r e " p r o b l e e m ; ^- T r a n s p o r t p r o c e s s e n : t r a n s p o r t v a n i m p u l s , s t o f , d e e l t j e s , t u r b u l e n t e v i s k o s i t e i t en d i f f u s i e , mengweg, Lagrangese b e s c h r i j v i n g ; 5. W a n d t u r b u l e n t i e ; s t r o m i n g i n k a n a l e n , b u i z e n , g r e n s l a g e n , s n e l h e i d s -v e r d e l i n g , t u r b u l e n t i e s t r u k t u u r , -v i s k e u z e s u b l a a g , s e k u n d a i r e s t r o m i n g ; ^' T i j d s a f h a n k e l i j k e g r e n s l a g e n ; i n v l o e d s t o r i n g e n , o s c i l l e r e n d e g r e n s -l a g e n ; I- V r i j e turbulêiitie: s t r a l e n , menglaag, z o g s t r o m i n g ;
8- Nieuwe t u r b u l e n t i e theorieën; aanpak " c l o s u r e p r o b l e m " , m o d e l l e r i n g b e w e g i n g s - en e n e r g i e v e r g e l i j k i n g .
1. P. BRADSHAW, 1 9 7 1 , An I n t r o d u c t i o n t o T u r b u l e n c e and i t s Measurement. Pergamon P r e s s , O x f o r d ( d u i d e l i j k e i n l e i d i n g )
2. A.S. MONIN, A.M. YAGLOM, 1 9 7 1 , S t a t i s t i c a l f l u i d mechanics: Mechanics o f T u r b u l e n c e , V o l . 1 .
M.I.T. P r e s s , Cambridge ( n a d r u k op s t a t i s t i s c h e b e h a n d e l i n g ,
b e h a n d e l t ook a t m o s f e r i s c h e t u r b u l e n t i e met d i c h t h e i d s v e r s c h i l l e n )
3. H. TENNEKES, J.L. LUMLEY, 1972, A f i r s t course i n t u r b u l e n c e . M.I.T. P r e s s , Cambridge
4. J.C. ROTTA, 1972, T u r b u l e n t e Strömungen. B.G. Teubner, S t u t t g a r t
5. J.O. HINZE, 1975, T u r b u l e n c e , 2nd Ed.
McGraw H i l l , New Y o r k (meest v o l l e d i g e handboek)
6. A.A. TOWNSEND, 1976, The s t r u c t u r e o f t u r b u l e n t shear f l o w , 2nd Ed. ( Cambridge U n i v e r s i t y Press
T i j d s c h r i f t e n
1. J o u r n a l o f F l u i d Mechanics ( b e l a n g r i j k s t e b r o n )
2. A m . I n s t , o f A e r o n a t i c s and A s t r o n a u t i c s J o u r n a l (A.I.A.A.J.) 3. P h y s i c s o f F l u i d s .
1
-KURSUS TURBULENTIE
1» I n l e i d i n g
De i n l e i d i n g w o r d t gegeven i n de v o r m v a n een a a n t a l d e f i n i t i e s en o m s c h r i j v i n g e n v a n b e g r i p p e n . Voor een samenhangende i n l e i d i n g z i e b i j l a g e 1 (pag. 1/15 u i t : J.O. H i n z e , " T u r b u l e n c e " ) .
1.1 D e f i n i t i e " t u r b u l e n t e s t r o m i n g " (
Een e x a c t e d e f i n i t i e i s n i e t m o g e l i j k ; v o l s t a a n moet worden met h e t geven v a n een a a n t a l kenmerken. H i n z e ( p a g . 2) g e e f t de v o l g e n d e d e f i n i t i e :
" T u r b u l e n t e v l o e i s t o f b e w e g i n g i s een o n r e g e l m a t i g e s t r o m i n g s k o n d i t i e , w a a r i n v e r s c h i l l e n d e g r o o t h e d e n een w i l l e k e u r i g e v a r i a t i e i n de t i j d en de r u i m t e v e r t o n e n , z o d a n i g d a t s t a t i s t i s c h b e p a a l d e g r o o t h e d e n
kunnen worden o n d e r s c h e i d e n " .
B e l a n g r i j k e kenmerken z i j n ( z i e ook Tennekes en Lumley, pag. 1 / 3 ) : ~ o n r e g e l m a t i g : dus geen d e t e r m i n i s t i s c h e maar een s t a t i s t i s c h e aanpak; - d r i e d i m e n s i o n a a l : t u r b u l e n t i e i s d r i e d i m e n s i o n a a l ook i n s i t u a t i e s ^ met een t w e e d i m e n s i o n a l e g e m i d d e l d e s t r o m i n g . Kenmerkend z i j n ook de
w e r v e l s t e r k t e f l u k t u a t i e s met komponenten i n a l l e koördinaatrichtingen. T w e e d i m e n s i o n a l e w e r v e l p a t r o n e n , b i j v o o r b e e l d w e r v e l s t r a t e n o f n e r e n worden n i e t t o t t u r b u l e n t i e g e r e k e n d ;
~ d i f f u s i e f : t y p e r e n d i s de s t e r k e menging v a n i m p u l s , warmte en massa; - d i s s i p a t i e f ; k i n e t i s c h e e n e r g i e v a n de t u r b u l e n t e s t r o m i n g w o r d t ge-d i s s i p e e r ge-d o n ge-d e r i n v l o e ge-d v a n ge-de v i s k o s i t e i t . Toevoer v a n e n e r g i e v i n d t p l a a t s t e n k o s t e v a n de h o o f d s t r o m i n g door i n t e r a k t i e v a n s c h u i f -s p a n n i n g en -snelheid-sgradiënten. 1.2 N o t a t i e s Koördinaten: i = 1 , 2, 3 i = 1 i n s t r o o m r i c h t i n g i = 2 l o o d r e c h t op de wand i = 3 e v e n w i j d i g aan de wand
Snelheden U, = U . + u , i = l , 2, 3
1 1 1 ' '
= f l u k t u e r e n d e s n e l h e i d s k o m p o n e n t
I n v e r g e l i j k i n g e n w o r d t de C a r t e s i a a n s e t e n s o r - n o t a t i e g e b r u i k t ; b i j h e r h a l i n g v a n een i n d e x w o r d t gesommeerd, b i j v . i n de kontinuïteits-v e r g e l i j k i n g : 9U. 9u^ ^ d^'^ d^'^ 'S^^ ^ o f i n de k o n v e k t i e v e t e r m u i t de b e w e g i n g s v e r g e l i j k i n g : 8U. pU j 9Xj 9U., 9U 9U^ i = 1 -> p{U, ~-± + n —± + Jl} 1 9x^ 12 9x2 3 9U„ 9U 9U„ 1 9x^ 2 9x2 3 I n t w e e d i m e n s i o n a l e s i t u a t i e s w o r d t ook g e b r u i k gemaakt v a n de n o t a t i e s X en z (= X 2 ) v o o r de koördinaatrichtingen en U en W v o o r de s n e l h e d e n . 1.3 Typen t u r b u l e n t e s t r o m i n g e n A f h a n k e l i j k v a n de g e o m e t r i e kunnen v e r s c h i l l e n d e t y p e n s t r o m i n g e n worden o n d e r s c h e i d e n : l:._ii°52SË2Ë_£H£^HlËB£iê* k a r a k t e r i s t i e k e eigenschappen o n a f h a n k e l i j k v a n de p l a a t s . I s a l l e e n m o g e l i j k b i j k o n s t a n t e snelheidsgradiënten. 2i_iË2£E°Eê_ËHE]ËHlêStiË• k a r a k t e r i s t i e k e eigenschappen o n a f h a n k e l i j k van de r i c h t i n g . Wordt b i j b e n a d e r i n g b e r e i k t b i j s t r o m i n g a c h t e r een u n i f o r m r o o s t e r .
