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UBC Theses and Dissertations

Combined free and forced convection from horizontal plates Classen, Lutz 1968

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COMBINED FREE AND FORCED CONVECTION FROM HORIZONTAL PLATES by LUTZ CLAASSEN B . A . S c , U n i v e r s i t y of B r i t i s h Columbia 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A .Sc . i n the Department of Mechan ica l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d s tandard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1968 In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. L. Claassen Department of Mechanical Engineering The University of B r i t i s h Columbia Vancouver 8 , B.C. i i ABSTRACT A t h e o r e t i c a l a n a l y s i s and e x p e r i m e n t a l r e s u l t s are presented f o r . f ree convect ion and combined f r e e and f o r c e d convect ion from a heated h o r i z o n t a l s u r f a c e . The p r i n c i p a l o b j e c t i v e was to i n v e s t i g a t e a laminar boundary layei? f low which had been shown, t h e o r e t i c a l l y o n l y , to form above a heated s u r f a c e . This boundary l a y e r f low i s fundamental ly d i f f e r -ent from f lows above i n c l i n e d or v e r t i c a l s u r f a c e s s i n c e the d r i v i n g fo rce or buoyancy f o r c e acts p e r p e n d i c u l a r to the pr imary boundary l a y e r mot ion . The f lows analyzed are those f o r which the system of p a r t i a l d i f f e r e n t i a l equat ions d e s c r i b i n g the f low can be reduced to s imul taneous t o t a l d i f f e r e n t i a l e q u a t i o n s . The method i n v o l v e s the i n t r o d u c t i o n of s i m i l a r i t y parameters and then the n u m e r i c a l i n t e g r a t i o n of the r e s u l t i n g s i m p l i f i e d system of t o t a l d i f f e r e n t i a l e q u a t i o n s . These s o l u t i o n s are r e s t r i c t e d , f o r 2 - d i m e n s i o n a l f l o w , to a s e m i - i n f i n i t e s u r f a c e , and f o r a x i a l l y - s y m m e t r i c a l f l o w , to an i n f i n i t e d i s c . In c o n j u n c t i o n w i t h the former on ly f r e e convec t ion i s cons idered w h i l e f o r the l a t t e r combined convect ion i s cons idered as w e l l . The f low was examined e x p e r i m e n t a l l y w i t h a s e m i - f o c u s i n g co lou r s c h l i e r e n system. From the photographs i t may be concluded that the s e m i - i n f i n i t e s u r f a c e a n a l y s i s would c o r r e c t l y p o r t r a y a p h y s i c a l f l o w . The f l o w , though, remains l a m i n a r f o r a short d i s t a n c e only and then breaks down i n t o an u n s t a b l e c e l l u l a r p a t t e r n . The a x i a l l y - s y m m e t r i c a l a n a l y s i s , a l though i t y i e l d e d a n a l y t i c a l l y a v a l i d boundary l a y e r s o l u t i o n , appears to have no p h y s i c a l p a r a l l e l above a d i s c of f i n i t e r a d i u s . i i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i i NOMENCLATURE i x I INTRODUCTION 1 II REVIEW OF PREVIOUS WORK 5 III FREE CONVECTION FROM A SEMI-INFINITE PLATE 8 3.1 Theoretical Solutions for an A r b i t r a r y 8 Pr Number 3.2 Asymptotic Solutions 19 3.2.1 Pr » 1 19 3.2.2 Pr « 1 22 IV FREE AND COMBINED CONVECTION FROM A DISC 31 4.1 Free Convection for an Ar b i t r a r y Pr Number 33 4.2 Combined Free and Forced Convection f o r an 38 Ar b i t r a r y Pr number and a Disc Temperature which i s Proportional to r^ 4.3 Numerical Integration 42 V EXPERIMENTAL APPARATUS 44 5.1 O p t i c a l Bench 44 5.2 Alignment of the Opt i c a l Benches 46 5.3 Colour F i l t e r 46 5.4 Rectangular Plates and Disc 47 VI EXPERIMENTAL PROCEDURE 49 6.1 Test Procedure 49 6.2 Experimental Results 49 Contents cont inued i v V I I DISCUSSION OF THE RESULTS 5 7 7.1 Numer ica l S o l u t i o n s 58 7.2 S c h l i e r e n Observat ions 60 V I I I CONCLUSIONS 64 REFERENCES 66 APPENDIX A SOME ASPECTS OF THE SCHLIEREN SYSTEM 73 APPENDIX B CALCULATIONS OF BOUNDARY LAYER DEPTH AND 77 CRITICAL GRASHOF NUMBER V LIST OF TABLES Table I Boundary Values f o r the S e m i - I n f i n i t e 68 I s o t h e r m a l P l a t e Table I I Asymptot ic Funct ions f o r Pr -*-°°(Inner S o l u t i o n ) 69 Table I l a Asymptot i c Funct ions f o r P r -»• °° (Outer S o l u t i o n ) 70 Table I I I Asymptot i c Funct ions f o r P r -*• 0 (Outer S o l u t i o n ) 71 Table I l i a Asympto t i c Func t ions f o r P r -> 0 ( Inner S o l u t i o n ) 72 VI LIST OF FIGURES F i g u r e 1 C a r t e s i a n Axes N o t a t i o n 8 F i g u r e 2 Stream Funct ions f o r 2 -D imens iona l Flow 15 F igure 3 V e l o c i t y Funct ions f o r 2 -D imens iona l 16 Flow F i g u r e 4 P ressure Funct ions f o r 2 -D imens iona l 17 Flow F igure 5 Temperature Funct ions f o r 2 - D i m e n s i o n a l 18 Flow F i g u r e 6 P ressu re Funct ions f o r 2 -D imens iona l 27 Flow i n Terms of Asymptot ic Coordinates f o r Pr -> °° F i g u r e 7 Temperature Funct ions f o r 2 -D imens iona l 28 f low i n Terms of Asymptot ic Coordinates f o r P r -»• 0 0 F igure 8 P ressure Funct ions f o r 2 -D imens iona l Flow 29 i n Terms of Asymptot ic Coord inates f o r P r -> 0 F i g u r e 9 Temperature Funct ions f o r 2 -D imens iona l 30 Flow i n Terms of Asymptot ic Coord inates f o r P r -+ 0 F igure 10 C y l i n d r i c a l - P o l a r Axes N o t a t i o n 32 F igure 11 Stream and V e l o c i t y Funct ions f o r A x i a l l y 36 Symmetr ica l F l o w - I s o t h e r m a l P l a t e F i g u r e 12 P ressu re and Temperature Funct ions f o r 37 A x i a l l y Symmetr ica l Flow - I so thermal P l a t e F i g u r e 13 V e l o c i t y Funct ions f o r A x i a l l y Symmetr ica l 40 Flow - P l a t e Temperature P r o p o r t i o n a l to r2 F i g u r e 14 Temperature Funct ions f o r A x i a l l y Symmetr ica l 41 Flow - P l a t e Temperature P r o p o r t i o n a l t o r^ F i g u r e 15 21 Inches Square P l a t e 48 F igure 16 7 Inches Diameter D i s c 48 v i i F igures cont inued F igure 17 S c h l i e r e n Photographs w i t h 21 i n c h P l a t e ; 51 a . AT = 23 .5 °F, Too = 71. °F, re f ' ' b. A T = 39. °F, Too = 71. °F, re f ' c. AT = 43.5 °F, Too = 71 . °F, r e f d. A T = 50 .5 °F, Too = 71. °F. ref ' F i g u r e 18 S c h l i e r e n Photographs w i t h 12 i n c h P l a t e ; 52 a . AT = 30. °F, T^ = 72.°F , r e f ' ' b. AT = 56 .5 °F, Too = 71.5 °F, re f c . A T = 81.0 °F, Too = 71.5 °F, r e f d. AT = 117.5°F, Too = 71.5 °F. r e f F igu re 19 S c h l i e r e n Photographs w i t h 7 i n c h D i s c : 53 a . A T = 52 .5 °F, Too = 72. °F, Si = 0 rpm, r e f r b. AT . » 62 .5 °F, Too = 72.0°F, SI = 3 2 . 8 rpm r e f c c. A T = 94 .5 °F, Too = 72.5°F, SI = 13.9 rpm r e f ' ' ^ d. A T r e f = 94 .5 °F, T^ = 72.5°F, SI = 18.6 rpm. F i g u r e 20 S c h l i e r e n Photographs w i t h 7 i n c h D i s c : 54 a . AT . - 91 .5 °F, T = 71. °F, SI = 0 rpm, re f ' oo » b. A T « 115. °F, Too = 71 °F, SI = 0 rpm. r e f ^ F igure 21 C r i t i c a l Gr f o r 21 i n c h Square P l a t e 55 F i g u r e 22 C r i t i c a l Gr f o r 12 i n c h Square P l a t e 55 F i g u r e 23 C r i t i c a l Gr f o r 7 i n c h D i s c f o r Free 56 Convect ion F i g u r e 24 C r i t i c a l Gr v s . Re f o r 7 i n c h D i s c 56 F i g u r e 25 Comparison of T h e o r e t i c a l P r e d i c t i o n s 63 and E x p e r i m e n t a l R e s u l t s F i g u r e 26 Elements of the O p t i c a l System 76 F i g u r e 27 Schematic of the Semi -Focus ing System 76 F i g u r e 28 A Q u a l i t a t i v e A n a l y s i s of the Number of Co lour 81 F r inges Observable i n the S c h l i e r e n Photographs ACKNOWLEDGEMENTS The author wishes to express h i s deep g r a t i t u d e f o r the i n v a l u -ab le adv ice and guidance g iven him throughout a l l s tages of the program by P r o f e s s o r Z. Rotem. S i n c e r e thanks are a l s o extended to Dr . E .G . Hauptmann f o r h i s many good comments about s c h l i e r e n photography. In a d d i t i o n the Author wishes to thank the e n t i r e s t a f f of the M e c h a n i c a l E n g i n e e r i n g Department, U n i v e r s i t y of B r i t i s h Co lumbia , f o r t h e i r p e r i o d i c a s s i s t a n c e and the Department f o r the use of i t s f a c i l i t i e s . S p e c i a l thanks are due to Mr. P. Hurren and to Mr. J . Hoar , c h i e f t e c h n i c i a n s , f o r t h e i r very important p r a c t i c a l a d v i c e . Without t h e i r a s s i s t a n c e and c o n s i d e r a t i o n of sometimes very urgent demands the work cou ld not have been completed. Support f o r t h i s research was p r o v i d e d by the N a t i o n a l Research C o u n c i l of Canada, through grant No. 2772. Computing t ime was granted f r e e of charge by the Computing Center through a g e n e r a l support by the N . R . C . . The Author i s g r a t e f u l f o r t h i s a s s i s t a n c e . IX NOMENCLATURE X,Y,Z,R Dimensional coordinates (see figures 1 and 10) x,y,z,r Dimensionless coordinates (see figures 1 and 10) u,v,w Dimensionless v e l o c i t y components (see figures 1 and 10) L Reference length (usually h a l f width of plate) AT . ref or Reference temperature difference (for constant surface AT. temperature i s equal to (T - T )) T Temperature of the surface w r T Temperature of the f l u i d f a r away from the surface U j. Reference v e l o c i t y ref J U £ L re f Re Reynolds number (= — - — ) Pr Prandtl number (= - ^ j ^ - ) Gr Grashof number (see equations (7) and (61)) Ra Rayleigh number (= Pr.Gr) Nu Nusselt number (= = - r~) k dy k Conductivity of the f l u i d Cp S p e c i f i c heat at constant pressure of the f l u i d h Heat transf e r c o e f f i c i e n t T - T w 0 0 8 Dimensionless temperature (= — ^ ) ref IT Dimensionless pressure (see equations (5) and (59)) 3 Expansion c o e f f i c i e n t (for a i r = 1/T") U,v Absolute and kinematic v i s c o s i t y of the f l u i d X Nomenc la ture -cont inued a Angle of i n c l i n a t i o n to the h o r i z o n t a l Angular v e l o c i t y S i m i l a r i t y v a r i a b l e s F A s t ream f u n c t i o n F' A v e l o c i t y f u n c t i o n G A p r e s s u r e f u n c t i o n H A temperature f u n c t i o n (= G' i n every case) T A* c i r c u l a t i o n f u n c t i o n Supers c r i p t s S i g n i f i e s a t ransformed v a r i a b l e S i g n i f i e s a s y m p t o t i c . v a r i a b l e f o r Pr » 1 ~ S i g n i f i e s asymptot i c v a r i a b l e f o r P r « 1 S u b s c r i p t s 1 S i g n i f i e s " i n n e r " reg ion 2 S i g n i f i e s " o u t e r " reg ion w S i g n i f i e s c o n d i t i o n s on the s u r f a c e 0 0 S i g n i f i e s c o n d i t i o n s f a r away from the s u r f a c e . 1 I. INTRODUCTION A heated s u r f a c e , when surrounded by an expanse of f l u i d which changes i t s d e n s i t y w i t h a change of the temperature , w i l l g ive r i s e to a f ree convect ion boundary l a y e r f low i f the temperature exceeds a c e r t a i n minimum v a l u e . The s i m p l e s t type of f low i s encountered about a v e r t i c a l s u r f a c e . In t h i s case a laminar boundary l a y e r may fo rm, which o r i g i n a t e s at the lower edge. The fo rmat ion of the boundary l a y e r i s a r e s u l t o f a t r a n s f e r of h e a t . f r o m the s u r f a c e . As f l u i d i n the p r o x i m i t y of the s u r f a c e i s h e a t e d , i t decreases i t s d e n s i t y . This imparts buoyancy to the f l u i d which causes i t to f low a long the p l a t e . The buoyancy fo rces i n t h i s s i t u a t i o n act i n a d i r e c t i o n p a r a l l e l to the mot ion . The boundary l a y e r f low about an i n c l i n e d s u r f a c e resembles q u a l i t a t i v e l y t h a t over a v e r t i c a l s u r f a c e , s i n c e a component of the buoyancy fo rces w i l l act i n a d i r e c t i o n p a r a l l e l to the mot ion . How-e v e r , ( e s p e c i a l l y at s m a l l i n c l i n a t i o n s to the h o r i z o n t a l ) the t r a n s -verse component of the buoyancy fo rce becomes l a r g e r and the f low may become i n t e r m i t t e n t l y u n s t a b l e . On the other hand , f o r a very n e a r l y h o r i z o n t a l s u r f a c e buoyancy a i d s the motion only i n d i r e c t l y . I t induces a t r a n s v e r s e p ressure g rad ien t of which the v a r i a t i o n a long the s u r f a c e d r i v e s the f l u i d . The c h a r a c t e r of t h i s f low i s , t h e r e f o r e , expected to be fundamental ly d i f f e r e n t from the f low over a s t e e p l y i n c l i n e d p l a t e . The h i s t o r i c a l aspect of i n v e s t i g a t i o n s i n t o t h i s f low w i l l be d i s c u s s e d i n s e c t i o n I I . 