UBC Theses and Dissertations

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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 , University  o f 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 t h e Department of Mechanical Engineering  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1968  In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study.  I further agree that permission for extensive  copying  of t h i s thesis f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives.  I t i s understood that  copying or p u b l i c a t i o n of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission.  L. Claassen  Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8 , B.C.  ii  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 p r e s e n t e d f o r . f r e e c o n v e c t i o n and combined f r e e and f o r c e d c o n v e c t i o n 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 t o i n v e s t i g a t e a l a m i n a r boundary layei? f l o w w h i c h had been shown, t h e o r e t i c a l l y o n l y , t o form above a heated s u r f a c e .  This boundary l a y e r flow i s  fundamentally  differ-  ent from f l o w s 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  driving  f o r c e o r buoyancy f o r c e a c t s p e r p e n d i c u l a r t o the p r i m a r y  boundary  layer motion. The f l o w s a n a l y z e d are those f o r w h i c h the s y s t e m of p a r t i a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g the f l o w can be reduced t o s i m u l t a n e o u s t o t a l d i f f e r e n t i a l equations.  The method i n v o l v e s  the  introduction  of s i m i l a r i t y parameters and t h e n the n u m e r i c a l i n t e g r a t i o n of r e s u l t i n g s i m p l i f i e d s y s t e m 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 .  the 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 o n l y f r e e c o n v e c t i o n i s c o n s i d e r e d w h i l e the l a t t e r combined c o n v e c t i o n i s  for  c o n s i d e r e d as w e l l .  The f l o w 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 c o l o u r s c h l i e r e n system.  From the photographs i t may be c o n c l u d e d t h a t  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 , t h o u g h , remains l a m i n a r f o r a s h o r t d i s t a n c e only and then b r e a k s 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 t h o u g h 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  layer  s o l u t i o n , appears t o 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 radius.  finite  iii TABLE OF CONTENTS  ABSTRACT  i i  LIST OF TABLES  v  LIST OF FIGURES  vi  ACKNOWLEDGEMENTS  viii  NOMENCLATURE  ix  I  INTRODUCTION  1  II  REVIEW OF PREVIOUS WORK  5  III  FREE CONVECTION FROM A SEMI-INFINITE PLATE  8  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 Pr Number  8  3.2  Asymptotic  3.2.1  Pr »  1  19  3.2.2  Pr «  1  22  IV  V  VI  Solutions  19  FREE AND COMBINED CONVECTION FROM A DISC  31  4.1  Free C o n v e c t i o n f o r an A r b i t r a r y P r Number  33  4.2  Combined Free and F o r c e d C o n v e c t i o n f o r an A r b i t r a r y P r number and a D i s c Temperature which i s P r o p o r t i o n a l t o r ^  38  4.3  Numerical I n t e g r a t i o n  42  EXPERIMENTAL APPARATUS  44  5.1  O p t i c a l Bench  44  5.2  Alignment  46  5.3  Colour F i l t e r  46  5.4  R e c t a n g u l a r P l a t e s and D i s c  47  o f the O p t i c a l Benches  EXPERIMENTAL PROCEDURE  49  6.1  T e s t Procedure  49  6.2  Experimental Results  49  iv Contents VII  VIII  continued  DISCUSSION OF THE RESULTS  57  7.1  Numerical Solutions  58  7.2  S c h l i e r e n Observations  60  CONCLUSIONS  REFERENCES  64 66  APPENDIX A  SOME ASPECTS OF THE SCHLIEREN SYSTEM  73  APPENDIX B  CALCULATIONS OF BOUNDARY LAYER DEPTH AND CRITICAL GRASHOF NUMBER  77  V  LIST OF TABLES Table  I  Boundary Values f o r t h e S e m i - I n f i n i t e Isothermal Plate  68  Table  II  Asymptotic Functions for Pr  -*-°°(Inner S o l u t i o n )  69  Table  Ila  Asymptotic Functions for Pr  -»• °° (Outer S o l u t i o n )  Table  III  Asymptotic Functions  f o r P r -*• 0 (Outer S o l u t i o n )  71  Table  Ilia  A s y m p t o t i c F u n c t i o n s f o r P r -> 0 ( I n n e r S o l u t i o n )  72  70  VI  LIST OF FIGURES Figure  1  C a r t e s i a n Axes N o t a t i o n  8  Figure  2  Stream F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow  15  Figure  3  V e l o c i t y Functions f o r 2-Dimensional Flow  16  Figure  4  Pressure Functions f o r 2-Dimensional Flow  17  Figure  5  Temperature F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow  18  Figure  6  Pressure Functions f o r 2-Dimensional Flow i n Terms of A s y m p t o t i c C o o r d i n a t e s f o r P r -> °°  27  Figure  7  Temperature F u n c t i o n s f o r 2 - D i m e n s i o n a l f l o w i n Terms of A s y m p t o t i c C o o r d i n a t e s f o r P r -»•  28  00  Figure  8  P r e s s u r e F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow i n Terms of A s y m p t o t i c C o o r d i n a t e s f o r P r -> 0  29  Figure  9  Temperature F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow i n Terms of A s y m p t o t i c C o o r d i n a t e s f o r P r -+ 0  30  Figure  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  Figure  11  Stream and V e l o c i t y F u n c t i o n s f o r A x i a l l y Symmetrical Flow-Isothermal P l a t e  36  Figure  12  P r e s s u r e and Temperature F u n c t i o n s f o r A x i a l l y S y m m e t r i c a l Flow - I s o t h e r m a l Plate  37  Figure  13  V e l o c i t y Functions f o r A x i a l l y Symmetrical Flow - P l a t e Temperature P r o p o r t i o n a l t o r  40 2  Figure  14  Temperature F u n c t i o n s f o r A x i a l l y S y m m e t r i c a l Flow - P l a t e Temperature P r o p o r t i o n a l t o r^  41  Figure  15  21 Inches Square P l a t e  48  Figure  16  7 Inches Diameter D i s c  48  vii Figures Figure  17  Figure  18  Figure  19  continued  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.  AT  b.  AT  c.  AT  d.  AT  ref ref ref  = 2 3 . 5 °F, ' = 3 9 . °F,  Too  = 4 3 . 5 °F,  Too  Too  51  = 71. °F, ' = 7 1 . °F, ' = 7 1 . °F,  = 5 0 . 5 °F, Too = 7 1 . °F. ref ' S c h l i e r e n Photographs w i t h 12 i n c h P l a t e ; a. AT = 30. °F, T^ = 7 2 . ° F , ref ' ' b. AT = 5 6 . 5 °F, Too = 71.5 °F, ref c. AT = 8 1 . 0 °F, Too = 7 1 . 5 °F, ref d. AT = 117.5°F, Too = 71.5 °F. ref S c h l i e r e n Photographs w i t h 7 i n c h D i s c : a. AT = 5 2 . 5 ° F , Too = 72. °F, Si = 0 rpm, ref b. AT . » 6 2 . 5 °F, Too = 7 2 . 0 ° F , SI = 3 2 . 8 rpm ref c. A T = 9 4 . 5 °F, Too = 72.5°F, SI = 1 3 . 9 rpm ref ' ' ^ d. A T = 9 4 . 5 °F, T^ = 72.5°F, SI = 18.6 rpm.  52  53  r  c  r  Figure  20  e  f  S c h l i e r e n Photographs w i t h 7 i n c h D i s c : a.  AT  b.  AT  54  . - 9 1 . 5 °F, T = 7 1 . °F, SI = 0 rpm, ref ' oo » « 115. °F, Too = 71 °F, SI = 0 rpm. ref  ^  Figure  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  Figure Figure  22 23  55 56  Figure  24  C r i t i c a l Gr f o r 12 i n c h Square P l a t e C r i t i c a l Gr f o r 7 i n c h D i s c f o r F r e e Convection C r i t i c a l Gr v s . Re f o r 7 i n c h D i s c  Figure  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 and E x p e r i m e n t a l R e s u l t s  63  Figure  26  Elements of the O p t i c a l System  76  Figure  27  Schematic of the S e m i - F o c u s i n g System  76  Figure  28  A Q u a l i t a t i v e A n a l y s i s o f the Number of C o l o u r F r i n g e s Observable i n the S c h l i e r e n Photographs  81  56  ACKNOWLEDGEMENTS The a u t h o r wishes t o e x p r e s s h i s deep g r a t i t u d e a b l e a d v i c e and guidance g i v e n h i m throughout by P r o f e s s o r Z. Rotem.  f o r the i n v a l u -  a l l s t a g e s of the program  S i n c e r e thanks are a l s o extended t o D r .  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 A u t h o r w i s h e s t o thank the e n t i r e s t a f f o f 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 Columbia, for  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  their  facilities.  S p e c i a l thanks are due t o Mr. P. Hurren and t o Mr. J . H o a r , 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 advice. 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 v e r y urgent  chief  Without  their  demands the  work  c o u l d not have been completed. Support f o r t h i s  r e s e a r c h was p r o v i d e d by t h e N a t i o n a l Research  C o u n c i l o f Canada, through grant No. 2772.  Computing time was  granted  f r e e o f charge by the Computing Center through a g e n e r a l s u p p o r t by N.R.C..  The A u t h o r i s  grateful for this assistance.  the  IX  NOMENCLATURE  X,Y,Z,R  Dimensional  x,y,z,r  Dimensionless c o o r d i n a t e s (see f i g u r e s 1 and 10)  u,v,w  Dimensionless v e l o c i t y  L  Reference  AT  c o o r d i n a t e s (see f i g u r e s 1 and 10)  components  length ( u s u a l l y h a l f width of p l a t e )  . ref or  AT.  Reference temperature temperature  difference  i s e q u a l t o (T  T w  Temperature o f the s u r f a c e  T  Temperature o f the f l u i d  U  (see f i g u r e s 1 and 10)  j. ref  ( f o r constant s u r f a c e  - T ))  r  Reference  f a r away from the s u r f a c e  velocity J  U  L re f £  Re  Reynolds number (= — - — )  Pr  P r a n d t l number (= - ^ j ^ -  Gr  Grashof number (see e q u a t i o n s  Ra  R a y l e i g h number (= Pr.Gr)  Nu  N u s s e l t number (= k  ) (7) and (61))  = - r~) dy  k  C o n d u c t i v i t y o f the f l u i d  Cp  S p e c i f i c heat at constant p r e s s u r e o f the f l u i d  h  Heat t r a n s f e r  8  Dimensionless temperature  IT  Dimensionless p r e s s u r e (see e q u a t i o n s  3  Expansion c o e f f i c i e n t  U,v  A b s o l u t e and k i n e m a t i c v i s c o s i t y o f the f l u i d  coefficient  T - T w (= — ^ ref  0 0  ) (5) and (59))  ( f o r a i r = 1/T")  X  Nomenclature-continued  a  A n g l e of i n c l i n a t i o n t o t h e Angular  Similarity  horizontal  velocity  variables  F  A stream function  F'  A velocity  function  G  A pressure  function  H  A t e m p e r a t u r e f u n c t i o n (= G' i n e v e r y case)  T  A* c i r c u l a t i o n  function  Supers c r i p t s Signifies  a transformed v a r i a b l e  Signifies asymptotic.variable ~  for Pr »  1  S i g n i f i e s asymptotic v a r i a b l e for Pr «  1  Subscripts 1  Signifies "inner"  region  2  S i g n i f i e s "outer"  region  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  00  S i g n i f i e s conditions  f a r away from the s u r f a c e .  1 I.  INTRODUCTION  A h e a t e d s u r f a c e , when surrounded by an expanse of f l u i d w h i c h changes i t s d e n s i t y w i t h a change of the t e m p e r a t u r e , w i l l g i v e t o a f r e e c o n v e c t i o n boundary l a y e r f l o w i f a c e r t a i n minimum v a l u e . about a v e r t i c a l s u r f a c e .  the temperature exceeds  The s i m p l e s t type of f l o w i s  encountered  I n t h i s case a l a m i n a r boundary l a y e r may  f o r m , w h i c h o r i g i n a t e s at the lower edge.  