2.i_ïêSÉ£HE]ËylÊ5£iÊ• s t r o m i n g d i e i n overwegende mate door een v a s t e wand w o r d t b e p a a l d . V o o r b e e l d e n :
C o u e t t e s t r o m i n g : s t r o m i n g t u s s e n e v e n w i j d i g e p l a t e n , w a a r b i j de s t r o m i n g w o r d t opgewekt door de p l a t e n t . o . v . e l k a a r t e bewegen, k a n a a l s t r o m i n g : s t r o m i n g t u s s e n twee e v e n w i j d i g e p l a t e n , opgewekt d o o r een drukgradiënt.
3
-C o u e t t e k a n a a l g r e n s l a a g
4 ^ _ v r i 2 e _ t u r b u l e n t i e : s t r o m i n g opgewekt d o o r s n e l h e i d s v e r s c h i l l e n zonder i n v l o e d v a n een v a s t e wand b i j v . s t r a l e n , menglaag, zog-s t r o m i n g .
s t r a a l menglaag zog
1.4 Gemiddelden
Gezien h e t s t o c h a s t i s c h e k a r a k t e r v a n t u r b u l e n t e s t r o m i n g e n i s h e t n o d i g met g e m i d d e l d e n t e werken. M i d d e l i n g k a n gebeuren o v e r s
- de t i j d ( v o o r s t a t i o n a i r e s t r o m i n g e n ) ; - de p l a a t s ( v o o r homogene t u r b u l e n t i e ) ;
- h e t ensemble v a n N h e r h a l i n g e n v a n een e x p e r i m e n t .
Voor stationaire^(èil homogene t u r b u l e n t i e mag worden aangenomen ( e r g o d i c i t e i t s h y p o t h e s e ) d a t deze m i d d e l i n g s p r o c e d u r e s t o t d e z e l f d e waarden l e i d e n . Problemen o n t s t a a n b i j t i j d s a f h a n k e l i j k e s t r o m i n g e n
( g e t i j d e n , g o l v e n ) , w a a r b i j de m i d d e l i n g s t i j d z o d a n i g moet worden ge-|' k o z e n d a t de v a r i a t i e s i n de gemiddelde waarden k l e i n b l i j v e n :
U = ^ ƒ U ( t + T ) d t ^ T ^
= t i j d s c h a a l v a n t u r b u l e n t i e
B i j m i d d e l l i n g v a n samengestelde g r o o t h e d e n g e l d e n de r e k e n r e g e l s : ( s t e l A = A + a B = B + b ) . a = b = O ( v o l g t u i t A = A + a = A + a = A + a ) Ab = A.b = O AB = (A + a) (B + b ) = AB + ab Ia 9a J^-J-^ ^ -, •TT" = J Ads = J Ads ds ds 1.5 T u r b u l e n t i e - i n t e n s i t e i t en k i n e t i s c h e e n e r g i e De s t e r k t e v a n de t u r b u l e n t e f l u k t u a t i e s o f t u r b u l e n t i e - i n t e n s i t e i t w o r d t g e d e f i n i e e r d met de " r o o t - m e a n - s q u a r e " ( r . m . s . ) waarde: u'. = 1 1 en de r e l a t i e v e t u r b u l e n t i e - i n t e n s i t e i t d o o r : r . = ^
(J
. ÏÏ. t
t-,-i U. ^ I' ^ 1 De som v a n de k w a d r a t e n v a n de t u r b u l e n t i e - i n t e n s i t e i t e n i s g e l i j k aan 2 maal de k i n e t i s c h e e n e r g i e v a n de t u r b u l e n t i e p e r massa-eenheid:I n h e t algemeen i s u | > u ^ ~ u^ De waarde v a n r ^ = uj^/Ü i s a f h a n k e l i j k v a n h e t t y p e t u r b u l e n t i e : i n v r i j e t u r b u l e n t i e v a n de o r d e 0,2 a 0,3, b i j w a n d t u r b u l e n t i e v a n de o r d e 0,2 v l a k b i j de wand t o t 0,03 v e r v a n de wand. 1.6 S c h a l e n Voor een g l o b a l e b e s c h r i j v i n g v a n t u r b u l e n t i e w o r d t h e t b e g r i p " s c h a a l " geïntroduceerd o.a. v o o r de l e n g t e , de s n e l h e i d en de t i j d . V o o r b e e l d e n : iê5S££Ë£ll§êl s c h a a l v a n de g r o t e w e r v e l s i n een p i j p s t r o m i n g i s v a n de o r d e v a n de p i j p d i a m e t e r 2) de K o l m o g o r o v - d i s s i p a t i e s c h a a l v o o r de k l e i n s t e w e r v e l s
n = (vVe)^
V = k i n e m a t i s c h e v i s k o s i t e i t e = e n e r g i e d i s s i p a t i e p e r massa-eenheid , . , ,5 -S2ÊlllêiËË5£lï5§l' 1) gemiddelde ( o v e r h e t p r o f i e l ) s n e l h e i d 2) U'' = (T / p ) ^ w U = w a n d s c h u i f s p a n n i n g s s n e l h e i d = w a n d s c h u i f s p a n n i n g Voor de t i j ^ d s c h a a l k a n de v e r h o u d i n g v a n een k a r a k t e r i s t i e k e l e n g t e -s c h a a l en T o e p a s s i n g v a n s c h a l e n v i n d t p l a a t s b i j beschouwingen o v e r de m o g e l i j k e v e r e e n v o u d i g i n g e n v a n de b e w e g i n g s v e r g e l i j k i n g e n ( z i e h o o f d s t u k 5 ) . U i t de aanname d a t de l e n g t e s c h a a l v a n de s t r o m i n g l o o d r e c h t op de s t r o o m r i c h t i n g k l e i n i s t e n o p z i c h t e v a n d i e i n de s t r o o m r i c h t i n g v o l g t een a a n z i e n l i j k e v e r e e n v o u d i g i n g v a n de b e w e g i n g s v e r g e l i j k i n g e n ( h y d r o s t a t i s c h e d r u k v e r d e l i n g l o o d r e c h t op de s t r o o m r i c h t i n g ) . 1.7 G e l i j k v o r m i g h e i d ( s e l f - p r e s e r v a t i o n )
Voor b e p a a l d e t u r b u l e n t e s t r o m i n g e n k a n g e b r u i k gemaakt worden v a n de h y p o t h e s e d a t de s t r u k t u u r i n a l l e doorsneden l o o d r e c h t op de s t r o m i n g s r i c h t i n g g e l i j k v o r m i g i s . Met deze h y p o t h e s e worden de b e -w e g i n g s v e r g e l i j k i n g e n s t e r k v e r e e n v o u d i g d en z i j n k o n k l u s i e s t e n
aan-z i e n v a n een a a n t a l s t r o m i n g s p a r a m e t e r s m o g e l i j k . Een ronde s t r a a l i s r e d e l i j k g e l i j k v o r m i g v o o r a f s t a n d e n v a n a f de s t r a a l m o n d g r o t e r dan ca. 30 s t r a a l d i a m e t e r s . A f g e l e i d k a n worden d a t dan de maximale s n e l -h e i d i n de s t r a a l e v e n r e d i g met x^''" a f n e e m t .
