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.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

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 IN OPEN CANALS by HENRY JAW-HERE WU B.Sc. ( C i v i l E n g i n e e r i n g ) P r o v i n c i a l Cheng-Kung U n i v e r s i t y , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n t h e Department o f C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s . i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f C i v i l Engineering The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date A p r i l 29, 1970. i i ABSTRACT T h i s s t u d y i n v e s t i g a t e s the p r o p a g a t i o n o f a su r g e wave i n a power c a n a l f o l l o w i n g l o a d r e j e c t i o n o r r e d u c t i o n . D i m e n s i o n l e s s r e l a t i o n -s h i p s a r e d e r i v e d t o p r e d i c t (a) t h e i n i t i a l wave h e i g h t , (b) t h e v a r i a t i o n o f the wave h e i g h t a l o n g the c a n a l and ( c ) t h e maximum s t a g e o f w a t e r d e p t h 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 c a n a l s o f 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 r o s s - s e c t i o n s . The e f f e c t s o f v a r i o u s p a r a m e t e r s , s u c h as v e l o c i t y and d e p t h o f i n i t i a l f l o w , f r i c t i o n a l c o e f f i c i e n t s , bed s l o p e , c r o s s - s e c t i o n o f t h e c a n a l , d i s t a n c e o f wave p r o p a g a t i o n and i n i t i a l wave h e i g h t o f the su r g e a r e s t u d i e d . A computer program i s d e v e l o p e d 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 . I t i s f o u n d t h a t , as a p o s i t i v e s u r g e p r o p a g a t e s a l o n g t h e c a n a l , the wave h e i g h t d e c r e a s e s l i n e a r l y w i t h d i s t a n c e f o r a s h o r t c a n a l , a c c o r d i n g t o an e x p o n e n t i a l f u n c t i o n f o r a l o n g c a n a l . An a p p r o x i m a t e l o g a r i t h m i c r e l a t i o n s h i p i s a l s o f o u n d between t h e v a r i a t i o n o f wave h e i g h t o f a p o s i t i v e s u r g e and c a n a l c r o s s - s e c t i o n a l p a r a m e t e r s . 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 downstream end o f t h e c a n a l i s n o t l i n e a r w i t h r e s p e c t t o t i m e . An a l m o s t l i n e a r r e l a t i o n s h i p between th e maximum wa t e r d e p t h a t t h e downstream end o f the c a n a l and the l e n g t h o f t h e c a n a l i s n o t e d . The d i m e n s i o n l e s s r e l a t i o n s h i p s d e r i v e d i n t h i s s t u d y may be us e d t o e s t a b l i s h d e s i g n c r i t e r i a f o r c r e s t e l e v a t i o n s o f t h e banks and w a l l s o f power c a n a l s t o a v o i d o v e r t o p p i n g . i i i TABLE OF CONTENTS page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES v NOTATION x • ACKNOWLEDGEMENTS x i i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 DETERMINATION OF WAVE HEIGHT AND VELOCITY FOR I N I T I A L 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- 6 . 'Wave H e i g h t and V e l o c i t y 2-2 I n i t i a l Surge Waves i n C a n a l s o f R e c t a n g u l a r 9 C r o s s - S e c t i o n 2-3 I n i t i a l Surge Waves i n C a n a l s o f T r i a n g u l a r 11 C r o s s - S e c t i o n 2- 4 I n i t i a l Surge Waves i n C a n a l s o f T r a p e z o i d a l 12 C r o s s - S e c t i o n CHAPTER 3 NUMERICAL CALCULATIONS FOR SURGE WAVE PROPAGATION 3- 1 G e n e r a l 16 3-2 B a s i c A s s u m p t i o n s 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 S o l u t i o n o f 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 f o r 18 a 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 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 23 P r o p a g a t i n g i n Power C a n a l s 3- 6 N u m e r i c a l C a l c u l a t i o n s f o r N e g a t i v e Waves 29 P r o p a g a t i n g i n Power C a n a l s CHAPTER 4 RESULTS OF ANALYSIS 4- 1 G e n e r a l 34 4-2 D i m e n s i o n l e s s R a t i o s 34 i v 4-3 V a r i a t i o n of P o s i t i v e Surge-Wave Height i n 36 Rectangular Canals 4-4 V a r i a t i o n of P o s i t i v e Surge-Wave Height i n 38 T r i a n g u l a r Canals 4-5 V a r i a t i o n of P o s i t i v e Surge-Wave Height i n 39 T r a p e z o i d a l Canals 4-6 Approximate Equations 40 4-7 Reduction of Wave Height A f t e r R e f l e c t i o n 46 a t a Rese r v o i r l o c a t e d at the Upper End of the Canal 4-8 Propagation of Negative Waves R e f l e c t e d 50 from a Reser v o i r at the Upper End of a P r i s m a t i c Power Canal 4-9 Maximum Water Depth a t Downstream End of a 51 Canal 1 CHAPTER 5 CONCLUSIONS 54 FIGURES 57 REFERENCES 109 APPENDIX A ( l ) 110 APPENDIX A(2) 113 APPENDIX B 1 1 6 FIG. 2-1 FIG. 2-2 FIG. 2-3 FIG. 2-4 FIG. 2-5 FIG; 2-6 FIG. 3-2 FIG. 3-3 LIST OF FIGURES Definition sketch: Surge wave in an open channel Definition sketch: Surge wave i n a rectangular canal I n i t i a l surge waves in 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 Definition sketch: Surge wave i n a trapezoidal canal FIG. 2-7(a) I n i t i a l surge waves in a trapezoidal canal for T — 0 FIG. 2-7(b) I n i t i a l surge waves in 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 in 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 in a trapezoidal canal for T = 0.9 FIG. 3-1 Definition sketch: ^and C - characteristic grids on the x-t plane Characteristic grids on the x-t plane D e f i n i t i o n s k e t c h : ( a ) u p s t r e a m end boundary and (b) downstream end boundary FIG. 3-4 A part ..of results of computer calculations i n example 3-1 v i FIG. 3-9 FIG. 3-10 FIG. 3-11 FIG. 3-5 A part of r e s u l t s of computer c a l c u l a t i o n s i n example 3-2 FIG. 3-6 Schematic presentation of the c h a r a c t e r i s t i c grids f o r a p o s i t i v e surge wave propagating upstream FIG. 3-7 Schematic presentation of the c h a r a c t e r i s t i c grids r f o r a p o s i t i v e surge wave propagating downstream FIG. 3-8 Flow chart of the computing program f or a p o s i t i v e wave propagating upstream along the canal D e f i n i t i o n sketch f o r example 3-3 D e f i n i t i o n sketch f o r example 3-3 The comparison of the method of c h a r a c t e r i s t i c s with Favre's method from example 3-3 FIG. 3-12 Summary of c a l c u l a t i o n i n example 3-3 by Favre's method FIG. 3-13 Results of c a l c u l a t i o n i n example 3-4 FIG. 3-14 (a) P o s i t i v e surge wave reaches the reservoir at the upper end of the canal. (b) Negative surge wave r e f l e c t e d from the reservoir at the upper end of the canal. FIG. 3-15 D e f i n i t i o n sketch f o r point R i n f i g u r e 3-16 FIG. 3-16 Schematic diagram of c h a r a c t e r i s t i c grids f o r the negative surge wave at point R FIG. 3-17 Schematic diagram of c h a r a c t e r i s t i c grids for the negative surge wave at point R v i i 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 Q = 0.20 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 wave propagating along the canal. FIG. 4-8 Schematic diagram f o r a p o s i t i v e wave reaches" the upper end of the canal FIG. 4-9 Reduction of the negative wave height reflected at the r e s e r v o i r at the upper end of the rectangular canal FIG. 4-10 (a) Variation of wave height of a negative surge propagating along a rectangular power canal for F 0 » 0.200, 0.125 and 0.075 FIG. 4-10 (h) Variation of wave height"of a negative surge propagating along a rectangular power canal for F Q = 0.175 and 0.100. FIG. 4-10 (c) Variation of wave height of a negative surge propagating along a rectangular power canal for F Q = 0.150 and 0.05 FIG. 4-11 (a) Variation of wave height of a negative surge propagating along a triangular power canal for F Q = 0.20, 0.125 and 0.75 „ v i i i F I G . 4-11 (b) V a r i a t i o n of wave height of a negative surge propagating along a t r i a n g u l a r power canal f o r F = 0.175 and 0.100 o F I G . 4-11 (c) V a r i a t i o n of wave height of a negative surge propagating along a t r i a n g u l a r power canal f o r F q = 0.150 and 0.050 F I G . 4-12 (a) V a r i a t i o n of wave height of a negative surge propagating along a trapezoidal power canal f o r r = 1.50 and F = 0.200, 0.125 and 0.075 o F I G . 4-12 (b) V a r i a t i o n of wave height of a negative surge propagating along a trapezoidal power canal f o r r = 1.50 and F = 0.175 and 0.100 o F I G . 4-12 (c) V a r i a t i o n of wave height of a negative surge propagating along a trapezoidal power canal f o r r = 1.50 and F = 0.150 and 0.050 o F I G . 4-13 Schematic diagram of the v a r i a t i o n of water surface f o r a p o s i t i v e wave propagating upstream along the canal F I G . 4-14 Schematic diagram of the v a r i a t i o n of water surface f o r a negative wave propagating downstream along the canal F I G . 4-15 V a r i a t i o n of water surface at the downstream end with respect to time f o r the rectangular canal of V q = 30 f t . , b = 30 f t . and n = 0.03095 o F I G . f-16 V a r i a t i o n of water surface at the downstream end with respect to time f o r the t r i a n g u l a r canal of y = 22.36 f t . , m = 2.0 and n = 0.03095 o F I G . 4-17 V a r i a t i o n of water surface at the downstream end with respect to time f o r the trapezoidal canal of m = 1.5, b = 25.5 f t . , n = 0.03095 and r = 1.50 o i x FIG. 4-18 Maximum water depth at the downstream end of a rectangular canal FIG. 4-19 Maximum water depth at the downstream end of a t r i a n -gular canal FIG. 4-20 Maximum water depth at the downstream end of a trapezoidal canal f o r r - 1.50 NOTATION A b b o c f g H k L hi IM n R P Q r S f S0 t V v w w x * X y c r o s s - s e c t i o n area " width of free surface bottom width of canal wave c e l e r i t y Froude number = V/yg«y g r a v i t y a c c e l e r a t i o n t o t a l head at r e s e r v o i r dimensionless parameter = b 0/m»y 0 canal length reference length = y 0 / S Q canal side slope r e s i s t a n c e c o e f f i c i e n t i n the Manning formula h y d r a u l i c radius wetted perimeter discharge dimensionless parameter = l+][+£ f r i c t i o n slope (energy gradient) slope of bottom time water v e l o c i t y absolute wave v e l o c i t y - V f c u n i t weight of water distance distance of wave propagation water depth c e n t r o i d a l depth of c r o s s - s e c t i o n area 2 note that some Authors use F = _Y_ as Froude number 8 - y f t f t f t f t /sec. 2 f t/sec» f t f t f t f t f t f t ^ / s e c . sec. f t / s e c . f t / s e c . l b / f t 3 f t f t f t f t Z surge wave height f t ^ dimensionless parameter = V W Q/V 0 0 dimensionless parameter = ZQ/yQ T discharge ratio = Q/Qo rQ unit shear force on bottom and sides of canal lb/ft^ M energy coefficient / X cf> dimensionless parameter = LR /r=l / \ "R / r=n S u b s c r i p t o r e f e r s to i n i t i a l steady s t a t e c o n d i t i o n X X I ACKNOWLEDGMENT The a u t h o r w i s h e s t o e x p r e s s h i s g r a t e f u l a p p r e c i a t i o n t o h i s s u p e r v i s o r Dr. E. Ruus, f o r h i s v a l u a b l e g u i d a n c e and e n c o u r a g e -ment d u r i n g t h e r e s e a r c h , p r e p a r a t i o n , and development o f t h i s t h e s 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 Mr. P. D o n n e l l y f o r h i s a d v i c e and s u g g e s t i o n s . 1 CHAPTER 1 INTRODUCTION The p r e d i c t i o n of the height and v e l o c i t y of.a surge wave, the maximum stage and the other flow c h a r a c t e r i s t i c s i n a canal due to large instantaneous changes i n discharge has been of i n t e r e s t to engineers f o r many years. The designer of a h y d r o - e l e c t r i c headrace canal, f o r example, must determine the maximum stage of water that could occur as a r e s u l t -of a sudden load r e j e c t i o n i n power output, considering both the surge wave c r e s t and the subsequent unsteady r i s e i n stage at any point. In h y d r o - e l e c t r i c head or' t a i l r a c e canals, large changes i n discharge are i n i t i a t e d by opening or c l o s i n g of turbine gates. These discharge changes cause p o s i t i v e or negative surge waves. A wave, which r e s u l t s from an increase i n water depth i s c a l l e d a p o s i t i v e wave. A wave, which r e s u l t s from a decrease i n water depth i s c a l l e d a negative wave. In a p o s i t i v e surge wave, the higher portions of the wave have a greater v e l -o c i t y of propagation than the lower ones. Hence the f r o n t of the wave tends to become steeper and steeper u n t i l the slope of the f r o n t i s enough to create a r o l l e r . The fr o n t of the wave then resembles a moving hydraulic jump. A f t e r a p o s i t i v e surge-wave fr o n t i n a head race canal passes a given section, the water l e v e l at that s e c t i o n does not, i n general, remain stationary, but continues to r i s e . The unsteady water surface r i s e behind the surge wave p e r s i s t s until;.interrupted by the negative wave, which r e s u l t s from the r e f l e c t i o n of the p o s i t i v e wave at the r e s e r v o i r at the upstream end of the canal. 2 The negative waves are not stable i n form, because the upper portions of the wave t r a v e l f a s t e r than the lower portions. This r e s u l t s i n a gradual f l a t t e n i n g of the wave front as i t moves along the canal. At s o l i d boundaries, such as closed gates, 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 negative wave as a negative wave. At the res e r v o i r , a p o s i t i v e wave i s r e f l e c t e d as a negative wave and a negative wave as a p o s i t i v e wave. In a short canal, the wave front may t r a v e l back and f o r t h several times before a new steady state i s reached. However, the f i r s t maximum stage i s higher than the subsequent peaks on the water surface o s c i l l a t i o n cycle and i s the most s i g n i f i c a n t f o r design considerations. This study includes four parts: ( 1 ) Determination of i n i t i a l surge wave height and v e l o c i t y (Chapter 2 ) , ( 2 ) The method of computation of the propagation of surge waves (Chap-ter 3 ) , ( 3 ) The v a r i a t i o n of wave heights f o r p o s i t i v e and for the r e f l e c t e d negative surges (Chapter 4 ) , and ( 4 ) The v a r i a t i o n of water depth at the downstream end of the canal (Chapter 4 ) . The study deals with: (a) I n i t i a l uniform steady flow with Froude numbers from 0 . 0 5 to 0 . 2 0 i n c l u s i v e . (b) Straight canals of constant l o n g i t u d i n a l slope, shape, siz e of cross-section and f r i c t i o n a l c o e f f i c i e n t throughout. Rectangular, t r i a n g u l a r and trapezoidal cross-sections are considered. ( c ) P o s i t i v e waves caused by an 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 u p s t r e a m t o w a r d t h e upper end o f the c a n a l . ( d) Downstream t r a v e l l i n g n e g a t i v e waves, r e s u l t i n g f r o m r e f l e c t i o n o f 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 upper end o f the c a n a l . 