sediment transport in rivers
accuracy of measurements in a curved open
channel
H. J. de Vriend
report on experimental and theoretical
investigations
R 657-VII M 1415 part III March 1978, t o w
toegepast onderzoek waterstaatSUMMARY page J I n t r o d u c t i o n 1 2 E l a b o r a t i o n p r o c e d u r e 2 2.1 Magnitude and d i r e c t i o n o f t h e v e l o c i t y v e c t o r 2 2.2 T a n g e n t i a l and r a d i a l v e l o c i t y components 3 2.3 A v e r a g i n g o v e r t h e d e p t h o f f l o w 3 2.4 Magnitude and d i r e c t i o n o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r 3
2.5 Main and h e l i c a l v e l o c i t y components 4
2.6 N o r m a l i z a t i o n 4 3 Summary o f measured q u a n t i t i e s 5 4 Summary o f e r r o r s t a k e n i n t o account 7 5 N u m e r i c a l s i m u l a t i o n 10 5. 1 G e n e r a l o u t l i n e o f t h e methbd 10 5.2 Q u a n t i f i c a t i o n o f i n p u t d a t a 10 5.3 R e s u l t s 11 6 T h e o r e t i c a l a n a l y s i s 12 6. 1 G e n e r a l 12 6.2 Magnitude and d i r e c t i o n o f t h e v e l o c i t y v e c t o r 13 6. 3 T a n g e n t i a l and r a d i a l v e l o c i t y component 17 6.4 A v e r a g i n g over t h e d e p t h o f f l o w 18 6.4.1 E r r o r due t o t h e a p p l i c a t i o n o f t h e t r a p e z o i d a l r u l e ... 18 6.4.2 E r r o r s i n t h e depth-averaged v e l o c i t y components 19 6.5 Magnitude and d i r e c t i o n o f t h e depth-averaged v e l o c i t y v e c t o r 20
6.6 N o r m a l i z e d main and h e l i c a l v e l o c i t y components 21
6.7 N o r m a l i z a t i o n o f t h e e r r o r s 21 6.8 G e n e r a l i z a t i o n t o any v e r t i c a l 23
CONTENTS ( c o n t i n u a t i o n ) page 6.9 Q u a n t i f i c a t i o n o f t h e i n f l u e n c e f a c t o r s and f u n c t i o n s 25 6.9.1 S y s t e m a t i c e r r o r s 26 6.9.2 Random e r r o r s 27 6.10 Q u a n t i f i c a t i o n o f t h e e r r o r s 27 7 C o n c l u s i o n s 30 7.1 T h e o r e t i c a l a n a l y s i s v e r s u s n u m e r i c a l s i m u l a t i o n 30 7.2 C o n c l u s i o n s r e g a r d i n g t h e p r e s e n t measurements 30 REFERENCES APPENDIX A The e r r o r s i n a v e r t i c a l APPENDIX B I n f l u e n c e f a c t o r s and f u n c t i o n s
I I Q u a n t i t i e s v a r y i n g randomly f r o m v e r t i c a l t o v e r t i c a l I I I z - i n d e p e n d e n t q u a n t i t i e s v a r y i n g f r o m p o i n t t o p o i n t IV z-dependent q u a n t i t i e s v a r y i n g f r o m o b s e r v a t i o n t o o b s e r v a t i V R e s u l t s o f t h e n u m e r i c a l e r r o r a n a l y s i s V I S y s t e m a t i c e r r o r s due t o t h e t r a p e z o i d a l i n t e g r a t i o n r u l e V I I I n f l u e n c e f a c t o r s o f E { d Z j } and E {dC,} V I I I I n f l u e n c e f a c t o r s o f t h e random e r r o r s I X N o r m a l i z e d s y s t e m a t i c e r r o r i n V^.^^ X N o r m a l i z e d s y s t e m a t i c e r r o r i n A X I V a r i a n c e o f t h e n o r m a l i z e d random e r r o r i n V^ ^ — t o t X I I V a r i a n c e o f t h e n o r m a l i z e d random e r r o r i n A X I I I N o r m a l i z e d s y s t e m a t i c e r r o r i n V' . - m a m XIV N o r m a l i z e d s y s t e m a t i c e r r o r i n V^^^ XV V a r i a n c e o f t h e n o r m a l i z e d random e r r o r i n V' . - i n a i n XVI V a r i a n c e o f t h e n o r m a l i z e d random e r r o r i n V/ , — h e l X V I I I m p o r t a n c e o f t h e v a r i o u s sources o f e r r o r s
LIST OF FIGURES 1 Random e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 2 I n f l u e n c e f u n c t i o n s o f t h e s y s t e m a t i c e r r o r s due t o t h e t r a p e z o i d a l i n t e g r a t i o n r u l e on t h e s y s t e m a t i c e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 3 I n f l u e n c e f u n c t i o n s o f E {dZ } on t h e s y s t e m a t i c e r r o r s i n t h e n o r m a l i z e d — 1 v e l o c i t i e s 4 I n f l u e n c e f u n c t i o n s connected t o E {dC } on t h e s y s t e m a t i c e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 5 I n f l u e n c e f u n c t i o n s o f E {dH} on t h e s y s t e m a t i c e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 6 I n f l u e n c e f u n c t i o n s o f Var {dZ } on t h e v a r i a n c e s o f t h e random e r r o s i n — 1 t h e n o r m a l i z e d v e l o c i t i e s
7 I n f l u e n c e f u n c t i o n s connectec t o Var {dC } on t h e v a r i a n c e s o f t h e random — 1 e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 8 I n f l u e n c e f u n c t i o n s o f Var {dH} on t h e v a r i a n c e s o f t h e random e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 9 I n f l u e n c e f u n c t i o n s o f Var { d Z } on t h e v a r i a n c e s o f t h e random e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s
10 I n f l u e n c e f u n c t i o n s o f Var { $ ' } and Var { d $ ' } on t h e v a i a n c e o f t h e random — k e r r o r i n t h e n o r m a l i z e d h e l i c a l v e l o c i t y 11 I n f l u e n c e f u n c t i o n s o f Var {N} on t h e v a r i a n c e s o f t h e random e r r o r s i n t h e n o r m a l i z e d v e l o c i t i e s 12 R e s u l t i n g s y s t e m a t i c e r r o r s 13 V a r i a n c e s o f t h e r e s u l t i n g random e r r o r s
a^ C o e f f i c i e n t i n t h e t r a p e z o i d a l i n t e g r a t i o n r u l e A V a l u e o f a computed f r o m t h e measured d a t a C o e f f i c i e n t i n t h e a p p r o x i m a t i o n o f t h e s y s t e m a t i c e r r o r due t o t h e t r a p e z o i d a l i n t e g r a t i o n r u l e C C h e z y - f a c t o r C j , C g , c^ C o e f f i c i e n t s i n t h e r a t i n g f o r m u l a f o r t h e v e l o c i t y meter Cg C o e f f i c i e n t i n t h e r a t i n g f o r m u l a f o r t h e d i r e c t i o n meter Cg V a l u e o f Cg a t t h e c a l i b r a t i o n t e m p e r a t u r e 0 Cj V a l u e o f c^ f o u n d f r o m c a l i b r a t i o n o f t h e v e l o c i t y meter C o v { x , y } C o v a r i a n c e between t h e random v a r i a b l e s x and y
dC^ D i f f e r e n c e between t h e a c t u a l and t h e c a l i b r a t e d v a l u e o f c j dH E r r o r i n t h e measured d e p t h o f f l o w
dz D e v i a t i o n i n t h e l e v e l o f a p o i n t
dZ E r r o r i n t h e measured v e r t i c a l d i s t a n c e between a p o i n t and t h e l o w e s t m e a s u r i n g p o i n t i n t h e r e l e v a n t v e r t i c a l
dZ' = d Z j + dZ E r r o r i n t h e measured l e v e l o f a p o i n t
dZj E r r o r i n t h e measured l e v e l o f t h e l o w e s t m e a s u r i n g p o i n t d$' E r r o r i n t h e measured f l o w d e v i a t i o n a n g l e due t o t h e keyway
b a c k l a s h E { x } E x p e c t a t i o n o f t h e random v a r i a b l e X f ^ ^ ^ ( 5 ) V e r t i c a l d i s t r i b u t i o n f u n c t i o n o f t h e e x p e c t e d f l o w d e v i a t i o n a n g l e f E x p e c t e d v a l u e o f f ^ ^ ^ ( C ) i n t h e m e a s u r i n g p o i n t i f . (C) V e r t i c a l d i s t r i b u t i o n f u n c t i o n o f t h e e x p e c t e d main v e l o c i t y m a i n ^ ^ ' f . Expected v a l u e o f f , (C) i n t h e m e a s u r i n g p o i n t i main£ ^ m a i n ^ ' f ^ ( 5 ) V e r t i c a l d i s t r i b u t i o n f u n c t i o n o f Var { N } f ^ , ( C ) V e r t i c a l d i s t r i b u t i o n f u n c t i o n o f Var } f ( C ) . . . f , , (C) I n f l u e n c e f u n c t i o n s f o r t h e e r r o r s i n v' . and v ' , 1 1 ^ mam h e l F j . . . F j j I n f l u e n c e f a c t o r s f o r t h e e r r o r s i n v ^ ^ ^ and a g A c c e l e r a t i o n due t o g r a v i t y h L o c a l d e p t h o f f l o w H Measured v a l u e o f h i Measuring p o i n t i n d e x
n Number o f p u l s e s t o be counted by t h e v e l o c i t y meter N Number o f p u l s e s counted by t h e v e l o c i t y meter
n ( z . ) V e r t i c a l d e r i v a t i v e o f n i n p o i n t i a t l e v e l z=z.