1.8 K o r r e l a t i e
Voor de k a r a k t e r i s e r i n g v a n de r u i m t e l i j k e e n t i j d s t r u k t u u r v a n de t u r b u l e n t i e worden k o r r e l a t i e s g e b r u i k t . De algemene vorm v o o r s t a t i o -n a i r e s t r o m i -n g i s : ^ ^ u ( 5 , t ) u . ( 5 + ?, t + T ) R ( x , r , T) = ~ u [ ( x ) . u ^ ( x + r ) X = p l a a t s v e k t o r r = a f s t a n d s v e k t o r T = t i j d s v e r s c h u i v i n g
\''"" VoorbeeljJen z i j n de l o n g i t u d i n a l e k o r r e l a t i e f u n k t i e v o o r i n homogène s t r o m i n g : R j ^ ^ ( l j : ^ , 0 , 0 ) j = u ( x , t ) u ( ( x + r J , t ) ,1 2 r^^ = a f s t a n d l a n g s x^-as en de a u t o - k o r r e l a t i e f u n k t i e : u ^ ( t ) . u ^ ( t + T ) u. I 2 U i t de d e f i n i t i e v o l g t d a t R.,. = 1 v o o r r = O, x = 0. A f g e l e i d kan 1 J g' i worden d a t de k o r r e l a t i e f u n k t i e i n homogene s t r o m i n g s y m m e t r i s c h t e n
o p z i c h t e van r = O i s en daar ook een h o r i z o n t a l e r a a k l i j n h e e f t :
Met b e h u l p v a n R kunnen twee l e n g t e ( o f t i j d ) maten g e d e f i n i e e r d
worden: 1. A = /~Rdr O de i n t e g r a a l s c h a a l d i e een maat i s v o o r de g r o t e w e r v e l s 2. A u i t — = - H — ) , 2 2^ r = 0 A^ dr' A i s de d o o r s n i j d i n g van de i n g e s c h r e v e n p a r a b o o l v o o r r = O en i s een maat v o o r de k l e i n e w e r v e l s ( n i e t de k l e i n s t e w e r v e l s ; h i e r v o o r g e l d t de Kolmogorov d i s s i p a t i e s c h a a l ) . Ter i l l u s t r a t i e e n i g e
7 -1.0 0.8 0.6 0.4 0.2 O 1.0 0.8 0.6 0.4 0.2 O 1.0 0.8 0,6 0.4 0.2 O ~ i — I — I — I — I — I — r -Rn(r,0,0) z/So=0.66
/?,i(0, O,/-) z/So=0.66
Rn(0,r,0) 2/60 = 0.52 «33(1-, 0 , 0 ) 2/S„ = 0.66 H «33(0, O, r) z/So = 0.66 R„(0,r,0) z/So=0.66 /?„(/•, O, 0) z/So = 0.69 . R „ ( 0 , 0 , r ) z/So=0.66 RniO.r.O) z/«o = 0.66 O 0.2 0.4 0.6 0.8 r/So 1.0 1.2 1.4
Normal components of the correlation function in the outer part of a xi •
boundary layer (after Grant 1958) Normal components of the correlation function in the inner part of a boundary layer (after Grant 1958).
I n d i e n de f l u k t u a t i e s k l e i n z i j n t e n o p z i c h t e v a n de gemiddelde s n e l h e i d i s h e t r e d e l i j k aan t e nemen ( T a y l o r ' s h y p o t h e s e ) d a t de t u r b u l e n t i e s t r u k t u u r n i e t s n e l v e r a n d e r t t i j d e n s h e t p a s s e r e n v a n een k l e i n i n t e r v a l r . I n d a t g e v a l z i j n de r u i m t e en t i j d k o r r e l a t i e i n de s t r o o m r i c h t i n g g e l i j k v o o r ; T = r/Ü_j^ 1.9 S p e k t r a
De t u r b u l e n t e f l u k t u a t i e s i n een p u n t kunnen ook g e a n a l y s e e r d worden n a a r v e r d e l i n g o v e r de f r e k w e n t i e , b i j v o o r b e e l d v o o r u, :
ÜT = ƒ E , ( n ) d n J- O 1 E ^ ( n ) = e n e r g i e d i c h t h e i d s s p e k t r u m v a n Er i s u i t e r a a r d een v e r b a n d t u s s e n h e t s p e k t r u m en de a u t o k o r r e l a t i e -f u n k t i e . Een s n e l l e a -f v a l v a n de k o r r e l a t i e d u i d t op h o o g -f r e k w e n t e f l u k t u a t i e s . De r e l a t i e i s een F o u r i e r - c o s i n u s t r a n s f o r m a t i e : E^(n) = 4 u 2 /°°R^^(t) cos(2TTnt)dt R , , ( t ) = ƒ E, ( n ) c o s ( 2 7 r n t ) d n O i Er b e s t a a n a n a l o g e r e l a t i e s t u s s e n de r u i m t e l i j k e k o r r e l a t i e en h e t r u i m t e l i j k ( g o l f g e t a l ) e n e r g i e s p e k t r u m ( z i e v e r d e r h f d s t . 3 ) . 1.10 Reynoldse spanningen B i j m i d d e l i n g v a n de N a v i e r S t o k e s v e r g e l i j k i n g e n o v e r de t i j d o n t -s t a a n nieuwe t e r m e n , b i j v o o r b e e l d pü^" en pu^u^ d i e h e t k a r a k t e r v a n r e s p . n o r m a a l s p a n n i n g e n en s c h u i f s p a n n i n g e n hebben. Voor een nadere beschouwing z i e h o o f d s t u k 2. A l s i n l e i d i n g w o r d t o n d e r s t a a n d f r a g m e n t u i t Bradshaw ( 1 9 7 1 , pag 10-12) gegeven.
1.5. The Reynolds Stresses
We now come to the explanation of how quite small velocity fluc-tuations can produce such large changes i n flow resistance and other properties. The rate at which x-component momentum passes through one of the faces dy dz of the elementary control volume of Fig. 1 is equal to the product of the mass flow rate gUdydz and the velocity U: we call this the momentum flux through the face. Now suppose that in addition to a mean velocity U there is a time-dependent (fluctuating) component of velocity i n the x direction, u—the mean (time average) value of M, denoted by «, being zero by definition. Then the x-component dy momentum flux through the face dy dz is
e(U + uy dy dz or ^(C/^ ^ Wu ^ «*) dy dz
equal in the mean to q{lJ^ + \?) dy dz where is the mean value of u^. Therefore, a fluctuation with zero mean, superimposed on the mean velocity, produces a mean momentum flux of its own, proportional to the mean square of the fluctuating velocity, because momentum flux is
9
-the product of mass flow and velocity and -the fluctuation contributes to both. This non-linearity ofthe relation between velocity and momentum flux appears in the Navier-Stokes equations and is the basic cause of their mathematical difficulty even in laminar flow.
I f we measure the mean velocity and pressure in turbulent flow we find that the mean motion is not that which would be produced by viscous forces alone: there is an extra apparent stress —QIP' normal to the face dydz (it is a compressive stress since tP is poshive: in the notation of Section 1.3 it is an addition to (r^^^). The equivalence of stress and momentum flux follows at once from Newton's second law. Similarly, there are extra normal stresses -QV^ and —QW^ in the y and
z directions: usually, u^, and differ by no more than a factor of 2 or 3.
Again, the rate at which x-component momentum passes through the face dx dz of the control volume is the product of the mass flow in the
y direction, Q(V + v) dx dz, and the velocity in the x direction, U + u:
the mean momentum flux is Q(UV + ïïü) dx dz and ff^y = -QÜv repre-sents an extra mean shear stress on the face dx dz. Note that the rate at which j'-component momentum passes through the face dy dz leads to a shear stress = (^xy on that face also, just as in the case of viscous stresses (which can, of course, exist at the same time as the extra turbulent stresses). These extra turbulent stresses are called the Reynolds stresses, in honour of Osborne Reynolds: the time-mean Navier-Stokes equations, in which these stresses appear, are called the Reynolds equations.
