sheet geometry (Moore 1979, Meiron et al. 1982). Experimental observations (e.g. Thorpe 1971) indicate that an initially sinusoidal perturbation of a plane vortex ...
www.moffatt.tc TURBULENCE AND CHAOTIC PHENOMENA IN FLUIDS T. Tatsumi (editor) Elsevier Science Publishers B.V. (North-Holland) 8 IUTAM, 1984
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SIMPLE TOPOLOGICAL ASPECTS O F TURBULENT VORTICITY DYNAMICS H.K.
Moffatt
Department of Applied Mathematics and T h e o r e t i c a l P h y s i c s U n i v e r s i t y o f Cambridge Cambridge, England
The p r o c e s s by which t u r b u l e n t v o r t i c i t y i s i n t e n s i f i e d i s d i s c u s s e d , i n r e l a t i o n t o t h e dimensiona l i t y o f t h e s u b s p a c e i n which v o r t i c i t y i s concent r a t e d . For a s i n g l e , randomly t w i s t e d v o r t e x f i l a m e n t , no v o r t e x s t r e t c h i n g can o c c u r and t h e e n s t r o p h y remains bounded f o r a l l t i m e . If the v o r t i c i t y i s d i s t r i b u t e d on a randomly c o n v o l u t e d s u r f a c e , t h e t o t a l h e l i c i t y of t h e flow b e i n g nonz e r o , t h e n t h e e n s t r o p h y c a n , and i n g e n e r a l w i l l , i n c r e a s e w i t h o u t l i m i t due t o l o c a l i n s t a b i l i t i e s o f Kelvin-Helmholtz t y p e , l e a d i n g t o s p i r a l s i n g u l a r i t i e s of t h e v o r t e x s h e e t . The r e l a t i o n between t h e s t r u c t u r e o f t h e s e s p i r a l s i n g u l a r i t i e s and t h e s p e c t r a l exponent i n t h e i n e r t i a l r a n g e of wavenumbers i s d e m o n s t r a t e d . INTRODUCTION
One o f t h e e s s e n t i a l f e a t u r e s of t h e p r o c e s s o f e n e r g y t r a n s f e r i n t u r b u l e n t f l o w from l a r g e t o small s c a l e s i s t h e a s s o c i a t e d tendency f o r e n s t r o p h y ( i . e . mean-square v o r t i c i t y ) t o i n c r e a s e . This increase i s u l t i m a t e l y b a l a n c e d ( i n a s t a t i s t i c a l l y s t e a d y s t a t e ) by v i s c o u s d i s s i p a t i o n o f e n s t r o p h y ; b u t i t is l e g i t i m a t e t o e n q u i r e what happens i n t h e i n v i s c i d l i m i t v -+ 0 , and i n p a r t i c u l a r t o i n v e s t i g a t e t h e s t r u c t u r e o f t h e v o r t i c i t y f i e l d t h a t develops i n t h i s l i m i t , a s t h e e n s t r o p h y becomes unbounded. This s t r u c t u r e ( i n p h y s i c a l space) presumably h a s a b e a r i n g on t h e e n e r g y spectrum t h a t i s e s t a b l i s h e d on s c a l e s a t which v i s c o u s e f f e c t s a r e n e g l i g i b l e . The e q u a t i o n f o r mean-square v o r t i c i t y i n homogeneous t u r b u l e n c e i s
where y = c u r l g. equation is
The most naPve c l o s u r e e n a b l i n g i n t e g r a t i o n of t h i s
