An unsteady heat transfer problem related to high power lasers

2 4 equat ions a r e o b t a i n e d f a r A and 8 by s u b s t i " * c " [ 3 26 6 ( 6 ) ' 2 + 1 ( q ] 8 6 , ( 3 ) s t i t u t i o n , a s i n g l e d i f f e r e n t i a l equa t ion i s 6 h ) c(-) '' ' ; c = 4.64 . ( 4 ) ob ta ined f o r A . The equa t ion i s so lved by us ing % a techn ique S i m i l a r t o Karman momentum i n t e g r a l urn i s t h e cons tan t f r e e stream v e l o c i t y , 6 techn ique. t h a t t h e thermal boundary layer t h i ckness inc reases and a are the kinematic and the thermal f e r c o e f f i c i e n t decreases w i t h i n c r e a s i n g u . Numerical r e s u l t s a r e ob ta ined by us ing the f o u r t h o r d e r R u m K u t t a method. I t i s found. W i t h u and t h e amp l i t ude of the w a l l heat t rans diffusivity of the gas reSpeCtively, The temperais the hydrodynamic boundary layer thickness, " t u r e T(x .y . t ) s a t i s f i e s t h e boundary c o n d i t i o n s . T = O a t y = O and T = T _ c o s u t as y + m . i n t r o d u c t i o n ( 5 ) A pu lsed c losed c y c l e e l e c t r i c d ischarge l a s e r i s one o f t h e dev ices c u r r e n t l y be ing developed fo r o b t a i n i n g h i g h energy i n t e n s i t y l a s e r beams. T h e . Assuming a s o l u t i o n o f the form, o p t i c a l q u a l i t y o f the beam i s d i r e c t l y r e l a t e d t o ach iev ing a h i g h medium homogeneity i n t h e c a v i t y . T = A(x,y) cos ut + B(x,y) s i n ut ( 6 ) Th is medium homogeneity depends on an e f f e c t i v e des ign of t h e flow loop. L we get , A complete d e s c r i p t i o n o f t h e f l o w loop and the assoc ia ted problems was c l e a r l y d iscussed by ( 7 ) aB aB a's Hunukut la and Venkataraman.' A c l a s s o f unsteady ax av ~ z -wA + u + v = u heat t r a n s f e r problems was i d e n t i f i e d by them. dY So lv ing these unsteady heat t r a n s f e r problems would r e s u l t i n a b e t t e r unders tand ing o f t h e f l o w loop design. Work was i n i t i a t e d by s o l v i n g the problem o f the response o f t h e w a l l heat t r a n s f e r t o a s t e p change i n t h e f l u i d b u l k temperature f o r t h e f l o w p a s t a f l a t p l a t e . and aA aA a2A uB + " + v = u a Y 2 ax ay Equat ion (8) can be used to e l i m i n a t e B ( x , y ) f rom ( 7 ) . The r e s u l t i n g equa t ion f o r A(x,y) i s g i v e n by In t h i s paper, t h e unsteady heat t r a n s f e r due t o the p e r i o d i c change i n t h e b u l k temperature o f t h e f l u i d i s eva lua ted . The p l a t e temperature i s assumed t o be cons tan t and the f l u i d temperature o u t s i d e the thermal boundary l a y e r i s assumed t o 2 a a2A aA aA ax ay a Y 2 -w A + u [a -u v -1 be o f the form T_ cos ut . ax Hathemat i c a l Fo rmu la t i on The gove rn ing equa t ion for unsteady twodimensional thermal boundary l a y e r i s g i ven by aT aT aT azT + o + " = a a t ax ay ay2 (9) *Center for t h e A p p l i c a t i o n o f Mathematics/OepartA procedure s i m i l a r t o Karman-Pohlhausen **Energy Research Center, Member AIAA equa t ion (9 ) . Assuming a thermal boundary layer ment o f Hathematics i n t e g r a l techn ique i s f o l l owed for the s o l u t i o n o f of h ( x ) , a p r o f i l e o f the form Copyright C! Arnerinn Inrlilule of Atronsutica and AIlmmUtics. he. . 1986. All righlr reserved. 4 I s se lec ted . The above p r o f i l e f o r A , s a t i s f i e s t h e c o n d i t i o n s , v S u b s t i t u t i n g t h e r e l a t i o n s (2-4) and (IO) i n equa t ion (9) and i n t e g r a t i n g t h i s equa t ion from y 0 t o y h , we g e t a d i f f e r e n t i a l equa t ion for h(x) as, Reynolds number based on boundary l a y e r th ickness 2000 , V P r a n d t l Number = 0.7 , Range o f t h e f requency parameter (F = %) = 0.33 33.33 . (12) u. , , A f t e r s o l v i n g equa t ion ( i 2 ) n u m e r i c a l l y and o b t a i n i n g h as a f u n c t i o n o f t h e boundary layer th ickness 6 , t h e f u n c t i o n s A and 8 can be eva lua ted . I t can be seen from equa t ion ( IO) t h a t t h e f u n c t i o n A i s independent o f w . However 8 depends on w . The f u n c t i o n s A and 8 a r e d i s p l a y e d I n F i g u r e I. F i g u r e 2 g i v e s a p l o t o f h/ho as a f u n c t i o n o f the f requency parameter F . Here h 15 t h e thermal boundary l a y e r t h i ckness for any F and h, 1s the thermal boundary layer th ickness cor respond ing t o F = 0 , which i s t h e steady s t a t e case. I t can be 5een t h a t the thermal boundary layer t h i ckness inc reases as t h e f requency of o s c i l l a t i o n inc reases . The local w a l l Nusse l t number Nu i s DTOp o r t i o n a i t o the d e r i v a t i v e o f t h e temperature, a A an . ay a y Nu = (cos w t + s i n u t ) 1 4 wx 2 h 4 96 5 + (-1 (5) = 'r 4 Equat ion (12) i s to be so l ved s u b j e c t t o t h e The r a t i o o f t h e amp l i t ude o f Nu a t any F i n i t i a l c o n d i t i o n t o the steady s t a t e case Nu i s p l a t t e d as a f u n c t i o n of F i n F i g u r e 30 I t can be seen t h a t the heat t r a n s f e r c o e f f i c i e n t decreases a5 the f requency o f o s c i l l a t i o n increa5es. ( 1 3 ) 1,(0) = 0 . I t i s seen f rom (12) t h a t t h e te rm I 14 wx 2 h 4 1.0 5 u , n u m e r i c a l l y and the i n i t i a l c o n d i t i o n s a r e gener(-1 (a) becomes smal l a t t h e o r i g i n x = 0 and equa t ion (12) degenerates t o t h e steady s t a t e F = 0 .33 equat ion . Cor respond ing ly e q u a t i o n (12) i s so l ved a t e d by cons ide r ing t h e steady s t a t e form o f t h e above. The r e s u l t s a re d iscussed i n t h e n e x t sec t ion . B