The f o r m a t i o n 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 . i n t h e 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  As  decreases i t s  T h i s i m p a r t s buoyancy to t h e f l u i d w h i c h causes i t plate.  rise  fluid  density.  t o flow a l o n g the  The buoyancy f o r c e s 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 motion. The boundary l a y e r flow 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 o v e r 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 f o r c e s 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 t o t h e m o t i o n . ever,  ( 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 t o t h e h o r i z o n t a l )  v e r s e component of the buoyancy  How-  the t r a n s -  f o r c e becomes l a r g e r and the f l o w 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 o t h e r h a n d , f o r a v e r y 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 o n l y i n d i r e c t l y .  It  induces a transverse  pressure  g r a d i e n t o f which t h e v a r i a t i o n a l o n g the s u r f a c e d r i v e s the  fluid.  The c h a r a c t e r of t h i s f l o w i s , t h e r e f o r e , e x p e c t e d t o be f u n d a m e n t a l l y different  from the f l o w o v e r 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  a s p e c t of i n v e s t i g a t i o n s i n t o t h i s f l o w w i l l be d i s c u s s e d i n s e c t i o n II.  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 m a i n l y to a moderate temperature d i f f e r e n c e between t h e 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 t h e e x c e p t i o n of the dependence of d e n s i t y upon t e m p e r a t u r e , and the e q u a t i o n s motion may be s i m p l i f i e d a c c o r d i n g l y .  If  of  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 a n d , t h e r e f o r e , p o s s e s s e s o n l y one l e a d i n g e d g e , the absence o f 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  solutions"  s h o u l d be o b t a i n a b l e . T h i s t h e s i s d i s c u s s e s 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 flows  (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 . S t e w a r t s o n (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  solution  and a l i m i t e d amount of n u m e r i c a l d a t a f o r 2 - d i m e n s i o n a l f l o w and f o r a P r a n d t l number (Pr)  equal to . 7 2 .  He i n c o r r e c t l y  claimed,  t h a t a boundary l a y e r would form below a h e a t e d s u r f a c e or above a c o o l e d s u r f a c e .  I n 1 9 6 5 , G i l l e t a l . (2)  however, conversely  corrected this  p r e t a t i o n , to the now a c c e p t e d f a c t , t h a t a boundary l a y e r forms above a h e a t e d s u r f a c e o r below a c o o l e d s u r f a c e . Stewartson's  results  to P r = 1 . 0 and 1 0 .  only  They a l s o extended  I n the p r e s e n t t h e s i s , i n  t h e s e r e s u l t s a r e recomputed and extended t o v a r i o u s  part,  o t h e r P r numbers.  The approach taken t o a r r i v e a t s i m i l a r i t y s o l u t i o n s , however, from p r e v i o u s  inter-  differs  methods and i s thought t o 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  carefully  defined. * Numbers i n b r a c k e t s a f t e r the names of a u t h o r s r e f e r to  references.  3 The l a m i n a r c o n v e c t i o n f l o w as d e s c r i b e d above i s a f u n c t i o n o f the parameter P r .  S i n c e 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 v e r y s m a l l v a l u e of P r , t h e system of e q u a t i o n s was examined f o r a s y m p t o t i c v a l u e s of t h i s p a r a m e t e r . 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 t o be "universal" solutions.  It  i s , moreover, shown i n t h i s work  that  e x t r e m e l y good a p p r o x i m a t i o n s are o b t a i n e d w i t h the a s y m p t o t i c s o l u t i o n s f o r P r < . 1 and P r > 10. F o r a x i a l l y - s y m m e t r i c a l flow pure f r e e c o n v e c t i o n and combined f r e e and f o r c e d c o n v e c t i o n are c o n s i d e r e d .  The former y i e l d s a number  o f s i m i l a r i t y s o l u t i o n s w h i l e t h e l a t t e r y i e l d s o n l y one t e m p e r a t u r e v a r y i n g w i t h r^).  (surface  A l t h o u g h t h e s e , s o l u t i o n s are  .identifiable  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 o b s e r v a b l e modes of f l o w  on a d i s c of f i n i t e r a d i u s s i n c e they  d e s c r i b e a boundary l a y e r t h a t 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 e v i d e n c e has been g i v e n  for  the e x i s t e n c e of the l a m i n a r f r e e c o n v e c t i v e , h o r i z o n t a l boundary In o r d e r to c o n f i r m the c o n t r a d i c t o r y  layer.  t h e o r y 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 c o l o u r t e c h n i q u e was b u i l t . I n 2 - d i m e n s i o n a l f l o w 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 i n w a r d s . t h e c e n t e r and r i s e i n a v e r t i c a l plume.  The boundary l a y e r s meet i n It  i s observed e x p e r i m e n t a l l y  t h a t t h e boundary l a y e r does not t r a n s form, smoothly t o a v e r t i c a l plume b u t i s t e r m i n a t e d by a r e g i o n o f t h e r m a l i n s t a b i l i t y .  The p o i n t  onset o f t h i s i n s t a b i l i t y i s observed to be e x t r e m e l y s e n s i t i v e d i s t u r b a n c e s i n the s u r r o u n d i n g s  of the p l a t e .  of to  With t h i s hindrance.  4  i n mind d a t a i s p r e s e n t e d to d e f i n e the l o c a t i o n of the  instability.  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 r e g i o n of the p l a t e v e c t i o n t a k e s 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  con-  particular,  n e a r the p l a t e the f l o w 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 low d e n s i t y " t u r b u l e n c e  bubbles".  Schlieren observations originates  o f a d i s c show a boundary l a y e r w h i c h  at the edge and grows i n w a r d s ,  t h e c e n t e r of the d i s c . over a c i r c u l a r a n n u l u s .  or  a g a i n r i s i n g as a plume over  The f l o w m a y , . h o w e v e r , be i n t e r p r e t e d as t h a t If  the annulus r a d i u s were made v e r y  large  compared to i t s r a d i a l e x t e n t so t h a t the r a d i u s would no l o n g e r 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 ) 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 .  the  flow  In t h i s sense the f l o w w h i c h  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 s y m m e t r i c a l f l o w i s o f a 2 d i m e n s i o n a l n a t u r e and does not r e l a t e d i r e c t l y for axially-symmetrical  flow.  to the t h e o r y  developed  5 II  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 t h e r m a l c o n v e c t i o n s a b o u t 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 F i s h e n d e n and Saunders (3) , Schmidt (4)  and Weise (5)  1930's.  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 t a k e n by Weise were thought t o be f i r s t e v i d e n c e the e x i s t e n c e o f a l a m i n a r boundary l a y e r . however,  for  The g e n e r a l consensus w a s ,  t h a t f o r a p l a t e h e a t e d on b o t h 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  u n d e r t a k e n 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)  t h e f r e e c o n v e c t i o n from a p l a t e o f f i n i t e w i d t h .  on  They assumed  t h a t t h e f l o w c o u l d be a p p r o x i m a t e d t o t h a t w h i c h e x i s t s around an i n f i n i t e l y long horizontal cylinder,  the c r o s s - s e c t i o n o f w h i c h 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  larger axis h o r i z o n t a l .  f l a t p l a t e flow w o u l d be approached when the e c c e n t r i c i t y i s t o approach i n f i n i t y .  allowed  T h i s f l o w may be e x p e c t e d t o t a k e p l a c e over  a f i n i t e p l a t e h e a t e d on b o t h s i d e s when the c h a r a c t e r i s t i c number i s  Grashof  small.  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 l o w s o f 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 S t e w a r t s o n ( 1 ) . al.  (2)  The  r e i n t e r p r e t e d and e x t e n d e d h i s r e s u l t s ,  P r = 1 . 0 and 10.  the  G i l l et  f o r P r = . 7 2 , to  By l o o k i n g at t h e i n d u c e d p r e s s u r e g r a d i e n t s a l o n g t h e  p l a t e t h e y , f u r t h e r m o r e , deduced t h a t a boundary l a y e r f l o w w o u l d  only  e x i s t above a h e a t e d p l a t e or below a c o o l e d p l a t e , s i n c e the f l o w must coexist with a favourable pressure gradient. d e n s i t y s t r a t i f i c a t i o n w h i c h makes i t  The f l o w then has a v e r t i c a l  gravitationally  unstable.  6  At some p o i n t i t  may t h e r e f o r e be e x p e c t e d t o s e p a r a t e , and i n t h e  case of a f i n i t e p l a t e , i t would r i s e n e a r 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 t h e flow i s  dominated by p r i m a r i l y v e r t i c a l m o t i o n s , w i t h some c e l l u l a r  structure.  The l a m i n a r flow may thus be assumed t o possess an upper l i m i t of the Grashof number; flow b e y o n d . t h i s l i m i t s u g g e s t s i n appearance t h a t 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 a t  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 t o gravitational  instability.  Combined f r e e and f o r c e d c o n v e c t i o n 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 t o a l e s s e r degree.  For t h i s reason i t i s  t o d e s c r i b e h e r e , 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 . 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 e t a l . ( 1 0 ) .  useful A very  They show t h a t  t h e parameter c o n t r o l l i n g the r e l a t i v e i m p o r t a n c e o f the f r e e and 2 f o r c e d c o n v e c t i o n terms i n . the e q u a t i o n s o f m o t i o n i s Gr/Re .  Further-  more, they f i n d t h a 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 o n l y when the f r e e s t r e a m m 2 m—1 velocity  and s u r f a c e temperature v a r y r e s p e c t i v e l y  where m can be chosen so as t o s u i t a boundary F o r the h o r i z o n t a l p l a t e M o r i (11) forced 2-dimensional laminar flow.  as x  and x  ,  condition.  c o n s i d e r s buoyancy e f f e c t s on  He e x p r e s s e s 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 flow 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 o f 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 c o n v e c t i o n i n a x i a l l y - s y m m e t r i c a l f l o w . 2 parameter i n t h i s case i s Re^ /Gr f r e e c o n v e c t i o n mode.  The  perturbation  4/5 where e x p a n s i o n occurs about the  I n s p i t e of the f a c t t h a t 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 l o w i t i s  interesting  7 to n o t e t h a t t h e 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 o f M o r i ' s . The p u r e l y  f o r c e d flow p r o b l e m o v e r a r o t a t i n g d i s c (von Karman  f l o w ) h a s , of course been e x t e n s i v e l y  analysed, c.f.  o b t a i n e d an e x a c t 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 .  