LIST OF SYMBOLS ( c o n t i n u a t i o n )
np Number o f m e a s u r i n g p o i n t s i n a v e r t i c a l R L o c a l r a d i u s o f c u r v a t u r e
T D u r a t i o n o f t h e o b s e r v a t i o n p e r i o d
temp^ Water t e m p e r a t u r e d u r i n g t h e c a l i b r a t i o n o f t h e v e l o c i t y meter temp^ Water t e m p e r a t u r e d u r i n g t h e measurements
v ^ ^ ^ H e l i c a l v e l o c i t y component ( i . e . , t h e h o r i z o n t a l component o f t h e v e l o c i t y p e r p e n d i c u l a r t o t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r ) ^ V,' , N o r m a l i z e d h e l i c a l v e l o c i t y = J^^^ ^ h e l ^ ^ h t o t
^main Main v e l o c i t y component ( i . e . , t h e h o r i z o n t a l component o f t h e v e l o c i t y i n t h e d i r e c t i o n o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r ) V . I ,. , , main V . N o r m a l i z e d mam v e l o c i t y = — mam v^ ^ t o t v ^ R a d i a l v e l o c i t y component ( i . e . , p e r p e n d i c u l a r t o t h e c h a n n e l a x i s ) v ^ Depth-average o f v ^ Vj. Measured v a l u e o f v ^ i n p o i n t i
V^'^ Depth-average o f V„ computed f r o m V^-, ( i = 1 ,. . . ,np) by use o f t h e t r a p e z o i d a l r u l e v ^ T a n g e n t i a l v e l o c i t y component ( i . e . , p a r a l l e l t o t h e c h a n n e l a x i s ) Vj. Depth-average o f v ^ V*. Measured v a l u e o f v. i n p o i n t i _ i L
Vj. Depth-average o f V^ computed f r o m V(- _ ( i = l ,. . . ,np) by use o f t h e t r a p e z o i d a l r u l e
v ^ ^ ^ T o t a l v e l o c i t y
Vj.^^ T o t a l d e p t h - a v e r aged v e l o c i t y V^ ^ Measured v a l u e o f v ^ ^ i n p o i n t i _tot£ t o t ^
V^^^ Depth-averaged v e l o c i t y computed f r o m V^ and V^
^ t o t ~ ^ t o t " ^ t o t D i f f e r e n c e between t h e t o t a l v e l o c i t y i n t h e h i g h e s t measur-np i n g p o i n t and v ^ ^ ^
Var { X } V a r i a n c e o f t h e random v a r i a b l e X
Var { N } Depth-average o f t h e v a r i a n c e o f t h e random v a r i a b l e N Var } Depth-average o f t h e v a r i a n c e o f t h e random v a r i a b l e $' z V e r t i c a l d i s t a n c e t o t h e f i x e d bed
a D e v i a t i o n a n g l e o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r 3, = b,/h^ Cg L e v e l a t w h i c h t h e t h e o r e t i c a l v e l o c i t y becomes e q u a l t o z e r o r)^ Dynamic v i s c o s i t y o f w a t e r a t t h e c a l i b r a t i o n t e m p e r a t u r e Dynamic v i s c o s i t y o f w a t e r a t t h e m e a s u r i n g t e m p e r a t u r e K Von Karman's c o n s t a n t u. Mean o f t h e random v a r i a b l e dC ^dc J — 1
E,. L e v e l between z = z._ and z = z.; d u r i n g t h e e l a b o r a t i o n : 5 = i ; ^ i > a.„ s t a n d a r d d e v i a t i o n o f t h e random v a r i a b l e dC, dCj — 1 a,r, S t a n d a r d d e v i a t i o n o f t h e random v a r i a b l e dZ dZ — O ,„ Standard d e v i a t i o n o f t h e random v a r i a b l e dZ, dZ J — 1 a„ S t a n d a r d d e v i a t i o n o f t h e random v a r i a b l e H n — a,, s t a n d a r d d e v i a t i o n o f t h e random v a r i a b l e ^ F l o w d e v i a t i o n a n g l e q)' Reading o f t h e f l o w d i r e c t i o n meter f Measured v a l u e o f (J)'
(t)J R e f e r e n c e r e a d i n g o f t h e f l o w d i r e c t i o n meter f o u n d when t h e vane i s p a r a l l e l t o t h e c h a n n e l a x i s
$J Measured v a l u e o f (j)^ <j)^ V e r t i c a l d e r i v a t i v e o f (j)'
SUMMARY
As a p a r t o f a r e s e a r c h programme on t h e f l o w and t h e bed t o p o g r a p h y i n r i v e r bends, a s e r i e s o f e x p e r i m e n t s on t h e v e l o c i t y d i s t r i b u t i o n i n s t e a d y f l o w
t h r o u g h a s h a l l o w c u r v e d open c h a n n e l has been e x e c u t e d a t t h e De V o o r s t La-b o r a t o r y o f t h e D e l f t H y d r a u l i c s L a La-b o r a t o r y _ 6 , 7 j . I n c o n n e c t i o n w i t h t h e s e e x p e r i m e n t s , t h e i n f l u e n c e o f v a r i o u s sources o f e r r o r s i n t h e measured d a t a on t h e a c c u r a c y o f t h e e l a b o r a t e d r e s u l t s i s i n v e s t i g a t e d i n t h e p r e s e n t r e -p o r t , b o t h by n u m e r i c a l s i m u l a t i o n and by t h e o r e t i c a l a n a l y s i s . The r e s u l t s o f t h e two methods a r e compared and show good agreement. S u g g e s t i o n s on how t o improve t h e a c c u r a c y o f t h e e l a b o r a t e d r e s u l t s a r e g i v e n .