Turbulence produces additional fluxes (rates of transport) of quantities other than momentum: for example, i f there are temperature fluctu-ations, d, in the flow, implying enthalpf fluctuations QC^O per unit volume (assuming Q and Cp to be constant for simphcity) then there is an extra rate of enthalpy transfer QC^du in the x direction, Qc^dv in the y direction and QCpOw in the z direction, in addition to enthalpy transfer by molecular conduction.
Now velocity fluctuations of +10 per cent about the mean, U, will produce Reynolds stresses of the order of 0-OOIQU^. In a pipe, say,
mean velocity gradients are of order Ujd, giving viscous stresses of order fiUjd. The ratio is 0-mUdlv, or 100 i f Udjv = 100,000. Indeed, in nearly all cases of interest, Reynolds stresses and other turbulent transport rates are much larger than viscous stresses and other molecular transport rates: this is why turbulence is of such great practical impor-tance as well as being a fascinating phenomenon. A t first sight, the mechanism by which turbulent eddies produce the Reynolds stresses seems quite similar to the mechanism by which random molecular motion produces viscous stresses, but there is no worth-while analogy between the two because
(1) turbulent eddies are continuous and contiguous whereas gas molecules are discrete and collide only at intervals;
(2) although molecular mean free paths are small compared to the dimensions of the mean flow, turbulent eddies are not.
I t follows that the turbulent transport rates are not determined by the mean gradient of the transported quantity as in molecular transport— that is to say, the turbulent diffusivities, defined analogously to the molecular diffusivities mentioned at the end of Section 1.3, are not
usually constants or even discoverable functions of the local variables, but depend on the previous history of the flow which carries the tur-bulent eddies (see Section 2.2). There are some special cases in which the diffusivities depend simply on the velocity and length scales of the flow when the latter are particularly simple {not because the eddy motion is any simpler than usual): we shall return to this subject in Sections 3.3 and 3.4.
Since the extra apparent stresses and transport rates are produced by the fluctuating motion itself, it appears that in order to understand them we must at least partly understand the behaviour of the fluctu-ations. Because we cannot solve the complete time-dependent Navier-Stokes equations for each case we have to appeal to experiment even for a qualitative understanding of turbulence, as well as for quantitative information for engineering purposes. However, it is possible to do theoretical work on turbulence at many levels, ranging from attempts to solve the Navier-Stokes equations in simplified form to elementary dimensional analvsis.
1.11 W e r v e l s t e r k t e en d i s s i p a t i e
T u r b u l e n t e s t r o m i n g e n b e z i t t e n w e r v e l s t e r k t e f l u k t u a t i e s ( i n d i s k r e t e vorm v o o r g e s t e l d a l s w e r v e l s ) . Door i n t e r a k t i e van gemiddelde s t r o m i n g en w e r v e l s v i n d t o v e r d r a c h t van w e r v e l s t e r k t e i n a l l e r i c h t i n g e n en naar k l e i n e s c h a l e n p l a a t s . Voor een i n l e i d i n g z i e o n d e r s t a a n d f r a g -ment (Bradshaw, pag. 1 2 - 1 7 ) .
1.6. Vortex Stretching* '
Turbulent eddies (and some laminar flows or "inviscid" flows) have both translational and rotational motion, familiar to anyone who has looked over a river bridge. The net rate of rotation (or average angular velocity) about the z axis of the fluid element shown in Fig. 2c is
1 / di; du \
(we use small letters to show that in general we are dealing with fluc-tuations). We define the z component of vorticity as twice this angular velocity, (dvldx) — (ÖM/ÖJ). The vorticity is to be distinguished from the rate of shear strain (duldy) + (dvldx): one is a measure of rotation, the other a measure of deformation (Fig. 5). Now if in addition to a rotation about the z-axis the fluid element is under the influence of a rate of linear strain in the z direction, dwjdz, the element will be stretched in the z direction and its cross-section in the x^-plane wiU get smaller. I f
(a) ^ = - ^ = e . s a y : vorticity = 2e, rate of shear strain = 0
/
does not change) (b) 1^ = = e, ; vorticity = 0, rate of shear strain'= 2e
DX oy
Fio. 5. The distinction between vorticity and rate of (shear) strain.
we take the case of an element of circular cross-section in the xy-plane
and neglect viscous forces for simplicity, we can see that conservation of angular momentum requires the product of the vorticity and the square of the radius to remain constant: more generaUy, the integral of the tangential component of velocity round the perimeter, called the circulation, remains constant in the absence of viscous forces (see pp. 93ff". of Batchelor^*'). During the stretching process the kinetic
11
-energy of rotation increases (at the expense of the kinetic -energy of the
w component motion that does the stretching) and the scale of the motion in the xj-plane decreases. Therefore an extension in one direction (the z direction here) can decrease the length scales and increase the velocity components in the other two directions (x and y) which in turn stretch other elements of fluid with vorticity components in these directions, and so on. The length scale of the motion that is augmented gets smaller at each stage. I f we draw out a family tree (Fig. 6) showing how stretching in the z direction intensifies the motion in the x and y directions, produc-ing smaller-scale stretchproduc-ing in x and y and intensifying the motion in the y, z and z, x directions respectively, and so on, we can see quaUtatively
Frequency of symbols at each generation
X I y I 2 0 0 1
x y y z y z z x y z z x z x x y
FIG, 6. "Family tree" showing how vortex stretching produces small-scale isotropy. The labels are the directions of stretching in each "generation": the
length scale decreases from one generation to the next.
that an initial stretching in one direction produces nearly equal amounts of (smaller-scale) stretching in each of the x, y and z directions after a few stages of the process. Thus the small-scale eddies in turbulence do not share the preferred orientation of the mean rate of strain: in fact they have a universal structure which makes their study easier (see Section 2.5). The "cascade" of energy ofthe turbulent motion continues to smaller and smaller scales (larger and larger velocity gradients): indeed, discontinuities of velocity would develop if it were not for the smoothing action of viscosity. Put another way, viscosity finally dis-sipates (into thermal internal energy, loosely called, "heat") the energy that is transferred to the smallest eddies, but it does not play any essential part in the stretching process as such.
In the above discussion i t was implied that the element of fluid considered was part of a line vortex with its axis in the z direction (see Fig. 7 for a brief revision of vortex properties). We can imagine any flow with vorticity to be made up of large nuinbers of infinitesimal slender vortices, "vortex lines": sometimes it is convenient to talk of
Circumferential velocity V Streamlines are all circular Radius r V c c r voc 1 1 " = 57 + Vorticity 1 or a
1
= 0 ifI r r o t a t i o n a l - ^ - * Vortex core Radius r »-flow
a "vortex sheet", which is a layer of locally parallel vortex Unes—for instance, laminar shear layers are often discussed as if they were made up of a stack of elementary vortex sheets, with the vortex lines parallel or nearly so.
Turbulence can be thought of as a tangle of vortex lines or partly rolled vortex sheets, stretched in a preferred direction by the mean flow (the mean vortex lines) and- in random directions by each other. Turbulence always has all three directions of motion even if the mean
velocity has only one or two components: i f the fluctuating velocity component in one direction were everywhere zero the vortex Unes would necessarily all lie in this direction and there would be no vortex stretch-ing, no transfer of fluctuation energy to smaller scales, and the motion would not be what we call turbulence. But for the diffusing effect of viscosity, vortex lines or sheets would move with the fluid: the effect of viscous diffusion is seen in the slow growth of laminar shear layers. In turbulent flow, viscous diffusion of vorticity is negligible except for the smallest eddies—those that dissipate the energy transferred from the larger eddies. Fluid that is initially without vorticity ("irrotational") can acquire it only by viscous diffusion but, once acquired, vorticity can be increased many orders of magnitude by vortex stretching. Pressure fluctuations do not directly affect vorticity in incompressible flow.
