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Free convective heat transfer from a heated horizontal downward facing surface Wu, Erh-Rong 1969

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F R E E C O N V E C T I V E HEAT T R A N S F E R FROM A H E A T E D H O R I Z O N T A L DOWNWARD F A C I N G S U R F A C E  by ERH-RONG WU  A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS  FOR T H E D E G R E E OF  M A S T E R OF A P P L I E D S C I E N C E in the Department of Mechanical  Engineering  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e requi red standard .  T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A November, 1969  In presenting  this thesis in p a r t i a l  fulfilment of the  requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference  and study.  I further agree that permission for  extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his  representatives.  It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written  Department of  MechanictK I  Engineering  The University of B r i t i s h Columbia Vancouver 8, Canada Date  October  , /Ml  permission  ABSTRACT A study  of laminar  plates of f i n i t e width the o t h e r  with  free convection  about h o r i z o n t a l  one s i d e heated  insulated i s presented  i s o t h e r m a l l y and  in this thesis.  i n v e s t i g a t i o n forms part o f a c o n t i n u i n g  This  program, and i t s  t e c h n o l o g i c a l o r i g i n and s i g n i f i c a n c e a r e d i s c u s s e d  i n the  i n t r o d u c t i on. The  governing  partial differential  equations  i n g t h e c o n t i n u i t y , momentum a n d e n e r g y e q u a t i o n s numerically  through  successive-over range from  relaxation technique  and  Prandtl  t h e change o f both  numbers i s d i s c u s s e d  i s o - v o r t i c i t y and i s o t h e r m a l number alone  distribution  and heat  theoretical  analysis places  of the boundary c o n d i t i o n s of the f i n i t e s cheme.  number vorticity  Grashof  p l o t s o b t a i n e d . The e f f e c t energy  transfer rate i s discussed, streamlines  f o r the three  number  on t h e b a s i s o f t h e s t r e a m -  on t h e momentum f i e l d ,  comparison of the isotherms, obtained  of Prandtl  The v a r i a t i o n o f the flow,  temperature f i e l d s with  curves  f o r a Rayleigh  0.22 t o 5 0 0 , a t t h r e e v a l u e s  and  of Prandtl  are solved  a f i n i t e d i f f e r e n c e method using a  ( 0 . 7 2 , 5.0 a n d 1 0 . 0 ) .  line,  compris-  by  a  and c o r r e l a t i o n  d i f f e r e n t Prandtl  numbers.  e m p h a s i s on t h e s i n g u l a r  nature  s p e c i f i e d a n d on t h e i n f l u e n c e  s i z e o f the domain o f the f i n i t e  The  difference  Some r e s u l t s f o r an u p w a r d f a c i n g h o r i z o n t a l i s o thermally heated  p l a t e o f f i n i t e width were a l s o o b t a i n e d ,  and were c o m p a r e d t o d a t a f o r t h e downward f a c i n g c a s e . A s e m i - f o c u s s i n g S c h l i e r e n c o l o u r system  was  order to i n v e s t i g a t e experimentally the flow behaviour a h o r i z o n t a l plate with the heated The  experimental  surface  used on  facing-downward  r e s u l t s sought were e v i d e n c e o f the non-  boundary  l a y e r nature of the flow.  The e v i d e n c e o f non-  boundary  l a y e r f l o w was o b t a i n e d c o n c l u s i v e l y .  ACKNOWLEDGEMENTS The  author wishes  P r o f e s s o r Zeev throughout wishes  t o thank  gratitude to  Rotem f o r h i s i n v a l u a b l e a d v i c e and g u i d a n c e  a l l stages o f t h e program.  ing Department  Also, the author  the entire staff of the Mechanical of the University of British  their assistance. Hurren  t o e x p r e s s h i s deep  In p a r t i c u l a r  and Mr. J . Hoar,  thanks  Engineer-  Columbia f o r  a r e d u e t o M r . P.  C h i e f T e c h n i c i a n s and t o t h e i r  staff f o r their help during construction of the experimental apparatus . S u p p o r t f o r t h i s r e s e a r c h was p r o v i d e d b y t h e National Computing The  Research  C o u n c i l o f Canada,  G r a n t No. 6 7 - 2 7 7 2 .  t i m e was p r o v i d e d b y t h e U . B . C . C o m p u t i n g  author i s grateful  f o r this  support.  Center.  T A B L E OF  CONTENTS  Chapter  Page  ABSTRACT  i i  ACKNOWLEDGEMENTS  v  L I S T OF T A B L E S  viii  L I S T OF F I G U R E S  ix  NOMENCLATURE I II  III  INTRODUCTION  1  NUMERICAL  6  SOLUTION  2.1  Analytical  2.2  Finite Difference Approximation  2.3  Technique of Numerical Computation  Formulation  6 11 ....  19  R E S U L T S AND D I S C U S S I O N  23  3.1  S t r e a m l i n e s and I s o t h e r m a l s  23  3.2  E f f e c t s of the Prandtl  26  3.3  C a l c u l a t i o n o f the Rates o f Heat Transfer Comparison with P r e v i o u s Heat T r a n s f e r Results  29  3.5  E f f e c t o f P r a n d t l N u m b e r on t h e R a t e o f Heat T r a n s f e r  56  3.6  S t a b i l i t y of the Numerical Calculation  3.4  IV  xii  EXPERIMENTAL  INVESTIGATION  Number  28  . .  61 62  4.1  Experimental Apparatus  63  4.2  Results'  64  vii  Page V  CONCLUSIONS  72  REFERENCES  74  APPENDIX  76  L I S T OF T A B L E S Table la  lb  2  Page N u s s e l t Numbers f o r H o r i z o n t a l P l a t e s H e a t e d on One S i d e a t P r a n d t l N u m b e r s 0.72 a n d 5.0  58  N u s s e l t Numbers f o r H o r i z o n t a l P l a t e s H e a t e d on One S i d e a t P r a n d t l N u m b e r 10.0  59  N u s s e l t Numbers f o r H o r i z o n t a l H e a t e d on B o t h S i d e s  60  Plate  L I S T OF  FIGURES  Figure  Page  1  Coordinate  System  8  2  Grid System  3  Domain and Boundary C o n d i t i o n s Calculation  4  Grid System  5  Streamlines f o r Horizontal Plate Downward a t Gr=0.5; Pr=0.72  6  I s o t h e r m a l s f o r P l a t e F a c i n g Downward Gr=0.5; Pr=0.72  7  V o r t i c i t y Contours f o r Plate Facing a t G r = 0 . 5 ; P r = 0.72  8  S t r e a m l i n e s f o r P l a t e F a c i n g Upward a t Gr=0.5; Pr=0.72  33  9  I s o t h e r m a l s f o r P l a t e F a c i n g Upward a t Gr=0.5; Pr=0.72  34  10  V o r t i c i t y Contours f o r Plate Facing at Gr=0.5; Pr=0.72  35  11  Streamlines f o r Plate Facing at Gr=50; Pr=0.72  Downward  12  Isothermals f o r Plate Facing at Gr=50; Pr=0.72  Downward  13  V o r t i c i t y Contours a t G r = 5 0 ; P r = 0.72  14  S t r e a m l i n e s f o r P l a t e F a c i n g Downward Gr=200; Pr=0.72  at  15  I s o t h e r m a l s f o r P l a t e F a c i n g Downward Gr=200; Pr=0.72  at  16  V o r t i c i t y Contours f o r Plate Facing a t G r = 2 0 0 ; P r = 0.72  17  S t r e a m l i n e s f o r P l a t e F a c i n g Downward Gr=0.5; Pr=10  12 in  Numerical  14 16  Facing at Downward  Upward  f o r Plate Facing  30 31 32  36 37  Downward  Downward at  38 39 40 41 42  X  Figure  Page  18  Isothermals f o r Plate Facing at Gr=0.5; Pr=10  Downward  19  V o r t i c i t y Contours f o r Plate Downward a t Gr=0.5; Pr=10  Facing  20  Streamlines f o r Plate at Gr=10; Pr=10  Facing  Downward  21  Isothermals f o r Plate Facing at Gr=10; Pr=10  Downward  22  V o r t i c i t y Contours f o r Plate Downward a t Gr=10; Pr=10  Facing  23  Streamlines f o r Plate Facing at Gr=20; Pr=10  Downward  24  Isothermals f o r Plate at Gr=20; Pr=10  Facing  Downward  25  Streamlines f o r Plate at Gr=40; Pr=5  Facing  Downward  26  Isothermals f o r Plate Facing at Gr = 40; Pr= 5  Downward  27  Streamlines f o r Plate Facing at Gr=100; Pr=5  Downward  28  Isothermals f o r Plate Facing a t Gr = 1 0 0 ; Pr=5  Downward  29  Isothermals f o r Plate Facing at Gr=100; Pr=5  Upward  30  C o r r e l a t i o n between N u s s e l t G r a s h o f Number  31a  Source Bench o f S c h l i e r e n  System  66  31b  Output Bench o f S c h l i e r e n System  66  32a  M i r r o r Type S c h l i e r e n  67  32b  Lens Type S c h l i e r e n System  33a  21 i n c h e s S q u a r e P l a t e a n d M i r r o r Schlieren Apparatus  Number  43 44 45 46 47 48 49 50 51 52 53 54  and  System  57  67 Type 68  xi Figure  Page  33b  21 i n c h e s S q u a r e P l a t e  68  33c  12 i n c h e s S q u a r e P l a t e  69  34a  S c h l i e r e n P h o t o g r a p h U s i n g 12 i n c h P l a t e , T =68.5°F ; T =89.5° F o  34b  w  Gr=3.6xl(T S c h l i e r e n P h o t o g r a p h o f 12 i n c h T =72.5°F ; T =132.5°F oo  Test  Plate:  W  Gr=10xl0 Schlieren  70  8  34c  Too=72°F  P h o t o g r a p h o f 12 i n c h W  Schlieren  Schlieren  Too=72°F  70  8  P h o t o g r a p h o f 21 i n c h  Too=66.5°F  Gr=22.4xl0 35b  Plate:  ; T =144°F  Gr=12.1xl0 35a  69  ; T =89°F  Plate:  W  71  8  P h o t o g r a p h o f 21 i n c h  ; T =144°F  Gr = 6 4 . 