I I n t r o d u c t i o n
D u r i n g t h e e x p e r i m e n t s on c u r v e d open c h a n n e l f l o w d e s c r i b e d i n P a r t s I and I I o f t h i s r e p o r t [ 6 , 7 ] , b o t h t h e m a i n v e l o c i t y f i e l d and t h e d i s t r i b u t i o n o f t h e h o r i z o n t a l component o f t h e secondary c i r c u l a t i o n were d e t e r m i n e d f r o m t h e measured m a g n i t u d e and d i r e c t i o n o f t h e t o t a l v e l o c i t y v e c t o r t h r o u g h o u t t h e
f l o w . As t h e secondary c i r c u l a t i o n was r a t h e r weak ( t h e t r a n s v e r s e v e l o c i t y component h a r d l y ever exceeded 5% o f t h e l o c a l main v e l o c i t y ) , h i g h demands had t o be made on t h e a c c u r a c y o f t h e measurements.
I n o r d e r t o f i n d o u t w h i c h o f t h e e r r o r s i n t h e measured d a t a a r e i m p o r t a n t t o t h e accuracy o f t h e e l a b o r a t e d r e s u l t s , t h e i n f l u e n c e o f v a r i o u s e r r o r s on t h i s accuracy has been a n a l y z e d i n t h i s p a r t o f t h e r e p o r t . T h i s was done b o t h a n a l y t i c a l l y , u s i n g a l i n e a r i z e d t h e o r y , and n u m e r i c a l l y , s i m u l a t i n g random e r r o r s by use o f a Monte C a r l o t e c h n i q u e , w h i c h p r o v i d e s t h e o p p o r t u n i t y t o compare t h e r e s u l t s o f t h e s e two e s s e n t i a l l y d i f f e r e n t approaches.
T h i s p a r t o f t h e r e p o r t was w r i t t e n by H.J. de V r i e n d o f t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y , w i t h t h e s u p p o r t , e s p e c i a l l y f o r t h e n u m e r i c a l a n a l y s i s , o f J.G. G r i j s e n o f t h e D e l f t H y d r a u l i c s L a b o r a t o r y .
2
-2 E l a b o r a t i o n p r o c e d u r e
The p r o c e d u r e used t o compute t h e main v e l o c i t y and t h e t r a n s v e r s e components o f t h e secondary v e l o c i t y ( t o be c a l l e d h e l i c a l v e l o c i t y component) f r o m t h e measured d a t a was d e s c r i b e d i n t h e f i r s t p a r t o f t h i s r e p o r t [_6J , b u t f o r t h e
sake o f c o m p l e t e n e s s , i t i s b r i e f l y summarized h e r e .
2.1 M a g n i t u d e and d i r e c t i o n o f t h e v e l o c i t y v e c t o r
The m a g n i t u d e o f t h e measured v e l o c i t y i s computed f r o m :
^ o t " ^1 ^ S
where: n = t h e number o f p u l s e s t o be c o u n t e d by t h e v e l o c i t y meter d u r i n g t h e o b s e r v a t i o n p e r i o d , t h i s number b e i n g e s t i m a t e d by t a k i n g t h e mean o f t h e numbers o f p u l s e s counted d u r i n g two o b s e r v a t i o n s i n
t h e r e l e v a n t measuring p o i n t T = d u r a t i o n o f t h e o b s e r v a t i o n p e r i o d c^ = s l o p e o f t h e v e l o c i t y r a t i n g l i n e s = s „ + % K -c, = v a l u e o f t h e v e l o c i t y c o r r e s p o n d i n g t o t h e p o i n t o f i n t e r s e c t i o n o f t h e v e l o c i t y r a t i n g l i n e and t h e v e l o c i t y a x i s c^ = c o e f f i c i e n t o f p r o p o r t i n a l i t y
= dynamic v i s c o s i t y o f w a t e r a t t h e measuring t e m p e r a t u r e temp^ = dynamic v i s c o s i t y o f w a t e r a t t h e c a l i b r a t i o n t e m p e r a t u r e temp^
The d i r e c t i o n o f t h e v e l o c i t y v e c t o r r e l e v a n t t o t h e c h a n n e l a x i s f o l l o w s from:
* = C3 ((])'-({>;) ( 2 )
where: c, = r a t i n g c o e f f i c i e n t ; i n t h e p r e s e n t case c, = 1 (J)' = r e a d i n g o f t h e f l o w d i r e c t i o n meter
= r e f e r e n c e r e a d i n g o f t h e f l o w d i r e c t i o n meter f o u n d when t h e vane i s p a r a l l e l t o t h e c h a n n e l a x i s
The v a l u e o f (J)' i s e s t i m a t e d by t a k i n g t h e mean o f t h e r e a d i n g s d u r i n g two ob-s e r v a t i o n ob-s i n t h e r e l e v a n t meaob-suring p o i n t .
2.2 T a n g e n t i a l and r a d i a l v e l o c i t y components The v e l o c i t y v e c t o r i s d i v i d e d i n t o a t a n g e n t i a l component v ^ ( p a r a l l e l t o t h e c h a n n e l a x i s ) and a r a d i a l component v ^ ( p e r p e n d i c u l a r t o t h e c h a n n e l a x i s ) : = V j . ^ ^ cos ^ ( 3 ) v ^ = v ^ ^ j . s i n (j) ( 4 ) 2.3 A v e r a g i n g o v e r t h e d e p t h o f f l o w
The v e l o c i t y components v ^ and v ^ a r e averaged o v e r t h e d e p t h o f f l o w , u s i n g t h e t r a p e z o i d a l r u l e w i t h z e r o v e l o c i t y a t t h e bed and z e r o v e r t i c a l d e r i -v a t i -v e a t t h e w a t e r s u r f a c e : np-1 z., - z. z + z „ 1 + 1 1 - 1 , r, np np-i-, ^ t " 2h \ . ^ - 2h > \ 1=1 1 np np-1 z, - z, z + z V 1-1 . r, np np-i-, \ • 2h \ . ^ - 2h ' \ 1=1 1 np
where: i = number o f t h e m e a s u r i n g p o i n t i n a v e r t i c a l ( i = 0 a t t h e bed; i = n p near t h e w a t e r s u r f a c e ) z. = t h e v e r t i c a l d i s t a n c e o f t h e measuring p o i n t i t o t h e bed (-0=0; \ p s h ) h = t h e l o c a l d e p t h o f f l o w 2.4 M a g n i t u d e and d i r e c t i o n o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r The m a g n i t u d e o f t h e t o t a l d e p t h - a v e r a g e d v e l o c i t y , v ^ ^ ^ , f o l l o w s f r o m : \ot = K'K^' (7: and t h e d i r e c t i o n a o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r w i t h r e s p e c t t o t h e c h a n n e l a x i s i s g i v e n by:
- 4 ~
a = a r c t a n (^^./v^) ( 8 )
2.5 Main and h e l i c a l v e l o c i t y components
The m a i n and t h e h e l i c a l v e l o c i t y components a r e f o u n d by d e c o m p o s i t i o n o f t h e measured v e l o c i t y v e c t o r i n a p o i n t i i n t o two components p a r a l l e l and p e r p e n -d i c u l a r t o t h e -d i r e c t i o n o f t h e -d e p t h - a v e r a g e -d v e l o c i t y v e c t o r .