The rate of supply of kinetic energy to the turbulence is, in the absence of body forces, the rate at which work is done by the mean rate of strain against the Reynolds stresses in the flow as it stretches the tur-bulent vortex Unes. In laminar flow, viscous stresses caused by molecular motion convert ("dissipate") mean flow kinetic energy directly into thermal internal energy: in turbulent flow the eddies extract energy from the mean flow and retain it for a while before it£eaches the small dissipating eddies. Turbulent kinetic energy, ^Q(t? + + w^) per unit volume, is introduced into the eddies that contribute to the Reynolds stresses in direct proportion to their contributions. The stress-producing eddies are the larger ones, which are best able to interact with the mean flow: we have already seen that vortex stretching tends to make the smaller eddies lose all sense of direction and become statistically isotropic (see Glossary) so that, for instance, their contribution to the Reynolds shear stress —QÜV is zero. The smaller eddies are much weaker than those that produce most of the Reynolds stress because most of the energy that reaches them is immediately passed on to the smallest eddies of all and there dissipated by viscosity. In the central part of a typical pipe flow at least half the turbulent kinetic energy and most of the Reynolds shear stress occurs in eddies with wavelengths greater than the pipe radius. The size of the dissipating eddies depends on the vis-cosity and the speed of flow as well: typically their wavelength is less than 1 per cent of the pipe radius (see the research paper by Laufer in Section 3.6 of the "Further Reading" list for more details).
Since the viscous stresses are usually so small compared with the turbulent stresses and since the parts of the eddy structure that depend on viscosity are so small and so weak compared with the stress-produc-ing part of the turbulence, we can for many purposes neglect viscosity in the study of turbulent flow, regarding it only as a property of the fluid that produces energy dissipation in very small eddies. Exceptions
13
-are flows m process of transition from laminar to turbulent and the flow very close to a solid surface (say within 0-5 mm in air flow in a pipe at 20 metres/sec). I n both these cases viscous and turbulent stresses can be of the same order and viscosity directly affects the eddies that produce the Reynolds stresses. This makes transition a very difficult problem but fortunately the flow in the viscous sub-layer close to a solid surface is a function of only a few variables, and dimensional analysis, plus a few empirical constants, "solves" the problem for engineering purposes. Apart from these cases, the behaviour of turbulence would be much the same whatever the dissipation mechanism in the fluid. The main characteristics of turbulence are the result of three-dimensional vortex stretching which, mathematically speaking, depends on the non-Unear terms in the Navier-Stokes equations that represent the acceleration of the fluid, and not upon the terms representing the viscous forces.
We can now define turbulence:
Turbulence is a three-dimensional time-dependent motion in which vortex stretching causes velocity fluctuations to spread to all wavelengths between a minimum determined by viscous forces and a maximum determined by the boundary conditions of the flow. It is the usual state of fluid motion except at low Reynolds numbers.
1.12 T u r b u l e n t e v i s k o s i t e i t (eddy v i s c o s i t y )
De a a n w e z i g h e i d v a n de Reynoldse s p a n n i n g e n ( e n met name de
s c h u i f s p a n n i n g e n z o a l s pu^u^) v o r m t h e t p r o b l e e m b i j h e t o p l o s s e n van t u r b u l e n t e s t r o m i n g s v r a a g s t u k k e n . Het i s d u i d e l i j k d a t v a n a f h e t b e g i n g e t r a c h t i s deze termen t e benaderen. B o u s s i n e s q i n t r o -duceerde h e t b e g r i p " s c h i j n b a r e " o f " t u r b u l e n t e " o f " w e r v e l " v i s k o s i t e i t g e d e f i n i e e r d door: 8 Ü . 3Ü. -u.u. = e ( ^ + ^ ) I J m dx^ n a a r a n a l o g i e met de v i s k e u z e s c h u i f s p a n n i n g e n . Z o a l s d o o r Bradshaw (1.10) i s aangegeven i s de a n a l o g i e zeer g e b r e k k i g en een v e r s c h u i v e n van de m o e i l i j k h e i d daar e a f h a n k e l i j k i s v a n h e t t u r b u l e n t i e v e l d . N i e t t e m i n g e e f t deze "gradiënt-type" b e n a d e r i n g i n een a a n t a l g e v a l l e n een r e d e l i j k e b e n a d e r i n g . Naar a n a l o g i e met de k i n e t i s c h e t h e o r i e v a n gassen k a n worden g e z i e n a l s h e t p r o d u k t v a n een s n e l h e i d en een l e n g t e .
P r a n d t l nam v o o r de s n e l h e i d u ^ d i e e v e n r e d i g met de snelheidsgradiënt dÜ^/dx2| en een l e n g t e (de mengweglengte) w e r d g e s t e l d ( v o o r de b e -p a l i n g v a n u^u^) met a l s r e s u l t a a t : I 2 du dx^ dU • < 4 > Ook h i e r i s h e t p r o b l e e m de b e p a l i n g v a n 1 .
1.13 T u r b u l e n t e diffusiekoëfficiënt (eddy d i f f u s i v i t y )
I n de t i j d s g e m i d d e l d e t r a n s p o r t v e r g e l i j k i n g e n v o o r een s c a l a i r (warmte, k o n c e n t r a t i e ) komen eveneens "probleem"termen v o o r v a n de vorm -u.y y = k o n c e n t r a t i e f l u k t u a t i e
1
H i e r v o o r k a n a n a l o o g worden g e s t e l d ; 8 r
-u.Y = e
i s g e r e l a t e e r d , maar n i e t g e l i j k aan e^. De v e r h o u d i n g ^^/^y i s r = gemiddelde k o n c e n t r a t i e
3 g e r e l a t e e r d , maar n i e t g e l i
h e t t u r b u l e n t e P r a n d t l o f Schmidt g e t a l ; P r ^ o f Sc^.
1.14 Overgang l a m i n a i r e ^ t u r b u l e n t e s t r o m i n g
Een a n a l y s e v a n de b e w e g i n g s v e r g e l i j k i n g v o o r een l a m i n a i r e s t r o m i n g t o o n t aan d a t de s t r o m i n g boven een b e p a a l d e s n e l h e i d i n s t a b i e l i s . De s n e l h e i d , w a a r b i j deze i n s t a b i l i t e i t o p t r e e d t i s a f h a n k e l i j k v a n de a a r d en de g r o o t t e v a n de s t o r i n g . B i j g r e n s l a g e n i s de initiële i n s t a b i l i t e i t i n de vorm v a n t w e e d i m e n s i o n a l e g o l v e n d i e e c h t e r z e l f i n s t a b i e l z i j n en d r i e d i m e n s i o n a l e vormen aannemen, d i e op hun b e u r t weer v e r v o r m e n . D i t p r o c e s g a a t zeer s n e l waardoor de s t r o m i n g o v e r een z e e r k o r t e a f s t a n d v o l l e d i g t u r b u l e n t w o r d t . De overgang l a m i n a i r t u r b u l e n t kan worden aangegeven met een Reynolds g e t a l .
I n p i j p s t r o m i n g l i g t de grens w a a r b i j initiële v e r s t o r i n g e n i n s t a n d b l i j v e n b i j Re = 2000.