8 x l 0  Plate:  W  8  71  NOMENCLATURE Acceleration of gravity Grashof Mesh  n u m b e r [=  (g3(T -TjL )/v ] 3  2  w  size  Counters Width  of the plate  Exponent N u s s e l t number  (-96/9y)  Static  pressure  Prandtl  n u m b e r (= v / a )  R a y l e i g h n u m b e r (= P r • G r ) Temperature Temperature  o f the heated  Temperature  o f t h e f l u i d f a r away f r o m  Dimensionless  surface the plate  v e l o c i t y components  Dimensional v e l o c i t y components i n t h e x and y directions respectively Dimensionless Dimensional Thermal  coordinates Cartesian  (= x / L , y / L )  coordinates  d i f f u s i v i t y of the fluid  Volumetric expansion c o e f f i c i e n t at constant p r e s s u r e ( f o r a i r = 1/T^) Dimensionless  vorticity  Dimensionless  temperature  [ ( T - T ) / ( T -T ) ]  xii i V  Kinematic v i s c o s i t y of the  p  Dens i t y  fluid  Dimensionless Stokes stream f u n c t i o n Laplacian operator in Cartesian coordinate system Subscripts x, y  P a r t i a l d e r i v a t i v e s with r e s p e c t to respecti vely  i, J  The i system  0,  A, B  t h  row a n d t h e j  t  h  column of the  C o n d i t i o n s a t a node o f the g r i d  x, y grid  system  R e f e r r i n g to p o i n t s f a r removed from p l a t e and f r o m t h e b o u n d a r i e s .  the  I. F o r many c a s e s necessity  INTRODUCTION  of technological  a r i s e s of having  to c a l c u l a t e rates o f heat ( o r  m a s s ) t r a n s f e r by f r e e c o n v e c t i o n plates  of f i n i t e extent.  electrochemical the  operating  under no-flow  i n v e s t i g a t i o n forms part o f a c o n t i n u i n g Engineering  For the f r e e - c o n v e c t i v e exact  equations  conditions, constants,  power r e a c t o r s , and so on.  Department of Mechanical  flow,  arise in the analysis  apparatus f o r measuring d i f f u s i o n  cooling of nuclear  sent  from one s i d e o f h o r i z o n t a l  Such cases  of hot f i l m anemometers  importance the  solutions  heat  ;  pre-  program i n the  at U.B.C, transfer in  o r even asymptotic  o f c o n t i n u i t y momentum  The  laminar  ones o f t h e  and energy a r e only  obtainable.  Therefore  large values  o f t h e G r a s h o f number a r e f o u n d t h r o u g h t h e  simplifying  most s o l u t i o n s  f o r the case  rarely  assumptions of boundary layer theory.  of very  These  b o u n d a r y l a y e r s o l u t i o n s may be r o u g h l y  g r o u p e d i n t o two  gories:  ( i i ) solutions  ( i ) s i m i l a r i t y s o l u t i o n s ; and  integrations  across  the boundary layer.  reduce the governing  partial  differential  system of simultaneous ordinary certain transformations formations  length  of variables.  plate's''.  occurs,  equations  differential  by  usually to  equations  These s i m i l a r i t y  a r e , however, l i m i t e d to cases  characteristic horizontal  The former  cate-  a through trans-  i n w h i c h no  c . f . that of semi-infinite  The l a t t e r method u s u a l l y assumes a c e r -  2  t a i n form o f the unknown f u n c t i o n s , i . e . the d i s t r i b u t i o n w i t h i n the boundary i n t e g r a t i o n of the  somewhat l i m i t e d and  of  On  l a y e r , and p r o c e e d s  a p p l i c a b l e due the  their computational  difficulties  number, boundary-1ayer  are  values  approximations  to the s i n g u l a r nature of the  are  i n t e g r a t i o n are regarded  solution.  The  as t h e two  p e r t u r b a t i o n m e t h o d , h o w e v e r , may  be  o f t e n be u n s u i t a b l e f o r t h e c a s e s  s i n g u l a r boundary title  case.  c o n d i t i o n s , such  Here the boundary  f a c e s a r e s i n g u l a r and  expansion  difficult  restricts a wider  undoubt-  i n the task of f i n d i n g the of asymptotic  range  inner-  expansions).  o f R a y l e i g h and G r a s h o f  using t h i s d i r e c t method, numerical  of values of the parameters  to  be  numbers. instability  Gr and  Pr.  More-  t o t h e l i m i t a t i o n on a v a i l a b l e s t o r a g e m e m o r y i n  e l e c t r o n i c computers to  present  u n f o r t u n a t e l y the e x t e n s i o n of the c a l c u l a t i o n to  range  o v e r , due  f o r the  method, i t i s n a t u r a l l y expected  a p p l i c a b l e to a wider However, even  Grashof  c o n d i t i o n s on t h e p l a t e s u r -  s o l u t i o n ( i n the sense  As t o t h e n u m e r i c a l  of  with  as t h o s e f o u n d  these s i n g u l a r i t i e s  edly a formidable d i f f i c u l t y  and  best t o o l s of  i f not i m p o s s i b l e to apply to the case of moderate n u m b e r a n d may  not  equations,  Both^methods of p e r t u r b a t i o n s (asymptotic expansions) numerical  the  are a l s o known, but t h e i r r o l e i s  the o t h e r hand, f o r small or moderate  the Grashof  to  equations.  Series expansions  great.  temperature  the numerical  the i n v e s t i g a t i o n of a f i n i t e  m e t h o d may  be  restricted  domain of l i m i t e d s i z e  or  3  to cases  w h i c h can  be m o d e l l e d a p p r o x i m a t e l y  f r o m an  i t e domain i n t o a f i n i t e  domain.  F u r t h e r m o r e , the  convergence of numerical  s o l u t i o n s of the governing  i s u s u a l l y f o u n d t o be r a t h e r s l o w .  Thus a  amount of computing time i s i n d i s p e n s a b l e . the  boundary layer approximation  w o u l d be p r e f e r a b l e governing  in looking  rate  of  equations  considerable As  a  consequence  method, whenever  applicable,  f o r the s o l u t i o n s of  the  equations. In t h i s t h e s i s , t h e a i m  of experimental  part  e s s e n t i a l l y q u a l i t a t i v e , namely, to i n v e s t i g a t e the pattern  infin-  of a i r from the underside  horizontal  p l a t e , and  large values  of Gr.  flow  o f an i s o t h e r m a l l y  heated  to examine whether a boundary  e x i s t s below a downward f a c i n g h e a t e d s u r f a c e The  existence  of a  boundary-1ayer about a heated or cooled  was  layer  at l e a s t f o r  free-convective horizontal  plate  had  2  been d i s c u s s e d  by S t e w a r t s o n :  due  t o an e r r o r o f  s i g n i n h i s d e r i v a t i o n he i n c o r r e c t l y c o n c l u d e d boundary l a y e r does indeed horizontal  f o r m on t h e s u r f a c e  algebraic  that  a  of a heated  p l a t e f a c i n g downward, or above such a  surface  3  which i s cooled.  Later,  i n 1965,  Stewartson's i n t e r p r e t a t i o n to a  Gill  contrary  namely that a boundary l a y e r forms only face or below a cooled  one.  The  et a l .  corrected  conclusion,  above a heated  existence  sur-  of a boundary  l a y e r above a h e a t e d h o r i z o n t a l p l a t e h a s b e e n e x p e r i m e n t a l l y v e r i f i e d by R o t e m 4 » 1 a n d by C l a a s s e n 5 u s i n g a s e r r n focussing colour-Schlieren system. Here i n order to  4  i n v e s t i g a t e the g e o m e t r i c a l l y r e v e r s e d of a boundary l a y e r below a heated The  same o p t i c a l s y s t e m  Schlieren heated  s u r f a c e was  a l r e a d y d e s c r i b e d was  colour photographs  flat plate.  s i t u a t i o n , the  absence  t o be used  tested.  to  take  o f t h e f l o w p a t t e r n under  the  S a t i s f a c t o r y r e s u l t s were o b t a i n e d  to  3  c o n f i r m the m o d i f i e d  conclusion  the non-boundary-1ayer nature p l a t e , even f o r l a r g e values  by G i l l  et al . , that i s  of the flow beneath of the Grashof  a  heated  number.  In t h e a n a l y t i c a l p a r t , f o l l o w i n g t h e r e a s o n i n g the p r e c e d i n g  paragraphs,  in the subsequent  which w i l l  sections, a successive-over  m e t h o d [ d e s c r i b e d by Y o u n g ^ ] was governing  be d i s c u s s e d  partial  differential  equations.  5.0  and  500.  O v e r a l l heat  o f P r a n d t l and numerical and  and  for a Rayleigh  Grashof  over  number,  number range  number are p r e s e n t e d  i n the form  and G r a s h o f  and  up  to  combinations  f i e l d s with v a r i a t i o n of the v a l u e  An e x p o n e n t i a l  values  of Prandtl  c h a n g e o f the momentum,  streamlines, i s o v o r t i c i t y curves  Nusselt  calcu-  numbers were a l s o computed, The  the  Numerical  transfer rates for various  integration.  temperature  Grashof  10.0,  detail  relaxation  employed to s o l v e  l a t i o n s were c a r r i e d out at t h r e e v a l u e s 0.72,  in  of  of graphs  through vorticity of  the  showing  isothermals.  r e l a t i o n s h i p between the o v e r a l l  n u m b e r s was  obtained  even at moderate  o f G r , by u s i n g a l e a s t m e a n s q u a r e s  the p o i n t s c a l c u l a t e d .  