% a i n £ = \ot. ^ * i
-V l i = ^ o t £ ("'i "
2.6 N o r m a l i z a t i o n
The m a i n and h e l i c a l v e l o c i t y components a r e n o r m a l i z e d by:
v' . = V , / v , ^ (11)
3 Summary o f measured q u a n t i t i e s
The q u a n t i t i e s measured d u r i n g o r i n r e l a t i o n t o t h e c u r v e d f l o w e x p e r i m e n t s can be d i v i d e d i n t o f o u r g r o u p s :
a G e n e r a l c o n s t a n t s ( i n v a r i a n t t h r o u g h o u t t h e e x p e r i m e n t s o r l a r g e p a r t s o f t h e m ) :
1/2 The two c o e f f i c i e n t s d e t e r m i n i n g t h e v e l o c i t y r a t i n g l i n e : c^ and c^; 3 t h e r a t i n g c o e f f i c i e n t s o f t h e f l o w a n g l e : c^;
4 t h e w a t e r t e m p e r a t u r e d u r i n g t h e c a l i b r a t i o n o f t h e v e l o c i t y m e t e r : temp^; and
5 t h e c o e f f i c i e n t o f p r o p o r t i o n a l i t y c^^ i n t h e c o r r e c t i o n t e r m o f c^ i n E q u a t i o n ( 1 ) , a c c o u n t i n g f o r t h e d e v i a t i o n f r o m temp^ o f t h e w a t e r
t e m p e r a t u r e temp^ d u r i n g t h e measurements. T h i s c o e f f i c i e n t was n o t d e t e r m i n e d f o r t h e p r e s e n t i n s t r u m e n t , b u t was e s t i m a t e d on t h e b a s i s o f e a r l i e r e x p e r i m e n t s 1 . b V e r t i c a l - c o n s t a n t s ( i n v a r i a n t i n each v e r t i c a l , b u t v a r i a b l e f r o m v e r t i c a l t o v e r t i c a l ) : 1 The r a d i a l c o o r d i n a t e R; 2 t h e l o c a l d e p t h o f f l o w h; and 3 t h e r e f e r e n c e a n g l e o f t h e d i r e c t i o n m e t e r (})^. c P o i n t - c o n s t a n t s ( i n v a r i a n t d u r i n g an o b s e r v a t i o n i n a p o i n t , b u t v a r i a b l e f r o m p o i n t t o p o i n t ) : 1 The v e r t i c a l d i s t a n c e t o t h e bed: ( i = 0 , 1 np; z^=0);
2 t h e d u r a t i o n o f t h e o b s e r v a t i o n p e r i o d : T, and as T was chosen c o n s t a n t (30 s; 60 s) f o r a l l m e a s u r i n g p o i n t s i n each p a r t o f t h e e x p e r i m e n t s , i t can be c o n s i d e r e d as a g e n e r a l c o n s t a n t ; and
3 t h e w a t e r t e m p e r a t u r e temp^, w h i c h d i d n o t p e r c e p t i b l y change d u r i n g t h e measurements i n a v e r t i c a l , and can t h e r e f o r e be c o n s i d e r e d as a v e r t i
-c a l - -c o n s t a n t .
d I n s t a n t a n e o u s l y v a r i a b l e q u a n t i t i e s (randomly v a r y i n g d u r i n g an o b s e r v a t i o n i n a p o i n t ) :
1 The i n s t a n t a n e o u s v e l o c i t y ; and 2 t h e i n s t a n t a n e o u s f l o w a n g l e .
6 As o n l y t h e t i m e a v e r a g e o f t h e s e q u a n t i t i e s i s c o n s i d e r e d , t h e y a r e r e -p l a c e d by: 1' The number o f p u l s e s n t o be c o u n t e d by t h e v e l o c i t y m e t e r d u r i n g t h e o b s e r v a t i o n p e r i o d ; and 2' t h e t i m e - a v e r a g e o f t h e f l o w d e v i a t i o n a n g l e ())'.
4 Summary o f e r r o r s t a k e n i n t o a c c o u n t M e a s u r i n g o r e s t i m a t i n g t h e q u a n t i t i e s m e n t i o n e d above i n t r o d u c e s s y s t e m a t i c and random e r r o r s , w h i l e e r r o r s a r e a l s o i n t r o d u c e d by u s i n g o b s e r v a t i o n s o f l i m i t e d d u r a t i o n t o d e t e r m i n e t h e t i m e - a v e r a g e d v e l o c i t i e s and f l o w d i r e c t i o n s . As a c c o u n t i n g f o r a l l p o s s i b l e e r r o r s w o u l d be v e r y l a b o r i o u s , t h e e r r o r s t h a t may be e x p e c t e d t o be o f m i n o r i m p o r t a n c e t o t h e r e s u l t s o f t h e e l a b o r a t i o n a r e n e g l e c t e d .
D i v i d i n g t h e e r r o r s i n t o t h e same f o u r groups as t h e q u a n t i t i e s t o be measured ( C h a p t e r 3) y i e l d s : a G e n e r a l s y s t e m a t i c e r r o r s ( i n v a r i a n t t h r o u g h o u t t h e e x p e r i m e n t s o r l a r g e p a r t s o f them). A l l o f these e r r o r s a r e v e r y s m a l l i n r e l a t i o n t o t h e t r u e v a l u e s o f t h e r e l e v a n t q u a n t i t i e s , e x c e p t t h e e r r o r i n c^ [ l ] . As t h e t e m p e r a t u r e o f t h e w a t e r d u r i n g t h e measurements d i d n o t d e v i a t e more t h a n 3°C f r o m t h e c a l i -b r a t i o n t e m p e r a t u r e , however, t h e c o r r e c t i o n t e r m f o r c^ i n E q u a t i o n ( 1 ) w i l l be s m a l l w i t h r e s p e c t t o c^ and even s m a l l e r w i t h r e s p e c t t o t h e measured v a l u e s o f v ^ ^ ^ . Hence a l l g e n e r a l s y s t e m a t i c e r r o r s a r e n e g l e c t e d . b E r r o r s t h a t a r e s y s t e m a t i c i n a v e r t i c a l ( i . e . , i n v a r i a n t i n a v e r t i c a l , b u t randomly v a r y i n g f r o m v e r t i c a l t o v e r t i c a l ) .
1 The e r r o r i n t h e measured l o c a l d e p t h o f f l o w H ^ , w h i c h i s assumed t o have a normal d i s t r i b u t i o n w i t h e x p e c t a t i o n 0 and s t a n d a r d d e v i a t i o n a,,.
H 2 The e r r o r i n t h e measured reference angle o f t h e d i r e c t i o n meter $^, as
f a r as i t i s due t o e r r o r s i n t h e d e t e r m i n a t i o n o f t h e zero r e f e r e n c e a n g l e and t h e p o s i t i o n i n g o f t h e i n s t r u m e n t i n a v e r t i c a l (see P a r t I , Appendix B ) . T h i s e r r o r i s assumed t o be n o r m a l l y d i s t r i b u t e d , t o o , w i t h e x p e c t a t i o n 0 and s t a n d a r d d e v i a t i o n a.,. 0 3 The e r r o r m t h e p o s i t i o n i n g o f t h e i n s t r u m e n t i n t h e l o w e s t p o i n t o f the v e r t i c a l ( i = 1 ) . As a l l h i g h e r p o i n t s a r e r e f e r r e d t o t h e l o w e s t p o i n t i n t h e r e l e v a n t v e r t i c a l , t h e e r r o r i n t h e d e t e r m i n a t i o n o f t h e v e r t i c a l d i s t a n c e o f t h i s p o i n t t o t h e bed must be c o n s i d e r e d as
be-i n g s y s t e m a t be-i c f o r t h a t v e r t be-i c a l . T h be-i s e r r o r , dZ^, be-i s assumed t o be
« )
-8-n o r m a l l y d i s t r i b u t e d w i t h e x p e c t a t i o -8-n y^^ a-8-nd s t a -8-n d a r d d e v i a t i o -8-n 0^^ . 4 The e r r o r i n t h e v e l o c i t y measurements due t o t h e g r a d u a l p o l l u t i o n o f
t h e p r o p e l l e r . Assuming t h e d e c r e a s e i n t h e measured v e l o c i t y caused by t h i s p o l l u t i o n t o be l i n e a r i n t i m e and t h e measurements t o be made w i t h c o n s t a n t t i m e i n t e r v a l s , t h i s e r r o r can be c o n s i d e r e d as an e r r o r i n t h e c o e f f i c i e n t o f t h e v e l o c i t y r a t i n g l i n e , being c o n s t a n t i n a v e r t i c a l . I t i s assumed t o have a l i m i t e d n o r m a l d i s t r i b u t i o n w i t h e x p e c t a t i o n y^^ and s t a n d a r d d e v i a t i o n o^^ , l i m i t e d t o dC^ <^ 0.
