UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Dimensionless ratios for surge waves in open canals Wu, Henry Jaw-Here 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1970_A7 W83.pdf [ 5.77MB ]
Metadata
JSON: 831-1.0050557.json
JSON-LD: 831-1.0050557-ld.json
RDF/XML (Pretty): 831-1.0050557-rdf.xml
RDF/JSON: 831-1.0050557-rdf.json
Turtle: 831-1.0050557-turtle.txt
N-Triples: 831-1.0050557-rdf-ntriples.txt
Original Record: 831-1.0050557-source.json
Full Text
831-1.0050557-fulltext.txt
Citation
831-1.0050557.ris

Full Text

DIMENSIONLESS RATIOS FOR SURGE WAVES I N OPEN CANALS  by  HENRY JAW-HERE B.Sc. Provincial  (Civil  WU  Engineering)  Cheng-Kung U n i v e r s i t y ,  1963  A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE  in  the Department of  Civil  We a c c e p t required  Engineering  this thesis  as conforming  to the  standard  THE U N I V E R S I T Y OF B R I T I S H COLUMBIA A p r i l 1970  In p r e s e n t i n g  this  thesis  an a d v a n c e d  degree  the  s h a l l make i t  I  Library  further  for  scholarly  by h i s of  agree  this  written  at  that  for  is  financial  of  Civil  Engineering  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  April  29,  1970.  by  the  understood  gain  Columbia  for  extensive  may be g r a n t e d It  British  available  permission.  Department  Date  freely  permission  purposes  for  fulfilment  the U n i v e r s i t y of  representatives. thes.is  in p a r t i a l  shall  the  requirements  Columbia, reference  copying  of  I agree and this  copying  or  for that  Study. thesis  Head o f my D e p a r t m e n t  that  not  of  or  publication  be a l l o w e d w i t h o u t  my  ii  ABSTRACT This  study  i n v e s t i g a t e s t h e p r o p a g a t i o n o f a s u r g e wave i n a p o w e r  canal  following load  ships  are derived  rejection or reduction.  to predict  v a r i a t i o n o f t h e wave h e i g h t of  relation-  (a) the i n i t i a l  wave h e i g h t ,  along  a n d ( c ) t h e maximum  the canal  (b)the  water depth a t t h e downstream end f o r s t r a i g h t p r i s m a t i c  rectangular, of  Dimensionless  various  t r i a n g u l a r and t r a p e z o i d a l c r o s s - s e c t i o n s .  parameters,  frictional  canals of  The e f f e c t s  such as v e l o c i t y and d e p t h o f i n i t i a l  coefficients,  bed slope,  o f wave p r o p a g a t i o n a n d i n i t i a l  flow,  cross-section of the canal,  wave h e i g h t  found t h a t , as a p o s i t i v e surge propagates along decreases  t o an e x p o n e n t i a l relationship positive  linearly  with  distance  f u n c t i o n f o r a long  i s also  the canal, the  f o ra short  canal.  It is  canal,  An a p p r o x i m a t e  respect  t o time.  An almost l i n e a r  of the canal  The d i m e n s i o n l e s s e s t a b l i s h design of  power  canals  i s not  r e l a t i o n s h i p between  t h e maximum w a t e r d e p t h a t t h e d o w n s t r e a m e n d o f t h e c a n a l length  of a  c r o s s - s e c t i o n a l parameters.  The v a r i a t i o n o f w a t e r d e p t h a t t h e d o w n s t r e a m e n d o f t h e c a n a l linear with  according  logarithmic  f o u n d b e t w e e n t h e v a r i a t i o n o f wave h e i g h t  surge and c a n a l  distance  of the surge a r e s t u d i e d .  A computer program i s developed f o r t h e c a l c u l a t i o n s r e q u i r e d .  wave h e i g h t  stage  and t h e  i s noted.  relationships derived criteria to avoid  i n this  f o rcrest elevations overtopping.  s t u d y may be u s e d t o o f t h e banks and w a l l s  iii  TABLE OF CONTENTS page ABSTRACT TABLE OF CONTENTS LIST  i  OF F I G U R E S  i  i  x  • ACKNOWLEDGEMENTS  x i i  CHAPTER 1  INTRODUCTION  CHAPTER 2  DETERMINATION OF WAVE HEIGHT AND V E L O C I T Y FOR I N I T I A L SURGE WAVES 2-1  1  Fundamental Equations Governing 'Wave H e i g h t a n d V e l o c i t y  t h e Surge-  6 .  2-2  I n i t i a l S u r g e Waves i n C a n a l s o f R e c t a n g u l a r Cross-Section  2-3  I n i t i a l S u r g e Waves i n C a n a l s o f T r i a n g u l a r Cross-Section  11  2- 4  I n i t i a l S u r g e Waves i n C a n a l s o f T r a p e z o i d a l Cross-Section  12  9  NUMERICAL CALCULATIONS FOR SURGE WAVE PROPAGATION 3- 1  General  16  3-2  B a s i c Assumptions  17  3-3  The E q u a t i o n s o f C h a r a c t e r i s t i c s  17  3-4  Solution of Equations o f C h a r a c t e r i s t i c s a G r a d u a l l y V a r i e d Unsteady Flow  3-5  N u m e r i c a l C a l c u l a t i o n s f o r P o s i t i v e Waves P r o p a g a t i n g i n Power C a n a l s  23  3- 6  Numerical Calculations  29  Propagating CHAPTER 4  i  v  NOTATION  CHAPTER 3  i  i n Power  f o r N e g a t i v e Waves  for  18  Canals  RESULTS OF A N A L Y S I S 4- 1  General  34  4-2  Dimensionless Ratios  34  iv  4-3  V a r i a t i o n o f P o s i t i v e Surge-Wave H e i g h t i n Rectangular Canals  36  4-4  V a r i a t i o n o f P o s i t i v e Surge-Wave H e i g h t i n T r i a n g u l a r Canals  38  4-5  V a r i a t i o n o f P o s i t i v e Surge-Wave H e i g h t i n Trapezoidal Canals  39  4-6  Approximate Equations  40  4-7  R e d u c t i o n o f Wave H e i g h t A f t e r R e f l e c t i o n a t a R e s e r v o i r l o c a t e d a t t h e Upper End of t h e Canal  46  4-8  P r o p a g a t i o n o f N e g a t i v e Waves R e f l e c t e d f r o m a R e s e r v o i r a t t h e Upper End o f a P r i s m a t i c Power C a n a l  50  4-9  Maximum Water Depth a t Downstream End o f a Canal  51  1  CHAPTER 5  CONCLUSIONS  54  FIGURES  57  REFERENCES  109  APPENDIX A ( l )  110  APPENDIX A(2)  113  APPENDIX B  1  1  6  LIST OF FIGURES Definition sketch: Surge wave i n an open channel  FIG. 2-1 FIG. 2-2 FIG. 2-3 FIG. 2-4 FIG. 2-5 FIG; 2-6  Definition sketch:  FIG. 2-7(a)  I n i t i a l surge waves in a trapezoidal canal for  T  —  FIG. 2-7(b)  I n i t i a l surge waves i n a trapezoidal canal for  T  = 0.1  FIG. 2-7(c)  I n i t i a l surge waves in a trapezoidal canal for  T  0.2  FIG. 2-7(d)  I n i t i a l surge waves in a trapezoidal canal for  T  - 0.3  FIG. 2-7(e)  I n i t i a l surge waves i n a trapezoidal canal for  T  = 0.4  FIG. 2-7(f)  I n i t i a l surge waves in a trapezoidal canal for  T  = 0.5  FIG.  2-7(g)  I n i t i a l surge waves in a trapezoidal canal for  T  = 0.6  FIG. 2-7(h)  I n i t i a l surge waves in a trapezoidal canal for  T  = 0.7  FIG. 2-7(i)  I n i t i a l surge waves in a trapezoidal canal for  T  = 0.8  FIG. 2-7(j)  I n i t i a l surge waves i n a trapezoidal canal for  T  = 0.9  FIG. 3-1  Definition sketch: ^and C  Definition sketch: Surge wave i n a rectangular canal I n i t i a l surge waves i n a rectangular canal Definition sketch: Surge wave i n a triangular canal I n i t i a l surge waves i n a triangular canal Surge wave i n a trapezoidal canal  -  characteristic grids on  the x-t plane  FIG. 3-2 FIG. 3-3 FIG. 3-4  Characteristic grids on the x-t plane Definition  sketch:  ( a ) upstream end boundary and (b) downstream end boundary  A part ..of results of computer calculations i n example 3-1  0  vi  FIG. 3-5  A p a r t of r e s u l t s example  F I G . 3-6  o f computer c a l c u l a t i o n s i n  3-2  Schematic p r e s e n t a t i o n o f the c h a r a c t e r i s t i c grids  F I G . 3-7  for a positive  surge wave p r o p a g a t i n g upstream  Schematic p r e s e n t a t i o n o f the c h a r a c t e r i s t i c f o r a positive  r  FIG. 3-8  surge wave p r o p a g a t i n g downstream  Flow c h a r t o f the computing wave p r o p a g a t i n g upstream  program f o r a p o s i t i v e  a l o n g the c a n a l  FIG.  3-9  Definition  s k e t c h f o r example  3-3  FIG.  3-10  Definition  s k e t c h f o r example  3-3  FIG.  3-11  The  comparison o f the method o f c h a r a c t e r i s t i c s  F a v r e ' s method from example FIG. 3-12  grids  Summary o f c a l c u l a t i o n  with  3-3  i n example 3-3 by F a v r e ' s  method FIG. 3-13 FIG. 3-14  Results of calculation (a)  Positive  i n example  3-4  surge wave reaches the r e s e r v o i r at the upper  end of the canal.  (b)  N e g a t i v e surge wave r e f l e c t e d from the r e s e r v o i r at the  upper end of the canal. FIG. 3-15  Definition  sketch f o r point R i n f i g u r e  FIG. 3-16  Schematic  diagram o f c h a r a c t e r i s t i c g r i d s  3-16 f o r the  n e g a t i v e surge wave a t p o i n t R FIG. 3-17  Schematic  diagram o f c h a r a c t e r i s t i c g r i d s  n e g a t i v e surge wave a t p o i n t R  f o r the  vii FIG. 4-1.  Definition sketch  FIG. 4-2  Variation of wave height of a positive surge propagating along a rectangular power canal  FIG. 4-3  Variation of wave height of a positive surge propagating along a triangular power canal  FIG. 4-4  Variation of wave height of a positive surge propagating along a trapezoidal power canal;pn an i n i t i a l flow F = 0.20 Q  FIG. 4-5  Variation of wave height of a positive surge propagating along a trapezoidal power canal for r = 1.50  FIG. 4-6  Variation of wave height of a positive surge propagating along a trapezoidal power canal for r = 1.75  FIG. 4-7  Effect of shape factor of a canal on the variation of wave height of a positive surge  FIG.  4-8  wave propagating along the canal.  Schematic diagram f o r a p o s i t i v e wave reaches" the upper end o f the c a n a l  FIG. 4-9  Reduction of the negative wave height reflected at the reservoir  FIG. 4-10 (a)  at the upper end o f the rectangular c a n a l  Variation of wave height of a negative surge propagating along a rectangular power canal for F » 0.200, 0.125 and 0  0.075 FIG. 4-10 (h)  Variation of wave height"of a negative surge propagating along a rectangular power canal for F = 0.175 and 0.100. Q  FIG. 4-10 (c)  Variation of wave height of a negative surge propagating along a rectangular power canal for F = 0.150 and 0.05 Q  FIG. 4-11 (a)  Variation of wave height of a negative surge propagating along a triangular power canal for F = 0.20, 0.125 and Q  0.75  „  viii  FIG.  4-11  (b)  V a r i a t i o n o f wave h e i g h t o f a n e g a t i v e propagating F  FIG.  4-11  (c)  o  = 0.175  a l o n g a t r i a n g u l a r power c a n a l f o r and  0.100  V a r i a t i o n o f wave h e i g h t o f a n e g a t i v e surge a l o n g a t r i a n g u l a r power c a n a l f o r F  FIG.  4-12  (a)  surge  q  propagating  = 0.150  V a r i a t i o n o f wave h e i g h t o f a n e g a t i v e surge  and  0.050  propagating  a l o n g a t r a p e z o i d a l power c a n a l f o r r = 1.50 and  F FIG.  4-12  (b)  o  = 0.200, 0.125 and 0.075  V a r i a t i o n o f wave h e i g h t o f a n e g a t i v e surge  propagating  a l o n g a t r a p e z o i d a l power c a n a l f o r r = 1.50 and F  FIG.  4-12  (c)  o  = 0.175  and  0.100  V a r i a t i o n o f wave h e i g h t o f a n e g a t i v e surge  propagating  a l o n g a t r a p e z o i d a l power c a n a l f o r r = 1.50 and  F FIG.  4-13  o  = 0.150 and 0.050  Schematic diagram o f t h e v a r i a t i o n o f water s u r f a c e f o r a p o s i t i v e wave p r o p a g a t i n g  FIG.  4-14  upstream along the c a n a l  Schematic diagram o f the v a r i a t i o n o f water s u r f a c e f o r a n e g a t i v e wave p r o p a g a t i n g  FIG.  4-15  downstream along  the c a n a l  V a r i a t i o n o f water s u r f a c e a t the downstream end w i t h r e s p e c t t o time f o r the r e c t a n g u l a r c a n a l o f V  b FIG.  f-16  o  q  = 30 f t . ,  = 30 f t . and n = 0.03095  V a r i a t i o n o f water s u r f a c e a t the downstream end w i t h r e s p e c t to time f o r the t r i a n g u l a r c a n a l o f  y FIG.  4-17  o  = 22.36 f t . , m = 2.0 and n = 0.03095  V a r i a t i o n o f water s u r f a c e a t the downstream end w i t h r e s p e c t t o time f o r the t r a p e z o i d a l c a n a l o f m = 1.5,  b  o  = 25.5 f t . , n = 0.03095 and r = 1.50  ix  FIG. 4-18  Maximum water depth a t the downstream end of a rectangular canal  FIG. 4-19  Maximum water depth a t the downstream end of a t r i a n g u l a r canal  FIG. 4-20  Maximum water depth a t the downstream end of a t r a p e z o i d a l canal f o r r - 1.50  NOTATION A  c r o s s - s e c t i o n area  b  width o f free surface  ft  bottom w i d t h o f c a n a l  ft  b  o  "  c  wave  f  Froude number = V/yg«y  g  gravity acceleration  H  t o t a l head a t r e s e r v o i r  k  dimensionless  L  canal length  hi  reference length  celerity  canal side  IM  f t /sec.  2  f t/sec»  parameter = b / m » y 0  ft 0  ft = y /S 0  ft  Q  slope  n  r e s i s t a n c e c o e f f i c i e n t i n t h e Manning  R  hydraulic  P  wetted  Q  discharge  r  dimensionless  S  S  0  formula  radius  ft  perimeter  ft  friction  f  ft  ft^/sec. parameter = l ][+£  slope  +  (energy  gradient)  s l o p e o f bottom  t  time  V  water v e l o c i t y  ft/sec.  a b s o l u t e wave v e l o c i t y - V f c  ft/sec.  v  w  sec.  w  u n i t weight o f water  x  distance  * X  y  d i s t a n c e o f wave  lb/ft ft  propagation  ft  w a t e r depth  ft  c e n t r o i d a l depth o f c r o s s - s e c t i o n a r e a  2 note t h a t some A u t h o r s use F =  _Y_  8-y  as Froude number  ft  3  Z  surge wave height  ^  dimensionless parameter = V /V  0  dimensionless parameter =  T  discharge r a t i o =  r  unit shear force on bottom and sides  Q  ft WQ  0  Z /y Q  Q  Q/Qo  of canal M cf>  lb/ft^  energy coefficient /  dimensionless parameter =  S u b s c r i p t o r e f e r s to i n i t i a l  X  R  L  /r=l /  \ "R  steady s t a t e c o n d i t i o n  /  r=n  XXI  ACKNOWLEDGMENT  The  author  wishes t o express  his grateful appreciation to  h i s s u p e r v i s o r D r . E. R u u s , f o r h i s v a l u a b l e g u i d a n c e a n d e n c o u r a g e ment d u r i n g  the research,  p r e p a r a t i o n , and development o f t h i s  A p p r e c i a t i o n i s a l s o e x t e n d e d t o M r . P. D o n n e l l y suggestions.  thesis.  f o r h i s a d v i c e and  1 CHAPTER 1  The  INTRODUCTION  p r e d i c t i o n o f the h e i g h t and v e l o c i t y o f . a surge wave, the maximum  s t a g e and the o t h e r f l o w c h a r a c t e r i s t i c s instantaneous many y e a r s .  i n a c a n a l due t o l a r g e  changes i n d i s c h a r g e has been o f i n t e r e s t  to engineers f o r  The d e s i g n e r o f a h y d r o - e l e c t r i c headrace c a n a l , f o r example,  must determine t h e maximum stage o f water t h a t c o u l d o c c u r as a r e s u l t -of a sudden l o a d r e j e c t i o n  i n power output,  wave c r e s t and the subsequent unsteady r i s e  c o n s i d e r i n g both  the surge  i n stage a t any p o i n t .  I n h y d r o - e l e c t r i c head or' t a i l r a c e c a n a l s , l a r g e changes i n d i s c h a r g e are i n i t i a t e d  by opening  or c l o s i n g of turbine gates.  changes cause p o s i t i v e o r n e g a t i v e  surge waves.  These  A wave, which  from an i n c r e a s e i n water depth i s c a l l e d a p o s i t i v e wave. r e s u l t s from a d e c r e a s e positive  results  A wave, which  i n water depth i s c a l l e d a n e g a t i v e wave.  In a  surge wave, the h i g h e r p o r t i o n s o f the wave have a g r e a t e r  o c i t y of propagation tends  discharge  than  the lower  ones.  t o become s t e e p e r and s t e e p e r u n t i l  enough t o c r e a t e a r o l l e r . moving h y d r a u l i c jump. r a c e c a n a l passes  vel-  Hence the f r o n t o f the wave the s l o p e o f the f r o n t i s  The f r o n t o f the wave then resembles a  A f t e r a p o s i t i v e surge-wave f r o n t  i n a head  a g i v e n s e c t i o n , the water l e v e l a t t h a t s e c t i o n does  n o t , i n g e n e r a l , remain s t a t i o n a r y , but c o n t i n u e s water s u r f a c e r i s e b e h i n d  to r i s e .  The unsteady  the surge wave p e r s i s t s u n t i l ; . i n t e r r u p t e d by  t h e n e g a t i v e wave, which r e s u l t s from the r e f l e c t i o n o f the p o s i t i v e wave a t the r e s e r v o i r a t the upstream end o f the c a n a l .  2  The of  n e g a t i v e waves a r e not s t a b l e i n form, because the upper p o r t i o n s the wave t r a v e l f a s t e r than the lower p o r t i o n s .  This results  g r a d u a l f l a t t e n i n g o f the wave f r o n t as i t moves along  At s o l i d boundaries,  the c a n a l .  such as c l o s e d g a t e s , a p o s i t i v e wave i s r e f l e c t e d  as a p o s i t i v e wave and a n e g a t i v e wave as a n e g a t i v e wave. reservoir,  in a  A t the  a p o s i t i v e wave i s r e f l e c t e d as a n e g a t i v e wave and a  n e g a t i v e wave as a p o s i t i v e wave.  I n a s h o r t c a n a l , the wave f r o n t may  t r a v e l back and f o r t h s e v e r a l times However, the f i r s t  maximum stage  b e f o r e a new steady  s t a t e i s reached.  