fitting  method  5  The effect of Prandtl number on the convective  flow  and the heat-transfer rate were examined by comparing the streamline, v o r t i c i t y and isothermal graphs as well as the calculated Nusselt numbers to those obtained for a "reference case" of given Rayleigh number. number was  The Prandtl  found to have some influence on the s t a b i l i t y of  one numerical  calculations as well.  As far as could be ascertained, no published results of either the flow, temperature and v o r t i c i t y or for the rate of heat transfer  fields  for the t i t l e case had  been available previous to this investigation.  II. 2.1  Analytical  NUMERICAL SOLUTION  Formulation  In a t w o - d i m e n s i o n a l  f i n i t e or i n f i n i t e domain, the  g o v e r n i n g e q u a t i o n s f o r s t e a d y l a m i n a r f r e e c o n v e c t i o n may (*)  be w r i t t e n as f o l l o w s 3u _ 3u u — + v — = 3x 3y v  . 1 3P p  1 3P  3V  3V  u — + v — = 3x 3y Continuity 3u —  3x  3x  p  3y  9  + vv u  (* * \  Or  ,  _ _ + g3(T-Tj + W^v ?  .  (2)  equations: 3v + —  =  0  .  (3)  3y  Energy equation: 3T 3T 2~ — + v — u —3x + v —3y = a V T  (4)  where the C a r t e s i a n c o o r d i n a t e system i s d e f i n e d i n F i g u r e 1  (*)  ' C e r t a i n p r e c o n d i t i o n s h a v e t o be f u l f i l l e d f o r t h e s e e q u a t i o n s t o be v a l i d ; t h e s e a r e d i s c u s s e d i n r e f e r ence [ 1 ] . v  (**)  'A. l i s t o f s y m b o l s i s i n c l u d e d a t t h e b e g i n n i n g o f thesis. K  this  It should all  be n o t e d  that in d e r i v i n g these  t h e f l u i d p r o p e r t i e s a r e a s s u m e d t o be  except  equations  constants  the d e n s i t y v a r i a t i o n i n v o l v e d i n the buoyancy  In a d d i t i o n , t h e h e a t - d i s s i p a t i o n t e r m equation  is neglected, while  g r a d i e n t terms in equations  e l i m i n a t e d by c r o s s d i f f e r e n t i a t i o n dimensional  variables will  independent  coordinates:  energy  the "decompression-work"  by i m p l i c a t i o n ( p -  assumed v a n i s h i n g l y small pressure  i n the  be  x x = - , L  is  constant).  (l)-(2) will and  term.  now  the f o l l o w i n g  The be non-  introduced,  y y = - , L  (5)  vorticity: L ?  =  2  -  v  stream  9v  9u  9x  9y  (  )  (6)  .  function  3iji  —  9y  L  =  v  9^  u,  —  9x  L  = - v,  (7)  v  temperature: T - T T  00  (8)  - T  w is a suitably defined reference 00  T  w  r  temperature  rendering  0  Face-Down  Y.V Insulated •> x , u  J Heated g Face-Up  Y,V Heated •» x , u  ] Insulated  Figure 1. C o d r d i n a t e  System  9  at most of order u n i t y , while  i s the . temperature  l o c a t i o n f a r removed from the boundaries. equations  c a n now  c  xx  ^xx -^e  +  The  at a  governing  be r e w r i t t e n as f o l l o w s ,  ^t y y  r  =  ^ y tx  (ii; c  V 4  -tf>  x^y  y  - Gr  )  6  x  v  (9)  '  (10)  yy  y x + ujx e y  =  K—  Pr  v  (e  xx  +e  ),  yy  (11)  where s u b s c r i p t s x and y r e f e r to t h e r e s p e c t i v e p a r t i a l d e r i v a t i v e s , and Gr, Pr a r e d i m e n s i o n l e s s p a r a m e t e r s Grashof  and P r a n d t l number r e s p e c t i v e l y ) . One  may  h a s now  t o c o n s i d e r two  a r i s e i n p r a c t i c e f o r the t i t l e  (i) a s t r i p - p l a t e with i t s heated s t r i p - p l a t e with i t s heated c o n d i t i o n s f o r t h e two  d i f f e r e n t cases  which  problem  thesis.  of this  face upward;  f a c e downward.  c a s e s , the heated  (ii) a  The  boundary  p l a t e f a c i n g down-  w a r d a n d f a c i n g u p w a r d as f o l l o w s :  Heated -\  (the  s u r f a c e f a c i n g downward  < x < \;  y = 0(+):  (or a l t e r n a t i v e l y y = O(-):  = 0,  ^  ^  = 0,  \p  9  = 1,  4J  9  y  i>  = ^  = 0  y  constant), X  =  0  =  ty  =  constant)  y  10 = 0  x , y -»- ±°° ;  ? = 0 tfj =  (13)  =0  Th  ^y  (ifj = c o n s t a n t ) .  '  Heated s u r f a c e f a c i n g upward  ?  i  x  <  y  °( )  =  +  :  (I|J = y =  O(-)  0 U  \b ^  >  =  constant) \b  (ty = 0,  x , y -*• ±»j  constant)  c  =  0  \b  =  \b  ^x  0  =  ^y  X  (14)  ^y  S  =0  (ij> = c o n s t a n t ) A n a l y t i c a l l y the problem  c o n s i d e r e d h e r e , i n an  solution  o r two  matching  '  infinite  d o m a i n , s h o u l d be t r e a t e d by s e e k i n g a u n i f o r m l y asymptotic  (15)  asymptotic  valid solutions in  t h e f o r m o f s o - c a l l e d i n n e r and o u t e r e x p a n s i o n s . n a t e l y , o w i n g t o t h e s i n g u l a r b o u n d a r y c o n d i t i o n s on  Unfortuthe  p l a t e s u r f a c e s , t h e t a s k o f f i n d i n g an i n n e r s o l u t i o n i s b o u n d t o be e x t r e m e l y  difficult.  Furthermore,  there is a  s i n g u 1 a r i t y . i n t h e m o m e n t u m f i e l d a t i n f i n i t y as p o i n t e d o u t g by Mahony i n t h a t t h e v e l o c i t y c o m p o n e n t i n t h e d i r e c t i o n  o p p o s i t e to the g r a v i t y f i e l d  i n c r e a s e s w i t h o u t bounds as  t h e d i s t a n c e away f r o m a l i n e s o u r c e o f h e a t i n c r e a s e s . Due to the d i f f i c u l t i e s j u s t mentioned, finite  difference  a n u m e r i c a l method by  a p p r o x i m a t i o n t o be undertaken over a ;  f i n i t e domain i s thought to be the most a p p r o p r i a t e approach to the p r e s e n t problem.  I t s h o u l d be emphasized  in view of the c h a r a c t e r i s t i c s of a s i n g u l a r problem, a s i n g u l a r i t y impossible to obtain  at i n f i n i t y  here t h a t  perturbation  in this case, i t is  a uniformly valid asymptotic  by a s i n g l e e x p a n s i o n s t a r t i n g  solution  at the o u t e r boundary of the  f i n i t e - d i f f e r e n c e scheme used i n the n u m e r i c a l  calculation.  T h e r e f o r e , i n this study numerical c a l c u l a t i o n s are  carried  out on the b a s i s of a f i n i t e - d i f f e r e n c e scheme i n a f i n i t e domain which  i s thought to be s u f f i c i e n t l y  as a p h y s i c a l sequences  2.2  situation  in reality  is concerned.  of this s i m p l i f i c a t i o n w i l l  Finite-Difference  l a r g e asf a r The  be d i s c u s s e d l a t e r  conv  .  Approximation  W i t h t h e s q u a r e g r i d s y s t e m as s h o w n i n F i g u r e 2 we may  write first  central-difference approximations for  b o t h f i r s t o r d e r and s e c o n d o r d e r d e r i v a t i v e s o f t h e function with respect to d i s t a n c e g as f o l l o w s , c . f . (  ,  v  'See Appendix.  i n t h e x and y d i r e c t i o n s ,  ,. . . • I + I . J - V I . J x i,j  2  stream  h  s ystem Figure 2 . G r i d  13  (16) 2h  xxi ,j  ^i + l ,j  +  V l ,j  "  ,j  (17)  yy i  >J  where the s u b s c r i p t s  k, j  column of the l a t t i c e  denote  the i ^  respectively,  row and the j * *  and h is  the mesh  1  size.  S i m i l a r approximations can be w r i t t e n f o r the v o r t i c i t y c and temperature 9. expressions  S u b s t i t u t i o n of these  i n t o equations  ( 9 ) to  1,3 + 1  + i 8  Gr h(e  . - e. , .) i-2,j  ( 1 1 ) leads  ^ i j - l ^ 3  finite-difference to  +  (18)  14  6.25  1  o O  1  e  I  1 la=o Figure 3  y-  =0  D o m a i n and b o u n d a r y c o n d i t i o n s f o r calculations  numerical  y,  y,-0 =0  T  The  i,j+1  extent  for these 3.  I t can  1,3-1  o f the g r i d s y s t e m and  4> a n d  On  may  are shown i n  a c c o u n t of the r i g h t  b o t h r i g h t - h a l f and  Figure  l e f t - h a l f planes  half-plane the values  s i g n w h e r e a s c has  (the y a x i s ) , are s p e c i f i e d  a l g e b r a i c a l s i g n r e l a t i o n s h i p s o f if;, order  the  C o n s e q u e n t l y , the boundary c o n d i t i o n s  axis of symmetry  higher  take  8 c a r r y t h e same a l g e b r a i c a l  to these  conditions  be s e e n t h a t i n v i e w o f t h e s y m m e t r y o f  different sign. the  the boundary  f i n i t e - d i f f e r e n c e equations  w h o l e f i e l d , we only.  (19)  1,1  boundary conditions  c,  on t h e u p p e r a n d  of  a along  according 8. lower  The sur-  faces  of  the p l a t e  are o b t a i n e d  Referring at point  to  Figure  n  *  —  so t h a t we  manner.  the stream  function  get,  X  »  A h  following  4 we expand  A i n the y - d i r e c t i o n  —  i n the  1  J+1  q  -J  h B  •  i-1  i+1  i  Figure  h A  On t h e  3^  1! 9y  0  h  4 d ty  41  3y  surface  of  x  = o, *  y  Grid  System  h  3 ty  h  3 \\>  2!  3y'  31  3y'  (21)  4~  the p l a t e we  ty = c o n s t a n t ,  *  4  say  = o,  have,  ty  = 0 ,  *  =  x x  0 .  }  (22)  17 Therefore  equation  (21) becomes  2 h/  2 3*>  h  21  3/  3!  3  3 ^  h  Sy "  4!  3  3  Inserting boundary conditions ( 1 0 ) we  ^xx^o  3%  4  (23)  3y  (22) i n t o  equations  (9)  and  obtain,  +  (24)  ^yy^o  y  yy  D i f f e r e n t i a t i n g now get, using  (25)  o equation  (10) with  respect  to y,  we  (22) 3 ^  3C  3  (26)  T  9y Substitute equation the  conditions  ( 1 0 ) i n t o (9).  (22), i . e . ^  =0 A  Then t a k i n g i n t o  account  on t h e p l a t e s u r f a c e ,  we  A  o b t a i n, 3 i>  i7  Gr  36 — 3X  (27)  18 S u b s t i t u t i o n o f e q u a t i o n s ( 2 5 ) , (26) and (27) i n t o  equation  (23) l e a d s t o  -  ?  o  +  —  U  A-?o  i-r  }  86  (28)  — Gr — 24 3x  and 3 ~2  h  h  fc C  6  M  A  +  J  A  —  8  u  36 Gr — 3x  H e n c e , f o r t h e h e a t e d p l a t e f a c i n g d o w n w a r d , we  (29)  have  approximately  3  h 2  h^  vrn  + —  C ) A  6  rt  8  rt  Gr  30 — 3x  (y(+))  (29a)  M->>  < >  on t h e u p p e r f a c e , a n d  3  *  ? 0  ^  h  ;<v  2  —  ?  B>  29B  on t h e l o w e r f a c e . F o r t h e h e a t e d p l a t e f a c i n g u p w a r d on t h e o t h e r h a n d  C  0  «  3 — h  ' h ,(* - — 6 A  2  C ) A  (y(+))  (30a)  19 on t h e u p p e r  face, while  (*n  on t h e  *n)  Gr  +  3G 9x  —  (30b)  (y(-))  underside.  P r o v i d e d G r i s n o t t o o l a r g e ( G r <_ 2 0 0 )  and t h e mesh  width  (*)  h i s s m a l l t h e l a s t t e r m c a n be n e g l e c t e d o f Gr w e r e f i x e d will  beforehand.  e n a b l e us t o d e t e r m i n e  approximation. numerical  As m e n t i o n e d  v  The  Only a completed  values  calculation  the p o i n t of breakdown of previously a uniformly  the  valid  s o l u t i o n i n an i n f i n i t e d o m a i n i s i m p o s s i b l e t o  o b t a i n and hence  the numerical  l i m i t e d to a f i n i t e domain. boundary  computations  A c c o r d i n g l y we  c o n d i t i o n s at i n f i n i t y  this numerical  c o n c e r n i n g the boundary  Technique  the of  is considered.  This idea  c o n d i t i o n s a t i n f i n i t y has been  be g i v e n  of Numerical  Equations  apply  be  to p r a c t i c a l a p p l i c a t i o n  c u s s e d by S u r i a n o a n d Y a n g ^ a n d by P a n t o n ^ . discussion will  have to  to the outer boundary  scheme w i t h a view  w h e r e an a c t u a l p h y s i c a l p r o b l e m  2.3  .  dis-  A detailed  later. Computation  ( 1 8 ) t o ( 2 0 ) a r e s o l v e d by a m e t h o d  of  i t e r a t i o n w i t h t h e a i d o f an o v e r - r e l a x a t i o n t e c h n i q u e . (*)  ' I t s i m p o r t a n c e w i l l be c o n f i n e d t o t h e of the l a t e r a l edges of a heated s t r i p . K  vicinity  Two  d i f f e r e n t computers  were used i n the c o u r s e o f  i n v e s t i g a t i o n to c a r r y out the computations: a n d an IBM 3 6 0 / 6 7 . numerical  an IBM  0 <_ x < 6 . 2 5 ,  the  following:  ( i ) A g r i d s y s t e m i s s e t up w i t h a r a n g e - 6 . 2 5 ,  7044  The d e t a i l s o f the p r o c e d u r e o f  computation are d e s c r i b e d i n the  <_ 6.25  this  <_ y  and a mesh s i z e h = 0 . 1 2 5 ,  covering the r i g h t - h a l f plane of the f i e l d  con-  s i d e r e d , s e e F i g u r e 2. ( i i ) A s e t o f i n i t i a l v a l u e s o f ty, 0, a n d c ' i s p r e s c r i b e d a t e v e r y i n t e r i o r g r i d p o i n t , as w e l l boundary  c o n d i t i o n s on t h e a x i s o f s y m m e t r y , on  the  outer boundary  the  lower s u r f a c e of the plate  the  heated face upward).  o f t h e g r i d s y s t e m , and  differ-  i n v o l v e d i n the  c o n d i t i o n s a r e c a r r i e d o u t by  5-point polynomial interpolation ( i i i ) Two  on  ( f o r the case of  The n u m e r i c a l  entiations for a l l derivatives boundary  as  computing  formulae,  l o o p s a r e s e t up i n t h e c o m p u t i n g  program;  the  major loop p r e s c r i b e s the range of j , while  the  m i n o r n e s t e d one p r e s c r i b e s t h e r a n g e o f i .  H e r e i , j r e p r e s e n t , as d e n o t e d b e f o r e , t h e i c o l u m n and t h e j Computations  o f new  row o f t h e g r i d  system.  v a l u e s o f ty, 0, c  at the  g r i d p o i n t s s t a r t from the lower l e f t hand c o r n e r o f t h e half  side  domain, proceeding to the  r i g h t hand c o r n e r a c c o r d i n g to the l o o p s .  upper  (iv)  Once t h e c a l c u l a t i o n reaches the p l a t e l i e s ,  the lower  t h e r o w on w h i c h  surface boundary  d i t i o n s a r e r e p l a c e d by t h e u p p e r boundary c o n d i t i o n s : analytical  surface  this i s the equivalent of  continuation i n numerical  computation  i s then  are completed.  con-  continued  This concludes  terms.  until  The  the loops  one i t e r a t i o n  over the f i e l d , ( v ) W h e n e v e r a new v a l u e o f a n y d e p e n d e n t v a r i a b l e at  a g r i d point i s c a l c u l a t e d , the computer i s  instructed to evaluate two  successive  the end o f each  the d i f f e r e n c e between  iterations (the residue).  At  i t e r a t i o n , the computer i s then  i n s t r u c t e d t o f i n d o u t t h e maximum r e s i d u e o f every (vi)  d e p e n d e n t v a r i a b l e , ( i p , 6, c ) .  By c h e c k i n g  t h e maximum r e l a t i v e e r r o r ( t h e r a t i o  o f maximum r e s i d u e t o t h e maximum v a l u e same i t e r a t i o n ) , e x p e r i m e n t a l the r a t e o f convergence  (vii)  to  determine  be  stopped.  I f the c a l c u l a t i o n s converge the numerical  tape  and a r e then  operating isothermals  investigations of  may b e u n d e r t a k e n  a t what stage  then  i n the  the computations  by t h i s  p l o t t e d by a s m a l l  " o f f l i n e " , i n the form  should  criterion,  r e s u l t s a r e s t o r e d on  and c o n s t a n t  so as  CDC  magnetic computer  of streamlines,  vorticity  lines.  22 In t h e c o u r s e been found 200,  of the numerical  t h a t when t h e v a l u e o f t h e R a y l e i g h number  the rate of convergence  becomes v e r y slow.  try to speed-up the rate of convergence, f a c t o r s were used. range  c a l c u l a t i o n s i t has  In o r d e r t o  several relaxation  F o r s m a l l e r R a y l e i g h numbers i n t h e  o f 0.3 - 5 0 , t h e o p t i m u m r e l a x a t i o n f a c t o r s w e r e  t o be b e t w e e n 1.0 a n d 0.76 f o r e q u a t i o n s 0.3 - 0.1  exceeds  f o r equation  found  (19) and ( 2 0 ) , and  (18);'for larger Rayleigh  numbers  (>200), the optimum r e l a x a t i o n f a c t o r s l i e i n the range 0.48 - 0.26 f o r e q u a t i o n s 0.001  f o r equation  r e q u i r e d was a b o u t and  ( 1 9 ) a n d ( 2 0 ) a n d 0.01  of  a r e as low as  ( 1 8 ) . The number o f i t e r a t i o n s t y p i c a l l y 300 f o r s m a l l t o m o d e r a t e R a y l e i g h  between 2000 and 3000 f o r l a r g e r R a y l e i g h  numbers,  numbers.  Moreover, i t i s of i n t e r e s t to point out that the v a l u e o f t h e P r a n d t l n u m b e r P r h a s s o m e i n f l u e n c e on t h e stability number. Grashof  of the numerical A t P r = 0.72  number  s o l u t i o n f o r a given  Rayleigh  ( i . e . f o r a i r ) , as t h e v a l u e o f t h e  i n c r e a s e s to 500, the e r r o r between s u c c e s s i v e  i t e r a t i o n s s t a r t s t o o s c i l l a t e , w h e r e a s f o r P r = 1 0 , G r = 40 (i.e.  Ra = 4 0 0 ) t h e o s c i l l a t i o n  appeared.  of residues  had not y e t  III. Numerical  R E S U L T S AND  DISCUSSION  c a l c u l a t i o n s have been c a r r i e d - o u t f o r  t h r e e v a l u e s o f the P r a n d t l number, 0.72, R a y l e i g h n u m b e r s up t o 5 0 0 .  The  numerical  p l o t t e d - o u t by m e a n s o f a c o n t o u r s e r i e s of graphs isovorticity In e a c h  5.0  and  10.0,  results  were  computer program.  contours  are presented  i n F i g u r e s (5) t o  the numerical  d a t a on t h e  ding to the c u r v e s , c o u n t i n g from  y = 0.0, The  (x >_ 0) d e n o t e d  b a s i s of the graphs  heat  correspon-  to the  by a s h o r t d o u b l e  line  outer  Streamlines  number w i l l  now  at  segment.  v a r i a t i o n o f t h e momentum and t h e t e m p e r a t u r e  As  (29).  r i g h t - h a n d h a l f of the p l a t e i s l o c a t e d  the change of Grashof  3.1  the innermost  and  right-  hand s i d e r e p r e s e n t the magnitude of the f u n c t i o n  The  A  c o n s i s t i n g of s t r e a m l i n e s , i s o t h e r m a l s  of these graphs,  contour.  and  fields  be d i s c u s s e d , on  the  obtained.  and  Isothermals  i s w e l l known, f o r v e r y s m a l l R a y l e i g h  number  t r a n s f e r i s e s s e n t i a l l y d o m i n a t e d by c o n d u c t i o n .  