2 The t h e o r e t i c a l i n v e s t i g a t i o n of t h i s t h e s i s a p p l i e s main ly to a moderate temperature d i f f e r e n c e between the f l u i d b u l k and the s u r f a c e . The f l u i d p r o p e r t i e s may thus be assumed t o be c o n s t a n t , w i t h the excep -t i o n of the dependence of d e n s i t y upon temperature , and the equat ions of motion may be s i m p l i f i e d a c c o r d i n g l y . I f i n a d d i t i o n the s u r f a c e i s a s e m i - i n f i n i t e p l a t e and, t h e r e f o r e , possesses only one l e a d i n g edge, the absence of a c h a r a c t e r i s t i c l e n g t h suggests that " s i m i l a r i t y s o l u t i o n s " shou ld be o b t a i n a b l e . This t h e s i s d i scusses thos'e f r e e and combined f r e e and f o r c e d c o n -v e c t i o n f lows (the l a t t e r f o r an a x i a l l y - s y m m e t r i c a l s i t u a t i o n ) f o r which s i m i l a r i t y s o l u t i o n s can be o b t a i n e d . Stewartson (1)* was f i r s t to p u b l i s h i n 1958 a s i m i l a r i t y s o l u t i o n and a l i m i t e d amount of n u m e r i c a l data f o r 2 - d i m e n s i o n a l f low and f o r a P r a n d t l number (Pr) e q u a l to . 7 2 . He i n c o r r e c t l y c l a i m e d , however, that a boundary l a y e r would form below a heated s u r f a c e or converse ly above a coo led s u r f a c e . In 1965, G i l l et a l . (2) c o r r e c t e d t h i s i n t e r -p r e t a t i o n , to the now accepted f a c t , t h a t a boundary l a y e r forms on ly above a heated s u r f a c e or below a coo led s u r f a c e . They a l s o extended S tewar tson ' s r e s u l t s to P r = 1.0 and 10. In the present t h e s i s , i n p a r t , these r e s u l t s are recomputed and extended to v a r i o u s o ther Pr numbers. The approach taken to a r r i v e at s i m i l a r i t y s o l u t i o n s , however, d i f f e r s from prev ious methods and i s thought to be more fundamental and s y s t e m a t i c . In p a r t i c u l a r the range of v a l i d i t y of these s o l u t i o n s i s c a r e f u l l y d e f i n e d . * Numbers i n b r a c k e t s a f t e r the names of authors r e f e r to r e f e r e n c e s . 3 The laminar convect ion f low as d e s c r i b e d above i s a f u n c t i o n of the parameter P r . S ince many f l u i d s can be grouped as h a v i n g e i t h e r a very l a r g e or a very s m a l l va lue of P r , the system of equat ions was examined f o r asymptot ic va lues of t h i s parameter . When thus expanded f o r an a s y m p t o t i c a l l y l a r g e or s m a l l P r number the s o l u t i o n s become independent of t h e . P r number and may be s a i d to be " u n i v e r s a l " s o l u t i o n s . I t i s , moreover, shown i n t h i s work that ext remely good approx imat ions are obta ined w i t h the asymptot i c s o l u -t i o n s f o r P r < . 1 and P r > 10. For a x i a l l y - s y m m e t r i c a l f low pure f r e e convect ion and combined f r e e and fo rced convect ion are cons ide red . The former y i e l d s a number of s i m i l a r i t y s o l u t i o n s w h i l e the l a t t e r y i e l d s on ly one ( su r face temperature v a r y i n g w i t h r^) . A l though t h e s e , s o l u t i o n s are . i d e n t i f i a b l e as boundary l a y e r s o l u t i o n s they are not r e a d i l y i n t e r p r e t e d as p h y s i -c a l l y observab le modes of f low on a d i s c of f i n i t e rad ius s i n c e they d e s c r i b e a boundary l a y e r that grows w i t h i n c r e a s i n g r a d i u s . H i s t o r i c a l l y , l i t t l e e x p e r i m e n t a l ev idence has been g iven f o r the e x i s t e n c e of the laminar f ree c o n v e c t i v e , h o r i z o n t a l boundary l a y e r . In o rder to c o n f i r m the c o n t r a d i c t o r y theory a s c h l i e r e n system u s i n g the s e m i - f o c u s i n g co lou r techn ique was b u i l t . In 2 - d i m e n s i o n a l f low a boundary l a y e r forms at e i t h e r end above a f i n i t e p l a t e and grows inwards . The boundary l a y e r s meet i n the cente r and r i s e i n a v e r t i c a l plume. I t i s observed e x p e r i m e n t a l l y that the boundary l a y e r does not t rans form, smoothly to a v e r t i c a l plume but i s te rminated by a r eg ion of thermal i n s t a b i l i t y . The po in t of onset of t h i s i n s t a b i l i t y i s observed to be extremely s e n s i t i v e to d i s t u r b a n c e s i n the surroundings of the p l a t e . With t h i s h indrance . 4 i n mind data i s p resented to d e f i n e the l o c a t i o n of the i n s t a b i l i t y . Beyond the c r i t i c a l p o i n t - i n the c e n t r a l reg ion of the p l a t e - c o n -v e c t i o n takes p l a c e i n an e s s e n t i a l l y v e r t i c a l d i r e c t i o n . In p a r t i c u l a r , near the p l a t e the f low i s c h a r a c t e r i z e d by c e l l u l a r f l u c t u a t i o n s or low d e n s i t y " t u r b u l e n c e b u b b l e s " . S c h l i e r e n observa t ions o f a d i s c show a boundary l a y e r which o r i g i n a t e s at the edge and grows inwards , again r i s i n g as a plume over the center of the d i s c . The f low may, .however , be i n t e r p r e t e d as that over a c i r c u l a r annulus . I f the annulus r a d i u s were made very l a r g e compared to i t s r a d i a l ex tent so tha t the rad ius would no longer be a c h a r a c t e r i s t i c dimension ( s i m i l a r i t y s o l u t i o n s r e q u i r e t h i s ) the f low would become e s s e n t i a l l y 2 - d i m e n s i o n a l . In t h i s sense the f low which i s observed e x p e r i m e n t a l l y f o r a x i a l l y symmetr i ca l f low i s of a 2 -d imens iona l nature and does not r e l a t e d i r e c t l y to the theory developed f o r a x i a l l y - s y m m e t r i c a l f l o w . 5 I I REVIEW OF PREVIOUS WORK Some r e s u l t s of i n v e s t i g a t i o n s i n t o thermal convect ions about f i n i t e h o r i z o n t a l p l a t e s were p u b l i s h e d i n the e a r l y 1 9 3 0 ' s . F ishenden and Saunders (3) , Schmidt (4) and Weise (5) p u b l i s h e d some d a t a , summarized p r i m a r i l y i n the form of n o n - d i m e n s i o n a l c o r r e l a t i o n s . S c h l i e r e n p i c t u r e s taken by Weise were thought to be f i r s t ev idence f o r the e x i s t e n c e of a laminar boundary l a y e r . The genera l consensus was , however, t h a t f o r a p l a t e heated on both i t s s i d e s the boundary l a y e r would form on the u n d e r s i d e . Subsequent a n a l y t i c a l work was undertaken by Sugarawa and M i c h i y o s h i (6) and by M i c h i y o s h i (7) on the f r e e c o n v e c t i o n from a p l a t e of f i n i t e w i d t h . They assumed that the f low cou ld be approximated t o tha t which e x i s t s around an i n f i n i t e l y l o n g h o r i z o n t a l c y l i n d e r , the c r o s s - s e c t i o n of which was an e l l i p s e of l a r g e e c c e n t r i c i t y , w i t h i t s l a r g e r a x i s h o r i z o n t a l . The f l a t p l a t e f low would be approached when the e c c e n t r i c i t y i s a l l o w e d to approach i n f i n i t y . Th is f l ow may be expected to take p l a c e over a f i n i t e p l a t e heated on both s i d e s when the c h a r a c t e r i s t i c Grashof number i s s m a l l . The f i r s t s o l u t i o n s f o r the l a m i n a r boundary l a y e r f lows o f the type d i s c u s s e d i n t h i s work were o b t a i n e d by Stewartson ( 1 ) . G i l l e t a l . (2) r e i n t e r p r e t e d and extended h i s r e s u l t s , f o r P r = . 7 2 , to P r = 1.0 and 10. By l o o k i n g at the induced p r e s s u r e g r a d i e n t s a l o n g the p l a t e t h e y , f u r t h e r m o r e , deduced tha t a boundary l a y e r f low would on ly e x i s t above a heated p l a t e or below a coo led p l a t e , s i n c e the f low must c o e x i s t w i t h a f a v o u r a b l e p r e s s u r e g r a d i e n t . The f low then has a v e r t i c a l d e n s i t y s t r a t i f i c a t i o n which makes i t g r a v i t a t i o n a l l y u n s t a b l e . 6 At some p o i n t i t may t h e r e f o r e be expected to s e p a r a t e , and i n the case of a f i n i t e p l a t e , i t would r i s e near the a x i s of symmetry of the p l a t e as a v e r t i c a l plume. W i t h i n t h i s c e n t r a l r e g i o n the f low i s dominated by p r i m a r i l y v e r t i c a l mot ions , w i t h some c e l l u l a r s t r u c t u r e . The laminar f low may thus be assumed to possess an upper l i m i t of the Grashof number; f low b e y o n d . t h i s l i m i t suggests i n appearance that above an i n f i n i t e l y l a r g e p l a t e . T r i t t o n (8 and 9) d i s c u s s e s at l e n g t h the t r a n s i t i o n of the boundary l a y e r on i n c l i n e d p l a t e s due to g r a v i t a t i o n a l i n s t a b i l i t y . Combined f r e e and f o r c e d convect ion from a h o r i z o n t a l s u r f a c e has been d i s c u s s e d to a l e s s e r degree. For t h i s reason i t i s u s e f u l to d e s c r i b e here, s o m e , r e s u l t s found f o r the v e r t i c a l p l a t e . A very e x t e n s i v e a n a l y s i s was p u b l i s h e d by Sparrow et a l . (10) . They show that the parameter c o n t r o l l i n g the r e l a t i v e importance of the f r e e and 2 fo rced convect ion terms i n . the equat ions of motion i s Gr/Re . F u r t h e r -more, they f i n d tha t s i m i l a r i t y s o l u t i o n s e x i s t on ly when the f ree st ream m 2 m—1 v e l o c i t y and s u r f a c e temperature vary r e s p e c t i v e l y as x and x , where m can be chosen so as to s u i t a boundary c o n d i t i o n . For the h o r i z o n t a l p l a t e Mor i (11) cons iders buoyancy e f f e c t s on f o r c e d 2 - d i m e n s i o n a l laminar f l o w . He expresses h i s r e s u l t s as a p e r t u r -b a t i o n s e r i e s about the p u r e l y f o r c e d f low mode. H i s p e r t u r b a t i o n 2 . 5 parameter i s Gr /Re * . x x P a r t IV of t h i s t h e s i s d i s c u s s e s a boundary l a y e r s o l u t i o n f o r combined convect ion i n a x i a l l y - s y m m e t r i c a l f l o w . The p e r t u r b a t i o n 2 4/5 parameter i n t h i s case i s Re^ /Gr where expansion occurs about the f r e e convect ion mode. In s p i t e of the f a c t that t h i s s o l u t i o n i s not r e a d i l y i d e n t i f i a b l e w i t h a p h y s i c a l mode.of f low i t i s i n t e r e s t i n g 7 to note that the p e r t u r b a t i o n parameter i s the r e c i p r o c a l of M o r i ' s . The p u r e l y fo rced f low problem over a r o t a t i n g d i s c (von Karman f low) h a s , of course been e x t e n s i v e l y a n a l y s e d , c . f . by Cochran (12) who obta ined an exact s o l u t i o n f o r i s o t h e r m a l f l o w . M i l l s a p s and Pohlhausen (13) f o l l o w e d w i t h an exact s o l u t i o n of the heat t r a n s f e r problem f o r the von Karman f l o w . In a l l of the fo rced f lows i n v e s t i g a t e d t h e o r e t i c a l l y , the c o n f i g u r a t i o n i n v o l v e d a d i s c of i n f i n i t e r a d i u s , a l i m i t a t i o n which i s a l s o i n h e r e n t i n the d e r i v a t i o n i n t h i s t h e s i s . In p r a c t i c e , a d i s c would n a t u r a l l y have f i n i t e dimensions (a c h a r a c t e r i s t i c length) and s o l u t i o n s of the type cons idered would not p e r t a i n . 