by Cochran (12) who  M i l l s a p s and Pohlhausen  (13) f o l l o w e d w i t h an e x a c t s o l u t i o n of the heat t r a n s f e r p r o b l e m f o r the von Karman f l o w . In a l l of the f o r c e d flows 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 w h i c h 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 w o u l d 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 and s o l u t i o n s of the t y p e c o n s i d e r e d would not p e r t a i n .  length)  8 III 3.1  FREE CONVECTION FROM A SEMI-INFINITE PLATE  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 r e s e n t e d 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  plate  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 a n g l e t o the h o r i zontal.  O s t r a c h (14) i n d i c a t e d t h a t i n a l l cases o f n a t u r a l  the d e n s i t y of the f l u i d may be assumed t o remain c o n s t a n t , its  dependence on t e m p e r a t u r e .  other properties  convection excepting  Furthermore i t i s assumed t h a t  all  of the f l u i d do not v a r y 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 .  Inclusive  e q u a t i o n s of momentum, c o n t i n u i t y ,  of these a s s u m p t i o n s , the s i m p l i f i e d and energy f o r s t e a d y f l o w are as  follows: 3u 3u u 3x + v 3y 3v , 3x  V  SjTT  3x  + V  i  +  u -  3v 3y  Gr 8 t a n a  (1)  Gr 0  (2)  (3) 36 u 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 .  Note:  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.  g x,u  Figure 1  C a r t e s i a n Axes N o t a t i o n  9 x and y are r e n d e r e d d i m e n s i o n l e s s through the use of a r e f e r e n c e l e n g t h L , w h i c h makes the l a r g e s t v a l u e of x of o r d e r u n i t y . velocity  The  components u and v are rendered d i m e n s i o n l e s s through  use of r e f e r e n c e v e l o c i t y v/L.  the  The d i m e n s i o n l e s s p r e s s u r e TT i s  d e f i n e d as f o l l o w s : P IT  ~ Poo  —  =  L  2  3  eL + -^-j  Poo^  .  (x s i n a + y cos a ) .  (5)  V  The d i m e n s i o n l e s s tempexature i s d e f i n e d as f o l l o w s ,  = (T -  AT  T ) ro  /A T  r e f  .  (6)  , i s a s u i t a b l e r e f e r e n c e temperature ref r  d i f f e r e n c e which  renders  t h e l a r g e s t v a l u e 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 unity. J  For the case of an i s o t h e r m a l boundary AT 3  ref  order  . would be (T - T ) . w °°  The p a r a m e t e r G r , t h e Grashof number a p p r o p r i a t e t o the s y s t e m , i s  defined  as Gr = ( g B L  3  AT  r e f  cos a ) / v ,  (7)  2  W i t h 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 s h o u l d be n o t e d t h a t f o r a h e a t e d 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 f o r c e s  act  i n the n e g a t i v e y d i r e c t i o n , and t h e 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 t e r m i n e q u a t i o n s ( 1 ) , ( 2 ) i s thus p o s i t i v e . r e v e r s e d when a h e a t e d p l a t e f a c i n g downwards i s The s o l u t i o n of e q u a t i o n s (1)  The s i g n i s  considered.  t o (4) has t o f u l f i l l t h e 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  y+oo  u=0  = 0  0=Cx 0=0  n  (8) (9)  10 where C i s a g i v e n p o s i t i v e  c o n s t a n t , and n a g i v e n  exponent.  Assuming now t h a t the angle of i n c l i n a t i o n a i s v e r y 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 v a l u e of Gr i s s u f f i c i e n t l y  large,  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 " c o o r d i n a t e s and v a r i a b l e s  are  introduced, ^ „ 1/5 y = y Gr ;  ~ _ -2/5 ^ „ -1/5 ^ _ -4/5 u = u Gr ; v = v Gr ; T T = TT Gr .  These c o n s t i t u t e the boundary The r e s u l t i n g e q u a t i o n s  layer  / \ (10) nrt  transformations.  are:  2 ^ 3u , ^ 3u 3ff , 3 u , -2/5 u — + v — = - — + — - + Gr 3x 3y 3x „^2 3y h  J  2 3 u + _ 1/5 — - - Gr 0 tan a . 2 3x  .  (la) N  _ - 2 / 5 , ~ 3v . „ 3v , 3tf.. _ - 2 / 5 3 v . _ - 4 / 5 3 v'•+ , Gr (u -r— + v -TTT ) = - -TT: + Gr — 7 + Gr — 7 - 0 (2a) ^ ^ 3y 3x 2  8 x  2  2  fi  2  ^- + -r^r = 0 dx  (3a)  dy  30 . 30 1 ,3 0 ^ _ -2/5 u - r - + v -rpc = — ( — r + Gr 3x 3y Pr „~2 ' 3y 2  3 0 . — 7 ) .. . 2 3x 2  (4a) N  As Gr becomes v e r y l a r g e , the s y s t e m o f e q u a t i o n s above w i l l y i e l d an ( "inner" simply  solution.  I n the p r e s e n t  case the " o u t e r " s o l u t i o n  reduces  to, u = 0  so t h a t the boundary  0=0 conditions  of the " i n n e r " s o l u t i o n .  (9)  may be a p p l i e d at the o u t e r edge  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 " o u t e r " 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 stream f u n c t i o n  (as  usual),  (11)  a similarity  transformation,  = A x  and a s i m i l a r i t y  B x  e  G(n)  q  n  (13)  may be shown t h a t the c o n s t a n t s A through D are u s e f u l o n l y f o r  equations.  (12)  (n)  H  x- s  a b s o r p t i o n o f o t h e r c o n s t a n t s i n the r e s u l t i n g o r d i n a r y  of  = c x  variable,  Tl = D y  It  ff =  F(n)  P  They may t h e r e f o r e  the  differential  a l l be s e t e q u a l t o u n i t y w i t h o u t  loss  generality. Inserting  (11)  through  (13) i n t o  (la)  -  (4a)  a s e t of  e q u a t i o n s are o b t a i n e d f o r the exponents m, q , p and s .  compatibility  These  conditions  -2/5 state x,  (i)  and- ( i i )  variable  the requirement  f o r terms of o r d e r Gr  not to i n c r e a s e w i t h  f o r a l l o t h e r terms to become f u n c t i o n s of the s i m i l a r i t y only.  One of the exponents may be chosen a r b i t r a r i l y ,  f o r convenience put q = 1.  and  Then,  m = 2/5 + 4/5 n  p = 3/5 + 1/5 n  s = 2/5 -  1/5 n .  >  J  (14)  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 t o be v a l i d ,  n must be > -  The e q u a t i o n s  (la)  through  5F'"  H"  to,  + (3 + n ) F F "  -  - (2 - n ) r , G '  -a|Gr-  -o|Gr  1  (15)  (4a) now reduce  +  G  3.  = - H +  + ^  i/  5  x  n+3 5  a|Gr"  (3 + n)  (1 + 2 n X ( F ' )  2 / 5  = 2 ( 1 + 2n) G  2  x "  2  /  5  (  3  +  n  (16)  )  t a n a|  2 / 5  x  - 2 / 5 ( 3 + n)  (17)  FH' - n P r F ' H = a | G r "  x"  2 / 5  Note t h a t the l a s t t e r m 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) small only provided  2 / 5 ( 3 + n )  |.  remains  that  a <a|tan  (Gr  _ 1  1  /  5  n+3 ~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 o f the b o u n d i n g face t o  the h o r i z o n t a l i s b e l i e v e d t o be r a t h e r more a p p r o p r i a t e  the one p r e v i o u s l y p r o p o s e d by S t e w a r t s o n ( 1 ) . subject  (18)  The boundary  sur-  than  conditions  t o w h i c h t h e s e e q u a t i o n s have t o be s o l v e d , a r e ,  r, = 0  F = F  1  = 0  H = G' = 1  y (20) r i + o o  F'  =  0  H  =  0,  G =  0  13  It was pointed out i n section II that a boundary layer flow can only e x i s t above a heated surface or equivalently below a cooled surface; the p o s i t i v e sign only w i l l , therefore, be retained i n 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 p l a t e , results i n two simultaneous  ordinary d i f f e r e n t i a l equations, the present case y i e l d s  three equations with a resultant complication i n numerical evaluation. As equation  (17) may  serve to eliminate H ,  the resultant equation i s  then of the t h i r d order, rather than the second, as i s the case for the v e r t i c a l plate. For the case of the isothermal plate n = 0 and the equations s i m p l i f y to,  5 F*"  + 3FF"  - ( F ' ) = 2(G - T)G')  (21)  2  H = G'  (22)  H' ' + 3/5 Pr F H'  with boundary conditions (20).  -  0  (23)  These equations were numerically integrated  and the results are given i n figures 2 to 5.  The l o c a l Nusselt number i s  obtained as follows,  N u - - f  |  - -x ^ " " ^Gr ^ 2  y  Q  3  1  H'<0)  so that the average value of the Nusselt number becomes,  (24)  14  C o n s i d e r i n g n e x t the case of c o n s t a n t , imposed f l u x at t h e bounding surface, instead of.equations  m = 2/3 The r e l e v a n t  equations  (14)  the f o l l o w i n g are o b t a i n e d ,  p = 2/3  s = 1/3  f o r t h e f u n c t i o n s F,  i n s e r t i n g n = 1/3 i n t o e q u a t i o n s  (16)  AT  L = - — ref K  6 Gr" — H'(0)  where Q i s the g i v e n f l u x p e r u n i t  (26)  G and H may be o b t a i n e d by  through  f o r t h e r e f e r e n c e temperature d i f f e r e n c e  .  (19).  A suitable  choice  is,  1 / 5  (27) U  area.  /  ;  gure 2  Stream F u n c t i o n s  f o r 2 - D i m e n s i o n a l Flow  16  Figure 3  V e l o c i t y F u n c t i o n s , f o r 2 - D i m e n s i o n a l Flow  17  Figure 4  P r e s s u r e F u n c t i o n s f o r 2-Dimensional  Flow  Figure 5  Temperature F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow  3.2  Asymptotic Solutions Equations  (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 t h o u t l i m i t , e q u a t i o n (18) seems to become s i n g u l a r , f o r a v a n i s h i n g P r the e q u a t i o n s seem to u n c o u p l e . subsections  Pr »  d e s c r i b e the f l o w  field.  1  The t h e r m a l boundary tive  following  the e q u a t i o n s w i l l be t r a n s f o r m e d and i t w i l l be shown t h a t  t h e r m a l " i n n e r " and " o u t e r " s o l u t i o n s 3.2.1  In the  whereas  l a y e r , w i t h i n w h i c h the c o n d u c t i v e and c o n v e c -  terms are of e q u a l o r d e r of magnitude, w i l l now be much n a r r o w e r . . -  than the momentum boundary seen i n f i g u r e  layer.  The v e l o c i t i e s , i n t h i s c a s e , as i s  3 w i l l be of v e r y s m a l l magnitude.  S i n c e the t h e r m a l and momentum boundary different  .  l a y e r s have t h i c k n e s s e s  o r d e r of magnitude the f l o w 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 " r e g i o n w i l l be the t h e r m a l r e g i o n  (virtually  the complete temperature drop t a k e s p l a c e w i t h i n t h i s r e g i o n ) w h i l e "outer"  of.  r e g i o n w i l l comprise the r e s t of the momentum boundary  the  layer.  The f l o w e q u a t i o n s w i l l now be. t r a n s f o r m e d i n a way such t h a t  terms  w h i c h are of s i g n i f i c a n c e w i t h i n one r e g i o n 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 t o g i v e them an o r d e r of magnitude of u n i t y .  In t h i s way  terms which are o f s i g n i f i c a n c e o n l y w i t h i n the o t h e r r e g i o n a t t a i n a much lower o r d e r o f magnitude and become n e g l i g i b l e . sequently  the s o l u t i o n of the r e s u l t i n g t r a n s f o r m e d e q u a t i o n s  will Conwill  d e s c r i b e o n l y 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 t o as an " i n n e r " or "outer" 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 v a l u e s  a t the edges of each r e g i o n .  Therefore,  the o u t e r boundary  values  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 o f the " o u t e r " s o l u t i o n . c i t i e s must match.  