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 . The methods s u g g e s t e d by R. D. J o h n s o n ^ ^ , and H. F a v r e ^ ^ a r e i n c u r r e n t u s e . A c c o r d i n g t o Johnson's method, the t o t a l l e n g t h o f c h a n n e l i s d i v i d e d i n t o s e v e r a l r e a c h e s . The s l o p e s i n the w a t e r s u r f a c e and t h e 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 drop i n the w a t e r s u r f a c e and i n t h e c a n a l f l o o r a t j u n c t i o n s between the r e a c h e s . The 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 l e v e l . When a wave r e a c h e s the j u n c t i o n , two component waves emerge, one o f w h i c h t r a v e l s u p s t r e a m and the o t h e r downstream. When t h e downstream t r a v e l l i n g component wave r e a c h e s t h e downstream end o f t h e c a n a l , i t w i l l be r e f l e c t e d and t r a v e l s u p s t r e a m . When t h e u p s t r e a m t r a v e l l i n g component 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 , i t w i l l be t r a n s m i t t e d and r e f l e c t e d a g a i n i n t o two component waves w h i c h t r a v e l i n o p p o s i t e d i r e c t i o n s . When two waves t r a v e l l i n g i n o p p o s i t e d i r e c -t i o n s meet, a n o t h e r p a i r o f waves i s g e n e r a t e d . The wave h e i g h t s and w a t e r d e p t h s a t any s e c t i o n o f c a n a l can be computed s t e p by 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 e q u a t i o n s f o r t h e corn-Numerals i n p a r e n t h e s e s r e f e r t o c o r r e s p o n d i n g i t e m s i n R e f e r e n c e s . 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 a l o n g a c a n a l . He made t h e assump-t i o n t h a t t h e l o n g i t u d i n a l p r o f i l e o f the w a t e r s u r f a c e b e h i n d t h e wave f r o n t i s a s t r a i g h t l i n e . 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 c o n s i d e r e d as one r e a c h . T h i s method may be a p p l i e d t o r e l a t i v e l y s h o r t c a n a l s 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 ap p r o x -i m a t e s o l u t i o n . Because t h e s e manual c o m p u t a t i o n a l methods a r e l a b o r i o u s t h e i r u s e f u l -n e s s i s r e s t r i c t e d . I n r e c e n t y e a r s , t h e advances i n computer t e c h -n o l o g y has s t i m u l a t e d t h e development and a p p l i c a t i o n o f more r i g o r o u s a p p r o a c h e s . On s u c h r i g o r o u s s o l u t i o n i s d e s c r i b e d i n t h i s s t u d y . I n t h i s t h e s i s , e q u a t i o n s o f momentum and 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 us e d t o s o l v e t h e h e i g h t and v e l o c i t y o f a s u r g e wave. The p a r t i a l d i f f e r e n t i a l e q u a t i o n s o f m o t i o n and c o n t i n u i t y g o v e r n i n g t h e 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 i n t h e f r o n t o f , o r b e h i n d , t h e wave r e g i o n a r e c o n v e r t e d i n t o t o t a l d i f f e r e n t i a l e q u a t i o n s 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 and s o l v e d by a f i n i t e d i f f e r e n c e t e c h -n i q u e . 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 w i t h c h a r a c t e r i s t i c g r i d s . U s i n g t h i s model, a computer program i s d e v e l o p e d 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 and v e l o c i t y a t any s e c t i o n o f t h e c a n a l as a s u r g e wave i s t r a v e l l i n g u p s t r e a m . S i m i l a r l y f l o w p a r a m e t e r s a t any s e c t i o n i n f r o n t o f and b e h i n d the wave r e g i o n , a r e o b t a i n e d f o r the downstream t r a v e l l i n g wave f r o m computer c a l c u l a t i o n s . 5 A l t h o u g h a s i m p l e and i d e a l i z e d c a n a l i s c o n s i d e r e d , i n t h i s study, t h e 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 d e r i v e d may be modi-f i e d f o r t h e c o m p u t a t i o n o f su r g e wave p r o p a g a t i o n i n a c o m p l i c a t e d r e a l power c a n a l . 6 CHAPTER 2 DETERMINATION OF WAVE HEIGHT AND VELOCITY FOR INITIAL SURGE WAVES 2.1 Fundamental Equations Governing the Surge-Wave Height and V e l o c i t y I f the v e l o c i t y of the water f l o w i n g i n a canal i s changed r a p i d l y , a wave i s generated w i t h a sudden change i n depth. 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 flow, due to a gate motion, that increases the water depth. The depth of f l o w i s always considered to be p o s i t i v e w i t h reference to the channel 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 are assumed to be p o s i t i v e i n the downstream d i r e c t i o n . The water between two c r o s s - s e c t i o n s , one j u s t upstream and the other j u s t downstream of the wave f r o n t , i s considered. The v e l o c i t y of the mass of water between these s e c t i o n s 1 and 2 i s decreased from V^ to V£, and the momentum i s decreased a c c o r d i n g l y . By Newton's second law of motion, the unbalanced f o r c e r e q u i r e d to change the momentum i s the product of the mass per u n i t time and the change i n v e l o c i t y . This unbalanced f o r c e i s equal to the d i f f e r e n c e between the h y d r o s t a t i c pressure f o r c e s a c t i n g on the area and A^ at s e c t i o n 2 and 1. I t f o l l o w s t h a t : J L . A . ( V - V ) . ( V. - V, ) = w.A9.y„ - W.A, . y. 2-1 ) g 1 i w L I L L 1 1 Where w = s p e c i f i c weight of water g = a c c e l e r a t i o n of g r a v i t y 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 V"w = absolute wave v e l o c i t y y = c e n t r o i d a l depth of c r o s s - s e c t i o n area. Subscripts 1 and 2 i n d i c a t e parameters at sections 1 and 2 r e s p e c t i v e l y . The equation of 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 0 - V ) = A n . (V n - V ) ( 2-2 ) 2 2 w 1 1 w By s u b s t i t u t i n g Eq. (1-2) into Eq. (1-1) and s o l v i n g f o r , V = V + c ( 2 - 3 ' ) w 1 where c = c e l e r i t y ( r e l a t i v e to flowing water), c - t l~ g' ( A 2 ^ 2 - A l ^ T T " ( 2 - 3 ) V A r ( l - A x/A 2 ) Eq. (2-3) i s the general expression f o r the absolute wave v e l o c i t y i n a power cana l . The sign i n fr o n t of the square-root term i n Eq. (2-3) depends on the d i r e c t i o n of wave propagation. A p o s i t i v e sign i s used i f the wave moves downstream, and a negative one i f the wave moves upstream. I f i s eliminated from Eqs. (2-1) and (2-2), The above equation represents the r e l a t i o n s h i p between v e l o c i t i e s and depths of flow at sections 1 and 2. From Eqs. (2-2) and (2-4) or may be determined by a process of t r i a l and er r o r , given the other three independent v a r i a b l e s . The magnitude of the wave height i s equal to (y^ - P o s i t i v e values of (y^ - y^) i n d i c a t e an increase i n depth while negative values i n d i c a t e a reduction i n depth. 8 A l t h o u g h the above e q u a t i o n s a r e d e r i v e d f o r the case shown i n F i g . 2-1, t h e y can be a p p l i e d t o p o s i t i v e o r n e g a t i v e waves t r a v -e l l i n g e i t h e r u p s t r e a m o r downstream. F o r i n i t i a l s u r g e waves, Eqs. (2-3) and (2-4) may be r e w r i t t e n a s : V = V + c wo v 0 °o ( 2-5 ) _+/ s-(. A • y - A o • yQ > A o ' < 1 - A o / A ) and A - » - A o ' A o A . y - A 0 . y 0 = _ . < v o - V ) . . . . . . ( 2 - 6 ) g • ( A - A 0 ) where s u b s c r i p t o i n d i c a t e s p a r a m e t e r s o f t h e u n d i s t u r b e d i n i t i a l f l o w . W i t h t h e s e e q u a t i o n s , t h e i n i t i a l wave p a r a m e t e r s , s u c h as c e l e r i t y , h e i g h t , e t c . , may be d e t e r m i n e d f o r any c r o s s - s e c t i o n o f t h e c a n a l . I f a wave t r a v e l s i n s t i l l w a t e r , V q i s e q u a l t o z e r o . Thus, i n t h i s c a s e , t h e a b s o l u t e wave v o l o c i t y i s i d e n t i c a l t o t h e c e l e r i t y . S o l v i n g Eqs. (2-5) and (2-6) by t r i a l and e r r o r i s l a b o r i o u s . 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 , a d i m e n s i o n l e s s f o r m o f t h e e q u a t i o n s i s g i v e n b e l o w f o r t h r e e t y p e s o f p r i s m a t i c c a n a l s . 9 2.2 I n i t i a l Surge Waves i n C a n a l s o f R e c t a n g u l a r C r o s s - S e c t i o n . F o r t h e r e c t a n g u l a r c a n a l ( F i g u r e 2-2), Eqs. (2-1) and (2-2) can be s i m p l i f i e d t o y 2 - y G = • y° • ( v 0 - v w o ) • ( v 0 - v ) • • • • ( 2 - 7 > and y0 ; ( v 0 - v w o ) = y . ( v _ V w Q ) _ ( 2 _ 8 } vrhere y i s t h e d e p t h o f f l o w E l i m i n a t i n g V from Eqs. (2-7) and (2-8) g i v e s • ( 1 ' " ' "° ^ ( 2-9 ) ° 8 < 7 - y „ > 3 D i v i d i n g Eq. (2-9) by y^ r e s u l t s i n y w y , , 2 . 2 . y V 2 V 2 ( J-+ ! )•( J _ - 1 f = — • — • - - • ( 1 " — ) ••• ( 2-10 ) y D y D s y D y 0 • v D L e t Z o , where Z q = y - y Q y o and d i s c h a r g e r a t i o , Q r = t h e n y , where Q = A-V and Q = A.V QD ' o o o y0 = 1 + )8 ( 2-11 ) and V T ( 2-12 ) 10 Using Eqs. (2-11) and (2-12), Eq. (2-10) gives the dimensionless equation 2 £ 2 - ( 1 + /3) . ( 2 +0) ' • F = 5 ( 2 - 1 3 > ° 2 ( 1 + /3 - r ) 2 where Fo = A / 7 • V '.O Eq.(2-13) shows that the i n i t i a l wave height i n canals of rectangu-l a r c r o s s - s e c t i o n i s a function of the Froude number of the i n i t i a l flow and the discharge r a t i o . For a constant discharge r a t i o , the i n i t i a l wave height i s only a function of the Froude number of the flow. The family of curves corresponding to Eq. (2-13) i s shown on log - l o g paper i n Figure 2-3. Given the Froude number of the i n i t i a l flow, and the change i n discharge a f t e r the gate opening or the gate closure, the i n i t i a l wave height can be obtained d i r e c t l y from t h i s graph. Solving Eq. (2-2) for the absolute wave v e l o c i t y , gives " y Q • v 0 - y . V v vwo y D - y or vwo y~-vn.( i -V = ° L ( 2 - 1 4 ) o y 0 - v o . ( i - J L ) S u b s t i t u t i o n of 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 - 1 5 ) 11 where ^ ' i s t h e r a t i o o f a b s o l u t e wave v e l o c i t y t o t h e i n i t i a l f l o w v e l o c i t y , \ = V /V . S u b s t i t u t i n g Eq. (2-15) i n t o Eq. wo o (2-13) and s i m p l i f y i n g g i v e s 2 <X - 1 + T ) •"( 2 X - 1 + T ) . F„ = o : 1 ( 2-16 ) ° 2 X 2 • ( 1 _ X ) 2 Eq. (2-16) g i v e s the r e l a t i o n s h i p between t h e wave v e l o c i t y V t h e Froude number F o f t h e i n i t i a l f l o w and t h e d i s c h a r g e wo' o r a t i o T . For a g i v e n v a l u e o f T , t h e i n i t i a l wave v e l o c i t y i s a f u n c t i o n o f t h e Froude number. Eq. (2-16) i s shown g r a p h i c a l l y i n F i g u r e 2-3. 2.3 I n i t i a l Surge Waves i n C a n a l s o f T r i a n g u l a r C r o s s - S e c t i o n . F o r a t r i a n g u l a r c a n a l ( F i g u r e 2-4), Eq. (2-6) may be s i m p l i f i e d t o : 2 2 — ( 3 3 yQ • y • ' v 7 3 " y ° } = ^77^77"' v o • ( 1 - ~ ) < 2"17 > o ' o I n a t r i a n g u l a r c a n a l • V Q/A T V Q " Q o / A o ( 1 + 0 ) 2 By u s i n g t h e r e l a t i o n above and Eq. (2-11), Eq. (2-17) can be r e w r i t t e n a s : — r<i+/5>3-il .H O.J_LI£2!_ . f i - _ L _ - l or 2 1 T 3 2 ( 1 +/9 ) 2- [ ( 1 +^ ) 2 - l ] • [( 1 +p ) " l l F = = ~ L =L ... ( 9-18 3 [ < l + / 3 ) 2 - T ^ ( 2 1 S 12 I f Eq. (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 , and a s i m i l a r s u b s t i t u t i o n o f t h e d i m e n s i o n l e s s p a r a m e t e r s , T and /3- i s u s e d , t h e n V W O ( 1 + / 3 > 2 . T - T ^ 2 - 1 ( 1 + £ ) V 2 o ( 1 + / 3 ) - l o r T - 1 ( 1 + /S )2 - l ( 2 - 1 9 ) Eq. (2-18) shows t h a t t h e i n i t i a l wave h e i g h t i n a s y m m e t r i c a l 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 f l o w and t h e d i s c h a r g e r a t i o . Eq. (2-19) shows t h a t t h e a b s o l u t e wave v e l o c i t y i s a f u n c t i o n o f the wave h e i g h t . I n o t h e r words, t h e r a t i o o f wave v e l o c i t y t o t h a t o f t h e i n i t i a l f l o w i s a l s o a f u n c t i o n o f t h e Froude number and t h e d i s c h a r g e r a t i o . The g r a p h i c a l f o r m o f Eqs. (2-18) and (2-19) i s shown on F i g u r e 2-5. 2.4 I n i t i a l Surge Waves i n C a n a l s o f T r a p e z o i d a l C r o s s - S e c t i o n . I n a t r a p e z o i d a l c a n a l ( F i g u r e 2 -6) , t h e c r o s s - s e c t i o n a l p a r a m e t e r s 3 b, A , y, a t s e c t i o n s u p s t r e a m and downstream o f t h e wave f r o n t a r e b x = b c + 2 m . y D A o = ?o ' ( b o + m ' y o > v = y G 3 b Q + 2 m . y P  y o 6 bn + m • y. and b„ = b + 2 m . y I o A = y . ( b Q + m . y ) y 3 b n + 2 m . y y = _ __ . 13 6 b + m . y o 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 Eq. (2-4) g i v e s 2 2 T~ "( 3 b 0 + 2 m • y ) - ^  ' ( 3 b o + 2 m. yo) Vp-yp-y - ( b o + m - y > - ( b o • H n - y 0 ) - ( 1 : v / v ° ) 2 g. ^ ( b Q + m.y ).y - ( b Q + m.y 0) . y 0 J D i v i d i n g by m-y 3 r e s u l t s i n — •( Z_ ) 2:cJ^o + _ 2 _ y _ ) . J_ . ( _3bo + 2 ) b y D m - y 0 y 0 & m.y0 2 b b v V o V Q y ( -_o_ + 1 ) . ( _ £ _ + J L ).( 1 - ) = . m«y„ m.y y^ V„ * f o i o 2 . . ( 2-20 m , y „ y m.y^ -'O ^O Jo J o L e t bp ( 2-21 ) k - — m.y o t h e n V T • ( k + 1 ) V 0 ( k + l + 0 ) . ( i + / 3 ) ( 2-22 ) S u b s t i t u t i n g Eqs. (2-21) and (2-22) i n t o Eq. (2-20) g i v e s 1 ( 1 + / 3 ) 2 J 3 k+2 ( l+]3 )] - _L C 3 k + 2 ) £ ( k + l + / 3 ) . ( l + £ ) - T . ( k + 1 ) J 2 j \ k + l + / 3 ) . ( l + / 3 ) - ( k + l ) J . ( k + l + / 3 ) . ( l + / 3 ) 2 ( K + 1 ) = F„ . F 2 { (1+/5)2 • [ 3 k+2 (1+/3)] - (3k+2) | . | (k+l+/3) . (1+/3) - (k+1)] . (1+/S). (k+l+/3) 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 , k, w h i c h i s a f u n c t i o n o f t h e b o t t o m w i d t h , s i d e s l o p e o f the c r o s s - s e c t i o n and t h e d e p t h o f t h e i n i t i a l f l o w , a p p e ars t o g e t h e r w i t h d i s c h a r g e r a t i o and Fr 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 wave h e i g h t . P h y s i c a l l y , (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 t h a t 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 and s i d e s l o p e . F o r a c o n s t a n t v a l u e o f k, £ i s i n d e p e n d e n t o f t h e i n d i v i d u a l v a l u e o f b o t t o m w i d t h , s i d e s l o p e o r t h e d e p t h o f i n i t i a l f l o w . I f t h e same r e l a t i o n i n Eqs. (2-11) ,and (2-22) a r e u s e d , t h e n from t h e c o n t i n u i t y e q u a t i o n , i t f o l l o w s t h a t (l+/S).(k+l+/3) r . ( k + l ) wo = ( k + l ) (k+l+fi ).(!+#) O . (k-(k+1) - 1 V Q q+/3) (k+l+ff) _ or x = - j l - r ) . ( k + l ) ( 2 _ 2 4 ) (l+/3).