8 I I I FREE CONVECTION FROM A SEMI-INFINITE PLATE 3 . 1 T h e o r e t i c a l S o l u t i o n s f o r an A r b i t r a r y P r Number. The a n a l y s i s p resented below d e s c r i b e s a s e m i - i n f i n i t e f l a t p l a t e i n an i n f i n i t e expanse of f l u i d , i n c l i n e d at a s m a l l angle to the h o r i -z o n t a l . Ost rach (14) i n d i c a t e d that i n a l l cases of n a t u r a l convec t ion the d e n s i t y of the f l u i d may be assumed to remain c o n s t a n t , e x c e p t i n g i t s dependence on temperature . Furthermore i t i s assumed that a l l o ther p r o p e r t i e s of the f l u i d do not vary a p p r e c i a b l y f o r a moderate temperature d i f f e r e n c e . I n c l u s i v e of these assumpt ions , the s i m p l i f i e d equat ions of momentum, c o n t i n u i t y , and energy f o r steady f low are as f o l l o w s : u 3u 3x + v 3u 3y SjTT 3x i + + V u - Gr 8 tan a (1) 3v , 3v Gr 0 (2) 3x V 3y (3) u 36 3x (4) where the axes n o t a t i o n i s d e f i n e d i n f i g u r e 1 . g N o t e : y always p o i n t s i n t o the f l u i d , i r r e s p e c -t i v e of the d i r e c t i o n o f g. x , u F i g u r e 1 C a r t e s i a n Axes N o t a t i o n 9 x and y are rendered d imens ion less through the use of a re fe rence l e n g t h L , which makes the l a r g e s t va lue of x of o rder u n i t y . The v e l o c i t y components u and v are rendered d imens ion less through the use of re fe rence v e l o c i t y v/L. The d imens ion less p ressure TT i s d e f i n e d as f o l l o w s : 3 P ~ Poo 2 eL IT = — L + -^ - j (x s i n a + y cos a ) . (5) Poo^ . V The d imens ion less tempexature i s d e f i n e d as f o l l o w s , = (T - T r o) /A T r e f . (6) AT , i s a s u i t a b l e re fe rence temperature - d i f f e r e n c e which renders r e f r the l a r g e s t va lue of 0 always p o s i t i v e i n a l g e b r a i c a l s i g n and of o rder u n i t y . For the case of an i s o t h e r m a l boundary AT . would be (T - T ) . J 3 r e f w °° The parameter Gr , the Grashof number appropr ia te to the sys tem, i s d e f i n e d as Gr = (gBL 3 A T r e f cos a ) / v 2 , (7) With the d i r e c t i o n of the axes as i n d i c a t e d , i t shou ld be noted tha t f o r a heated p l a t e f a c i n g upwards, the g r a v i t a t i o n a l fo rces act i n the n e g a t i v e y d i r e c t i o n , and the a l g e b r a i c a l s i g n a s s o c i a t e d w i t h the buoyancy term i n equat ions ( 1 ) , ( 2 ) i s thus p o s i t i v e . The s i g n i s reversed when a heated p l a t e f a c i n g downwards i s c o n s i d e r e d . The s o l u t i o n of equat ions (1) to (4) has to f u l f i l l the f o l l o w i n g boundary c o n d i t i o n s , when the temperature of the p l a t e i s s p e c i f i e d , y = 0 ; x > 0 u = v = 0 0 = C x n (8) y + o o u = 0 0 = 0 (9) 10 where C i s a g iven p o s i t i v e c o n s t a n t , and n a g iven exponent. Assuming now that the angle of i n c l i n a t i o n a i s ve ry s m a l l , and t h a t the c h a r a c t e r i s t i c va lue of Gr i s s u f f i c i e n t l y l a r g e , the f o l l o w i n g a s y m p t o t i c a l l y " s t r e t c h e d " coord inates and v a r i a b l e s are i n t r o d u c e d , ^ „ 1/5 ~ _ - 2 / 5 ^ „ - 1 / 5 ^ _ - 4 / 5 / n r t \ y = y Gr ; u = u Gr ; v = v Gr ; TT= TT Gr . (10) These c o n s t i t u t e the boundary l a y e r t r a n s f o r m a t i o n s . The r e s u l t i n g equat ions a r e : 2 2 ^ 3u , ^ 3u 3ff , 3 u , h - 2 / 5 3 u + _ 1/5 . N u — + v — = - — + — - + Gr — - - Gr 0 tan a ( l a ) 3x 3y 3x „^2 . 2 J 3y 3x _ - 2 / 5 , ~ 3v . „ 3v , 3tf.. _ - 2 / 5 3 2 v . _ - 4 / 5 32v'•+ fi , Gr (u -r— + v -TTT ) = - -TT: + Gr — 7 + Gr — 7 - 0 (2a) 8 x ^ ^ 3y 2 3x 2 ^- + -r^r = 0 (3a) dx dy 30 . 30 1 , 3 2 0 ^ _ - 2 / 5 3 2 0 . N u - r - + v -rpc = — ( — r + Gr — 7 ) . . (4a) 3x 3y P r „~2 . 2 ' 3y 3x As Gr becomes very l a r g e , the system of equat ions above w i l l y i e l d an ( " i n n e r " s o l u t i o n . In the p resent case the " o u t e r " s o l u t i o n reduces s i m p l y t o , u = 0 0 = 0 so that the boundary c o n d i t i o n s (9) may be a p p l i e d at the outer edge of the " i n n e r " s o l u t i o n . The f o l l o w i n g are now d e f i n e d and s u b s t i t u t e d : * The concept of " i n n e r " and "outer" s o l u t i o n s i s e x p l a i n e d i n s e c t i o n 3 . 2 . 1 . 9 11 A st ream f u n c t i o n (as u s u a l ) , (11) a s i m i l a r i t y t r a n s f o r m a t i o n , = A x P F(n) ff = B x G(n) e = c x n H (n) (12) and a s i m i l a r i t y v a r i a b l e , Tl = D y q x - s (13) I t may be shown that the constants A through D are u s e f u l only f o r the a b s o r p t i o n of o ther constants i n the r e s u l t i n g o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . They may t h e r e f o r e a l l be se t equa l to u n i t y w i thout l o s s of g e n e r a l i t y . I n s e r t i n g (11) through (13) i n t o ( l a ) - (4a) a se t of c o m p a t i b i l i t y equat ions are o b t a i n e d f o r the exponents m, q , p and s . These c o n d i t i o n s - 2 / 5 s t a t e ( i ) the requirement f o r terms of o rder Gr not to i n c r e a s e w i t h x , and- ( i i ) f o r a l l o ther terms to become f u n c t i o n s of the s i m i l a r i t y v a r i a b l e o n l y . One of the exponents may be chosen a r b i t r a r i l y , and f o r convenience put q = 1. Then, m = 2/5 + 4/5 n p = 3/5 + 1/5 n > (14) s = 2/5 - 1/5 n . J 12 The v a l u e of the exponent n i s imposed by the boundary c o n d i t i o n s . For the t r a n s f o r m a t i o n to be v a l i d , n must be > - 3 . (15) The equat ions ( l a ) through (4a) now reduce t o , 5 F ' " + (3 + n ) F F " - (1 + 2nX ( F ' ) 2 = 2(1 + 2n) G - (2 - n ) r , G ' - a | G r - 2 / 5 x " 2 / 5 ( 3 + n ) + i/5 n+3 -o|Gr x 5 tan a| (16) G1 = - H + a | G r " 2 / 5 x - 2 / 5 ( 3 + n) (17) H " + ^ (3 + n) FH' - n P r F 'H = a | G r " 2 / 5 x " 2 / 5 ( 3 + n ) | . (18) Note t h a t the l a s t term on the r i g h t - h a n d s i d e of e q u a t i o n (16) remains s m a l l o n l y p r o v i d e d that n+3 a < a | t a n _ 1 (Gr 1 / 5 ~x~5~ )| (19) T h i s l i m i t i n g c o n d i t i o n upon the p e r m i s s i b l e i n c l i n a t i o n of the bounding s u r -face t o the h o r i z o n t a l i s b e l i e v e d to be r a t h e r more a p p r o p r i a t e than the one p r e v i o u s l y proposed by Stewartson ( 1 ) . The boundary c o n d i t i o n s s u b j e c t to which these equat ions have to be s o l v e d , a r e , r, = 0 F = F 1 = 0 H = G' = 1 r i + o o F' = 0 H = 0, G = 0 y (20) 13 It was pointed out in section II that a boundary layer flow can only exist above a heated surface or equivalently below a cooled surface; the positive sign only w i l l , therefore, be retained in equations (16) and (17). It i s seen from the above that, whereas an analysis resembling this one, performed on a v e r t i c a l freely convecting plate, results in two simultaneous ordinary di f f e r e n t i a l equations, the present case yields three equations with a resultant complication in numerical evaluation. As equation (17) may serve to eliminate H , the resultant equation is then of the third order, rather than the second, as is the case for the ver t i c a l plate. For the case of the isothermal plate n = 0 and the equations simplify to, 5 F*" + 3FF" - (F') 2 = 2(G - T)G') (21) H = G' (22) H' ' + 3/5 Pr F H' - 0 (23) with boundary conditions (20). These equations were numerically integrated and the results are given in figures 2 to 5. The local Nusselt number is obtained as follows, N u - - f | y Q - - x 2 ^ 3 " " ^ G r 1 ^ H'<0) (24) so that the average value of the Nusselt number becomes, 14 C o n s i d e r i n g next the case of c o n s t a n t , imposed f l u x at the bounding s u r f a c e , i n s t e a d o f . e q u a t i o n s (14) the f o l l o w i n g are o b t a i n e d , m = 2/3 p = 2/3 s = 1/3 . (26) The r e l e v a n t equat ions f o r the f u n c t i o n s F, G and H may be obta ined by i n s e r t i n g n = 1/3 i n t o equat ions (16) through (19) . A s u i t a b l e cho ice f o r the re fe rence temperature d i f f e r e n c e i s , L 6 G r " 1 / 5 AT = - — — (27) r e f K H' (0) U / ; where Q i s the g i ven f l u x per u n i t a r e a . gure 2 Stream Funct ions f o r 2 - D i m e n s i o n a l Flow 16 F i g u r e 3 V e l o c i t y Funct ions , f o r 2 - D i m e n s i o n a l Flow 17 Figure 4 Pressure Functions f o r 2-Dimensional Flow F i g u r e 5 Temperature Funct ions f o r 2 - D i m e n s i o n a l Flow 3.2 Asymptot ic S o l u t i o n s Equat ions (16) through (18) c o n t a i n a parameter (Pr) . As P r i n c r e a s e s w i thout l i m i t , equat ion (18) seems to become s i n g u l a r , whereas f o r a v a n i s h i n g P r the equat ions seem to uncouple . In the f o l l o w i n g subsec t ions the equat ions w i l l be t ransformed and i t w i l l be shown that thermal " i n n e r " and " o u t e r " s o l u t i o n s desc r ibe the f low f i e l d . 3 . 2 . 1 P r » 1 The thermal boundary l a y e r , w i t h i n which the conduct ive and convec -t i v e terms are of e q u a l o rder of magnitude, w i l l now be much n a r r o w e r . . -than the momentum boundary l a y e r . The v e l o c i t i e s , i n t h i s c a s e , as i s seen i n f i g u r e 3 w i l l be of very s m a l l magnitude. . S ince the thermal and momentum boundary l a y e r s have t h i c k n e s s e s o f . d i f f e r e n t order of magnitude the f low f i e l d may be thought of as. c o m p r i s i n g two r e g i o n s . The " i n n e r " reg ion w i l l be the thermal reg ion ( v i r t u a l l y the complete temperature drop takes p l a c e w i t h i n t h i s reg ion) w h i l e the " o u t e r " reg ion w i l l comprise the r e s t of the momentum boundary l a y e r . The f low equat ions w i l l now be. t ransformed i n a way such that terms which are of s i g n i f i c a n c e w i t h i n one reg ion w i l l be made e i t h e r l a r g e r or t. s m a l l e r to g ive them an order of magnitude of u n i t y . In t h i s way terms which are of s i g n i f i c a n c e only w i t h i n the other reg ion w i l l a t t a i n a much lower order of magnitude and become n e g l i g i b l e . Con-sequent l y the s o l u t i o n of the r e s u l t i n g t ransformed equat ions w i l l desc r ibe only one r e g i o n , and i n t h i s sense w i l l be r e f e r r e d to as an " i n n e r " or " o u t e r " s o l u t i o n . The boundary c o n d i t i o n s w i l l be the va lues at the edges of each r e g i o n . T h e r e f o r e , the outer boundary va lues 20 of the " i n n e r " s o l u t i o n must match w i t h the i n n e r boundary va lues of the " o u t e r " s o l u t i o n . S p e c i f i c a l l y the temperatures and v e l o -c i t i e s must match. Desp i te the c a t e g o r i z a t i o n s of the s o l u t i o n s to " i n n e r " o r " o u t e r " i t i s to be remembered t h a t each s o l u t i o n i s u n i f o r m l y v a l i d across the e n t i r e f low f i e l d ! F o r the " i n n e r " s o l u t i o n p u t , ,(*) n 1 = n P r H = H 1/5 3/5 F1 = F' P r n Note that the temperature f u n c t i o n must remain of o rder u n i t y , and t h e r e f o r e shou ld not be a s y m p t o t i c a l l y " s t r e t c h e d " . Fo r convenience the d e r i v a t i o n of the i s o t h e r m a l p l a t e only i s p r e s e n t e d . Then, s u b s t i t u t i n g i n t o equat ions (16) through (18) and l e t t i n g n = 0 the f o l l o w i n g equat ions r e s u l t : > (28) 5 F | " - 2 (5 - ri Gp = o | P r _ 1 |+ a|(Ra x J ) 3N-2/5 - 0| (Ra x 3 ) 1 / 5 tan a I (23) fl1 - G| - 0|(Ra x 3 ) 2 / 5 (30) Hp + 3/5 t 3 ^ = 0| (Ra x 3 ) 2 / 5 j (31) (*) For a r e f e r e n c e to the nomenclature used the reader i s r e f e r r e d t o " P e r t u r b a t i o n Methods i n F l u i d Dynamics" , M. Van-Dyke, Academic P r e s s , 1964. 