The mathematical and physical modeling of turbulent recirculating flows
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1. INTRODUCTION THERE are numerous metals processing operations in which a m o l t e n m e t a l pool held in a ladle o r mold u n d e r g o e s t u r b u l e n t r e c i r c u l a t i n g m o t i o n . T h e mold r e g i o n in continuous casting, a r g o n - s t i r r e d l a d l e s , induction s t i r r e d v e s s e l s , e l e c t r o s l a g r e f i n i n g , and oxygen s t e e l m a k i n g p r o c e s s e s m a y be cited as typical e x a m p l e s . In r e c e n t y e a r s , a g r e a t deal of work has b e e n done on the m a t h e m a t i c a l m o d e l l i n g of t h e s e s y s t e m s , through the s t a t e m e n t of the t u r b u l e n t N a v i e r - S t o k e s equations and u s i n g a c o m p u t a t i o n a l a l g o r i t h m that was p r o p o s e d by Spalding. x'2 The p r e d i c t i o n s b a s e d on this m o d e l have b e e n c o m p a r e d with e x p e r i m e n t a l m e a s u r e m e n t s obtained u s i n g both m o l t e n m e t a l s and aqueous systems.3"~ It was found that the e x p e r i m e n t a l l y m e a s u r e d and the t h e o r e t i c a l l y p r e d i c t e d v e l o c i t i e s w e r e in r e a s o n a b l e a g r e e m e n t . It has to be s t r e s s e d , however, that the m e a s u r e m e n t s a v a i l a b l e up to the p r e s e n t were not c o m p r e h e n s i v e . Some of the data obtained, such as the r e s u l t s of t r a c e r d i s p e r s i o n t e s t s , p r o v i d e only a n ind i r e c t m e a s u r e of the v e l o c i t y field and of the t u r b u lence l e v e l s . A r e c e n t study, 3 u s i n g a w a t e r m o d e l for t e s t i n g the m a t h e m a t i c a l r e p r e s e n t a t i o n , was not c o n c l u s i v e b e c a u s e of both the u n c e r t a i n t i e s r e g a r d ing the b o u n d a r y conditions when m o d e l l i n g a n a r g o n s t i r r e d ladle, and the i n h e r e n t e x p e r i m e n t a l i n a c c u r a cies involved in d e t e r m i n i n g low liquid v e l o c i t i e s , u s ing a hot f i l m a n e m o m e t e r . In p a r t i c u l a r , no r e l i a b l e v e l o c i t y m e a s u r e m e n t s were p r o d u c e d in the v i c i n i t y of the w a l l s . In the work done up to the p r e s e n t , r e l a t i n g to m e t a l l u r g i c a l s y s t e m s , the n e a r wall r e g i o n has not b e e n studied, notJ. SZEKELY is Professor of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. A. H. DILAWARIis Visiting Engineer, on leave from the Institute of Chemical Engineeringand Technology, University of the Punjab, Lahore-l, Pakistan, and R. METZ, formerly Graduate Student Department of Chemical Engineering, State University of New York at Buffalo, is now Research Engineer, Bethlehem Steel Corp., Homer Research Laboratory, Bethlehem, PA 18016. Manuscript submitted March 18, 1977. METALLURGICALTRANSACTIONSB
w i t h s t a n d i n g its i m p o r t a n c e in heat and m a s s t r a n s f e r phenomena. M o r e o v e r , a l l the work done up to the
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