(k+l+/3) - ( k + l ) I t may be n o t e d t h a t t h e shape 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 . Eqs. (2-23) and (2-24) a r e shown g r a p h i c a l l y on F i g u r e s 2-7 (a) - ( j ) . I t s h o u l d be n o t e d t h a t t h e r e c t a n g u l a r and the t r i a n g u l a r c a n a l s a r e t h e p a r t i c u l a r o r l i m i t i n g c a s e s o f the t r a p e z o i d a l c a n a l . A t r a p e z o i d a l c a n a l , when i t s s i d e s l o p e s become v e r t i c a l Ck = OC ), i s a r e c t a n g u l a r c a n a l . When k = oC , Eq. (2-23) i s i d e n t i c a l t o Eq. (2-13), and Eq. (2-24) t o Eq. (2-15). When t h e bottom w i d t h o f a t r a p e z o i d a l c a n a l i s r e d u c e d t o z e r o (k = 0), i t becomes a t r i a n g u l a r c a n a l . F o r k = 0, Eq. (2-23) i s i d e n t i c a l t o Eq. (2-18), and Eq. (2-24) t o Eq. (2-19). T h e r e f o r e , t h e r e c t a n g u l a r and the t r i a n g u l a r c r o s s - s e c t i o n s a r e t h e two l i m i t s o f the t r a p e -z o i d a l c r o s s - s e c t i o n . T h i s can a l s o be seen f r o m F i g u r e 2-7 (a) -Thus E q s . (2-23) and (2-24) a r e a l s o v a l i d f o r b o t h 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 . CHAPTER 3 NUMERICAL CALCULATIONS FOR SURGE-WAVE PROPAGATION G e n e r a l When an i n i t i a l s u r g e wave i s g e n e r a t e d a t t h e u p s t r e a m o r down-s t r e a m end o f a c h a n n e l , i t t r a v e l s i m m e d i a t e l y away f r o m t h e s o u r c e o f g e n e r a t i o n w i t h i t s v e l o c i t y o f p r o p a g a t i o n . T h i s v e l o c i t y o f p r o p a g a t i o n i s u s u a l l y c o n s i d e r a b l y i n e x c e s s o f t h e mean s t e a d y - f l o w v e l o c i t y . A p o s i t i v e wave c a u s e d by a r a p i d change o f d i s c h a r g e has a p r o f i l e w i t h a s t e e p f r o n t s i m i l a r t o a moving h y d r a u l i c jump. I f t h e i n i t i a l p r o f i l e o f t h e n e g a t i v e s u r g e wave formed w i t h a s t e e p f r o n t , i t w i l l soon f l a t t e n o u t as t h e s u r g e wave moves a l o n g t h e c h a n n e l . A r i g o r o u s s o l u t i o n f o r t h e c a l c u l a t i o n o f t h e p r o p a g a t i o n o f s u r g e waves does n o t e x i s t . A r a p i d a p p r o a c h by t h e a i d o f Com-p u t e r i s i n t r o d u c e d i n t h i s c h a p t e r . I n t h i s a p p r o a c h , t h e s u r g e wave i s d e t e r m i n e d 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 ( s e e C h a p t e r 2), and t h e f l o w v e l o c i t i e s and d e p t h s a t t h e u p s t r e a m and t h e downstream o f wave f r o n t b o u n d a r i e s a r e . c a l -c u l a t e d u s i n g t h e u n s t e a d y - f l o w e q u a t i o n s . These u n s t e a d y - f l o w e q u a t i o n s 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 by w h i c h t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s a r e t r a n s f o r m e d i n t o p a r t i c u l a r t o t a l d i f f e r e n t i a l e q u a t i o n s . The n u m e r i c a l s o l u t i o n i s o b t a i n e d by a f i r s t o r d e r f i n i t e - d i f f e r e n c e t e c h n i q u e . 17 3.2 B a s i c A s s u m p t i o n s The a p p l i c a t i o n o f t h e method o f c h a r a c t e r i s t i c s t o u n s t e a d y -f l o w i n an open c h a n n e l i s b a s e d on t h e f o l l o w i n g s i m p l i f y i n g a s s u m p t i o n s . ( 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 s e diment movement a r e c o n s i d e r e d . (b) L a t e r a l f l o w i s n e g l e c t e d , i . e . , one d i m e n s i o n a l f l o w i s c o n s i d e r e d . ( c ) F r i c t i o n due t o t h e m o t i o n o v e r a r o u g h c h a n n e l bed obeys Manning's f o r m u l a . (d) C h a n n e l s l o p e i s s m a l l enough so t h a t cos 6 — 1 , (e) V e r t i c a l component o f t h e a c c e l e r a t i o n has a n e g l i g i b l e e f f e c t on t h e p r e s s u r e . ( f ) The p r e s s u r e d i s t r i b u t i o n a l o n g a v e r t i c a l l i n e i s h y d r o s t a t i c . 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 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 o f (3) c h a r a c t e r i s t i c s . The e x p r e s s i o n s d e v e l o p e d by V. L. S t r e e t e r a r e u s e d i n t h i s s t u d y . A c c o r d i n g t o S t r e e t e r , two p a i r s o f 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 , w r i t t e n f o r a 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 i n an open c h a n n e l , a r e ( 3-2 ) ( 3-1 ) c dV f g . B . dy d t " A/ A 18 ar* + - w T i r • g - s o = 0 . . . . " . ' ( 3-3 ) dx / g-A~ ( 3_A \ where w = the s p e c i f i c w e i g h t o f w a t e r , A = t h e c r o s s - s e c t i o n a l a r e a , B = t h e s u r f a c e w i d t h o f c r o s s - s e c t i o n , R = the h y d r a u l i c r a d i u s , S = s l o p e o f b o t t o m T 0 = t h e u n i t s h e a r f o r c e on t h e bottom and s i d e s o f a c h a n n e l , g-A = the c e l e r i t y o f an i n f i n i t e s i m a l wave. The f i r s t group o f e q u a t i o n s a r e c a l l e d p o s i t i v e c h a r a c t e r i s t i c s o r C e q u a t i o n s , w h i l e t h e second ones a r e c a l l e d n e g a t i v e c h a r -a c t e r i s t i c s o r C e q u a t i o n s . The f i r s t e q u a t i o n o f each group i s v a l i d o n l y when th e second e q u a t i o n o f t h e group i s s a t i s f i e d . 3.4 S o l u t i o n o f 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 f o r a G r a d u a l l y V a r i e d  Unsteady F l o w The e q u a t i o n s can be r e p r e s e n t e d g r a p h i c a l l y on an x - t p l a n e , as shown i n F i g u r e 3-1. The p o i n t P r e p r e s e n t s t h e p o s i t i o n o f ^ t h e c a n a l s e c t i o n under c o n s i d e r a t i o n a t t i m e t and the p o i n t s P R and S r e p r e s e n t , r e s p e c t i v e l y , t h e p o s i t i o n o f c e r t a i n u p s t r e a m and downstream s e c t i o n s a t t i m e t and t . The v e l o c i t y o f wave R .s p r o p a g a t i o n can be r e p r e s e n t e d by t h e s l o p e o f t h e l i n e s c o n s t r u c t e d on t h e x - t p l a n e u s i n g Eqs. (3-2) and (3-4). P o i n t 1 9 R r e p r e s e n t s t h e p o s i t i o n o f t h e u p s t r e a m s e c t i o n f r o m w h i c h an i n f i n i t e s i m a l wave, once d e v e l o p e d , w i l l a r r i v e a t s e c t i o n P a f t e r t h e t i m e i n t e r v a l A t = t - t,.. p R S i m i l a r l y , p o i n t S r e p r e s e n t s t h e p o s i t i o n o f t h e downstream s e c t i o n f r o m w h i c h a wave once d e v e l o p e d w i l l a r r i v e a t s e c t i o n P a f t e r A t = t - t . V a l u e s o f V and y a t t h e P S 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 s o l v i n g Eqs. ( 3 - 1 ) and ( 3 - 3 ) s i m u l t a n e o u s l y . I n o r d e r t o a c c o m p l i s h t h i s , t h e f i r s t o r d e r f i n i t e d i f f e r e n c e t e c h n i q u e i s u s e d . 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 a r e t r a n s f o r m e d as f o l l o w s : . . . j V p " VR + G R ' ( y p " y R ) + GNR • ( t p - t ) = 0 . . . ( 3 - 5 ) C 1 - x R = ( V R + RG ) • ( t p - t R ) .;. ( 3 - 6 ) ( V p - V g - GS . ( y p - y g ) + GNS . ( t p - t g ) = 0 . . . ( 3 - 7 ) . ° (. Xp - x s - ( Vg - SG ) • ( t p - t s ) " .;. ( 3 - 8 ) v h e r e t h e s u b s c r i p t s i n d i c a t e t h e v a r i a b l e s a t t h e c o r r e s p o n d i n g p o i n t s , and GR = / £ RG = A R , and GNR = g . ( S R - S Q ) GNS = g . ( S g - S Q ) S and S a r e t h e f r i c t i o n a l s l o p e s a t p o i n t s R and S r e s p e c t i v e l y . R S A c c o r d i n g t o t h e Manning f o r m u l a , t h e f r i c t i o n a l s l o p e c a n be expressed as s =. .. "2-v- I vj 2.21 R 4 / 3 where n i s the c o e f f i c i e n t of roughness on the canal bottom and sides (or Manning's n), and V i s the mean v e l o c i t y of flow. V.|v| i n d i c a t e s that the f r i c t i o n a l r e s i s t a n c e i s always i n the opposite d i r e c t i o n of the flow. Knowing the v a r i a b l e s , x, t, V and y, at points R and S , the four unknown v a r i a b l e s x, t, V and y, at point P, can be found using equations, Eqs. (3-5) through (3-8). A g r i d of c h a r a c t e r i s t i c s i s established to f a c i l i t a t e a computer s o l u t i o n . A p a r t i c u l a r section of the canal may be a r b i t r a r i l y chosen. For s i m p l i c i t y , the canal of length L i s divided into a number of equal lengths. The procedure for c a l c u l a t i o n follows. 3.4.1. Preliminary computation Compute the i n i t i a l v e l o c i t y and depth of the flow by using Manning' s formula, y = * «486 0/3 1/2 and n. • • «• t> store the known values of x, t, V, and y at points P (m, 1), m = 1, 3, 5, etc., (figure 3-2). 3.4.2 Computation for the flow in the channel Based on the values of x, t, V and y at p o i n t R, the c h a r a c t e r i s t i c l i n e RP for a short distance can be l a i d 21 o u t a t a s l o p e o f 1/(V +RG) from Eq. (3-6). S i m i l a r l y , b a s e d on t h e v a l u e s o f x, t , V and y a t p o i n t S, t h e c h a r a c t e r i s t i c l i n e SP can be drawn t o r e p r e s e n t Eq. (3-8). These two c h a r a c t e r i s t i c l i n e s from R and S i n t e r s e c t a t p o i n t P. The v a l u e s o f x, t , V and y a t P a r e t h e n o b t a i n e d by s o l v i n g Eqs. (3-5) t h r o u g h (3-8) s i m u l t a n e o u s l y , i . e . , _ X S - X B + t R . ( V R + RG) - t S . ( V S - S G ) c • — — _ _ — _ — . — _ _ — . . . . . V J - y j V R + RG - Vg + SG x p = x R + ( V R.+ RG ) • ( t p - t R ) ( 3 -10) V -V +GR•y +GS•y -GNR.(t - t )+GNS.(t - t ) Y ="" R s ^ _ R _ _ J L _ s i ...( 3-n ) P GR + GS and V p = V R - G R - ( y p - y R ) - G N R . < t p - t R ) ( 3-12 ) 3.4.3. C o m p u t a t i o n f o r t h e f l o w a t b o u n d a r i e s A t e i t h e r end o f t h e c h a n n e l , o n l y one group o f t h e c h a r a c t e r -i s t i c e q u a t i o n s i s a v a i l a b l e . F o r an u p s t r e a m boundary ( F i g u r e 3-3-a), Eqs. (3-7) and (3-8) h o l d , and f o r a downstream boundary ( F i g u r e 3-3-b), Eqs. (3-5) and (3-6) a r e v a l i d . T h e r e f o r e two a u x i l i a r y e q u a t i o n s d e r i v e d f r o m t h e g i v e n boundary c o n d i t i o n s a t e i t h e r boundary a r e needed so t h a t f o u r unknowns can be s o l v e d . Some examples a r e s o l v e d and i l l u s t r a t e d a s f o l l o w s : 22 Example 3-1. A r e c t a n g u l a r c h a n n e l 1000 f t . l o n g and 12 f t . w i d e c a r r i e s 720 c f s w a t e r a t normal d e p t h . S = 0.001 and o Manning's n = 0.014. A t t h e u p s t r e a m end, t h e f l o w i s g i v e n by Q = 720 + 180. s i n (0.03 t ) and a t t h e downstream end t h e d e p t h i s m a i n t a i n e d c o n s t a n t a t y = y . C a l c u l a t e t h e f l o w c o n d i t i o n s i n t h e c h a n n e l . S o l u t i o n : The computer program i n IBM 7044 computer system f o r t h i s p r o b l e m i s g i v e n i n A p p e n d i x A ( l ) . A p o r t i o n o f t h e s o l u t i o n i s shown i n F i g u r e 3-4. Example 3-2. A r e c t a n g u l a r c h a n n e l 20 f t . w i d e and 10,000 f t . l o n g i s d i s c h a r g i n g under s t e a d y - u n i f o r m f l o w c o n d i t i o n a t y = 6.0 f t . Channel s l o p e S i s 0.0016. A t t i m e t = 0, o o t h e f l o w i s . i n c r e a s e d a t the u p s t r e a m end l i n e a r l y u n t i l i t has d o u b l e d i n 20 m i n u t e s . The f l o w i s t h e n d e c r e a s e d l i n e a r l y u n t i l i t i s o n e - h a l f t h e o r i g i n a l f l o w i n 10 a d d i t i o n a l m i n u t e s . F o r Manning's n = 0.0185, f i n d t h e v e l o c i t y and d e p t h i n t h e c h a n n e l f o r t h e f i r s t 40 m i n . o f u n s t e a d y f l o w . The g a g e - h e i g h t - d i s c h a r g e c u r v e a t t h e 3/2 downstream end i s Q = 132 (y - 2.32) S o l u t i o n : The computer program f o r t h i s p r o b l e m i s l i s t e d i n A p p e n d i x A(2), and a p o r t i o n o f t h e s o l u t i o n i s g i v e n i n F i g . 3-5. The s o l u t i o n s o f examples 1 and 2 have been checked w i t h t h e Example 15.5 and Example 15.6 i n r e f e r e n c e (3). A good agreement was o b t a i n e d . 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  C a n a l s . A c o n t i n u o u s u n s t e a d y f l o w i n an open c h a n n e l c a n be s o l v e d by t h e method o f c h a r a c t e r i s t i c s o n l y by making t h e a s s u m p t i o n t h a t t h e v e r t i c a l a c c e l e r a t i o n o f f l o w i s s m a l l and t h e r e f o r e c a n be neg-l e c t e d . F o r s u r g e p r o b l e m s , t h e s t e e p s t e p i n t h e wave r e g i o n c o n s t i t u t e s a d i s c o n t i n u i t y , and t h e v e r t i c a l a c c e l e r a t i o n o f t h e f l o w c a n n o t be n e g l e c t e d . A r i g o r o u s t h e o r y f o r t h e c o m p u t a t i o n o f s u r g e waves has n o t come t o t h e a u t h o r ' s k n o w l e d g e . However, a s u r g e wave r e s e m b l e s a moving h y d r a u l i c jump i n an open c h a n n e l . I n t h e r e g i o n s u p s t r e a m o r downstream o f t h e wave f r o n t , t h e f l o w i s s t e a d y , 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 c a n be s o l v e d by t h e method o f c h a r a c t e r i s t i c s i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n s . The wave f r o n t r e g i o n i t s e l f r e p r e s e n t s r a p i d l y v a r i e d u n s t e a d y f l o w and i t s s o l u t i o n must be b a s e d on 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, i , e . , E q . ( 2 - 1 ) and ( 2 - 2 ) . The c a l c u l a t i o n p r o c e d -u r e s f o r a s u r g e wave c a u s e d by a l o a d r e j e c t i o n a t t h e downstream end f r o m an i n i t i a l s t e a d y f l o w f o l l o w s . 3.5,1 I n i t i a l f l o w c o n d i t i o n The o r i g i n o f a b s c i s s a x on t h e x - t p l a n e i s c h o s e n a t t h e u p s t r e a m end ( F i g u r e 3-6). Assume t h a t t h e i n i t i a l f l o w c o n d i t i o n s a r e known t h r o u g h o u t t h e c a n a l a t t = 0. A c a n a l i s d i v i d e d i n t o r e a c h e s o f e q u a l l e n g t h s , and t h e known v a l u e s o f x, t , V and y a t p o i n t s P ( l , l ) , P ( 3 , l ) , ... a r e s t o r e d as x ( l , l ) , x ( 3 , l ) , x(5.1), 2 4 t ( l , l ) , t ( 3 , l ) , t ( 5 , l ) , ... , V ( l , l ) , V ( 3 , l ) , V ( 5 , l ) , ... y ( l , l ) , y ( 3 , l ) , y ( 5 , l ) , ... . The v a l u e s o f x, t , V and y a t p o i n t s P ( 2 , 2 ) , P ( 4 , 2 ) , .... a r e o b t a i n e d from t h e p o i n t s P ( l , l ) , P ( 3 , l ) , .... e t c . , by the method o f c h a r a c t e r i s t i c s . F o r example, t h e v a l u e s o f x, t , V and y a t p o i n t P ( 6 , 2 ) a r e o b t a i n e d from 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 w h i c h p a s s t h r o u g h P ( 5 , l ) and P ( 7 , l ) . 3 . 5 . 2 I n i t i a l p o s i t i v e surge wave a t downstream end o f c a n a l The e q u a t i o n s o f c o n t i n u i t y and momentum, t o g e t h e r w i t h 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 u s e d t o d e t e r m i n e t h e i n i t i a l wave h e i g h t and wave v e l o c i t y . x = L ( F i g u r e 3 - 6 ) , and t i s o b t a i n e d by t h e manner A A m e n t i o n e d i n s e c t i o n 3 . 4 . 3 . Thus, t h e v a r i a b l e s x, t , V and y a t p o i n t A a r e d e t e r m i n e d . 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 ) , P ( 3 , 3 ) , a r e d e t e r m i n e d 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 A t o B ( F i g u r e 3 - 6 ) ( a ) D e t e r m i n e t h e p o i n t B: Assume t h a t t h e sur g e wave t r a v e l s from A t o B w i t h t h e v e l o c i t y e q u a l t o i t s i n i t i a l v e l o c i t y a t A as t h e f i r s t a p p r o x i m a t i o n . t„ and x D a r e t h e n o b t a i n e d by s o l v i n g e q u a t i o n s B B X B - X E = ( V E + ( / ~ ^ ) ' ( t B " C E > O " 13) E XB - X A ' » < - v • < v y ( 3 - i 4 ) where t h e s u b s c r i p t s A, B, E i n d i c a t e t he v a l u e s o f V, A, 25 B, x and 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) D e t e r m i n e p o i n t C 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, d e t e r m i n e th e w a t e r v e l o c i t y and d e p t h b e h i n d t h e wave f r o n t a t B. For t h e f i r s t a p p r o x i m a t i o n t h e s e a r e n o t d i f f e r e n t f rom ones f o u n d a t A. t , V and y a r e o b t a i n e d from Eqs. (3-5) and (3-6) and boundary c o n d i t i o n s i n w h i c h x = L. ( c ) D e t e r m i n e p o i n t D F i r s t , t h e v e l o c i t y and d e p t h a t p o i n t D a r e e s t i m a t e d , t h e n th e v a r i a b l e s a t B can be o b t a i n e d from t h o s e a t D by 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 an i n f i n i t e s i m a l wave t r a v e l l i n g a l o n g t h e c h a r a c t e r i s t i c l i n e DB and r e a c h i n g p o i n t B a t t h e same t i m e as a su r g e wave t r a v e l l i n g f r om A t o B a l o n g AB. i s o b t a i n e d f o r t h e 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 a t B a r e d e t e r m i n e d i n above 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 i n t e r p o l a t i o n from t h o s e a t A and C, ass u m i n g t h a t t h e r a t e o f change o f w a t e r s u r f a c e a t t h e downstream end d u r i n g t . to., t ^ i s l i n e a r w i t h A D r e s p e c t t o t i m e . 