21 The angle of i n c l i n a t i o n to the h o r i z o n t a l i s now l i m i t e d by the f o l l o w -i n g c o n d i t i o n , a = a I t a n - 1 (Ra x 3 ) 1 / 5 I (32) For s u f f i c i e n t l y l a r g e v a l u e s of P r and Ra x the r i g h t hand terms i n the equat ions above become n e g l i g i b l y s m a l l . The equat ions may then be i n t e g r a t e d n u m e r i c a l l y to o b t a i n a f i r s t o rder s o l u t i o n f o r the " i n n e r " r e g i o n . The boundary c o n d i t i o n s f o r the equat ions above are now, n 1 = o n1 °° F = F' 1 1 F|' = 0 = 0 H, = H, = G ; = i G 1 = 0. > (33) Values of the r e s u l t a n t asymptot ic f u n c t i o n s are given i n Table I I . F i g u r e s 6 and 7 g ive the f u n c t i o n s G and H, p r e v i o u s l y determined f o r v a r i o u s l a r g e va lues of P r , when r e p l o t t e d i n asymptot i c coord ina tes (G^ and H^, versus n^) . The " i n n e r " reg ion as s t a t e d p r e v i o u s l y determined e n t i r e l y the heat t r a n s f e r c h a r a c t e r i s t i c s . The v a l u e of the l o c a l N u s s e l t number i s t h u s , Nu = - x 2 / 5 .. R a 1 / 5 (0) (0) = - 0 . 4 6 0 1 (34) and the v a l u e of the mean N u s s e l t number over the range x = 0 to x = 1 i s 5/3 of t h i s v a l u e . 22 In the " o u t e r " reg ion both the convect i ve and v i s c o u s terms i n the equat ion of motion must be r e t a i n e d , whereas the temperature has a l r e a d y dropped to i t s asymptot ic v a l u e ( i . e . to z e r o ) . There -f o r e , f o r the " o u t e r " s o l u t i o n p u t , n 2 = n P r H = 0. -3/10 r2 = f = F ' P r 3/5 > (35) S u b s t i t u t i n g i n equat ions (16) through (18) and s e t t i n g n = 0, the equat ions reduce to 5 F * " + 3 f F " 2 2 2 H2 = 0. (F') 2 = 0 > (36) The boundary c o n d i t i o n s are^ F2(0) = 0; F 2'(0) o f^(oo) = 1.1522; 7^ (») = 0. (37) Values of the asymptot i c f u n c t i o n s f o r the outer p r o f i l e are g iven i n Table I I a . 3.2.2 P r « 1 The thermal boundary l a y e r i n t h i s case w i l l be much w i d e r than the momentum boundary l a y e r . Consequent ly , the i n f l u e n c e o f the v i s c o u s terms i n the momentum equat ion upon the convec t i ve process shou ld b e -come n e g l i g i b l y s m a l l at a s m a l l d i s t a n c e from the boundary , w h i l e the i n e r t i a terms s h o u l d remain of the same o rder as the buoyancy te rms. Thus i t i s the " o u t e r " r e g i o n which p r i m a r i l y determines the convec -t i o n p r o c e s s , w h i l e the " i n n e r " r e g i o n ensures the d isappearance o f the v e l o c i t y at the boundary. F o r the " o u t e r " s o l u t i o n p u t , n 2 - n P r 2 / 5 \ - F' P r 1 / 5 fl2 = H. A g a i n , the temperature has to remain of o rder u n i t y and thus i s not a f f e c t e d by " s t r e t c h i n g " . C o n s i d e r i n g the i s o t h e r m a l p l a t e o n l y , equat ions (16) through (18) reduce t o , 3 F 2 F 2 " - ( F 2 ) Z + 2 ( r i 2 G* - G £ ) = a|Pr | + a [ P r / 3 ( G r x J ) - o| P r 1 / 5 R a 1 / 5 x 3 / 5 tan H 2 - \ = 0 | P r " 1 / 5 . ( R a x 3 ) _ 2 / 5 | H ' ' + 3/5 F 2 H 2 = 0|(Pr Ra x 3 ) " 2 / 5 | . 24 The angle of i n c l i n a t i o n to the h o r i z o n t a l i s now l i m i t e d by the c o n d i t i o n , a = a | t a n _ 1 ( P r Ra x 3 ) ~ 1 / 5 |. The boundary c o n d i t i o n s upon these equat ions a r e , n 2 = o F 2 = 0 H 2 = G 2 = 1.0 F - = 0 H 2 = G 2 = 0 . y (42) The c o n d i t i o n of F 2 (0) = 0 , as i s seen above, i s not f u l f i l l e d by the s o l u t i o n f o r the " o u t e r " r e g i o n . This i s a requirement upon the s o l u t i o n f o r the " i n n e r " r e g i o n ; i t must ensure tha t the n o - s l i p c o n d i t i o n at the boundary i s s a t i s f i e d . Values of the r e s u l t a n t asymptot i c f u n c t i o n s are g iven i n Table I I I . In f i g u r e s 8 and 9 , the v a l u e s of G and H as p r e v i o u s l y determined f o r v a r i o u s s m a l l va lues of P r are r e p l o t t e d i n the asymptot ic c o o r d i n a t e s of the " o u t e r " s o l u t i o n . The l o c a l N u s s e l t number i s : N u = - x ' 2 / 5 G r 1 / 5 P r 2 / 3 H ' ( 0 ) H' (0) = - 0 . 5 7 7 0 > (43) The mean v a l u e of Nu i s aga in found by i n t e g r a t i o n from x = 0 to x = 1 to be 5/3 of the v a l u e above. W i t h i n the " i n n e r " r e g i o n the v i s c o u s and i n e r t i a terms must be r e t a i n e d . S ince t h i s r eg io n i s ext remely narrow the temperature w i l l remain almost constant and of o rder u n i t y throughout . The v e l o c i t y at 25 the outer edge must match w i t h the i n n e r edge of the " o u t e r " s o l u t i o n . T h e r e f o r e , f o r the " i n n e r " s o l u t i o n p u t , nn = n P r -1/10 = F' P r 1/5 > (44) E± = 1.0 S u b s t i t u t i n g i n t o equat ion (16) f o r the i s o t h e r m a l p l a t e , the f o l l o w i n g e q u a t i o n r e s u l t s , 5 F ^ " + 3 F1 F|* - (Fj_) = 2 P r ± / Z (G 1 - T) G' ± ) (45) From equat ions (44) G^ = + constant (K) (46) There fo re 1/2 R . H . S . (equ. 45) = 2 P r ' K (47) Now G 1 (0 ) = a|G(0) Pr - 1 / 1 0 | = a | P r " 2 / 5 . P r " 1 / 1 0 | , (48) where the a|G(0) was taken from the " o u t e r " s o l u t i o n . Then, K = a P r - 1 / 2 , (49) T h e r e f o r e , as e x p e c t e d , G^ remains approx imate ly constant as P r ->• 0 , and from e q u a t i o n (47) R . H . S . (eq. 45) = f i n i t e constant ( C ) , (50) 26 and equat ion (45) becomes, 5 F J " + 3 I F p - ( F p 2 = C . ( 5 1 ) The boundary c o n d i t i o n s a r e , F 1 ( 0 ) = 0 F J ( 0 ) = 0 F^ (oo) = F 2 ( 0 ) . ( 5 2 ) From the l a s t c o n d i t i o n of (52) C = - ( F 2 ( 0 ) ) 2 = ( 1 . 5 7 2 3 ) 2 . (53) Values of the asymptot ic f u n c t i o n s f o r the " i n n e r " s o l u t i o n are given i n Table ( I I Ia ) . F igure 6 P ressu re Funct ions f o r 2 -D imens iona l Flow i n Terms o f Asymptot ic Coord inates f o r P r °° F igure 7 Temperature Funct ions f o r 2 - D i m e n s i o n a l Flow i n Terms o f Asymptot ic Coord inates f o r P r -*- 0 0 29 F i g u r e 8 P r e s s u r e Func t ions f o r 2 - D i m e n s i o n a l Flow i n Terms o f Asymptot ic Coord inates f o r P r -»• 0 30 31 IV FREE AND COMBINED CONVECTION FROM A DISC The a n a l y s i s p resented i n t h i s s e c t i o n i s f o r a steady a x i a l l y -s ymmet r i ca l f l o w . The boundary l a y e r assumptions a p p l i c a b l e t o t h i s f low are s i m i l a r t o those of the 2 - d i m e n s i o n a l f low case and may be taken from the i n t r o d u c t o r y remarks of s e c t i o n I I I . To be somewhat h o r i z o n t a l s u r f a c e . An est imate of the p e r m i s s i b l e angle of i n c l i n a t i o n to the h o r i z o n t a l may be obta ined by c o n s i d e r i n g the f low above a c o n i c a l enve lope , and a d j u s t i n g the p ressure and buoyancy terms a c c o r d i n g l y . The system of equat ions i n c y l i n d r i c a l p o l a r coord inates i s then as f o l l o w s ; c o n t i n u i t y , 1^  + ^+1^=0 (54 3r r dz momentum i n the r a d i a l d i r e c t i o n , more s p e c i f i c the a n a l y s i s w i l l however, be r e s t r i c t e d to a p e r f e c t l y + a | Gr' - 2 / 5 (55) momentum i n the a x i a l d i r e c t i o n , 3£ 9£ = 0 - a|Gr" - 2 / 5 +a Gr - 4 / 5 (56) momentum i n the az imutha l d i r e c t i o n , dv , uv u -r— + dr r (57) 32 energy , ~ 30 . ^ 30 3r 3z 3z (58) The axes n o t a t i o n i s d e f i n e d i n f i g u r e 10. A l l v a r i a b l e s are rendered d imens ion less through the use of a re fe rence length L o r a re fe rence v e l o c i t y v/L. z ,w r , u F igure 10 C y l i n d r i c a l - P o l a r Axes N o t a t i o n S ince no i n c l i n a t i o n i s cons idered the d imens ion less p ressure i s s imp l y , p - poo T 2 gLT 77 = —2 v~~ z •;. Poo v (59) 33 the d imens ion less temperature i s , T - T AT re f and the Gr number i s , Gr = (g $ L 3 A T r e f ) / v 2 . Inc luded i n the equat ions above are a l s o t h e : f o l l o w i n g boundary l a y e r t r a n s f o r m a t i o n s , z = z Gr 1/5 u = u Gr - 2 / 5 v = v Gr - 2 / 5 _ - 1 / 5 * „ . - 4 / 5 w = w Gr : ft = IT Gr (60) (61) V (62) The f o l l o w i n g subsec t ions d i scuss s p e c i f i c s i m i l a r i t y s o l u t i o n s f o r the equat ions (54) to (58) . 4 . 1 Free Convect ion f o r an A r b i t r a r y P r Number This a n a l y s i s w i l l r e s u l t i n a boundary l a y e r type of s o l u t i o n f o r a x i a l l y - s y m m e t r i c a l f low from a h o r i z o n t a l s u r f a c e , which i s induced by buoyancy fo rces i n the v e r t i c a l d i r e c t i o n . In t h i s case there i s no az imutha l component of the v e l o c i t y and equat ion (57) need not be cons ide red . The equat ions a p p l i c a b l e are t h e r e f o r e (54) , (55) , (56) and 0 8 ) . As f o r the 2 - d i m e n s i o n a l f low the f o l l o w i n g are now d e f i n e d . A S t o k e s ' stream f u n c t i o n , 1 U u = 7 3? w = -1 3^-r 3r (63) 34 S i m i l a r i t y t r a n s f o r m a t i o n s , i> - r P F(n) 5 TT = r m G(ri) n „ , s q - s = r H (n) ; n = z H r . > (64) When these f u n c t i o n s are s u b s t i t u t e d i n t o equat ions (55) , (56) and (58) a s e t of c o m p a t i b i l i t y equat ions f o r the exponents r e s u l t s which are analogous to equat ions (14) . Again one exponent may be d e f i n e d a r b i t r a r i l y ( c . f . q ) , q = 1 m = 2/5 + 4/5 n p = 8/5 + 1/5 n s = 2/5 - 1/5 n. > (65) The v a l u e of the exponent n i s imposed by the s u r f a c e temperature . The l i m i t a t i o n s on the a l l o w a b l e va lues f o r t h i s exponent are i d e n t i c a l t o (15 ) . Equat ions (65) t o g e t h e r w i t h (63) and (64) when s u b s t i t u t e d i n t o (55) , (56) and (58) produce the f o l l o w i n g s e t of t o t a l d i f f e r -e n t i a l equat ions which may be n u m e r i c a l l y i n t e g r a t e d f o r an a r b i t r a r y P r and an a r b i t r a r y s u r f a c e tempera tu re : I 35 5 F " * + (8 + n) F F " - (1 + 2n) ( F ' ) 2 = 2(1 + 2n)G - (2 - n) n G' + a ! G r _ 2 /V 2 / 5 ( n + 3 ) ,r _ u ^ „ ,^„ -2/5 _ . -2/5(n + 3) G' = H + o Gr ' r (66) (67) H 1 1 + (n + 8) F H ' - n P r F' H = a ' |cr 2 / 5 r 2 / 5 ( n + 3 ) | . (68) For the case of an i s o t h e r m a l s u r f a c e these equat ions s i m p l i f y t o , 5 F " ' + 8 F F " - ( F ' ) 2 = 2(G -^G,) (69) G' = H (70) H " + 8/5 P r F H' = 0 . (71) N u m e r i c a l r e s u l t s f o r P r = 1 are p r e s e n t e d i n f i g u r e s 11 and 12. S o l u t i o n s may be o b t a i n e d f o r asymptot i c va lues o f the P r number i n a manner s i m i l a r to that demonstrated f o r the 2 - d i m e n s i o n a l f low prob lem. A l though a boundary l a y e r type of s o l u t i o n has been a r r i v e d a t , i t s p h y s i c a l p e c u l a r i t i e s remain to be j u s t i f i e d . The s o l u t i o n y i e l d s a f low t h a t p rogresses r a d i a l l y outwards which i s of course not i d e n t i -f i a b l e w i t h any n a t u r a l l y o c c u r r i n g f lows over a p l a t e of f i n i t e r a d i u s . I t does p r e d i c t a f low observed i n c e l l s of i n s t a b i l i t y over h o r i z o n t a l s u r f a c e s by Chandrasekhar . F i g u r e 11 Stream and V e l o c i t y Funct ions f o r A x i a l l y Symmetr ica l F l o w - I s o t h e r m a l P l a t e F i g u r e 12 P ressu re and Temperature Funct ions f o r A x i a l l y Symmetr ica l Flow - I so thermal P l a t e i 38 4 .2 Combined Free and Forced Convect ion f o r an A r b i t r a r y P r Number and a D i s c Temperature 2 Which i s P r o p o r t i o n a l to r . The a n a l y s i s presented h e r e i n w i l l develop a boundary l a y e r type of s o l u t i o n f o r a x i a l l y - s y m m e t r i c a l f low above an e x a c t l y h o r i z o n t a l s u r f a c e which i s r o t a t i n g at a constant speed. The equat ions a p p l i c a b l e f o r t h i s f low are the equat ions (54) through (58) . As f o r equat ions (64) a set of s i m i l a r i t y t r a n s f o r m a t i o n s are d e f i n e d which l e a d to a se t of c o m p a t i b i l i t y equat ions f o r the exponents . Gen-e r a l i t y c a n , however, not be mainta ined i n t h i s i n s t a n c e and only the f o l l o w i n g s i m i l a r i t y t rans fo rmat ions are p o s s i b l e . 