values  S p e c i f i c a l l y t h e temperatures and v e l o -  D e s p i t e t h e c a t e g o r i z a t i o n s o f 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 t o be remembered t h a t each s o l u t i o n i s f i e l d,(*) !  u n i f o r m l y v a l i d a c r o s s the e n t i r e f l o w For the " i n n e r " s o l u t i o n p u t ,  n  1  = n Pr  3/5  1/5  F  1  = F' P r  n  >  H  (28)  = H  Note t h a t t h e t e m p e r a t u r e f u n c t i o n must remain o f o r d e r u n i t y , and t h e r e f o r e s h o u l d 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 " . F o r convenience t h e d e r i v a t i o n o f the i s o t h e r m a l p l a t e i s presented.  Then, s u b s t i t u t i n g i n t o e q u a t i o n s (16) t h r o u g h (18)  and l e t t i n g n = 0 t h e f o l l o w i n g e q u a t i o n s  5 F|"  - 2 (5 - ri  Gp = o | P r  _ 1  result:  3 -2/5  |+ a | ( R a x ) J  - 0| (Ra x ) 3  fl  1  -  G| -  0|(Ra x )  Hp + 3/5 t  (*)  only  3  2  /  1 / 5  N  tan a I  (23)  (30)  5  3 ^ = 0| (Ra x ) 3  2  / 5  j  F o r a r e f e r e n c e t o t h e n o m e n c l a t u r e used the r e a d e r 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 D y n a m i c s " , M. V a n - D y k e , Academic P r e s s , 1964.  (31)  21  The angle of i n c l i n a t i o n t o the h o r i z o n t a l i s now l i m i t e d by the ing  follow-  condition,  a =aItan  - 1  (Ra x ) 3  1  / 5  I  (32)  F o r s u f f i c i e n t l y l a r g e v a l u e s o f P r and Ra x  the r i g h t hand terms i n  t h e e q u a t i o n s above become n e g l i g i b l y s m a l l .  The e q u a t i o n s may then  be i n t e g r a t e d n u m e r i c a l l y "inner" region.  to obtain a f i r s t order s o l u t i o n for  the  The boundary c o n d i t i o n s f o r the e q u a t i o n s above are  now, F  n1 = o  1  = F' = 0 1  H, = G ; = i >  n1  F|'  °°  = 0  H, = G  1  (33)  = 0.  V a l u e s o f the r e s u l t a n t a s y m p t o t i c f u n c t i o n s are g i v e n i n Table F i g u r e s 6 and 7 g i v e the f u n c t i o n s G and H , p r e v i o u s l y  II.  determined f o r  v a r i o u s l a r g e v a l u e s of P r , when r e p l o t t e d i n a s y m p t o t i c c o o r d i n a t e s (G^  and H^, v e r s u s n^) . The " i n n e r "  r e g i o n as s t a t e d p r e v i o u s l y  heat t r a n s f e r c h a r a c t e r i s t i c s . is  determined e n t i r e l y  the  The v a l u e o f the l o c a l N u s s e l t number  thus,  Nu = - x  2  /  5  ..  Ra  1 / 5  (0)  (0) = - 0 . 4 6 0 1  and the v a l u e of the mean N u s s e l t number o v e r the range x = 0 to x = 1 is  5/3 o f t h i s  value.  (34)  22  I n the " o u t e r "  r e g i o n b o t h the c o n v e c t i v e and v i s c o u s  terms  i n the e q u a t i o n of motion must be r e t a i n e d , whereas the temperature has a l r e a d y dropped t o i t s a s y m p t o t i c v a l u e ( i . e . fore,  to z e r o ) .  There-  f o r the " o u t e r " s o l u t i o n p u t ,  n  = n P r-3/10  2  H =  2  > (35)  0.  S u b s t i t u t i n g i n e q u a t i o n s (16) e q u a t i o n s reduce  r = f = F ' P3/5 r  through (18)  The boundary  the  to  5 F2* " + 3 f2 2F " H  and s e t t i n g n = 0,  2  =  (F') = 0 2  > (36)  0.  c o n d i t i o n s are^  F (0) = 0; 2  F '(0) 2  o f^(oo)  = 1.1522;  7^  (») = 0.  V a l u e s o f t h e a s y m p t o t i c f u n c t i o n s f o r t h e o u t e r p r o f i l e are g i v e n Table  3.2.2  (37)  in  IIa.  P r «  1  The t h e r m a l 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 t h e momentum boundary l a y e r .  C o n s e q u e n t l y , the i n f l u e n c e o f the v i s c o u s  terms i n t h e momentum e q u a t i o n upon the c o n v e c t i v e p r o c e s s s h o u l d b e come n e g l i g i b l y s m a l l a t a s m a l l d i s t a n c e from t h e b o u n d a r y , w h i l e t h e i n e r t i a terms s h o u l d remain o f the same o r d e r as t h e buoyancy t e r m s . Thus i t i s t h e " o u t e r " r e g i o n w h i c h p r i m a r i l y determines t h e c o n v e c t i o n p r o c e s s , w h i l e the " i n n e r " region ensures the disappearance of the v e l o c i t y  at t h e b o u n d a r y .  F o r the " o u t e r " s o l u t i o n p u t ,  n  fl  2  - n Pr  \  2 / 5  -  F' P r  1 / 5  = H.  2  A g a i n , t h e t e m p e r a t u r e has t o remain o f o r d e r u n i t y and thus i s not affected  by  "stretching".  C o n s i d e r i n g t h e i s o t h e r m a l p l a t e o n l y , e q u a t i o n s (16) through (18) reduce t o ,  3F  2  F "  -  2  (F ) 2  Z  + 2(ri  2  G* -  G ) = a|Pr | + a [ P r £  - o| P r  H  2  - \  = 0|Pr"  H ' ' + 3/5 F  2  H  2  1 / 5  .(Ra x ) 3  _ 2 / 5  |  = 0|(Pr Ra x ) " 3  2 / 5  |.  / 3  (Gr  1 / 5  Ra  x ) J  1 / 5  x  3 / 5  tan  24  The angle o f 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 ( P r Ra x ) ~ _ 1  3  1 / 5  |.  The boundary c o n d i t i o n s upon these e q u a t i o n s a r e ,  n  2  = o  F  F  The c o n d i t i o n of F  2  2  -  0  H  = 0  H  =  = G  2  2  2  =  1.0  y (42)  = G = 0 . 2  (0) = 0 , as i s seen above, i s n o t f u l f i l l e d  by t h e s o l u t i o n f o r the " o u t e r "  region.  T h i s 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 t h a t the n o - s l i p  c o n d i t i o n at the boundary i s  satisfied.  V a l u e s of the r e s u l t a n t a s y m p t o t i c f u n c t i o n s are g i v e n i n Table In f i g u r e s  8 and 9 , the v a l u e s o f G and H as p r e v i o u s l y  III.  determined f o r  v a r i o u s s m a l l v a l u e s of P r are r e p l o t t e d i n the a s y m p t o t i c c o o r d i n a t e s of t h e " o u t e r " s o l u t i o n .  Nu= - x '  2 / 5  The l o c a l N u s s e l t number i s :  Gr  1 / 5  Pr  2 / 3  H'(0) >  H' (0) = - 0 . 5 7 7 0  The mean v a l u e of Nu i s a g a i n found by i n t e g r a t i o n from x = 0 to x = 1 t o be 5/3 o f t h e 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 retained.  S i n c e t h i s r e g i o n i s e x t r e m e l y narrow the temperature w i l l  remain almost c o n s t a n t and of o r d e r u n i t y t h r o u g h o u t .  The v e l o c i t y  at  (43)  25 the o u t e r edge must match w i t h the i n n e r edge of t h e " o u t e r " Therefore,  f o r the " i n n e r " s o l u t i o n  nn = n  -1/10  put,  = F'  Pr  solution.  Pr  1/5  > (44) E  = 1.0  ±  S u b s t i t u t i n g i n t o e q u a t i o n (16) f o r the i s o t h e r m a l p l a t e , t h e f o l l o w i n g equation r e s u l t s ,  5 F^"  From e q u a t i o n s  + 3 F  F|* -  1  (Fj_)  = 2 Pr  (G  ± / Z  1  - T) G' ) ±  (45)  (44)  G^ =  + constant  (K)  (46)  1/2 45) = 2 P r ' K  (47)  Therefore  R.H.S.  (equ.  Now G (0) 1  = a|G(0)  where the a|G(0)  P r-  1 / 1 0  | = a|Pr"  2 / 5  . Pr"  1 / 1 0  was t a k e n from t h e " o u t e r " s o l u t i o n .  |,  (48)  Then,  -1/2, K = a Pr Therefore,  (49)  as e x p e c t e d , G^ remains a p p r o x i m a t e l y c o n s t a n t as P r ->• 0 ,  and from e q u a t i o n R.H.S.  (47) (eq.  45) = f i n i t e c o n s t a n t  (C),  (50)  26  and e q u a t i o n (45) becomes,  +3  5 F J "  The boundary  I  conditions  F (0) 1  =  (F  0  2  ( F p  FJ  (0))  (0)  (IIIa) .  (51)  C.  =  0  F^  (oo)  =  F (0) 2  .  (52)  (52)  2  = (1.5723)  Values of the a s y m p t o t i c f u n c t i o n s i n Table  =  2  are,  From the l a s t c o n d i t i o n of  C = -  -  Fp  2  .  f o r the " i n n e r " s o l u t i o n are  (53)  given  Figure 6  P r e s s u r e F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow i n Terms of Asymptotic Coordinates f o r Pr °°  Figure 7  Temperature F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow i n Terms o f A s y m p t o t i c C o o r d i n a t e s f o r P r -*-  00  29  Figure 8  P r e s s u r e F u n c t i o n s f o r 2 - D i m e n s i o n a l Flow i n Terms o f A s y m p t o t i c C o o r d i n a t e s 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 r e s e n t e d i n t h i s s e c t i o n i s f o r a s t e a d y symmetrical flow.  axially-  The boundary l a y e r assumptions a p p l i c a b l e t o t h i s  flow 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 l o w 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  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 t o a p e r f e c t l y horizontal surface.  An  estimate o f 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  t o the h o r i z o n t a l may be o b t a i n e d by c o n s i d e r i n g the f l o w above a c o n i c a l e n v e l o p e , and a d j u s t i n g the p r e s s u r e and buoyancy terms  accordingly.  The s y s t e m of e q u a t i o n s i n c y l i n d r i c a l p o l a r c o o r d i n a t e s i s as  then  follows; continuity,  (54  13r^ +r ^+1^=0 dz momentum i n the r a d i a l d i r e c t i o n ,  -2/5  + a | Gr'  (55)  momentum i n the a x i a l d i r e c t i o n ,  3£ 9£  -2/5  =  0 - a|Gr"  +a  Gr  -4/5  (56)  momentum i n the a z i m u t h a l d i r e c t i o n , dv , u -r— + dr  uv r  (57)  32  energy,  ~ 30 . ^ 30 3r 3z  (58)  3z  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 variables  are  rendered d i m e n s i o n l e s s through the use o f a r e f e r e n c e l e n g t h L o r a reference v e l o c i t y  v/L.  z ,w  r,u  F i g u r e 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 i n c e no i n c l i n a t i o n i s c o n s i d e r e d t h e d i m e n s i o n l e s s p r e s s u r e  is  s imp l y , p - poo 2 T  77  =  —2  Poo  v  gLT  v~~  z  •;.  (59)  33  t h e d i m e n s i o n l e s s temperature T -  is,  T (60)  AT r e f and t h e Gr number i s , Gr = (g $ L  3  AT  r e f  )/v .  (61)  2  I n c l u d e d i n the e q u a t i o n s above are a l s o t h e : f o l l o w i n g layer  boundary  transformations,  z = z Gr  1/5  -2/5 u = u Gr  v = v Gr  -2/5 V  _ -1/5 w = w Gr :  ft  *  The f o l l o w i n g s u b s e c t i o n s d i s c u s s s p e c i f i c s i m i l a r i t y f o r the e q u a t i o n s  4.1  (54)  to  (62)  „. - 4 / 5 = IT Gr  solutions  (58).  Free C o n v e c t i o n f o r an A r b i t r a r y  P r Number  T h i s 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  solution  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 from a h o r i z o n t a l s u r f a c e , w h i c h i s i n d u c e d by buoyancy  f o r c e s i n the v e r t i c a l d i r e c t i o n .  case t h e r e i s no a z i m u t h a l component of the v e l o c i t y (57) need not be c o n s i d e r e d . (54) , (55) , (56)  In  this  and e q u a t i o n  The e q u a t i o n s a p p l i c a b l e are  therefore  and 0 8 ) .  As f o r t h e 2 - d i m e n s i o n a l flow the f o l l o w i n g are now d e f i n e d . A Stokes' stream f u n c t i o n ,  1  u  =  7  U  3?  1 3^w = - r 3r  (63)  34  Similarity  transformations,  i> -  r  = r  P  F(n)  5  TT  n „ , s H (n) ;  = r  n = z  q H  G(ri)  m  r  -s  > (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 e q u a t i o n s and (58)  (55) ,  (56)  a s e t of c o m p a t i b i l i t y e q u a t i o n s f o r the exponents  w h i c h are analogous t o e q u a t i o n s defined a r b i t r a r i l y  (c.f.  (14).  results  A g a i n one exponent may be  q),  q = 1 m = 2/5 + 4/5 n > (65)  p = 8/5 + 1/5 n s = 2/5 -  1/5 n .  