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 th 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 l i n e BD. C o n t i n u e t h i s i t e r a t i o n u n t i l t h e two s u c c e s s i v e r e l a t i v e 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 l e s s 26 than 0.001. (d) Determine and by using c h a r a c t e r i s t i c equation, Eq. (3-7), along DB and the equations of momentum and continuity at B. (e) Compute the new value of wave v e l o c i t y at B (Eq. (2-3) ), and average the wave v e l o c i t i e s at A and B. This i s the new wave v e l o c i t y with which the wave t r a v e l s from A to B. (f ) With the new wave v e l o c i t y obtained from (e) repeat the procedures of c a l c u l a t i o n from (a) to (e) u n t i l the dif f e r e n c e s of two successive calculated r e l a t i v e values of V and y at B, C, and D are l e s s than 0.001. (g) Determine the v a r i a b l e s at a l l necessary points upstream of the wave front (at point B) by the method of c h a r a c t e r i s t i c s . 3.5.3.2. Wave t r a v e l s from B to G, L e t c . Using the c a l c u l a t i o n process s i m i l a r to that i n section 3.5.3.1. the va r i a b l e s at G and F are obtained. The values of V and y at C are given from the previous set of computation. The new variables at H are obtained from those at G and C by equations of c h a r a c t e r i s t i c s . I t i s necessary i n t h i s step to carry on the c a l c u l a t i o n for the other points at the downstream end, such as I (Figure 3-6) i n th i s step, 27 a l o n g t h e 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 f o r wave p r o p a g a t i o n . By t h i s 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 the s u r g e wave a l o n g t h e p r i s m a t i c c a n a l i s o b t a i n e d and t e r m i n a t e s when t h e wave f r o n t r e a c h e s t h e u p s t r e a m r e s e r v o i r . I t s h o u l d be n o t e d t h a t , f o r a p o s i t i v e s u r g e wave, the v e l o c i t y o f wave p r o p a g a t i o n i s alw a y s g r e a t e r t h a n 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 t h a n 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 h i n d t h e f r o n t o f t h e s u r g e wave, i . e . , t h e i n v e r s e s l o p e o f the l i n e KE i s l e s s t h a n t h e i n -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 i s a g a i n l e s s t h a n t h e i n v e r s e s l o p e o f l i n e CH. E v e n t u a l l y , as 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 a l o n g t h e c a n a l , t h e s e l i n e s would c o n v e r g e and i n t e r s e c t each o t h e r . Because p o s i t i v e s u r g e waves have an a b r u p t f r o n t , t h e c h a r a c t e r i s t i c l i n e s 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 th e o t h e r . When t h e c h a r a c t e r i s t i c l i n e s b e f o r e and b e h i n d the wave r e g i o n and the t r a c e o f wave p r o p a g a t i o n meet, (see f i g u r e 3 - 6 ) , f l o w p a r a m e t e r s a t M a r e d e t e r m i n e d f r o m P, and 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 and 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 c o m p u t a t i o n i s shown on F i g u r e 3-8. F o r a p o s i t i v e wave o c c u r r i n g a t t h e u p s t r e a m end and t r a v e l l i n g t o t h e downstream end, t h e c h a r a c t e r i s t i c l i n e s w i l l be s i m i l a r t o t h e s e i n F i g u r e 3 - 7 . The p r o c e s s o f c o m p u t a t i o n f o r a p o s i t i v e s u r g e wave s t a r t i n g at the downstream end.and propagating upstream i n the canal has been programmed in FORTRAN IV f o r the IBM 7044 computer. Examples have been selected to examine the computer program. P a r a l l e l hand ca l c u l a t i o n s . b y the Favre method are shown f o r comparison. Examp1e 3.3. A rectangular channel 38,800 f t . long, 45 f t . wide and 41.175 f t . deep c a r r i e s a flow with a v e l o c i t y at 7.101 f t / s e c . S q = 0.0002376. Suddenly, the flow i s complete; stopped at the downstream end by c l o s i n g a gate. Compute the i n i t i a l wave height, and wave height when i t reaches the upstream end. So l u t i o n : The r e s u l t s of computer computation using method of c h a r a c t e r i s t i c s give Z^ = 8.39 f t . and Z = 4.23 f t . , where Z and Z are the downstream and upstream wave heights o ...respectively, ( Figures 3-9 and 3-10.). D e t a i l s of the computations are shown g r a p h i c a l l y on Figure 3-11. _ The a l t e r n a t i v e c a l c u l a t i o n f o r t h i s example c a r r i e d out by the Favre method gave Z = 4.15 f t . f o r the wave reaching the upstream end of canal. D e t a i l s are 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, a f a i r l y good agreement i s observed. It should be noted that, i n the Favre method, the water surface i n the downstream area i s assumed s t r a i g h t l i n e when the wave front reaches 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 , t he a c t u a l w a t e r s u r f a c e f o r t he c a s e o f c o m p l e t e c l o s u r e i s f a i r l y c l o s e t o a h o r i z o n -t a l l i n e i n t h e l o w e r end o f c a n a l , w h i l e i t i s somewhat s l o p e d i n t h e upper end. From F i g u r e 3-11, i t can be seen 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 c h a n n e l i s n e a r l y l i n e a r and t h e r e f o r e F a v r e ' s a s s u m p t i o n i s j u s t i f i e d . Example 3.4 A r e c t a n g u l a r c h a n n e l 1000 f t . l o n g and 12 f t . w ide 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 and o Manning's n = 0.014. The f l o w a t t h e downstream end i s sudden-l y s h u t o f f . C a r r y 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 p r o p a g a t i o n o f s u r g e wave a l o n g t h e c h a n n e l . S o l u t i o n : The r e s u l t s o f c a l c u l a t i o n a r e shown on F i g u r e 3-13, and have been compared w i t h t h e example on page 258, r e f e r e n c e (3). 3.6 N u m e r i c a l C a l c u l a t i o n s f o r N e g a 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 When a p o s i t i v e wave r e a c h e s t h e upper end 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 wave and p r o c e e d s towards the downstream end. I f t h e wave h e i g h t i s s m a l l o r moderate compared t o t h e d e p t h , i t can be assumed 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 d u r i n g t h e t r a v e l and t h e e q u a t i o n s d e r i v e d f o r a p o s i t i v e wave can be a p p l i e d w i t h o u t i n t r o d u c i n g s i g n i f i c a n t e r r o r s . The h e i g h t o f 30 the negative wave r e f l e c t e d from the r e s e r v o i r i s us u a l l y r e l a t i v e l y small. On the assumption that the p r o f i l e of a negative wave w i l l not change s i g n i f i c a n t l y , , a scheme for computer computa-tio n s i s used as follows. • 3.6.1 Determination of the i n i t i a l r e f l e c t e d negative wave Assuming that there i s no entrance l o s s and no v e l o c i t y head recovery (see section 4-7), the water l e v e l immediately behind the wave front of the r e f l e c t e d negative wave a t the extreme upper end of channel i s the same as that i n the r e s e r v o i r . It follows that » " y . * < 3-16 ) where y and V are the water depth and v e l o c i t y of the o o i n i t i a l steady flow ( F i g . 3-14 (a) ). At the r e s e r v o i r , which i s the upstream boundary of the canal, XD = 0 and y = H. The v e l o c i t y of the negative wave i s - K K according to Eq. (2-3). g.(A 2.y 2 - A 1 > y i ) V = _V, . + ... / ' L , X 1 • . . • ( 3-17 ) ^1 / A2 w 1 A,.(1 - A,/A- ) where and (Figure 3-15) are given i n the l a s t step of the c a l c u l a t i o n i n the propagation of the p o s i t i v e wave. By using the equation of c o n t i n u i t y , i s obtained, i . e . : V < V 2 - Vw > + v . . . . ( 3-18 ) 1 Ai  w and the height of the negative wave i s ' where = H Here Z has n e g a t i v e v a l u e w h i c h i n d i c a t e s t h e n e g a t i v e wave. 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 t o P ( F i g u r e 3-16.) (a) Assume f o r t h e f i r s t a p p r o x i m a t i o n , t h a t the wave t r a v e l s from 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 v e l o c i t y e q u a l t o i t s i n i t i a l v e l o c i t y a t R. Det e r m i n e t and x by s o l v i n g e q u a t i o n s Then, c a l c u l a t e t h e v a r i a b l e s downstream o f the wave f r o n t , y^ and , by l i n e a r e x t r a p o l a t i o n from t h o s e a t E and S. (b) D e t e r m i n e t h e w a t e r d e p t h and v e l o c i t y b e h i n d t h e wave f r o n t , y^ and 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 and v a l u e s e x t r a p o l a t e d above. ( 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 a t P by Eq. ( 3 - 1 7 ) . F u r t h e r 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 p r o p a g a t i n g from R t o P by a v e r a g i n g t h e v e l o c i t i e s a t R and P. (d) Repeat t h e p r o c e d u r e s ( a ) t h r o u g h ( c ) u n t i l t h e d e s i r e d a c c u r a c y i s r e a c h e d . 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 A l o n g t h e Channel P P ) • ( t p - t E ) / ... ( 3 - 2 1 ) (3-20) 32 Ce) D e t e r m i n e t h e v a r i a b l e s a t D by 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 f r o m p o i n t s R and P. ( f ) D e t e r m i n e t h e v a r i a b l e s a t Q by 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 t h r o u g h D Q and the boundary c o n d i t i o n s . 3.6.2.2. 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 f r o m R t o P, e t c . ( F i g u r e 3-17) (a) D e t e r m i n e t h e v a r i a b l e s y„ and V , a t E, by e q u a t i o n s o f b E c h a r a c t e r i s t i c s . ( b) D e t e r m i n e the wave h e i g h t and v e l o c i t y a t P by u s i n g t h e p r o c e d u r e s m e n t i o n e d above i n 3.6.2.1.a t h r o u g h 3.6.2.1.f. ( c ) D e t e r m i n e t h e v a r i a b l e s a t p o i n t s F, Q, K by 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 . ( d) C o n t i n u e t h e s e computing p r o c e s s e s u n t i l t h e n e g a t i v e wave r e a c h e s t h e downstream end. I t s h o u l d be n o t e d 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 o f wave p r o p a g a t i o n i s alw a y s l e s s t h a n t h a t f o r 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 g r e a t e r t h a n t h a t f o r an i n f i n i t e s i m a l wave i n the r e g i o n b e h i n d the s u r g e wave, 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 3-17) i s g r e a t e r t h a n RP w h i c h i s g r e a t e r t h a n t h a t o f l i n e RF. T h e r e -f o r e , as a n e g a t i v e s u r g e t r a v e l s downstream c o n t i n u o u s l y a l o n g t h e c a n a l , t h e s e l i n e s w o u l d d i v e r g e . The i n v e s t i g a t i o n shows t h a t t h e u n s t e a d y r i s e b e h i n d t h e p o s i t i v e s u r g e o r i n t h e f r o n t o f a n e g a t i v e s u r g e v a r i e s n e a r l y l i n e a r l y . T h e r e f o r e , i t i s l o g i c a l 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 and d e p t h 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 l i n e a r w i t h r e s p e c t t o t i m e , i n a s h o r t t i m e p e r i o d f r o m t„ t o E t p , w i t h o u t i n t r o d u c i n g a p p r e c i a b l e e r r o r s . 34 CHAPTER .4 RESULTS OF ANALYSIS 4.1 G e n e r a l In p r e v i o u s c h a p t e r s e q u a t i o n s and t e c h n i q u e s have been d e v e l -oped t o c a l c u l a t e t h e magnitude o f t h e i n i t i a l wave h e i g h t and t h e wave h e i g h t a t any p o i n t as t h e s u r g e wave p r o p a g a t e s a l o n g t h e c a n a l . The p u r p o s e o f t h i s c h a p t e r i s t o f i n d ap-p r o p r i a t e d i m e n s i o n l e s s r e l a t i o n s h i p s between the v a r i a b l e s g o v e r n i n g t h e v a r i a t i o n o f wave h e i g h t , t h e d i s t a n c e o f p r o p a -g a t i o n a n d " o t h e r f l o w p a r a m e t e r s . 4.2.. D i m e n s i o n l e s s R a t i o s When s o l v i n g h y d r a u l i c t r a n s i e n t p r o b l e m s , i t i s c o n v e n i e n t t o make u s e o f d i m e n s i o n l e s s r a t i o s . The i n t r o d u c t i o n o f s u c h r a t i o s u s u a l l y r e d u c e s 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 p r o b l e m and s i m p l i f i e s t h e s o l u t i o n . I n t h e p r o b l e m o f t h e v a r i a t i o n o f wave h e i g h t , Z , o f a s u r g e wave p r o p a g a t i n g a l o n g 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 t h e d e p t h y , v e l o c i t y V q o f t h e i n i t i a l f l o w , s i d e s l o p e 1/m and b o t t o m w i d t h b Q , c a n a l l o n g i t u d i n a l bed s l o p e S q , t h e d i s t a n c e x o f wave p r o p a g a t i o n , g r a v i t a t i o n a l a c c e l e r a t i o n g, and t h e i n i t i a l wave h e i g h t Z , i . e . , ° Z = f, (g,y , V ,S ,m,b ,x*,Z ) (4-1) 1 o w o o o o o It s h o u l d be n o t e d t h a t t h e f r i c t i o n a l e f f e c t i s i n c l u d e d i n Eq. (4-1) b e c a u s e , g i v e n t h e p a r a m e t e r s in t h i s e q u a t i o n , t h e f r i c t i o n f a c t o r can be c a l c u l a t e d f r o m Manning's f o r m u l a . 3 5 The e f f e c t o f g r a v i t y i s r e p r e s e n t e d by a r a t i o o f i n e r t i a l f o r c e s t o t h o s e o f g r a v i t y . T h i s r a t i o i s g i v e n by t h e Froude number, d e f i n e d as F o =_Zo_ where and y ^ a r e t h e mean v e l o c i t y and d e p t h o f t h e i n i t i a l f l o w , g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n . T h i s i s n o r m a l l y t h e most i m p o r t a n t d i m e n s i o n l e s s r a t i o i n open c h a n n e l p r o b l e m s and i s w e l l - k n o w n as F r o u d e " s Law. The r a t i o x*/L i s a d o p t e d f o r t h e d i m e n s i o n l e s s a b s c i s s a o f 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 , where x* i s 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 su r g e wave f r o m the downstream end, and L i s K. t h e r e f e r e n c e l e n g t h o f c h a n n e l , i . e . , P h y s i c a l l y , L i s t h e l e n g t h o f a h o r i z o n t a l l i n e p a s s i n g K 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 . 