Voor Re < 2000 worden initiële s t o r i n g e n gedempt. B i j een zeer z o r g -v u l d i g e i n s t r o m i n g i s h e t m o g e l i j k de s t r o m i n g l a m i n a i r t e houden t o t Re = 100,000 B i j g r e n s l a g e n i s de k r i t i e k e waarde v a n Re; Re = 10^ a 3.10^ a f h a n k e l i j k v a n de mate v a n t u r b u l e n t i e i n de s t r o m i n g b u i t e n de g r e n s l a a g . B i j v r i j e t u r b u l e n t i e v i n d t de o v e r g a n g a l b i j z e e r k l e i n e Reynolds g e t a l l e n p l a a t s Rej^ = ^ = 10 a 100 D = s t r a a l d i a m e t e r
BIJLAGE 1
1
1-1 D E F I N I T I O N O F T U R B U L E N C E A N D I N T R O D U C T O R Y
C O N C E P T S
The notion of turbulence is generally accepted nowadays, and, broadly speaking, its
meaning is understood, at least by technical people. Yet it is curious to note that the
use of the word "turbulent" to characterize a certain type of flow, namely, the
counterpart of streamline motion, is comparatively recent. Osborne Reynolds, one of
the pioneers in the study of turbulent flows, named this type of motion "sinuous
motion."
Turbulence is rather a familiar notion; yet it is not easy to define in such a
way as to cover the detailed characteristics comprehended in it and to make the
definition agree with the modern view of it held by professionals in this field of
applied science.
According to Webster's "New International Dictionary," turbulence means:
agitation, commotion, disturbance... . This definition is, however, too general, and
does not suffice to characterize turbulent fluid motion in the modern sense. In
irregular motion which in general makes its appearance in fluids, gaseous or liquid,
when they flow past solid surfaces or even when neighboring streams of the same
fluid flow past or over one another." According to this definition, the flow has to
satisfy the condition of irregularity.
Indeed, this irregularity is a very important feature. Because of irregularity,
it is" impossible to describe the motion in all details as a function of time and
space coordinates. But, fortunately, turbulent motion is irregular in the sense that
it is possible to describe it by laws of probability. It appears possible to indicate
distinct average values of various quantities, such as velocity, pressure, temperature,
etc., and this is very important.
Therefore, it is not sufficient just to say that turbulence is an irregular
motion and to leave it at that. Perhaps a definition might be formulated somewhat
more precisely as follows: "Turbulent fluid motion is an irregular condition of
flow in which the various quantities show a random variation with time and space
coordinates, so that statistically distinct average values can be discerned."
The addition "with time and space coordinates" is necessary; it is not sufficient
to define turbulent motion as irregular in time alone. Take, for instance, the case
in which a given quantity of a fluid is moved bodily in an irregular way; the
motion of each part of the fluid is then irregular with respect to time to a
stationary observer, but not to an observer moving with the fluid. Nor is turbulent
motion a motion that is irregular in space alone, because a steady flow with an
irregular flow pattern might then come under the definition of turbulence. Though
the two cases of irregular motions may be useful for studying theoretically certain
aspects of turbulence. It may be remarked that in the first case the Eulerian velocity
at a point with respect to a stationary coordinate system is a random function
of time, in the second case it is the Lagrangian velocity of a fluid particle that is a
random function of time.
As Taylor and Von Karman have stated in their definition, turbulence can be
generated by friction forces at fixed walls (flow through conduits, flow past bodies)
or by the flow of layers of fluids with different velocities past or over one another.
As will be shown in what follows, there is a distinct difference between the kinds of
turbulence generated in the two ways. Therefore it is convenient to indicate turbulence
generated and continuously affected by fixed walls by the designation "wall
turbulence" and to indicate turbulence in the absence of walls by "free turbulence,"
the generally accepted term.
In the case of real viscous fluids, viscosity effects will result in the conversion
of kinetic energy of flow into heat; thus turbulent flow, like all flow of such fluids,
is dissipative in nature. If there is no continuous external source of energy for the
continuous generation of the turbulent motion, the motion will decay. Other effects
of viscosity are to make the turbulence more homogeneous and to make it less
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 3
dependent on direction. In the extreme case, the turbulence has quantitatively the
same structure in all parts of the flow field; the turbulence is said to be homogeneous.
The turbulence is called isotropic if its statistical features have no preference for any
direction, so that perfect disorder reigns. As we shall see later, no average shear
stress can occur and, consequently, no gradient of the mean velocity. This mean
velocity, if it occurs, is constant throughout the field. A more complete definition
of isotropic turbulence will be given later.
In all other cases where the mean velocity shows a gradient, the turbulence
will be nonisotropic, or anisotropic. Since this gradient in mean velocity is associated
with the occurrence of an average shear stress, the expression "shear-flow turbulence"
is often used to designate this class of flow. Wall turbulence and anisotropic free
turbulence fall into this class.
Von Karman^ has introduced the concept of homologous turbulence for the
case of constant average shear stress throughout the field, for instance, in plane
Couette flow.
Frequently the expression "pseudo turbulence", is used; this refers to the
hypothetical case of a flow field with a regular pattern that shows a distinct
constant periodicity in time and space. The difference between pseudo and real
turbulence becomes clear if we compare pictures made of the two types. The first
picture shows a regular flow pattern with constant periodicities throughout the field,
whereas the second can show the condition only at one instant—the next instant
the pattern may have changed in shape and magnitude. Pseudo-turbulent flow fields
may be very useful for simulating real turbulent fields, for they can be made more
accessible to theoretical treatment; it is relatively easy, for instance, to calculate the
dissipation of kinetic energy by viscous effects in such a field. In his book "The
Structure of Turbulent Shear Flow," Townsend suggests a few types of
pseudo-turbulent flows that are suitable for studying various characteristics typical of real
turbulent flows. On the other hand, in using a pseudo turbulence in a theoretical
study to show some of the features of real turbulence, one often has to be very
careful in interpreting the results. For instance, serious errors might result if one
calculated transport and diffusion by turbulence from an assumed pseudo-turbulent
flow pattern, since these processes are mainly, if not entirely, determined by the
irregularity and randomness of the real turbulent motions.
This disorderliness and randomness of turbulence is clearly shown by the
following case. Consider an oscillogram of the velocity fluctuations at a point in a
flow field. If we determine from this oscillogram the number of amplitudes that
have an assigned value, and so the probability of amplitudes, for isotropic turbulence
a Gaussian distribution is obtained. For turbulent shear flow, generally, the
distribution will be more or less skew.
As we pointed out in connection with our definition of turbulence, average
values of quantities exist with respect to time and space. Mere observation of
turbulent flows and of oscillograms of quantities varying turbulently shows that
these average values exist, because:
more or less regularly in time.
2. At a given instant a distinct pattern is repeated more or less regularly in
space; so turbulence, broadly speaking, has the same over-all structure throughout
the domain considered.
If we compare different turbulent motions in each of which a distinct pattern
can be discerned, we shall observe differences, for instance, in the size of the patterns.
This means that, to describe a turbulent motion quantitatively, it is necessary to
introduce the notion of scale of turbulence: a certain scale in time and a certain scale
in space. The magnitude of these scales will be determined by the dimensions of and
the velocities within the apparatus in which the turbulent flow occurs. For turbulent
flow through a pipe, for instance, one may expect a time scale of the order of
magnitude of the ratio between pipe diameter and mean-flow velocity and a space
scale of the order of magnitude of the diameter of the pipe.
It is apparent that it is insufficient to characterize a turbulent motion by its
scale alone, since to do so does not tell us anything about the violence of the
motion. One cannot take the average value of the velocity as a measure of this
violence, because the violence ofthe fluctuations with respect to this average velocity
is just what one wants to know.