i s h i g h e r than the subsequent peaks on  the water s u r f a c e o s c i l l a t i o n c y c l e and i s the most s i g n i f i c a n t f o r design considerations.  T h i s study  includes four parts:  (1)  Determination  (2)  The method o f computation o f the p r o p a g a t i o n o f surge waves (Chapter  (3)  of i n i t i a l  (Chapter 2 ) ,  3 ) ,  The v a r i a t i o n o f wave h e i g h t s f o r p o s i t i v e and f o r the r e f l e c t e d n e g a t i v e surges  (4)  surge wave h e i g h t and v e l o c i t y  (Chapter 4 ) , and  The v a r i a t i o n o f water depth a t the downstream end o f the c a n a l (Chapter 4 ) .  The (a)  study  deals with:  I n i t i a l uniform  steady  f l o w w i t h Froude numbers from 0 . 0 5 to 0 . 2 0  inclusive. (b)  S t r a i g h t canals of constant c r o s s - s e c t i o n and f r i c t i o n a l  longitudinal coefficient  s l o p e , shape, s i z e o f throughout.  Rectangular,  t r i a n g u l a r and t r a p e z o i d a l c r o s s - s e c t i o n s a r e c o n s i d e r e d .  (c)  P o s i t i v e waves c a u s e d b y a n i n s t a n t a n e o u s l o a d r e j e c t i o n a t t h e downstream end and t r a v e l l i n g upstream the  (d)  t h e upper  end o f  canal.  Downstream t r a v e l l i n g of  toward  n e g a t i v e waves, r e s u l t i n g  from  reflection  p o s i t i v e waves r e a c h i n g t h e r e s e r v o i r a t t h e u p p e r  end o f t h e  canal. S e v e r a l a p p r o x i m a t e methods f o r c o m p u t a t i o n o f t h e p r o p a g a t i o n o f surge waves  i n c a n a l s have been d e v e l o p e d by p r e v i o u s i n v e s t i g a t o r s .  m e t h o d s s u g g e s t e d b y R. D. J o h n s o n ^ ^ use.  A c c o r d i n g t o Johnson's  divided  into  several  , a n d H. F a v r e ^ ^ a r e i n c u r r e n t  method, t h e t o t a l  reaches.  length of channel i s  The s l o p e s i n t h e w a t e r  c a n a l bed i n each r e a c h a r e r e p r e s e n t e d by a v e r t i c a l s u r f a c e and i n t h e c a n a l f l o o r  s u r f a c e and t h e drop  i n the water  a t j u n c t i o n s between t h e r e a c h e s .  w a t e r s u r f a c e and c a n a l f l o o r w i t h i n each r e a c h a r e assumed When a wave r e a c h e s t h e j u n c t i o n , which t r a v e l s upstream travelling will  The  level.  two c o m p o n e n t w a v e s e m e r g e , o n e o f  and t h e o t h e r downstream.  When t h e d o w n s t r e a m  c o m p o n e n t wave r e a c h e s t h e d o w n s t r e a m e n d o f t h e c a n a l , i t  be r e f l e c t e d  and t r a v e l s upstream.  When t h e u p s t r e a m  c o m p o n e n t wave r e a c h e s a n o t h e r j u n c t i o n f a r t h e r u p s t r e a m , t r a n s m i t t e d and r e f l e c t e d in  The  opposite directions.  again into  travelling  i s generated.  w a t e r depths a t any s e c t i o n o f c a n a l  be  travel  i n opposite direc-  T h e wave h e i g h t s a n d  c a n be c o m p u t e d s t e p b y s t e p .  F a v r e i n t r o d u c e d a method u s i n g two e m p i r i c a l  Numerals i n parentheses r e f e r  i twill  two c o m p o n e n t w a v e s w h i c h  When two waves  t i o n s m e e t , a n o t h e r p a i r o f waves  travelling  e q u a t i o n s f o r t h e corn-  to corresponding items i n References.  p u t a t i o n o f a s u r g e wave p r o p a g a t i n g tion  along  a canal.  He made t h e a s s u m p -  that the l o n g i t u d i n a l p r o f i l e of the water surface  front  i s a straight line.  considered  as one r e a c h .  short canals imate  behind  I n a d d i t i o n , t h e whole l e n g t h o f c a n a l i s T h i s m e t h o d may be a p p l i e d t o r e l a t i v e l y  w i t h a s t r a i g h t and moderate s l o p e ,  t o o b t a i n an approx-  solution.  Because these  manual c o m p u t a t i o n a l  ness i s r e s t r i c t e d .  I n recent  methods a r e l a b o r i o u s t h e i r u s e f u l -  years,  t h e advances i n computer  n o l o g y h a s s t i m u l a t e d t h e d e v e l o p m e n t a n d a p p l i c a t i o n o f more approaches.  In this  On s u c h r i g o r o u s  thesis,  equations  solution i s described  partial  differential  equations  rigorous  study.  into  the method o f c h a r a c t e r i s t i c s  and v e l o c i t y o f a s u r g e wave.  o f motion and c o n t i n u i t y  the g r a d u a l l y - v a r i e d , unsteady f l o w wave r e g i o n a r e c o n v e r t e d  i n this  tech-  o f momentum a n d c o n t i n u i t y f o r a r a p i d l y  v a r i e d f l o w a r e used to solve the height  The  t h e wave  total  governing  i n the f r o n t o f , o r behind, the differential  equations  and s o l v e d by a f i n i t e  by u s i n g  difference tech-  nique.  A m a t h e m a t i c a l model i s s e t up on an x - t p l a n e grids.  Using  characteristic  t h i s model, a computer program i s developed f o r c a l c u l a -  t i o n o f t h e wave h e i g h t s u r g e wave i s t r a v e l l i n g section  with  and v e l o c i t y upstream.  i n f r o n t o f and behind  downstream t r a v e l l i n g  a t any s e c t i o n o f t h e c a n a l as a Similarly  f l o w p a r a m e t e r s a t any  t h e wave r e g i o n , a r e o b t a i n e d  wave f r o m c o m p u t e r c a l c u l a t i o n s .  f o r the  5  Although  a s i m p l e and i d e a l i z e d  method and t e c h n i q u e s  employed and r e l a t i o n s h i p s  fied  f o r the computation  real  power  canal.  canal i s considered, i n this d e r i v e d may  o f s u r g e wave p r o p a g a t i o n  study, t h e be m o d i -  i n a complicated  6 CHAPTER 2  DETERMINATION OF WAVE HEIGHT AND VELOCITY FOR INITIAL SURGE WAVES  2.1  Fundamental  E q u a t i o n s G o v e r n i n g t h e Surge-Wave H e i g h t and V e l o c i t y  I f t h e v e l o c i t y o f t h e water f l o w i n g i n a c a n a l i s changed a wave i s g e n e r a t e d w i t h a sudden change i n d e p t h .  rapidly,  F i g u r e 2-1  i l l u s t r a t e s a surge wave r e s u l t i n g from a sudden change i n f l o w , due t o a g a t e m o t i o n , t h a t i n c r e a s e s t h e water d e p t h . of  The depth  f l o w i s always c o n s i d e r e d t o be p o s i t i v e w i t h r e f e r e n c e t o t h e  c h a n n e l bottom.  Wave v e l o c i t y , c e l e r i t y and water v e l o c i t y a r e  assumed t o be p o s i t i v e i n t h e downstream d i r e c t i o n .  The w a t e r  between two c r o s s - s e c t i o n s , one j u s t upstream and t h e o t h e r j u s t downstream o f t h e wave f r o n t , i s c o n s i d e r e d . The v e l o c i t y o f t h e mass o f water between t h e s e s e c t i o n s 1 and 2 i s d e c r e a s e d from V^ t o V£, and t h e momentum i s d e c r e a s e d a c c o r d i n g l y .  By Newton's  second l a w o f m o t i o n , t h e u n b a l a n c e d f o r c e r e q u i r e d t o change t h e momentum i s t h e p r o d u c t o f t h e mass p e r u n i t time and t h e change in velocity.  This unbalanced f o r c e i s equal to the d i f f e r e n c e  between t h e h y d r o s t a t i c p r e s s u r e f o r c e s a c t i n g on t h e a r e a A^ a t s e c t i o n 2 and 1. J L .  g  A  . ( V 1  i  - V w  I tfollows  that:  ) . ( V. - V, ) = w.A .y„ - W.A, . y. L I L L 1 1  Where w = s p e c i f i c w e i g h t o f water g = acceleration of gravity A = c r o s s - s e c t i o n a l area V = water v e l o c i t y  and  9  2-1 )  V"  = a b s o l u t e wave v e l o c i t y  y  = centroidal  w  Subscripts  depth o f c r o s s - s e c t i o n  area.  1 and 2 i n d i c a t e parameters a t s e c t i o n s  1 and 2  respectively.  The e q u a t i o n o f c o n t i n u i t y  between s e c t i o n  1 and 2 i s  A„ . ( V - V ) = A . (V - V ) 2 2 w 1 1 w 0  n  By s u b s t i t u t i n g V  w  where  Eq. (1-2) i n t o Eq. (1-1) and s o l v i n g  = V  1  t l~ V  Eq.  for  ,  + c  (2-3')  c = celerity  c -  ( 2-2 )  n  g' A  (relative  ( A  r  2 ^ 2 -  A  l  ( l - A /A x  to f l o w i n g  water),  ^TT" 2  (2  - 3 )  )  (2-3) i s t h e g e n e r a l e x p r e s s i o n f o r the a b s o l u t e wave v e l o c i t y  i n a power c a n a l . (2-3)  The s i g n  i n f r o n t o f the square-root  term i n E q .  depends on the d i r e c t i o n o f wave p r o p a g a t i o n . A p o s i t i v e  sign  i s used i f the wave moves downstream, and a n e g a t i v e one i f the wave moves u p s t r e a m .  If  i s eliminated  from E q s . (2-1) and ( 2 - 2 ) ,  The above e q u a t i o n r e p r e s e n t s t h e r e l a t i o n s h i p between and  depths o f f l o w a t s e c t i o n s  (2-4)  or  From E q s . (2-2) and  may be determined by a p r o c e s s o f t r i a l  g i v e n the o t h e r wave h e i g h t  1 and 2.  velocities  t h r e e independent v a r i a b l e s .  i s equal to ( y ^ -  indicate  an i n c r e a s e  reduction  i n depth.  and e r r o r ,  The magnitude o f the  P o s i t i v e values of  i n depth w h i l e n e g a t i v e v a l u e s  ( y ^ - y^)  indicate a  8 A l t h o u g h t h e above e q u a t i o n s a r e d e r i v e d Fig.  2-1,  elling Eqs.  t h e y c a n be a p p l i e d  V  =  wo  v  surge  waves,  + c °o  V 0  A  and -  For i n i t i a l  waves t r a v -  a n d (2-4) may be r e w r i t t e n a s :  ( 2-5 )  _+/ s - ( .  A  to p o s i t i v e or negative  e i t h e r upstream o r downstream.  (2-3)  f o r t h e c a s e shown i n  »  A.y-A .y =  where s u b s c r i p t  •  - o  y  A  ' <  A  0  o  A  o  '  1  - o  o  >  Q  / A  A  )  o  A  _ g • ( A - A  0  • y  . < v 0  -  o  V  )  ......  (2-6)  )  i n d i c a t e s parameters of the undisturbed  initial  flow.  With these equations, height, If  the i n i t i a l  case, the absolute  Solving  celerity,  e t c . , may be d e t e r m i n e d f o r a n y c r o s s - s e c t i o n o f t h e c a n a l .  a wave t r a v e l s i n s t i l l  this  wave p a r a m e t e r s , s u c h a s  E q s . (2-5)  and  water,  V  i s given  i s equal  to zero.  wave v o l o c i t y i s i d e n t i c a l  (2-6)  by t r i a l  To s i m p l i f y c a l c u l a t i o n s , t h e r e f o r e , equations  q  below f o r three  Thus, i n  to the c e l e r i t y .  and e r r o r i s l a b o r i o u s .  a dimensionless  types of p r i s m a t i c  form of the canals.  9 2.2  Initial For  S u r g e Waves i n C a n a l s o f R e c t a n g u l a r  the rectangular  canal  ( F i g u r e 2-2),  Cross-Section.  E q s . (2-1)  and  (2-2)  can be s i m p l i f i e d t o  y  - y  2  • y° • ( v  = G  0  - v  • ( v  )  w o  - v)  0  ••••  ( 2  >  7  and  y  ; ( v  vrhere  y  0  - v  0  ) =  w o  .  y  i s the depth of  Eliminating V  (  v  _  f r o m E q s . (2-7)  y  w  ( Jy  y  ! )•( J _  +  y  D  3  by y^  ,  ,  2  - 1 f  D  (  2  _  8  }  (  a n d (2-8)  gives  ' " ' "° ^  1  ( 2-9  )  < 7 - y „ >  8  D i v i d i n g E q . (2-9)  _  flow  • °  )  V w Q  results i n  .  2  = —  s  .  V 2  y  •--  • —  y  D  y  0  2  V  • ( 1 " — • v  ) ••• ( 2-10 D  Let Z  y  and  , where Z  o  q  = y - y  Q  o  discharge  Q r = Q  ratio, , w h e r e Q = A-V ' o o o  D  and Q =  A.V  then  y y  =  1 + )8  ( 2-11  )  0  and V  T  ( 2-12  )  )  10 U s i n g Eqs. (2-11) and (2-12), Eq. (2-10) g i v e s the d i m e n s i o n l e s s equation £ - ( 1 + /3) 2  2  F  . ( 2  = °  2 ( 1 +  /3  +0) ' • 5  -r )  ( 2 - 1 3  >  2  where  o  F  =  •  V  A/7  '.O  Eq.(2-13) shows t h a t the i n i t i a l lar  wave h e i g h t i n c a n a l s o f r e c t a n g u -  c r o s s - s e c t i o n i s a f u n c t i o n o f the Froude number o f t h e i n i t i a l  f l o w and the d i s c h a r g e r a t i o . F o r a c o n s t a n t d i s c h a r g e r a t i o , the initial  wave h e i g h t i s o n l y a f u n c t i o n o f the Froude number o f  the f l o w . is  The f a m i l y o f curves c o r r e s p o n d i n g  shown on l o g - l o g paper i n F i g u r e 2-3.  number o f the i n i t i a l the gate opening  to Eq. (2-13)  G i v e n the Froude  flow, and the change i n d i s c h a r g e  o r the gate c l o s u r e , the i n i t i a l  can be o b t a i n e d d i r e c t l y  from t h i s  Q  • v  v wo  y  v  D  wave h e i g h t  graph.  S o l v i n g Eq. (2-2) f o r t h e a b s o l u t e wave v e l o c i t y , " y  after  gives  - y . V  0  - y  or v  wo  V  o  =  y~-v .( i n  y - o.( v  0  i -  °  L  ( 2-14 )  JL)  S u b s t i t u t i o n o f Eqs. (2-11) and (2-12), Eq. (2-14) can be s i m p l i f i e d to r - l  fi  •.  < 2-15 )  11 where ^'is t h e r a t i o o f a b s o l u t e flow v e l o c i t y ,  = V  \  /V . S u b s t i t u t i n g  wo  (2-13)  and s i m p l i f y i n g  (2-15)  into Eq.  o gives  o : 2X •(1 _ )  °  (  1  2  2  .  2-16 )  X  Eq.  (2-16)  V  t h e F r o u d e number F o f t h e i n i t i a l o  wo'  ratio is  Eq.  to the i n i t i a l  <X - 1 + T ) •"( 2 X - 1 + T )  2 F„ =  T  .  gives the r e l a t i o n s h i p  For a given value  between  Initial  i n Figure  S u r g e Waves  For a t r i a n g u l a r  t h e wave  velocity  flow and the discharge  o f T , the i n i t i a l  a f u n c t i o n o f t h e F r o u d e number.  graphically  2.3  wave v e l o c i t y  Eq.  wave  (2-16)  velocity  i s shown  2-3.  i n Canals o f T r i a n g u l a r  canal  Cross-Section.  ( F i g u r e 2-4), E q . (2-6) may b e  simplified to:  2 —  (  y  3  3  2 •y  • v'  v  " ° ^77^77"' o • ( - ~ )  3  y  Q  }=  1  o '  In a t r i a n g u l a r  •  V V  By u s i n g rewritten  —  Q /A  "  o  < 2  o  " > 17  canal  Q/A  Q  7  T  (1+0)  o  2  t h e r e l a t i o n above and Eq.  (2-11),  Eq.  (2-17)  can be  as:  r<i /5> -il .H .J_LI£2!_ 3  +  O  . f i - _ L _ - l  or  2 1 T ( 1 +/9 ) - [ ( 1 + ^ ) - l ] • [ ( 1 +p = ~ L 2  2  2  F  =  3 [ <l /3) +  2  - T ^  3 ) " l l  =L ... (  ( 9-18 2  1  S  12 If  E q . (2-2) i s s o l v e d  f o r a b s o l u t e wave v e l o c i t y ,  s u b s t i t u t i o n o f thedimensionless  V  W  (  O  1 + / 3 >  2  ( 1+/3 )  o  T (  (2-18)  Eq.  then  1  ( 1 + £ )  V  or  p a r a m e t e r s , T a n d /3- i s u s e d ,  . T - T ^ 2 -  2  and a s i m i l a r  -  1 (  + /S  1  - l  )  2  2 - 1 9  )  - l  shows t h a t t h e i n i t i a l  wave h e i g h t  in a  symmetrical  c a n a l o f t r i a n g u l a r c r o s s - s e c t i o n i s a f u n c t i o n o f the Froude number o f t h e i n i t i a l Eq.  (2-19)  shows t h a t t h e a b s o l u t e  t h e wave h e i g h t .  I n other  that of thei n i t i a l and  the discharge  (2-19)  2.4  Initial  f l o w and t h edischarge  ratio.  wave v e l o c i t y  i s a function of  w o r d s , t h e r a t i o o f wave v e l o c i t y t o  f l o w i s a l s o a f u n c t i o n o f t h e F r o u d e number  ratio.  The g r a p h i c a l f o r m o f E q s .  i s shown o n F i g u r e  and  2-5.  S u r g e Waves i n C a n a l s o f T r a p e z o i d a l ( F i g u r e 2-6),  In a t r a p e z o i d a l canal  (2-18)  Cross-Section.  thec r o s s - s e c t i o n a l parameters  b , A , y , a t s e c t i o n s u p s t r e a m a n d d o w n s t r e a m o f t h e wave f r o n t a r e b  = b  x  o  A  y  o  y  2m  +  ?o '  =  =  v  c  G  (  b  o  +  . y  6  ' o >  m  3 b  y  +  Q  b  n  and b„ = b + I o  2m  D  .y  2m  . y  + m • y.  P  3  A  (b  = y .  y = __ _ . 6  y  13  + m .y )  Q  3 b  + 2 m . y  n  bo + m . y J  S u b s t i t u t i o n o f t h e s e r e l a t i o n s i n t o E q . (2-4)  2  gives  2  T~ " (  3  + 2  b 0  m •y) -^  Vp-yp-y g.  -  ^ (b  Dividing  ( b  o  +  -y>-  m  m D  - y  +  •  H n  Q  -y  ) 0  -  (  :  1  0  o  v  + m.y ) . y  _2_y_  ) . J_ .  y  &  0  2 =  o  y)  2 m.  o+  /  0  v  °  )  2  J  3  y  V  b  b  by m-y r e s u l t s i n 2  b  (  + m.y ) . y - ( b  Q  •( Z_ ) :cJ^o  —  (3  '  0  b  Q  .  m.y  b  m  *  «y„ fo y„  m.y i o y  m ,  -'O  ^O  )  2  0  V  v  ( -_o_ + 1 ) . ( _ £ _ +  y  _3bo +  (  o  J  o  J L ).( 1 y^  ) V„ 2  . . ( 2-20  m.y^ J  o  Let b  ( 2-21 )  p  k - —  m.y  o t hV en  ( k + l + 0 ) . ( i  0  +  / 3  ( 2-22 )  )  V T • ( k+ 1 ) S u b s t i t u t i n g E q s . (2-21) a n d (2-22) i n t o E q . (2-20) g i v e s 1  ( 1+ /3) J3  k+2  2  2  = F„ .  (  K  j \ k + l  { (1+/5) • [ 2  F  1  +  3  )  ( l + ] 3 )]  - _L C 3 k + 2 )  £ ( k + l + / 3 ) . ( l + £ )  + /3).(l+/3)  - T .( k + 1 ) J  2  - ( k + l ) J . ( k + l + / 3 ) . ( l + / 3 )  k+2 (1+/3)] - (3k+2) | . | (k+l+/3) . (1+/3) - (k+1)] . (1+/S). (k+l+/3)  2  6(k+1). ['(k+l+/3).(l+j3) - r . ( k + l ) J  2  ( 2-23 )  14  I n a t r a p e z o i d a l c r o s s - s e c t i o n , one a d d i t i o n a l v a r i a b l e , is  a f u n c t i o n of the bottom width,  and  the depth of the i n i t i a l  ratio  flow,  side  slope  of the c r o s s - s e c t i o n  appears together  and F r o u d e number, t o d e t e r m i n e t h e i n i t i a l  Physically,  k, w h i c h  with  wave  discharge  height.  (1 + k ) i s a r a t i o o f t h e a r e a o f a t r a p e z o i d a l  c r o s s - s e c t i o n t o that o f a t r i a n g u l a r c r o s s - s e c t i o n w i t h the same d e p t h a n d s i d e  slope.  