4 - 1 ) . The r a t i o Z / Z q i s u s e d t o r e p r e s e n t t h e r e l a t i v e v a l u e o f t h e wave h e i g h t a t any 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 v a l u e a t t h e downstream end. The shape f a c t o r k r e p r e s e n t s the c r o s s -s e c t i o n c h a r a c t e r i s t i c s , where k = b /m.y . I t s h o u l d be o o n o t e d , t h a t t h e i n i t i a l wave h e i g h t Z Q i s a f u n c t i o n o f t h e Fr o u d e number F and w a t e r d e p t h y o f t h e o o 3 6 i n i t i a l f l o w , s e c t i o n 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 r » Thus, t h e Eq. ( 4 - 1 ) may be r e d u c e d t o = f 2 ( F q , x * / L R , k ) ( 4 - 2 ) Z o It i s t h u s assumed 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 o and x / L n f o r a c o n s t a n t v a l u e o f k. The a n a l y s i s f o r t h e v a r i -R a t i o n o f s u r g e 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 be d i s c u s s e d f o r c a n a l s o f 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 r o s s -s e c t i o n s , s e p a r a t e l y , 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 R e c t a n g u l a r C a n a l s 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 a r e a i s g o v e r n e d by t h e w i d t h and d e p t h o f t h e f l o w . I n t h i s c a s e , m i s e q u a l t o z e r o ( v e r t i c a l w a l l ) and t h e f a c t o r k i s e q u a l t o i n f i n i t y . E q . ( 4 - 2 ) i s t h e r e d u c e d t o = f _ ( F , x * / 0 ( 4 - 3 ) 3 o R Z o In o r d e r t o d e t e r m i n e the r e l a t i o n s h i p s i n Eq. ( 4 - 3 ) , F , y , b o o o and S were k e p t as i n d e p e n d e n t v a r i a b l e s and V , n and Z as o o o de p e n d e n t v a r i a b l e s . V q i s d e t e r m i n e d by F r o u d e 1 s Jaw and t h e f r i c t i o n a l c o e f f i c i e n t , n, i s d e t e r m i n e d f r o m Manning's f o r m u l a . The a n a l y s e s were c a r r i e d o u t by v a r y i n g t h e v a l u e s o f F^, y o > b Q and s y s t e m a t i c a l l y i n a d d i t i o n t o g i v i n g v a l u e s o f t h e i n i t i a l c h a nge 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 f o r t h e co m p u t e r program. When y ^ v a r i e s f r o m 5 f t . t o 4 0 f t . a t 5 f t . i n t e r v a l s , t h e r e s u l t s o f computer r u n s f o r g i v e n v a l u e s o f F , o 37 b and S , w h i c h r e s u l t e d 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 o o th e downstream end and t r a v e l l e d u p s t r e a m , a r e p l o t t e d on a d i m e n s i o n l e s s p l a n e w i t h x*/L_, as a b s c i s s a and Z / Z as R o o r d i n a t e . The p l o t shows t h a t the r e l a t i o n s h i p x*/L v e r s u s Z / Z ^ does n o t v a r y , a l t h o u g h t h e v a l u e o f v a r i e s . S i m i l a r l y , f o r g i v e n v a l u e s o f F , y and S , the v a l u e o f b i s v a r i e d o o o o fr o m 1 f t . t o 60 f t . a t 5 f t . i n t e r v a l s , and f o r g i v e n v a l u e s o f S 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 . A g a i n , t h e o r e l a t i o n s h i p , x-'/L^. v e r s u s Z / Z does n o t v a r y . However, when • R o F i s v a r i e d , w h i l e k e e p i n g y , S and b c o n s t a n t , t h e r e l a t i o n -o o o o s h i p c u r v e does =vary. I t i s t h e r e f o r e c o n c l u d e d t h a t t h e r e -l a t i o n c u r v e Z / Z a g a i n s t x*/L v a r i e s o n l y w i t h F and i s o K o i n d e p e n d e n t o f the i n d i v i d u a l v a l u e s o f 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 r u n s were c a r r i e d o u t f o r t h e v a l u e s o f F fro m 0.050 t o 0.200 a t 0.025 i n t e r v a l s . R e s u l t s o a r e p l o t t e d on F i g . 4-2. I t c a n be seen from t h i s f i g u r e t h a t , f o r a g i v e n F. , t h e c u r v e i s n e a r l y a s t r a i g h t l i n e a t h i g h e r o v a l u e s o f Z / Z . The f i g u r e shows t h a t the r a t e o f v a r i a t i o n o f o Z / Z q i s i n i t i a l l y r a p i d and n e a r l y c o n s t a n t and t h e n ..decreases g r a d u a l l y . T h e o r e t i c a l l y , a wave w o u l d need t o p r o p a g a t e t o i n f i n i t y b e f o r e i t c o m p l e t e l y d i s a p p e a r s . I n t h i s s t u d y , t h e l o w e r l i m i t o f Z / Z ^ i s s e t t o 0.025. Thus, i n summary: ( a ) The r e l a t i o n c u r v 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 o n l y . 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 v a l u e s o f o t h e r p a r a m e t e r s such as the c r o s s - s e c t i o n , l o n g i t u d i n a l s l o p e o f the c h a n n e l , and the r o u g h n e s s o f t h e c h a n n e l . 3 8 (b) F o r a c o n s t a n t v a l u e o f Z / Z q , the h i g h e r t h e v a l u e o f F , t h e h i g h e r w i l l be t h e v a l u e o f x*/L . T h i s does o K n o t i m p l y , t h a t t h e wave t r a v e l s a l o n g e r d i s t a n c e t o r e a c h a g i v e n r e d u c t i o n i n wave h e i g h t , because 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 al w a y s a s s o c i a t e d e i t h e r w i t h a l a r g e r l o n g i t u d i n a l s l o p e S q and a s m a l l e r v a l u e o f L , o r - w i t h a s m a l l e r roughness o f c h a n n e l and a K 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„. c Jo R 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 C a n a l s A c a n a l w i t h a t r i a n g u l a r 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 p r a c t i c e . I t i s a l i m i t i n g c a s e o f t h e t r a p e z o i d a l c r o s s -s e c t i o n where t h e bottom w i d t h i s z e r o . Two v a r i a b l e s , d e p t h and s i d e s l o p e , a r e i n v o l v e d i n c o n t r o l l i n g t h e c r o s s - s e c t i o n a r e a . I n t h i s c a s e , t h e shape f a c t o r k i s e q u a l t o z e r o . Eq. ( 4 - 2 ) i s t h e n a g a i n r e d u c e d t o Eq. ( 4 - 3 ) . 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 t o t h a t f o r r e c t a n g u l a r c a n a l s i n s e c t i o n 4 - 3 was u s e d . S i m i l a r r e l a t i o n c u r v e s were o b t a i n e d . Eq. ( 4 - 3 ) i s v a l i d f o r a t r i a n g u l a r c a n a l . 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 Z / Z a g a i n s t x*/L i n o K s e c t i o n 4 - 3 ( a ) and (b) a l s o h o l d . The c u r v e does n o t depend on s i d e s l o p e o f t h e c a n a l . The r e s u l t s o f the v a r i a t i o n o f 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 i a n g u l a r c a n a l w i t h F q f r o m 0 . 0 5 t o 0 . 2 0 a t 0 . 0 2 5 i n t e r v a l s a r e p l o t t e d i n F i g . 4 - 3 . 3 9 4.5 V a r i a t i o n of P o s i t i v e Surge Wave Height"in Trapezoidal Canals 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 of the most common shapes used f o r canals. For t h i s case, the c r o s s - s e c t i o n a l area i s governed by depth y , bottom width b and side slope 1/m. The o o shape f a c t o r k can vary between 0 and cc . For given values of F^ and k, a systematic a n a l y s i s s i m i l a r to that f o r rectangular and t r i a n g u l a r canals i s used, i . e . , the value of one v a r i a b l e i s changed and the others held constant. The r e s u l t s show that the curve Z / Z against x*/L does not change f o r f i x e d values of O K F q and k. Therefore, Eq. ( 4 - 2 ) i s v a l i d f o r a surge v a r i a t i o n i n a t r a p e z o i d a l canal. For given values of F , and for values of k = 0 , 0 . 3 3 3 , 0 . 5 0 0 , 1 . 0 0 0 , 1.667, 2 . 5 0 0 , 5 , 0 0 0 and cc surge c a l c u l a t i o n s were made. Results f o r F = 0 . 2 0 0 , are p l o t t e d on F i g . 4 - 4 . From t h i s family o of curves, i t can be seen that the higher the value of k, the higher i s the value of x*/L R for a given Z / Z q . The curves with k = oc and k = 0 are two l i m i t i n g curves. These c o i n c i d e with those drawn f o r a rectangular and a t r i a n g u l a r canal r e s p e c t i v e l y . For a given k, the r e l a t i o n curve Z/Z versus x*/L , v a r i e s with o K F^. For each given value of k, there i s a family of curves corresponding to various values of F . For r = l+ l / ( l + k ) having 40 v a l u e s o f 1.50 and 1.75, t h e r e s u l t s f o r F v a r y i n g f r o m 0.05 o t o 0.20 a t 0.025 i n t e r v a l s a r e shown on F i g s . 4-5 and 4-6 r e s p e c t -i v e l y , where r i s g i v e n i n Eq. (4-6). The r e m a r k s i n s e c t i o n 4.3(b) a l s o a p p l y f o r a t r a p e z o i d a l c a n a l . 4.6 A p p r o x i m a t e E q u a t i o n s As i t has been m e n t i o n e d e a r l i e r , t h e r e l a t i o n c u r v e Z / Z q a g a i n s t x*/L i s n e a r l y a s t r a i g h t l i n e a t h i g h e r v a l u e s o f Z / Z . T h e r e -R o f o r e f o r Z/Z v a l u e s g r e a t e r t h a n 0.6, t h e c u r v e may be a p p r o x -o i m a t e d by a s t r a i g h t l i n e w i t h o u t i n t r o d u c i n g s i g n i f i c a n t e r r c r a . When Z/Z i s l e s s t h a n 0.6, t h e c u r v e may be e x p r e s s e d by an o e x p o n e n t i a l f u n c t i o n . The a p p r o x i m a t e d f o r m u l a s r e p r e s e n t i n g t h e b e s t f i t by t h e " l e a s t s q u a r e " method a r e g i v e n i n t h e f o l l o w i n g s e c t i o n s : 4 . 6 . 1 A p p r o x i m a t e e q u a t i o n s f o r t h e 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 t h e e q u a t i o n s u g g e s t e d i s Z / Z_ = 1.0 + 0.1876 Z 1.0 - 1.0 0.3899 F - 0.00037 \ o ( X * ) LR (4-4a) (b) F o r JZ_ l e s s t h a n 0.60 t h e e q u a t i o n s u g g e s t e d i s Z o Z = A + B .exp ( C . x * / L D ) (4-4b) Z~ R o 41 where A - 0.01589 26.54 F - 6.7334 o B = 1.167 + 0.6728 F C = 3.045 - 1.702 F o -0.7985 Eqs. (4-4a) and (4-4b) are p l o t t e d on F i g . 4-2 f o r comparison . I t can be seen that the values estimated from the equations are good approximations. When a surge wave travels along the channel with a given i n i t i a l flow f o r which the Froude number i s d i f f e r e n t than those shown on F i g . (4-2), the v a r i a t i o n of the wave height may be obtained from the above equations. 4.6.2 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 surge wave propagating along a t r i a n g u l a r canal. (a) For _Z_ greater than 0.60 use equation Z o Z_ = 1.0 + 3.041 Z 1.0 1.0 0.4358 F + 0.000092 o ' (x* ) (4-5a) (b) For _Z_ l e s s than 0.60 use equation Z o Z = A + B .exp (C.x*/L ) (4-5b) where A = - (0.0352 + 1.479; . F ) o B = 0.642 . F + 1.205 o C = (0.7485 . F -1.135 0.6106) 42 E q s . (4-5a) and (4-5b) a r e a l s o p l o t t e d on F i g . 4-3. F o r a s u r g e 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 c a n a l , t h e v a r i a t i o n o f wave h e i g h t may be p r e d i c t e d f r o m t h e s e e q u a t i o n s . 4.6.3 A p p r o x i m a t e e q u a t i o n s f o r t h e 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 t r a p e z o i d a l c a n a l . From F i g . 4-4, i t i s seen t h a t t h e v a l u e o f k v a r i e s f r o m z e r o t o i n f i n i t y f o r v a r i o u s s i z e s o f t r a p e z o i d a l c r o s s - s e c t i o n c a n a l s . To s i m p l i f y m a t h e m a t i c a l t r e a t m e n t , t h e p a r a m e t e r r was u s e d t o s u b s t i t u t e f o r k, where, r = 1 + 1 + k (4-6) Thus r = 1 when k = OC , and r = 2 when k = 0. The v a l u e o f r v a r i e s between 1 and 2 f o r t r a p e z o i d a l c r o s s - s e c t i o n s . F o r a g i v e n F and Z/Z , t h e v a r i a t i o n o f x*/L_ v e r s u s k i s o o R n o n l i n e a r . A p a r a m e t e r may be d e f i n e d , s u c h t h a t ( ~ ) <f> = R r = n (4-7) (—) i n w h i c h ( x ) . i s t h e v a l u e o f ( x ) T r = 1 ~ R R a p p r o p r i a t e t o the 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 *<. c a n a l : and ( x ) i s t h a t f o r t h e c u r v e o f r = n o f a g i v e n t r a p e z o i d a l c a n a l f o r a g i v e n Z/Z^. V a l u e s o f were p l o t t e d a g a i n s t r on l o g - l o g p a p e r , F i g . 4-7, and i t was f o u n d t h a t t h e p o i n t s l o c a t e d a p p r o x i m a t e l y on a s t r a i g h t l i n e . T h e r e f o r e , i t 43 may be c o n c l u d e d t h a t t h e r e l a t i o n s h i p between tp and r i s a l o g a r i t h m i c f u n c t i o n . I t may be assumed t o be <f>; = c r " d (4-8) where c and d 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 . T h i s r e l a t i o n s h i p h o l d s f o r v a r i o u s v a l u e s o f Z/Z and F . Because o o a l l c u r v e s have t o pass t h e p o i n t o f x /L^ = 0 and Z/Z = 1.0, R o i . e . , when r = 1.0 and tp = 1.0, c i s e q u a l t o 1.0. Thus Eq. (4-8) i s s i m p l i f i e d t o <t> = r ~ d . . . (4-9) where the v a l u e o f d i s e q u a l t o t h e n e g a t i v e s l o p e o f t h e c u r v e on F i g . (4-7). The exponent d, d e t e r m i n e d by the l e a s t s q u a r e method i s e q u a l t o 0.565. Eq. (4-8) t h e n becomes * = r - ° ' 5 6 5 (4-10) U s i n g t h i s e q u a t i o n , the v a l u e o f <p may be o b t a i n e d p r o v i d e d t h e shape f a c t o r k i s g i v e n . From Eq. (4-7), 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 s u r g e 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 wave 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 ( ) r = n = * { ) r = 1 (4-11) where / x \ may be o b t a i n e d f r o m Eqs. (4-4a) o r (4-4b) ( L R ) r = provided F q i s given. Comparison of the estimate from Eqs. (4-4a), (4-4b) and (4-11) with the computer r e s u l t s are shown 44 on F i g s . 4-5 and 4-6. Two examples a r e shown t o i l l u s t r a t e t he a p p l i c a t i o n o f t h e s e f o r m u l a s . Example 4 - 1 G i v e n F = 0.1, S = 0.001, m = 1.0, b = 10 f t , y = 10 f t . o o o o F o r t h e c a s e o f t o t a l c l o s u r e and f o r Z / Z q = 0.3,find 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 su r g e moving a l o n g a t r a p e z o i d a l c a n a l . S o l u t i o n L- = y o = 10,000 f t . R S~ o k = ^ o _ = 1.0 m.y o r = 1 + 1 = 1.5 1 + k when F = 0.1, and Z / Z = 0.3, i t i s f o u n d from F i g . 4-4 o o ° * = 0.1715 l = r - ° ' 5 6 5 = 0.7958 / x _ \ = 0.1715 . (0.7958) = 0.1365 U R i r = 1.5 x * = 1365 f t . From F i g . 4-5, / x^ \ = 0.1370 ) r = 1.1 x * = 1370 f t . I t may be seen 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 f o r m u l a i s f a i r l y a c c u r a t e . 45 The above e q u a t i o n s may a l s o be u s e d t o e s t i m a t e the wave h e i g h t o r 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 su r g e t r a v e l l i n g i n a t r i -a n g u l a r c a n a l i n w h i c h r = 2.0 and <f> = 0.6761. Example 4-2 G i v e n F = 0.125, Z/Z = 0.30, S = 0.001, Y = 10 f t . , o o o o f i n d 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 su r g e wave t r a v e l l i n g a l o n g a t r i a n g u l a r c a n a l . S o l u t i o n : L 10 = 10,000 f t . 0.001 ( a ) From F i g . 4-2, / x ^ \ =0.2160 U R j r = 1 By u s i n g Eq. (4-11) = 0.6761 • (0.2160) = 0.1465 x* = 1465 f t . ( b ) From F i g . 4-3, i t i s f o u n d / x * \ = 0.1480 x * = 1480 f t . ( c ) Using Eq. (4-5b), A = 0.583, B = 1.2850 and C = 8.5296 = 0.1498 LR x* = 1498 f t . P e r c e n t a g e e r r o r o f ( a ) = 1480 - 1465 = 1.6% 1480 P e r c e n t a g e e r r o r o f ( c ) = 1480 - 1498 = -1.2% 1480 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 l o c a t e d  a t t h e Upper End o f the C a n a l When w a t e r e n t e r s a m i l d s l o p e c h a n n e l , t h e d e p t h y ( ( F i g . 4-8) i s r e l a t e d t o t h e s t a t i c r e s e r v o i r l e v e l by t h e energy e q u a t i o n . The r e l a t i o n between t h e d e p t h s y and y c a n be e x p r e s s e d by >i + + { 4 _ 1 2 ) where y^ i s t h e head l o s s due t o f r i c t i o n a t e n t r a n c e and may be e x p r e s s e d i n terms o f t h e v e l o c i t y head a t the e n t r a n c e o f the c a n a l , t h a t i s V 2 y = C . 6 6 2g . . . . (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 depends on the c o n d i t i o n s a t e n t r a n c e . 