A l l o t h e r e r r o r s o f t h i s t y p e ( f o r i n s t a n c e i n R) a r e n e g l e c t e d , c E r r o r s t h a t a r e s y s t e m a t i c i n a p o i n t ( i . e . , i n v a r i a n t d u r i n g an o b s e r v a -t i o n i n a p o i n -t , b u -t randomly v a r y i n g f r o m p o i n -t -t o p o i n -t ) : 1 The e r r o r dZ i n Z^-Z^ ( i = 2 , 3 , . . . , n p ) , w h i c h i s assumed t o be n o r m a l l y d i s t r i b u t e d w i t h e x p e c t a t i o n 0 and s t a n d a r d d e v i a t i o n o^^ ( c o n s t a n t t h r o u g h o u t a v e r t i c a l ) . 2 The e r r o r i n t h e f l o w d e v i a t i o n a n g l e f ' due t o t h e b a c k l a s h i n t h e k e y -way (see P a r t I , Appendix B ) . As t h i s e r r o r can o n l y be i n t r o d u c e d when
d i s p l a c i n g t h e vane t o a n o t h e r measuring p o i n t , i t i s c o n s t a n t d u r i n g an o b s e r v a t i o n . T h i s i s e x p e c t e d t o be one o f t h e most s e v e r e e r r o r s i n t h e p r e s e n t e x p e r i m e n t s , so s p e c i a l a t t e n t i o n must be p a i d t o i t s p r o b a b i l i -t y d i s -t r i b u -t i o n .
A normal d i s t r i b u t i o n w i l l n o t be adequate h e r e , because t h e e r r o r i s l i m i t e d due t o t h e l i m i t e d b a c k l a s h and i f t h e r e i s a maximum i n t h e p r o b a b i l i t y d e n s i t y f u n c t i o n , i t w i l l o c c u r a t b o t h ends o f t h e b a c k l a s h r a t h e r t h a n somewhere i n between. T h e r e f o r e i n t h e n u m e r i c a l s i m u l a t i o n two d i f f e r e n t p r o b a b i l i t y d e n s i t y f u n c t i o n s were a p p l i e d , v i z . a u n i f o r m d i s t r i b u t i o n between t h e 1 i m i t s d $ ^ = ± a and a d i s t r i b u t i o n c o n s i s t i n g o f two D i r a c - f u n c t i o n s i n d$^ = -a and +a, r e s p e c t i v e l y .
3/4 The e r r o r s i n N and $' due t o t h e l i m i t e d d u r a t i o n o f t h e o b s e r v a t i o n p e r i o d . These e r r o r s v r i l l be t r e a t e d i n c o m b i n a t i o n w i t h t h e e f f e c t o f t h e e r r o r s i n t h e v e l o c i t y and d i r e c t i o n measurements mentioned i n Category d.
A l l o t h e r e r r o r s o f t h i s t y p e a r e n e g l e c t e d .
d I n s t a n t a n e o u s l y v a r i a b l e e r r o r s .
The i n s t a n t a n e o u s l y v a r i a b l e e r r o r s t h a t a r e i n t r o d u c e d when m e a s u r i n g t h e i n s t a n t a n e o u s l y v a r i a b l e v e l o c i t i e s and f l o w d e v i a t i o n a n g l e s a r e n o t
im-p o r t a n t f o r t h e t i m e - a v e r a g e d v a l u e s o f t h e s e q u a n t i t i e s i f t h e o b s e r v a t i o n p e r i o d i s s u f f i c i e n t l y l o n g .
The mean o v e r t h e o b s e r v a t i o n p e r i o d o f a random v a r i a b l e w i t h zero expect a expect i o n w i l l expect h e n approach expect h i s e x p e c expect a expect i o n . A s e p a r a expect e e m p i r i c a l i n v e s expect i g a -t i o n o f -t h i s -t y p e o f e r r o r i s h a r d l y p o s s i b l e , u n l e s s a n o -t h e r m e a s u r i n g d e v i c e o f f a r b e t t e r q u a l i t y a t t h i s p o i n t i s a v a i l a b l e . By l a c k o f such a d e v i c e , t h e a p p r o x i m a t e t i m e - a v e r a g e d q u a n t i t i e s N ( o r r a t h e r N/T) and were i n v e s t i g a t e d , o n l y y i e l d i n g i n f o r m a t i o n r e g a r d i n g t h e combined e f f e c t o f e r r o r s due t o t h e i m p e r f e c t i o n o f t h e m e a s u r i n g system ( i n s t a n t a n e o u s l y v a r i a b l e ) and e r r o r s due t o t h e l i m i t e d d u r a t i o n o f t h e o b s e r v a t i o n p e r i o d . These combined e r r o r s must be c o n s i d e r e d as b e i n g o f t y p e c and r e p l a c i n g t h e e r r o r s m e n t i o n e d under c 3/4.
F u r t h e r i n v e s t i g a t i o n s t o t h e s e e r r o r s (see a l s o Ref. 4 ) have shown t h a t t h e v a r i a n c e s o f t h e random v a r i a b l e s N ( o r r a t h e r N/T) and v a r y a l o n g a v e r t i c a l , h a v i n g a maximum a t t h e bed and d e c r e a s i n g when moving upward u n t i l , near t h e s u r f a c e , a s m a l l i n c r e a s e o c c u r r e d , a g a i n . Moreover, t h e c o v a r i a n c e between t h e e r r o r s i n N and $' t u r n e d o u t t o be n e g l i g i b l e . I n a d d i t i o n t o t h e e r r o r s i n t r o d u c e d by m e a s u r i n g o r e s t i m a t i n g t h e i n p u t d a t a o f t h e e l a b o r a t i o n p r o c e d u r e , t h i s p r o c e d u r e i n t r o d u c e s an i m p o r t a n t e r r o r i t -s e l f when u -s i n g t h e t r a p e z o i d a l r u l e (5,6) f o r a v e r a g i n g t h e t a n g e n t i a l and r a d i a l v e l o c i t y - c o m p o n e n t s o v e r t h e d e p t h o f f l o w . T h i s e r r o r depends on t h e number o f p o i n t s t a k e n i n a v e r t i c a l . An i n d i c a t i o n o f i t s magnitude i s o b t a i n e d by u s i n g ( 5 ) t o average t h e b a s i c l o g a r i t h m i c c u r v e ( 7 0 ) , h a v i n g t h e e x a c t average v a l u e 1. Both i n t h e l O - p o i n t and i n t h e 6 - p o i n t case t h e v a l u e s f o u n d f r o m ( 5 ) a r e about 3% t o o s m a l l . The s m a l l d i f f e r e n c e between t h e s e two cases can be e x p l a i n e d f r o m t h e i r s i m i l a r i t y near t h e bed, t h e most c r i t i c a l r e g i o n o f t h e v e r t i c a l i n t h i s r e s p e c t .