For a constant value  independent o f the i n d i v i d u a l value slope Eqs.  or the depth o f i n i t i a l  (2-11) ,and (2-22)  equation,  i t follows  V  side  I f t h e same r e l a t i o n i n  a r e used, then from t h e c o n t i n u i t y  that  (k+l)  =  o f bottom width,  flow.  (l+/S).(k+l+/3) wo  o f k, £ i s  r  .(k+l)  ( k + l + f i ).(!+#)  q + / 3O) .. ( k(k-+ l + f f )  Q  - 1  _  (k+1) or x  =  - j l - r ).(k+l)  (  2  (l+/3).(k+l+/3) - ( k + l ) I t may b e n o t e d t h a t t h e s h a p e f a c t o r , k, a l s o a p p e a r s i n t h e wave v e l o c i t y e q u a t i o n . graphically  on F i g u r e s  that the rectangular  Eqs. 2-7  (2-23)  (2-24)  a r e shown  I t should  be n o t e d  and  (a) - ( j ) .  and the t r i a n g u l a r c a n a l s  particular or limiting  cases of the t r a p e z o i d a l  are the canal.  A  _  2 4  )  trapezoidal is Eq.  c a n a l , when i t s s i d e s l o p e s  a rectangular  (2-13),  canal.  and Eq.  (2-24)  of a t r a p e z o i d a l canal triangular  (2-18),  canal.  and Eq.  When  For  (2-24)  k=  to Eq.  k = 0, to Eq.  cross-section.  Thus E q s .  (2-23)  and  This  (2-24)  and t r i a n g u l a r c a n a l s .  Ck = OC ) ,  oC , E q .  (2-23)  (2-15).  When t h e b o t t o m  i s reduced t o zero Eq.  (2-23)  (2-19).  and t h e t r i a n g u l a r c r o s s - s e c t i o n s zoidal  become v e r t i c a l  i s identical  width  (k = 0), i t becomes a i s identical  Therefore,  t o Eq.  the rectangular  a r e t h e two l i m i t s  of the trape-  c a n a l s o be s e e n f r o m F i g u r e are also valid  to  f o r both  2-7  (a) -  rectangular  CHAPTER  3  NUMERICAL CALCULATIONS FOR SURGE-WAVE  PROPAGATION  General When a n i n i t i a l  surge  wave i s g e n e r a t e d  stream  end o f a c h a n n e l ,  source  of generation with i t s velocity of propagation.  ity  of propagation  i ttravels  a t t h e u p s t r e a m o r down-  i s usually  steady-flow velocity.  immediately  c o n s i d e r a b l y i n excess  A p o s i t i v e wave c a u s e d  d i s c h a r g e has a p r o f i l e w i t h a steep f r o n t hydraulic  jump.  I f the i n i t i a l  formed w i t h a steep wave moves a l o n g  front,  i t will  by a r a p i d  s i m i l a r t o a moving  A rapid  approach  wave  b y t h e a i d o f Com-  I n t h i s approach, of continuity  the surge and  2), and t h e f l o w v e l o c i t i e s and depths  t h e u p s t r e a m a n d t h e d o w n s t r e a m o f wave f r o n t culated using the unsteady-flow  is  change o f  soon f l a t t e n o u t as t h e surge  by u s i n g t h e e q u a t i o n s  momentum ( s e e C h a p t e r  equations.  boundaries  These  t o t a l d i f f e r e n t i a l equations.  o b t a i n e d by a f i r s t  unsteady-flow  are transformed The n u m e r i c a l  order f i n i t e - d i f f e r e n c e  at  are.cal-  a r e s o l v e d by u s i n g t h e method o f c h a r a c t e r i s t i c s  the p a r t i a l d i f f e r e n t i a l equations  particular  o f t h e mean  p r o f i l e o f the n e g a t i v e surge  i s introduced i n this chapter.  wave i s d e t e r m i n e d  which  veloc  f o r the c a l c u l a t i o n of the propagation of  surge waves does n o t e x i s t .  equations  This  the channel.  A rigorous solution  puter  away f r o m t h e  by  into solution  technique.  17 3.2  Basic Assumptions The  a p p l i c a t i o n o f the method o f  f l o w i n an open c h a n n e l  characteristics  i s b a s e d on  the  to unsteady-  following simplifying  assumptions. (a)  Homogeneous w a t e r , n e i t h e r d e n s i t y c u r r e n t s n o r movement a r e  (b)  Lateral  sediment  considered.  flow i s neglected,  i . e . , one  dimensional  flow i s  considered. (c)  Friction  due  Manning's (d)  Channel  (e)  Vertical  The  a rough channel  bed  obeys  formula.  slope  e f f e c t on (f)  to the motion over  i s s m a l l e n o u g h so t h a t  cos 6 — 1 ,  component o f t h e a c c e l e r a t i o n has the  pressure  a  negligible  pressure. d i s t r i b u t i o n along  a vertical  line  is  hydrostatic.  3.3  The  Equations  of C h a r a c t e r i s t i c s  V a r i o u s ways have been u s e d f o r d e r i v i n g the e q u a t i o n s  of  (3) characteristics. are used equations  in this  The  expressions  study.  According  developed  by  V.  L.  t o S t r e e t e r , two  Streeter p a i r s of  the  of c h a r a c t e r i s t i c s , w r i t t e n f o r a gradually v a r i e d  unsteady-flow  i n an o p e n c h a n n e l ,  are  ( 3-1  )  ( 3-2  )  18 dV dt  c  fg.B " A/ A  dx  .  dy  ar*  +  -wTir • - o g  s  =  . . . . " . ' ( 3-3  0  )  ( 3_A  / g-A~  \  where w  =  the s p e c i f i c weight  of  A =  the c r o s s - s e c t i o n a l  area,  B =  the s u r f a c e width of  R =  the h y d r a u l i c r a d i u s ,  S =  slope of  bottom  T  the u n i t  shear  0  =  a  g-A  The or  =  first C  group of equations  or C  equations. second  S o l u t i o n of Equations  as ^  infinitesimal  ones are c a l l e d n e g a t i v e  The  first  char-  equation of each group i s  equation of the group i s  of C h a r a c t e r i s t i c s  satisfied.  for a Gradually Varied  c a n be  represented graphically  shown i n F i g u r e 3-1.  The  on an x - t  S represent, respectively,  downstream s e c t i o n s a t time  c a n be  c o n s t r u c t e d on  t P  and  t  and  t .  r e p r e s e n t e d by  The  of  the p o i n t s  the p o s i t i o n of c e r t a i n velocity  upstream o f wave  .s  R  propagation  plane,  p o i n t P r e p r e s e n t s the p o s i t i o n  the canal s e c t i o n under c o n s i d e r a t i o n a t time  and  characteristics  Flow  equations  R and  s i d e s of  wave.  are c a l l e d p o s i t i v e  e q u a t i o n s , w h i l e the second  Unsteady  t h e b o t t o m and  channel,  v a l i d o n l y when t h e  The  cross-section,  f o r c e on  the c e l e r i t y o f an  acteristics  3.4  water,  the  slope of the  the x - t p l a n e u s i n g Eqs.  (3-2)  and  lines  (3-4).  Point  19  R r e p r e s e n t s the p o s i t i o n of the upstream an P  infinitesimal after  wave, once d e v e l o p e d ,  the time  interval At  = t  from which  section  P after  At  = t  - t . S  Values  i n t e r s e c t i o n p o i n t P a r e o b t a i n e d by (3-3)  simultaneously.  first  order  equations  of  ...  "  .  V p  1  °  V  +  p  - V  g  (.  Xp  - x  s  ( V  =  R  V  points,  '  G R  R  - x  the  difference  ( y  RG  . ( y  - GS -  +  R  ( Vg  subscripts  " y  p  p  - SG  )  indicate  solving  ) + GNR  g  p  - t  R  ) + GNS  • ( t  p  this,  s  the  follows:  ) = 0  - t  p  and  The  as  ) p  -  t g  )  the v a r i a b l e s  ... .;.  . ( t  - t  the  (3-1)  i s used.  • ( t  at  y at  Eqs.  transformed  • ( t  - y )  o f V and  technique  R  section  downstream  will arrive  to accomplish  c h a r a c t e r i s t i c s are  (  vhere  GR  finite  In order  which  R  a wave once d e v e l o p e d  P  C  w i l l arrive at  p o i n t S r e p r e s e n t s the p o s i t i o n of the  section  j  from  - t,..  p Similarly,  section  ) = 0 "  a t the  (3-5 ( 3-6  /  £ A  RG  R  (3-7  )  .;.  ( 3-8  )  corresponding  =  ,  GNR  = g.(S  GNS  = g.(Sg  R  -  S )  -  S )  Q  and  S  R  and  S  S  According  are  the  Q  frictional  s l o p e s a t p o i n t s R and  to the Manning formula,  the  frictional  S  respectively.  s l o p e can  )  .. .  and  =  )  be  expressed  as  s =. .. " -v2  2.21  I vj R  4 / 3  where n i s the c o e f f i c i e n t of roughness on sides  (or Manning's n ) , and  V.|v|  i n d i c a t e s t h a t the f r i c t i o n a l  opposite  d i r e c t i o n o f the  and y, a t p o i n t s R and y, a t p o i n t P,  can be  S,  the c a n a l bottom  V i s the mean v e l o c i t y of  flow. the  and  flow.  r e s i s t a n c e i s always i n the  Knowing the v a r i a b l e s , x,  f o u r unknown v a r i a b l e s x,  found u s i n g e q u a t i o n s ,  Eqs.  (3-5)  t, V  t, V  and  through  (3-8).  A g r i d of c h a r a c t e r i s t i c s solution. chosen.  A particular For  s e c t i o n o f the c a n a l may  simplicity,  number o f e q u a l  i s e s t a b l i s h e d to f a c i l i t a t e a computer  lengths.  the  be  arbitrarily  canal of length L i s d i v i d e d i n t o a  The  procedure f o r  calculation  follows.  3.4.1.  Preliminary  computation  Compute the  initial  velocity  u s i n g Manning' s formula,  y  3.4.2  3,  5,  Computation f o r the Based on  0/3 «•  t , V, and  e t c . , (figure  flow  the v a l u e s o f x,  characteristic line  depth o f the  * «486 n. • •  =  s t o r e the known v a l u e s o f x, ( m , 1), m = 1,  and  RP  i n the  flow  1/2 t>  by  and  y at points P 3-2).  channel  t , V and y a t p o i n t R,  the  f o r a s h o r t d i s t a n c e can be  laid  21 out  a t a slope  o f 1/(V +RG) f r o m E q .  (3-6).  Similarly,  based on t h e values  o f x , t , V a n d y a t p o i n t S, t h e  characteristic  SP c a n b e d r a w n t o r e p r e s e n t  line  T h e s e two c h a r a c t e r i s t i c p o i n t P. by  The v a l u e s  _  X S - X B +  R  through  . ( V  R  +  (3-8)  = x  p  R  +  S  S  obtained i.e.,  -SG)  _ _ — _ — . — _ _ — .  ....VJ  -  y  j  + RG - Vg + SG  R  ( V .+  RG ) • ( t  R  V -V +GR•y +GS•y  =""  Y  simultaneously,  RG) - t . ( V  • — —  V  x  t  from R and S i n t e r s e c t a t  o f x, t , V and y a t P a r e then  s o l v i n g E q s . (3-5)  c  lines  (3-8).  Eq.  R  - t  p  -GNR.(t  R  (3  )  - t )+GNS.(t  -t )  ^ _ R _ _ J L _ s i  s  -10)  ...( 3-n  )  GR + GS  P  and V  3.4.3.  p  = V -GR-(y -y )-GNR.<t -t ) R  p  R  p  ( 3-12 )  R  Computation f o r the flow a t boundaries At  e i t h e r end o f t h e channel,  istic  equations  i s available.  ( F i g u r e 3-3-a), E q s . (3-7) downstream boundary are v a l i d .  are  F o r an upstream  boundary  a n d (3-8) h o l d , a n d f o r a  two a u x i l i a r y  boundary  needed so t h a t  one group o f t h e c h a r a c t e r -  ( F i g u r e 3-3-b), E q s . (3-5)  Therefore  from the given  only  equations  (3-6)  derived  c o n d i t i o n s a t e i t h e r boundary  f o u r unknowns c a n be  Some e x a m p l e s a r e s o l v e d  and  and i l l u s t r a t e d  solved.  as f o l l o w s :  22 E x a m p l e 3-1.  A  wide c a r r i e s  720  r e c t a n g u l a r channel  1000  f t . l o n g and  c f s water a t normal depth.  S  12 f t .  = 0.001  and  o  = 0.014.  Manning's n by  Q =  720  + 180.  i n the  Solution:  The  solution  ft. at y  long o  at y = y  .  C a l c u l a t e the  7044  i n Appendix A ( l ) .  i s shown i n F i g u r e  r e c t a n g u l a r channel  f t .  20  Channel  slope  S  o  doubled  i n 20 m i n u t e s .  a d d i t i o n a l minutes. and  of unsteady  The  i t i s one-half  flow.  The  At  the o r i g i n a l  channel  time  t  = 0,  decreased  flow i n  = 0.0185,  f o r the  10,000  linearly until i t  f l o w i s then  For Manning's n  depth i n the  system  flow condition  0.0016.  is  has  velocity  flow  A p o r t i o n of  f t . w i d e and  f l o w i s . i n c r e a s e d a t t h e u p s t r e a m end  until  the  3-4.  the  linearly  computer  i s d i s c h a r g i n g under steady-uniform  = 6.0  flow i s given  a t t h e downstream end  c o m p u t e r p r o g r a m i n IBM  the  A  the  channel.  t h i s problem i s given  3-2.  t ) and  constant  for  Example  the u p s t r e a m end,  s i n (0.03  depth i s maintained conditions  At  find  first  gage-height-discharge  40  curve  10 the min. at  the  3/2 downstream end  i s Q = 132  Solution:  computer program f o r t h i s problem i s  listed given  The  The  i n A p p e n d i x A(2), in Fig.  (y -  and  a p o r t i o n of the  solution  is  3-5.  s o l u t i o n s o f examples 1 and  t h e E x a m p l e 15.5  2.32)  and  A g o o d a g r e e m e n t was  E x a m p l e 15.6 obtained.  2 have been checked i n reference  (3).  with  23 3.5  N u m e r i c a l C a l c u l a t i o n s f o r P o s i t i v e Waves P r o p a g a t i n g  i n Power  Canals.  A  continuous  method of  unsteady  f l o w i n an open c h a n n e l can  c h a r a c t e r i s t i c s o n l y by  vertical  a c c e l e r a t i o n of  lected.  For  constitutes  surge problems,  a  neglected.  the  s u r g e waves has  not  A  the  i s s m a l l and steep  a d i s c o n t i n u i t y , and  f l o w c a n n o t be of  flow  making  come t o t h e  t h e r e f o r e can  step  the  regions  theory  author's  is  steady,  f o r the  wave f r o n t r e g i o n  f l o w and and ures end  3.5,1  characteristics itself  introduced represents  i t s s o l u t i o n m u s t be  momentum, i , e . , E q .  b a s e d on and  ( 2 - 1 )  f o r a s u r g e wave c a u s e d by f r o m an  initial  Initial  flow  The  o r i g i n of  end  ( F i g u r e 3-6).  steady  wave f r o n t , t h e f l o w  points P ( l , l ) ,  can  be  solved  equations The  by  sections.  unsteady  of c o n t i n u i t y  calculation  load r e j e c t i o n at the  proced-  downstream  flow follows.  a b s c i s s a x on  equal  channel.  condition  the  x-t plane  Assume t h a t t h e  known t h r o u g h o u t t h e reaches of  a  However,  open  rapidly varied  ( 2 - 2 ) .  the  computation  i n the p r e v i o u s  the  neg-  region  knowledge.  the  the  the  a c c e l e r a t i o n of  o r g r a d u a l l y v a r i e d u n s t e a d y f l o w and  the method of The  upstream or downstream of  be  i n t h e wave  s u r g e wave r e s e m b l e s a m o v i n g h y d r a u l i c j u m p i n a n  In  s o l v e d by  assumption that  the v e r t i c a l  rigorous  be  canal  lengths,  P(3,l),  ...  at and are  initial  t = 0. the  i s chosen at  A  flow conditions  canal  known v a l u e s  s t o r e d as  the  is divided of  x(l,l),  x,  upstream are  into  t , V and  y  x ( 3 , l ) , x(5.1),  at  24  t(l,l),  t(3,l),  y(l,l),  y(3,l), y(5,l),  at  points  t(5,l),  P ( 2 , 2 ) ,  P(3,l),  P(l,l),  V(3,l),  , V(l,l),  ... .  The v a l u e s o f x, t , V and y  P ( 4 , 2 ) ,  .... a r e o b t a i n e d  from  3.5.2  from  P ( 6 , 2 )  the equations o f c h a r a c t e r i s t i c s which  through  P ( 5 , l ) and P ( 7 , l ) .  Initial  positive  The  the points  .... e t c . , b y t h e m e t h o d o f c h a r a c t e r i s t i c s .  For example, t h e v a l u e s o f x, t , V and y a t p o i n t obtained  V ( 5 , l ) , ...  ...  are  pass  s u r g e wave a t downstream e n d o f c a n a l  equations o f continuity  a n d momentum, t o g e t h e r  with  t h e g i v e n change i n d i s c h a r g e due t o a l o a d r e j e c t i o n , a r e used t o determine x  A  wave h e i g h t a n d wave  velocity.  = L ( F i g u r e 3 - 6 ) , a n d t i s o b t a i n e d by t h e manner A  mentioned and  the i n i t i a l  i n section  Thus, t h e v a r i a b l e s x, t , V  3 . 4 . 3 .  y a t p o i n t A a r e determined.  The v a r i a b l e s a t o t h e r  p o i n t s u p s t r e a m o f t h e wave r e g i o n , P ( l , 3 ) , are determined  P ( 3 , 3 ) ,  by t h e method o f c h a r a c t e r i s t i c s .  3.5.3  Wave p r o p a g a t i o n a l o n g t h e c h a n n e l  3 . 5 . 3 . 1  Wave t r a v e l s  from  toB (Figure  A  3 - 6 )  ( a ) D e t e r m i n e t h e p o i n t B: A s s u m e t h a t t h e s u r g e wave t r a v e l s velocity  equal  t oi t s i n i t i a l  from A t o B w i t h t h e  velocity  a t A asthe f i r s t  approximation. t„ a n d x a r e t h e n o b t a i n e d b y s o l v i n g B B  equations  D  X  X  B -  B  -  X  X  E  =  (  V  E  +  A ' » < - v  (  / ~ ^  • <  )  ' E  v  (  t  B  " E > C  y  w h e r e t h e s u b s c r i p t s A , B, E i n d i c a t e  O  " 13)  ( 3 - i 4 ) t h e v a l u e s o f V, A,  25 B, x a n d t a t t h e c o r r e s p o n d i n g p o i n t s . (b)  Determine  By u s i n g  point  C  the equations of continuity  the  water v e l o c i t y  For  the f i r s t  a n d d e p t h b e h i n d t h e wave f r o n t a t B.  approximation these are not d i f f e r e n t  f r o m o n e s f o u n d a t A. (3-5) (c)  a n d (3-6) Determine  First, the  a n d momentum, d e t e r m i n e  t , V  and boundary point  the v e l o c i t y  and y  a r e o b t a i n e d from Eqs.  conditions  i n which x  = L.  D and depth a t p o i n t D a r e e s t i m a t e d , t h e n  v a r i a b l e s a t B c a n be o b t a i n e d f r o m t h o s e a t D b y t h e  e q u a t i o n s o f c h a r a c t e r i s t i c s by c o n s i d e r i n g a n wave t r a v e l l i n g  along the c h a r a c t e r i s t i c  line  infinitesimal DB a n d  r e a c h i n g p o i n t B a t t h e same t i m e a s a s u r g e wave from A t o B a l o n g AB.  travelling  i s obtained f o r the f i r s t  a p p r o x i m a t i o n by t h e e q u a t i o n  B where t h e a p p r o x i m a t e v a l u e s o f p a r a m e t e r s x, t , V and y at  B are determined  i n a b o v e p r o c e d u r e s , and x ^ = L.  