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 r e w r i t t e n v2 y R = y-L + M • _JL (4-14) 2g When t h e w a t e r e m p t i e s i n t o a r e s e r v o i r , an amount o f k i n e t i c V 2 e n e r g y e q u a l t o _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 e x p e c t e d t o be r e s t o r e d as a p o t e n t i a l e n e r g y . Thus, t h e r e s -e r v o i r l e v e l s h o u l d be h i g h e r by t h i s amount t h a n the w a t e r l e v e l a t t h e u p s t r e a m end o f t h e c a n a l . T h i s e n e r g y , however i s u s u a l l y d i s s i p a t e d i n e d d i e s and w h i r l s . I n p r a c t i c a l c a l c u l a t i o n s i t may be i g n o r e d , and y may be r e g a r d e d as e q u a l t o y (no v e l o c i t y 1 K head r e c o v e r y ) . < 47 A p o s i t i v e s u r g e wave r e a c h i n g t h e r e s e r v o i r a t the u p s t r e a m end o f the c a n a l w i l l be r e f l e c t e d as a n e g a t i v e wave w h i c h t h e n p r o c e e d s downstream. The f l o w d i r e c t i o n a t the e n t r a n c e a f t e r t h e wave i s r e f l e c t e d , w i l l depend on t h e l e n g t h and s l o p e o f the c h a n n e l . I n g e n e r a l , i n a s h o r t c h a n n e l , o f m i l d s l o p e w a t e r v e l o c i t y w i l l be r e v e r s e d i . e . , w a t e r f l o w s i n t o t h e r e s e r v o i r . I n a l o n g c a n a l however, the v e l o c i t y o f f l o w i s r e d u c e d b u t u s u a l l y t h e r e i s no change i n d i r e c t i o n ( w a t e r s t i l l f l o w s i n t o t h e c a n a l ) . The amount o f r e d u c t i o n o f wave h e i g h t w i l l t h e r e f o r e , depend on t h e d i r e c t i o n o f the f l o w a t t h e s e c t i o n o f e n t r a n c e i m m e d i a t e l y a f t e r t he n e g a t i v e wave p a s s e s . I f the f l o w d i r e c t i o n changes ( r e v e r s e t o the i n i t i a l d i r e c t i o n ) , t he r e d u c t i o n o f wave h e i g h t < i Z i s : A Z = y 2 - y 1 ~ y R - v± o r a „ - V l (4-15) I f 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 h e i g h t i s : 2 AZ = y R - yn - M'. V2 (4-16) n x 2g v 2 Assuming t h e v a l u e o f i s s m a l l and ^ . _2 i s n e g l i g i b l e , 2g Eq. (4-16) i s i d e n t i c a l t o Eq. (4-15), and AZ i s a f u n c t i o n o f i n i t i a l v e l o c i t y . T h e r e f o r e , t h e 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 the e n t r a n c e o f t h e c a n a l can be o b t a i n e d a p p r o x i m a t e l y by Eq. (5-15) r e g a r d l e s s t h e d i r e c t i o n o f t h e f l o w a f t e r r e f l e c t i o n a t t h a t s e c t i o n . 4 8 I t h as been 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 wave h e i g h t a t t h e downstream end depends on the Fr o u d e number o f the 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 Fr o u d e number o ( s e e F i g . 4 - 1 0 ) . The a p p r o x i m a t e e q u a t i o n s f o r c a l c u l a t i n g A Z / Z ^ 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 a r e f o u n d i n t h e f o l l o w i n g . 4 . 7 . 1 E q u a t i o n f o r r e c t a n g u l a r c a n a l s F o r r e c t a n g u l a r C a n a l s , A Z / Z . v a r i e s w i t h F r o u d e number, and o t h e d i s t r i b u t i o n i s l i n e a r on l o g - l o g p a p e r w i t h F^ as a b s c i s s a and A Z / Z as o r d i n a t e . U s i n g t h e l e a s t s q u a r e method f o r b e s t o f i t , t h e e q u a t i o n o b t a i n e d i s A Z = 0 . 4 5 3 3 F °* 9 6 8 ( 4 - 1 7 ) o 4 . 7 . 2 E q u a t i o n f o r t r i a n g u l a r c a n a l s F o r t r i a n g u l a r c a n a l s , t h e c o r r e s p o n d i n g e q u a t i o n i s A Z = 0 . 6 6 3 1 F °' 9 7 9 ( 4 - 1 8 ) o 4 . 7 . 3 E q u a t i o n f o r t r a p e z o i d a l c a n a l s F o r t r a p e z o i d a l c a n a l s , i t i s f o u n d t h a t A Z / Z q v a r i e s n o t o n l y w i t h F r o u d e number b u t a l s o w i t h t h e p a r a m e t e r r . F o r a g i v e n v a l u e o f r , t h e r e l a t i o n s h i p between ^ \ Z / Z and F i s s i m i l a r o o to t h a t f o r t h e 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 . The e q u a t i o n is o f t h e f o r m A Z = P . F ? 2 ( 4 - 1 9 ) Z o where p.. and 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.. and 49 are the functions of r . They are determined as p = 0.4533 r 0 * 5 4 9 0 (4-20) P 2 = 0.9680 r 0 ' 0 1 6 2 8 . . . . . . . . . (4-21) Therefore Eq. (4-19) becomes A Z = 0.4522 r ° - 5 4 9 0 .F 0 ' 9 6 8 0 r 0 - 0 1 - 2 8 - ' ( A " 2 2 ) o Z o Eq. (4-22) i s v a l i d f o r the values of r from 1.0 to 2.0 which means that Eq. (4-22) i s also v a l i d f o r rectangular and t r i a n g u l a r canals because when the values of r are equal to 1.0 and 2.0, the Eq. (4-22) i s i d e n t i f i c a l to 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 also can be found by an a l t e r n a t i v e procedure based on d e r i v a t i o n s - i n C h a p t e r 2 . Dividing both sides of the equation A z = ^ 2g by Z , i t becomes o TJ2 F 2 F 2 .AZ = v l = 12 . = _1£L_ . . . . (4-23) Z 2g.Z o o where ft = Z /y P o c 2Zo y 0 The values of .A Z/Z f o r various flows i n rectangular, t r i -o angular and trapezoidal canals can be found from Eq. (4-23) with Eqs. (2-13), (2-18) and (2-23) r e s p e c t i v e l y . 50 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 the Upper End o f a P r i s m a t i c Power C a n a l N e g a t i v e waves a r e n o t s t a b l e i n f o r m because p o i n t s on t h e upper p a r t o f t h e wave t r a v e l f a s t e r t h a n t h o s e on t h e l o w e r p a r t . I f t h e h e i g h t o f t h e wave i s r e l a t i v e l y l a r g e , compared w i t h t h e d e p t h o f 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 f r o m p o i n t t o p o i n t a l o n g t h e wave f r o n t . The i n i t i a l p r o f i l e o f n e g a t i v e s u r g e waves h a v i n g ' a s t e e p f r o n t w i l l f l a t t e n o u t as t h e waves t r a v e l a l o n g t h e c a n a l . I f t h e h e i g h t o f t h e wave i s moderate o r s m a l l , t h e e q u a t i o n s i n C h a p t e r 2, d e r i v e d f o r a p o s i t i v e wave, can be a p p l i e d t o d e t e r -mine a p p r o x i m a t e l y t h e h e i g h t and v e l o c i t y o f n e g a t i v e waves. The n e g a t i v e waves r e f l e c t e d f r o m t h e r e s e r v o i r may be s m a l l , com-p a r e d w i t h t h e d e p t h . F o r a wave w i t h a h e i g h t o f l e s s t h a n 207o o f the d e p t h t h e e r r o r i n t r o d u c e d by assuming t h a t t h e n e g a t i v e wave p r o c e e d s 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 be i n -s i g n i f i c a n t . I n t h i s s t u d y t h e c a l c u l a t i o n s f o r n e g a t i v e waves a r e b a s e d on t h e above a s s u m p t i o n . The e q u a t i o n s f o r p o s i t i v e waves ( C h a p t e r 2) a r e t h e n assumed t o be v a l i d f o r n e g a t i v e waves. The b a s i c e x p r e s s i o n s f o r c a l c u l a t i n g t h e p r o p a g a t i o n o f n e g a t i v e waves, w h i c h r e s u l t f r o m r e f l e c t i o n a t the upper end o f t h e c a n a l a r e p r e s e n t e d i n C h a p t e r 3. S i m i l a r l y t o t h e c a l c u l a t i o n s f o r a p o s i t i v e s u r g e wave, sy s t e m -a t i c v a r i a t i o n o f t h e v a r i a b l e s f o r d i f f e r e n t shapes o f c a n a l s was u s e d as t h e i n p u t t o t h e computer program. R e s u l t s o b t a i n e d f o r t h e n e g a t i v e waves a r e p l o t t e d on t h e same d i m e n s i o n l e s s p l a n e 51 t o g e t h e r w i t h t h o s e f o r p o s i t i v e waves. The v a r i a t i o n o f the r e l a t i v e h e i g h t , Z/Z , o f the n e g a t i v e wave i n term o f x /L o K i s shown i n F i g u r e 4-10a t h r o u g h 4-12g. From t h e s e f i g u r e s t h e f o l l o w i n g p o i n t s a r e o b s e r v e d : 1. When a wave i s r e f l e c t e d a t a g i v e n v a l u e o f x /L ( o r a K g i v e n v a l u e o f Z/Z^) the shape o f t h e c u r v e i s i n d e p e n d e n t o f the i n i t i a l p a r a m e t e r s y , V and S f o r a g i v e n F and r . o o o o 2. F o r a g i v e n F q and r the shape o f c u r v e f o r a r e f l e c t e d wave JL depends on t h e l o c a t i o n ( x " / L R ) o f t h e p o i n t o f r e f l e c t i o n . 3. The 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 d i m i n i s h e s g r a d u a l l y as t h e wave t r a v e l s downstream. 4.9 Maximum Water D e p t h a t Downstream End o f C a n a l When a s u r g e wave o c c u r s a t t h e downstream end o f t h e c a n a l due t o t h e l o a d r e j e c t i o n , t h e w a t e r d e p t h a t t h e downstream end i n c r e a s e s s u d d e n l y by t h e amount e q u a l t o the i n i t i a l wave h e i g h t . As t h e wave f r o n t t r a v e l s u p s t r e a m , 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 s t a t i o n a r y . T h i s i s because t h e d i s c h a r g e o f the f l o w i s r e d u c e d , t h e v e l o c i t y i s a l s o r e d u c e d , and t h e k i n e t i c energy o f t h e f l o w i s c o n v e r t e d t o p o t e n t i a l energy i n the f o r m o f an i n c r e a s e d h e i g h t o f t h e w a t e r s u r f a c e a t t h e downstream end. F i g s . 4-13 and 4-14 i l l u s t r a t e t h e e l e m e n t a r y f a s h i o n o f t h i s v a r i a t i o n o f w a t e r d e p t h i n t h e p o s i t i v e and n e g a t i v e wave p r o p a g a t i n g c y c l e . The d e p t h o f w a t e r a t the downstream end i n c r e a s e s w i t h t i m e . The r a t e o f i n c r e a s e i s i n i t i a l l y a p p r o x i m a t e l y l i n e a r w i t h t i m e . 52 I f t he d e p t h i s e x p r e s s e d by t h e r a t i o y/y Q> t h e n t h e r a t e o f v a r i a t i o n o f y / y Q w i t h r e s p e c t t o t i m e i n c r e a s e s w i t h l a r g e r v a l u e s o f t h e Fro u d e number. F o r a f l o w w i t h a g i v e n Froude number, t h e r a t e o f i n c r e a s e i n y / y Q a l s o v a r i e s w i t h c a n a l c h a r a c -t e r i s t i c s , s u c h 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 r o u g h n e s s and shape. I f t h e c r o s s - s e c t i o n o f t h e c a n a l i s g i v e n , t h e r a t e o f i n c r e a s e i n y / y Q i n c r e a s e s w i t h t h e i n c r e a s i n g o f S Q . As an example, f o r a r e c t a n g u l a r c a n a l , w i t h b Q = 30 f t . , = 30 f t . and n = 0.03095, the 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 numbers a r e p l o t t e d i n F i g . 4-15. S i m i l a r c u r v e s f o r t h e t r i -a n g u l a r c a n a l o f y Q = 22.36 f t . , m = 2.0 and n = 0.03095 and t h e t r a p e z o i d a l c a n a l o f b Q = 25.5 f t . , m =*1.5, n = 0.03095 and r = 1.5 a r e p l o t t e d i n F i g . 4-16 and F i g . 4-17 r e s p e c t i v e l y . I t i s i n t e r e s t i n g t o n o t e t h a t t h e s l o p e s o f c u r v e s i n c r e a s e as the v a l u e s o f c r o s s - s e c t i o n a l shape f a c t o r r d e c r e a s e . A f t e r g a t e c l o s u r e t h e d e p t h a t t h e downstream end i n c r e a s e s c o n t i -n u o u s l y u n t i l t h i s 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 wave, r e f l e c t e d f r o m t h e upper end o f t h e c a n a l and t h e n d e c r e a s e s r a p i d l y by an amount o f about t w i c e t he n e g a t i v e wave h e i g h t . The 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 a t t h e ti m e i m m e d i a t e l y b e f o r e t h e a r r i v a l o f t h e n e g a t i v e wave f r o n t . As has been men-t i o n e d b e f o r e , t h i s maximum w a t e r d e p t h i s one o f t h e most impor-t a n t i t e m s o f i n f o r m a t i o n i n t h e d e s i g n o f t h e c a n a l . I n t h i s s t u d y , s y s t e m a t i c v a r i a t i o n o f i n d i v i d u a l v a r i a b l e s was 53 used as input to the computer program. Results are p l o t t e d on a dimensionless plane wit h y /y as o r d i n a t e and 2L/L„ as a b s c i s s a , r max ''o R where L i s the canal l e n g t h . For r e c t a n g u l a r canals, i t i s found that the curve of y /y against 2L/Ln does not change w i t h the Jmax Jo ° R ° various values of b , y and S i f the Froude number of i n i t i a l o o o flow, F q i s f i x e d . This r e l a t i o n a l s o holds f o r t r i a n g u l a r c a n a l s . For the 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 introduced. The curve y /y against 2L/L„ v a r i e s not only w i t h F but a l s o max •'o R o w i t h r . For a given Froude number, the value of y Iy increases - max o w i t h i n c r e a s i n g value of r . For a given r, the value of y /y • ° Jmax Jo v a r i e s w i t h F . The r e s u l t s of computer c a l c u l a t i o n s f o r a maximum stage developed at the downstream end due to a sudden r e d u c t i o n 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) canals are 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 CONCLUSIONS The primary o b j e c t i v e of t h i s study was to derive dimensionless r a t i o s to describe the i n i t i a t i o n and propagation of a surge wave i n a power canal following a load reduction or r e j e c t i o n . The i n i t i a l surge-wave height, r e s u l t i n g from an instantaneous reduction of discharge, v a r i e s with the shape and dimensions of a canal and with the amount of the i n i t i a l flow. This study shows that, i n dimensionless terms, the r e l a t i v e i n i t i a l wave height i s a function of the shape fac-tor k, discharge r a t i o x and the Froude number of the i n i t i a l flow F . o For a sudden t o t a l closure, i . e . , T = 0, f£ i s only a function of F q i n a rectangular and t r i a n g u l a r canal, and i s a function of F^ and k i n a trapezoidal canal. A dimensionless equation, Eq. (2-23), derived i n t h i s study can be applied to pr e d i c t an i n i t i a l surge wave height due to a sudden t o t a l or p a r t i a l change i n discharge i n canals of rectang-u l a r , t r i a n g u l a r and trapezoidal c ross-sections. The r e s u l t s of c a l c u l a t i o n s from the mathematical model developed i n t h i s study are i n close agreement with those from the methods proposed by previous i n v e s t i g a t o r s , i n p a r t i c u l a r , with the r e s u l t s from the Favre method applied to short canals. When a p o s i t i v e surge wave i s i n i t i a t e d at the downstream end of a canal and propagates upstream, the wave height decreases gradually. The rate of decrease of wave height depends on canal parameters such as the f r i c t i o n a l c o e f f i c i e n t s , the bed slope, the shape and dimensions of the c r o s s - s e c t i o n and the i n i t i a l flow. T h i s d e c r e a s e o f wave h e i g h t i s a p p r o x i m a t e l y l i n e a r f o r a d i s t a n c e , f o r w h i c h Z/Z i s s t i l l g r e a t e r t h a n 0.6. When Z/Z becomes l e s s t h a n o o 0.6, t h e v a r i a t i o n o f wave h e i g h t i s more an 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 w a t e r s u r f a c e p r o f i l e i n c r e a s e s w i t h i n c r e a s i n g d i s t a n c e f r o m t h e 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. The w r i t e r has d e r i v e d d i m e n s i o n -l e s s e q u a t i o n s f r o m w h i c h t h e wave h e i g h t o f a p o s i t i v e s u r g e a t any s e c t i o n o f 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 may be p r e d i c t e d . Eq. (4-4) i s f o r r e c t a n g u l a r c a n a l s , and Eq. (4-5) i s f o r t r i a n g u l a r c a n a l s . The i n f l u e n c e o f shape and 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 on t h e v a r i a t i o n o f wave h e i g h t o f a p o s i t i v e s u r g e , may be ex-p r e s s e d by a l o g a r i t h m a t i c f u n c t i o n . E q s . (4-10) and (4-11) g i v e t h e 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 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 c r o s s - s e c t i o n . U s i n g t h e s e two e q u a t i o n s , t h e v a r i a t i o n o f wave 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 can be p r e d i c t e d f r o m Eq. (4-4). I n a l o n g c a n a l where a n e g a t i v e s u r g e -wave, r e f l e c t e d f r o m t h e r e s e r -v o i r a t t h e up p e r 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 i t s h e i g h t i s i n i t i a l l y r a p i d . T h i s 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 . I n a v e r y l o n g c a n a l , a n e g a t i v e wave may p r o p a g a t e downstream f o r a l o n g d i s t a n c e w i t h l i t t l e a t t e n u a t i o n a f t e r t h i s i n i t i a l r e d u c t i o n . The r i s e o f t h e w a t e r s u r f a c e b e h i n d t h e wave f r o n t a t t h e downstream end o f t h e c a n a l i s n o t l i n e a r w i t h r e s p e c t t o t h e t i m e . The r a t e o f t h i s r i s e i n t h e w a t e r s u r f a c e i n c r e a s e s w i t h l a r g e r v a l u e s o f t h e Fro u d e number F o f i n i t i a l f l o w . The 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 c a n a l , c a u s e d by a r e d u c t i o n i n d i s c h a r g e , o c c u r s i m m e d i a t e l y b e f o r e t h e a r r i v a l o f a n e g a t i v e s u r g e wave t h a t r e s u l t e d f r o m r e f l e c t i o n a t t h e upper end o f t h e c a n a l . T h i s maximum d e p t h depends on t h e s l o p e , l e n g t h and c r o s s - s e c t i o n o f t h e c a n a l and t h e i n i t i a l f l o w . The d i m e n s i o n l e s s r e l a t i o n s h i p s d e r i v e d i n t h i s s t u d y may be u s e d t o e s t a b l i s h d e s i g n c r i t e r i a f o r c r e s t e l e v a t i o n s o f t h e banks and w a l l s o f power c a n a l s t o a v o i d o v e r t o p p i n g . I n t h i s c r i t e r i a some a l l o w a n c e must be made f o r s e c o n d a r y s u r g e s n o t a n a l y s e d i n t h i s t h e s i s and f o r a minimum d e s i r e d f r e e b o a r d . w Oi != o •-a o w H FIG. 2-1 DEFINITION SKETCH : SURGE WAVE IN AN OPEN CHANNEL Vo 2^ A i = bo-yi y x = y l / 2 • 2-2 DEFINITION SKETCH : SURGE WAVE IN A RECTANGULAR CANAL F I G . 