2 = r F(n) n = z - r 2 G (n) 9 = r 2 H (n) (72) r v = a r 2 T (n) • Here "F " i s a c i r c u l a t i o n v a r i a b l e . The constant " a " i s d e f i n e d by the n o - s l i p c o n d i t i o n on the s u r f a c e and i s equal t o , a = - ^ T T (73) where and Gr 2 ' 5 V AT = (T - T V @r = 1. (74) r e f w 0 0 39 Upon s u b s t i t u t i n g the a b o v e . i n t o the equat ions (55) t o (58) the f o l l o w i n g t o t a l d i f f e r e n t i a l equat ions r e s u l t , M 2 = 2 J l F " 1 + 2 F ' ' F - (F 1 ) = 2G - a T G' = H r" + 2 (FT" - F' D = 0 H " + 2 P r (FH' - F' H) = 0. (75) (76) (77) (78) For P r = 1 t h i s s e t of equat ions reduces t o two t h i r d order t o t a l d i f f e r -e n t i a l equat ions which must be i n t e g r a t e d s i m u l t a n e o u s l y . The boundary c o n d i t i o n s a r e , n = o F = 0 F = 0 H = 1 r = I F' = 0 G = G ' = r = 0 > (79.) The r e s u l t s of the n u m e r i c a l i n t e g r a t i o n are g iven i n f i g u r e s 13 and 14 f o r va lues of a = 0 a n d . 9 . The s t a r t i n g p r o f i l e f o r the i n t e g r a -t i o n of the l a t t e r was taken from Cochran ( 1 2 ) . 1 T g= 0 g=9 F(oo) .793 1.578 FlO) 1.164 16.488 F i g u r e 13 V e l o c i t y Funct ions f o r A x i a l l y Symmetr ica l Flow - P l a t e Temperature P r o p o r t i o n a l t o r Figure 14 Temperature Functions f or A x i a l l y Symmetrical Flow - Plate Temperature Proportional to r^ 42 4 . 3 N u m e r i c a l I n t e g r a t i o n As d i s c u s s e d i n the r e s p e c t i v e s e c t i o n s , the s e t s of equat ions which were n u m e r i c a l l y i n t e g r a t e d a r e : S e m i - i n f i n i t e p l a t e ( i s o t h e r m a l s u r f a c e ) , A r b i t r a r y P r - equat ions (21) to (23) . Pr » 1 " o u t e r r e g i o n " - equat ions (29) to (31) " i n n e r r e g i o n " - equat ions (36) P r << 1 " i n n e r r e g i o n " - equat ions (39) t o (41) " o u t e r r e g i o n " - equat ions (51) . I n f i n i t e d i s c , Free c o n v e c t i o n , i s o t h e r m a l s u r f a c e - equat ions (69) to (71) 2 Combined convect ion (T « r ) - equat ions (75) to (78) . w E x c e p t i n g equat ions (36) and ( 5 1 ) , the se ts of equat ions may i n . every case be reduced to two s imultaneous t h i r d order t o t a l - d i f f e r e n t i a l e q u a t i o n s . These e q u a t i o n s , s u b j e c t to the boundary c o n d i t i o n s s t a t e d , cannot be s o l v e d i n c l o s e d form. T h e r e f o r e , i n t e g r a t i o n has to proceed n u m e r i c a l l y . The i n t e g r a t i o n r e q u i r e s an i n i t i a l assumption of the complete form of one of the f u n c t i o n s . Fur thermore , the m i s s i n g i n i t i a l c o n d i t i o n had to be assumed and was then i t e r a t e d upon so as to f u l f i l l the c o n d i t i o n s at the o ther end. The d i r e c t i o n of i n t e g r a t i o n was always f rom the end at which two i n i t i a l c o n d i t i o n s were known. In t h i s manner one f u n c t i o n would be generated w h i l e va lues f o r the o ther were t r a n s f e r r e d 43 f rom the assumed form. The technique was then r e v e r s e d , i n t e g r a t i n g the o ther e q u a t i o n , u n t i l each f u n c t i o n had converged w i t h i n a s p e c i f i e d l i m i t o f e r r o r . The i n t e g r a t i o n techn ique employed was a f o u r t h order Runge -Kut ta fo rward i n t e g r a t i o n r o u t i n e . U n f o r t u n a t e l y , t h i s method o f s o l u t i o n would converge t o a c o r r e c t r e s u l t on ly when reasonab ly c l o s e e s t i m a t e s o f the m i s s i n g i n i t i a l c o n d i t i o n and of the complete form of the o t h e r f u n c t i o n were a v a i l a b l e . In o r d e r to o b t a i n these v a l u e s an approximate i n t e g r a t i o n a c c o r d i n g to the von Karmian - Pohlhausen techn ique was per formed; In g e n e r a l the e s t i m a t e s had to be much b e t t e r f o r the extreme P r numbers. Only a s m a l l d e v i a t i o n i n the m i s s i n g i n i t i a l v a l u e w o u l d , f o r i n s t a n c e , cause the s o l u t i o n t o o s c i l l a t e and d i v e r g e . A l s o , convergence f o r the extreme P r numbers was ex t remely s low and a techn ique ave rag ing c o n s e c u t i v e p r o f i l e s was employed. Moreover , i n some cases the equat ions possess a s i n g u l a r i t y at the s t a r t i n g boundary . For these equat ions i n t e g r a t i o n was s t a r t e d at a ve ry s m a l l v a l u e of the independent v a r i a b l e i n s t e a d o f f rom z e r o . I n i t i a l v a l u e s as o b t a i n e d from the i n t e g r a t i o n f o r the i s o t h e r m a l s e m i - i n f i n i t e p l a t e are summarized i n Table I. L a s t l y , f o r the combined f r e e and f o r c e d - convect ion case f o r a r o t a t i n g d i s c the s t a r t i n g f low p r o f i l e s were taken from Cochran 's (12) t a b u l a t e d data f o r a s p i n n i n g i s o t h e r m a l d i s c . 44 V EXPERIMENTAL APPARATUS The purpose of the experimental work was to e s t a b l i s h the existence of a laminar boundary layer above a heated h o r i z o n t a l surface, and to obtain some measure of the distance over which i t : e x t e n d s . The existence of the boundary layer was doubted because, as discussed previously, i t extends i n an adverse gravity f i e l d . An o p t i c a l approach was chosen since it,provides a method free of interference with the flow f i e l d f o r a q u a l i t a t i v e j u s t i f i c a t i o n of the analysis and also s u f f i c i e n t l y accurate data to obtain some quantitative r e s u l t s . Since the laminar boundary layer was expected to e x i s t under moderate temperature differences only, the o p t i c a l system was required to have a high s e n s i t i v i t y . Moreover i t had to be of the limited-focusing type such that attention could be r e s t r i c t e d to a s l i c e of the area above the h o r i -zontal surface. A s c h l i e r e n semi-focusing colour system was thought to f u l f i l l a l l the requirements as w e l l as p o s s i b l e , at a r e l a t i v e l y moderate cos t. 5.1 O p t i c a l Bench. The o p t i c a l system consisted of two separate benches : on the f i r s t were mounted the source, the source lenses, and the "upstream" s c h l i e r e n mirror; and on the second the "downstream" s c h l i e r e n mirror, the f o c a l plane cplour f i l t e r , the object lenses, and the camera. The benches were separated s u f f i c i e n t l y to allow an interference free l o c a t i o n of the h o r i z o n t a l t e s t plates. The separation must be of at least two f o c a l lengths. A schematic drawing of the system i s given i n figure (26). 45 F i g u r e 27 shows the o p e r a t i o n of the s e m i - f o c u s i n g t e c h n i q u e , of which a b r i e f account i s g iven i n Appendix A. A l s o d e s c r i b e d are the s e n s i -t i v i t y and depth of f i e l d c r i t e r i a . The s p e c i f i c a t i o n s of the system were as f o l l o w s : Source - 500 watt tungsten f i l a m e n t lamp Condensing lens - coa ted , a n a s t i g m a t i c , f : 3 . 5 , 5 inches f o c a l l e n g t h . Source s l i t - .006 inches w i d t h by .5 inches l e n g t h . S c h l i e r e n lenses - 8 i n c h diameter p a r a b o l i c m i r r o r s , 63 .5 inches f o c a l l e n g t h . F o c a l p lane l i g h t c u t - o f f - graded 4 c o l o u r f i l t e r w i t h bands .009 i n s . w i d e . Object lens - Schneider "Symmar" f : 5 . 6 , 210 mm f o c a l l e n g t h . Camera - 35 mm S . L . R . camera body w i t h f o c a l - p l a n e s h u t t e r . Whi le a t tempt ing to a t t a i n an o p t i m a l r e s o l u t i o n from the apparatus the f o l l o w i n g genera l c o n s i d e r a t i o n s were found to be very impor tan t . For a h i g h s e n s i t i v i t y system the lenses shou ld be of a good q u a l i t y and c o r r e c t e d f o r most o p t i c a l a b b e r a t i o n s . Whi le exper iment ing w i t h lenses o f a l e s s e r q u a l i t y than used i n the f i n a l v e r s i o n of the a p p a r a t u s , a s t i g -matism was severe and t h i s a c c o r d i n g l y r e s t r i c t e d the p e r m i s s i b l e s e n s i t i -v i t y o f the system. Fur thermore , as the s e n s i t i v i t y was i n c r e a s e d a l i g n -ment and i s o l a t i o n from v i b r a t i o n e x c i t a t i o n became more c r i t i c a l . The l a t t e r , i n p a r t i c u l a r , was very much more important than at f i r s t appre -c i a t e d . The band dimensions of the c o l o u r f i l t e r s p e c i f i e d above were found to be an optimum f o r the system s t r i k i n g a ba lance between the d e s i r a b i l i t y of h i g h s e n s i t i v i t y and low d i f f r a c t i o n b l u r r i n g . 46 5 .2 Al ignment of the O p t i c a l Benches As trie i n v e s t i g a t i o n was to be performed on p e r f e c t l y h o r i z o n t a l p l a t e s , al ignment of the s c h l i e r e n beam was c r i t i c a l . A t h e o d o l i t e was used i n c o n j u n c t i o n w i t h a h e l i u m gas l a s e r beam (make: S p e c t r a P h y s i c s , 0 . 3 mW continuous o u t p u t ) . The l a s e r beam was passed a long the o p t i c a l a x i s o f the system and component p o s i t i o n s were ad jus ted u n t i l they agreed w i t h the c o r r e c t . r e a d i n g of the t h e o d o l i t e , s i m u l t a n e o u s l y w i t h e q u a l i n l e t and o u t l e t angles of the o f f - c e n t e r s c h l i e r e n system. The p l a t e s were c a r e f u l l y l e v e l l e d w i t h a s e n s i t i v e mechanica l l e v e l (make: S t a r r a t t ) . The supports had f i n e adjustment screws f o r t h i s purpose. 5 . 3 Co lour F i l t e r A c o n s i d e r a b l e number o f attempts were made to a r r i v e at a s a t i s f a c t o r y graded f i l t e r . At f i r s t f i l t e r s were made from Kodak g e l a t i n paper by c u t t i n g narrow s t r i p s and a f f i x i n g them next to each o t h e r . These f i l t e r s proved to be very good and had good c o l o u r r e n d i t i o n f o r s t r i p w idths above approx imate ly .025 i n c h e s . Below t h i s w i d t h , i n t e r f e r e n c e , d u e to improper l y cut edges became r a p i d l y s e v e r e . The f i l t e r f i n a l l y used i n the apparatus was.made from High Speed Ektachrome type B f i l m , which was a l s o the f i l m u s e d . f o r a l l of the s c h l i e r e n p i c t u r e s . The procedure fo r making the f i l t e r c o n s i s t e d of p roduc ing f i r s t a 4 x 5 i n c h . m a s t e r g r a t i n g which had s u c c e s s i v e c l e a r and t r i p l e w i d t h opaque bands. The master was then p l a c e d onto a d i f f u s e l y i l l u m i n a t e d s u r f a c e and indexed w h i l e reexpos ing a s i n g l e frame s u c c e s s i v e l y w i t h y e l l o w , b l u e , red and green f i l t e r s mounted on the camera. 5 . 