The v a l u e of the exponent n i s imposed by the s u r f a c e t e m p e r a t u r e . The l i m i t a t i o n s on t h e a l l o w a b l e v a l u e s f o r t h i s exponent are i d e n t i c a l to  (15). Equations  into  (55) , (56)  (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  and (58) produce the f o l l o w i n g s e t of t o t a l  differ-  e n t i a l e q u a t i o n s w h i c h 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  surface temperature:  35  I  5 F " * + (8 + n) F F "  -  (1 + 2n)  (F')  2  = (66)  2(1 + 2n)G - (2 - n) n G' + a ! G r V _2/  2 / 5 ( n+ 3)  ,r _ ^ „ ,^„-2/5 _ . - 2 / 5 ( n + 3) G' = H + o Gr ' r  (67)  u  H  1  1  +  (n + 8) F H '  - n P r F' H =  a  ' |cr  2 / 5  r  2  /  5  (  n  +  | .  3 )  F o r the case o f an i s o t h e r m a l s u r f a c e t h e s e e q u a t i o n s s i m p l i f y  5 F"'  + 8 FF"  -  (F')  2  (68)  to,  = 2(G -^G )  (69)  ,  G' = H  (70)  H"  (71)  + 8/5 P r F H' = 0 .  Numerical r e s u l t s f o r P r = 1 are presented i n f i g u r e s  11 and 1 2 .  S o l u t i o n s may be o b t a i n e d f o r a s y m p t o t i c v a l u e s o f the P r number i n a manner s i m i l a r t o t h a t demonstrated f o r the 2 - d i m e n s i o n a l f l o w p r o b l e m . A l t h o u g h 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 its  p h y s i c a l p e c u l a r i t i e s remain t o be j u s t i f i e d .  flow that progresses  The s o l u t i o n y i e l d s  r a d i a l l y outwards w h i c h i s of course not  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 l o w s o v e r a p l a t e o f radius.  It  identi-  finite  does p r e d i c t a f l o w o b s e r v e d i n c e l l s of i n s t a b i l i t y  h o r i z o n t a l s u r f a c e s by C h a n d r a s e k h a r .  at,  over  a  F i g u r e 11  Stream and V e l o c i t y F u n c t i o n s f o r A x i a l l y Symmetrical Flow-Isothermal P l a t e  F i g u r e 12  P r e s s u r e and Temperature F u n c t i o n s f o r A x i a l l y S y m m e t r i c a l Flow - I s o t h e r m a l P l a t e  i  38 4.2  Combined Free and Forced C o n v e c t i o n f o r an Arbitrary  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 t o r . The a n a l y s i s  p r e s e n t e d h e r e i n w i l l develop a boundary  t y p e 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 l o w above an e x a c t l y s u r f a c e w h i c h i s r o t a t i n g at a c o n s t a n t s p e e d . f o r t h i s f l o w are the e q u a t i o n s As f o r e q u a t i o n s  (64)  (54) through  The e q u a t i o n s  layer horizontal applicable  (58).  a s e t 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  w h i c h l e a d t o a s e t of c o m p a t i b i l i t y e q u a t i o n s f o r the e x p o n e n t s . erality  c a n , however, n o t be m a i n t a i n e d i n t h i s i n s t a n c e and o n l y  defined Genthe  f o l l o w i n g 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 p o s s i b l e .  2  -  r  n=  F(n)  = r  2  r v = a r  G  (n)  2  T (n) •  z  9 = r  Here " F " i s a c i r c u l a t i o n v a r i a b l e .  2  H (n)  The c o n s t a n t " a " i s  (72)  defined  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 e q u a l t o ,  a = -^TT  Gr ' 2  (73) 5  where  V  and AT  ref  = (T  w  -  TV 00  @r = 1.  (74)  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 e q u a t i o n s t o t a l d i f f e r e n t i a l equations  F"  result,  ( F )M 2 = 2G -  + 2F'' F -  1  =  1  (55) t o (58) the f o l l o w i n g  a  2  T  Jl  (75) (76)  G' = H  r" + H"  2 (FT"  -  F'  D =0  (77) (78)  + 2 P r (FH' - F' H) = 0.  F o r P r = 1 t h i s s e t of e q u a t i o n s reduces t o two t h i r d o r d e r t o t a l  differ-  e n t i a l e q u a t i o n s w h i c h 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  n = o  F = 0  F  F' = 0  are,  = 0  H = 1  r =  I  > (79.)  G=G'=r=0  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 i v e n i n f i g u r e s 13 and 14 f o r v a l u e s of a = 0  and.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 t a k e n from Cochran  (12).  1  T  0  g=9  .793  1.578  1.164  16.488  g=  F(oo)  FlO)  F i g u r e 13  V e l o c i t y Functions f o r A x i a l l y Symmetrical 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 14  Temperature F u n c t i o n s f o r A x i a l l y Symmetrical Flow - P l a t e Temperature P r o p o r t i o n a l t o r ^  42  4.3  Numerical Integration 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 , t h e s e t s of e q u a t i o n s  w h i c h 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 plate (isothermal  surface),  A r b i t r a r y P r - e q u a t i o n s (21) Pr »  to (23) .  1  "outer region" - equations  (29)  to  (31)  "inner region" -  equations  (36)  equations  (39) t o  (41)  P r << 1 "inner region" -  " o u t e r r e g i o n " - e q u a t i o n s (51) . Infinite  disc,  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 - e q u a t i o n s (69) Combined c o n v e c t i o n (T  w  «  2 r )  - equations  (75)  to  (71)  t o (78) .  E x c e p t i n g e q u a t i o n s (36) and ( 5 1 ) , the s e t s of e q u a t i o n s may i n . e v e r y case be reduced to two s i m u l t a n e o u s t h i r d o r d e r equations.  These e q u a t i o n s , s u b j e c t t o 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. numerically.  total-differential  Therefore,  i n t e g r a t i o n has t o p r o c e e d  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  complete f o r m of one of the f u n c t i o n s .  F u r t h e r m o r e , 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 t o be assumed and was then i t e r a t e d upon so as to t h e c o n d i t i o n s a t the o t h e r e n d .  the  fulfill  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 r o m the end at w h i c h two i n i t i a l c o n d i t i o n s were known.  I n t h i s manner  one f u n c t i o n would be generated w h i l e v a l u e s f o r the o t h e r were  transferred  43  f r o m the assumed f o r m .  The t e c h n i q u e was then r e v e r s e d ,  integrating  t h e o t h e r 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 of e r r o r .  The i n t e g r a t i o n t e c h n i q u e employed was a f o u r t h  Runge-Kutta forward i n t e g r a t i o n Unfortunately, correct  order  routine.  t h i s method o f s o l u t i o n w o u l d converge t o a  r e s u l t o n l y when r e a s o n a b l y 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 o f t h e complete f o r m 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 . I n o r d e r to o b t a i n t h e s e v a l u e s an a p p r o x i m a t e i n t e g r a t i o n a c c o r d i n g t o t h e von Karmian -  P o h l h a u s e n t e c h n i q u e was p e r f o r m e d ;  In g e n e r a l t h e  e s t i m a t e s had t o be much b e t t e r f o r t h e extreme P r numbers.  Only a  s m a l l d e v i a t i o n i n t h e 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 t h e extreme  P r numbers was e x t r e m e l y s l o w and a t e c h n i q u e a v e r a g i n g  consecutive  p r o f i l e s was employed. M o r e o v e r , i n some cases t h e e q u a t i o n s p o s s e s s a s i n g u l a r i t y the s t a r t i n g b o u n d a r y .  at  F o r t h e s e e q u a t i o n s i n t e g r a t i o n was s t a r t e d  at a v e r y s m a l l v a l u e of the i n d e p e n d e n t v a r i a b l e  i n s t e a d of from 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 f r o m the i n t e g r a t i o n f o r t h e 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 T a b l e Lastly,  I.  f o r the combined f r e e and f o r c e d - c o n v e c t i o n 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 l o w p r o f i l e s were t a k e n f r o m C o c h r a n ' s for a spinning isothermal disc.  (12)  tabulated data  44 V  The  EXPERIMENTAL APPARATUS  purpose of the e x p e r i m e n t a l work was  to e s t a b l i s h the  existence  of a l a m i n a r boundary l a y e r above a h e a t e d h o r i z o n t a l s u r f a c e , and o b t a i n some measure of the d i s t a n c e of the boundary l a y e r was  o p t i c a l approach was  i n t e r f e r e n c e w i t h the analysis  and  :  doubted because, as d i s c u s s e d  extends i n an adverse g r a v i t y An  over which i t e x t e n d s .  flow  The  to  existence  previously, i t  field. chosen s i n c e i t , p r o v i d e s a method f r e e o f  field  also s u f f i c i e n t l y  for a qualitative justification  accurate  of  the  d a t a t o o b t a i n some q u a n t i t a t i v e  results. Since  the  l a m i n a r boundary l a y e r was  temperature d i f f e r e n c e s o n l y , high s e n s i t i v i t y .  zontal surface. fulfill  a l l the  the o p t i c a l system was  Moreover i t had  t h a t a t t e n t i o n c o u l d be  expected to e x i s t under moderate  restricted  to be  of the  required  to have a  l i m i t e d - f o c u s i n g t y p e such  to a s l i c e of the a r e a above the h o r i -  A s c h l i e r e n semi-focusing  c o l o u r system was  thought  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  to  moderate  cos t.  5.1  O p t i c a l Bench.  The  o p t i c a l system c o n s i s t e d o f two  s e p a r a t e benches :  were mounted the s o u r c e , the s o u r c e l e n s e s , and mirror;  and  plane cplour separated  the  f i l t e r , the  sufficiently  object  t o allow The  l e n s e s , and  the  camera.  must be  focal  The benches were  an i n t e r f e r e n c e f r e e l o c a t i o n of  separation  first  the "upstream" s c h l i e r e n  on the second the "downstream" s c h l i e r e n m i r r o r , the  horizontal test plates. lengths.  on  of at l e a s t two  A s c h e m a t i c drawing of the system i s given i n f i g u r e  the focal (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 w h i c h a brief  account i s g i v e n 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  follows:  Source - 500 watt t u n g s t e n f i l a m e n t lamp Condensing l e n s Source s l i t -  coated, anastigmatic, f:  3 . 5 , 5 inches f o c a l length.  .006 i n c h e s w i d t h by . 5 i n c h e s  S c h l i e r e n lenses -  length.  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 focal length.  Focal plane l i g h t  cut-off -  graded 4 c o l o u r f i l t e r w i t h bands .009 i n s . wide.  Object lens Camera -  S c h n e i d e r "Symmar" f :  35 mm S . L . R .  5 . 6 , 210 mm f o c a l l e n g t h .  camera body w i t h f o c a l - p l a n e  shutter.  W h i l e a t t e m p t i n g t o 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 g e n e r a l c o n s i d e r a t i o n s were found t o be v e r y For a h i g h s e n s i t i v i t y  important.  system the l e n s e s s h o u l d be of a good q u a l i t y  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 .  While experimenting 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 , matism was s e v e r e and t h i s a c c o r d i n g l y v i t y o f the s y s t e m .  Furthermore,  and  astig-  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 -  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 . l a t t e r , i n p a r t i c u l a r , was v e r y much more i m p o r t a n t than a t f i r s t  The appre-  ciated. 