2-3 I N I T I A L SURGE WAVES IN 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 IN A TRIANGULAR CANAL FIG. 2-7 ( c ) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR r = 0 . 2 0 FIG. 2-7 (d) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR r = 0.30 FIG. 2-7 (e) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR T = 0.40 UIO"5 2 3 4 5 6 7 8 9 IxlO-* 2 3 4 5 6 7 8 9 1x10"' 2 3 4 5 6 7 8 9 10 F 2 F I G . 2-7 ( f ) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR T = 0.50 UIO"5 2 3 4 5 6 7 8 9 UK)"* 2 3 4 5 6 7 8 9 IxiO"' 2 3 4 5 6 7 8 9 10 FIG. 2-7 (g) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR r = 0.60 FIG. 2-7 (h) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR T = 0.70 UIO"3 2 3 4 5 6 7 8 9 UICT8 2 3 4 5 6 7 8 9 1x10"' 2 3 4 5 6 7 8 9 I O FIG. 2-7 ( i ) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR T = 0.80 to* FIG. 2 - 7 ( j ) I N I T I A L SURGE WAVES IN A TRAPEZOIDAL CANAL FOR r. = 0.90 t i 71 F I G . 3-1 DEFINITION SKETCH : C"1" AND C~ CHARACTERISTIC GRIDS ON THE x - t PLANE FIG. 3-3 DEFINITION 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(nr i - l ,n+l) V(m+l,n+l) y(m+l,n+l) P(m+1 ,n+l) P(m+2,n) x(m sn) t(m,n) V(m,n) y(m,n) x(m+2,n) t(m+2,n) V(m+2,n) y(m+2,n) FIG. 3-2 DEFINITION SKETCH CHARACTERISTIC GRIDS ON THE x - t PLANE EXECUTION I I 2 3 4 5 6 7 8 9 10 11 V 7.615 7.635 7.635 7. 635 7.635 7 .535 7.635 7.635 7.635 7.635 7.635 V 7. 859 7.859 7.859 7.859 7. 839 7.359 1.8 V-> 7.:3 59 ' 7.H5V 7."59 7.359 X 0.000 100.000 200.000 300.000 400.000 600.000 600.000 700.000 800.000 900.000 1000.000 T 0.000 O.OCO 0.000 0.000 0.000 0.000 0.000 0.000 0.000 C.000 C.000 2 V 7. 634 7 . 634 7. 634 7. 634 7. 6 34 7. 634 7 . 6 34 7 . 634 7. 634 7.634 Y 7. 859 7. 859 7. 85 9 7. 8 59 7. 859 7. 8 59 7 . 859 7. 8 59 7. 85 9 7.859 X 73. 998 173. 998 27 3. 998 373.998 473. 9 9.5 573. 4 98 67 3 . 9 98 77 ). 9 9 8 .-I 7 3 . 9 9 8 97 3.998 T 3. 14 3 3. 143 3. 143 3. 143 3. [43 3 . 143 3 .143 3 . 14 i 1. 14 3 3. 1 4 3 3 V 8.083 7. 633 7. 63 3 7. 633 7. 63 3 7. 63 3 7 . 63 * 7. 6 3 3 7 . 633 7. 6 3 3 7. 6 34 Y 8.081 7. 859 7. 859 7. 859 7. 859 7. 8 59 7 . 859 7. 859 7 . 8 59 7. 859 7. 859 X 0.000 147. 994 247. 994 347. 994 44 7. 994 547. 9 9 3 647 . 9 9 J 74 7 . 99 3 84 7 . 9 9 3 94 7. 99 3 1000 . 000 T 12.ot r 6. 286 6. 286 6. 286 6. 286 6. 2 86 6. 286 6. 786 i> • 286 6. 78 6 4 . 248 4 V 8. 076 7. 632 7. 632 7. 632 7 . 6 32 7. 6~3~2 .6 32 T. ~63~7~ 7. 632 7.633 Y R . 0 79 7. 859 7. 8 59 7. 859 7. 859 7. 8 59 7 . H 5 9 7. 85 9 7 . 8 5 4 7.859 X 74. 527 221 . 98 5 321. 985 421.98 5 52 1 . 985 621. 98 5 72 1 82 1 . '•85 92 1. 9 8 6 9 7 3. 994 T 15. 165 9. 430 9. 430 9. 4 30 9. 4 30 9. 4 30 9 .4 30 •i. 4 30 9. 4 30 7. .3 9 1 5 V 8.453 8. 070 7. 631 7. 6 3 1 7. 63 1 7. 531 7 . 631 7. 631 7 . 631 7 . .5 3 2 7. 632 Y 8. 284 8. 0 76 7. 859 7. 859 7. 859 7 . 3 59 7 . 85 9 7. 859 7. 8 5 i 7. 3 5 9 7. H59 X 0.000 149. 044 295. 974 3 95. 974 495. 97 4 59 5. 974 695. 9 74 795. • 74 89 5. 9 7'. 94 7 . H'4 1 OOC. 000 T 24.420 18. 244 12. 573 12. 5 73 12. 57 3 12. 5 73 12 . 57 1 1?. 57 3 1 2 . 57 i 10 . 634 8 . 496 6 V 8. 443 8. 064 7. 630 7. 630 7. 6 30 7. 6~36~~ _ . _ 763~0- 7. 630 7. 63 1 7.631 Y 8. 280 8. 074 7. 8 59 7. 359 7. 8 59 7. 8 59 7 .859 7. 859 7. 8 59 7. 8 59 X 74. 931 223. 553 369. 959 469.959 5 59. 959 669. 959 769 .9 59 86 9 . 95 9 9 2 1. 9 72 <• 7 3. 989 T 27. 443 21 . 323 15. 716 15.716 15. 7 16 15. 7 16 1 5 .716 1 5 . 7 1 6 1 1. 6 7 I 1 1.639 7 V 8.699 a. 433 8. 053 7. 629 7. 629 7. 629 7 . 629 7. 62 9 7. 6 30 7. 6 30 7. 63 1 Y 8.440 8. 2 75 8. 071 7. 8 59 7. 85 9 7. 859 I. 8 59 i. 859 7. 85 ) 7. •1 5 9 7. 859 X 0.000 149. 850 298. 053 443. 941 54 3. 94 1 64 3. 94 1 74 3 . 94 1 ii 4 3 . ••4 1 8 9 5. 956 94 7. 9 7 6 1 000. 000 T 36.946 30. 46 8 24. 40 3 18. 859 18. 859 13. 8 59 18. 85 9 18. 8 59 15 . 821 14. 7 8 2 12. 744 8 V 8. 60 7 8 . 423 8. 052 7. 628 7. 628 7. 628 7 .6 28 7 . 62-' 7. 62 9 7.630 Y 8. 4 34 8 . 270 8. 069 7.859 7. 859 7. 8 59 7 .8 59 !. 859 7. 8 5 9 7. 6 59 X 75. 180 224. 756 372. 544 51 7. 92 1 617. 92 1 717. 921 8 1 7 .971 8 6 J . 93 7 92 1 . )6 9 973.985 T 39. 931 33. 49 3 2 7. 484 22.002 21. 002 22. C02 22 .002 1 9 . 9 64 1 7. 97 6 15.687 9 V 8. 789 8. 675 8. 414 8. 04 7 7.62 7 7. 62 7 1. 627 7. 52 8 7. 62 8 7. 62 9 7. 6 30 Y 8.528 8. 42 7 8. 265 8. 067 7. 85 9 7 . 8 59 7 . 8 5 9 7. 3 59 7 . 859 7. 85 9 7. 859 , X 0.000 150. 345 299. 650 447.026 59 1. 897 691. 89 7 791 . 89 7 84 3. > 1 b 895 . 9 3-) 94 7. 96 7 1000. COO T 49.579 42. 918 36. 520 30. 565 25. 146 25. 1 46 25 . 146 ? 3. 107 2 1 . 06 3 1 9. 030 16. 992 J FIG. 3-4 A PART OF RESULTS OF COMPUTER CALCULATIONS IN EXAMPLE 3-1 TIME , SECONDS U. 75 F I G . 3-6 SCHEMATIC PRESENTATION OF CHARACTERISTIC GRIDS FOR A POSITIVE 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 (3) COMPUTE Z„,V,,„ o wo 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 FIG. 3-10 DEFINITION SKETCH FOR EXAMPLE 3-3 79 SECTION 0 SECTION 1 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 h1=41.175 hx+=41.175 h^=41.175+4.27 F -41.175x45 0 FQ=49.567x45 Fp=54.644x45 F1-41.175x45 F1+=41.175x45 =45.445 Q0-292x45 Qo°° Q'o=° Q1=292x45 0.^=292x45 F|=45.445x45 V0=7.101 V0-0 Vj=7.105 V .105 0^ = 154.7x45 B0=45 B Q«45 PQ-144.134 R'=49.567x45 144.134 B''-45 P"=154.288 0 R'Q=15.89 B1-45 Bj+=45 AQJ--137.3x45 Vf=3.41 B^ =45 -15.45 Ah x = 0 Pi=135.89 Z0-8.393 A ?l = 0 R[=15.05 y 0-45 A Ql = 0 ZJ=4.27 a.Q--34.84 a1=-32.14 C Q—41.94 AQQ--292 C^-39.25 AF^-8.393x45 A F|=4.27x45 Assume Z^- 5.077 (Horizontal) y--45 AF-'-Z-'.Y'' -5.077x45 AQo'O 2 (V 2) 1.486" #3 L -Summary of t r i a l s : +0.00002525 - 38800 Bm = 45 m A B = 0 AQ - 0 . Aa - 2.699 a - 33.49 a R - 15.47 in I - 0.000237 n V =1.705 m (V 2) =5.81 No. of t r i a l Guess Zi Water Volume »i Water Volume -Vi 1 4.27 339000 x 45 2 4.00 342000 x 45 3 4.10 339500 x 45 4 4.15 339000 x 45 344000 x 45 334000 x 45 337500 x 45 339200 x 45 *1< V 2 *1> V2 *1> V2 Y i * v 2 F I G . 3-12 SUMMARY OF CALCULATIONS IN EXAMPLE 3-3 FOLLOWING FAVRE METHOD X , tl. F I G . 3 - 1 3 R E S U L T S OF C A L C U L A T I O N S IN E X A M P L E 3 - 4 RESERVOIR (a) r* — — . RESERVOIR H y T ± (b) F I G . 3-14 (a) POSITIVE SURGE REACHES THE RESERVOIR AT THE UPPER END OF THE CANAL, (b) NEGATIVE SURGE REFLECTED FROM THE RESERVOIR. J W 1 r- a t p o i n t R i n F i g . 3-16 FI G . 3-15 DEFINITION SKETCH FOR POINT R IN FIG. 3-16 surge F I G . 3-17 SCHEMATIC DIAGRAM OF CHARACTERISTIC GRIDS FOR THE NEGATIVE WAVE AT POINT R 84 FIG. 4-1 DEFINITION 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 9 0 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 POSITIVE 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 0-3 0-4 0-5 0-6 FIG. 4-10 (a) VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A RECTANGULAR POWER CANAL 01 0-2 0-3 0 4 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 0-3 0 4 0-5 0 6 FIG. A-10 (c) VARIATION OF WAVE HEIGHT OF A NEGATIVE SURGE PROPAGATING ALONG A RECTANGULAR POWER CANAL (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 0 4 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 800 I 900 I 1000 —I— 1100 1200 1300 O U> FIG. 4-15 t . second VARIATION OF WATER SURFACE AT THE DOWNSTREAM END WITH RESPECT TO TIME FOR THE RECTANGULAR CANAL OF y Q - 30 f t . . h Q - 30 f t . AND n - 0.03095 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 FIG. 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 , 1959-7. 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 in 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 Listing C E X A M P L E 1 5 . 6 C O P E N C H A N N E L S O L U T I O N BY C H A R G R I D METHOD IN R E C T A N G U L A R C H A N N E L D I M E N S I O N V ( 2 2 t l O O ) , Y ( 2 2 , 1 0 0 ) , X ( 2 2 , 1 0 0 ) , T ( 2 2 , 1 0 0 ) W R I T E ( 6 , 1 0 0 ) W R I T t ' ( 6 , 1 0 1 ) R E A D ( 5 , 3 0 0 ) Q O , B , G » H L , D X , H N , SO Y 0 = 8 . R O = Y O * B / { B + 2 . * Y 0 ) V 0 = l . 4 8 6 * R Q * * 0 „ 6 6 7 * S Q R T ( S 0 ) / H N Y O N E W = Q O / ( B*V0) I F ( ( Y O N E W - Y O ) . L T . 1 . 0 E - 3 ) GO TO 3 YO=YONEW T~ GO TO 2 YO=YONEW V O = Q O / ( B * Y O ) DO 10 1 = 1 , 2 1 , 2 V{ I , I ) = V O Y ( I , I ) = Y O i d T ( I , I ) = 0 . . . . . . . C O N T I N U E X ( I , I ) = 0 . DO 11 1 = 3 , 2 1 , 2 J = l I M = I - 2 . .X ( I , 1 ) = X ( I M , 1 )+DX _ ..... C O N T I N U E W R I T E ( 6 , 1 0 2 ) J , ( V ( K , 1 ) , K = 1 , 2 1 , 2 ) W R I T E { 6 , 1 0 3 ) ( Y ( K , l ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 4 ) ( X ( K , 1 ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 5 ) ( T ( K , 1 ) , K = l , 2 1 , 2 ) £._ F I N D THE S O L U T I O N IN THE M I D D L E . S E C T I O N OF C H A N JEL J = 2 * 8 0 DO 4 0 1 = 2 , 2 0 , 2 I M = I - i J M = J - 1 I N= I + I G R = S O R T ( G / Y ( I M » J M ) ) GS = S f J R T ( G / Y ( I N , J M ) ) R G = S Q R T ( G * Y ( I M , J M ) ) S G = S O R T ( G * Y ( I N , J M ) ) S S N U M = H N * H N * V ( I N , J M ) * A B S ( V ( I N , J M ) ) RS = B*Y{ I N , J M ) / ( B + 2 . * Y ( I N , J M ) ) S S D E N 0 = 2 . 2 1 * R S * * l . 3 3 3 G N S = G * ( S S N U M / S S O E N O - S O ) R R = 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 . 3 3 3 J- 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 40 _ 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) C 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 ) 18 VP=Q/(B*YP) VPS=VP-V ( 2, J )  YPNEW=Y(2,J)MVPS+GNS*TPS)/GS IF(ABS(YPNEW-YP) .LT. l.OE-3) GG TO 19 YP = YPNEW . _ _ _ _ _ _ _ _ GO TO 18 19 Y(1,JJ)=YPNEW V(1,JJ)=Q/(B*Y( 1, JJ) )  C 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( IN,JM ) -X( IM , JM ) +T( IM,JM)*(V(IM,JM)+KG)-T(IN,JM)*(V( IN,JM)-LIZ 1 S G ) 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 T P R = T ( I , J J ) - r ( I M , J M ) X ( i ; j J } = X ( I M , J M ) + ( V ( I M , J M ) + R G ) * T P R  T P S = T ( I , J J ) - T ( I N , J M ) V R S = V ( I M , J M ) - V ( I N , J M ) 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.± GNS *T P. S ) / ( G R.+ G S ) Y P R = 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 5 0 C O N T I N U E  C F I N D T H E S O L U T I O N AT T H E D O W N S T R E A M E N D OF C H A N N E L X ( 2 1 , J J ) = H L 1 = 2 1 ._ _ I M = I - 1 R G = S Q R T ( G * Y ( I M , 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 ) ) T P R = T ( 2 1 , J J ) - T ( I M , J ) RR = B * Y ( I M , J ) / ( B + 2 . * Y { I M, J ) ) . R R D E N O = 2 . 2 1 * R R * * 1 . 3 3 3 R R N U M = H N * H N * V ( I M , J ) * A B S ( V ( I M , J ) > G N R - G * ( R R N U M / R R D E N O - S O )  Y ( 2 l , J J ) = Y 0 Y P R = Y ( 2 1 , J J ) - Y ( I M , J ) V ( 2 1 , J J ) = V ( I M , J 1 - G R * Y P R - G N R * T P R .. .. W R I T E ( 6 , 1 0 2 ) J J , ( V ( K , J J ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 3 ) ( Y ( K , J J ) , K = 1 , 2 1 , 2 ) W R I T E ( 6 , 1 0 4 ) ( X ( K , J J ) , K = 1 , 2 I , 2 )  W R I T E ( 6 , 1 0 5 ) ( T ( K , J J ) , K = l , 2 l , 2 ) I F ( J . G T . 3 2 ) GO TO 7 0 . J = J + 2 ; .. ; „.* GO TO 8 0 3 0 0 F O R M A T ( 7 F 8 . 0 ) 1 0 0 F O R M A T ( 4 X , I H I , 9X , 1 H L, 9 X , 1 H2 , 9 X , 1H3 , 9 X , I H 4 , 9 X , 1.H5 , 9 X , 1H6 , 9 X , 1 H 7 , 9 X , 1 I H 8 , 9 X , 1 H 9 , 8 X , 2 H 1 0 , 8 X , 2 H 1 1 ) 1 0 L F O R M A T ( I X , 1 H J / / ) J L 0 2 . F 0 R M A T ( I 3 , 4 X , I H V , 1 1 F 1 0 . 3 ) . 1 0 3 F O R M A T ( 7 X , 1 H Y , 1 I F 1 0 . 3 ) 1 0 4 F O R M A T { 7 X , 1 H X , I I F 1 0 . 3 ) 1 0 5 F 0 R M A T ( 7 X , l H r t l l F 1 0 . 3 //)  1 0 6 F O R M A K 1 3 , 4 X , 1H V , 6 X , 1 OF 10 . 3 ) 1 0 7 F O R M A T ( 7 X , 1 H Y , 6 X , 1 OF 1 0 . 3 ) 1 0 8 F O R M A T ( 7 X , 1 H X , 6 X , 1 0 F 1 0 . 3 ) _._ . 1 0 9 F O R M A T ( 7 X , 1 H T , 6 X , 1 0 F I O . 3 / / ) ' 7 0 S T O P E N D $ E N T R Y 113 APPENDIX A(2) Program Listing C E X A M P L E 1 5 . 5 C O P E N C H A N N E L S O L U T I O N DY C H A R G R I D M E T H O D I N R E C T A N G U L A R C H A N N E L D I M E N S I O N V ( 2 2 f ' 1 0 0 ) , Y ( 2 2 , 1 0 0 ) , X ( 2 2 , 1 0 0 ) , T ( 2 2 , l O O ) W R I T E ( 6 , 1 0 0 ) W R I T E ( 6 , 1 0 1 ) R E A D ( 5 » 3 0 0 ) Y O , B , G, H L , D X , A O , P O , H N , SO Q O = 1 3 2 . * { Y O - 2 . 3 2 ) * * 1 . 5 V O = 0 0 / A O DO 1 0 1 = 1 , 2 1 , 2 V ( I , l ) = V O Y ( I , 1 ) = Y O T ( I , I ) = 0 . 1 0 C O N T I N U E X ( 1 , 1 ) = 0 . DO 1 1 1 = 3 , 2 1 , 2 J = l I M = I - 2 X ( I , I ) = X ( I M , 1 ) + DX 1 1 C O N T I N U E W R I T E ( 6 , 1 0 2 ) J , ( V ( K , 1 ) , K =1, 2 1 , 2 ) W R I T E ( 6 , 1 0 3 ) ( Y ( K , 1 ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 4 ) ( X ( K , l ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 5 ) ( T ( K , 1 ) , K M , 2 1 , 2 ) C 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 J = 2 8 0 DO 4 0 1 = 2 , 2 0 , 2 I M = I - 1 J M = J - 1 I N = I + l G R = S O R T ( G / Y ( I M , J M ) ) G S = S Q R T ( G / Y { I N , J M ) ) R G = S Q R T ( G * Y ( I M , J M ) ) S G = S Q R T { G * Y ( I N , J M ) ) S S N U M = 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 ) ) S S D E N O = 2 . 2 1 * R S * * l . 3 3 3 G N S = G * ( S S N U M / S S D E N O - S O ) RR = B * Y ( I M , J M ) / ( B + 2 . * Y ( I M , J M ) ) R R D E N 0 = 2 . 2 1 * R R * * 1 . 3 3 3 R R N U M = H N # H N * V l I M , J M ) * A B S l V ( I M , J M ) ) G N R = G * ( R R N U M / R R D E N O - S O ) T P N U M = X ( I N , J M ) - X U M , J M ) + T ( I M , J M ) * ( V ( I M , J M J + R G ) - T ( I N , J M ) * ( V ( I N , J M ) -1 S G ) T P D E N O = V ( I M , J M ) + R G - V ( I N , J M ) + S G T ( I , J ) = T P N U M / T P D E N O T P R = T ( I , J ) - T ( I M . J M ) X ( I , J ) = X ( I M , J M ) + ( V ( I M , J M J + R G ) * T P R T P S = T ( I , J ) - T { I N , J M ) V R S = V ( I M , J M ) - V ( I N . J M ) JL JLH-X { I , J J ) = X ( I M , J M ) + ( V ( I M , J M ) + R G ) # T P R T P S = T ( I , J J ) - T ( I N , J M ) V R S = V ( I M , J M ) - V ( I N , J M ) Y ( I , J J ) = ( V R S + G R * Y ( I M , J M ) + G S * Y ( I N , J M ) - G N R * T P R + G N S * T P S ) / ( G R + G S ) Y P R = 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 5 0 C O N T I N U E .... c F I N D T H E S O L U T I O N . AT. T H E . D O W N S T R E A M . EN.D_.OF C H A N N E L '_ ... X ( 2 1 , J J ) - H L 1 = 2 1 I M = I - 1 RG= S Q R T ( G * Y ( I M , J ) ) T ( 2 1 , J J ) = T ( I M , J ) + ( H L - X [ I M , J ) ) / ( V ( I M , J ) + R G ) G R = S Q R T ( G / Y ( I M , J ) ) T P R = T ( 2 1 , J J ) - T ( I M , J ) RR = B * Y M M , J ) / ( B + 2 . * Y ( I M , J ) ) R R D E N 0 = 2 . 