4 Rectangular P l a t e s and D i s c Two r e c t a n g u l a r p l a t e s were made, 12 inches square and 21 inches s q u a r e , and one round p l a t e , 7 inches i n d iameter . The top s u r f a c e s of the p l a t e s were r e s p e c t i v e l y 3/8 inches and 3/4 inches aluminum p l a t e and 1/4 inches copper p l a t e . The su r faces were backed by sandwiched (*) mica-n ichrome w i r e h e a t e r s . Maximum heat outputs of the heaters were as f o l l o w s : 12 i n c h p l a t e - 100 w a t t s ; 21 i n c h p l a t e - 300 w a t t s ; 6 i n c h d i s c 25 w a t t s . I n v a r i a b l y , c o n s i d e r a b l e p r e c a u t i o n was taken t o avo id heat leakage through the bottom and s i d e s . The bases were made of wood f i l l e d w i t h f i b e r g l a s i n s u l a t i o n and s u r f a c e d on the o u t s i d e w i t h 1/4 i n c h cork i n s u l a t i o n . The p l a t e s were i n i t i a l l y made w i t h sharp l e a d i n g edges but were l a t e r equipped w i t h rounded edges. These edges were t h i n ho l low h a l f -rounds of wood f i l l e d w i t h i n s u l a t i o n to ensure t h a t the edge s tayed as c o o l as p o s s i b l e . The p l a t e s are p i c t u r e d i n f i g u r e s 15 and 16. Temperatures of the s u r f a c e s were measured u s i n g b u r i e d , copper -constantan thermocouples . They were spaced at 1 i n c h i n t e r v a l s a long a r a d i u s of the d i s c and at 2 i n c h i n t e r v a l s a long a center l i n e of the square p l a t e s . A l l thermocouples were c a r e f u l l y c a l i b r a t e d p r i o r t o i n s t a l l a t i o n . (*) Except the 12 x 12 inches p l a t e , which was heated by a Pyrex pannel t h i n l y coated w i t h a un i fo rm l a y e r of z i n c o x i d e . F igure 16 7 Inches Diameter D i s c 49 VI EXPERIMENTAL PROCEDURE 6 . 1 Test Procedure The p l a t e s were l o c a t e d and mounted i n a manner such tha t d i r e c t i n t e r f e r e n c e from the support or any s u r f a c e s around the p l a t e s cou ld be d i s c o u n t e d . No s p e c i a l p r e c a u t i o n s were t a k e n , on the o ther h a n d , to i s o l a t e p o s s i b l e . f r e e convect ion cu r rents s e t up w i t h i n the l a b o r a t o r y . To reduce t h i s e f f e c t , a f a i r l y lengthy qu iescent s t a t e was observed p r i o r to the t a k i n g of any s c h l i e r e n photographs. In a d d i t i o n , the p l a t e s u r f a c e temperatures were p e r m i t t e d to s t a b i l i z e be fo re p i c t u r e s were taken . T y p i c a l l y , thermocouples l o c a t e d a long a rad ius of the d i s c and a long a cente r l i n e of the square p l a t e s d i s p l a y e d a v a r i a t i o n of l e s s than one degree Fahrenhe i t f o r the maximum t e m p e r a t u r e - d i f f e r e n c e c o n s i d e r e d . In genera l the s u r f a c e temperature would i n c r e a s e a l i t t l e from the o u t s i d e edge to the centre of the p l a t e s . For every separate group of p i c t u r e s t a k e n , a c a l i b r a t i o n photograph was taken as w e l l . This c o n s i s t e d of a t ransparent s c a l e p l a c e d at the c e n t e r l i n e of the p l a t e s . The q u a n t i t a t i v e data was read o f f the p i c t u r e s by p r o j e c t i n g them onto a l a r g e graded s c r e e n . This was p o s i t i o n e d i n a way such that the g r i d l i n e d up w i t h the s c a l e markings of the c a l i b r a t i o n t ransparency . 6 .2 Exper imenta l R e s u l t s . Examples of s c h l i e r e n photographs are given i n f i g u r e s 17 through 20. As was s t a t e d i n the i n t r o d u c t o r y s e c t i o n s , the f low has a g r a v i t a -t i o n a l l y u n s t a b l e s t r a t i f i c a t i o n and may thus be t r i p p e d to i n s t a b i l i t y 50 by any spur ious l a b o r a t o r y c u r r e n t s . In essence the f lew behaved i n the form of laminar s p e l l s which v i s i b l y became more s t a b l e w i t h the e l i m i n a t i o n of some i n t e r f e r e n c e s ( fan c o o l i n g of the s o u r c e , w a l k i n g about and even b r e a t h i n g i n the v i c i n i t y o f . t h e p l a t e ) . A l l p i c t u r e s were taken w i t h s t r i c t adherence to these p r e c a u t i o n a r y measures. The p o i n t of i n s t a b i l i t y i s p l o t t e d i n f i g u r e s 21 to 24 f o r the v a r i o u s p l a t e s . As may be expected from analogy w i t h o ther t r a n s i t i o n a l f l o w s , the onset of i n s t a b i l i t y may only be s t i p u l a t e d to occur over a c e r t a i n range of the c h a r a c t e r i s t i c parameter . Examining f i g u r e s 21 and 2 2 , i t i s observed t h a t t r a n s i t i o n does not occur at e x a c t l y the same average Grashof number f o r the two d i f f e r e n t l y s i z e d p l a t e s . The onset of i n s t a b i l i t y i s probably a f f e c t e d by the , s i z e of the thermal j e t r i s i n g above the cente r of the p l a t e s which has a s t a b i l i z i n g e f f e c t upon the f low . Fur thermore , the onset of i n -s t a b i l i t y i s seemingly a f f e c t e d by the f i n i t e n e s s of the p l a t e s . Th is i s seen by the f a l l i n g - o f f of the data at lOw AT ^ i n f i g u r e s 22 and 23 . F igures 23 and 24 d e f i n e the p o i n t of i n s t a b i l i t y f o r the s t a t i o n a r y and r o t a t i n g d i s c r e s p e c t i v e l y . In t h i s c a s e , the Gr number i s b a s e d , as f o r the square p l a t e s , upon the d i s t a n c e from the o u t s i d e edge, w h i l e the Reynolds number i s based upon the d i s t a n c e from the c e n t e r . These d e f i n i t i o n s are t h e o r e t i c a l l y j u s t i f i e d s i n c e the f r e e convec t i ve f low and the fo rced f low are b a s i c a l l y i n oppos i te d i r e c t i o n s . L a s t l y , observab le i n the photographs f o r the square p l a t e s i s a s m a l l s e p a r a t i o n bubble near the l e a d i n g edge. In c o n t r a s t , f o r a sharp l e a d i n g edge temporary s e p a r a t i o n was more pronounced, and f o r the h i g h e r temper-atures the r e s u l t was the e n t i r e s e p a r a t i o n of the f l o w . F i g u r e 17 S c h l i e r e n Photographs w i t h 21 i n c h P l a t e ; a ' A T r e f = 2 3 - 5 ° F ' T o o = 7 1 - ° F ; b. AT - 3 9 . ° F , 7 1 . - F ; c . AT =43.5°F, 1^=71. °F; d. AT =50.5°F, T =71. °F. 52 1*4 o II o al u < 1) 4-1 nj o r H LO PL. • r H U II d r 8 CM " r - l PM 0 J ! O 4-1 • • H i H 5 0 0 II CO ^ X ! 0) 6 >-i cfl H M < txO O • 4J O o xi ••> PM C U 0) • H r H 00 oo m r H . a) m M II d « H • H rJ PM H < II o 00 H o F i g u r e 19 S c h l i e r e n Photographs w i t h 7 i n c h D i s c : a- A T r e f = 5 2 . 5 ° F , T r o = 72. % 0=0 rpm; b. A T r e f = 6 2 . 5 ° F , Too=72.0°F, 0=32.8 rpm; c A T r e f = 9 4 . 5 ° F , T a >= 72.5°F, 0=13.9 rpm; d A l r e f = 9 4 . 5 ° F , T ^ 7 2 . 5 « F , 0=18.6 rpm. F i g u r e 20 S c h l i e r e n Photographs w i t h 7 i n c h Dis a . AT r = 9 1 . 5 ° F , T =71.°F , f i =0 rpm, b. AT =115.°F, T =71°F, ii =0 rpm. r e f 0 0 ^ 5 5 40 60 AT r ef~°F Figure 21 C r i t i c a l Gr for 21 Inch Square Plat* 20 40 60 80 AT r e f ~°F 100 Figure 22 C r i t i c a l Gr for 12 inch Square Plata 60 80 A T r e f ~ « F Pig. 23 C r i t i c a l Gr for 7 inch Diac for Frae Convaction 57 V I I DISCUSSION OF THE RESULTS Thermal convec t ion above a h o r i z o n t a l s u r f a c e had h i t h e r t o been e x p l a i n e d by two d i f f e r e n t t h e o r i e s . ( i ) When the s u r f a c e i s of an i n f i n i t e ex tent c o n v e c t i o n must i n the mean take p l a c e i n an e s s e n t i a l l y v e r t i c a l d i r e c t i o n . Consequent ly near the s u r f a c e i t would be dominated by some type of c e l l u l a r f l o w . ( i i ) On the o ther hand , when the s u r f a c e i s f i n i t e , a boundary l a y e r theory of r e l a t i v e l y recent o r i g i n , has l e a d to the b e l i e f tha t c o n v e c t i o n occurs through a l a m i n a r boundary l a y e r f low which grows f rom the edges. The boundary l a y e r s would meet i n the middle o f the s u r f a c e , t u r n up , and r i s e as a plume. S i n c e the plume would cover on ly a ve ry s m a l l p o r t i o n of the p l a t e s u r f a c e , (near the s u r f a c e i t would have a w i d t h of o rder 25 a c c o r d i n g to Stewartson (1)^ i t was p o s t u l a t e d tha t the heat t r a n s f e r r a t e would be governed by the 5th root o f the Grashof number. As the f i n i t e s u r f a c e i s e n l a r g e d i t becomes ev ident tha t at some, p o i n t t h i s dependence must break down, f o r the f l ow i n the c e n t r a l r e g i o n must become i n c h a r a c t e r s i m i l a r t o t h a t above an i n f i n i t e s u r f a c e . In f a c t , s i n c e the l a m i n a r boundary l a y e r as d i s c u s s e d p r e v i o u s l y , i s p o t e n t i a l l y u n s t a b l e i t may be expected t h a t the l a m i n a r f low would extend f o r only a s h o r t d i s t a n c e . As the c h a r a c t e r i s t i c parameter of the f low i s the G r a s -hof number, t h i s d i s t a n c e s h o u l d be e x p r e s s i b l e i n terms o f t h i s parameter , as a " c r i t i c a l Grashof number". The c r i t i c a l Grashof number then enables the d e t e r m i n a t i o n of a p e r m i s s i b l e p l a t e s i z e or a t e m p e r a t u r e - d i f f e r e n c e between the p l a t e and f l u i d b u l k f o r which the f low over v i r t u a l l y the e n t i r e p l a t e w i l l be l a m i n a r . 58 S o l u t i o n s of the equat ions of f low can u n f o r t u n a t e l y only be ob -t a i n e d f o r a s e m i - i n f i n i t e p l a t e or a d i s c o f . i n f i n i t e l y l a r g e r a d i u s . In t h i s respect the p l a t e s used f o r the exper imenta l observa t ions of , the f low do not a f f o r d an exact model f o r the t h e o r e t i c a l s o l u t i o n s . F u r t h e r -more, s i n c e the s o l u t i o n s are i n the form of s i m i l a r i t y s o l u t i o n s t h e i r v a l i d i t y i s l i m i t e d by c e r t a i n a p r i o r i assumpt ions . The f o l l o w i n g subsec t ions r e s p e c t i v e l y d i s c u s s the t h e o r e t i c a l s o l u t i o n s and t h e i r l i m i t a t i o n s , and the e x p e r i m e n t a l d a t a . 7 . 1 Numer ica l S o l u t i o n s The h o r i z o n t a l boundary l a y e r f l o w , as e x p l a i n e d i n the i n t r o d u c t i o n , i s expected to be fundamenta l ly d i f f e r e n t from the f low tha t occurs above an i n c l i n e d s u r f a c e . The a p p l i c a b i l i t y of the s o l u t i o n s m u s t , . . t h e r e -f o r e , be l i m i t e d to a very s m a l l i n c l i n a t i o n of the s u r f a c e t o . t h e - h o r i -z o n t a l . The p e r m i s s i b l e angle of i n c l i n a t i o n as d e r i v e d f o r the s e m i -i n f i n i t e s u r f a c e i s (equat ion ( 1 9 ) ) , For Gr = 10 the i n c l i n a t i o n i s thus found to be l e s s than a b o u t . 5 ° . In a d d i t i o n to the l i m i t a t i o n upon the i n c l i n a t i o n , s i m i l a r i t y s o l u t i o n s are i n genera l r e s t r i c t e d t o c e r t a i n reg ions of the f low f i e l d as terms which are of a s m a l l e r o rder are n e g l e c t e d . Outs ide the reg ions of good a p p r o x i m a t i o n , the terms which were assumed to be s m a l l w i l l become s i g n i f i c a n t . The c o n t r o l l i n g term f o r the s e m i - i n f i n i t e s u r f a c e a n a l y s i s which thus must remain l a r g e i s a < a | tan- — (Gr . -1/5 x 5 (Gr x 3+n,2/5 59 In c o n j u n c t i o n w i t h t h i s term and the p e r m i s s i b l e i n c l i n a t i o n , the imposed w a l l temperature v a r i a t i o n must be such tha t the exponent n remains between - 3 and +2. In s e c t i o n 3.2 n u m e r i c a l s o l u t i o n s are d e r i v e d f o r an a s y m p t o t i c a l l y l a r g e and an a s y m p t o t i c a l l y s m a l l P r number. These s o l u t i o n s have a s i g n i f i c a n t p r a c t i c a l importance as they may be a p p l i e d w i t h a good approx imat ion down to P r = 1 0 , and up to P r = . 