the energy Laplace's  e q u a t i o n may  then approximately  s h o u l d be s y m m e t r i c a l  C a r t e s i a n axes of c o o r d i n a t e s . Pr = 0.72,  illustrates  to  Rayleigh  with r e s p e c t to  the  F i g u r e 6 f o r G r = 0.5  this tendency  As  be r e d u c e d  e q u a t i o n , the i s o t h e r m a l s i n the small  number range  with  to symmetry of  and the  isothermals. upper  The  constant  f a c e of t h e p l a t e s h i f t  parison with  t h o s e below t h e  be e x p l a i n e d  by t h e f a c t  number a buoyancy both  the f l o w  vicinity  force  and t h e  surface  since  (the  the upper  temperature closely:  upper with  surface.  Rayleigh  influence  in the  near  surface  of  coming from t h e  the p l a t e ) .  of the p l a t e  is  In  for  addition  insulated,  computation.  face.  All  condition Figure  this  of zero  is  the  bunch t o g e t h e r  In e f f e c t  evidently  temperature  a  and P r a n d t l  m a g n i t u d e of t h e s t r e a m f u n c t i o n  numbers.  indicates  that  very of  correct to  the  consistent gradient  5 shows t h e c o r r e s p o n d i n g  a t t h e same G r a s h o f  are  heat-  d e t a i l s w o u l d n e c e s s i t a t e the usage  grid  the b o u n d a r y  on  i s o t h e r m a l s above the p l a t e  n e a r the u p p e r s u r f a c e  fine  com-  The s h i f t may  w o u l d have the i s o t h e r m a l s t e r m i n a t e n o r m a l l y  the u p p e r  are  fields  the  upward i n  some s m a l l  of the f l o w  insulated surface.  lines  lower  above  t h a t even a t such a low  The  surface  the f i n e r  an e x t r e m e l y result  lower  lines  a little  temperature  of t h e p l a t e .  curves  only  has s t i l l  i n t h e downstream r e g i o n ing  temperature  on  stream-  The  small  velocities  smal1. In v i e w  horizontal  plate  upward n u m e r i c a l latter  case.  seen t h a t  the  of  a c o m p a r i s o n of t h e two c a s e s o f a h e a t e d  f a c i n g downward and t h a t  c a l c u l a t i o n s were a l s o c a r r i e d  From F i g u r e  9 (plate  temperature  distribution  almost i d e n t i c a l  of a p l a t e  to t h a t  for  out f o r  f a c i n g upward) far  facing  it  the  can be  from the p l a t e  the case of the heated  side  is  25 f a c i n g d o w n w a r d , p r o v i d e d Gr i s s m a l l . comparing  F i g u r e s 5 and 7 w i t h  Furthermore,  F i g u r e s 8 and  l i n e a n d v o r t i c i t y d i s t r i b u t i o n s o f t h e two to  be n e a r l y i d e n t i c a l  illustrates small  to each  F o r P r = 5.0 number range, resemble  10 t h e cases  o t h e r as w e l l .  and  This  found again  this  insignificant. Pr = 10.0,  i n the small  t h e s e t s o f i s o t h e r m a l s and  those  stream-  are  the f a c t t h a t the c o n v e c t i o n e f f e c t at  R a y l e i g h number i s  by  o f Pr = 0.72.  Rayleigh  streamlines  As t h e R a y l e i g h n u m b e r i s i n -  c r e a s e d , the c o n v e c t i o n e f f e c t becomes p r o g r e s s i v e l y more important. thermals  F r o m F i g u r e s 18 t o 28 i t i s s e e n become a s y m m e t r i c a l  corresponding  about  values of stream  c e n t e r of the main v o r t e x i s found  circulation.  Also,  here  15 f o r G r = 200  as t y p i c a l  and  Pr = 0.72,  then  R a y l e i g h number (about immediately  b e l o w and  gether, resembling  150),  temperature  the s t r e a m l i n e s  above the p l a t e bundle  a thermal  assumed t h a t t h i s r i s i n g flow f i e l d .  are  examples, in order to d i s p l a y the field  c o m p a r e d t o t h a t i n t h e low R a y l e i g h number r a n g e .  plume or a j e t .  column of hot f l u i d  At  (Figure  this 14)  closely toI t may  be  dominates  the  At f i r s t , t h i s column moves upward, i t then  t u r n s t o r i g h t and due  the  t o s h i f t upward and  m a r k e d d i f f e r e n c e s o f f l o w p a t t e r n and  and  the  outwards. F i g u r e s 14 a n d  chosen  t h e (x) - a x i s and  iso-  function increase, indicating  the i n c r e a s e i n the r a t e of f l u i d  slightly  t h a t the  left  i n the v i c i n i t y of the top  boundary,  t o t h e s u d d e n c o o l i n g f l o w s down r a p i d l y a l o n g  the  26 outer lateral  boundaries.  to  the r i s i n g flow.  feed again  a vortex, a little  Finally, i t is entrained  the center o f which  This c i r c u l a t i o n generates (as seen  i n F i g u r e 14) s h i f t s  upward above the p l a t e and toward  side boundary.  This shift of vortex  d i s t o r t i o n of the isothermals  inward  the right-hand  center a l s o causes  the  i n the f a r r e g i o n from the  p l a t e , F i g u r e 15. F i g u r e 15 s h o w s t h e t e m p e r a t u r e above the p l a t e are almost and to  f a r from  vertical.  I t c a n be o b s e r v e d  flow.  i n F i g u r e 27 t h a t a s t h e R a y l e i g h  500 ( G r = 100, P r = 5 ) , t h e s t r e a m l i n e s  increase of jet-flow velocity.  (in  i s seen  t o move f u r t h e r a w a y f r o m with  the center the plate  the l o c a t i o n of the  c e n t e r i n F i g u r e 25. F i g u r e 28 s h o w s t h e t e m p e r a t u r e  values  r e v e a l i n g an  A t t h e same t i m e  t h e x - d i r e c t i o n ) by c o m p a r i s o n  vortex  stretch out,  a r e d i s t o r t e d due  a b o v e t h e p l a t e move e v e n c l o s e r t o g e t h e r ,  of the vortex  isothermals  They then  the plate the isothermals  t h e e f f e c t o f t h e downward  number r e a c h e s  field;  of the parameters.  field  The i s o t h e r m a l s  r i s e up f u r t h e r a b o v e t h e p l a t e .  f o r t h e same  near  the plate  This i s obviously the  result of the i n c r e a s i n g l y fast j e t flow.  3.2  E f f e c t of the Prandtl  Number  Of p a r t i c u l a r i m p o r t a n c e numerical  among t h e r e s u l t s o f t h e s e  c a l c u l a t i o n s i s the e f f e c t of the Prandtl  number  on t h e v e l o c i t y t e m p e r a t u r e the heat  transfer rate.  and  vorticity fields  No p r e v i o u s  on t h i s , e v e n f o r t h e p l a t e h e a t e d will  be n e c e s s a r y  to take  and  temperature  isothermals will  on b o t h  n u m b e r 200 and  was  chosen  as s u c h  o f 0.72,  o r i n o t h e r w o r d s when t h e c o n v e c t i o n  15,  downwards.  F i g u r e s 23 a n d  24,  and  t h a t the l o c a t i o n of the v o r t e x is highest in e l e v a t i o n . found  f o r Gr = 40,  a  numbers.  "standard"  5.0  and  becomes more  important,  and  then  An e x a m i n a t i o n  towards  of  F i g u r e s 26 a n d  27  c e n t e r f o r Gr = 20,  Pr =  t h a t f o r Gr = 200,  Pr =  Furthermore,  the squeezing  gether  of s t r e a m l i n e s  among t h e t h r e e c a s e s . and  l e a s t f o r Gr = 20,  ponding of f l u i d  stream  Pr = 0.72,  to-  intense  I t i s l e s s so f o r Gr = 40, The  0.72  lowest.  above the p l a t e i s the most  P r = 10.  10  vortex  i s the  and  Figures reveals  ( R a y l e i g h number e v e n s m a l l e r t h a t 200) a t G r = 200  10.0  increased,  Next i n o r d e r comes the  P r = 5, a n d  and  number i s  the c e n t e r o f v o r t i c i t y moves upward f i r s t  14 a n d  flow-patterns  As a l r e a d y m e n t i o n e d i n  s e c t i o n , when t h e R a y l e i g h  a little  It  n u m b e r as a  the d i s t r i b u t i o n s of s t r e a m l i n e s  be c o m p a r e d t o t h a t r e s u l t .  t h e r i g h t and  is available  its sides.  of the v a r i a t i o n s of  f o r P r a n d t l number v a l u e s  the preceding  upon  d i s t r i b u t i o n s for d i f f e r e n t Prandtl  Here a R a y l e i g h of comparison;  information  a certain Rayleigh  c r i t e r i o n f o r comparison  and  Pr = 5  magnitude of the  corres-  f u n c t i o n a l s o i n d i c a t e s that the c i r c u l a t i o n  f o r Gr = 200,  t i o n a l l y , a comparison  P r = 0.72  i s the most r a p i d .  of the temperature  fields  of  AddiFigures  1 5 , 26 a n d 28 a l s o r e v e a l s t h a t t h e s m a l l e r t h e P r a n d t l number  becomes, the f u r t h e r the isotherms  the more p r o n o u n c e d t h e downward-flow  3.3  C a l c u l a t i o n s of the Rates  of Heat  A v e r a g e N u s s e l t numbers based an a v e r a g e  heat  spread  o u t , and  becomes.  Transfer on t h e p l a t e w i d t h  t r a n s f e r c o e f f i c i e n t were c a l c u l a t e d  and  from  t h e d e f i n i t i o n o f Nu,  (31) 1/2 for the heated  p l a t e s u r f a c e f a c i n g downward, and  (32)  f o r the s u r f a c e f a c i n g upward. temperature polynomial  gradients  In t h e c o m p u t a t i o n  the  i n the i n t e g r a l were e v a l u a t e d  i n t e r p o l a t i o n formula  f o r numerical  by a  differen-  t i a t i o n o f computed d a t a , and a l l i n t e > g r a t i o n s were c a r r i e d o u t by S i m p s o n ' s r u l e . Results 10.0 a r e g i v e n  f o r t h r e e P r a n d t l n u m b e r s , 0 . 7 2 , 5.0 i n Table (1).  