If the momentary value of the velocity is written
U=Ü + u
where the overscore denotes the average value, so that by definition ö = 0, kmight
be possible to take the average ofthe absolute values ofthe fluctuation, i.e., |u|, as a
measure of this violence. However, it is not usual to do it in this way. For reasons
which will become obvious later on, it has been usual, since Dryden and Kuethe^^
introduced this definition in 1930, to define the violence or intensity of the turbulence
fluctuations by the root-mean-square value
The relative intensity will then be defined by the ratio
j / t Ü
Average values can be determined in various ways. If the turbulence flow
field is quasi-steady, or stationary random, averaging with respect to time can be
used. In the case of a homogeneous turbulence flow field, averaging with respect
to space can be considered. It is not always possible, however, to take time-mean
or space-mean values if the flow field is neither steady nor homogeneous. In such
cases we may assume that an average is taken over a large number of experiments
t Many investigator.s, including Dryden and Kuethe,'" use the designation 'intensity" or "degree of turbulence" or "turbulence level" for the relative intensity just defined.
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 5
that have the same initial and boundary conditions. We then speak of an
ensemble-mean value.
If we use the Eulerian description of the flow field, one of the above three
methods of averaging may have to be applied to a varying quantity at any point
in the flow field.
If we want to study turbulent transport or diffusion processes, it is often
convenient to apply the Lagrangian description of the paths of separate fluid
particles. In this case averaging may be carried out with respect to a large number
of particles that have either the same starting time but different origins (this requires
on the average a homogeneous flow field) or the same origin but different starting
times (this requires on the average a quasi-steady flow field). Of course we may also
consider an ensemble average.
Expressed in mathematical form, the three methods of averaging applied, for
instance, to the Eulerian velocity U, are:
Time average for a stationary turbulence:
1 1
C'^^U{xo) = lim dt Uixo.t)
T-^oo J - T
Space average for a homogeneous turbulence
1
0(to) =''lim
+X
Ensemble average of N identical experiments:
dx V{x,to)
-XN
e_ Y."U„iXo.to)
Ü{xo,to) =
-By introducing a probability density function ''^{U), which in normalized form
satisfies
dU'^SiU)
the ensemble average may be also expressed as follows
+ COD{xoJo)= dVU'^iU)
^ -«.
For a stationary and homogeneous turbulence we may expect and assume that
the three averaging procedures lead to the same result.
t s e D = D^Ü
fore, for practical reasons, we cannot carry out the averaging procedures with
respect to time or with respect to space for infinite values of T or Z respectively,
but only for finite values. However, certain conditions then have to be satisfied.
Let us for instance, consider averaging with respect to time of the Eulerian
velocity of a turbulent flow. The flow may contain very slow variations that we do
not wish to regard as belonging to the turbulent motion of the flow. Take, for
instance, the case where the tui;bulent flow through a duct shows a slight pulsation
of low frequency—or, in meteorology, where we wish to distinguish between the
average wind speed during certain periods of the day and the average speed
during much longer periods.
Therefore we take T to be a finite time interval. This interval must be
sufficiently large compared with the time scale Ti of the turbulence or else, since
this corresponds to a certain quasi periodicity, with the main period of change in
flow pattern. On the other hand it must be small compared with the period T2
of any slow variations in the field of flow that we. do not wish to regard as
belonging to the turbulence. It is clear that there is a certain arbitrariness in the
choice of the fluctuations that we do wish to consider. Fortunately, in practice, such a
choice can be made without too much difficulty. If we take an oscillogram of a
turbulent flow, it is usually easy to discern some average main period of the change
in flow pattern. Furthermore it may be helpful to keep in mind that, for a given
mean velocity, the order of magnitude of such main periods corresponds to the
size of the turbulence-generating object or of the apparatus in which the turbulent
flow is studied.
Taking for T a finite value, we now define the average value by
with the condition Ti « T « T2.
The average value should be independent of the origin t of the averaging
procedure, provided t < T2. Thus dÜldt should be either zero or, in the case of a
slightly varying main flow, negligibly small.
In the foregoing an average value has been designated by an overscore. In this
book any averaging procedure will be denoted by such an overscore. In the study
of turbulence we often have to carry out an averaging procedure not only on single
quantities but also on products of quantities. Here the overscores have the following
properties.
Let A = A + a and B = B + h. In any further averaging procedure A and B
may be treated as constants. Thus,
U =
T
dx Uit +
T )A=A+a=A+a=A+a
whence a = 0
AB^AB--=AB
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 7
Similarly,
Ba = Ba = Ba = 0 since 5 = 0
AB = (A + a)iB +
b) =
M^Ab +
Ta + ab
= AB^d)
We have already mentioned the concepts of space scale and time scale and,
corresponding to them, the quasi periodicity in turbulence. A study of photographs
of turbulent flows or of oscillograms of velocity fluctuations will reveal that, properly
speaking, it is not permissible to speak of the quasi periodicity or the scale of
turbulence. It is possible to speak of an average maximum quasi periodicity or
scale determined principally by the dimensions of the apparatus. But besides this
there are many smaller quasi periodicities and others smaller still. Turbulence
consists of many superimposed quasi-periodic motions.
The counterpart of the scale or quasi periodicity is the quasi frequency. Hence
many quasi frequencies are present.
The characteristic features of turbulence: irregularity and disorderliness, involve
the impermanence of the various frequencies and also of the various periodicities
and scales. For this reason we have used the adjective "quasi." Henceforward, if
we keep this impermanent character in mind, we can for convenience leave out the
"quasi."
It is said that turbulence consists of the superposition of ever-smaller periodic
motions—or, since a periodicity in velocity distribution involves the occurrence of
velocity gradients which correspond to a certain vortex motion, the extent of which is
determined by the periodicity, we may also say that turbulence consists of the
super-position of eddies of ever-smaller sizes. But can this go on indefinitely? Intuitively,
one would expect not. In real fluids viscosity effects prevent this from happening.
The smaller an eddy, the greater also in general the velocity gradient in the eddy
and the greater the viscous shear stress that counteracts the eddying motion. Thus,
in each turbulent flow, there will be a statistical lower limit to the size of the smallest
eddy; there is a minimum scale of turbulence that corresponds to a maximum
frequency in the turbulent motion.
All these various-sized eddies of which a turbulent motion is composed have a
certain kinetic energy, determined by their vorticity or by the intensity of the
velocity fluctuation of the corresponding frequency. An interesting question which
soon arises when the more detailed features of turbulence are being studied is how
the kinetic energy of turbulence will be distributed according to the various
fre-quencies. Although, as stated, in real turbulence a distinct frequency is not
per-manently present, yet it is possible on the average to allocate a certain amount of
the total energy to a distinct frequency. Such a distribution of the energy between
the frequencies is usually called an energy spectrum. It can be established by means
of suitable instruments. Though a harmonic analysis of the velocity fluctuations
can be carried out, this fact is no proof that, conversely, the turbulent fluctuations
are composed of these harmonics. Compare the similar problem in the case of
a number of harmonics). Burgers^ has drawn attention to the similar controversy
in the case of light, an old one in the theory of optics, namely, whether the colors
of the spectrum can be said to be present originally in white light or whether they
are produced by the spectroscope.
In the foregoing we have spoken about turbulent motion, which can be
assumed to consist of the superposition of eddies of various sizes and vorticities
with distinguishable upper and lower limits. The upper size limit of the eddies is
determined mainly by the size ofthe apparatus, whereas the lower limit is determined
by viscosity effects and decreases with increasing velocity of the average flow, other
conditions remaining the same. Within these smallest eddies the flow is of a strong
viscous nature, where molecular effects are dominant. Now the reader may wonder
whether these smallest eddies might not become so small that the flow within them
could no longer be treated as a continuum flow. In other words, what is the size
of these smallest eddies compared with the mean free path of the molecules? The
following figures may help to convey an idea of the problem.