5 N u m e r i c a l s i m u l a t i o n
5.1 G e n e r a l o u t l i n e o f t h e method
The s i m u l a t i o n method can be summarized as f o l l o w s p J :
a The p r o b a b i l i t y d e n s i t y f u n c t i o n s o f t h e random i n p u t d a t a o f t h e s i m u l a t e d p r o c e s s b e i n g g i v e n , a s e t o f v a l u e s o f t h e s e d a t a i s g e n e r a t e d u s i n g a random number g e n e r a t o r .
b The o p e r a t i o n s on t h e i n p u t d a t a c o n s t i t u t i n g t h e p r o c e s s a r e c a r r i e d o u t and t h e r e s u l t s t o be s t u d i e d a r e s t o r e d .
c The above r o u t i n e ( a - b ) i s r e p e a t e d many t i m e s , y i e l d i n g a l a r g e number o f v a l u e s f o r each o f t h e o u t p u t q u a n t i t i e s t o be c o n s i d e r e d .
d C o n s i d e r i n g t h e v a l u e s o b t a i n e d f o r a c e r t a i n o u t p u t q u a n t i t y as samples o f a random v a r i a b l e , t h e y a r e a n a l y s e d t o y i e l d t h e p r o b a b i l i t y d e n s i t y f u n c -t i o n o f -t h i s q u a n -t i -t y .
Thus t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f each o f t h e r e s u l t s o f t h e process can be d e t e r m i n e d . F u r t h e r d e t a i l s o f t h e method and i t s e v a l u a t i o n can be f o u n d i n 2 .
I n terms o f t h e p r e s e n t p r o b l e m , i t i m p l i e s t h a t :
- A l a r g e number o f s e t s o f "measured d a t a " i s g e n e r a t e d u s i n g a random num-b e r g e n e r a t o r ; t h e e l a b o r a t i o n p r o c e d u r e d e s c r i b e d i n S e c t i o n 2 i s a p p l i e d t o each s e t o f measured d a t a , y i e l d i n g a l a r g e number o f v a l u e s o f t h e q u a n t i t i e s t o be s t u d i e d , i . e . , v^.^^, a, v|^^^^_ and v ^ ^ ^ , ( i = l ,2,. . . ,np) ; and f r o m t h e s e v a l u e s t h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f each o f t h e s e q u a n t i -t i e s i s d e r i v e d . 5.2 Q u a n t i f i c a t i o n o f i n p u t d a t a The v a l u e s a t t r i b u t e d t o t h e v a r i o u s q u a n t i t i e s p l a y i n g a p a r t i n t h e e r r o r a n a l y s i s a r e summarized i n Tables I - I I I . I n T a b l e I V t h e v a l u e s o f t h e e x p e c t a t i o n s o f N and $' between t h e g i v e n p o i n t s a r e e s t i m a t e d by l i n e a r i n t e r p o l a t i o n , t h e s t a n d a r d d e v i a t i o n s b e i n g assumed t o be c o n s t a n t about each p o i n t . A l l random q u a n t i t i e s a r e assumed t o be u n c o r r e l a t e d .
5.3 R e s u l t s
A g e n e r a l c o n c l u s i o n t o be drawn f r o m t h e r e s u l t s o f t h e n u m e r i c a l s i m u l a t i o n (based on 300 g e n e r a t e d s e t s of "measured d a t a " ) i s t h a t a l l r e s u l t a n t r a n -dom e r r o r s a r e n o r m a l l y d i s t r i b u t e d , even though t h e e r r o r s i n t h e i n p u t d a t a , e s p e c i a l l y a r e n o t . T h i s i m p l i e s t h a t each o f t h e r e s u l t s i s f u l l y c h a r -a c t e r i z e d by i t s me-an -and i t s s t -a n d -a r d d e v i -a t i o n . I n T a b l e V t h e r e s u l t s o f t h e n u m e r i c a l e r r o r a n a l y s i s a r e g i v e n , t o g e t h e r w i t h t h e v a l u e s o f t h e q u a n t i t i e s w i t h o u t e r r o r s . I n a d d i t i o n , t h e r e s u l t s f o r v ' . and v ' , a r e shown i n F i g u r e 1. main h e l ^ As t o T a b l e V, i t s h o u l d be n o t e d t h a t t h e s e r e s u l t s a r e based on a l i m i t e d number (300) o f g e n e r a t e d s e t s o f measured d a t a . T h i s i m p l i e s t h a t t h e mean
--v a l u e s a r e n o t e x a c t , t h e s t a n d a r d d e --v i a t i o n o f t h e e r r o r b e i n g 2 about 300 ^
t i m e s t h e s t a n d a r d d e v i a t i o n o f t h e r e l e v a n t q u a n t i t y . Hence a ( v f . ^ ^ ) « 0 . 0 0 3
ï-o^ean
and a(a ) 0.03. T h i s e x p l a i n s p a r t o f t h e d i f f e r e n c e s between t h e v a l u e s mean ^
under "mean" and those under "no e r r o r s " i n t h i s T a b l e . On t h e o t h e r hand, t h e s e d i f f e r e n c e s a r e p a r t l y due t o t h e s y s t e m a t i c e r r o r s i n t h e i n p u t d a t a . From t h e r e s u l t s o f t h i s n u m e r i c a l s i m u l a t i o n i t i s c o n c l u d e d t h a t t h e most s e v e r e e r r o r s occur i n a and v ^ ^ ^ and t h a t t h e i n f l u e n c e o f d$^ i s predominant i n V,' 1 b u t n o t i n a.
h e l
An a d d i t i o n a l r u n w i t h $g = 0 y i e l d e d a much s m a l l e r s t a n d a r d d e v i a t i o n i n a, whereas v ^ ^ ^ was n o t i n f l u e n c e d a t a l l , f r o m w h i c h i t i s c o n c l u d e d t h a t r a -t h e r -t h a n d$,' i s -t h e main source o f e r r o r s i n a.