2 1 * R R # * l . 3 3 3 R R N U M = H N * H N * V ( I M , J ) * A B S ( V ( I M , J ) ) G N R = G * ( R R N U M / R R D E N O - S O ) Y P = Y ( I M , J ) . 2 7 Y P 3 = ( Y P - 2 . 3 2 ) * * 3 V P = 1 3 2 . * S Q R T ( Y P 3 ) / ( B * Y P ) V R P = V ( I M , J ) - V P Y P N E W = Y ( I M , J ) + ( V R P - G N R * T P R ) / G R I F ( A B S ( Y P N E W - Y P ) . L T . 1 . 0 E - 3 ) GO TO 2 6 Y P = Y P N £ W GO TO 2 7 2 6 Y ( 2 1 , J J ) = Y P N E W Y P 3= ( Y ( 2 1 , J J ) - 2 . 3 2 ) * * 3 V ( 2 1 , J J ) = i 3 2 . * S Q R T ( Y P 3 ) / { B * Y ( 2 i , J J ) ) W R I T E ( 6 , 1 0 2 ) J J , ( V ( K , J J ) , K = l , 2 1 , 2 ) W R I T E ( 6 , 1 0 3 ) ( Y ( K , J J ) , K = 1 , 2 1 , 2 ) WRT T E ( 6 , 1 0 4 ) .( X ( K, J J ) , K= 1 , 2 1 ,2 ) W R I T E ( 6 , 1 0 5 ) ( T ( K , J J ) , K = l , 2 1 , 2 ) I F ( T ( 1 , J J ) . G T . 2 4 0 0 . ) GO TO 7 0 J = J + 2 GO TO 8 0 3 0 0 F O R M A T ( 9 F 8 . 0 ) 1 0 0 F O R M A T ( 4 X , i H I , 9 X , 1 H 1 , 9 X , 1 H 2 , 9 X , 1 H 3 , 9 X , 1 H 4 , 9 X , I H 5 , 9 X , 1 H 6 , 9 X , 1 H 7 , 9 X , 1 1 H 8 . 9 X , I H 9 , 8 X , 2 H 1 0 , 8 X , 2 H 1 1 ) 1 0 1 F O R M A T ( I X , 1 H J / / ) 1 0 2 F 0 R M A T ( I 3 , 4 X , 1 H V , 1 1 F 1 0 . 3 ) 1 0 3 F O R M A T ! 7 X , 1 H Y , 1 I F 1 0 . 3 ) 1 0 4 F O R M A T ( 7 X , 1 H X , U F 1 0 . 3 ) 1 0 5 F O R M A K 7 X , 1 H T , 1 1 F 1 0 . 3 / /) 1 0 6 F O R M A T ( 1 3 , 4 X , 1 H V , 6 X , 1 0 F 1 0 . 3 ) . 1 0 7 F O R M A T ( 7 X , 1 H Y , 6 X , I O F 1 0 . 3 ) 1 0 8 F O R M A T ( 7 X , 1 H X , 6 X , 1 0 F 1 0 . 3 ) 1 0 9 F O R M A T ! 7 X , 1 H T , 6 X , 1 0 F 1 0 . 3 / / ) . . 7 0 S T O P E N D S E N T R Y 115 i Y ( I , J ) = ( V R S + G R * Y ( I M , J M ) + G S * Y { I N , J M ) - G N R * T P R + G N S * T P S ) / ( G R + G S ) Y P R = Y ( I , J ) - Y ( I M , J M ) V ( I , J ) = V ( I M , J M ) - G R * Y P R - G N R * T P R 40 C O N T I N U E W R I T E ( 6 , 1 0 6 ) J , ( V ( K , J ) , K = 2 , 2 0 , 2 ) W R I T E C 6 . 1 0 7 ) ( Y ( K , J ) , K = 2 , 2 0 , 2 ) W R I T E ( 6 , 1 0 8 ) ( X ( K , J ) , K = 2 , 2 0 , 2 ) • W R I T E { 6 , 1 0 9 ) ( T ( K , J ) , K = 2 , 2 0 , 2 ) C F I N D T H E S O L U T I O N AT THE U P S T R E A M END OF C H A N N E L J J = J + l X ( 1 , J J ) = 0 . 1 = 1 I N = I + i SG= S Q R T { G * Y ( I N , J ) ) T ( 1 , J J ) = T ( 2 , J ) - X ( 2 , J ) / { V ( 2 , J ) - S G ) I F { T ( i , J J ) . G T . 1 7 9 9 . ) GO TO 17 D Q = 0 0 * T { 1 , J J ) / 1 2 0 0 . I F ( T ( 1 , J J ) . L T . 1 1 9 9 . ) GO TO 16 DO = Q O * ( T ( I , J J ) - 1 2 0 0 . ) / 1 2 0 0 . Q = 2 . * Q 0 - D Q _ _ . _ GO TO 17 1 6 Q=QO+DQ 1 7 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 ) ) .... - — — S S D E N O = 2 . 2 1 * R S * * 1 . 3 3 3 _ S S N U M = H N * H N * V ( 2 , J ) * A B S ( V ( 2 , J ) ) G N S = G * ( S S N U M / S S D E N O - S O ) Y P = Y ( 2 , J ) 1 8 V P = Q / ( B * Y P ) V P S = V P - V ( 2 , J ) Y P N E W = Y { 2 , J ) + ( V P S + G N S * T P S ) / G S I F { A B S ( Y P N E W - Y P ) . L T . l . O E - 3 ) GO TO 19 Y P = Y P N E W GO TO 18 19 Y ( I , J J ) = Y P N E W V( 1 , J J ) = Q / ( R * Y ( 1 , J J ) ) ; C F I N D THE 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 I M = I - 1 J M = J J - 1 I N = I + .l G R = S Q R T ( G / Y ( I M , J M ) ) G S = S O R T t G / Y ( I N , J M ) ) ' _ _ _ _ _ R G = S Q R T ( G * Y ( I M , J M ) ) S G = S Q R T ( G * Y ( I N , J M ) ) S S N U M = 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 ) ) S S D E N O = 2 . 2 i * R S * * l . 3 3 3 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 ) ) R R D E N 0 = 2 . 2 l * R R * * l . 3 3 3 R R N U M = H N * H N * V ( I M , J M ) * A B S ( V ( I M , J M ) ) ;  G N R = G * ( R R N U M / R R D E N O - S O ) 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 ) -I S G ) _ _ ;. ._ 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 T P R = T ( I , J J ) - T ( I M , J M )  1 1 6 APPENDIX B Program Listing C . SURuE. WAVES I N THE POWER CANALS C WFP I S THE CONTROL PARAMETER IE P O S I T I V E OR N E G A T I V E W A V E C B U = W 1 ' L) T H OF W . S . AT YO S E C T I O N C b= M lLJTII OF GHANNcL BOTTOM DIMENSION 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 » R O » h K E I » G A M A » H I T N » R E F P T » X M » l 3 » W F P 101 F u R i - - i A T ( S F l O . O ) I r ( K O . L T . 0 . 0 ) SI OP J u I i - i - O 0 Q - 0 . 0 H u E G = - 1 . 0 UPE<C>=-1. 0 G = 3 2 . 2 N I M = K O * * C . 6 6 6 7 / H K E i IF (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 - X H * Y O / G A M A Gu TO 1 1 5 2 1 1 5 1 I r (A. ' - ' . L E . U . U ) bO TO 1 1 5 3 Y G = _ . 0 * R O * S U K T ( i . u + X M * X M ) / X M Gu TO 1 1 5 2 1 1 5 3 Y u = K O * B / ( D - 2 . 0 * K O ) 1 1 5 2 Vu=F*5QRTlG*YO) 5 u = l V 0 / ( 1 . 4 6 6 + H K E _ ) )**2 B u - o + 2 . 0 * Y O*XM .YOBAR=YO* 180+2• 0 * _ ) / (3• 0 * ( 8 0 + B ) ) A u = l!fi + Y O * X M ) ' * Y O GC-VO+AO Po=b+2.0*Y0*SGRT ( i .0+XM*XM) X L = Y O / S O r iLF = 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 140 FuRi-iAT ( 5X t 3 i i F = > F b . 3f2 X » 4 H RO= t F 5 • 1» 2X t 3l i K = t F6 . 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 REFPT= » F 5 . 2 r <+H XM=» F<+* 1» 5X* 3H B = » 2 H 6 . 1 » 2 X » 4 H Y G = r F o . 2 » 2 X » 4 H V 0 = » F 6 « 2 / / > C N E G A T I V E WAVE R E F L E C f A T RESERVOIR ( CONSTSTANT VALUES OF X / L O ) C Ti iE Nuf-iBER U F H i T N nAS TO BE CHANGED AS HL CHANGE:. C H I T i j-uO. O F REACHES O F CHANNEL D I V I D E D C 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 H I _ = A L + R E F P T I O - I i l O + L ; . 1 IuN- I .O+1 I u M = 1 0 - l I ^ A - I O N I r ( i C N . G T . 2 0 ) GU '10 4 K i = l Gu 10 5 4 K i = i O N - 2 0 5 Ki*=r\I+-l Xu=u.u Du 10 1 = 1 r I UNr 2 I I / V ( I » 1 ) = V 0 Y d ' 1 ) = Y 0 T i l » l ) = 0 . 10 C O N T I N U E X ( l » 1 ) = 0 . NST=0 J = l C LOCATION OF WAVES NWPT Nv,P"l = ION+i C h E C K = 3 . DO i l 1 = 3» IONr 2 i,-,= i - ;_ X i I t l ) - A ( I M r 1 ) + D X 11 CuNlINUE J=2 NJ=u 80 Ti_ST = - 1 . 0 Do o l LL=2f 10' 2 l = I O - L L + 2 I i v= l -1 l N = i + l J I » 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 ) 81 CALL MID ( X t Si t T f Y t RG » SG » GS » GR * GNRf GNS » I * J » IN » J M * iM» JM ) C FIND THE WAVE AT DGWwST. END J = J + l I = IuN l M = i - l l n = _ + l J i ' i = v J - l XU»J)=HL V( I »J)=0.J C=SGRT(G*YO) 2u = u . <4*Y0 BP=oO+2.0*Z0*XM bi-,=U . 5* ( B G + D P ) 71 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 Y p B A R = Y P * ( B P+2 . u + o ) / ( b . U * ( B P + B ) ) C w U M = ( A P * r P b A R - M O * Y G B A R ) * 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 . 0.0U1) GO TO 72 C=0.5*(CNLW+C) G O r o 71 72 YlI»J)=YP v . . = c - v o V.*0=VW Zo=Y ( I»J ) -YO 3 1 3 N,*p:i = i N.,Av = I X P R = X ( I t J ) - X ( I M tJM) BK=O+2.0 * Y ( i M tJM)*XM 1 1 8 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 7X»2H 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 * 4FlU•t»K12•£ » Flu • 4 rFd. irF8.3 ) C PuNt-H 137 t F» Ur L..E i Al » Y O P Zf \/W» L>» GAMAfHKEl f XX C F I N I J 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+bIP)+ 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 / V L L - GNRS ( 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 I i w + 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 JDtN0=V ( iM» JM) + Ru-V (IN # J M ) +SG T£ I ' J) = T P I M U I ' : . / I P U E N O IF( TP .Gl'. T d ' J ) ) GO TO 22 45 K-N,vAV-i L-ION+3-(K-NST)+N3T-NJ*2 1 62 IF i NWPT .NE. iON ) GO TO 24 C H L L . W A V O G N (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 24 C M L L . 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 L-l»I-l» 1 s j - i t VW i NST t XM f M O » Y O B M R ) 25 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) C 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 NnPi "= I 35 IK=NWAV-Nrt-PT I F ( i K .LT. 1) GO TO 34 Do 06 I D - I t 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 C O N T I N U E 34 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 K N = K+1 LN= L-1 C M L L 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 ) C M L U 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 32 CONiINUE C FONo THE UNSTEADY F'LOw AT DOWNSTREAM END 33 K-IoN L=L+1 X(K»L)=HL V ( K t L ) = o . 0 C M L L . DOv.N(XrY»T»V»K»L»B»G»Hi->l»SO»XM) TT=T(Kf D-TTO Y t — i ( K » L ) / Y U W K I "I E (6 t 398 ) Y ( K # L ) i Y Y » TT 398 FuRi-'iAT (10UX» 3Flu . b ) C FIND T H E FLOW IN i'H_ FRONT OF WAVE 61 I r ( NWPi .Eo. 2) uO To 51 IF(UPEND .GT. 0.0) GO TO 200 1 = 1-2 64 IF( I .LT. NWPT ) 60 TO 63 1 = 1-2 Gv^  TO 64 63 I F ( Y U + l r J - D - G i . YG) GO TO 45 Go TO 26 22 Hu-J 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 C 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 C A L L G N R S ( I » J * G » V » Y » X f T » t 3 » H N t S O r I M » J M » G R » R G t G N R f X M ) C A L L 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 r G N S » X M ) 2 7 . C A L L 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 ) J c V L N = - l J o D D = 2 C F I N o I H E F L O W I N T H E U P P E R S T R E A M E N D 1 = 1 X ( 1 1 J ) = 0 . u Y ( I > J ) = Y O C A L L U P S T ( G » X » Y » V » T r I r J » B i H N t S O F X M ) G o 1 0 3 1 2 6 J t _ V t - N = 2 J o D U = - l L L P = I 0 + 2 - I DO 2 9 L L = L L P t 1 0 1 2 I = I G - L L + 2 l> ,= i - l l . . = i + l J M = J - 1 C A L L . G N R S ( I » J » G r V » Y » X » T » 0 » H N r S O F I M » J M t G R » R G F G N R > X M ) C M L L 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 ) 2 9 C A L L M I D ( A » V » T » Y t R G »so F G S F G R » G N K » G N S » I » J » I N * J M H M » J M ) 3 1 J = J + 1 I = I O N + 3 - J - N J * 2 C G O Q A C K T O T H E v « A v E F R O N T C C H A N G E T H E V A L U E S O F J I F I J . L E . 6 ) GO TO 3 9 N o =T-4 J + l C J - J - 2 l K = i 0 N + 3 - ( 6 + ( N J - l J * 2 ) D o 4 2 K = 2 F I K F 2 4 2 C A L L V Y X T ( K # 6 » V » Y » X » T > K = I K L = 6 K K = I \ + 1 D o 4 3 K = K K » I O N L = L + 1 I F ( L . G E . J D I M ) GO T O 6 3 0 4 3 C K L L V Y A T ( K » L » V f Y » X » T ) 6 3 0 I r v = l O N + 5 - I 5 + ( N J - 1 ) * 2 ) D o o 7 K = 1 » I K » _ 3 7 C M L L V Y X T v K » 5 ' V » Y r X » T ) K = I K L=5 Kr \= i \ + 1 D o 3 8 K = K K » I O N L = L + 1 1 F ( L . G E . J D I M ) G o T O 6 4 0 3 8 C M L L V Y A T ( K » L » V I Y » X » T ) 6 4 0 L = J + ( I i J * 2 ) K = I G N + 3 - L K T = i O N - K - , v i S T I F I K S . L T . K l " ) G o T O 3 9 D o 4 7 K X = r \ T » K S K = I 0 M - K X 4 L - I O N + 3 - ( K + ( N J - 1 ) * 2 - M b T ) + N S T M - K N = L 6 8 I F ( A B S ( Y ( M - 1 » N - 1 ) - Y 0 ) . L T . O . D G O T O 6 9 C A L L V Y X T ( M - 1 » vi-1' V » Y » X » T ) |V|i = h - l N l = u - 1 M = M - 1 N - N - l G O T O 6 6 6 9 L L = u D O 4 7 K = K » I O N L - L + L L I i - ( L . GEo J b I M ) G O T O 39 L L = i 4 7 C A L L V Y X T ( K » L » V » Y r X » T ) 39 I=: Iu|-! + 3-( J + N J + 2 ) I F I ( N W A v ' - N S T ) . L E . I ) GO T O 6 1 K - I L - J Gc T O 6 2 5 1 I - l Ut-Ei\0 = l . U J = J + 1 G u T O 4 5 C N E G A j I V E W A V E S C R E S i _ T T H E V A L U E S A T P T . S 2 0 0 K = N . J A V L - l O N + 3 - ( K - N S T ) + N 3 T - N J * 2 H i \ E v i = - 1 . 0 iPR = l It=3 I b = b l f L - 2 I_ ;=4 I u - b V ( l b » 2 ) - v ( K + 2 ' L ) Y ( 1 S > 2 ) = Y ( K + 2 » L ) X ( I S r 2 ) = x ( K + 2 » i _ ) T i I » 2 ) = T ( K + 2 » L ) C R E S c T T H E V A L U E S M L O N G C + P A S S I N G R E F L E C T P T . K K = i C N - N W A V I J K L . = 0 D O 2 0 1 1 = 3 >5 r 2 I O K L = I J K L + 1 K , \ i = f\K + l Dc 2 0 1 L K = 1 » K N M = K + L K - i + I J K L - 1 N = L + L K - l - i J K L + 1 I F ( . M . G T . J D I M ) G o T O 2 U 2 J=L<\ V ( I t J ) = V ( M i N ) Y l l r J ) = Y ( i - i f i j ) X ( I » J ) = X ( | . i » N ) 2 0 1 T I 1 t J ) = T ( M » i \ i ) C F I N u I H E I N I T I A L N E G A T I V E W A V E S Gu T O 2 0 3 122 202- K ix——- K — 1 4 203 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 ( l r l ) = X ( 3 f 1 ) Til»l)=T(3»l) X ( I F L » J ) - X ( I PR » J ) T ( I PL *J)=T(IPRiJ) Y(IPLr J ) = Y O + v O * v O /(2.U*G) BPR= D+2.0* Y ( I P R f J ;*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 * ( B P R + B ) ) B P L = B + 2 • 0 * Y ( 1 P L , J ) * X M A P L = ( B + b P L ) + Y ( I P L , J ) / _ . 0 YPLoAR=Y(IPL» J ) * ( L > P L + 2. U * b ) / (3. U* ( B P L + B > ) V f i = V ( I P R t o ) + S O R T ( b * ( A P L * Y P L B A R - A P R * Y P R 3 M R ) * A R L / I A P R * ( A P L - A P R ) ) ) V ( I r , L » J ) - ( V ( I P R * J ) - V W ) * A P R / A R L + VW Z=Y(IPL» J ) - Y (iPRrb) D = A D S ( Z / Z O ) E = V w / V W G X A = H L + X ( I P R » J ) B c T A i = X X / x L T T - i ' ( I P h » d ) - T 7 0 WKliE(6»lb3) Z » V W» D» E r X X f B E T A 1 » TT f Y ( I P R »J) C 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 V C + » D = V ( _ » 1 ) Yit»l)=Y(2»i) X(4»D=X(2»i) Tl4»l)=T(_»i) 210 J = J + 1 C H L L . G N R S (I»J»G»V»Y»X»TrB» HN r S O r IS f J t GS» SG t GNS * XM) JK = vj-i C M L U G N R S 1 1 » J r G» V » Y » X » T # 3» HN r SO rIE» JR » GR » RG»GNR tXM) C M L L M I D ( X » V » T f Y I KG r SG F GS t G R F G N K t GNSF IE' Jr I S F J» X E F J R ) G FlNu X r T »AT # PT. P B P R = B + 2 . 0 * Y ( I P R 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*(BPR+B)) B P L = B + 2 • 0 * Y ( I PL > J , < ) * X N A P L = ( B + b P L ) * Y ( I P L tJR)/2.0 Y R L b A R = Y ( I P L F J P J * ( l i P L + 2 . 0 + B ) / ( 3 . 0 * ( B P L + B ) ) V , . - v ( 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 (IS»J) * (V (IS» b) -SG) ) / ( V wl 1 - V d S » J ) t S G ) 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)) C 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 ( I P R»J)=Y(lSrJ)+(Y(I_»J)-Y(IS » J))*(T ( I P R»J ) - T(iS»J ) ) / ( T(IE»J)-i T(iS»J)) v a d b J ) = v(IS»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 VC IPLr J ) - \ i i ipLr J - i ) X l I r ' L » J ) = X ( i P K » J ) T ( I P L r J ) = T ( 1 P R » J ) BPR=B+2.0*YUPRr J ) * X M A P R - ( B + B F M ) * Y ( I P R » J )/2.0 220 A P L = ( V ( I P R » J ) - V . ; ) * A P R / ( V ( I P L » J ) — V w ) I F ( O A M A . L E . 0.0) GO TO 1181 V ( IKL r J) = (-b+SGi<T ( B * B + ' + . 0 * X M * A P L ) ) / (2. 0 + XM) Go 10 1182 1181 Ii-'UM . L E . 0.0) Go TO 1133 Y(IPL»J)=SQRT(APL/XM) GO TO 1182 1183 Y l I P L f J ) = H P L / B C DETERMINE THE pr. OF Q 1182 I r ( J .GT. 2) GO To 221 H=Yu+VO*VO/(2.0*G) X ( I o r J ) = 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 ( I0 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)+(Vt iDrJ) -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 = Y I I P L » J ) - Y ( I P K » J ) D=AbS(Z/ZO) E-V../VW0 XA=i-iL + XUPRr J) B I _ T A 1 = X A / X L T~i = 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)-TV0 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 .oT . u.u) G O 10 1 Ir(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 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(IPLrjR ) * X i ' i ArJL=(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* ( APL-APR ) ) ) T(IPR » J) = I X(IPR » J)-X(iPrt» JR))/Vri + T(IPR tJR) YIIPR»J)=Y(IS»J-l)+(TtiPRrJ)-T(ISiJ-D)*(Y(IE»J ) - Y ( I S ' J - l ) ) / ( T ( I E 1 , J ) - T l I b r J - l ) ) Hi\lEG=l.U Go TO 27 0 137 F U R ; M A T ( F 6 . 3 ' 5 F 8 . < 4 F 3 F B . 2 # F 1 0 . 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}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            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:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0050557/manifest

Comment

Related Items