As t h e m a i n e m p h a s i s o f t h e p r e s e n t downward f a c i n g heated numbers  and  study  p l a t e c a s e , o n l y a few  f o r t h e c a s e o f an u p w a r d f a c i n g h e a t e d  i s on t h e  Nusselt surface  were  c a l c u l a t e d , f o r comparison  purposes.  In t h e  R a y l e i g h number r a n g e , the N u s s e l t numbers-for c l o s e to each o t h e r . situation  smaller both cases  T h i s i s c o n s i s t e n t with the p h y s i c a l  i n t h a t R a y l e i g h number r a n g e :  the  dominant  mechanism of heat t r a n s f e r i s pure c o n d u c t i o n , which course independent  of p l a t e o r i e n t a t i o n .  temperature  f o r l a r g e r R a y l e i g h n u m b e r s i t may  graphs  s e e n by c o m p a r i n g  are  i s of  R e f e r r i n g to the  F i g u r e s 28 a n d 29 t h a t t h e g a p s  be  between  the a d j a c e n t i s o t h e r m a l s i n the v i c i n i t y of the p l a t e are w i d e r f o r the upward f a c i n g case than f o r the  downward  f a c i n g c o n f i g u r a t i o n . T h e r e f o r e the N u s s e l t numbers of the f o r m e r c a s e a r e much s m a l l e r t h a n t h o s e f o r t h e l a t e r This also falls  i n t o l i n e w i t h S u r i a n o and Y a n g ' s ^  f o r p l a t e s h e a t e d on b o t h s i d e s , t h o u g h certainly  i n a c c u r a t e as c o m p a r e d  case. results  those results  are  with the experimental 12  r e s u l t s o f B u s n i k and B e z l o m t ' z e v  .  Indeed,  Suriano  and  Yang'^ o c c a s i o n a l l y compute v a l u e s o f the N u s s e l t number s m a l l e r than u n i t y , a p h y s i c a l i m p o s s i b i l i t y . 3.4  Comparison  with P r e v i o u s Heat T r a n s f e r R e s u l t s  F o r p l a t e s h e a t e d on a s i n g l e s i d e t h e r e a r e no  pre-  v i o u s l y p u b l i s h e d r e s u l t s a v a i l a b l e , f o r the R a y l e i g h number range c o n s i d e r e d i n t h i s s t u d y . have on  been  both  Numerical  results  o b t a i n e d by S u r i a n o a n d Y a n g ' ^ f o r p l a t e s h e a t e d  t h e i r s i d e s , and t h e s e have  been  d i s c u s s e d by  Figure 5  S t r e a m l i n e s f o r H o r i z o n t a l P l a t e F a c i n g Downward a t Gr=0.5; Pr=0.72  Figure 6  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t Gr=0.5; Pr=0.72  G r z 0.5 Pr  = 0.72  1.50-  1.30  b  1.00  c  0.90  d  0.70  e  0.50  f  0.30-  -<5L25  6L25 Figure 7 V o r t i c i t y Contours f o r Plate Facing a t G r = 0 . 5 ; Pr=0.72  Downward  -a  Figure 8  S t r e a m l i n e s f o r P l a t e F a c i n g Upward a t G r = 0 . 5 ; Pr=0.72  Figure 9  I s o t h e r m a l s f o r P l a t e F a c i n g Upward a t Gr=0.5; Pr=0.72  F i g u r e 10  V o r t i c i t y f o r P l a t e F a c i n g Upward a t G r = 0 . 5 ; Pr=0.72  F i g u r e 11  S t r e a m l i n e s f o r P l a t e F a c i n g Downward a t Gr=50; Pr=0.72  Gr = 50 Pr  0.82 0.80 0.75 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.08  F i g u r e 12  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t Gr=50; Pr=0.72  =0.72  38  6.25  Gr  = 50  P rz0.72  "1.5 0 -1.3 0 -1.1 0 -1.00 -0.70 -0.50 -0.20  F i g u r e 13  V o r t i c i t y Contours f o r Plate Facing a t G r = 5 0 ; Pr=0.72  Downward  F i g u r e 14  S t r e a m l i n e s f o r P l a t e F a c i n g Downward a t G r = 2 0 0 ; Pr=0.72  F i g u r e 16  V o r t i c i t y Contours f o r Plate Facing at, G r = 2 0 0 ; . Pr=0.72  Downward  F i g u r e 17  S t r e a m l i n e s f o r P l a t e F a c i n g Downward a t G r = 0 . 5 ; Pr=10  43  Gr  =0.5  Pr = 10  0.8 0  0.75 0.70 0.6 0 0.5 0 0.4 0 0.2 0 0.1 0 0.0 9  0.06 0.04 0.0 2  F i g u r e 18  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t G r = 0 . 5 ; Pr=10  44  Gr=0.5  P r: 1 0  -1.42 -1.20  -1.0  0  -0.90 -0.70  -0.5  0  -0.3 0  6.25 F i g u r e 19  V o r t i c i t y C o n t o u r s f o r P l a t e F a c i n g Downw a r d a t G r = 0 . 5 ; Pr=10  -0.2  0  -0.1  0  F i g u r e 20  Streamlines f o r Plate Facing a t G r = 1 0 ; Pr=10  Downward  F i g u r e 22  V o r t i c i t y C o n t o u r s f o r P l a t e F a c i n g Downw a r d a t G r = 1 0 ; Pr=10  Figure  23  Streamlines for Gr=20; Pr=lO  Plate  Facing  Downward  at  F i g u r e 24  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t G r = 2 0 ; Pr=10  6.25  Figure  25  Streamlines for G r = 4 0 ; Pr=5  Plate  Facing  Downward  at  F i g u r e 26  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t G r = 4 0 ; Pr=5  F i g u r e 27  S t r e a m l i n e s f o r P l a t e F a c i n g Downward a t G r = 1 0 0 ; Pr=5  F i g u r e 28  I s o t h e r m a l s f o r P l a t e F a c i n g Downward a t G r = 1 0 0 ; Pr=5  X  F i g u r e 29  I s o t h e r m a l s f o r P l a t e F a c i n g Upward a t G r = 1 0 0 ; Pr=5  Panton tal  .  Buznik and B e z l o m t ' z e v  results for that configuration.  r e s u l t s b y an e m p i r i c a l  Nu  have o b t a i n e d  = ' Nu  experimen-  They summarized  their  correlation,  + 1/2 G r 0 . 2 5 P r 0 . 2 5 + 0.01  G r 0.4 P r 0.4  (33a)  or  Ra  + 0.01  Ra 0.4  (33b)  In t h i s c o r r e l a t i o n N u „ i s t h e N u s s e l t n u m b e r f o r o p u r e c o n d u c t i o n , t h a t i s 1.0 b y d e f i n i t i o n , w h i l e t h e s e c o n d term i s immediately i d e n t i f i e d with the f r e e - c o n v e c t i o n term from a  vertical  plate the value of the exponent Prandtl  boundary-1ayer  plate:  for a horizontal  s h o u l d be 1 / 5 .  n u m b e r h a s t h e same e x p o n e n t  Also the  a s G r o n l y w h e n P r >>>  see c . f . r e f e r e n c e ( 1 ) . The l a s t term g i v e s presumably i n f l u e n c e o f c r e e p i n g f l o w , though have e x p e c t e d the exponent  t h e r e a g a i n one  1,  the  would  t o be 1 / 2 , r e f e r e n c e ( 1 3 ) .  I f o n e t r i e s t o e x p r e s s t h e d a t a f o r t h e mean N u s s e l t number o b t a i n e d i n t h e p r e s e n t i n v e s t i g a t i o n i n t h e form o f a "power law" c o r r e l a t i o n  through the use o f a least-mean-  s q u a r e s f i t on c a l c u l a t e d p o i n t s , o n e o b t a i n s a c o r r e l a t i o n of the type Nu  oc  Ra  (34)  for  t h e range  Ra > T O O , F i g u r e ( 3 0 ) . T h e v a l u e o f t h e  e x p o n e n t m i s 0 . 4 2 2 , 0 . 4 8 3 a n d 0 . 4 3 2 f o r P r = 0 . 7 2 , 5.0 a n d 10.0 r e s p e c t i v e l y .  3.5  E f f e c t o f P r a n d t l Number on t h e R a t e o f Heat T r a n s f e r In F i g u r e 30 t h e c o r r e l a t i o n c u r v e s  Prandtl  number a r e above t h o s e f o r s m a l l e r P r a n d t l  Within  the  curves  a r e almost  The  f o rlarger  curve  lower R a y l e i g h  number.  n u m b e r range (Ra < 1 0 0 ) , a l l t h r e e  horizontal  (parallel  t o one a n o t h e r ) .  f o r P r = 0.72 h a s a l r e a d y r e a c h e d  i t s asymptotic  v a l u e o f Nu = 1.0 f o r G r -> 0 , w h i l e f o r l a r g e r P r t h e c u r v e s are s t i l l  s l o p i n g downwards.  data i n Table range, number.  R e f e r r i n g to these  curves and  1, i t i s c l e a r t h a t i n t h e l o w R a y l e i g h  number  t h e l a r g e r t h e P r a n d t l number, t h e g r e a t e r t h e N u s s e l t This i s i n accord with  t h e p h y s i c a l s i t u a t i o n and  f l u i d p r o p e r t i e s , s i n c e f o r very low R a y l e i g h number, t r a n s f e r i s dominated by c o n d u c t i o n : s m a l l e r P r a n d t l number p o s s e s s e s  heat  and f l u i d with t h e  higher c o n d u c t i v i t y , thus  d i s s i p a t i n g heat with a s m a l l e r buoyant  flow.  On t h e o t h e r h a n d , f o r h i g h e r R a y l e i g h n u m b e r a n d h i g h P r i t c a n be s e e n of  from T a b l e s  l a and l b that t h e value  N u s s e l t number i s i n v e r s e l y d e p e n d e n t upon t h e P r a n d t l  number.  As R a y l e i g h number i n c r e a s e s , c o n v e c t i o n  predominant.  The v e l o c i t y o f f l u i d s with  becomes  low v i s c o s i t y  ( s m a l l e r v a l u e o f t h e P r a n d t l number) i s l a r g e r than  that  Nu Gr  Pr  f a c i n g downward facing upward  0.3  0.72  1.0737  0.5  0.72  1.0769  1.0  0.72  1.0789  10.0  0.72  1.09 3 6  50.0  0.72  1.4342  100.0  0.72  2.0 8 7 5  200.0  0.72  2.5 9 0 4  500.0  0.72  3 . 