•12-6 T h e o r e t i c a l a n a l y s i s
6.1 G e n e r a l '
C o n s i d e r a n o n - l i n e a r f u n c t i o n £ ( x ^ , '' * " n ^ p h y s i c a l q u a n t i t i e s X j , Ü J ' * * " ' — n '^^^y^^g randomly i n t i m e and h a v i n g mean v a l u e s and cova-r i a n c e s Y^j ( i , j = 1,2,..,,n). On t a k i n g samples o f X j '''* ^—-^ by c o n t i -nuous m e a s u r i n g d u r i n g a p e r i o d o f l i m i t e d d u r a t i o n ( i n t r o d u c i n g s y s t e m a t i c and random m e a s u r i n g e r r o r s ) , t h e timemean v a l u e s o f t h e s e samples f o r m a n -o t h e r s e t -o f rand-om v a r i a b l e s X ,X ,...,X h a v i n g mean v a l u e s M, and c-ova-
cova-—1 — 2 ' — n ° 1
r i a n c e s F^j ( i , j = l , 2 , , . . , n ) . I t s h o u l d be n o t e d t h a t c o n s i s t s o f two components, one caused by e s t i m a t i n g t h e mean v a l u e s o f t h e random v a r i a b l e s XJ 2En t a k i n g t h e mean o f a l i m i t e d sample and one caused by t h e random m e a s u r i n g e r r o r s . These two components w i l l n o t be s e p a r a t e d , however, as i n most cases i t i s e a s i e r t o d e t e r m i n e T^^ ( i . e . , t h e c o v a r i a n c e o f t h e measured d a t a ) as a w h o l e . Suppose t h e f u n c t i o n f^ (x^,x^»• • • JX^^) a d m i t s a l i n e a r a p p r o x i m a t i o n by t a k i n g t h e f i r s t o r d e r terms o f a T a y l o r s e r i e s e x p a n s i o n about t h e p o i n t ( y ^ , i J ^ , . . . , 1 ( 2 i i . Ü 2 ' - - " ^ n ) = f V " <9ï7>y ( ^ i - V (>3) 1=1 1 1=1 1 3 f i n w h i c h (^—) denotes t h e p a r t i a l d e r i v a t i v e o f f w i t h r e s p e c t t o x. , e v a l u -o x £ y 1 a t e d a t t h e p o i n t ,y2, • • • , \ ) •
T a k i n g e x p e c t a t i o n s i n (13) and (14) and s u b t r a c t i n g y i e l d s t h e f o l l o w i n g ex-p r e s s i o n f o r t h e s y s t e m a t i c e r r o r i n f_ (X^,X^,. . . ,X^) as a f u n c t i o n o f t h e s y s t e m a t i t i c e r r o r s (M^ - y.) i n X^ (i«=l ,2,. . . , n ) :
E { f ( X j , X 2 ' * ' - ' V ^ - E { f ( x , , X 2 " - - ' ^ n ^ ^ ~ ( ^ ) ( M . - y . ) ( 1 5 ) i = l i y
From (14) t h e f o l l o w i n g a p p r o x i m a t i o n o f t h e v a r i a n c e o f f^ ^ i ^ i — n ^ be d e r i v e d : 1=1 j = l 1 ^ I f g ( x ,x ,...,x ) denotes a n o t h e r n o n - l i n e a r f u n c t i o n o f x ,x ,...,x a l s o &. ' _ 2 ' — 1 ' — 2 ' '—n a d m i t t i n g a l i n e a r a p p r o x i m a t i o n about t h e p o i n t ( y ^ , . . . , y ^ ) , t h e c o v a r i a n c e between £ '2^2 '' ' ''—n^ and ^ '2^2 — a p p r o x i m a t e d by:
Gov { f U ,X,....,X,), 1 ( X , . X 2 ^ n ) > ^ . f , J , ( | r> y ^ " ^ V ^ i j ('^> X J 1 J The l i n e a r i z e d a p p r o x i m a t i o n s (15) t h r o u g h (17) a r e assumed t o be a p p l i c a b l e t o a l l n o n l i n e a r f u n c t i o n s o f random v a r i a b l e s e n c o u n t e r e d d u r i n g t h e e l a b o -r a t i o n o f t h e measu-red d a t a . i 6.2 M a g n i t u d e and d i r e c t i o n o f t h e v e l o c i t y v e c t o r I
The m a g n i t u d e and t h e d i r e c t i o n o f t h e v e l o c i t y v e c t o r a r e measured t w i c e i n each g r i d p o i n t . The r e s u l t s o f t h e s e two o b s e r v a t i o n s a r e averaged t o y i e l d t h e measured magnitude and d i r e c t i o n o f t h e v e l o c i t y v e c t o r t o be used i n t h e f u r t h e r e l a b o r a t i o n .
I f n ( z ^ ) i s t h e (random) number o f p u l s e s t o be c o u n t e d by t h e v e l o c i t y m e t e r d u r i n g an o b s e r v a t i o n o f d u r a t i o n T i n a p o i n t i a t d i s t a n c e f r o m t h e bed
(so n ( z ^ ) denotes t h e p h y s i c a l q u a n t i t y t o be measured), t h e n i n a f i r s t o r -der a p p r o x i m a t i o n :
n ( z , + dz) » n ( z . ) + n ( z . ) dz (18)
_ _ _2 X —
i n w h i c h dz denotes a random d e v i a t i o n f r o m z. and n ( z . ) denotes t h e v e r t i -— 1 -—z ^ i '
c a l d e v i a t i v e o f n e v a l u a t e d a t t h e p o i n t i . Thus n ( z ^ + dz) i s a p p r o x i m a t e d by a n o n - l i n e a r f u n c t i o n o f t h e random v a r i a b l e s n ( z . ) , n ( z . ) and dz. I f n ( z . ) and n ( z . ) a r e t h e mean v a l u e s o f n ( z . ) and n ( z . ) , t h e n a T a y l o r
-14-s e r i e -14-s e x p a n -14-s i o n about t h e p o i n t { n ( z , ) j n ( z , ) , 0 } c u t o f f a f t e r t h e f i r -14-s t X IZ X o r d e r terms y i e l d s a l i n e a r a p p r o x i m a t i o n o f t h i s f u n c t i o n . n ( z . + dz) « n ( z . ) + { n ( z . ) - n ( z . ) } + n ( z . ) dz ( 1 9 ) — X X X X Z X Assuming n ( z . ) t o be e q u a l t o - 1 ^ z X 0 z known f u n c t i o n . , (19) can be e v a l u a t e d i f n ( z , ) i s a Z £ 1
I f N ( z ^ ) and dZ' a r e t h e measured v a l u e s o f t h e p h y s i c a l q u a n t i t i e s n ( z ^ ) and dz, N ( z ^ ) and dZ' h a v i n g mean v a l u e s N ( z ^ ) and dZ', t h e n :
N ( z . + d Z ' ) « n ( z . ) + { N ( z . ) - n ( z . ) } + n ( z . ) dZ' ( 2 0 ) — 1 1 — 1 1 z 1 — — i s a l i n e a r a p p r o x i m a t i o n o f N ( z ^ + d Z ' ) . T a k i n g e x p e c t a t i o n s i n ( 2 0 ) and (19) and s u b t r a c t i n g t h e r e s u l t s y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r t h e s y s t e m a t i c e r r o r i n N ( z ^ + d Z ' ) : E { N ( z . + d Z ' ) } - E { ( n ( z . ) } « N ( z . ) - n ( z . ) + n ( z . ) dz' ( 2 1 ) _ ^ ' X X X Z X The v a r i a n c e o f N ( z ^ + dZ') i s a p p r o x i m a t e d t o t h e f i r s t o r d e r by:
Var { N ( z . + d Z ' ) } ^ { n ( z . ) } ^ Var { d Z ' } + Var { N ( Z . ) } + 2n ( z . )
— 1 z 1
—
1 z 1Gov { N ( z ^ ) , dZ'} ( 2 2 )
T a k i n g account o f t h e s i m p l i f y i n g assumptions ( C h a p t e r 4) and r e w r i t i n g dZ' as d Z j + dZ, these e x p r e s s i o n s reduce t o :
E { N ( z . + d Z ' ) } - E { n ( z . ) } «< n ( z . ) dZ, ( 2 3 ) X X Z X 1
Var { N ( z . + d Z ' ) } « { n ( z . ) } ^ Var { d Z } + Var { N ( Z . ) } ( 2 4 ) X —~~— 2 2_ "' •" X
On t a k i n g t h e mean o f two samples o f N ( z ^ + dZ') i n s t e a d o f a s i n g l e sample, t h e e x p r e s s i o n f o r t h e s y s t e m a t i c e r r o r (23) remains unchanged, whereas t h e v a r i a n c e i s reduced by a f a c t o r 2:
Var { | N (z^ + dZ') N ( z ^ + dZ') i { n ( z . ) } ^ Var { d Z } + 2 2_ ——.