1 r e s p e c t i v e l y . When the equat ions were thus expanded w i t h the P r a n d t l number the f low f i e l d at each extreme was d e s c r i b e d by two r e g i o n s , an " i n n e r " and an " o u t e r " . , For a l a r g e P r number the thermal boundary l a y e r i s much s m a l l e r than the v e l o c i t y boundary l a y e r and i s , t h e r e f o r e , p h y s i c a l l y con ta ined w i t h i n an " i n n e r " r e g i o n . The " o u t e r " reg ion then conta ins only the remainder of the.momentum boundary l a y e r . For a s m a l l P r a n d t l number the thermal boundary l a y e r i s much l a r g e r than the momentum boundary l a y e r . T h e r e f o r e , the e n t i r e temperature drop takes p l a c e w i t h i n an " o u t e r " r e g i o n . In the " i n n e r " reg ion the temperature remains almost constant throughout w h i l e the the v e l o c i t y drops to zero on the boundary. M a t h e m a t i c a l l y the fo rmat ion of these reg ions may be s a i d to be the r e s u l t of u s i n g only the f i r s t term of a s i n g u l a r p e r t u r -b a t i o n expansion i n P r . T h e r e f o r e , d e s p i t e the f a c t tha t the s o l u t i o n s i n each case d e s c r i b e on ly one r e g i o n , they extend across the e n t i r e f low f i e l d and the expanded v a r i a b l e r) measures i n every case from the s u r f a c e . L a s t l y , s i m i l a r i t y s o l u t i o n s are presented f o r a x i a l l y s ymmet r i ca l f l o w . These s o l u t i o n s p r e d i c t a f low that proceeds outwards above an i n f i n i t e d i s c . The f low which was e x p e r i m e n t a l l y observed above a f i n i t e s u r f a c e grows inwards and thus does not comply w i t h the theory . For l a c k of an exact unders tand ing of the t h e o r e t i c a l s o l u t i o n s i t may be 60 p o s t u l a t e d that a p h y s i c a l f low a long the l i n e s of the theory may occur under some o ther than constant s u r f a c e temperature c o n d i t i o n . For the combined f low a s i m i l a r i t y s o l u t i o n e x i s t s on ly f o r a s u r -2 face temperature that i n c r e a s e s w i t h r . In t h i s case the c h a r a c t e r i s -t i c parameter , c o n t r o l l i n g the r e l a t i v e importance of f r e e convect ion 2 4/5 to f o r c e d c o n v e c t i o n , i s Re^ /Gr . Th is parameter agrees w i t h the 2 - d i m e n s i o n a l a n a l y s i s , Mor i ( 1 1 ) , f o r which a p h y s i c a l s i t u a t i o n e x i s t s . 7.2 S c h l i e r e n Observat ions The amount of e x p e r i m e n t a l data that had been a v a i l a b l e f o r c o m p a r i -son w i t h the p r e d i c t i o n s of the theory i s very s m a l l . Schmidt (4) took some s c h l i e r e n p i c t u r e s of the f low about a h o r i z o n t a l s u r f a c e but i n h i s case both s i d e s o f . t h e p l a t e were heated . The upper s u r f a c e was, t h e r e f o r e , i n the l e e of a " t h e r m a l j e t " from below the p l a t e . F ishenden and Saunders (3) and r e c e n t l y R e i l l y e t a l . (15) p u b l i s h e d some data f o r the heat t r a n s f e r r a t e from a h o r i z o n t a l s u r f a c e . However, they d i d not d e f i n e t h e i r c o n f i g u r a t i o n a c c u r a t e l y enough to s t a t e p o s i t i v e l y whether convect ion was by l a m i n a r f low or n o t . The i r data i s r e p l o t t e d i n f i g u r e 25 t o g e t h e r w i t h the t h e o r e t i c a l da ta f o r the constant temperature s e m i -i n f i n i t e p l a t e . The mean t h e o r e t i c a l N u s s e l t number and the Grashof number a r e , i n t h i s c a s e , based upon the t o t a l p l a t e w i d t h , o r , Nu = - | . 2 2 / 5 G r 1 / 5 H' ( 0 ) . The s l o p e s of t h e i r data seem to imply that convect ion occured through a c e l l u l a r mot ion . This o b s e r v a t i o n i s q u i t e r e a l i s t i c as t h e i r 61 data was probably obta ined at l a r g e temperature d i f f e r e n c e s . This o b s e r v a t i o n i s a d d i t i o n a l l y supported by the f a c t that t h e i r data l i e s h i g h e r , i . e . they measured a h i g h e r heat t r a n s f e r r a t e . Stewartson ( 1 ) , on the o ther hand, a l s o compared h i s data to Fishenden and Saunders ' and c la imed that the o n e - f i f t h power r e l a t i o n s h i p p rov ides a good f i t . As mentioned e a r l i e r , he was however, m i s l e d by c l a i m i n g that h i s boundary l a y e r s o l u t i o n p o r t r a y e d a f low beneath a heated s u r f a c e and consequent ly compared h i s t h e o r e t i c a l r e s u l t to the wrong exper imenta l d a t a . T h e r e f o r e , i n order to j u s t i f y e x p e r i m e n t a l l y the e x i s t e n c e of laminar c o n v e c t i o n , account must be taken of the c r i t i c a l Grashof number f o r the f l o w . S c h l i e r e n p i c t u r e s p resented i n t h i s t h e s i s support the content ion that a laminar boundary l a y e r does e x i s t . However, t h i s content ion i s c o n d i t i o n a l . For the s u r f a c e s examined the c h a r a c t e r i s t i c Gr number x must be of o rder l e s s thati 1 0 ^ , but l a r g e enough (say g reate r than 500) to form a boundary l a y e r f l o w . The su r round ing space must be s u f f i c i e n t l y f ree of spur ious a i r cu r ren ts so as not to t r i p the f l o w , which i s g r a v i -t a t i o n a l l y ext remely u n s t a b l e . The average c r i t i c a l Grashof number (Gr^) f o r the 12 i n c h p l a t e was measured to be 8 x 10^ (from f i g u r e 2 2 ) . In appendix B i t i s shown that the e q u i v a l e n t Grashof number ( Gr ) based on the boundary l a y e r 4 4 t h i c k n e s s i s 3.2 x 10 . The R a g i s then 2 . 3 x 10 (Pr = .72 fo r a i r ) . This c r i t i c a l R a y l e i g h number may be compared w i t h the c r i t i c a l va lues ob ta ined from the c l a s s i c a l R a y l e i g h s t a b i l i t y s t u d i e s . For the case of a h o r i z o n t a l l a y e r of f l u i d i n an open v e s s e l , i n s t a b i l i t y , f o r an i n i t i a l l y s t a t i o n a r y f l u i d , s e t s i n at a Ra^ (based on the depth of the f l u i d ) of 1100. I t may be reasoned that the much h i g h e r va lue ob ta ined f o r t r a n s i t i o n 62 o f the laminar boundary l a y e r of the type cons idered i n t h i s t h e s i s i s due to the s t a b i l i z i n g e f f e c t of the l a t e r a l f low f e e d i n g the c e n t r a l plume. VI I I CONCLUSIONS This i n v e s t i g a t i o n has l e d to the f o l l o w i n g r e s u l t s : Two-d imensional f low ( f r e e convect ion) 1 . T h e o r e t i c a l s o l u t i o n s f o r the s e m i - i n f i n i t e p l a t e have been d e r i v e d f o r a "power r e l a t i o n " type of w a l l temperature v a r i a t i o n . The constant f l u x c o n d i t i o n i s obta ined as a s p e c i a l case. Numer ica l data i s presented f o r the i s o t h e r m a l s u r f a c e . 2 . Asymptot ic s o l u t i o n s f o r P r approaching i n f i n i t y and zero have been d e r i v e d . These s o l u t i o n s are a good approx imat ion when P r > o r < . l . Numer ica l data i s presented f o r the i s o t h e r m a l s u r f a c e . 3. L i m i t s upon the a p p l i c a b i l i t y of the theory are d e f i n e d . 4. The onset of unstab le f l o w , above the p l a t e s has been examined and found to occur at a Gr number of order 10"*. Below t h i s Gr number the laminar boundary l a y e r f low may extend over almost the e n t i r e s u r f a c e and the heat t r a n s f e r r a t e w i l l depend upon the f i f t h root of the Gr number. A x i a l l y symmetr i ca l f low ( f ree convect ion) 5 . T h e o r e t i c a l s o l u t i o n s f o r the i n f i n i t e d i s c have been d e r i v e d . These p r e d i c t an outward f low and are thus not a p p l i c a b l e to a d i s c of f i n i t e d iameter . A x i a l l y symmetr i ca l f low (combined convect ion) . .._ . 2 4/ 6. The c h a r a c t e r i s t i c parameter f o r combined convect ion i s Re^ /Gr when expansion occurs about the f ree convect ion mode. 65 7. The t h e o r e t i c a l s o l u t i o n i s f o r a f low which i s a ided by buoyancy. The s i m i l a r i t y approach y i e l d s only t h i s one s o l u t i o n which i s 2 f o r a s u r f a c e temperature that i n c r e a s e s w i t h r . 8 . The f low e x p e r i m e n t a l l y examined i s of the opposing t ype . 9. A curve i s p resented which approx imate ly l o c a t e s the t r a n s i t i o n r eg io n f o r a f low that i s p r i m a r i l y f r e e l y c o n v e c t i n g t o a f low that i s p r i m a r i l y b e i n g f o r c e d . 66 REFERENCES 1. K. S tewar tson , "On f ree convect ion from a h o r i z o n t a l p l a t e " , Z . A . M . P . , 9 a , 1958, 276 -282 . 2 . W.N. G i l l , D.W. Zeh and E. de l -Casal•, . "F ree convect ion on a h o r i -z o n t a l p l a t e " , Z . A . M . P . . 16 , 1965, 5 3 9 - 5 4 1 . 3. M. Fishenden and O.A. Saunders , "The c a l c u l a t i o n of convect ion heat t r a n s f e r " , p a r t 2 , E n g i n e e r i n g 130, 193-194; a l s o i n "An I n t r o d u c t i o n  to Heat T r a n s f e r " , Oxford U . P . , 1950, pp. 9 5 - 9 6 . 4. E-. Schmidt , . "Schl ierenaufnahmen des Temperaturefe ldes i n der Nahe warmeabgebender K o r p e r , V . D . I . Forschung 3 ( 4 ) , 1932, 181 -189 . 5 . R. We ise , "Warmeubergang durch f r e i e Konvekt ion an quadrat i schen P l a t t e n " , V . D . I . Forschung, 6., 1935, 281 -292 . 6. S . Sugawara and I. M i c h i y o s h i , "Heat t r a n s f e r from a h o r i z o n t a l f l a t p l a t e by n a t u r a l c o n v e c t i o n " . T rans . Japan Soc. Mech. E n g r s . , 2 1 , 1955, 651 -657. 7. I. M i c h i y o s h i , "Heat t r a n s f e r from an i n c l i n e d t h i n f l a t p l a t e by n a t u r a l c o n v e c t i o n " , B u l l e t i n J . S . M . E . , 7 ( 2 8 ) , 1964, 745-750. 8. D . J . T r i t t o n , "Turbu lent f r e e convect ion above a heated p l a t e . i n c l i n e d at a s m a l l angle to the h o r i z o n t a l " , J . F l u i d M e c h . , 16 , 1963, 282 -312 . 9 . D . J . T r i t t o n , " T r a n s i t i o n to tu rbu lence i n the f r e e convect ion boundary l a y e r s on an i n c l i n e d heated p l a t e " , J . F l u i d M e c h . , 16 , 1963, 417-435. 10. E .M. Sparrow, R. E i c h h o r n , J . L . Gregg, "Combined f r e e and fo rced convect ion i n a boundary l a y e r f l o w " , The P h y s i c s of F l u i d s , 2_ ( 3 ) , 1959, 319-328. • 11. Y . M o r i , "Buoyancy e f f e c t s i n f o r c e d laminar convect ion f low over a h o r i z o n t a l f l a t p l a t e " , J . Heat T r a n s f e r , 8 3 , 1961, 479-482. 12. W.G. Cochran, "The f low due to a r o t a t i n g d i s c " , P r o c . Camb. P h i l .  Soc. , 30 , 1934, 368-375. 13. K. M i l l s a p s and K. Poh lhausen , "Heat t r a n s f e r by l a m i n a r f low from a r o t a t i n g p l a t e " , J . A e r o n a u t i c a l S c i e n c e s , 1 9 , 1952, 120-126. 14. S . O s t r a c h , "An a n a l y s i s of laminar f r e e - c o n v e c t i o n f low and heat t r a n s f e r about a f l a t p l a t e p a r a l l e l to the d i r e c t i o n of the genera t ing body f o r c e " , N . A . C . A . , TR - 1111, 1953. 67 15. 1 . 6 . R e i l l y , Ch i T i e n and M. Adelman, " E x p e r i m e n t a l study of n a t u r a l c o n v e c t i o n heat t r a n s f e r i n a non-Newtonian f l u i d " , Canad. J . Chem. Eng. 4 4 , 1966, 6 1 - 6 3 . 16. D.A. D i d i o n , Youn Hwan Oh, "A q u a n t i t a t i v e s c h l i e r e n - g r i d method f o r temperature measurement i n a f r e e c o n v e c t i o n f i e l d " , Defense Documenta- t i o n C e n t e r , AD 637012, 1966. 