Then, y ^ i s o b t a i n e d by l i n e a r  interpolation  from  those  at  A a n d C, a s s u m i n g t h a t t h e r a t e o f c h a n g e o f w a t e r  at  t h e downstream end d u r i n g t . to., t ^ i s l i n e a r A D  respect the  to time.  with  W i t h t h i s new v a l u e o f y ^ , d e t e r m i n e  new v a l u e o f t ^ a l o n g c h a r a c t e r i s t i c  Continue t h i s  surface  iteration until  line  BD.  t h e two s u c c e s s i v e  relative  v a l u e s o f t ^ meet t h e r e q u i r e m e n t , s a y , t h e d i f f e r e n c e  less  26  than 0.001. (d)  Determine  Eq.  and  by u s i n g c h a r a c t e r i s t i c  ( 3 - 7 ) , a l o n g DB and t h e e q u a t i o n s  equation,  o f momentum and  c o n t i n u i t y a t B. (e)  Compute t h e new v a l u e o f wave v e l o c i t y a t B (Eq.  (2-3)  ), and average the wave v e l o c i t i e s a t A and B.  T h i s i s t h e new wave v e l o c i t y w i t h which t h e wave t r a v e l s from A to B. (f)  W i t h t h e new wave v e l o c i t y o b t a i n e d from (e) r e p e a t  the p r o c e d u r e s  o f c a l c u l a t i o n from (a) t o (e) u n t i l the  d i f f e r e n c e s o f two s u c c e s s i v e c a l c u l a t e d r e l a t i v e  values  o f V and y a t B, C, and D a r e l e s s than 0.001. (g)  Determine t h e v a r i a b l e s a t a l l n e c e s s a r y  upstream o f the wave f r o n t  points  ( a t p o i n t B) by t h e method o f  characteristics.  3.5.3.2.  Wave t r a v e l s Using  from B to G, L e t c .  the c a l c u l a t i o n  process  similar  to t h a t i n s e c t i o n  3.5.3.1. the v a r i a b l e s at G and F a r e o b t a i n e d . The v a l u e s of V and y a t C a r e g i v e n from the p r e v i o u s s e t o f The  new v a r i a b l e s at H a r e o b t a i n e d  C by e q u a t i o n s  of c h a r a c t e r i s t i c s .  from those  at G and  I t i s necessary i n  t h i s s t e p t o c a r r y on the c a l c u l a t i o n at  computation.  f o r the o t h e r p o i n t s  the downstream end, such as I ( F i g u r e 3-6) i n t h i s  step,  27  along  the c h a r a c t e r i s t i c  l i n e GH t o p r o v i d e t h e i n f o r m a t i o n f o r  c a r r y i n g on t h e n e x t s t e p o f c a l c u l a t i o n By  this  along  f o r wave p r o p a g a t i o n .  p r o c e d u r e , a t r a c e o f t h e p r o p a g a t i o n o f t h e s u r g e wave  the prismatic  f r o n t reaches  c a n a l i s o b t a i n e d a n d t e r m i n a t e s when t h e wave  the upstream  for a positive  reservoir.  s u r g e wave, t h e v e l o c i t y  I t s h o u l d be n o t e d  that,  o f wave p r o p a g a t i o n i s  always g r e a t e r than t h a t o f an i n f i n i t e s i m a l  wave i n t h e r e g i o n  b e f o r e t h e f r o n t o f t h e s u r g e wave, and l e s s  than t h a t o f an  infinitesimal  wave i n t h e r e g i o n b e h i n d t h e f r o n t o f t h e s u r g e  w a v e , i . e . , t h e i n v e r s e s l o p e o f t h e l i n e KE i s l e s s v e r s e s l o p e o f t h e l i n e AB, w h i c h slope of l i n e along  CH.  the canal,  other.  Eventually,  the o t h e r .  lines  than the inverse  a s a s u r g e wave t r a v e l s c o n t i n u o u s l y  these l i n e s would converge  Because p o s i t i v e  characteristic  i s again less  than the i n -  and i n t e r s e c t  each  s u r g e waves h a v e an a b r u p t f r o n t , t h e  c a n n o t p r o j e c t f r o m one s i d e o f t h e s u r g e t o  When t h e c h a r a c t e r i s t i c  lines  b e f o r e and b e h i n d t h e  wave r e g i o n a n d t h e t r a c e o f wave p r o p a g a t i o n m e e t , ( s e e f i g u r e 3-6),  f l o w parameters  a t M a r e d e t e r m i n e d f r o m P, a n d t h o s e a t P  a r e o b t a i n e d f r o m p o i n t s 0 a n d Q.  The  f l o w c h a r t o f t h e computer program f o r t h i s  shown o n F i g u r e 3-8. end will  The  and t r a v e l l i n g be s i m i l a r  computation i s  F o r a p o s i t i v e wave o c c u r r i n g  a t the upstream  t o t h e downstream end, t h e c h a r a c t e r i s t i c  to these i n F i g u r e 3 - 7 .  process of computation f o r a p o s i t i v e  s u r g e wave  starting  lines  a t the downstream  end.and p r o p a g a t i n g upstream i n the c a n a l  has been programmed i n FORTRAN IV f o r the IBM  7044 computer.  Examples have been s e l e c t e d to examine the computer Parallel  hand c a l c u l a t i o n s . b y  the F a v r e method  program.  a r e shown f o r  comparison.  Examp1e 3.3.  A r e c t a n g u l a r channel 38,800 f t .  wide and 41.175 f t . 7.101  ft/sec.  S  q  l o n g , 45 f t .  deep c a r r i e s a f l o w w i t h a v e l o c i t y a t  = 0.0002376.  stopped a t the downstream  Suddenly, the f l o w i s complete;  end by c l o s i n g a g a t e .  Compute the  i n i t i a l wave h e i g h t , and wave h e i g h t when i t r e a c h e s  the  u p s t r e a m end.  Solution:  The r e s u l t s o f computer computation u s i n g  o f c h a r a c t e r i s t i c s g i v e Z^ = 8.39 Z  o  and Z a r e the downstream  ft.  and Z = 4.23  method  ft.,  where  and upstream wave h e i g h t s  ...respectively, ( F i g u r e s 3-9 and 3-10.). D e t a i l s o f the  _ The the  computations a r e shown g r a p h i c a l l y  alternative  calculation  F a v r e method  f o r t h i s example  gave Z = 4.15 f t .  upstream end o f c a n a l .  on F i g u r e  3-11.  c a r r i e d out by  f o r the wave r e a c h i n g the  D e t a i l s a r e shown on  Fig.3-12 .  Comparing the r e s u l t s  from the two d i f f e r e n t methods,  fairly  good agreement  i s observed.  i n the  F a v r e method,  I t s h o u l d be n o t e d  a that,  the water s u r f a c e i n t h e downstream  area i s assumed s t r a i g h t l i n e when the wave f r o n t r e a c h e s the  upstream end.  29  When t h e wave r e a c h e s t h e r e s e r v o i r ,  the a c t u a l water  for  the case o f complete c l o s u r e i s f a i r l y  tal  line  sloped  i n t h e lower  end o f c a n a l , w h i l e  surface  close to a horizoni t i s somewhat  i n t h e u p p e r e n d . F r o m F i g u r e 3-11,  i t c a n be s e e n  t h a t t h e v a r i a t i o n o f wave h e i g h t when i t t r a v e l s a l o n g t h e channel  i snearly linear  and t h e r e f o r e Favre's  assumption i s  justified.  E x a m p l e 3.4  A r e c t a n g u l a r channel  1000  f t . l o n g a n d 12 f t .  wide c a r r i e s  720 c f s w a t e r a t n o r m a l d e p t h .  S  = 0.001 a n d o  = 0.014.  Manning's n ly  shut  off.  propagation Solution: 3-13,  Carry  The f l o w a t t h e downstream e n d i s sudden-  o u t t h e c a l c u l a t i o n o f f l o w c o n d i t i o n s and  o f s u r g e wave a l o n g  the channel.  T h e r e s u l t s o f c a l c u l a t i o n a r e shown o n F i g u r e  and have been compared w i t h  t h e example o n page  258,  r e f e r e n c e (3).  3.6  Numerical  C a l c u l a t i o n s f o r Negative  Waves P r o p a g a t i n g  i n Power  Canals When a p o s i t i v e wave r e a c h e s t h e u p p e r e n d o f t h e c a n a l , i t i s r e f l e c t e d as a n e g a t i v e end.  I f t h e wave h e i g h t  wave a n d p r o c e e d s t o w a r d s t h e d o w n s t r e a m i s s m a l l o r moderate compared t o t h e  depth,  i t c a n be a s s u m e d t h a t t h e s t e e p wave f r o n t i s r e t a i n e d  during  the travel  be  applied without  and the equations  d e r i v e d f o r a p o s i t i v e wave c a n  introducing significant  e r r o r s . The h e i g h t o f  30 the n e g a t i v e wave r e f l e c t e d relatively wave w i l l tions  3.6.1  small.  from the r e s e r v o i r  i s usually  On the assumption t h a t the p r o f i l e o f a n e g a t i v e  n o t change s i g n i f i c a n t l y , , a scheme f o r computer  computa-  i s u s e d as f o l l o w s . •  Determination  o f the i n i t i a l  r e f l e c t e d n e g a t i v e wave  Assuming t h a t t h e r e i s no e n t r a n c e recovery behind  l o s s and no v e l o c i t y  ( s e e s e c t i o n 4-7), t h e water l e v e l  head  immediately  the wave f r o n t o f t h e r e f l e c t e d n e g a t i v e wave a t the  extreme upper end o f channel reservoir.  » " where y initial  o  i s the same as t h a t i n the  I t follows that  . *  y  < 3-16  and V a r e the water depth and v e l o c i t y o steady  flow  ( F i g . 3-14  )  o f the  (a) ) . A t the r e s e r v o i r ,  which i s t h e upstream boundary o f the c a n a l , X  - K D  = 0 and y  = H.  K  according to  The v e l o c i t y o f t h e n e g a t i v e wave i s  Eq. (2-3). g.(A .y - A . + ... / ' , A , . ( 1 - A,/A^1 2 ) 2  V  = _V,  w  1  2  1  L  >  X  y  i  ) • . . • ( 3-17 )  1  /A  where last  and  ( F i g u r e 3-15)  a r e g i v e n i n the  step o f the c a l c u l a t i o n i n the p r o p a g a t i o n o f t h e  p o s i t i v e wave.  By u s i n g the e q u a t i o n o f c o n t i n u i t y ,  i s obtained, i . e . :  V < 2  -w >  V  1  and  ....  V  i  A  +  v  w  t h e h e i g h t o f the n e g a t i v e wave i s  ( 3-18 )  ' where  = H  Here Z has n e g a t i v e v a l u e which  indicates  R e f l e c t e d N e g a t i v e Wave P r o p a g a t e s  Along  t h e n e g a t i v e wave.  t h e Channel  D e t e r m i n a t i o n o f t h e n e g a t i v e wave t r a v e l l i n g  from  R to P  ( F i g u r e 3-16.) (a)  Assume f o r t h e f i r s t  travels  P  t h a t t h e wave  f r o m p o i n t R t o p o i n t P ( F i g u r e 3-16) w i t h  a velocity t  approximation,  equal  to i t s i n i t i a l  a n d x by s o l v i n g P  velocity  a t R.  Determine  equations  (3-20)  )  Then, c a l c u l a t e y^ (b)  and  p  - t  E  )  / ... ( 3 - 2 1 )  t h e v a r i a b l e s d o w n s t r e a m o f t h e wave  , by l i n e a r  e x t r a p o l a t i o n from  Determine the water  f r o n t , y^ a n d  • ( t  depth  front,  t h o s e a t E a n d S.  and v e l o c i t y  behind  t h e wave  by u s i n g t h e e q u a t i o n s o f c o n t i n u i t y and  momentum a n d v a l u e s e x t r a p o l a t e d a b o v e . (c)  D e t e r m i n e t h e new v a l u e o f wave v e l o c i t y  (3-17).  Further determine  propagating  from  a t P by E q .  t h e new v a l u e o f wave  R t o P by a v e r a g i n g  velocity  the v e l o c i t i e s  a t R and  P. (d)  Repeat t h e procedures  accuracy  i s reached.  (a) through  (c)u n t i l  the desired  32 Ce)  Determine the v a r i a b l e s  istics (f)  f r o m p o i n t s R a n d P.  Determine the v a r i a b l e s  istics  3.6.2.2.  a t D by e q u a t i o n s o f c h a r a c t e r -  a t Q by e q u a t i o n s o f c h a r a c t e r -  through D Q and the boundary c o n d i t i o n s .  D e t e r m i n a t i o n o f t h e n e g a t i v e wave t r a v e l l i n g (Figure (a)  f r o m R t o P, e t c .  3-17)  D e t e r m i n e t h e v a r i a b l e s y„ a n d V b  E  , a t E, b y e q u a t i o n s o f  characteristics. (b)  D e t e r m i n e t h e wave h e i g h t a n d v e l o c i t y  procedures (c) of  m e n t i o n e d above i n  Determine the v a r i a b l e s  Continue  reaches  these  3.6.2.1.f.  a t p o i n t s F, Q,  K by  equations  computing processes  s h o u l d be n o t e d  of  wave p r o p a g a t i o n  surge  i s always  as a n e g a t i v e surge  t h a t f o r an  infinitesimal  wave a n d g r e a t e r  wave i n t h e r e g i o n b e h i n d t h e  the unsteady  i s logical  i s g r e a t e r t h a n t h a t o f l i n e RF. travels  rise  behind  3-17) There-  downstream c o n t i n u o u s l y a l o n g  l i n e s would d i v e r g e .  f r o n t o f a n e g a t i v e surge  depth  than  w a v e , i . e . , t h e i n v e r s e s l o p e o f t h e l i n e RE ( F i g u r e  the c a n a l , these that  less  the f r o n t of the surge  t h a t f o r an i n f i n i t e s i m a l  g r e a t e r t h a n RP w h i c h  fore,  t h e n e g a t i v e wave  t h a t , f o r a n e g a t i v e s u r g e wave, t h e v e l o c i t y  wave i n t h e r e g i o n b e f o r e than  until  t h e downstream end.  It  it  through  characteristics.  (d)  is  3.6.2.1.a  a t P by u s i n g t h e  The i n v e s t i g a t i o n shows  the p o s i t i v e  surge  varies nearly linearly.  or i n the Therefore,  t o assume t h a t t h e v a r i a t i o n o f v e l o c i t y a n d  o f t h e f l o w i n t h e f r o n t o f t h e n e g a t i v e wave a t P i s  33  linear with respect t , p  without  to time,  i n a s h o r t time  introducing appreciable  errors.  p e r i o d f r o m t„ E  to  34 CHAPTER .4 RESULTS OF ANALYSIS 4.1  General In p r e v i o u s  chapters  oped t o c a l c u l a t e  equations  and techniques  t h e magnitude o f t h e i n i t i a l  t h e wave h e i g h t a t a n y p o i n t a s t h e s u r g e along  the canal.  The p u r p o s e o f t h i s  dimensionless  governing  t h e v a r i a t i o n o f wave h e i g h t ,  4.2.. D i m e n s i o n l e s s When s o l v i n g  flow  relationships  between t h e v a r i a b l e s the distance o f propa-  h y d r a u l i c t r a n s i e n t problems, i t i s convenient t o  u s u a l l y reduces  ratios.  The i n t r o d u c t i o n o f s u c h  t h e number o f v a r i a b l e s i n v o l v e d i n t h e the solution.  v a r i a t i o n o f wave h e i g h t ,  Z  I n the problem o fthe  , o f a surge  wave  propagating  a power c a n a l , t h e i n d e p e n d e n t v a r i a b l e s i n v o l v e d a r e  the depth  y , velocity  V  q  of the i n i t i a l  and b o t t o m w i d t h b , c a n a l l o n g i t u d i n a l Q  distance x and  i s t o f i n d ap-  Ratios  problem and s i m p l i f i e s  along  propagates  parameters.  make u s e o f d i m e n s i o n l e s s ratios  wave h e i g h t a n d  wave  chapter  propriate  g a t i o n and"other  have been d e v e l -  o f wave p r o p a g a t i o n ,  the i n i t i a l  flow, side slope bed slope S  gravitational  q  ,  1/m  the  a c c e l e r a t i o n g,  wave h e i g h t Z , i . e . ,  ° Z = f, ( g , y , V ,S ,m,b ,x*,Z ) 1 It  should  o  w  o  be n o t e d  o  factor  o  (4-1)  o  that the f r i c t i o n a l  Eq. (4-1) b e c a u s e , g i v e n friction  o  effect  i s included i n  t h e p a r a m e t e r s in t h i s  c a n be c a l c u l a t e d  from Manning's  equation, the formula.  3 5  The to  effect of gravity those  i s represented  of gravity.  This r a t i o  by a r a t i o o f i n e r t i a l  forces  i s g i v e n by t h e F r o u d e number,  d e f i n e d as  F  =_Zo_  o  where flow,  a n d y ^ a r e t h e mean v e l o c i t y g i s the g r a v i t a t i o n a l  most i m p o r t a n t  dimensionless  and depth o f t h e i n i t i a l  acceleration. ratio  This i s normally the  i n open channel  and  i s w e l l - k n o w n a s F r o u d e " s Law.  The  r a t i o x*/L i s adopted f o r the d i m e n s i o n l e s s K  t h e d i s t a n c e o f wave p r o p a g a t i o n , propagation  problems  abscissa of  where x* i s the d i s t a n c e o f  o f a s u r g e wave f r o m t h e downstream  end, and L  K.  i s  the r e f e r e n c e l e n g t h o f channel, i . e . ,  Physically,  L  i s the length of a h o r i z o n t a l  K  line  passing  t h r o u g h t h e w a t e r s u r f a c e a t t h e downstream end and t h e c a n a l b o t t o m a t some p o i n t u p s t r e a m ( s e e F i g .  The  ratio  Z/Z  i s used  q  to represent  4-1).  the r e l a t i v e value of the  wave h e i g h t a t a n y s e c t i o n o f c a n a l t o i t s i n i t i a l the downstream end.  The s h a p e f a c t o r k r e p r e s e n t s  section characteristics, noted, of  the  that  the  Froude  w h e r e k = b /m.y . o o initial wave h e i g h t Z  number  F  o  and  water  value a t the cross-  I t s h o u l d be Q  is  depth  a y  o  function of  the  36  initial  Z  = f  to  (F , x*/L ,k)  2  r »  f a c t o r k o f t h e c a n a l and d i s c h a r g e r a t i o  t h e E q . ( 4 - 1 ) may b e r e d u c e d  Thus,  It  flow, section  q  (4-2)  R  o  i s thus  a s s u m e d t h a t t h e v a r i a t i o n o f Z/Z  i6 a f u n c t i o n o f F o  and  x /L  R  f o r a c o n s t a n t v a l u e o f k.  n  a t i o n o f surge for  o  The a n a l y s i s  f o rthe v a r i -  waves f o r d i f f e r e n t v a l u e s o f k w i l l  canals of rectangular, triangular  be d i s c u s s e d  and t r a p e z o i d a l  cross-  sections, separately, 4.3  Variation of Positive  S u r g e Wave H e i g h t  i n Rectangular  In a r e c t a n g u l a r c a n a l , t h e c r o s s - s e c t i o n a l the w i d t h zero Eq.  and depth  of the flow.  In this  ( v e r t i c a l w a l l ) and t h e f a c t o r k ( 4 - 2 ) i s the reduced  a r e a i s governed by  case, m i s equal to  i s equal  to i n f i n i t y .  to  = f_ (F , x * / 0  Z o  3  In o r d e r and  S  o  o  to determine  were k e p t  frictional The and  i n Eq. ( 4 - 3 ) ,  the r e l a t i o n s h i p s  V  coefficient,  variables  i s determined  q  and V  o  by F r o u d e  n, i s d e t e r m i n e d  a n a l y s e s were c a r r i e d systematically  (4-3)  R  as independent  dependent v a r i a b l e s .  