The physical and mathematical modelling of the flow field in the mold region in continuous casting systems: Part II. The
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useful b a s i s of c o m p a r i s o n for t e s t i n g the v a l i d i t y of t h e o r e t i c a l p r e d i c t i o n s . The t h e o r e t i c a l d e s c r i p t i o n of the v e l o c i t y (and t e m p e r a t u r e ) fields in the mold r e g i o n , to be given in the s u b s e q u e n t s e c t i o n s of the p a p e r , is r i g o r o u s , in a s e n s e that it is b a s e d on the solution of the t u r b u l e n t equation of m o t i o n . However, in viewing this m a t t e r m o r e r e a l i s t i c a l l y , the d e s c r i p tion should also be r e g a r d e d as a model b e c a u s e of the s i m p l i f y i n g a s s u m p t i o n s i n t r o d u c e d in the s t a t e m e n t c o n c e r n i n g the b o u n d a r y conditions and in the r e p r e s e n t a t i o n of the Reynolds s t r e s s e s . It is noted that until quite r e c e n t l y the solution of m u l t i d i m e n s i o n a l t u r b u l e n t flow p r o b l e m s would have r e p r e s e n t e d a f o r m i d a b l e , if not i m p o s s i b l e u n d e r t a k ing. However, r e c e n t l y Spalding and c o w o r k e r s d e veloped a methodology for the n u m e r i c a l solution of t w o - d i m e n s i o n a l t u r b u l e n t flow p r o b l e m s ;la the t e c h nique to be used in the p r e s e n t p a p e r is a m o d i f i c a tion of Spalding's approach. Since t h e r e e x i s t a n u m b e r of t u r b u l e n t flow p r o b l e m s in m e t a l s p r o c e s s i n g o p e r a t i o n s , the a n a l y s i s of which has yet to be a t t e m p t e d , it is hoped that this work m a y p r o v i d e a s t i m u l u s for r e s e a r c h in this a r e a . R e g a r d i n g the o r g a n i z a t i o n of this p a p e r , the f o r m u l a t i o n is in Section 1 and the technique of solution in Section 2. The c o m p u t e d r e s u l t s a r e given in Section 3 and f i n a l l y , the concluding r e m a r k s a r e c o n tained in Section 4.
1) FORMULATION Let us c o n s i d e r a r e c t a n g u l a r or a c y l i n d r i c a l mold into which m o l t e n s t e e l is b e i n g p o u r e d f r o m a nozzle, e . g . , such as sketched in Fig. 1. Let us a s s u m e , f u r t h e r m o r e , that in the upper mold r e g i o n the t h i c k n e s s of the s o l i d i f i e d l a y e r is s m a l l enough so that any d i s t o r t i o n in the g e o m e t r y of the liquid pool m a y be n e glected. The p r o b l e m is then to c a l c u l a t e v e l o c i t y d i s t r i b u t i o n and the t e m p e r a t u r e field within the pool for a given mold and nozzle g e o m e t r y , c a s t i n g r a t e , VOLUME 4, MAY 1973-1379
i n l e t t e m p e r a t u r e of the m e t a l and a s p e c i f i e d h e a t flux at the w a l l s ,
of the c a s t i n g v e l o c i t y ) w h e r e a s x 2 is p e r p e n d i c u l a r to the m o l d w a l l s .
1.1) G e n e r a l Equations
EQUATION
F o r t u r b u l e n t flow of an i n c o m p r e s s i b l e fluid the g e n e r a l f o r m of the c o n s e r v a t i o n equations is give
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