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 to be an optimum f o r the system s t r i k i n g a b a l a n c e between the 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  blurring.  found  desirability  46  5.2  Alignment 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 t o be performed on p e r f e c t l y p l a t e s , alignment of the s c h l i e r e n beam was c r i t i c a l .  A theodolite  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: P h y s i c s , 0 . 3 mW continuous o u t p u t ) .  horizontal  Spectra  The l a s e r beam was passed a l o n g  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  adjusted  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 s y s t e m . 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 m e c h a n i c a l l e v e l (make:  Starratt).  The s u p p o r t s had f i n e adjustment screws f o r  this  purpose. 5.3  Colour  Filter  A c o n s i d e r a b l e number o f attempts were made t o 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 n e x t t o each o t h e r .  These f i l t e r s  p r o v e d t o be v e r y 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 i d t h s above a p p r o x i m a t e l y . 0 2 5 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  i m p r o p e r l y c u t edges became r a p i d l y  to  severe.  The f i l t e r f i n a l l y used i n the apparatus was.made f r o m H i g h Speed Ektachrome type B f i l m , w h i c h 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 pictures.  The procedure f o r making the f i l t e r c o n s i s t e d o f p r o d u c i n g  first  a 4 x 5 i n c h . m a s t e r g r a t i n g w h i c h 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 m a s t e r 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 i n d e x e d w h i l e r e e x p o s i n g 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 , r e d and green f i l t e r s mounted on the camera.  5.4  R e c t a n g u l a r 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 i n c h e s square and 21 i n c h e s s q u a r e , and one round p l a t e , 7 i n c h e s i n d i a m e t e r . of the p l a t e s were r e s p e c t i v e l y and 1/4 i n c h e s copper p l a t e .  The top s u r f a c e s  3/8 i n c h e s and 3/4 i n c h e s aluminum p l a t e  The s u r f a c e s were backed by  sandwiched  (*) mica-nichrome wire h e a t e r s . as f o l l o w s :  Maximum heat outputs of the h e a t e r s were  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 t a k e n t o a v o i d heat l e a k a g e through the b o t t o m 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 t h e o u t s i d e w i t h 1/4 i n c h cork insulation. 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 h o l l o w  half-  rounds o f wood f i l l e d w i t h i n s u l a t i o n t o ensure t h a t the edge s t a y e d as c o o l as p o s s i b l e . Temperatures  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  o f the s u r f a c e s were measured u s i n g b u r i e d ,  constantan thermocouples.  copper-  They were spaced at 1 i n c h i n t e r v a l s  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 square p l a t e s .  15 and 16.  along  along a center l i n e of  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  the  to  installation.  (*)  E x c e p t the 12 x 12 i n c h e s p l a t e , w h i c h was h e a t e d by a P y r e x p a n n e l t h i n l y coated w i t h a u n i f o r m l a y e r of z i n c o x i d e .  F i g u r e 16  7 Inches Diameter D i s c  49  VI 6.1  EXPERIMENTAL PROCEDURE  Test Procedure The p l a t e s were l o c a t e d and mounted i n a manner such t h a t  interference  from the support  be d i s c o u n t e d .  or any s u r f a c e s around the p l a t e s  direct could  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 t h e r h a n d , t o  i s o l a t e p o s s i b l e . f r e e c o n v e c t i o n c u r r e n t s s e t up w i t h i n the To reduce t h i s e f f e c t ,  a fairly  to t h e t a k i n g of any s c h l i e r e n  laboratory.  l e n g t h y q u i e s c e n t s t a t e was observed  prior  photographs.  I n a d d i t i o n , t h e 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 b e f o r e p i c t u r e s were t a k e n .  Typically,  thermocouples  located  a l o n g a r a d i u s of the d i s c and a l o n g a c e n t e 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 F a h r e n h e i t f o r the maximum temperature-difference would i n c r e a s e a l i t t l e  considered.  In g e n e r a l t h e s u r f a c e  from the o u t s i d e edge t o the c e n t r e of the p l a t e s .  For e v e r y s e p a r a t e group of p i c t u r e s was taken as w e l l .  temperature  taken, a calibration  photograph  T h i s c o n s i s t e d of a t r a n s p a r e n t s c a l e p l a c e d a t the  c e n t e r l i n e of t h e p l a t e s . The q u a n t i t a t i v e d a t a was read o f f t h e p i c t u r e s by p r o j e c t i n g onto a l a r g e graded s c r e e n .  T h i s was p o s i t i o n e d i n a way such t h a t  g r i d l i n e d up w i t h the s c a l e markings o f t h e c a l i b r a t i o n  6.2  them the  transparency.  Experimental Results. Examples o f s c h l i e r e n photographs  are g i v e n i n f i g u r e s  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 l o w 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  17 through 2 0 .  instability  50  by any s p u r i o u s l a b o r a t o r y  currents.  I n essence t h e f l e w behaved i n  the form o f l a m i n a r s p e l l s w h i c h v i s i b l y became more s t a b l e w i t h 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  ( f a n 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 t h e v i c i n i t y were taken w i t h s t r i c t  the  of.the plate).  A l l pictures  adherence to t h e s e 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 t h e various p l a t e s .  As may be e x p e c t e d from analogy w i t h o t h e r  transitional  f l o w s , the onset o f i n s t a b i l i t y may o n l y be s t i p u l a t e d t o o c c u r 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 p a r a m e t e r . E x a m i n i n g 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 n o t  o c c u r at e x a c t l y the same average Grashof number f o r t h e two sized plates.  differently  The onset of i n s t a b i l i t y i s p r o b a b l y a f f e c t e d by the  s i z e of the t h e r m a l j e t  ,  r i s i n g above t h e c e n t e r o f the p l a t e s w h i c h  has a s t a b i l i z i n g e f f e c t upon the f l o w .  F u r t h e r m o r e , the onset of i n -  s t a b i l i t y i s s e e m i n g l y 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 . i s seen by t h e f a l l i n g - o f f o f the d a t a at lOw AT  This  ^ i n f i g u r e s 22 and 2 3 .  F i g u r e s 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 t h e d i s t a n c e from the o u t s i d e e d g e , w h i l e the Reynolds number i s based upon t h e d i s t a n c e from the c e n t e r . 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 c o n v e c t i v e  These flow  and the f o r c e d flow are b a s i c a l l y i n o p p o s i t e d i r e c t i o n s . L a s t l y , o b s e r v a b l e i n t h e photographs s e p a r a t i o n b u b b l e n e a r the l e a d i n g edge.  f o r the square p l a t e s i s a s m a l l I n 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 p r o n o u n c e d , and f o r the h i g h e r a t u r e s 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 o f the f l o w .  temper-  a  '  A T  ref  = 2 3  - ° ' 5  F  T  oo= -°F; 7 1  b.  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 ; AT -39.°F, 71.-F; c . AT = 4 3 . 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. • rH  II  U  d8 r  CM r-l  J!  " PM 0 O  4-1  •H  5  •  iH 00  II  CO X!  ^ 0)  6 >-i cfl H M < txO  O 4J  • O  o  xi ••> PM C U  0) •H rH  00  oo m rH  .  a) m  M  d  •H PM  II  «H  rJ  H  <  II  o 00  H o  a-  AT  r e f  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 : =52.5°F, = 72. % 0 = 0 r p m ; b. A T = 6 2 . 5 ° F , Too=72.0°F, 0=32.8 r p m ;  c  AT  r e f  =94.5°F,  Tro  Ta>  r e f  = 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 = 9 1 . 5 ° F , T = 7 1 . ° F , f i =0 rpm, r  b.  AT  ref  = 1 1 5 . ° F , T =71°F, ii =0 rpm. ^ 00  55  40 60 AT f~°F re  Figure 21  20  40  C r i t i c a l Gr for 21 Inch Square Plat*  60 80 AT ~°F  100  ref  Figure 22  C r i t i c a l Gr for 12 inch Square Plata  60 AT Pig. 23  80 r e f  ~«F  C r i t i c a l Gr f o r 7 inch Diac f o r Frae Convaction  57 VII  DISCUSSION OF THE RESULTS  Thermal c o n v e c t i o n 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 e x p l a i n e d by two d i f f e r e n t i n f i n i t e extent  theories.  When t h e s u r f a c e i s of an  c o n v e c t i o n must i n the mean t a k e p l a c e i n an e s s e n t i a l l y  vertical direction.  C o n s e q u e n t l y n e a r the s u r f a c e i t would be dominated  by some type o f c e l l u l a r f l o w . is  (i)  been  f i n i t e , a boundary  to the b e l i e f t h a t  (ii)  l a y e r theory  On t h e o t h e r h a n d , when t h e of r e l a t i v e l y  r e c e n t o r i g i n , has l e a d  c o n v e c t i o n o c c u r s through a l a m i n a r boundary  w h i c h grows f r o m the edges.  The boundary  surface  layer  flow  l a y e r s would meet i n t h e m i d d l e  o f the s u r f a c e , t u r n u p , and r i s e as a plume.  S i n c e the plume would  cover o n l y a v e r y s m a l l p o r t i o n of t h e p l a t e s u r f a c e ,  ( n e a r the s u r f a c e  i t w o u l d have a w i d t h o f o r d e r 25 a c c o r d i n g t o S t e w a r t s o n (1)^  i t was  p o s t u l a t e d t h a t t h e h e a t t r a n s f e r r a t e w o u l d be governed by the 5 t h r o o t o f t h e G r a s h o f 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 e v i d e n t point  t h a t at some,  t h i s dependence must b r e a k down, f o r t h e f l o w i n t h e c e n t r a l  region  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 . f a c t , s i n c e the l a m i n a r boundary unstable i t short  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 ,  is  In potentially  may be e x p e c t e d t h a t the l a m i n a r f l o w w o u l d e x t e n d f o r o n l y a  distance.  As t h e c h a r a c t e r i s t i c parameter o f t h e f l o w i s the G r a s -  h o f 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 as a " c r i t i c a l G r a s h o f number".  parameter,  The c r i t i c a l Grashof number then e n a b l e s  the d e t e r m i n a t i o n o f a p e r m i s s i b l e p l a t e s i z e o r a t e m p e r a t u r e - d i f f e r e n c e between t h e p l a t e and f l u i d b u l k f o r w h i c h the f l o w o v e r v i r t u a l l y p l a t e w i l l be l a m i n a r .  the  entire  58 S o l u t i o n s of the e q u a t i o n s of f l o w can u n f o r t u n a t e l y 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  o n l y be o b large  radius.  In t h i s r e s p e c t the p l a t e s used f o r the e x p e r i m e n t a l o b s e r v a t i o n s  of , the  f l o w do not a f f o r d an e x a c t 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 .  