1  intervals,  as  o  s Jaw a n d t h e  from Manning's  formula.  o u t b y v a r y i n g t h e v a l u e s o f F^, y > b  i n addition  When y ^ v a r i e s  the results  F , y , b o o o  , n and Z  o  Q  to giving values of the i n i t i a l  change i n d i s c h a r g e and c a n a l l e n g t h as t h e i n p u t d a t a computer program.  Canals  from  o f computer runs  f o r the  5 f t . to 40 f t . at 5 f t . f o rg i v e n values o f F , o  37  b  o  and  S , which resulted o  t h e d o w n s t r e a m end dimensionless ordinate. Z/Z^  and  i n a p o s i t i v e wave, t h a t o c c u r r e d a t  t r a v e l l e d upstream, are p l o t t e d Z/Z o  p l a n e w i t h x*/L_, a s a b s c i s s a a n d R  The p l o t  on a as  shows t h a t t h e r e l a t i o n s h i p x * / L  does n o t vary, a l t h o u g h  the v a l u e o f  versus  varies.  Similarly,  for given values o fF , y andS , the v a l u e o f b i s v a r i e d o o o o f r o m 1 f t . t o 60 f t . a t 5 f t . i n t e r v a l s , a n d f o r g i v e n v a l u e s o f S  o  i s v a r i e d f r o m 0.0001 t o 0.01 a t 0.0005 i n t e r v a l s .  relationship, • F  o  Z / Z does n o t v a r y . o  x-'/L^. v e r s u s R  i s v a r i e d , while keeping y  ship curve  d o e s =vary.  , S and o o  Z / Z a g a i n s t x*/L  l a t i o n curve  o  H o w e v e r , when  b constant, o  I t i s therefore concluded  Again, the  the  relation-  t h a t the  varies only with F K  re-  and i s  o  independent o f the i n d i v i d u a l v a l u e s of y , b and S , as i n d i c a t e d o o o i n Eq.  (4-3).  S i m i l a r computer runs were c a r r i e d out  v a l u e s o f F f r o m 0.050 t o 0.200 a t 0.025 i n t e r v a l s . o are p l o t t e d  o n F i g . 4-2.  I t can be seen from t h i s  f o r a g i v e n F. , t h e c u r v e o values o f Z / Z . o Z/Z  q  is initially  gradually. infinity lower  and  Theoretically,  before  limit (a)  rapid  is  s  e  t  nearly constant  disappears.  t o 0.025.  t h e n ..decreases  Inthis  study, t h e  Z / Z a g a i n s t x^/L^, v a r i e s w i t h F o R o  I t i s n o t e f f e c t e d by the change o f i n d i v i d u a l  longitudinal the  and  T h u s , i n summary:  v a l u e s o f o t h e r parameters such  of  a t higher  a wave w o u l d n e e d t o p r o p a g a t e t o  The r e l a t i o n c u r v e ,  only.  that,  shows t h a t t h e r a t e o f v a r i a t i o n o f  i t completely  o f Z/Z^  the  Results  figure  i s nearly a straight line  The f i g u r e  for  as the c r o s s - s e c t i o n ,  slope o f the channel,  channel.  and the  roughness  38  (b)  For a constant F  o  , the higher w i l l  not  imply,  reach  of Z / Z , the higher  value  q  the value of  be t h e v a l u e o f x * / L . K  t h a t t h e wave t r a v e l s  a longer  This  does  distance  to  a g i v e n r e d u c t i o n i n wave h e i g h t , b e c a u s e a f l o w  w i t h a h i g h e r v a l u e o f F^ i s a l w a y s a s s o c i a t e d with a larger longitudinal  slope S  q  either  and a s m a l l e r  of L , o r - w i t h a s m a l l e r roughness of channel K  value  and a  s m a l l e r d e p t h o f f l o w y , and thus w i t h a s m a l l e r L„. o R c  4.4  J  V a r i a t i o n o f P o s i t i v e Surge-Wave H e i g h t A canal with a triangular practice.  I t i s a limiting  case of the t r a p e z o i d a l c r o s s -  Eq.  In this  case,  ( 4 - 2 ) i s then  analysis 4-3  i s zero.  Two v a r i a b l e s , d e p t h  side slope, are involved i n controlling  area.  similar  was u s e d .  is valid  the cross-section  t h e shape f a c t o r k i s e q u a l  again reduced to Eq. ( 4 - 3 ) .  Similar relation  curves  canal.  to zero. A  to that f o r rectangular canals  f o r a triangular  systematic  i n section  were o b t a i n e d .  Z/Z  a g a i n s t x*/L o  on  4-3(a)  and  (b) a l s o h o l d .  side slope of the canal.  p o s i t i v e wave p r o p a g a t i n g 0.05  to  0.20  at  0.025  Eq. ( 4 - 3 )  The c h a r a c t e r i s t i c s o f t h e  v a r i a t i o n o f r e l a t i v e wave h e i g h t  section  Canals  c r o s s - s e c t i o n i s n o t o f t e n used i n  s e c t i o n where t h e bottom w i d t h and  i n Triangular  in K  The c u r v e  does n o t depend  The r e s u l t s o f t h e v a r i a t i o n o f a  along  intervals  a triangular canal with F are  plotted  in Fig.  4 - 3 .  q  from  39  4.5  V a r i a t i o n o f P o s i t i v e Surge Wave H e i g h t " i n T r a p e z o i d a l The  t r a p e z o i d a l c r o s s - s e c t i o n i s one o f the most common shapes  used f o r c a n a l s .  For t h i s case,  the c r o s s - s e c t i o n a l a r e a i s  governed by depth y , bottom w i d t h b  and s i d e s l o p e 1/m. The  o  o  shape f a c t o r k can v a r y between 0 and cc of  Z/Z  curve  a g a i n s t x*/L  values  to t h a t f o r r e c t a n g u l a r  show t h a t t h e  does n o t change f o r f i x e d v a l u e s o f  T h e r e f o r e , Eq. ( 4 - 2 )  and k.  The r e s u l t s  K  O q  For given  t r i a n g u l a r c a n a l s i s used, i . e . , the v a l u e o f one v a r i a b l e i s  changed and t h e o t h e r s h e l d c o n s t a n t .  F  .  F^ and k, a s y s t e m a t i c a n a l y s i s s i m i l a r  and  Canals  i svalid  f o r a surge  variation  i n a trapezoidal canal. For  g i v e n v a l u e s o f F , and f o r v a l u e s o f k = 0 , 0 . 3 3 3 ,  1.000,  1.667, 2 . 5 0 0 ,  Results for F of  curves,  higher k  For  5 , 0 0 0 and cc surge c a l c u l a t i o n s were made.  = 0.200,  a r e p l o t t e d on F i g . 4 - 4 .  i t c a n be seen t h a t the h i g h e r  i s the v a l u e o f x * / L  = oc  those  o  R  for a given Z / Z . q  and k = 0 a r e two l i m i t i n g c u r v e s .  From t h i s f a m i l y  t h e v a l u e o f k, the The c u r v e s  drawn f o r a r e c t a n g u l a r and a t r i a n g u l a r c a n a l  a g i v e n k, the r e l a t i o n c u r v e  Z/Z  For  with  These c o i n c i d e w i t h respectively.  v e r s u s x*/L , v a r i e s w i t h  o F^.  0.500,  K  each g i v e n v a l u e o f k, t h e r e i s a f a m i l y o f c u r v e s  corresponding  to various values of F .  For r  = l + l / ( l + k ) having  40  v a l u e s o f 1.50  a n d 1.75,  the r e s u l t s  f o rF  v a r y i n g from  0.05  o  0.20  to  ively,  0.025  at  where  intervals  r  a r e shown o n F i g s .  4.6  Approximate As  fore  i s nearly  R  f o r Z/Z  o  exponential best  respect-  apply f o r a trapezoidal  the relation  canal.  t h e c u r v e may be  line without introducing  t h a n 0.6,  function.  f i t by t h e " l e a s t  curve Z / Z  q  against  a t h i g h e r values of Z/Z . o  line  v a l u e s g r e a t e r t h a n 0.6,  i s less  o  earlier,  a straight  i m a t e d by a s t r a i g h t When Z/Z  4-6  Equations  i thas been mentioned  x*/L  and  (4-6).  i s g i v e n i n Eq.  T h e r e m a r k s i n s e c t i o n 4.3(b) a l s o  4-5  There-  approx-  significant  errcra.  t h e c u r v e may b e e x p r e s s e d b y a n  The a p p r o x i m a t e d  formulas representing the  square" method a r e g i v e n i n t h e f o l l o w i n g  sections: 4.6.1  Approximate equations f o r the v a r i a t i o n o f a p o s i t i v e s u r g e wave p r o p a g a t i n g a l o n g a r e c t a n g u l a r c a n a l (a)  F o r Z_  g r e a t e r t h a n 0.60  the equation suggested i s  Z  / Z_ Z  = 1.0 + 0.1876  1.0 \  1.0 0.3899 F - 0.00037 o  (4-4a) (b)  F o r JZ_  Z Z  Z~ o  less  t h a n 0.60  the equation suggested i s  o = A + B .exp ( C . x * / L ) D  R  (4-4b)  X * )  ( L  R  41  where A  - 0.01589  26.54 F - 6.7334 o  B = 1.167 + 0.6728 F C = 3.045 - 1.702 F Eqs. It  (4-4a)  and  (4-4b)  o  -0.7985  a r e p l o t t e d on F i g .  4-2  f o r comparison .  can be seen t h a t the v a l u e s e s t i m a t e d from the e q u a t i o n s a r e  good a p p r o x i m a t i o n s .  When a surge wave t r a v e l s a l o n g the  channel w i t h a g i v e n i n i t i a l is different  f l o w f o r which the Froude number  than those shown on F i g .  (4-2),  the v a r i a t i o n o f  the wave h e i g h t may be o b t a i n e d from the above e q u a t i o n s .  4.6.2  Approximate  e q u a t i o n s f o r the v a r i a t i o n o f a p o s i t i v e  surge wave p r o p a g a t i n g a l o n g a t r i a n g u l a r (a)  canal.  F o r _Z_ g r e a t e r than 0.60 use e q u a t i o n Z  o  Z_ Z  1.0  = 1.0 + 3.041  1.0 0.4358 F + 0.000092 o  (x* )  '  (4-5a) (b)  F o r _Z_ Z  l e s s than 0.60 use e q u a t i o n  o  Z  = A + B .exp (C.x*/L  where A  (4-5b)  )  = - (0.0352 + 1.479  ;  .F  o  )  B = 0.642 . F + 1.205 o  C=  (0.7485 . F -1.135  0.6106)  42  (4-5a)  Eqs. surge  (4-5b)  and  propagating  w a v e h e i g h t may  4.6.3  are also plotted  on F i g .  4-3.  For a  along a t r i a n g u l a r canal, the v a r i a t i o n of  be p r e d i c t e d f r o m  these  equations.  Approximate equations f o r the v a r i a t i o n of a p o s i t i v e wave p r o p a g a t i n g a l o n g a t r a p e z o i d a l c a n a l . F r o m F i g . 4-4, to  infinity  i t i s seen t h a t the v a l u e o f k v a r i e s  mathematical  substitute  f o r k, w h e r e ,  zero  treatment,  t h e p a r a m e t e r r was u s e d t o  1 +  =  1  +  (4-6)  k  T h u s r = 1 w h e n k = OC varies  from  f o rvarious sizes of trapezoidal cross-section canals.  To s i m p l i f y  r  surge  , a n d r = 2 when k = 0.  between 1 and 2 f o r t r a p e z o i d a l  a given F  a n d Z/Z  o  nonlinear .  cross-sections.  , t h e v a r i a t i o n o f x*/L_  A parameter  versus  R  may be d e f i n e d , s u c h  k  For is  that  )  (~  <f> =  o  The v a l u e o f r  (4-7)  R r = n  (—) i n which  ( x  T  ) r  R  appropriate  =  .  i  s the value of ( x  ~  1  )  R  t o t h e c u r v e o f r = 1, i . e . , f o r a r e c t a n g u l a r *<.  canal:  and ( x  trapezoidal  )  i s that f o r the curve  c a n a l f o r a g i v e n Z/Z^.  a g a i n s t r on l o g - l o g paper, points  located approximately  F i g . 4-7,  Values  of r = n of a given  of  were  a n d i t was f o u n d  on a s t r a i g h t l i n e .  plotted that the  Therefore, i t  43  may  be c o n c l u d e d  logarithmic  that the r e l a t i o n s h i p  function.  <f>; = c r "  I t may  all  tp a n d r i s a  be a s s u m e d t o be  (4-8)  d  where c and d a r e t h e c o e f f i c i e n t s relationship  between  holds  t o be d e t e r m i n e d .  f o r v a r i o u s v a l u e s o f Z/Z  o  This  and F . B e c a u s e o  c u r v e s h a v e t o p a s s t h e p o i n t o f x / L ^ = 0 a n d Z/Z = R o  i . e . , when r = 1.0  a n d tp = 1.0,  (4-8) i s s i m p l i f i e d  <t> =  r ~  . . .  d  (4-7).  Using  = r -°' this  Thus Eq.  (4-9)  to the negative  The e x p o n e n t d, d e t e r m i n e d  method i s e q u a l  *  t o 1.0.  to  where the v a l u e o f d i s equal on F i g .  c i s equal  1.0,  t o 0.565.  E q . (4-8)  then  slope of the curve  by t h e l e a s t  square  becomes  (4-10)  5 6 5  equation,  the value of  the shape f a c t o r k i s g i v e n . p r o p a g a t i o n of a surge  <p may  From Eq.  be o b t a i n e d  (4-7),  provided  the d i s t a n c e of  f o r a c e r t a i n amount o f r e d u c t i o n o f w a v e  h e i g h t i n a t r a p e z o i d a l c a n a l c a n be p r e d i c t e d by  (4-11) (  ) r = n  where / x ( L  R  provided F  = *{  \ ) r = q  ) r = 1  may  i s given.  ( 4 - 4 a ) , (4-4b) and  be o b t a i n e d f r o m  Eqs.  (4-4a) o r (4-4b)  Comparison o f the e s t i m a t e from  (4-11) w i t h the computer r e s u l t s  Eqs.  are shown  44  on F i g s .  4-5  a n d 4-6.  a p p l i c a t i o n o f these  Example Given  of  formulas.  4-1  = 0.1,  F o  For  Two e x a m p l e s a r e shown t o i l l u s t r a t e t h e  = 0.001, m = 1.0,  S o  the case o f t o t a l  b o  = 10  c l o s u r e and f o r Z / Z  p r o p a g a t i o n o f a surge moving along  q  f t , y o  = 10  f t .  = 0.3,find t h e d i s t a n c e  a trapezoidal  canal.  Solution  L- = o  = 10,000 f t .  y  S~ o  R  k  = ^o_ m.y o  r  = 1 +  when F  o  = 1.0  1 1+k  = 0.1,  = 1.5 and Z / Z  o  = 0.3,  i t i s found  from F i g . °  4-4  * = 0.1715  l  = r-°'  =  5 6 5  /x_\  0.7958  = 0.1715  U i r = 1.5  . (0.7958) = 0.1365  R  x*  = 1365 f  From F i g .  t .  4-5,  ) r = x*  = 1370  I t may is  = 0.1370  / x^ \ 1.1  f t .  be s e e n t h a t t h e d i s t a n c e e s t i m a t e d f r o m p r o p o s e d  fairly  accurate.  formula  45  The or  above e q u a t i o n s  may a l s o be u s e d  to estimate  the d i s t a n c e o f p r o p a g a t i o n o f a surge  angular canal i n which  Example Given F find  r  = 2.0  and  <f>  t h e wave h e i g h t  travelling  in a  tri-  = 0.6761.  4-2 o  = 0.125,  Z/Z  = 0.30,  o  S  o  = 0.001,  Y  t h e d i s t a n c e o f p r o p a g a t i o n o f a surge  o  = 10  ft.,  wave t r a v e l l i n g  a triangular canal. Solution:  L  10 0.001  = 10,000 f t .  (a)  From F i g .  4-2, / x ^ \  U By u s i n g E q .  R  j  =0.2160 r  =1  (4-11) = 0.6761 • (0.2160) = 0.1465  x*  (b)  x*  (c)  = 1465  f t .  From F i g .  = 1480  4-3,  i t i s found  = 0.1480  / x* \  f t .  Using E q . (4-5b), A = 0.583, B = 1.2850 and C = 8.5296 = 0.1498 L  R  x* = 1498 f t .  Percentage  error of (a)  = 1480  - 1465  =  1.6%  =  -1.2%  1480 Percentage  error of (c)  = 1480  - 1498  1480  along  46  4.7  R e d u c t i o n o f Wave H e i g h t A f t e r R e f l e c t i o n a t a R e s e r v o i r a t t h e Upper End o f t h e C a n a l When w a t e r e n t e r s a m i l d  slope channel, the depth y  is  reservoir  related  to the static  The r e l a t i o n b e t w e e n t h e d e p t h s y  >i  expressed canal,  and y  c a n be e x p r e s s e d by  { 4  where y ^ i s t h e head  loss  due t o f r i c t i o n  i n terms o f t h e v e l o c i t y  ( F i g . 4-8)  (  l e v e l by t h e energy e q u a t i o n .  +  +  located  head  _  1 2 )  a t e n t r a n c e a n d may b e  a t the entrance of the  that i s  V  = C .  y 6  2  2g  6  ....  (4-13)  i n w h i c h C^ i s a c o e f f i c i e n t w h i c h d e p e n d s o n t h e c o n d i t i o n s a t entrance. Assuming  t h e e n t r a n c e l o s s y^ i s n e g l i g i b l e ,  Eq.  (4-12)  c a n be  rewritten  v2 y  = y-L + M • _JL 2g  R  When t h e w a t e r e m p t i e s i n t o  V energy  equal to  (4-14) a reservoir,  a n amount o f k i n e t i c  2  _1, c a r r i e d w i t h t h e f l o w i n g w a t e r , i s  • 2g expected ervoir  t o be r e s t o r e d  level  s h o u l d be h i g h e r by t h i s  at the upstream dissipated  end o f the c a n a l .  i n e d d i e s and w h i r l s .  may b e i g n o r e d , a n d y head  as a p o t e n t i a l  recovery).  energy.  Thus, t h e r e s -  amount t h a n t h e w a t e r  T h i s energy, however i s u s u a l l y In practical  calculations i t  may b e r e g a r d e d a s e q u a l t o y 1  level  (no v e l o c i t y  K  47  <  A positive of  s u r g e wave r e a c h i n g  thecanal w i l l  be r e f l e c t e d  p r o c e e d s downstream.  I n general,  velocity  will  In a long usually the  a t t h e upstream end  as a n e g a t i v e  wave w h i c h  The f l o w d i r e c t i o n  t h e wave i s r e f l e c t e d , channel.  thereservoir  will  channel,  canal).  of mild slope  i . e . , water flows  c a n a l however, t h e v e l o c i t y  there  a t the entrance  into  o f flow i s reduced but  i s no c h a n g e i n d i r e c t i o n  (water  still  depend on t h e d i r e c t i o n o f t h e f l o w a t t h e s e c t i o n  changes height  If  (reverse  y  =  „  a  thenegative  wave p a s s e s .  to the i n i t i a l  direction),  flows will  of  into  therefore,  entrance  I f the flow  direction  t h e r e d u c t i o n o f wave  <iZ i s :  A Z  or  after  water  the reservoir.  T h e amount o f r e d u c t i o n o f wave h e i g h t  immediately  after  depend on t h e l e n g t h and s l o p e o f t h e  i na short  be r e v e r s e d  then  2  -  - y  ~  1  y  - v±  R  l  V  (4-15)  t h e f l o w does n o t change d i r e c t i o n ,  t h e r e d u c t i o n o f wave  height i s : 2  AZ =  y  R  n  - yn - ' . M  V  2  (4-16)  2g  x  v Assuming t h e value  of  i s small and  ^ .  2  _2  i s negligible,  2g Eq. of  (4-16) i s i d e n t i c a l t o E q . (4-15), initial  after  velocity.  Therefore,  r e f l e c t i o n a t the entrance  approximately flow after  AZ  i sa function  t h e r e d u c t i o n o f wave of the canal  b y E q . (5-15) r e g a r d l e s s  r e f l e c t i o n a t that  and  section.  height  c a n be o b t a i n e d  the direction of the  48  I t h a s b e e n shown i n C h a p t e r 2 t h a t t h e i n i t i a l the  wave h e i g h t  at  d o w n s t r e a m end depends on t h e F r o u d e number o f t h e f l o w .  