7 8 81  1.0605  1.0904  1.55 3 5  Nu Gr  Pr  f a c i n g downward facing upward  0.1  5.0  1.3276  1.0  5.0  1.5684  1 0.0  5.0  1.7670  2 0.0  5.0  2.005 1  40.0  5.0  2.64 7 2  60.0  5.0  3.3 3 2 9  80.0  5.0  4.074 1  10 0.0  5.0  4.1 3 2 4  1.22 8 4  169 77  2.1 0 2 7  T a b l e l a . N u s s e l t numbers f o r h o r i z o n t a l p l a t e s h e a t e d on o n e  s i d e only  59  Table lb. Nusselt heated  Gr  numbers f o r on o n e  horizontal  plates  side  Nu  Pr facing  downward facing  upward  0.5  1 0.0  1.8 74 9  1 0.0  1 0.0  2.1 0 2 4  1.6077  2 0.0  1 0.0  2.63 2 7  1.871 1  3 0.0  1 0.0  3.1 1 0 0  1.90 1 0  4 0.0  1 0.0  3.5 5 5 4  6 0  i  to TJ  Buz  c  o  •*E  TJ  o \J  0  N  4)  p-  CO  CN •  •  K K  o  CO CK  •  •  n. CN  K  wo  CO  CO  O  CN  CN  ON  CO  CN  CO  OQ  _Q  c  0)  o  CO  0) 0  o  O  0)  at  o  •  K  o  —»  •  o  CO  r-  CN O  K  CN  O  CN K  •  •  O  O  CK  O  CO  o  CO  CN CO •  CO  11  o>  C L  c  "a  0  >-  c o  N  z  0  —  3  0)  -O.  to  3  •  CK  o  U0  •  CN •  •  «/»  CN  d) o  w  c o  o  o  CN  K  in  O  O  Ov  r— CN  CO  CN  CO  •  CO  a>  C  o o  >_  0) CO u. < • >  CN K •  O 11 Q.  a? o 3  o  t—^  «ft  0) v . " CN 0) o •O  ^ «- O• 0 3  E 3  o  •  •—  o  o  w— CN •  ON  CO  •o  CK  CN  o  •  o o  O  CN  CO  CO  o  K  CO  •  K •  CN  •  o  O  CO  •  1 1  1  1  •  >o m  3  _Q  0 OC  o  •  o  o •  in•  o d  O d  o d  CN "O K  o•  CO  o d o  *"~  o o  1  1 1  o  C  0)  1  •  O •  o  o o  CN  CO  1  1  1 1  o o o  •  o o o  •  61 of f l u i d s w i t h l a r g e r P r a n d t l number:  thus here the r a t e  of heat t r a n s f e r f o r f l u i d s o f s m a l l Pr w i l l than t h a t o b t a i n a b l e f o r l a r g e Pr. Pr i s not uniform i n t h i s range.  be  larger  Thus the i n f l u e n c e of N a t u r a l l y , n o t t o o much  r e l i a n c e s h o u l d be p l a c e d on t h i s p h y s i c a l i n t e r p r e t a t i o n .  3.6  S t a b i l i t y of the Numerical C a l c u l a t i o n In t h e n u m e r i c a l  was  calculation, numerical  instability  d e t e c t e d to o c c u r at R a y l e i g h number l a r g e r than  W i t h i n our u n d e r s t a n d i n g ,  an e x a c t f o r m o f  500.  stability  c r i t e r i o n , e s p e c i a l l y f o r the case of a s i n g u l a r boundary c o n d i t i o n as i n t h e p r e s e n t p r o b l e m , The  only practical  instability  i s not always p o s s i b l e .  means f o r p o s t p o n i n g  i s the decrease  the o c c u r r e n c e  o f mesh s i z e .  H o w e v e r , due  l i m i t e d c o r e s t o r a g e o f an e l e c t r o n i c c o m p u t i n g c a n o n l y be d o n e a t t h e p r i c e o f a d e c r e a s e of the computational present problem  field.  of  machine,  of the  extent  As d i s c u s s e d p r e v i o u s l y , t h e  is ( s t r i c t l y speaking)  a singular pertur-  bation case; a uniformly valid numerical  solution is  unobtainable. The boundaries problems  numerical chosen,  judged  results will  be a f f e c t e d by  see r e f e r e n c e (11).  i t w o u l d be an a d v a n t a g e  large enclosure.  to the  Thus a d e c r e a s e  For  the  practical  to r e t a i n a reasonably i n m e s h s i z e was  t o be a p r a c t i c a l p r o p o s i t i o n .  not  this  IV. The  EXPERIMENTAL  experimental  INVESTIGATION  p a r t o f t h e p r e s e n t s t u d y was  con-  5  c e i v e d as a c o r o l l a r y t o t h e w o r k o f C l a a s s e n . t i g a t e d n a t u r a l c o n v e c t i o n about heated  numbers.  p l a t e i n the range  of large  boundary Grashof  As s t a t e d i n t h e i n t r o d u c t o r y s e c t i o n , t h e  boundary l a y e r i s not expected facing heated  p l a t e even  the main purpose  furnish evidence heated  inves-  an u p w a r d f a c i n g h o r i z o n t a l  p l a t e , and showed t h e e x i s t e n c e o f a l a m i n a r  l a y e r above the heated  sense,  He  fluid  f o r l a r g e v a l u e s of Gr. of this experiment  a h o r i z o n t a l heated  In  this  i s , t h e r e f o r e , to the  surface, for all  Ra.  In o r d e r t o f u l f i l l semi-focussing  a downward  of the non-boundary l a y e r nature of  under  v a l u e s o f Gr o r  to e x i s t beneath  laminar  the purpose  c o l o u r - S c h 1 i e r e n system  mentioned was  above, a  utilized  to  under-  take a q u a l i t a t i v e o b s e r v a t i o n of the flow p a t t e r n under test plates.  As a d e t a i l e d d e s c r i p t i o n o f t h e p r i n c i p l e o f  S c h l i e r e n system  and o f t h e a p p a r a t u s  o n l y a s h o r t resume o f i t s use w i l l The examination The  semi-focussing  has  been  be g i v e n  behind  Thus the instrument  the  published^, here.  Schlieren instrument  of only a small depth  d i s t u r b a n c e s a h e a d and  blurred-out.  the  enables  of the f l o w - f i e l d  examined.  the f o c u s s e d r e g i o n makes p o s s i b l e the  i n a t i o n of t h a t s e c t i o n of the f l o w - f i e l d where the  the  are examflow  63 most c l o s e l y approaches  two-dimensional  conditions.  The  use o f c o l o u r bands e n a b l e s e a s y and r a p i d e v a l u a t i o n o f boundary-1ayer  4.1  t h i c k n e s s and t e m p e r a t u r e  Experimental  profile.  Apparatus  T h e S c h l i e r e n s y s t e m u s e d c o n s i s t s o f two optical  benches.  show t h e f i r s t  Photographs  b e n c h on w h i c h  l e n s e s , the source s l i t , m i r r o r ) were mounted.  i n F i g u r e s (31a) and  the l i g h t s o u r c e , the source  and the c o l l i m a t o r  ( a parabolic  used i n c o n j u n c t i o n with a theo-  the beam, check  i t s h o r i z o n t a l i t y and a d j u s t  t h e i n c i d e n c e and e x i t a n g l e s o f t h e beam t o values.  The s e c o n d b e n c h , shown i n p h o t o g r a p h  the S c h l i e r e n head  (31b)  A l s o seen i s a l a s e r d e v i c e p l a c e d  on t h e t r a c k , w h i c h was d o l i t e to align  separate  identical (b), carried  (another p a r a b o l i c mirror), a plane  t u r n i n g m i r r o r , the f o c a l - p l a n e c o l o u r "knife edges", o b j e c t l e n s e s , a n d t h e s c r e e n o r a 35 mm photographs.  working of the Two  camera f o r t a k i n g  A schematic diagram of the S c h l i e r e n 14  i n F i g u r e ( 3 2 ) , w h i c h was  reproduced from  the  system  , illustrates  system.  aluminum p l a t e s of s i z e  21x21x3/4 i n c h e s were used. h e a t e d by s a n d w i c h e d  The  mica-nichrome  12x12x3/8 inches  and  larger square plate  was  wire heater which  was  b a k e d on t h e p l a t e s u r f a c e w h i l e t h e s m a l l e r o n e was  heated  by a P y r e x p a n e l t h i n l y c o a t e d w i t h a u n i f o r m l a y e r o f oxi de.  the  zinc  The by u s i n g one  temperatures  buried  of the p l a t e s u r f a c e were measured  copper-constantan  inch i n t e r v a l s along The  (33b),  The  the sharp  of focus  mirror  The parallel  t h e i r heated  face  anchored i n the  plane,  Figures  o p t i c a l benches.  ceiling  over  chosen  (33a)  In o r d e r  and to  obtain  the working s e c t i o n the  (the S c h l i e r e n head).  source  to both  s l i t was  (33a)  plate  to  the  arrangement is  (33c).  set in a horizontal d i r e c t i o n ,  the p l a t e s u r f a c e  colour knife  The  and  the c o l o u r bands  of  edge.  Results S c h l i e r e n p h o t o g r a p h s o f t h e two  various (34c)  plates.  l o c a t i o n f o r a t e s t p l a t e was  shown i n the p l a t e s , F i g u r e s  4.2  at  l o c a t e d as c l o s e as p o s s i b l e t o t h e f o c a l p o i n t o f  second  the  each,  focussing  b e t w e e n t h e two  a s h o r t depth was  steel cables  laboratory.  straddling  l i n e of the  p l a t e s were suspended with  d o w n w a r d on t h r e e of the  a center  thermocouples spaced  Grashof  and  numbers are p r e s e n t e d  (35a),  plates tested  in Figures  at  (34a)  to  (35b).  F r o m t h e p h o t o g r a p h s , i t c a n be s e e n t h a t t h e l a y e r of f l u i d under the p l a t e i s almost of thickness boundaries,  near  o f t h e p l a t e as n e a r  order  the  r a t h e r l i k e a momentum b o u n d a r y l a y e r n e a r  front stagnation thickness  the middle  o f t h e same  heated  p o i n t of a b l u n t body.  of the c o l o u r  Moreover,  l a y e r under the p l a t e  the  the  increases  as  65  G r a s h o f number i s i n c r e a s e d :  This  to b o u n d a r y l a y e r p r e d i c t i o n s , see Therefore,  i t may  photographs that laminar a horizontal G r a s h o f and  now  contradictory  c.f. reference  be c o n c l u d e d  from the  heated p l a t e , even f o r l a r g e values Rayleigh  no a t t e m p t was  flow  of  the  be e m p h a s i z e d h e r e t h a t t h e  d i r e c t e d to the q u a l i t a t i v e  f i e l d under a heated surface,  made t o t a k e  on t h e e x p e r i m e n t a l  Schlieren  numbers.  o f t h i s e x p e r i m e n t was of the  (1).  boundary l a y e r does not e x i s t under  F i n a l l y , i t should  gation  is quite  plates.  