Further-  more, s i n c e the s o l u t i o n s are i n the form o f s i m i l a r i t y s o l u t i o n s  their  validity  i s l i m i t e d by c e r t a i n a p r i o r i  The f o l l o w i n g s u b s e c t i o n s  assumptions.  respectively  d i s c u s s the  theoretical  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  Numerical Solutions 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 s e x p e c t e d t o be f u n d a m e n t a l l y d i f f e r e n t above an i n c l i n e d s u r f a c e .  from the f l o w t h a t  occurs  The a p p l i c a b i l i t y o f the s o l u t i o n s  f o r e , be l i m i t e d to a v e r y 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 zontal.  introduction,  must,..there-  to.the-hori-  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 surface i s  (equation  (19)),  a < a | tan- — For Gr = 10  5  .-1/5 (Gr x  the i n c l i n a t i o n i s thus found t o be l e s s than a b o u t . 5 ° .  In a d d i t i o n t o 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  solutions  are i n g e n e r a l r e s t r i c t e d t o c e r t a i n r e g i o n s of the f l o w f i e l d as terms w h i c h are o f a s m a l l e r o r d e r are n e g l e c t e d .  O u t s i d e the  regions  of good a p p r o x i m a t i o n , the terms w h i c h 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 w h i c h thus must remain l a r g e (Gr  x  3+n,2/5  is  59  In c o n j u n c t i o n w i t h t h i s term and t h e p e r m i s s i b l e i n c l i n a t i o n , t h e imposed w a l l t e m p e r a t u r e v a r i a t i o n must be such t h a t the exponent n remains between - 3 and +2. I n 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 a p p r o x i m a t i o n down t o P r = 1 0 , and up t o P r = . 1 r e s p e c t i v e l y . When the e q u a t i o n s were thus expanded w i t h the P r a n d t l number the f l o w 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 t h e r m a l boundary l a y e r much s m a l l e r than t h e 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 , c o n t a i n e d w i t h i n an " i n n e r " r e g i o n .  is physically  The " o u t e r " r e g i o n then c o n t a i n s  o n l y 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 t h e r m a l boundary l a y e r i s much l a r g e r than t h e momentum boundary  layer.  Therefore,  w i t h i n an " o u t e r " r e g i o n .  t h e e n t i r e temperature drop t a k e s p l a c e I n the " i n n e r " r e g i o n t h e temperature remains  almost c o n s t a n t throughout w h i l e t h e the v e l o c i t y drops to z e r o on the boundary.  M a t h e m a t i c a l l y the f o r m a t i o n of these r e g i o n s may be s a i d  t o be the r e s u l t of u s i n g o n l y the f i r s t b a t i o n expansion i n P r .  Therefore,  term of a s i n g u l a r  pertur-  d e s p i t e the f a c t t h a t the s o l u t i o n s  i n each case d e s c r i b e o n l y one r e g i o n , they e x t e n d a c r o s s the e n t i r e  flow  f i e l d and the expanded v a r i a b l e r) measures i n e v e r y case from t h e 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 p r e s e n t e d f o r a x i a l l y s y m m e t r i c a l flow.  These s o l u t i o n s p r e d i c t a f l o w t h a t proceeds outwards above an  infinite disc.  The f l o w w h i c h 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 t h e o r y .  For  l a c k of an e x a c t u n d e r s t a n d i n g 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 t h a t a p h y s i c a l f l o w a l o n g the l i n e s of the t h e o r y may o c c u r under some o t h e r  than c o n s t a n t s u r f a c e temperature  condition.  F o r the combined f l o w 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 o n l y f o r a s u r 2 f a c e temperature t h a t i n c r e a s e s w i t h r .  I n t h i s case the c h a r a c t e r i s -  t i c p a r a m e t e r , c o n t r o l l i n g the r e l a t i v e i m p o r t a n c e o f f r e e 2 4/5 t o f o r c e d c o n v e c t i o n , i s Re^  /Gr  2-dimensional a n a l y s i s , Mori (11), 7.2 S c h l i e r e n O b s e r v a t i o n s  .  convection  T h i s p a r a m e t e r agrees w i t h the  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 .  The amount o f e x p e r i m e n t a l d a t a t h a t 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 o f the t h e o r y i s v e r y s m a l l .  Schmidt (4)  some s c h l i e r e n p i c t u r e s of the f l o w about a h o r i z o n t a l s u r f a c e b u t h i s case b o t h s i d e s o f . t h e p l a t e were h e a t e d . 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 " and Saunders (3)  took in  The upper s u r f a c e w a s ,  from below the p l a t e .  Fishenden  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 d a t a f o r  the h e a t 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 t o s t a t e p o s i t i v e l y c o n v e c t i o n was by l a m i n a r f l o w o r n o t .  whether  Their data i s replotted i n  figure  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 d a t a f o r the c o n s t a n t t e m p e r a t u r e s e m i infinite plate.  The mean t h e o r e t i c a l N u s s e l t number and the  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 ,  Nu = - | . 2  2 / 5  Gr  1 / 5  H'  or,  (0).  The s l o p e s of t h e i r d a t a seem t o i m p l y t h a t c o n v e c t i o n through a c e l l u l a r m o t i o n .  Grashof  occured  T h i s 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  their  61  d a t a was p r o b a b l y o b t a i n e d 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 s u p p o r t e d by the f a c t t h a t t h e i r data lies higher, 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 t h e r hand, a l s o compared h i s d a t a t o Fishenden and S a u n d e r s ' and c l a i m e d t h a t the o n e - f i f t h power r e l a t i o n s h i p p r o v i d e s a good f i t . 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 t h a t h i s  As  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 l o w beneath a h e a t e d s u r f a c e and c o n s e q u e n t l y compared h i s t h e o r e t i c a l r e s u l t t o the wrong e x p e r i m e n t a l d a t a . Therefore,  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  l a m i n a r c o n v e c t i o n , account must be t a k e n of t h e c r i t i c a l Grashof number f o r the  flow.  S c h l i e r e n p i c t u r e s p r e s e n t e d i n t h i s t h e s i s support the t h a t a l a m i n a r boundary l a y e r does e x i s t . conditional.  contention  However, t h i s c o n t e n t i o n i s  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 o f o r d e r l e s s thati 1 0 ^ , b u t l a r g e enough (say g r e a t e r than 500) t o form a boundary l a y e r f l o w .  The s u r r o u n d i n g space must be s u f f i c i e n t l y  f r e e of s p u r i o u s a i r c u r r e n t s so as not to t r i p t a t i o n a l l y extremely  the f l o w , w h i c h i s  gravi-  unstable.  The average c r i t i c a l Grashof number (Gr^) was measured t o be 8 x 10^ (from f i g u r e 2 2 ) .  f o r the 12 i n c h p l a t e In appendix B i t  i s shown  t h a t 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 i s then 2 . 3 x 10 (Pr = .72 f o r a i r ) . g  T h i s 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 v a l u e s o b t a i n e d 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  o f 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 stationary of 1100. I t  f l u i d , s e t s i n at a Ra^ (based on the depth o f the  may be reasoned t h a t the much h i g h e r v a l u e o b t a i n e d f o r  fluid)  transition  62  o f the l a m i n a r boundary l a y e r of the type c o n s i d e r e d i n t h i s  thesis  i s due t o 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 l o w f e e d i n g  the  plume.  central  VIII  CONCLUSIONS  T h i s i n v e s t i g a t i o n has l e d t o the f o l l o w i n g r e s u l t s : Two-dimensional flow (free 1.  convection)  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 " t y p e of w a l l temperature variation. case.  2.  The c o n s t a n t f l u x c o n d i t i o n i s o b t a i n e d as a s p e c i a l  N u m e r i c a l d a t a i s p r e s e n t e d f o r the i s o t h e r m a l s u r f a c e .  A s y m p t o t i c s o l u t i o n s f o r P r a p p r o a c h i n g i n f i n i t y and zero have been d e r i v e d . or  <.l.  These s o l u t i o n s are a good a p p r o x i m a t i o n when P r >  N u m e r i c a l d a t a i s p r e s e n t e d 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 t h e o r y are d e f i n e d .  4.  The onset of u n s t a b l e f l o w , above the p l a t e s has been examined and found t o occur at a Gr number of o r d e r 10"*.  Below t h i s  Gr  number the l a m i n a r boundary l a y e r f l o w may extend o v e r 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 r o o t of the Gr number.  A x i a l l y symmetrical flow (free 5.  convection)  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 l o w and are thus not a p p l i c a b l e t o a d i s c of f i n i t e d i a m e t e r .  A x i a l l y s y m m e t r i c a l f l o w (combined c o n v e c t i o n ) . .._ . 2 6.  The c h a r a c t e r i s t i c parameter f o r combined c o n v e c t i o n i s Re^ /Gr when e x p a n s i o n o c c u r s about the f r e e c o n v e c t i o n mode.  4/  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 l o w w h i c h i s a i d e d by  buoyancy.  The s i m i l a r i t y approach y i e l d s o n l y t h i s one s o l u t i o n w h i c h i s 2 f o r a s u r f a c e temperature t h a t i n c r e a s e s w i t h r . 8.  The f l o w e x p e r i m e n t a l l y examined i s of the o p p o s i n g t y p e .  9.  A curve i s p r e s e n t e d w h i c h a p p r o x i m a t e l y l o c a t e s the t r a n s i t i o n region f o r a flow that i s p r i m a r i l y f r e e l y that i s p r i m a r i l y being forced.  convecting t o a flow  66  REFERENCES  1.  K. S t e w a r t s o n , "On f r e e c o n v e c t i o n 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. d e l - C a s a l • , . " F r e e c o n v e c t i o n on a h o r i z o n t a l p l a t e " , Z . A . M . P . . 16, 1965, 539-541.  3.  M. Fishenden and O.A. S a u n d e r s , "The c a l c u l a t i o n of c o n v e c t i o n 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 1 3 0 , 1 9 3 - 1 9 4 ; a l s o i n "An I n t r o d u c t i o n t o Heat T r a n s f e r " , Oxford U . P . , 1950, p p . 9 5 - 9 6 .  4.  E-. S c h m i d t , . " S c h l i e r e n a u f n a h m e n des T e m p e r a t u r e f e l d e s i n der Nahe warmeabgebender K o r p e r , V . D . I . F o r s c h u n g 3 ( 4 ) , 1 9 3 2 , 1 8 1 - 1 8 9 .  5.  R. W e i s e , "Warmeubergang durch f r e i e K o n v e k t i o n an q u a d r a t i s c h e n P l a t t e n " , V . D . I . F o r s c h u n g , 6., 1 9 3 5 , 2 8 1 - 2 9 2 .  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 r a n s . Japan Soc. Mech. E n g r s . , 2 1 , 1 9 5 5 , 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 (28), 1964, 745-750.  8.  D . J . T r i t t o n , " T u r b u l e n t f r e e c o n v e c t i o n above a h e a t e d p l a t e . i n c l i n e d at a s m a l l angle t o t h e h o r i z o n t a l " , J . F l u i d M e c h . , 1 6 , 1 9 6 3 , 2 8 2 - 3 1 2 .  