Thus t h e r a t i o .A Z/Z  i s a f u n c t i o n o f the Froude  number  o Fig. 4-10).  (see A  Z/Z^  found 4.7.1  i n rectangular,  for calculating  t r i a n g u l a r and t r a p e z o i d a l c a n a l s a r e  i n the f o l l o w i n g .  Equation For  The a p p r o x i m a t e e q u a t i o n s  f o r rectangular  rectangular  canals  AZ/Z  Canals,  .varies with  F r o u d e number, and  o the  distribution  and  A Z/Z  fit,  o  as  i s linear ordinate.  the equation  A Z  = 0.4533  F  on l o g - l o g paper w i t h Using  F^ a s  abscissa  t h e l e a s t square method f o r b e s t  obtained i s  °*  (4-17)  9 6 8  o 4.7.2  Equation For  f o r t r i a n g u l a r canals  triangular canals,  A Z  = 0.6631  F  °'  the corresponding  equation i s  (4-18)  9 7 9  o 4.7.3  Equation For  f o r trapezoidal  trapezoidal canals,  with value  canals  i t i s found  F r o u d e number b u t a l s o w i t h  that A Z/Z  ^\Z/Z  and F  o  is  varies not only  the parameter r .  o f r , the r e l a t i o n s h i p between  to t h a t f o r t h e r e c t a n g u l a r  q  For a given i s similar  o  and t r i a n g u l a r c a n a l s .  The  equation  of the form  A Z  Z  = P  .  F  ?  2  (4-19)  o  w h e r e p.. a n d p„ a r e t h e c o e f f i c i e n t s  t o be d e t e r m i n e d .  P.. a n d  49  are  the f u n c t i o n s o f r .  p  P  = 0.4533 r *  5  4  9  0  = 0.9680 r '  0  1  6  2  0  0  2  T h e r e f o r e Eq. (4-19)  (4-22) i s v a l i d  (4-20)  8  . . . . . . . . .  (4-21)  becomes  AZ = 0.4522 r ° Z o Eq.  They a r e determined as  5 4 9 0  .F o  0  '  9  6  8  0  r  0  -  0  1  -  2  8  -  '  (  A  "  2  2  )  f o r the v a l u e s o f r from 1.0 t o 2.0  which means t h a t Eq. (4-22) i s a l s o v a l i d  f o r r e c t a n g u l a r and  t r i a n g u l a r c a n a l s because when the v a l u e s o f r a r e equal to 1.0 and 2.0, the Eq. (4-22) i s i d e n t i f i c a l t o Eq. (4-20) and (4-21) r e s p e c t i v e l y .  4.7.4  A l t e r n a t i v e procedure ^Z/Z  a l s o c a n be found by an a l t e r n a t i v e p r o c e d u r e based on d e r i v a t i o n s  -in Chapter 2 .  A z  .  =  D i v i d i n g both s i d e s o f the e q u a t i o n  ^ 2g  by Z , i t becomes o F 2  TJ2  .AZ Z o  =  l  v  =  12 2Zo  2g.Z o  y  where ft = Z /y P o c The v a l u e s o f .A Z/Z  o  . =  F 2  _1£L_ . . . . (4-23)  0  f o r v a r i o u s flows i n r e c t a n g u l a r , t r i -  a n g u l a r and t r a p e z o i d a l c a n a l s c a n be found from Eq. w i t h Eqs.  (2-13), (2-18) and (2-23) r e s p e c t i v e l y .  (4-23)  50 4.8  Propagation of Negative Upper End  o f a P r i s m a t i c Power  Negative  waves a r e n o t  p a r t of  t h e wave t r a v e l  the h e i g h t of of  faster  than  The  initial  b e c a u s e p o i n t s on  2,  n e g a t i v e waves r e f l e c t e d  pared  w i t h the depth.  the depth  the e r r o r  For  from  compared w i t h the  from  p o i n t to p o i n t  t h e waves t r a v e l  along  waves.  t h e r e s e r v o i r may  small,  be less  In this  The  2)  a r e t h e n a s s u m e d t o be v a l i d  basic expressions f o r calculating  waves, w h i c h  result  are presented  Similarly  from  i n Chapter  waves  f o r negative  the p r o p a g a t i o n of  waves.  negative  of the  canal  3.  to the c a l c u l a t i o n s f o r a p o s i t i v e  the  be i n -  for positive  r e f l e c t i o n a t t h e u p p e r end  a t i c v a r i a t i o n of the v a r i a b l e s u s e d as  for negative  equations  com-  negative  significant.  The  in  t h a n 207o  downstream w i t h an u n c h a n g i n g p r o f i l e w i l l  waves ( C h a p t e r  the c a n a l .  a p p l i e d to deter-  i n t r o d u c e d by a s s u m i n g t h a t t h e  the above a s s u m p t i o n .  having'  v e l o c i t y of negative  the c a l c u l a t i o n s  depth  along  waves  wave p r o c e e d s  a r e b a s e d on  If  the equations  a wave w i t h a h e i g h t o f  study  upper  large,  d e r i v e d f o r a p o s i t i v e w a v e , c a n be the h e i g h t and  the  the lower p a r t .  p r o f i l e of n e g a t i v e surge  f l a t t e n o u t as  mine a p p r o x i m a t e l y  the  t h o s e on  t h e h e i g h t o f t h e wave i s m o d e r a t e o r s m a l l ,  Chapter  of  i n form  t h e f l o w , t h e wave v e l o c i t y v a r i e s  a steep f r o n t w i l l  The  stable  a Reservoir at  Canal  t h e wave i s r e l a t i v e l y  t h e wave f r o n t .  If  Waves R e f l e c t e d f r o m  s u r g e wave,  for different  i n p u t to the computer program.  t h e n e g a t i v e waves a r e p l o t t e d on  system-  shapes of c a n a l s  was  Results obtained for  t h e same d i m e n s i o n l e s s  plane  51 together w i t h those  f o r p o s i t i v e waves.  The v a r i a t i o n o f t h e  r e l a t i v e h e i g h t , Z/Z , o f t h e n e g a t i v e wave i n t e r m o f x / L o K is  4-10a t h r o u g h 4-12g.  shown i n F i g u r e  From these 1.  figures  the f o l l o w i n g p o i n t s a r e observed:  When a wave i s r e f l e c t e d g i v e n v a l u e o f Z/Z^) the  initial  t h e shape o f t h e c u r v e  and r t h e shape o f curve  q  depends on t h e l o c a t i o n 3.  K  JL  (x"/L )  i s independent of  for a reflected  wave  of the point of r e f l e c t i o n .  R  T h e r a t e o f r e d u c t i o n i n h e i g h t o f a n e g a t i v e wave g r a d u a l l y a s t h e wave t r a v e l s  4.9  (ora  parameters y , V and S f o r a g i v e n F and r . o o o o  For a given F  2.  a t a g i v e n v a l u e o f x /L  diminishes  downstream.  Maximum W a t e r D e p t h a t D o w n s t r e a m E n d o f C a n a l When a s u r g e the  wave o c c u r s  load rejection,  suddenly  a t t h e d o w n s t r e a m e n d o f t h e c a n a l due t o  the water depth  by t h e amount e q u a l  wave f r o n t  a t t h e downstream end i n c r e a s e s  to the i n i t i a l  wave h e i g h t .  t r a v e l s upstream, t h e w a t e r s u r f a c e does n o t r e m a i n  stationary.  This  the v e l o c i t y  i s a l s o reduced,  is  to potential  converted  i s because t h e d i s c h a r g e o f the f l o w i s reduced, and t h e k i n e t i c  energy of t h e f l o w  energy i n t h e form o f an i n c r e a s e d  h e i g h t o f t h e water s u r f a c e a t t h e downstream end. 4-14  illustrate  depth  As t h e  the elementary  F i g s . 4-13 a n d  f a s h i o n of t h i s v a r i a t i o n o f water  i n t h e p o s i t i v e a n d n e g a t i v e wave p r o p a g a t i n g  The  depth  The  rate of increase i s i n i t i a l l y  cycle.  o f water a t t h e downstream end i n c r e a s e s w i t h approximately  time.  linear with  time.  52 If  the depth i s expressed  by t h e r a t i o  v a r i a t i o n of y / y with respect Q  values  o f t h e F r o u d e number.  number, t h e r a t e o f i n c r e a s e teristics, shape.  Q  then the r a t e o f  t o time increases w i t h  larger  For a flow with a given  Froude  i n y / y also varies with canal  such as l o n g i t u d i n a l bed s l o p e , f r i c t i o n a l  i ny/y increases Q  i sgiven,  with the increasing of S  an example, f o r a r e c t a n g u l a r  charac-  Q  I f the cross-section of the canal  increase  As  y/y >  canal, with b  Q  Q  roughness and  the rate of .  = 30 f t . ,  = 30 f t . a n d n = 0.03095, t h e c u r v e s w i t h v a r i o u s g i v e n F r o u d e n u m b e r s a r e p l o t t e d i n F i g . 4-15. angular  canal of y  trapezoidal r It  = 1.5  Q  Q  = 25.5 f t . , m =*1.5, n = 0.03095 a n d  a r e p l o t t e d i n F i g . 4-16  the v a l u e s  for the t r i -  = 22.36 f t . , m = 2.0 a n d n = 0.03095 a n d t h e  canal of b  i sinteresting  S i m i l a r curves  and F i g . 4-17  respectively.  t o note that the slopes o f curves  o f c r o s s - s e c t i o n a l shape f a c t o r r  i n c r e a s e as  decrease.  A f t e r g a t e c l o s u r e t h e depth a t t h e downstream end i n c r e a s e s nuously  until  reflected  this  i s i n t e r r u p t e d by t h e a r r i v i n g n e g a t i v e  from t h e upper end o f t h e c a n a l and then  r a p i d l y b y a n amount o f a b o u t t w i c e  the negative  maximum w a t e r d e p t h t h e r e f o r e o c c u r s before  the a r r i v a l  tioned before, tant items  In this  of the negative  study,  decreases  immediately A s h a s b e e n men-  t h i s maximum w a t e r d e p t h i s o n e o f t h e m o s t  of information i n the design  systematic  wave,  wave h e i g h t . T h e  a t t h e time  wave f r o n t .  conti-  impor-  of the canal.  variation of individual  v a r i a b l e s was  53  used as i n p u t t o t h e computer program. dimensionless  R e s u l t s a r e p l o t t e d on a  plane with y /y as o r d i n a t e and 2L/L„ as a b s c i s s a , max ''o R r  where L i s t h e c a n a l l e n g t h .  F o r r e c t a n g u l a r c a n a l s , i t i s found  that the curve of y /y a g a i n s t 2L/L does n o t change w i t h t h e max o ° R ° v a r i o u s v a l u e s o f b , y and S i f t h e Froude number o f i n i t i a l o o o flow, F i s f i x e d . This r e l a t i o n also holds f o r t r i a n g u l a r canals. n  J  J  q  F o r t h e t r a p e z o i d a l c a n a l , one more v a r i a b l e , r , i s i n t r o d u c e d . The c u r v e y /y a g a i n s t 2L/L„ v a r i e s n o t o n l y w i t h F b u t a l s o max •'o R o w i t h r . F o r a g i v e n Froude number, t h e v a l u e o f y Iy increases max o w i t h i n c r e a s i n g value of r . F o r a g i v e n r , the value o f y /y • ° max o J  varies with F . maximum s t a g e  J  The r e s u l t s o f computer c a l c u l a t i o n s f o r a  developed a t t h e downstream end due t o a sudden  reduction i n discharge  i n r e c t a n g u l a r , t r i a n g u l a r and t r a p e z o i d a l  ( f o r r = 1.5) c a n a l s a r e p l o t t e d i n F i g s . 4 - 18, 4 - 1 9 and 4 - 20.  c  54  CHAPTER 5 The to  primary o b j e c t i v e o f t h i s  CONCLUSIONS  study was t o d e r i v e d i m e n s i o n l e s s  ratios  d e s c r i b e the i n i t i a t i o n and p r o p a g a t i o n o f a surge wave i n a power  canal f o l l o w i n g a load reduction or r e j e c t i o n .  The of  initial  surge-wave h e i g h t , r e s u l t i n g from an i n s t a n t a n e o u s r e d u c t i o n  d i s c h a r g e , v a r i e s w i t h the shape and dimensions  the amount o f the i n i t i a l terms, the r e l a t i v e tor  flow.  initial  o f a c a n a l and w i t h  T h i s study shows t h a t , i n d i m e n s i o n l e s s  wave h e i g h t i s a f u n c t i o n o f the shape f a c -  k, d i s c h a r g e r a t i o x and the Froude number of the i n i t i a l  F o r a sudden t o t a l  c l o s u r e , i . e . , T = 0,  flow F . o  f£ i s o n l y a f u n c t i o n o f F  q  in  a r e c t a n g u l a r and t r i a n g u l a r c a n a l , and i s a f u n c t i o n o f F^ and k i n a trapezoidal this to  A dimensionless  e q u a t i o n , E q . (2-23), d e r i v e d i n  study can be a p p l i e d to p r e d i c t an i n i t i a l  a sudden t o t a l o r p a r t i a l  ular,  The  canal.  surge wave h e i g h t due  change i n d i s c h a r g e i n c a n a l s o f r e c t a n g -  t r i a n g u l a r and t r a p e z o i d a l c r o s s - s e c t i o n s .  r e s u l t s o f c a l c u l a t i o n s from the mathematical  model developed i n  t h i s study a r e i n c l o s e agreement w i t h those from the methods by p r e v i o u s i n v e s t i g a t o r s ,  proposed  i n p a r t i c u l a r , w i t h the r e s u l t s from the  F a v r e method a p p l i e d to s h o r t c a n a l s .  When a p o s i t i v e surge wave i s i n i t i a t e d a t the downstream end o f a c a n a l and propagates  upstream, the wave h e i g h t decreases  g r a d u a l l y . The  r a t e o f decrease o f wave h e i g h t depends on c a n a l parameters such as the frictional  coefficients,  the bed s l o p e , the shape and dimensions o f  the c r o s s - s e c t i o n and the i n i t i a l  flow.  This for  d e c r e a s e o f wave h e i g h t w h i c h Z/Z  0.6,  o  is still  i s approximately  greater  t h a n 0.6.  t h e v a r i a t i o n o f wave h e i g h t  linear  When Z/Z  o  f o r a distance, becomes l e s s  than  i s more a n e x p o t e n t i a l f u n c t i o n o f  * x /L .  The d e v i a t i o n f r o m F a v r e ' s a s s u m p t i o n f o r a s t r a i g h t l i n e  water surface p r o f i l e  increases with  p o i n t o f t h e i n i t i a t i o n o f t h e wave. less  equations  i n c r e a s i n g d i s t a n c e from the The w r i t e r h a s d e r i v e d  f r o m w h i c h t h e wave h e i g h t  dimension-  o f a p o s i t i v e surge a t any  s e c t i o n o f a r e c t a n g u l a r a n d a t r i a n g u l a r c a n a l may be p r e d i c t e d . Eq.  (4-4)  canals.  i s f o r rectangular  c a n a l s , a n d E q . (4-5)  i s for triangular  The i n f l u e n c e o f s h a p e a n d s i z e o f t h e c r o s s - s e c t i o n o f a  c a n a l o n t h e v a r i a t i o n o f wave h e i g h t  of a p o s i t i v e surge,  pressed  Eqs.  by a l o g a r i t h m a t i c f u n c t i o n .  r e l a t i o n s h i p o f s u r g e wave h e i g h t s cross-section. in  Using  these  a trapezoidal canal  In a long voir its  (4-10)  give the  t h e v a r i a t i o n o f wave  c a n be p r e d i c t e d f r o m E q .  height  (4-4).  s u r g e -wave, r e f l e c t e d  from the r e s e r -  a t t h e upper end o f t h e c a n a l , t r a v e l s downstream, t h e r e d u c t i o n o height  is initially  rapid.  This  becomes g r a d u a l l y s m a l l e r as t h e  wave t r a v e l s t o w a r d t h e g r e a t e r w a t e r d e p t h . negative  this  initial  long  canal, a  r i s e of the water surface behind  end  of the canal  F r o u d e number F  t h e wave f r o n t a t t h e d o w n s t r e a m  i s not linear with respect  i n the water surface of i n i t i a l  little  reduction.  The  rise  In a very  wave may p r o p a g a t e d o w n s t r e a m f o r a l o n g d i s t a n c e w i t h  attenuation after  this  (4-11)  i n a t r a p e z o i d a l and r e c t a n g u l a r  two e q u a t i o n s ,  c a n a l where a n e g a t i v e  and  may be e x -  to the time.  increases with  flow.  The r a t e o f  larger values  of the  T h e maximum w a t e r d e p t h a t t h e down-  56  s t r e a m end  of  the  canal,  c a u s e d by  immediately before  the  a r r i v a l of  from r e f l e c t i o n  the  u p p e r end  d e p e n d s on initial  The  the  at  slope,  dimensionless  power c a n a l s  m u s t be  and  a negative of  the  i n discharge,  s u r g e wave t h a t  canal.  This  c r o s s - s e c t i o n of  occurs resulted  maximum d e p t h  the  canal  and  the  flow.  relationships derived  e s t a b l i s h design of  length  a reduction  criteria to a v o i d  s t u d y may  f o r c r e s t e l e v a t i o n s of overtopping.  made f o r s e c o n d a r y s u r g e s n o t  a minimum d e s i r e d  in this  freeboard.  In this analysed  the  used  b a n k s and  criteria in this  be  to  walls  some a l l o w a n c e t h e s i s and  for  w  Oi !=  o •-a o  w H  FIG. 2-1  DEFINITION SKETCH :  SURGE WAVE IN AN OPEN CHANNEL  Vo ^2  A  i  y  • 2-2  x  = o-yi b  = y  l  /  2  DEFINITION SKETCH :  SURGE WAVE IN A RECTANGULAR CANAL  F I G . 2-3  I N I T I A L SURGE WAVES I N A RECTANGULAR CANAL  59  FIG. 2-6 DEFINITION SKETCH :  SURGE WAVE IN A-TRAPEZOIDAL  CANAL  o  F I G . 2-5  I N I T I A L SURGE WAVES I N A TRIANGULAR CANAL  F I G . 2-7  (c)  I N I T I A L SURGE WAVES I N A T R A P E Z O I D A L CANAL FOR  r  =0.20  F I G . 2-7  (d)  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR r =  0.30  F I G . 2-7  (e)  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR  T=  0.40  UIO"  5  2  3  4  5 6 7 8 9  IxlO * -  2  3  F F I G . 2-7 ( f )  4  5 6 7 8 9  1x10"'  2  3  2  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR T = 0.50  4  5 6 7 8 9  10  UIO"  5  2  3  4  5 6 7 8 9  F I G . 2-7 ( g )  UK)"*  2  3  4  5 6 7 8 9  IxiO"'  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR  2  3  r = 0.60  4  5  6  7 8 9 10  F I G . 2-7  (h)  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR T =  0.70  UIO"  3  2  3  4  5 6 7 8 9  F I G . 2-7 ( i )  UICT  8  2  3  4  5 6 7 8 9  1x10"'  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR  2  T  3  = 0.80  4  5 6 7 8 9 I O  to* FIG. 2-7(j)  I N I T I A L SURGE WAVES I N A TRAPEZOIDAL CANAL FOR  r. =  0.90  71  t  i  F I G . 3-1  D E F I N I T I O N SKETCH  F I G . 3-3  :  C"" AND C~ CHARACTERISTIC GRIDS ON THE x - t PLANE 1  D E F I N I T I O N SKETCH : ( a ) UPPER END BOUNDARY, (b) DOWNSTREAM  END BOUNDARY.  P(l,l)  P(3,l)  P(5,l)  P(7,l)  x(m+l,n+l) t(nri-l ,n+l) V(m+l,n+l) y(m+l,n+l) P(m+1 , n + l )  P(m+2,n)  x(m n) t(m,n) V(m,n) y(m,n) s  FIG.  3-2  D E F I N I T I O N SKETCH  x(m+2,n) t(m+2,n) V(m+2,n) y(m+2,n)  CHARACTERISTIC GRIDS ON THE x - t  PLANE  I  I  V V X T  2  7.615 7. 