For moderate flow velocities, that is, not much greater than, say, 1(X) m/s, the
smallest space scale or eddy will hardly be less than about 1 mm; this value is still
very large compared with the mean free path in gases under atmospheric conoitions,
which is of the order of 10 ""^ mm. One cubic millimeter of air under atmospheric
conditions contains roughly 2.7 x 10^^ molecules. Thus gases under atmospheric
conditions and certainly liquids also may be treated as continua in the study of
turbulent flow of moderate speed.
Relevant values of turbulent fluctuations are roughly 10 per cent of average
velocity and are between, say, 0.01 and 10 m/s. These values must be compared with
the mean velocity of molecules, which for air is of the order of 500 m/s. Turbulence
frequencies vary between, say, 1 and 10,000 s~S whereas molecular-collision
fre-quencies for air are about 5 x 10^ s" ^
Jhe domain of turbulent magnitudes is, therefore, sufficiently far away from
the domain of molecular magnitudes.
We will conclude this first introduction by discussing a few photographs of
fluid motion which will serve to elucidate the specific character of turbulent flow.
Figure 1-1 shows the flow pattern just downstream of a circular cylinder at
low values of the Reynolds number."^ The general flow pattern is so regular that it
hardly falls within the definition of real turbulence, that is, the condition in which
randomness prevails. At most this might be considered a pseudo turbulence.
Figure 1-2 shows a similar flow pattern, but one pertaining to a higher value
of the Reynolds number.^ Up to downstream distances 30 to 40 times the cyUnder
diameter, the general flow pattern is still fairly regular; the more detailed patterns—
and, beyond this distance, the general flow pattern also—gradually become more
and more turbulent. The detailed patterns become more turbulent as the Reynolds
number increases; this is seen clearly in Fig. 1-3, which shows a close-up of the
flow pattern close behind the cylinder."^ Within the region of the large regular eddies
G E N E R A L I N T R O D U C T I O N A N D CONCEPTS 9
FIGURE 1-1
Flow pattern downstream of a cylinder. Low Reynolds numbers.
the flow pattern is distinctly turbulent, with a space scale much smaller than those
of the large regular eddies.
The regularity and irregularity of the flow in the wake of a cyhnder are well
illustrated by velocity oscillograms taken at different locations in the wake flow.
Figure 1-4 shows such oscillograms together with one taken in the turbulent flow
through a windtunnel. An oscillogram taken at a point on the line through the
centers of the vortices of each row (that is, at a point eccentric with respect to the
center of the wake flow) shows a preference for a distinct frequency; an oscillogram
taken centrally behind the cylinder (that is, on the center line of the wake flow) also
shows a preference for a distinct frequency, which is, however, equal to twice the
previous frequency (this is the effect of the vortices, which are separated from either
side of the cylinder alternately). Compare these osciflograms with that for the real
turbulent flow through a windtunnel, and the difference is clear.
Figure l-5a to ƒ shows a series of flow patterns corresponding to increasing
distances downstream from a grid (mesh M = 45 mm, rod diameter d= 15 mm)
which is towed at a speed of 66.5 mm/s through a stagnant liquid. Whereas
FIGURE 1-2
FIGURE 1-3
Flow pattern close behind a cylinder. High Reynolds number.
Fig. l-5a shows a distinct and regular pattern, the patterns shown subsequently
become less and less distinct. Figure U5d to ƒ shows, in addition to the gradually
increasing turbulent character, the decay of the eddies; the contours become less
sharp.
The foregoing photographs have shown the generation of turbulence through
the flow of a fluid relative to a solid body. The following photographs show this
generation and development of turbulence when two neighboring streams of the
same fluid flow past each other.
Figure 1-6 shows the initial vortices produced at the boundary of a free jet
(half-jet boundary). During the development of these vortices farther downstream,
there is a gradual transition into irregular turbulent flow. Figure 1-7 shows the
Turbulence in windtunnel, u'/Ü = 0.0085. Time — 0.4 s. Relative amphfication = 64.
Center of wake. 63 mm behind 21 mm cylinder. Time — 0.3 s. Relative amplification = 1.
10 mm laterally from center of wake. Time — 0.3 s. Relative amplification = 1.
38 mm laterally from center of wake. Time — 0.3 s. Relative amplification = 8. FIGURE 1-4
Oscillograms of turbulence in a windtunnel and in the wake 63 mm behind a 21 mm
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 11
FIGURE 1-5
Flow pattern behind a grid. U =
Prandtl, LP)
FIGURE 1-6
Vortices at boundary of a half-jet. (From: Fliigel, G.,* by permission of the Verlag des Vereins Deutscher Ingenieure.)
turbulent character of the flow in a free jet, with the separate eddying domains
at the boundary region still distinguishable.
Figure 1-8 shows a "Schlieren" photograph of a jet at just the moment when
it was issuing.^ In the first stages distinct separate vortices are seen, in the later
stages complete turbulent flow.
As Fig. 1-7 shows, the flow in the central region of a jet is different from
the flow in the boundary region: the boundary-region flow is not continuous but
becomes more and more intermittent toward the outside. This difference in character
will of course find expression also in oscillograms taken at different points in the
jet. This is shown by Fig. 1-9. This difference in character will be discussed
FIGURE 1-7
Flow pattern i n a free jet. (Photograph by Van der Hegge Zijnen; taken at Royal/Dutch-Shell Laboratory, Delft.)
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 13
extensively in Chap. 6. At the moment it will be sufficient to mention that this
intermittence, which is typical of turbulence in free boundaries, indicates the presence
of large-scale eddies.
Finally, Fig. 1-10<7 and b shows the flow through a channel as photographed
with a moving camera."^ With a view to a more precise definition of scale of
FIGURE 1-9
Oscillograms of velocity fluctuations in a free jet at different distances from the axis.
(From: Corrsin. S.p° by permission of the National Advisory Committee for Aeronautics.)
FIGURE 1-10
Flow pattern in a channel. (Nikwadse, in Prandtl, L. and O. Tietjens^)
turbulence, to be given later, it is important to note that there seems to be a
correlation between the velocities within a region which extends from the center of
the channel to about midway between the center and the wall (this is particularly
strongly suggested by Fig. 1-lOb).
In the foregoing we have mentioned and discussed briefly certain features of
turbulent flows. I t may be useful to summarize them..
Thus, turbulence is a random phenomenon which shows a quasi-permanency
and quasi-periodicity both in time and in space. I n a sense this may be considered
as being an assumption, due to the fact that our information concerning the
turbulent flow at any instant is incomplete, so that henceforth it is impossible to
predict in detail the future behavior of the flow field.
Another assumption is that turbulence is a continuum phenomenon, an
assumption applicable to liquids and gases under atmospheric conditions. But caution
is dictated in the case of ultra-high supersonic or hypersonic turbulent flows.
The turbulence is characterized by a strong diffusive nature with respect to
any transferable property. And as all flows of real fluids it is dissipative due to
viscous actions; and therefore rotational. This dissipation takes place mainly in the
region of the smallest eddies of the hierarchy of eddies of many different sizes of
which the turbulence may be composed. There is a strong interaction between these
eddies due to the nonlinear and three-dimensional character of turbulence. For, a
G E N E R A L I N T R O D U C T I O N A N D C O N C E P T S 1 5
nonhomogeneous vortex flow pattern can only be either one-dimensional or
three-dimensional.
These nonlinear and spatial interactions of vortices resulting in transfer of
kinetic energy between them, the diffusive nature of turbulence, and another important
property not mentioned hitherto, namely the memory behavior, will be discussed in
more detail in later chapters.
K U R S U S W E T E N S C H A P P E L I J K E S T A F
T u r b u l e n t e s t r o m i n g e n
2. S t o c h a s t i s c h e b e s c h r i j v i n g v a n t u r b u l e n t i e