9.  D . J . T r i t t o n , " T r a n s i t i o n t o t u r b u l e n c e i n t h e f r e e c o n v e c t i o n boundary l a y e r s on an i n c l i n e d h e a t e d p l a t e " , J . F l u i d M e c h . , 1 6 , 1 9 6 3 , 4 1 7 - 4 3 5 .  10.  E . M . Sparrow, R. E i c h h o r n , J . L . G r e g g , "Combined f r e e and f o r c e d c o n v e c t i o n 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, 3 1 9 - 3 2 8 . •  11.  Y . M o r i , "Buoyancy e f f e c t s i n f o r c e d l a m i n a r c o n v e c t i o n f l o w o v e r 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 , 1 9 6 1 , 4 7 9 - 4 8 2 .  12.  W.G. C o c h r a n , "The f l o w 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. , 3 0 , 1 9 3 4 , 3 6 8 - 3 7 5 .  13.  K. M i l l s a p s and K. P o h l h a u s e n , "Heat t r a n s f e r by l a m i n a r f l o w 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 l a m i n a r f r e e - c o n v e c t i o n f l o w 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 t o t h e d i r e c t i o n of the g e n e r a t i n g body f o r c e " , N . A . C . A . , TR - 1 1 1 1 , 1953.  67 15.  1 . 6 . R e i l l y , C h i T i e n and M. Adelman, " E x p e r i m e n t a l s t u d y of n a t u r a l c o n v e c t i o n h e a t t r a n s f e r i n a n o n - N e w t o n i a n f l u i d " , Canad. J . Chem. Eng. 44, 1966, 61-63.  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 t e m p e r a t u r e measurement i n a f r e e c o n v e c t i o n f i e l d " , Defense Documentat 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 M e t h o d s " , 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. S c . 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 , L o n d o n , 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 c o n v e c t i o n heat t r a n s f e r t o a f l u i d i n t h e r e g i o n o f 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 o f T e c h n o l o g y , 1 9 6 6 , a p p e n d i x 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 I I  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  4.7305  1.1518  -0.0001  0.0003  -0.0011  5.01  :  TABLE  Ha  UNIVERSAL FUNCTIONS FOR P r - * »  F* 2  ^2  *2  0.00  0.00  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  III  UNIVERSAL FUNCTIONS FOR P r R5 n  2  as  as  F  2  F  0  h  !*5  2  G  2  0.002  0.00008  1.5723  -1.2456  1.000  - 0 . 5 7 70  0.50  0.6026  0.9731  -0.8163  0.7216  -0.5220  1.00  1.0054  0.6585  -0.5162  0.4881  -0.4075  1.50  1.2773  0.4424  -0.3180  0.3148  - 0 . 2 881  2.00  1.4590  0.2935  -0.1922  0.1963  -0.1906  2.50  1.5788  0.1921  -0.1148  0.1196  -0.1206  3.00  1.6567  0.1243  -0.0679  0.0717  -0.0741  3.50  1.7069  0.0795  -0.0399  0.0426  -0.0447  4.00  1.7388  0.0504  -0.0233  0.0251  -0.D267  4.50  1.7590  0.0316  -0.0136  0.0148  -0.0158  5.00  1.7716  0.0196  -0.0079  0.0086  -0.0093  5.50  1.7793  0.0120  -0.0045  0.0050  -0.0054  6.00.  1.7840  0.0072  -0.0026  0.0029  -0.0032  ,  TABLE  Ilia  UNIVERSAL FUNCTIONS  ^l  F  l  FOR P r  F  0  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 t h r o u g h the " i n n e r " r e g i o n .  73 APPENDIX A SOME ASPECTS OF THE SCHLIEREN SYSTEM General Discussion The p r i n c i p l e s and o p e r a t i o n o f the s c h l i e r e n o p t i c a l s y s t e m 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 . f o r i n f o r m a t i o n are ( 1 6 ) ,  (17)  and ( 1 8 ) .  Three v e r y good s o u r c e s 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 c o n t a i n s the items w h i c h were thought to be i m p o r t a n t by the a u t h o r i n the d e s i g n o f h i s s y s t e m .  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 2 7 .  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 i v e n  by,  S = f/a  (A-l)  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 may be t a k e n t o be o n e - h a l f o f the s o u r c e h e i g h t .  (l.^) and " a "  Therefore,  increasing  t h e 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 o r d e c r e a s i n g the s o u r c e h e i g h t improves the s e n s i t i v i t y .  In t h i s c o n t e x t s e n s i t i v i t y i s a measure  of the o b s e r v a b l e c o n t r a s t of the d i s p l a c e d s o u r c e image ( c . f . edge of b l u e f r i n g e on t h e s c h l i e r e n photographs) background ( l a r g e y e l l o w  outside  w i t h r e s p e c t t o the  area).  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 t o t h e number o f c o l o u r f r i n g e s w h i c h appear i n t h e s c h l i e r e n image. it  i s , however,  In t h i s  respect,  the range of the s y s t e m w h i c h 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 t h e maximum d e f l e c t i o n , fe  v  = ka  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 t h e 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 a  Q  and  i s the w i d t h 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 e n g t h (L ) i s g i v e n by the e x p r e s s i o n s e  Y  v  = L  .  s  -|~ 9Y  n  —  (A-3)  where n i s the r e f r a c t i v e i n d e x of Substituting i n (2), k =  Ia  L  s  .  light.  the number of f r i n g e s i s g i v e n  1  n  by,  .  9Y  (A-4)  This expression i s s i m i l a r to ( A - l )  and d e f i n e s the  sensiti-  v i t y i n terms of the number of c o l o u r f r i n g e s t h a t 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 e q u a t i o n (A-4)). Depth o f F i e l d  Criteria  G e n e r a l r e q u i r e m e n t s 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 high s e n s i t i v i t y  and a s h o r t depth of f i e l d .  These are opposed t o  each o t h e r s i n c e a s h o r t depth of f i e l d can o n l y be o b t a i n e d w i t h an extended source.  If  the system i s r e q u i r e d t o be s e n s i t i v e i n one  d i r e c t i o n o n l y t h e n , o f c o u r s e , b o t h of the above requirements  are  met when the s o u r c e 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  required.  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 , t h e depth o f is  given by, AX =  2 c  < / Yf X  f  >  field  75 where £ i s the d i a m e t e r of the " c i r c l e of c o n f u s i o n " , 0 . 2 5 mm.  If  the image f o c u s e s s l i g h t l y  approximately  i n f r o n t of or b e h i n d the  image p l a n e , the c i r c l e of c o n f u s i o n i s the l a r g e s t d i s c of l i g h t  at  the image p l a n e w h i c h does n o t 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 a n g l e of the extended s o u r c e .  T h e r e f o r e , t o o b t a i n a s h o r t depth o f f i e l d ,  t h e source  slit  s h o u l d be l o n g , and the s c h l i e r e n f i e l d s h o u l d be l o c a t e d as c l o s e as p o s s i b l e to t h e f o c a l p o i n t o f the second m i r r o r . Lastly,  the l e n g t h o f the s l i t w h i c h may be used i s l i m i t e d by  the occurence o f o p t i c a l a b e r r a t i o n s In t h i s respect  i n an o f f - c e n t e r s c h l i e r e n s y s t e m .  a u n i - a x i a l system u s i n g l e n s e s r a t h e r than concave  mirrors is superior  to the one d e s c r i b e d h e r e .  remark t h a t an 8 i n c h l e n s system u s i n g f u l l y involve prohibitive  cost.  It  i s needless  to  c o r r e c t e d components would  Condenser  FIG. 26  ELEMENTS OF SCHLIEREN SYSTEM  Color  Filter Film 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 t h e boundary  l a y e r depth i s v e r y  o f t e n t h e 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 . boundary l a y e r d e p t h . v a r i e s  S i n c e the  o n l y v e r y s l i g h t l y w i t h the d i s t a n c e a l o n g  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  relationship.  The s i m i l a r i t y a n a l y s i s o f s e c t i o n 3 . 1 y i e l d s  the  following  relationship for 2-dimensional flow: 2/5 Y, = T| X  (  g B  A T  ref —  V  )  "  1 / 5  (B-l)  2  where X and Y are t h e d i m e n s i o n s a l o n g the p l a t e and p e r p e n d i c u l a r :  i t , respectively,  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 ^ ^  1 For a i r S = — , and P r = . 7 2 . •Leo  d a t a ( f i g u r e 6) y i e l d s  is  (T  w  -  to Too).  T — T At a p o i n t where TT — = .02 n u m e r i c a l ^  ref  the v a l u e n = 5 . 2 .  From e q u a t i o n (1)  G r , - n3  the f o l l o w i n g can thus be deduced,  |  fi  Gr  v  x  2 / 5  (B-2)  78 whence A  Gr. = 3 . 2 x 10 5  Furthermore,  (B-3)  u s i n g the v a l u e of the k i n e m a t i c v i s c o s i t y  the f o l l o w i n g boundary  AT  (12 i n c h p l a t e ) .  = 70°F,  l a y e r depths were c a l c u l a t e d :  = 50°F  ref  for  Y . = .795 i n s . 6 1  > AT  _ = 100°F ref  YL 1  o  (B-4)  = .625 i n s .  Approximate A n a l y s i s o f the Agreement o f 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 n a t u r e i n t h a t some s i m p l i f y i n g assumptions may be made.  The p a t h o f l i g h t  s c h l i e r e n f i e l d i s assumed t o be p a r a l l e l to the p l a t e s u r f a c e . a s s u m p t i o n ' 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 s c h l i e r e n path (L )  be s m a l l .  g  The r e f r a c t i v e  i n d e x of l i g h t  r e l a t e d to t h e 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 (e^)>  a  t  a p a r t i c u l a r distance  A and r e f e r e n c e s  (Y^)  from t h e s u r f a c e by (see  (n)  order through This the is  then  direction appendix  (16) , (17) , and (18)),  SY.I  Y,  "  L  •  ( B  "  5 )  S  The t o t a l d e f l e c t i o n  D e  e x p r e s s e d as a d i s p l a c e m e n t o f the  l i g h t beam a c r o s s a graded c o l o u r f i l t e r  focussed  by,  ka  e  Y  -  ~f  (B-6)  the  79  where a  Q  i s the w i d t h o f 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  o f the s c h l i e r e n l e n s e s .  -21  Now,  = 22-  9n '  dY  3Y  K  and the r e l a t i o n s h i p between the t e m p e r a t u r e and t h e  fK-7i J  refractive  index, - 1  T  T Too  =  nn - 1  "  ( B  8 )  and a l s o s u b s t i t u t i n g f o r the c o n s t a n t s o f 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  results. 2 k(Y)  = -  1 . 7 3 (^r)  S  ^  T  w  )  (12 i n . p l a t e ) .  (B-9)  T h i s 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 t e m p e r a t u r e d a t a from f i g u r e 6 at P r = . 7 2 .  The number o f 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 t h e 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 a p p r o x i m a t e l y 0 . 6 5 i n s . .  d e p t h was c o n s t a n t f o r a AT measured ( 1 2 0 ° F ) . Therefore,  This  ^ f r o m a p p r o x i m a t e l y 50°F t o the h i g h e s t  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 t o be e x p e c t e d .  t h e 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 t e m p e r a t u r e s  o n l y d i s p l a y a p p r o x i m a t e l y the f u l l boundary l a y e r d e p t h . 6 = . 6 2 5 , and AT T«. = 70°F) ,  ref  = 100°F, d i r e c t l y  Substituting  i n t o t h e e x p r e s s i o n f o r the Gr, (at o  80  Gr  5  =  A T  r e  v  f  f i 3  —  4 = 3 . 2 x 10 .  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).  (B-10)  81  

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