any  quantitative  aim  investitherefore,  measurements  F i g u r e 31b  Output Bench of S c h l i e r e n  System  Condenser  FIG.32-Q  ELEMENTS OF SCHLIEREN SYSTEM  F i g u r e 33a  21 i n c h e s S q u a r e Apparatus  P l a t e and M i r r o r - T y p e  F i g u r e 33b  21  Plate  inches Square  Schlieren  F i g u r e 34a  S c h l i e r e n P h o t o g r a p h o f 12 i n c h P l a t e T = 68.5°F; T =89.5°F; Gr=3.6xlQ8  F i g u r e 34b  S c h l i e r e n P h o t o g r a p h o f 12 i n c h P l a t e T T,=132.5°F; Gr=lQ9  =72.5°F;  F i g u r e 35a  S c h l i e r e n P h o t o g r a p h o f 21 i n c h P l a t e T T =89°F; Gr=2.24xl0 5  =66.5°F;  72  CONCLUSIONS  T h i s t h e s i s i s c o n c e r n e d w i t h the f r e e - c o n v e c t i v e f l o w and h e a t t r a n s f e r from a h e a t e d , f i n i t e , h o r i z o n t a l s t r i p - p l a t e f a c i n g downwards i n a t w o - d i m e n s i o n a l f l o w - f i e l d . I t was e s t a b l i s h e d exper i m e n t a l l y t h a t even f o r very l a r g e v a l u e s o f the G r a s h o f number Gr, or  the R a y l e i g h number Ra, no b o u n d a r y - l a y e r t y p e o f f l o w  t a k e s p l a c e under such a p l a t e . T h i s was found t h r o u g h S c h l i e r e n obs e r v a t i o n s o f the f l o w f i e l d : an i n c r e a s e i n the t h i c k n e s s o f the l a y e r o f h e a t e d f l u i d was o b s e r v e d when the R a y l e i g h number o f t h e p l a t e was i n c r e a s e d , i n s t e a d o f a r e d u c t i o n o f t h i s t h i c k n e s s which would have o c c u r r e d had the flow been o f t h e b o u n d a r y - l a y e r t y p e . T h i s r e s u l t a g r e e s w i t h p r e v i o u s i n v e s t i g a t i o n s and p r e d i c t i o n s .  A n u m e r i c a l study o f the f u l l e q u a t i o n s o f v o r t i c i t y and energy t r a n s p o r t was then u n d e r t a k e n . A f i n i t e d i f f e r e n c e scheme was used, and an e n c l o s u r e o f f i n i t e dimensions had t o be chosen f o r q u i t e o b v i o u s reasons f o r t h i s i n v e s t i g a t i o n . An optimum between the s m a l l e s t p r a c t i c a b l e mesh s i z e and the [ d e s i r a b l e ) l a r g e s t e x t e n t o f the computation f i e l d had t o be s t r u c k . The b o u n d a r y c o n d i t i o n s obt a i n i n g a t i n f i n i t y , i . e . the v a n i s h i n g o f the v o r t i c i t y , the stream f u n c t i o n and t h e t e m p e r a t u r e , were a p p l i e d on the b o u n d a r i e s o f t h e e n v e l o p e o f f i n i t e e x t e n t . V a l u e s o f the s t r e a m f u n c t i o n , the v o r t i c i t y and the t e m p e r a t u r e were computed f o r a range o f G r a s h o f numb e r s from 0.5 t o 200, and o f P r a n d t l numbers from 0.72 t o 10. Conver-  73  gence o f the r e s u l t s was d e f i n e d and i n v e s t i g a t e d i n a h e u r i s t i c manner. The n u m e r i c a l v a l u e s o b t a i n e d a p p e a r t o y i e l d s t a b l e v a l u e s f o r t h e range o f t h e d i m e n s i o n l e s s parameters i n v e s t i g a t e d . N u s s e l t numbers were d e t e r m i n e d through n u m e r i c a l i n t e g r a t i o n o f t h e (dimens i o n l e s s ) h e a t f l u x o v e r t h e w i d t h o f the p l a t e . The v a l u e s o b t a i n e d were compared t o those found when the h e a t e d s i d e o f t h e p l a t e was assumed t o be t u r n e d upwards. As e x p e c t e d , i t was found t h a t the i n t e n s i t y o f h e a t t r a n s f e r ( i . e . Nu) was g r e a t e r f o r t h e downward f a c i n g p l a t e . The r e s u l t s were then compared t o t h e few v a l u e s a v a i l a b l e from the l i t e r a t u r e f o r p l a t e s h e a t e d on both t h e i r s i d e s . B e t t e r agreement w i t h t h e e x p e r i m e n t a l r e s u l t s known f o r t h a t case was o b t a i n e d than t h a t a c h i e v e d by o t h e r n u m e r i c a l d a t a p u b l i s h e d .  74  REFERENCES 1.  Z. R o t e m a n d L . C l a a s s e n , " N a t u r a l c o n v e c t i o n above unconfined horizontal surfaces", J . of Fluid Mechanics 38, 1 9 6 9 .  2.  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 f r o m a h o r i z o n t a l p l a t e " , Z.A.M.P. 9_a, 1 9 5 8 , 2 7 6 - 2 8 2 .  3.  W.N. G i l l , D.W. on a h o r i z o n t a l  4.  Z. R o t e m , " F r e e c o n v e c t i o n b o u n d a r y l a y e r f l o w o v e r h o r i z o n t a l d i s c s and p l a t e s I: s i m i l a r s o l u t i o n s near p l a t e " , Proceedings 1 s t Canad. Congress o f Applied M e c h a n i c s 2 b , 1 9 6 7 , 3 0 9 - 3 1 0.  5.  L. 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V a n N o s t r a n d C o . , New Y o r k , 1 9 6 1 .  10.  F . J . S u r i a n o and K-T. Yang, " L a m i n a r f r e e convection about v e r t i c a l and h o r i z o n t a l p l a t e s a t small and m o d e r a t e G r a s h o f numbers", I n t e r n . J . Heat and Mass T r a n s f e r 1J_, 1 9 6 8 , 4 7 3 - 4 9 0 .  11.  R. P a n t o n , " N o t e on l a m i n a r f r e e c o n v e c t i o n a l o n g a v e r t i c a l plate a t extremely small Grashof numbers", I n t . J . Heat and Mass T r a n s f e r 1 1 , 1 9 6 8 , 615-616.  Zeh and E. d e l - C a s a l , "Free convection p l a t e " , Z.A.M.P. 1_6 , 1 9 6 5 , 5 3 9 - 5 4 1 .  Grashof  numbers",  75 12.  V.M. B u z n i k a n d K.A. B e z 1 o m t z e v , "A g e n e r a l i z e d e q u a t i o n f o r the h e a t e x c h a n g e o f n a t u r a l and f o r c e d convection during external flow about bodies", Izv. U y s s h . U c h e b . Z a v e d . _2, 1 960 , 6 8 - 7 4 ; S u m m a r y i n A p p l i e d M e c h a n i c s R e v i e w s 2_, 1 9 6 3 , 480.  13.  Z. R o t e m , " C o u r s e i n t r a n s p o r t p h e n o m e n a " , o f B r i t i s h C o l u m b i a , 1967, unpublished.  14.  D.W. H o l d e r and 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 , N o t e s on A p p l S c . N_o. 31, H.M.S.O. , L o n d o n , 1 9 6 3 .  15.  S. S u g a w a r a a n d I . M i c h i y o s h i , " H e a t t r a n s f e r f r o m 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 convection", T r a n s . J a p a n Soc. Mech. E n g r s . 2 1 , 1955, 6 5 1 - 6 5 7 .  16.  J.O. W i l k e s a n d S.W. C h u r c h i l l , "The f i n i t e - d i f f e r e n c e computation of natural convection in a r e c t a n g u l a r e n c l o s u r e " , A. I . C h . E . J . 1 2 , 1 966 , 1 61 -1 6 6 .  17.  Z. Rotem, "Otjqjugate f r e e c o n v e c t i o n between c i r c u l a r c y l i n d e r s " , s u b m i t t e d f o r p u b l i c a t i o n , 1969.  1  University  76  APPENDIX  The n u m e r i c a l c a l c u l a t i o n c a r r i e d - o u t i n t h i s t h e s i s i s l i m i t e d t o a f l o w - f i e l d o f f i n i t e s i z e , F i g u r e 3. The boundary c o n d i t i o n s o f z e r o v o r t i c i t y and z e r o values o f the s t r e a m f u n c t i o n and o f the temperat u r e , which i n r e a l i t y o b t a i n a t a d i s t a n c e i n f i n i t e l y l a r g e from the p l a t e , were a p p l i e d a t the b o u n d a r i e s o f the f i n i t e f i e l d . The f a c t t h a t the i n f i n i t e f i e l d c a n n o t be modelled by a f i n i t e f i e l d w i t h o u t v i o l a t i o n o f a fundamental p r i n c i p l e has been d i s c u s s e d i n C h a p t e r I I . Though the n u m e r i c a l v a l u e s o f the N u s s e l t number o b t a i n e d approximate t h o s e f o r an u n c o n f i n e d p l a t e , the n a t u r e o f t h i s a p p r o x i m a t i o n cannot be u n i f o r m and a p h y s i c a l c o n s e r v a t i o n p r i n c i p l e i s v i o l a t e d . T h i s i s o f c o u r s e due t o the f a c t t h a t the s i n g u l a r problem was t r e a t e d as a r e g u l a r one i n a l i m i t e d f i e l d , through the computation t e c h n i q u e adopted. In the p r e s e n t case one f i n d s i n p a r t i c u l a r t h a t (as s h o u l d be e x p e c t e d ) the t o t a l h e a t f l u x o v e r the e n t i r e o u t e r boundary does n o t agree e x a c t l y w i t h the t o t a l d i s s i p a t i o n from the p l a t e , as i t s t r i c t l y s h o u l d f o r s t e a d y c o n d i t i o n s . N e i t h e r would the i n t e g r a t e d v o r t i c i t y f l u x e s a g r e e . Panton ^'has d i s c u s s e d t h i s p o i n t as w e l l . The analogue i n flows w i t h o u t h e a t c o n v e c t i o n o f the e f f e c t s d e s c r i b e d are S t o k e s and Whitehead's  1  paradoxes.  A f u r t h e r p o i n t t o note i s t h a t f o r the c o n f i g u r a t i o n s t u d i e d the f l o w - f i e l d i s doubly c o n n e c t e d . T h e r e f o r e " i n s t a b i l i t y " , t h a t i s a  77  c o n v e c t i v e f l o w s h o u l d a r i s e no m a t t e r how s m a l l the h e a t i n g ( o r i n o t h e r words, f o r a l l v a l u e s o f the R a y l e i g h number). T h i s agrees w i t h the r e s u l t s c a l c u l a t e d i n t h i s t h e s i s , and w i t h the p r i n c i p l e s g i v e n in reference  

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