17. D.W. H o l d e r , R . J . N o r t h , " S c h l i e r e n Methods" , N a t i o n a l P h y s i c a l  L a b o r a t o r y , Notes App. Sc . No. 3 1 , Her M a j e s t y ' s S t a t i o n a r y O f f i c e , London, 1963. 18. E .G . Hauptmann, "An e x p e r i m e n t a l i n v e s t i g a t i o n of f o r c e d convec t ion heat t r a n s f e r to a f l u i d i n the r e g i o n of i t s c r i t i c a l p o i n t , " P h . D . T h e s i s , C a l i f o r n i a I n s t i t u t e of Technology , 1966, appendix D. TABLE I BOUNDARY VALUES FOR THE SEMI-INFINITE ISOTHERMAL PLATE Pr F(~) F"(0) G(0) H' (0) 0.10 7.04147 2.03014 -3.3648 -0.19681 0.30 3.77414 1.36178 -2 . 29 39 -0.27868 0.50 2.8405 1.12619 -1.9421 -0.32396 0.72 2.33450 0.97998 -1.7290 -0.35909 1.00 1.97860 0.86611 -1.5658 -0.39204 2.00 1.43923 0.66616 -1.2832 -0.46901 5.00 1.00826 0.47366 -1.0134 -0.58816 10.00 0.79423 0.36638 -0.85915 -0.69069 TABLE II UNIVERSAL FUNCTIONS FOR Pr-*» F l G l H l H l 0.00 0.0 0.0 -1.2691 1.000 -0.4602 0.51 0.1161 0.4332 -0.8187 0.7664 -0.4541 0.99 0.3984 0. 7252 -0.5022 0.5548 -0.4221 1.50 0.8244 0.9289 -0.2714 0.3564 -0.3496 2.01 1.3314 1.0475 -0.1309 0.2028 -0.2506 2.49 1.8498 1.1062 -0.0588 0.1053 -0.1578 3.00 2.4224 1.1352 -0.0220 0.0455 -0.0817 3.51 3.0047 1.1465 -0.0071 • 0.0168 -0.0354 3.99 3.5561 1.1503 -0.0022 0.0057 -0.0137 4.50 4.1432 1.1515 -0.0005 0.0015 -0.0042 5.01 : 4.7305 1.1518 -0.0001 0.0003 -0.0011 TABLE H a UNIVERSAL FUNCTIONS FOR P r - * » ^2 *2 F* 2 0 .00 0 . 0 0 1.1522 0 .50 0.4902 0.8207 1.00 0.8334 0.5647 1.50 1.0666 0.3787 2.00 1.2216 0.2495 2.50 1.3231 0.1624 3.00 1.3889 0.1049 3.50 1.4313 0.0674 4.00 1.4585 0.0431 4.50 1.4759 0.0276 5 .00 1.4870 0.0176 6 .00 1.4986 0 .0071 H„ = 0 throughout the " o u t e r " r e g i o n . 71 TABLE I I I UNIVERSAL FUNCTIONS FOR P r 0 R5 n 2 as F 2 as F 2 !*5 G 2 h 0.002 0.00008 1.5723 - 1 . 2 4 5 6 1.000 - 0 . 5 7 70 0 .50 0.6026 0.9731 - 0 . 8 1 6 3 0.7216 - 0 . 5 2 2 0 1.00 1.0054 0.6585 - 0 . 5 1 6 2 0.4881 - 0 . 4 0 7 5 1.50 1.2773 0.4424 - 0 . 3 1 8 0 0.3148 - 0 . 2 881 2.00 1.4590 0.2935 - 0 . 1 9 2 2 0.1963 - 0 . 1 9 0 6 2.50 1.5788 0.1921 - 0 . 1 1 4 8 0.1196 - 0 . 1 2 0 6 3 .00 1.6567 0.1243 - 0 . 0 6 7 9 0.0717 - 0 . 0 7 4 1 3.50 1.7069 0.0795 , - 0 . 0 3 9 9 0.0426 - 0 . 0 4 4 7 4 .00 1.7388 0.0504 - 0 . 0 2 3 3 0 .0251 - 0 . D 2 6 7 4.50 1.7590 0.0316 - 0 . 0 1 3 6 0.0148 - 0 . 0 1 5 8 5 .00 1.7716 0.0196 - 0 . 0 0 7 9 0.0086 - 0 . 0 0 9 3 5 .50 1.7793 0.0120 - 0 . 0 0 4 5 0.0050 - 0 . 0 0 5 4 6.00. 1.7840 0.0072 - 0 . 0 0 2 6 0.0029 - 0 . 0 0 3 2 TABLE I l i a  UNIVERSAL FUNCTIONS FOR P r 0 ^ l F l F i 0.00 0 .00 0 .00 0 .50 0.1427 0 .5501 1.00 0.5285 0.9711 1.50 1.0918 1.2609 2.00 1.7698 1.4336 2.40 2.3597 1.5081 3.00 3.2817 1.5561 3.50 4.0633 1.5682 4 .00 4.8484 1.5714 4 .50 5.6344 1.5722 5.00 6.4205 1.5723 6.00 7.9927 1.5723 H ' a 1.0 through the " i n n e r " r e g i o n . 73 APPENDIX A SOME ASPECTS OF THE SCHLIEREN SYSTEM Genera l D i s c u s s i o n The p r i n c i p l e s and o p e r a t i o n of the s c h l i e r e n o p t i c a l system have been d i s c u s s e d at l e n g t h by many a u t h o r s . Three very good sources f o r i n f o r m a t i o n are ( 1 6 ) , (17) and (18) . The f o l l o w i n g d i s c u s s i o n i s by no means complete but con ta ins the i tems which were thought to be impor tant by the author i n the des ign of h i s system. The system i s s c h e m a t i c a l l y shown i n f i g u r e s 26 and 27 . S e n s i t i v i t y o f System The s e n s i t i v i t y i s g iven b y , where f i s the f o c a l l e n g t h of the second s c h l i e r e n m i r r o r (l.^) and " a " may be taken to be o n e - h a l f o f the source h e i g h t . T h e r e f o r e , i n c r e a s i n g the f o c a l l e n g t h o f the s c h l i e r e n m i r r o r or d e c r e a s i n g the source he igh t improves the s e n s i t i v i t y . In t h i s context s e n s i t i v i t y i s a measure of the observab le c o n t r a s t of the d i s p l a c e d source image ( c . f . o u t s i d e edge of b l u e f r i n g e on the s c h l i e r e n photographs) w i t h respect to the background ( l a r g e y e l l o w a r e a ) . S e n s i t i v i t y may, a l s o , a l t e r n a t i v e l y be connected to the number of c o l o u r f r i n g e s which appear i n the s c h l i e r e n image. In t h i s r e s p e c t , i t i s , however, the range of the system which i s b e i n g c o n s i d e r e d . The range i s s i m p l y the maximum d e f l e c t i o n , S = f / a ( A - l ) f e v = k a o (A-2) 74 where i s the t o t a l d e f l e c t i o n of the l i g h t beam i n the Y d i r e c t i o n , k i s the number of c o l o u r bands t r a v e r s e d by the d e f l e c t e d image and a Q i s the w id th of one band (assuming a l l bands have the same w i d t h ) . Now the t o t a l d e f l e c t i o n of a l i g h t beam as i t passes through the s c h l i e r e n f i e l d of l eng th (L ) i s g iven by the e x p r e s s i o n s ev = L . — -|~ (A-3) Y s n 9Y where n i s the r e f r a c t i v e index of l i g h t . S u b s t i t u t i n g i n ( 2 ) , the number of f r i n g e s i s g iven b y , k = I L . 1 . (A-4) a s n 9Y This e x p r e s s i o n i s s i m i l a r to ( A - l ) and de f ines the s e n s i t i -v i t y i n terms of the number of c o l o u r f r i n g e s that w i l l be d i s p l a y e d (see appendix B f o r a f u r t h e r e x p l a n a t i o n of equat ion (A-4)). Depth of F i e l d C r i t e r i a Genera l requirements f o r a s e m i - f o c u s s i n g s c h l i e r e n system are a h i g h s e n s i t i v i t y and a shor t depth of f i e l d . These are opposed to each o ther s i n c e a s h o r t depth of f i e l d can on ly be obta ined w i t h an extended source . I f the system i s r e q u i r e d to be s e n s i t i v e i n one d i r e c t i o n on ly t h e n , of c o u r s e , both of the above requirements are met when the source i s a narrow s l i t o r i e n t e d p e r p e n d i c u l a r t o the d i r e c t i o n i n which the s e n s i t i v i t y i s r e q u i r e d . R e f e r r i n g to f i g u r e 2 7 , f o r X l a r g e r than f , the depth o f f i e l d i s g iven b y , AX = 2 c < X / f> Yf 75 where £ i s the d iameter of the " c i r c l e of c o n f u s i o n " , approx imately 0 .25 mm. I f the image focuses s l i g h t l y i n f r o n t of or behind the image p l a n e , the c i r c l e of confus ion i s the l a r g e s t d i s c of l i g h t at the image p lane which does not r e s u l t i n a p p r e c i a b l e b l u r r i n g of the image, y i - s the i n c l u d e d angle of the extended s o u r c e . T h e r e f o r e , to o b t a i n a shor t depth of f i e l d , the source s l i t should be l o n g , and the s c h l i e r e n f i e l d shou ld be l o c a t e d as c l o s e as p o s s i b l e to the f o c a l p o i n t of the second m i r r o r . L a s t l y , the leng th of the s l i t which may be used i s l i m i t e d by the occurence of o p t i c a l a b e r r a t i o n s i n an o f f - c e n t e r s c h l i e r e n system. In t h i s respect a u n i - a x i a l system u s i n g lenses r a t h e r than concave m i r r o r s i s s u p e r i o r to the one d e s c r i b e d h e r e . I t i s need less to remark that an 8 i n c h lens system u s i n g f u l l y c o r r e c t e d components would i n v o l v e p r o h i b i t i v e c o s t . Condenser FIG. 26 ELEMENTS OF SCHLIEREN SYSTEM Color F i l t e r F i lm Plane FIG. 27 SCHEMATIC OF SEMI-FOCUSING SCHLIEREN SYSTEM APPENDIX B CALCULATIONS OF BOUNDARY LAYER DEPTH AND CRITICAL GRASHOF NUMBER C a l c u l a t i o n of Gr A Grashof number based upon the boundary l a y e r depth i s very o f t e n the more u s e f u l parameter i n s t a b i l i t y s t u d i e s . S ince the boundary l a y e r d e p t h . v a r i e s only very s l i g h t l y w i t h the d i s t a n c e a long the p l a t e , a measure of the depth j u s t p r i o r to t r a n s i t i o n would l e a d to a very i n a c c u r a t e p o r t r a y a l of the c r i t i c a l Grashof number. The c r i t i c a l Gr^ i s , t h e r e f o r e , c a l c u l a t e d from the measured c r i t i c a l Gr^ by means of t h e i r t h e o r e t i c a l r e l a t i o n s h i p . The s i m i l a r i t y a n a l y s i s of s e c t i o n 3 . 1 y i e l d s the f o l l o w i n g r e l a t i o n s h i p f o r 2 - d i m e n s i o n a l f l o w : 2/5 g B A T r e f " 1 / 5 Y, = T| X ( — ) ( B - l ) 2 V where X and Y are t h e : d i m e n s i o n s a long the p l a t e and p e r p e n d i c u l a r to i t , r e s p e c t i v e l y , n i s the s i m i l a r i t y v a r i a b l e and A T ^ ^ i s (Tw - Too). 1 T — T For a i r S = — , and P r = . 7 2 . At a p o i n t where TT — = .02 n u m e r i c a l •Leo ^ r e f data ( f i g u r e 6) y i e l d s the va lue n = 5 . 2 . From equat ion (1) the f o l l o w i n g can thus be deduced, G r , - n3 | f i G r v 2 / 5 (B-2) x 78 whence A Gr.5= 3 .2 x 10 (12 i n c h p l a t e ) . (B-3) Fu r thermore , u s i n g the va lue of the k i n e m a t i c v i s c o s i t y f o r = 70°F, the f o l l o w i n g boundary l a y e r depths were c a l c u l a t e d : AT = 50°F Y . = .795 i n s . r e f 1 6 AT _ = 100°F Y L = .625 i n s . r e f 1 o > (B-4) Approximate A n a l y s i s of the Agreement of the Theory w i t h the  S c h l i e r e n Photographs T h i s a n a l y s i s i s to be regarded as q u a l i t a t i v e i n na tu re i n o rder that some s i m p l i f y i n g assumptions may be made. The path of l i g h t through the s c h l i e r e n f i e l d i s assumed to be p a r a l l e l to the p l a t e s u r f a c e . This assumpt ion ' r e q u i r e s t h a t the d e n s i t y v a r i a t i o n and the l e n g t h o f the s c h l i e r e n path ( L g ) be s m a l l . The r e f r a c t i v e i n d e x of l i g h t (n) i s then r e l a t e d to the t o t a l d e f l e c t i o n o f the l i g h t beam i n the Y d i r e c t i o n (e^)> a t a p a r t i c u l a r d i s t a n c e (Y^) from the s u r f a c e by (see appendix A and r e f e r e n c e s (16) , (17) , and (18)), SY.I " L • ( B " 5 ) Y, S The t o t a l d e f l e c t i o n D e expressed as a d isp lacement of the focussed l i g h t beam across a graded c o l o u r f i l t e r b y , k a e Y - ~f (B-6) 79 where a Q i s the w i d t h of a s i n g l e c o l o u r band and f i s the f o c a l l e n g t h of the s c h l i e r e n l e n s e s . Now, -21 = 22- f K - 7 i dY 9n ' 3Y K J and the r e l a t i o n s h i p between the temperature and the r e f r a c t i v e i n d e x , T - 1 T = n 1 ( B " 8 ) oo n -  and a l s o s u b s t i t u t i n g f o r the constants of the s c h l i e r e n s y s t e m , the f o l l o w i n g e x p r e s s i o n r e l a t i n g the number of c o l o u r f r i n g e s and the temperature r e s u l t s . 2 k(Y) = - 1 .73 (^r) S ^ T w ) (12 i n . p l a t e ) . (B-9) Th is e x p r e s s i o n i s p l o t t e d i n f i g u r e 28 f o r temperature data from f i g u r e 6 at Pr = . 7 2 . The number of c o l o u r f r i n g e s o f the p l a t e may be compared w i t h the number on the s c h l i e r e n p i c t u r e s . The boundary l a y e r t h i c k n e s s , as measured from the s c h l i e r e n photographs f o r the square p l a t e s , i s approx imate ly 0 .65 i n s . . Th is depth was constant f o r a AT ^ f rom approx imate ly 50°F to the h i g h e s t measured (120°F) . F i g u r e 28 i n d i c a t e s t h a t t h i s r e s u l t i s to be expected . T h e r e f o r e , the s c h l i e r e n p i c t u r e s f o r the h i g h e r p l a t e temperatures on ly d i s p l a y approx imate ly the f u l l boundary l a y e r depth . S u b s t i t u t i n g 6 = . 6 2 5 , and AT = 100°F, d i r e c t l y i n t o the e x p r e s s i o n fo r the Gr, (at r e f o T«. = 70°F) , 80 A Tr e f f i 3 4 G r 5 = — = 3 .2 x 10 . (B-10) v The Grashof number based on 6 thus agrees e x p e r i m e n t a l l y w i t h the t h e o r e t i c a l e x p r e s s i o n ( B - 3 ) . 81 

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