859 0.000 0.000  V Y X  V Y X  T 4  12.ot r  8. 076  R .0 79 74. 527 15. 165  X  T 5  V Y X  T 6  8.453 8. 284 0.000 24.420  8. 443 8. 280  X  74. 931 27. 443  T  7  V Y X  T  8  8.699 8.440 0.000 36.946  8. 60 7 8. 4 34  75. 180 39. 931  T  V  Y X  T  8. 789 8.528 0.000 49.579  8. 675 8. 42 7 150. 345 42. 918  6  7  8  7. 635 7.859 300.000 0.000  7.635 7. 839 400.000 0.000  7 .535 7.359 600.000 0.000  7.635 1.8 V-> 600.000 0.000  7.635 7. 3 59 700.000 0.000  7. 628 7.859 51 7. 92 1 22.002  8. 04 7 8. 067 447.026 30. 565  7. 629 7. 859 64 3. 94 1 13. 8 59  7. 628 7. 859 617. 92 1 21. 002  7.62 7 7. 85 9 59 1. 897 25. 146  7. 6~36~~ 7. 8 59 669. 959 15. 7 16  7 .629 74 3 .94 1 18. 85 9  7. 62 7 7 .8 59 691. 89 7 25. 1 46  -  7. 62 9  ii 4 3•.•4 1 18. 8 59  7. 6 30 7. 85 ) 8 9 5.956 15 . 821  7 . 62-' !.859 8 6J . 93 7 19 . 9 64  7. 52 8 7. 3 59 84 3. > 1 b ? 3.107  7. 62 8 7 .859 895 . 9 3-) 2 1 .06 3  7. 632 7. H59 1 OOC. 000 8 .496  7.631 7. 8 59 <• 7 3. 989 1 1.639  7.6 30 7. •1 5 9 94 7. 9 7 6 14. 7 8 2  7. 62 9 7. 8 5 9 92 1 .)6 9 1 7.97 6  7. 6 34 7. 859 1000 . 000 4 . 248  7.633 7.859 9 7 3. 994 7. .3 9 1  7 ..5 3 2 7. 3 5 9 94 7 .H'4 10 . 634  7. 63 1 7. 8 59 9 2 1.9 72 1 1. 6 7 I  7.635 7.359 1000.000 C.000  7.634 7.859 97 3.998 3. 1 4 3  7. 6 3 3 7. 859 94 7. 99 3 6. 78 6  7. 632 7 .8 5 4 92 1. 9 8 6 9. 4 30  7 .631 7. 8 5 i 89 5. 9 7'. 1 2 57 . i  7. 630 7. 859 86 9 . 95 9 15 .7 1 6  i. 859  7 .6 28 7.8 59 8 1 7.971 22 .002  1. 627 7 .8 5 9 791 . 89 7 25 . 146  7. 85 9 82 1 .'•85 •i. 4 30  7. 631 7. 859 795. • 74 1?. 57 3  763~0 7 .859 769 .9 59 1 5.716  I. 8 59  7. 628 7. 8 59 717. 921 22. C02  T.~63~7~  11  7.635 7."59 900.000 C.000  7. 634 7. 85 9 .-I 7 3 9 . 98 1. 14 3  7 . 633 7 . 8 59 84 7 . 9 9 3 i> • 286  7. 6 3 3 7. 859 74 7 .99 3 6. 786  _._  10  7.635 ' 7.H5V 800.000 0.000  7 .634 7. 8 59 77 ).9 9 8 3 . 14 i  .6 32 7. H 5 9 72 1 9 .4 30  7 .631 7 .85 9 695. 9 74 12 . 57 1  9  :  7 . 6 34 7 . 859 67 3 . 9 98 3 .143  7 .63 * 7 .859 647 .9 9 J 6. 286  7. 6~3~2 7. 8 59 621. 98 5 9. 4 30  7. 531 7 .3 59 59 5. 974 12. 5 73  7. 6 30 7. 8 59 5 59. 959 15. 7 16  7. 629 7. 85 9 54 3.94 1 18. 859  7. 634 7. 8 59 573. 4 98 3 .143  7. 63 3 7. 8 59 547. 9 9 3 6. 2 86  7 .6 32 7. 859 52 1 .985 9. 4 30  7. 63 1 7. 859 495. 97 4 12. 57 3  7. 630 7. 359 469.959 15.716  7. 629 7. 8 59 443. 941 18. 859  7. 6 34 7. 859 473. 9 9.5 3. [43  7. 63 3 7. 859 44 7. 994 6. 286  7. 632 7. 859 421.98 5 9. 4 30  7. 6 3 1 7. 859 3 95.974 12. 5 73  8. 052 8. 069 372. 544 2 7.484  8. 414 8. 265 299. 650 36. 520  7. 634 7. 8 59 373.998 3. 143  7. 633 7. 859 347. 994 6. 286  7. 630 7. 8 59 369. 959 15. 716  8. 053 8. 071 298. 053 24. 40 3  8 .423 8 .270 224. 756 33. 49 3  5  7. 632 7. 8 59 321. 985 9. 430  7. 631 7. 859 295. 974 12. 573  8. 064 8. 074 223. 553 21 . 323  a. 433 8. 2 75 149. 850 30. 46 8  V Y X  9  8. 0 76 149. 044 18. 244  4  7. 634 7. 85 9 27 3.998 3. 143  7. 63 3 7. 859 247. 994 6. 286  7. 632 7. 859 221 . 98 5 9. 430  8. 070  V Y  7.635 7.859 200.000 0.000  7 .634 7. 859 173. 998 3. 143  7. 633 7. 859 147. 994 6. 286  8.083 8.081 0.000  V Y  EXECUTION 3  7.635 7.859 100.000 O.OCO  7. 634 7. 859 73. 998 3. 14 3  T 3  2  7. 63 1 7. 859 1 000.000 12. 744  7.630 7. 6 59 973.985 15.687  7. 62 9 7. 85 9 94 7. 96 7 1 9.0 3 0  7. 6 30 7. 859 , 1000. COO 16. 992  J  FIG. 3-4  A PART OF RESULTS OF COMPUTER CALCULATIONS IN EXAMPLE 3-1  TIME , SECONDS  U.  75  FIG.  3-6  SCHEMATIC PRESENTATION OF C H A R A C T E R I S T I C GRIDS FOR A P O S I T I V E SURGE WAVE PROPAGATING UPSTREAM  X  FIG.  3-7  SCHEMATIC PRESENTATION OF CHARACTERISTIC GRIDS FOR A POSITIVE SURGE WAVE PROPAGATING DOWNSTREAM  START WITH GIVEN I . S . F . SET x t . V . y J-J+l CALL MID  (2)  J-J+l I-ION CALL WAVMID COMPUTE Z„,V,,„ o wo  (3)  I-I-2  CALL MID  i  CALL WAVMID  FIG. 3-8  FLOW CHART OF THE COMPUTER PROGRAM FOR A POSITIVE WAVE PROPAGATING UPSTREAM ALONG THE POWER CANAL  F I G . 3-10  D E F I N I T I O N SKETCH FOR EXAMPLE 3-3  79  SECTION 1  SECTION 0  At time t-0 before wave Is formed  At time t-0 Immediately after wave Is formed  At time t - t ^  At time t=0 when wave occurs ln Section 0  At time t=t^ before wave reaches Section 1  At time t = t immediately after wave reaches Sec.l  hg-41.175  hg=49.567  h£=54.644  h =41.175  hx =41.175  h^=41.175+4.27  F -41.175x45 0 Q -292x45  FQ=49.567x45  Fp=54.644x45  F -41.175x45  F =41.175x45  Qo°°  Q'o=°  Q =292x45  0.^=292x45  F|=45.445x45  V =7.101  V -0  Vj=7.105  V  0^ = 154.7x45  B =45  B «45  B''-45  B -45  Bj =45  PQ-144.134  P"=154.288 0  Vf=3.41  R'=49.567x45 144.134  R'Q=15.89  B^=45  0  0  Q  0  0  Pi=135.89  A ?l  0  R[=15.05  =  A Ql  y -45 0  AQJ--137.3x45  Ah = x  Z -8.393  .105 +  1  -15.45  =45.445  +  1  1  1  0  0  +  1  ZJ=4.27  = 0  a  a. --34.84  1  Q  =-32.14  C —41.94 Q  C^-39.25  AQ --292 Q  A F|=4.27x45  AF^-8.393x45  Assume Z^- 5.077 (Horizontal)  m m  B  =  45  y--45  A B = 0  AF-'-Z-'.Y''  AQ - 0 .  -5.077x45  AQo'O 2 1.486"  Aa  - 2.699  a - 33.49 a (V ) #3 2  +0.00002525  Rin- 15.47 I n - 0.000237 V =1.705 m (V ) =5.81 2  L  - - 38800  Summary of t r i a l s : No. of trial 1 2 3 4  Guess Z i Water Volume »i 4.27 339000 x 45 4.00 342000 x 45 4.10 339500 x 45 4.15 339000 x 45  F I G . 3-12  Water Volume -Vi 344000 x 45 334000 x 45 337500 x 45 339200 x 45  *1< V  2  *1> 2 *1> 2 V  V  Y  i*v  2  SUMMARY OF CALCULATIONS I N EXAMPLE 3-3 FOLLOWING FAVRE METHOD  X, FIG.  3-13  tl.  RESULTS OF  CALCULATIONS  IN  EXAMPLE  3-4  RESERVOIR  (a)  r*  RESERVOIR  H  — —  .  y  T  ± (b)  F I G . 3-14  (a) (b)  P O S I T I V E SURGE REACHES THE RESERVOIR AT THE UPPER END OF THE CANAL, NEGATIVE SURGE REFLECTED FROM THE RESERVOIR.  r- a t p o i n t R i n F i g . 3-16  J W 1  F I G . 3-15  D E F I N I T I O N SKETCH FOR POINT R I N F I G . 3-16  surge  F I G . 3-17  SCHEMATIC DIAGRAM OF C H A R A C T E R I S T I C GRIDS FOR THE NEGATIVE WAVE AT POINT R  84  F I G . 4-1  D E F I N I T I O N SKETCH  85  FIG. 4-3  VARIATION OF WAVE HEIGHT OF A POSITIVE SURGE PROPAGATING ALONG A TRIANGULAR POWER CANAL  00  FIG. 4-4  VARIATION OF WAVE HEIGHT OF A POSITIVE SURGE PROPAGATING ALONG A TRAPEZOIDAL POWER CANAL IN AN INITIAL FLOW F, = 0 . 2 0  90  FIG. 4-7  EFFECT OF SHAPE FACTOR OF A CANAL ON THE VARIATION OF WAVE HEIGHT OF A POSITIVE SURGE WAVE PROPAGATING ALONG THE CANAL  91  FIG. 4 - 8  SCHEMATIC DIAGRAM FOR A P O S I T I V E WAVE REACHING . THE UPPER END OF THE  CANAL  92  FIG. 4-9  REDUCTION OF THE NEGATIVE WAVE HEIGHT REFLECTED AT THE RESERVOIR AT THE UPPER END OF THE RECTANGULAR CANAL  01  0-2  F I G . 4-10 ( a )  0-3  0-4  V A R I A T I O N OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A RECTANGULAR POWER CANAL  0-5  0-6  01  0-2  0-3  04  0-5  0-6  4>  FIG. 4-10 (b)  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A RECTANGULAR POWER CANAL  01  02  FIG. A-10 (c)  0-3  04  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A RECTANGULAR POWER CANAL  0-5  0 6  (a)  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A TRIANGULAR POWER CANAL  FIG. 4-11 (b)  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A TRIANGULAR POWER CANAL  oo FIG. 4-11 (c)  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A TRIANGULAR POWER CANAL  vO vO  01  0-2  0-3  04  0-5  0-6  O  FIG. 4-12 (c)  VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A TRAPEZOIDAL POWER CANAL FOR r = 1.50  102  FIG. 4-13  SCHEMATIC DIAGRAM OF THE VARIATION OF WATER SURFACE FOR A POSITIVE WAVE PROPAGATING UPSTREAM ALONG THE CANAL  FIG. 4-14  SCHEMATIC DIAGRAM OF THE VARIATION OF WATER SURFACE FOR A NEGATIVE WAVE PROPAGATING DOWNSTREAM ALONG THE CANAL  I— 100  ~T~ 200  I 300  400  I 500  600  700  —I  I  800  900  I  —I—  1000  1100  t . second FIG. 4-15  VARIATION OF WATER SURFACE AT THE DOWNSTREAM END FOR THE RECTANGULAR CANAL OF  y - 30 f t . . h Q  Q  WITH RESPECT TO TIME  - 30 f t . AND n - 0.03095  1200  1300  O U>  t> second  o FIG. 4-16  VARIATION OF WATER SURFACE AT THE DOWNSTREAM END WITH RESPECT TO TIME FOR THE TRIANGULAR  CANAL OF y  = 22.36 f t .  , m - 2.0  AND n = 0.03095  F I G . 4-19  MAXIMUM WATER DEPTH AT THE DOWNSTREAM END OF A TRIANGULAR CANAL  References 1.  Johnson R.D.:  The correlation of momentum and energy changes i n  steady flow with varying velocity and the application of the former to problems of unsteady flow or surges, i n open channels, engineers and engineering (The engineers club of Philadelphia), July, 1922. 2.  Favre, H.:  Ondes de Translation, Dunod, Paris, 1935.  3.  Streeter V.L. and Wylie E.B.: , "Hydraulic Transients", McGraw-Hill Book Company, New York, 1967,  4«  Rich G.R.:  "Hydraulic Transients", Dover Publications, Inc.,  New York, 1961. 5.  Jaeger C.: "Engineering Fluid Mechanics", Blackie and Son Limited, . London, 1956.  6.  Chow V.T.:  "Open-channel Hydraulics", McGraw-Hill Book Company,  New York , 19597.  Sandover J.A. and Zienkiewiez O.C.:  Experiments on surge waves,  water power, London, Vol. 9, No. 11, November, 1957. 8.  Haws E.T.:  Surges and Waves i n Open Channels, Water Power, Vol. 6  No. 11 November, 1954. 9.  U.S. Army Corps of Engineers:  Hydraulic Design:  surges i n canals  C i v i l Works Construction, Engineering Manual, March 1949. 10.  Henderson F.M.:  "Open Channel Flow", MacMillan Book Company, 1966  APPENDIX A(l) Program L i s t i n g  C C  T~  i d  £._ 80  EXAMPLE 1 5 . 6 O P E N C H A N N E L S O L U T I O N BY CHAR G R I D METHOD I N R E C T A N G U L A R C H A N N E L DIMENSION V ( 2 2 t l O O ) , Y(22,100), X(22,100), T(22,100) WRITE(6,100) WRITt'(6, 101) R E A D ( 5 , 3 0 0 ) Q O , B , G » H L , D X , H N , SO Y0 = 8 . RO=YO*B/{B+2.*Y0) V0=l.486*RQ**0„667*SQRT(S0)/HN YONEW=QO/(B*V0) IF((YONEW-YO) . L T . 1 . 0 E - 3 ) GO TO 3 YO=YONEW GO TO 2 YO=YONEW VO=QO/(B*YO) DO 1 0 1 = 1 , 2 1 , 2 V{ I , I ) = V O Y ( I , I)=YO T ( I , I ) =0. . . . . . . CONTINUE X(I,I)=0. DO 1 1 1 = 3 , 2 1 , 2 J = l IM=I-2 ..X( I, 1 ) = X ( I M , 1 )+DX _ ..... CONTINUE WRITE(6,102) J , (V(K,1), K=1, 2 1 , 2 ) WRITE{6,103) (Y(K,l), K=l,21,2) WRITE(6,104) (X(K,1), K=l,21,2) WRITE(6,105) (T(K,1), K=l,21,2) F I N D T H E S O L U T I O N I N THE M I D D L E . S E C T I O N OF C H A N J E L J =2 DO 4 0 1 = 2 , 2 0 , 2 I M=I- i JM=J-1 I N= I + I GR=SORT(G/Y(I M » JM)) GS = S f J R T ( G / Y ( I N , J M ) ) RG=SQRT(G*Y(IM,JM)) SG=SORT(G*Y(IN,JM)) SSNUM=HN*HN*V(IN,JM)*ABS(V(IN,JM)) R S = B*Y{ I N , J M ) / ( B + 2 . * Y ( I N , J M ) ) SSDEN0 = 2 . 2 1 * R S * * l . 3 3 3 GNS=G*(SSNUM/SSOENO-SO) RR=B*Y( I M , J M ) / ( B + 2 . * Y { I M , J M ) ) K R D E N 0 = 2 . 2 l * R R * * I . 333  *  J-  40  C  „  18  19 C  JL  J.  RRNUM=HN*HN*V(IM,JM)*ABS(V(IM,JM)) GNR=G*(RRNUM/RRDENO-SO) TPNUM=X(IN,JM)-XCIM,JM)+T(IM,JM)*{V(IM,JM)+RG)-T(IN,JM)*(V(IN,JM)1SG) TPDENO=V ( IM, JM ) +RG-V( I N, JM) + SG T(I,J)=TPNUM/TPDENO TPR = T ( I , J ) - T ( I M , JM) X{ I,J) = X( IM,JM) + (V( IM, JM)+RG)_*IPR TPS=T{I,J)-T(INtJM) VRS=V{IM,JM)-V(IN,JM) Y ( I , J ) = (VRS+GR*Y( IM,JM)+GS*Y(IN,JM)-GNR*TPR + GMS*TPS)/(GR + GS ) YPR = Y( I , J )-Y( IM,JM) V( I» J ) = V( IM,JM)-GR*YPR-GNR*TPR _ CONTINUE WRITE(6,106) J , (V<(k,Ji, K = 2 ,20,2 ) " " WRITE(6,107) ( Y ( K , J ) , K=?,20,2) WRITE(6,108) ( X ( K , J ) , K=2,20,2) WRITE(6,109) ( T ( K , J ) , K=2,20,2) FIND THE SOLUTION AT THE UPSTREAM END OF CHANNEL JJ=J+1 . ; '_ X(1,JJ)=0. 1=1 IN= I + l ' SG = SQRT(G*Y( IM,J ) ) T(1»JJ)=T(2»J)-X{2»J)/(V(2»J)-SG) Q=720.+180.*SIN(0.03*T( 1,JJ) ) GS=SQRT(G/Y(2, J ) ) TPS=T(1,JJ)-T(2,J) RS=S*Y(2,J)/(B+2.*Y(2,J) ) SSDENO=2.21*RS**l.333 SSNUM=HN*HN*V(2,J)#ABS(V(2»J)) GNS=G*( SSNUM/SSDENO-SO) YP = Y ( 2 , J ) VP=Q/(B*YP) VPS=VP-V ( 2, J ) YPNEW=Y(2,J)MVPS+GNS*TPS)/GS IF(ABS(YPNEW-YP) . L T . l.OE-3) GG TO 19 YP = YPNEW . _ _ _ _ _ _ _ _ GO TO 18 Y(1,JJ)=YPNEW V ( 1 , J J ) = Q / ( B * Y ( 1, J J ) ) FIND THE SOLUTION IN THE MIDDLE SECTION OF CHANNEL DO 50 1=3,19,2 IM=I-1 _ JM=JJ-1 INM + 1 GR= SORT(G/Y( IM,JM) ) GS=SQRT(G/Y(IN,JM)) RG=SQRT(G*Y(IM,JM)) SG=SQRT(G*Y( IN, JM) ) SSNUM=HN*HN*V(IN,JM)*ABS(V( IN, JM) ) RS=B*Y(IN,JM)/(B+2.*Y{IN,JM)) SSDEN0=2.21*RS**1.3 33 GNS=G*(SSNUM/SSOENO-SG) RR=B*Y([M,JM)/(B+2.*Y(IM,JM)) RRDEN0 = 2.-21*RR**1 . 333 . . _. RRNUM=HN*HN*V( IM,JM)*ABS(V(I M, JM) ) GNR=G*(RRNUM/RRDENO-SO) TPNUM=X( I N , J M ) - X ( IM , JM ) +T( I M , J M ) * ( V ( I M , J M ) + K G ) - T ( I N , J M ) * ( V ( IN,JM)-  LIZ  1SG) TPDENO=V(IM,JM)+RG-V(IN,JM)+SG T(I,JJ)=TPNUM/TPDENO TPR=T(I,JJ)-r(IM,JM) X(i;jJ}=X(IM,JM)+(V(IM,JM)+RG)*TPR TPS=T(I,JJ)-T(IN,JM) VRS=V(IM,JM)-V(IN,JM) Y ( I , J J ) = ( V R S + G R * Y ( I M , J M ) + G S * Y ( I N , J M ).-GN R * T P R.± G N S * T P. S ) / ( G R.+ G S ) YPR=Y(I,JJ)-Y(IM,JM) V{ I , J J ) = V ( I M , J M 50 C  CONTINUE FIND THE  }-GR*YPR-GNR*TPR  SOLUTION  AT  THE  DOWNSTREAM  END  OF  CHANNEL  X(21,JJ)=HL 1 = 2 1 ._ IM=I-1  _  RG=SQRT(G*Y(IM,J)) T ( 2 1 t J J ) = T ( I M , J ) + ( H L - X ( I M , J ) ) / ( V ( I M , J ) + RG) GR = S Q R T { G / Y ( I M , J ) ) TPR = T ( 2 1 , J J ) - T ( I M , J ) RR = B * Y ( I M , J ) / ( B + 2 . * Y {  I M, J ) )  .  RRDENO=2.21*RR**1.333 RRNUM=HN*HN*V(IM,J)*ABS(V(IM,J)> GNR-G*(RRNUM/RRDENO-SO)  Y(2l,JJ)=Y0 YPR=Y(21,JJ)-Y(IM,J) V ( 2 1 , J J ) = V( I M , J 1 - G R * Y P R - G N R * T P R WRITE(6, 102) J WRITE(6, 103) WRITE(6, 104) WRITE(6, 105) IF( J .GT. 32 J= J+2 GO TO 80 FORMAT(7F8.0)  . 300 100  ..  ..  J , ( V ( K , J J ) , K=l,21,2) (Y(K,JJ),K=1,21,2) ( X ( K , J J ) , K = 1 , 2 I ,2) (T(K,JJ) ,K=l,2l,2) ) GO TO 70 ; ..  ;  „.*  F O R M A T ( 4 X , I H I , 9X , 1 H L , 9 X , 1 H 2 , 9 X , 1 H 3 , 9 X , I H 4 , 9 X , 1.H5 , 9 X , 1 H 6 , 9 X , 1 H 7 , 9 X , 1IH8,9X, 1H9,8X,2H10,8X,2H11  10L JL02 . 103  FORMAT(IX,1HJ  )  //)  F0RMAT(I3,4X,IHV,11F10.3)  104  FORMAT{7X,1HX, IIF 10.3 )  105  F 0 R M A T ( 7 X , l H r l l F 1 0 . 3 //)  106  F O R M A K 1 3 , 4 X , 1H V , 6 X , 1 O F 1 0 . 3 )  t  107  F O R M A T ( 7 X , 1 H Y , 6 X , 1 OF  108  F O R M A T ( 7 X , 1 H X , 6 X , 1 0 F 1 0 . 3)  109  FORMAT (7X, 1HT, 6 X , 1 0 F I O . 3 / / ) '  70  STOP END  $ENTRY  .  FORMAT(7X,1HY, 1 IF 10.3)  10.3) _._  .  113  APPENDIX  A(2)  Program L i s t i n g  C C  EXAMPLE 15.5 OPEN CHANNEL SOLUTION DY C H A R G R I D METHOD IN RECTANGULAR DIMENSION V(22f'100), Y(22,100), X ( 2 2 , 1 0 0 ) , T(22,lOO) WRITE(6,100) WRITE(6,101) READ(5»  300)  YO,  B,  G,  HL,  DX,  AO,  PO,  HN,  CHANNEL  SO  QO=132.*{YO-2.32)**1.5 VO=00/AO DO 10 1 = 1 , V(I,l)=VO Y( I ,1)=YO  21,2  T( I , I ) =0. 10  11  C 80  CONTINUE X(1,1)=0. DO 11 1=3, 21, 2 J = l IM=I-2 X ( I , I )= X ( I M , 1 ) + DX CONTINUE WRITE(6,102) J , (V(K,1), K=1, 21, 2 ) WRITE(6,103) (Y(K,1), K=l,21,2) WRITE(6,104) ( X ( K , l ) , K=l,21,2) WRITE(6,105) (T(K,1), K M , 2 1 , 2 ) F I N D THE SOLUTION I N THE MIDDLE SECTION  OF  CHANNEL  J = 2 DO 40 1=2, 20, 2 IM=I-1 JM=J-1 IN= I + GR=SORT(G/Y(IM,JM)) GS=SQRT(G/Y{IN,JM)) RG=SQRT(G*Y(IM,JM)) SG=SQRT{G*Y(IN,JM)) SSNUM=HN*HN*V(IN,JM)*ABS(V(IN,JM)) RS=B*Y(IN,JM)/(B+2.*Y(IN,JM)) S S D E N O = 2 . 2 1 * R S * * l . 3 33 GNS=G*(SSNUM/SSDENO-SO) RR = B * Y ( I M , J M ) / ( B + 2 . * Y ( I M , J M ) ) RRDEN0=2.21*RR**1. 333 RRNUM=HN#HN*VlIM,JM)*ABSlV(IM,JM)) GNR=G*(RRNUM/RRDENO-SO)  l  TPNUM=X(IN,JM)-XUM,JM) + T(IM,JM)*(V(IM,JMJ+RG)-T(IN,JM)*(V(IN,JM)1SG) TPDENO=V(IM,JM)+RG-V(IN,JM)+SG T( I ,J ) = T P N U M / T P D E N O TPR=T(I,J)-T(IM.JM) X(I,J)=X(IM,JM)+(V(IM,JMJ+RG)*TPR TPS=T(I,J)-T{IN,JM) VRS=V(IM,JM)-V(IN.JM)  JL JLH-  X{I,JJ)=X(IM,JM)+(V(IM,JM)+RG)#TPR TPS=T(I,JJ)-T(IN,JM) VRS=V(IM,JM)-V(IN,JM) Y(I,JJ)=(VRS+GR*Y(IM,JM)+ GS* Y(IN,JM)-GNR*TPR+GNS*TPS)/(GR+GS) YPR=Y( I , J J ) - Y ( I M , J M ) V( I , J J ) = V ( I M , J M ) - G R * Y P R - G N R * T P R 50  CONTINUE F I N D THE  .... c  S O L U T I O N . A T . T H E . D O W N S T R E A M . EN.D_.OF  CHANNEL  '_ ...  X(21,JJ)-HL 1= 21 IM=I-1 RG= S Q R T ( G * Y ( I M , J ) ) T(21,JJ)=T(IM, J)+ (HL-X[IM,J))/(V(IM,J)+RG) GR=SQRT(G/Y(IM,J)) TPR=T(21,JJ)-T(IM,J) RR = B * Y M M , J ) / ( B + 2 . * Y ( I M , J ) ) RRDEN0=2.21*RR#*l.333 RRNUM =HN*HN*V ( I M , J ) * A B S ( V ( I M , J ) ) GNR=G*(RRNUM/RRDENO-SO) YP =Y ( I M , J )  .  YP3=(YP-2.32)**3 VP=132.*SQRT(YP3)/(B*YP)  27  V R P =V ( I M , J ) - V P YPNEW=Y(IM,J)+(VRP-GNR*TPR)/GR IF(  ABS(YPNEW-YP)  . L T . 1.0E-3)  GO  TO  26  YP=YPN£W GO TO 2 7 Y(21,JJ)=YPNEW YP 3= ( Y ( 2 1 , J J ) - 2 . 3 2 ) * * 3  26  V ( 2 1 , J J ) = i 3 2 . * S Q R T ( Y P 3 ) / { B * Y ( 2 i , J J ) ) WRITE(6,102) J J ,( V ( K , J J ) , K=l,21,2) WRITE(6, 103) (Y(K,JJ),K=1,21,2) WRT T E ( 6 ,  1 0 4 ) .( X ( K , J J ) , K = 1 , 2 1 , 2 )  WRITE(6, 105) ( T ( K , J J ) , K = l , 2 1 , 2 ) IF( T ( 1 , J J ) . G T . 2 4 0 0 . ) GO TO 7 0 J =J+ 2 GO TO 8 0 FORMAT(9F8.0) FORMAT(4X,iHI,9X,1H1,9X,1H2,9X,1H3,9X,1H4,  300 100 101 102  .  11H8.9X,IH9,8X,2H10,8X,2H11 FORMAT( IX ,1 H J //) F0RMAT(I3,4X,1HV,11F10.3)  103 104  FORMAT! 7X, 1HY,1 I F 1 0 . 3 ) FORMAT(7X,1HX,UF10.3)  105 106 107  FORMAK  108  FORMAT(7X,1HX,6X,10F10.3)  109  FORMAT!7X,1HT,6X,10F10.3  . 70  //)  FORMAT(13,4X,1HV,6X,10F10.3) FORMAT(7X,1HY,6X,IOF10.3)  STOP END  SENTRY  7X,1HT, 1 1 F 1 0 . 3  )  //)  .  9X,IH5,9X,1H6,9X,1H7,9X,  115 i  40  •  C  16 17  —  .... - —  18  19  ;  C  Y(I,J)=(VRS+GR*Y(IM, JM)+GS*Y{IN,JM)-GNR*TPR+GNS*TPS)/(GR+GS) YPR=Y(I,J)-Y(IM,JM) V(I,J)=V(IM,JM)-GR*YPR-GNR*TPR CONTINUE WRITE(6,106) J , (V(K,J), K=2,20,2) WRITEC6.107) (Y(K,J), K=2,20,2) WRITE(6,108) (X(K,J), K=2,20,2) WRITE{6,109) (T(K,J), K=2,20,2) F I N D T H E S O L U T I O N A T THE U P S T R E A M E N D OF C H A N N E L JJ=J+l X(1,JJ)=0. 1=1 IN=I+i SG= S Q R T { G * Y ( I N , J ) ) T(1,JJ)=T(2,J)-X(2,J)/{V(2,J)-SG) IF{ T(i,JJ) .GT. 1 7 9 9 . ) GO TO 1 7 DQ=00*T{1,JJ)/1200. IF( T(1,JJ) .LT. 1 1 9 9 . ) GO TO 1 6 DO = Q O * ( T ( I , J J ) - 1 2 0 0 . ) / 1 2 0 0 . Q =2.*Q0-DQ _ _._ GO TO 1 7 Q=QO+DQ GS = S Q R T ( G / Y ( 2 , J ) ) T P S = T ( 1 , J J ) - T ( 2 , J) RS = B * Y ( 2 , J ) / ( 8 + 2 . * Y ( 2 , J ) ) SSDENO=2.21*RS**1.333 _ SSNUM=HN*HN*V(2,J)*ABS(V(2,J)) GNS=G*(SSNUM/SSDENO-SO) YP=Y(2,J) VP=Q/(B*YP) VPS =V P - V ( 2 , J ) YPNEW=Y{2,J)+(VPS+GNS*TPS)/GS IF{ABS(YPNEW-YP ) .LT. l.OE-3) GO TO 1 9 YP=YPNEW GO TO 1 8 Y(I,JJ)=YPNEW V( 1 , J J ) = Q / ( R * Y ( 1 , J J ) ) F I N D T H E S O L U T I O N I N THE M I D D L E S E C T I O N OF C H A N N E L ... DO 5 0 1 = 3 , 1 9 , 2 IM=I-1 JM=JJ-1 I N = I + .l G R = S Q R T ( G / Y ( I M , J M ) ) GS=SORT t G/Y ( I N , JM) )  '  _  _  _  _  _  R G = S Q R T ( G * Y ( I M , J M ) ) S G = S Q R T ( G * Y ( I N , J M ) ) SSNUM =H N » H N » V ( I N , J M ) * A B S ( V (  I N , J M )  )  R S = B * Y ( I N , J M ) / { B + 2 . * Y ( I N , J M ) )  SSDENO=2.2i*RS**l.333 G N S = G * ( S S N U M / S S D E N O - S O ) R R = B * Y ( 1 M , J M ) / ( B + 2 . # Y ( I M , J M ) )  RRDEN0=2.2l*RR**l.333 RRNUM=HN*HN*V(  I M , J M ) * A B S ( V ( I M , J M ) )  ;  GNR=G*(RRNUM/RRDENO-SO) T P N U M = X ( I N , J M ) - X ( I M , J M ) + T ( I M , J M ) * ( V ( I M , J M ) + R G ) - T ( I N , J M ) * ( V ( I N , J M ) ISG) T P D E N O = V ( I M , J M ) +  __ R G - V ( I N , J M ) + S G  T ( I , J J ) = T P N U M / T P D E N O TPR =T ( I , J J ) - T ( I M , J M )  ;.  ._  116  APPENDIX B Program L i s t i n g C . C C C  SURuE. WAVES I N THE POWER C A N A L S WFP I S THE CONTROL P A R A M E T E R I E P O S I T I V E OR N E G A T I V E B U = W 1 ' L ) T H OF W . S . AT YO S E C T I O N b = l L J T I I OF G H A N N c L BOTTOM D I M E N S I O N V ( 7 0 t 5 6 ) t Y ( 7 0 >56)X(70 » 5 8 ) T ( 7 0 >50) 1 R _ A u ( 5 r l 0 l ) F»RO»hKEI»GAMA»HITN»REFPT»XM»l3»WFP 101 F u R i - - i A T ( S F l O . O )  W A V E  M  I r ( K O  1151  1153 1152  .LT.  0.0)  SI  OP  JuIi-i-O 0 Q-0.0 HuEG=-1.0 UPE<C>=-1. 0 G=32.2 NIM=KO**C .6667/HKEi I F (GAMA . L E . u . 0 ) GO TO 1 1 5 1 YuNuM=RO* ( Xi'-,/6A,-;A + 2 . 0+SQRT ( i . G+XM**2 ) ) Yu= fONUM/ ( XM* ( 1 . 0 + 1 . 0 / GAMA ) ) ' B-XH*YO/GAMA Gu TO 1 1 5 2 I r (A.'-' . L E . U . U ) bO TO 1 1 5 3 YG=_.0*RO*SUKT(i.u+XM*XM)/XM Gu TO 1 1 5 2 Yu=KO*B/(D-2. 0*KO) Vu=F*5QRTlG*YO) 5u=lV0/(1.466+HKE_) )**2 Bu-o+2.0*YO*XM .YOBAR=YO* 180+2• 0 * _ ) / ( 3 • 0 * ( 8 0 + B ) ) A = l!fi + Y O * X M ) ' * Y O GC-VO+AO P o = b + 2 . 0 * Y 0 * S G R T ( i .0+XM*XM) u  X L = Y O / S O  C C C C  riLF = 0 . W i \ I T E ( 6 r l 4 0 ) F r Ru 11 i K c_ I t G A M A t H I TN ' R E F P T * X M t B » Y O » V O 1 4 0 FuRi-iAT ( 5X t 3 i i F = > F b . 3 f 2 X » 4 H RO= t F 5 • 1» 2X t 3l i K = t F 6 . 1 t 2X * 6H G A M A = » 1 F < + . l » 2 X » o H H i T N = » F 6 . i » 2 X » 7 H R E F P T = » F 5 . 2 r <+H XM=» F<+* 1» 5X* 3H B = » 2 H6.1»2X»4H YG=rFo.2»2X»4H V0=»F6«2 // > N E G A T I V E WAVE R E F L E C f A T R E S E R V O I R ( CONSTSTANT V A L U E S OF X / L O ) TiiE Nuf-iBER U F H i T N n A S TO BE CHANGED AS HL CHANGE:. H I T i j - u O . O F R E A C H E S O F CHANNEL D I V I D E D R E F i - ' T - P T . O F N E G A T I V E W A V E R E F L E C T I N G E X P R E S S E D BY X / L O D A = A L / H I T N  H I 0 = 2 . 0*HITi i * R E F P i HI_=AL + R E F P T  4 5  I O - I ilO+L;. 1 IuN-I.O+1 IuM=10-l I^A-ION I r ( i C N .GT. Ki = l Gu 10 5 Ki=iON-20 Ki*=r\I+-l  Xu=u.u  20)  Du 10 1 = 1 r I UNr  GU '10 4  2  I I /  V(I»1)=V0 Y d ' 1)=Y0 T i l » l ) = 0 . 10  C  11  80  C O N T I N U E  X(l»1)=0. NST=0 J=l LOCATION OF WAVES NWPT Nv,P"l = ION+i ChECK=3. DO i l 1 = 3» IONr 2 i,-,= i - ; _ XiItl)-A(IMr1)+DX CuNlINUE J=2 NJ=u Ti_ST = - 1 . 0 Do o l LL=2f 10' 2 l=IO-LL+2 Iiv=l-1 lN=i+l JI»I «J-1 C A L u GI-iKS ( I f J » G f V r Y » X » T r ti r H N r SO » IM r J M t GR f RG t GNR » X M ) C/-.LL GNRS ( I » J » G » V » Y r X » T » 3 » l I N * SO » IN » J M r GS » SG t GNS r X M ) CALL MID ( X t Si t T f Y t RG » SG » GS » GR * GNRf GNS » I * J » I N » J M * iM» JM ) FIND T H E WAVE AT DGWwST. END J=J+l I = IuN lM=i-l ln=_+l Ji'i=vJ-l XU»J)=HL V( I »J)=0.J C=SGRT(G*YO) 2 u = u . <4*Y0 BP=oO+2.0*Z0*XM bi-,=U . 5* ( B G + D P ) V„=t-VO ,Zo=(GO-G)/(BM*Vrt) Bt-=u0 + 2. 0*ZU-*XM Yr=YG+ZO Br-,= u . 5* ( Bu + bP ) Ap=AO+ZO*oM YpBAR=YP*(BP+2.u+o)/(b.U*(BP+B)) CwUM=(AP*rPbAR-MO*YGBAR)*G CuEi J O = A O * ( 1 . 0 - A J / M P ) CuEw = SOt<T (Ci'jUM/CDENO) I r ( M B S ( C N E W - C ) . L i . 0 . 0 U 1 ) GO TO 72 C=0.5*(CNLW+C) G O r o 71 YlI»J)=YP v..=c-vo V.*0=VW Zo=Y(I»J)-YO N,*p i = i N.,Av = I X P R = X ( I t J ) - X ( I M tJM) B K = O + 2 . 0 * Y ( i M tJM)*XM =  81 C  71  72  313  :  118  Au = I E> + Y ( IM * JM ) * XM ) * Y ( IM r J M ) Ro=-oGRTlG + AR/L5R) Ti I ' J)=-7 ( I H » J M ) + A P R / ( V ( I M 'JM)+RG) •Xx=riL-X( I»J) 0- ZO/20 Z-20 E-VwO/VWO T'i'0=T(I0N»3) TcS1=2.0 BcTAl=XX/XL T j — i ( I r J)-TTO Wi<I I E ( 6 »1J1) 131 F O R M A T (IX » lrllr 7 X » 2 H Z»8X»3H VW» 8X r 2H D»9X»2H E#1UX>3H XX# 1 oil BETA1 ibX»3H rT»5x»3H Y Y #3X»13M DEP AT 0 EnD »4X»2H Yr_X»5M Y i/V0»5X»2H T / ) W K I I'E (6» 133) Zt M\'i> Di E> XXruETAl t TTf Y ( I t J) 133 FoR."iAT (5X * 4 F l U • t » K 1 2 • £ » F l u • 4 r F d . i r F 8 . 3 ) C PuNt-H 137 t F» Ur L..E i Al » Y O P Zf \/W» L>» GAMAfHKEl f XX C FINIJ 1 HE FLOW lu MlJDLE CHHL C CHECv i HE WAvE LOCATION 1- I-2 28 BiP-B+2•0*Y(1+2'J)* X M Y x B M R = Y I I +2» J i * (31P + 2. 0*B J / ( 3 . 0* (bIP+B) ) A _ P = ( B + b I P ) + YI 1+2 » J ) / 2 . 0 Ci-aUf--i= (AIP*YlBAR-Au*YOBAR) *G CuEiiO=AO* ( 1. 0-AO/AlP ) C-SuRT (CNuM/CJEiiO; Vi. = C-V0 • • Ii.,= i-1 J|vi=vJ-l C/VLLGNRS ( I » J » G f V r Y r X » T # B » HN » SO » IM t JM i GR » RG r GNR » XM) T P N U K = X ( I+2f J) - A ( iM » JM ) + T ( IM t JM ) * ( V ( IM t JM ) +RG) + I ( 1 + 2 * J ) * V r f TPDcK0=V ( i M t Jv\) + Ro + VW Tp=iPNUM/TPL>ENO Iiw + l CALL GNUS (If J>GrV»YrX'T»Bt HN r SO t IN » J M r GS F SG t GNS» XM) TK»NOM=X< li'o JM) -X ( IM i JM) + T(IM r J M ) * ( V ( IM» JM) + K6 ) - f ( IN * JM) * ( V ( IN» JM )  1 _G>  Tt DtN0=V ( iM» JM) + Ru-V (IN # J M ) +SG J  T£  I ' J)  =TPIMUI' ./IPUENO :  IF( TP .Gl'. T d ' J ) ) GO TO 22 K-N,vAV-i L-ION+3-(K-NST)+N3T-NJ*2  45  1  62  IF  i NWPT .NE.  24  1  25  C  iON  )  GO TO 24  (K»L»G»V»Y»X»T»B» HN i SO » VO t YO » VP » YP » XP » l'P r K+l» L - l » I - l . 1 VW» n L » XM'AO tYOBAR) G^ iO 25 CMLL. W A VMID ( K » L » G f V » Y r X » T # B »I IN r SO r VO » YO » VP» YP» XH » TP » K +1r CHLL.  sj-it  W A V O G N  VW i NST  t XM f M O » Y O B M R )  N..'Arf=K Z=Y iKrL)-YO D-Z/ZO E=V;./VV.'0 XA=nL-XlKfD UcTM. = XX/XL T'i = I (K»L) -TTO Wi< I IE ( 6 t 133 ) Z » V W r D t E » X X t BET A1 r 1 T r Y ( K » L) Puf-H-M 137» F r u r BE f A1 r Y 0 r Z » VW r B F GAMA t \ IKE 1 r XX  IF(0 .LT. 0.020) GO To 1 Ii-dNiWPT .LE. 1) Go TO 35  35  NnPi"=I IK=NWAV-Nrt-PT  IF(iK  .LT. 1) GO TO 34  Do  I D - I t  06  IK  M-K-IlJ N=L-ID VlM»N)=V(KrL) YIM»N)=Y(K»L) X(M»N)=X(K»L) T(MrN)=T(KrL) IF v M . E _ j . 1 ) Go TO 34 36  34  C O N T I N U E  I F ( MWAV .EQ. Iu) GO TO 33 K=Nv.AV L = IoN+3- ( K-iNST ) +NST-No*_ Kk=i-iWAV+l Do o2 K = KKr 10 L=L + 1 Ki-l=K-l KN=K+1 LN=L-1  GNP.S ( K t L t G 1 V , Y t X » T r 3 t HN t SO * KM t LM f GR » RG t GNR » XM ) GNRS(K tL tG» V tY » X tT f 8 » H N r S O »KN tLM r G$»SG > GNS tXM) C ALL M I D ( A » V » T f I f t\G r SG t GS » GR t GNR » GNS »KrL» KN t LM » KM » LM) IF (L .GE. JD1M) oO To 61 CONiINUE FONo THE UNSTEADY F'LOw AT DOWNSTREAM END K-IoN L=L+1 X(K»L)=HL V (K tL )=o .0 C M L L . DOv.N(XrY»T»V»K»L»B»G»Hi->l»SO»XM) TT=T(Kf D-TTO Yt — i ( K » L ) /YU W K I "I E (6 t 398 ) Y ( K # L ) i Y Y » TT 398 FuRi-'iAT (10UX» 3 F l u . b ) FIND T H E FLOW IN i'H_ FRONT OF WAVE I r ( NWPi .Eo. 2) uO To 51 IF(UPEND .GT. 0.0) GO TO 200 1 = 1-2 I F ( I .LT. NWPT ) 60 TO 63 1 = 1-2 Gv^ TO 64 I F ( Y U + l r J - D - G i . YG) GO TO 45 Go TO 26 Hu-J C M L L  CMLU  32 C 33  C 61 64 63 22  C  Ho2=HJ/2•U  J_.=nJ2+Q .01 Ho2i=(HJ+1.0)/2.0 J_1=HJ21+U.U1 IF(o2 .EQ. J21) Go TO 2o IF o2=J21 J IS A L V C N NUMBER Li_P=I0+2-i IoM=I0-l D-j 27 LL=LLP t I0M»2 I=IO-LL+2  I .••.]= 1 - 1 I.,=  1+1  J>i=J-l G N R S ( I » J * G » V » Y » X f T » t 3 » HN t SO r I M » J M » GR » R G t GNR f X M )  CALL 27  .  CALL  G N R S ( I » J r G » V r Y » X » T r B r H N F S O » I N F J M f G S tSG  CALL  M I D ( X f V f T » Y M<G f S G r G S » G R r G N R F G N S F I F J F 1 N F J M F I M F J M )  r G N S » XM)  J c V L N = - l JoDD=2 C  FINo  I HE  FLOW  IN  THE  UPPER  STREAM  E N D  1 = 1 X ( 1 1 J ) = 0 . u Y ( I > J ) = Y O U P S T ( G » X » Y » V » T r I r J » B i H N t S O F X M )  CALL Go 26  10  31  Jt_Vt-N=2 JoDU=-l L L P = I 0 + 2 - I  DO  LL=LLP t  29  101  2  I = I G - L L + 2 l>,=  i - l  l..= i + l  JM=J-1 G N R S ( I » J » G r V » Y » X » T » 0 » HN r SO F I M » J M t G R » RG FGNR > XM) G N R S I l » J F 6 tV r Y » X F T F 3 11 I N » S 0 » I N » J M » G S , S G r G N S , X M )  CALL. C M L L 29 31 C C  C A L L M I D ( A » V » T » Y t R G »so J=J+1 I=ION+3-J -NJ*2 GO Q A C K TO THE v«AvE CHANGE THE VALUES OF  I J .LE.  IF No  =T-4  6)  GO  FG S F G R» G N K » G N S » I » J » I N * J M  FRONT J TO  39  J + l  CJ-J-2 l K = i 0 N + 3 - ( 6 + ( N J - l J Do 42  42  CALL  K = 2F  I K F  *2)  2  V Y X T ( K # 6 » V » Y » X » T >  K = IK L=6 KK=I\ + 1 Do  43  K=KK»  ION  L=L+1 43 630  IF(L  . G E .  C K L L  VYAT  GO  TO  630  I r v = l O N + 5 - I 5+ ( N J - 1 ) * 2 ) Do  37  JDIM)  ( K » L » V f Y » X » T )  o7  C M L L  K = 1 » I K » _ V Y X T v K » 5 ' V » Y r X » T )  K=IK  L=5 Kr\=i\ + 1 Do  38  K=KK»  ION  L=L+1 1F(L 38 640  C M L L  .GE.  JDIM)  Go  TO  64 0  V Y A T ( K » L » V I Y » X » T )  L=J+(IiJ*2) K=IGN+3-L KT=iON-K-,viST IF Do  IKS .LT. Kl") 4 7 KX = r\T»KS  Go  TO  39  H M » J M )  K=I0M-KX  4  L-ION+3-(K+(NJ-1)*2-MbT)+NST M-K N=L 68  I F ( A B S ( Y ( M - 1  » N - 1 ) - Y 0 )  CALL V Y X T ( M - 1 » |V|i = h - l  vi-1'  . L T .  O . D  GO  TO  69  V » Y » X » T )  N l = u - 1 M=M-1 N - N - l GO 69  TO  66  LL=u DO  47  K=K»  ION  L - L + L L Ii-(L  . GEo  JbIM)  GO  TO  39  LL=i 47  C A L L  39  I = : I u | - ! + 3-( J + N J + 2 ) IF  V Y X T ( K » L » V » Y r X » T )  I  (NWAv'-NST)  . L E .  I)  GO  TO  61  K-I L - J  Gc 51  62  TO  I-l  Ut-Ei\0 = l . U J=J+1 Gu C C 20 0  TO  45  NEGA j I V E RESi_T  WAVES  THE  VALUES  AT  PT .  S  K=N.JAV L-lON+3-(K-NST)+N3T-NJ*2 Hi\Evi=-1.0  iPR = l  It=3 Ib=b l f L - 2 I_;=4 Iu-b V ( l b » 2  ) - v ( K + 2 ' L )  Y(1S>2  ) = Y ( K + 2 » L )  X ( I S r 2  ) = x ( K + 2»i_)  T i I C  »2  REScT  ) = T ( K + 2 » L )  THE  VALUES  MLONG  C+  PASSING  KK=iCN-NWAV IJKL.= 0 DO  201  1 = 3 >5 r 2  I O K L = I J K L + 1 K,\i =  f\K + l  Dc  201  LK=1»KN  M=K+LK-i+IJKL-1 N = L + L K - l - i J K L + 1 IF(.M  .GT.  JDIM)  Go  TO  2U2  J=L<\ V ( I t J ) = V ( M i N ) Y l l r J ) = Y  (i-ifij)  X ( I » J)=X(|.i»N) 201 C  T I 1 t J)=T(M»i\i) FINu  Gu  IHE TO  INITIAL 20  3  NEGATIVE  WAVES  REFLECT  PT.  122  4  202K ix——- K — 1 203 J = l  I F C w F P .GE. 0.0) G O T O i V(l»D=V(3»i) Yil#l)=Y(3»l) X(lrl)=X(3f1)  Til»l)=T(3»l) X ( I F L » J ) - X ( I PR »  J )  *J)=T(IPRiJ) Y(IPLr J ) = Y O + v O * v O / ( 2 . U * G ) BPR=D+2.0*Y(IPRfJ;*XM A p R = ( o + B P i < ) * Y ( I P R » J ) /2 . 0 Y P R b A R = Y ( I P R r J ) * ( u P R +2.0*3)/(3.0*(BPR+B) ) B P L = B + 2 • 0 * Y ( 1 P L , J ) * XM APL=(B+bPL)+Y(IPL,J)/_.0 YPLoAR=Y(IPL» J ) * ( L > P L + 2. U * b ) / (3. U * ( B P L + B > ) T ( I PL  Vfi=V(IPR to)+SORT(b*(APL*YPLBAR-APR*YPR3MR)*ARL/I  V ( I r L » J ) - ( V ( I P R *J ) -VW) Z=Y(IPL»J)-Y(iPRrb) ,  * A P R / A R L  A P R * ( A P L - A P R ) ) )  + VW  D = A D S ( Z / Z O )  E=Vw/VWG X A = H L + X ( I P R » J )  BcTAi=XX/xL TT-i'(IPh»d)-T7 0 C  WKliE(6»lb3) Z » V W» D» E r X X f B E T A 1 » TT f Y ( I P R »J) P u r J L H 137 > F »D r D E f A1» Y0» Z» V'rt»B»GAMA r H K E 1 1 XX VC+»D=V(_»1) Yit»l)=Y(2»i) X(4»D=X(2»i)  210  Tl4»l)=T(_»i) J=J+1  CHLL.  G N R S  JK = vj-i C M L U C M L L  G  t  GS» SG t GNS * XM)  G N R S 1 1 » J r G» V » Y » X » T # 3» HN r SO rIE» JR » GR » RG»GNR tXM) M I D ( X » V » T f Y I KG r SG F GS t G R F G N K t GNSF IE' J r I S F J» X E F J R )  FlNu X r T »AT # PT. P  BPR=B+2.0*Y(IPR tJR)*XM A p R = ( B + 3 P H ) * Y ( I P R » JR)/2.0 YPRuAR=Y(IPR t J R ) * ( B P R + 2.0 * B ) / ( 3 . 0 * ( B P R + B ) ) B P L = B + 2 • 0 * Y ( I PL > J , < ) * X N APL=(B+bPL)*Y(IPL tJR)/2.0 YRLbAR=Y( I P L F J P J *(liPL+2.0+B)/(3.0*(BPL +B) ) V,.-v  C  (I»J»G»V»Y»X»TrB» HN r S O r IS f J  ( I P R » jR ) + S Q u T ( G * ( M P L * Y P L b A R - A P R * Y P R B A R ) * A P L / ( A P R * ( A P L - A P R  ) ) )  T ( I PR' J ) = ( X ( I S t J ) - X ( I PR t J K ) + T ( I PR * JR ) * V ir.—T ( I S » J ) * (V (IS» b) -SG) ) / ( 1 -VdS»J)tSG) X l l P R r J ) = A < I S » J J + l V ( I S » J ) - S G ) * ( T ( I P R tJ)-T(IS tJ)) F l N u T H E V Y AT PT. P B Y M E A N S O F I N T E R P O L A T I O N Y(IPR»J)=Y(lSrJ)+(Y(I_»J)-Y(IS»J))*(T(IPR»J)-T(iS»J))/(T(IE»J)-  i T(iS»J))  v a d b J ) = v ( I S » J ) + {V(lL»J)-V(ISfJ))*(T(lPFbJ)-T(iSiJ))/(T(lErJ)-[(I  iS»J)) C ASSuMc THE V A L U E O F V ( I P L » J ) = V ( I P L t J - l ) 270 V C IPLr J ) - \ i i i p L r J - i )  220  XlIr'L»J)=X(iPK»J) T(IPLrJ)=T(1PR»J) BPR=B+2.0*YUPRrJ ) * X M A P R - ( B + B F M ) * Y ( I P R » J )/2.0 APL= (V( I PR » J ) - V . ; ) *A P R / ( V ( I P L» J ) —Vw )  I F ( O A M A  .LE.  0.0)  GO  TO  1181  V ( IKL r J) = (-b+SGi<T ( B * B + ' + . 0 * X M * A P L ) Go 10 1182 1181 Ii-'UM . L E . 0.0) Go TO 1133 Y(IPL»J)=SQRT(APL/XM) GO TO 1182  )  / (2. 0 + XM)  YlIPLfJ)=HPL/B  1183  C  DETERMINE THE p r . OF Q 1182 I r ( J . G T . 2) GO To 221 H=Yu+VO*VO/(2.0*G) X(IorJ)=X(2»1) YUufJ)=H C M L L . OMRS I I PL * J f G i V r Y > X t T , B r HN r _>0 r I PL r J > GP r Po r GNP » X M ) T U o r J)=T( IPL » JJ + I X C Iu» J ) - X ( IPLr J) ) / ( V ( I PL r J )-Po ) V I Ivi» J) = V ( IPL# J)+oP*( t ( i Q r J ) - Y ( i P L r J) )-GNP* (T( Io r J ) - T ( I PL r J) ) Go iO 222 221 C M Li- GNRS (10rJrGrVrYrXrTr6r H{.| r SO r I Q r JR » GM * HG # GNiir XM) C n U GNK'S ( 1 PL » J r G r V r Y » X r T » 5 r HN r _>0 r I PL r J r 6P r PG » G.-J.P r XM ) C ALu M10(X r V » T r I r i iGt PoiGP * GHtGNH ' GNP f I Q ' J » IPL r J r IO rJR) C D E T L R H I H E Tl-iE P T . OF D 222 C A L L . GNUS I IUr JrGr \/r Y» Xr T»b»HN?SO» ID r JR t G R t RG , GNi\ r XM ) CaLL OMRSIIo » J r Grv tY r X r T rb r H N r G O »1Q r J rGO r Q G rGNQ rXM) T ( Iu t J ) = ( X ( I Q r J i -X ( i D r Jr<) + T ( I u t J R ) * I V ( I L> t JR ) + R G ) - T ( 1G r J ) * ( V ( IO r J ) I -Go))/(V(IJroR)+RG-VIiQrJ)+GG) X ( l U r J) =X ( I'Ur J R ) + ( V ( lor JR) + R G ) * (T ( IL) r J ) ~T ( I J r J R ) ) V i I o i J ) = (V(ID r J K ) - V ( I o r J ) + G R + Y(ID»JR)+GG*Y(IQ r J ) - G N R * ( T ( I O ? J ) - T ( 1 0 1  , Ji<)  ) + G N G * ( T ( I Or o ) - T  ( lor  J)  ) ) / ( oR + G O )  V ( Iu r J) = V ( Ii) r oR ) -GK* ( Y ( I 0 r J ) -Y ( 1D t J R ) ) -GNR* ( T ( I u t J) - T ( ID r J R ) ) C FlNu THE NEw VALUE OF VPL 3Y MEANS O F INTERPOLATION O F POINTS QrDr C AND p VPL=V(IOrO)+(VtiDrJ)-V(IQ#J))*(T(IPLrJ)-T(lGrJ))/(T(IDrJ)-T(IGrJ)) JF' ( H B S ( VPL-V ( I P L r J) ) . L f . 0.001) GO T O 250 V ( IPL i J ) : u o * ( VPL + V II PL » J ) ) Go 10 220 250 Z=YIIPL»J)-Y(IPK»J) D=AbS(Z/ZO) E-V../VW0 XA=i-iL + XUPRr J) BI_TA1 = XA/XL  = i'( IPR r j ) - T 10 V»ixl i E ( 6 1153) Z r VW r D r E» XX t BETAi r T T * Y ( IPR * J ) C PuNv.H 137 r rt uroE i A1 r YOr Zr Vvi'r or oAMAr HKEI r XX Ir (o . L T . Ki\) Go iO 2o0 YiR=Y(IPRrJ)/Y0 YYL=Y(IPL r J ) / i 0 T . T - T ( IPRr J ) - T T G T T T I - K I Pu» J ) - T V 0 W K I I E ( 6 » 2 9 9 ) Y ( iPKr J) » Y i ll, T T T » Y ( IPL t J ) r YYLrTTTT 299 FoRr-iAT (2Xr23H MAX DEPTH A T DOrtNiT. END r 3F1U . 3 r D X » 3F10 . 3 / / / ) 280 IMnNEG . o T . u . u ) G O 10 1 I r ( u . L T . 0.U2) Go T O 1 IF(o . L i . KK) GO TO 2l0 C NEGATIVE WAVE REACHclS DoW,4STREAM L.ND J=J+1 T~i  J,< =  o-l  XiIPR»J)=HL VilPRrJ)=u.O br-R=B + 2.0*Y I IPRr J K ) +XM  *Prt=(B+BPR) *Y( IP>< r JR)/2.U YPR-AR=Y(IPR tJR)* iBPR+2.0*3)/(3.0*(BPR + B)) BPL=B + 2 . 0 * Y ( I P L r j R ) * X i ' i Ar L=(B+t3PL)+-Y(lPL»JR)/2.0 YPL3AR=Y(1PL , JR J * i 8PL + 2.0*8)/(3.0*(BPL + B)) V ,•, = •«/• ( iPRf JR) +SQRT ((5* ( ARL* YPLBAR-APR*YPRBAR ) *APL/ ( APR* ( APLT(IPR » J) = I X ( I P R » J)-X(iPrt» JR))/Vri + T(IPR tJR) J  YIIPR»J)=Y(IS»J-l)+(TtiPRrJ)-T(ISiJ-D)*(Y(IE»J ) - Y ( I S ' J - l ) ) / ( T ( I E  1 , J )-TlIbrJ-l)) Hi\lEG=l.U Go TO 27 0 137  FUR;MAT(F6.3'5F8.<4F3FB.2#F10.2) E;„D  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050557/manifest

Comment

Related Items