+ I Var { N ( Z ^ ) } (25)
I f ( z ^ ) denotes t h e measured v a l u e o f t h e p h y s i c a l q u a n t i t y ^' ( z ^ ) i n a p o i n t i a t d i s t a n c e z^ f r o m t h e bed, ^' ( z ^ ) and ^' ( z ^ ) h a v i n g mean v a l u e s
( z ^ ) and ^' ( z ^ ) , t h e n a s i m i l a r r e a s o n i n g l e a d s t o t h e s y s t e m a t i c e r r o r and t h e v a r i a n c e o f t h e random e r r o r i n t h e mean o f two samples o f ^' ( z ^ + d Z ' ) : E { i i ' ( Z j + dZ') i ' ( Z j + dZ') } - E ( z . ) } « (j)' ( z . ) d Z j + + $• ( z ^ ) - (f)' ( z . ) (26) Var $• ( z . + d Z ' ) J 1 $' ( z . + d Z ' ) J =« i {(|)^ ( z ^ ) } 2 Var { d Z } + + I Var { $ • ( z ^ ) } (27) i n w h i c h ^' ( z . ) denotes t h e v e r t i c a l d e r i v a t i v e o f (j)' i n p o i n t i . The m a g n i t u d e o f t h e t o t a l v e l o c i t y i n a p o i n t i , v , i s r e l a t e d t o t h e num-™*t O t £ b e r o f p u l s e s t o be counted i n t h a t p o i n t by t h e f o r m u l a : - t o t ^ T n ( z . ) . c + c^ ( n ^ - n ^ ) (28) ( c f . E q u a t i o n ( 1 ) ) . Then, a c c o u n t i n g f o r t h e s i m p l i f y i n g a s s u m p t i o n s , t h e f o r -mula f o r t h e c o m p u t a t i o n o f t h e " o b s e r v e d " t o t a l v e l o c i t y i n t h e p o i n t i , ^ t o t i ' ^ t o t , N ( z ^ + dZ') N ( z ^ + dZ') } + c 2 2 c a l i b r a t e d + ( n - r i o ) (29)
"16-The e r r o r s m V ^ a r e t h e n c h a r a c t e r i z e d by; — t o t . ^ 1 n ( z . ) 8v t o t , ,„ . * dCj _ _ _ _ _ t o t . 1 1 " ^ i E ^ ^ t o t . > - ^ ^ ^ t o t . > - -T \ = 1 c, (30) 3v. c. Var { V , > « I ( - # ^ ) ^ Var { d Z } + — V a r { N ( z . ) } ( 3 1 ) - t o t ^ ' 8z 1 — ~ ^ ^ \ o t i n w h i c h ( — k ) , denotes t h e v e r t i c a l d e r i v a t i v e o f t h e mean o f t h e t o t a l d z 1 v e l o c i t y i n t h e p o i n t i and v ^ ^ ^ = c^ n ( z ^ ) / T . i The a n g l e o f d e v i a t i o n o f t h e v e l o c i t y v e c t o r f r o m t h e d i r e c t i o n o f t h e chan-n e l a x i s i chan-n t h e r e l e v a chan-n t c r o s s - s e c t i o chan-n , , f o l l o w s f r o m : i i = i ' ( - i ) - (32) and t h e " o b s e r v e d " v a l u e $. f o l l o w s f r o m : — 1 i i = 1' ( z . +dZ^) - 3>; ( 3 3 ) i n w h i c h and $^ a r e c o n s t a n t s i n a v e r t i c a l , $^ v a r y i n g randomly f r o m v e r t i c a l t o v e r t i c a l . I n t h e p r e s e n t case (})^ i s t a k e n e q u a l t o z e r o f o r a l l v e r -t i c a l s . Hence t h e e r r o r s i n $^ a r e c h a r a c t e r i z e d b y : E { $ i } - E { i . } ^
(||).
dZj - ( 3 4 )Var 1 {^)\ Var { d Z } + | Var W ( z . ) } ( 3 5 )
i n w h i c h (tt^). denotes t h e v e r t i c a l d e r i v a t i v e o f t h e mean o f t h e d e v i a t i o n dz 1
a n g l e i n p o i n t i .
- The v a r i a n c e o f t h e random m e a s u r i n g e r r o r ,
t h e v a r i a n c e o f t h e e r r o r due t o t h e l i m i t e d d u r a t i o n o f t h e o b s e r v a t i o n p e r i o d , and
t h e v a r i a n c e o f t h e e r r o r due t o t h e b a c k l a s h i n t h e keyway.
6.3 T a n g e n t i a l and r a d i a l v e l o c i t y component
The d e c o m p o s i t i o n o f t h e v e l o c i t y v e c t o r i n t o a t a n g e n t i a l and a r a d i a l com-p o n e n t i n a com-p o i n t i i s r e a l i z e d by: v^ = ^ cos <b. — t . — t o t . -*-L 1 1 (36) V = V,. ^ s i n (b. — r . — t o t , — 1 1 1 (37) i n w h i c h v and v d e n o t e t h e randomly v a r i a b l e " p h y s i c a l " v a l u e s o f t h e s e ^ i ^ i components.
Hence t h e v e l o c i t y components V and V t o be d e r i v e d f r o m t h e measured d a t a i ^ i a r e n o n - l i n e a r f u n c t i o n s o f t h e random v a r i a b l e s V. . and $., t o w h i c h t h e — i,Oi.£ — i g e n e r a l laws d e r i v e d i n S e c t i o n 6.1 can be a p p l i e d . T h i s y i e l d s : 9v dC E { V ^ . } - E { v ^ _ } . ( ~ ^ ) . dZ^ cos 0).
.
v*^^_ — cos ^. + 1 1 t o t , 1 i-s i n (j), (38) 9v dC E { V ^ } - E { v ^ _ } . ( - ^ ) . d Z j s i n c^. . v*^^^ — s i n ^. . 1 1 1 1 + v. t o t . 1 cos (j) . (39) Var {V^ } i I cos^ (J). r 9v c ( t o t ^ 2 { d z } + — Var {N ( z . ) } dz 1—
^2 - 1 _ + I v j ^ t . ^ i " ' * i (|^)? Var { d Z } + Var { $ ' ( z . ) } (40)-18-Var {V } i I s i n ^ (j). ( ~ ^ ) ? Var { d Z } + Var { N ( Z . ) } dz 1
—
^2 - 1 . 1 Var { d Z } + Var ( z . ) } dz 1 — — 1 (41) Gov {V|._, V^_} i i 5 s i n (^^ cos (J)^ ( t o t 2 ^^^^ + - i Var {N ( z . ) } c3z 1 „2 — — 1 5 s i n (j). cos (J). t o t (|^)? Var { d Z } + Var {(j) ( z . ) } dz 1 — 1 (42) 6.4 A v e r a g i n g o v e r t h e d e p t h o f f l o w 6.4.1 E r r o r due t o t h e a p p l i c a t i o n o f t h e t r a p e z o i d a l r u l e I f f ( z ) i s a f u n c t i o n o f z and f i t s d e p t h - a v e r a g e d v a l u e , t h e n : f =i
ƒ f ( z ) dz ^ 0 np " " i h i- L ƒ f ( z ) dz + i- ƒ f ( z ) dz h . , h 1=1 z. , z 1-1 np (43) A d o p t i n g t h e t r a p e z o i d a l r u l e , the i n t e g r a l s over t h e i n t e r v a l s ( z ^ _ ^ , z^) a r e a p p r o x i m a t e d by: 1 z. - z,_ ( z . - z._,) ƒ f ( z ) dz « ' 2 ^ { f ( z . ) + f ( z . _ j ) } - ^ ^> ' f " ( q ) ^ i - 1 (44) i n w h i c h t h e l a s t t e r m e s t i m a t e s t h e e r r o r , f " ( ^ ^ ) d e n o t i n g t h e second d e r i -v a t i -v e o f f w i t h r e s p e c t t o z i n a p o i n t z = E. on t h e i n t e r -v a l ( z . ,, z . ) . ^ 1 1-1 ' 1 Assuming f t o v a n i s h a t t h e bed (z=0) and - r - t o v a n i s h a t t h e w a t e r s u r f a c edz ( z = h ) , s u b s t i t u t i o n o f (44) i n t o (43) y i e l d s t h e t r a p e z o i d a l r u l e ( 5, 6 ) and an e s t i m a t e o f i t s e r r o r : np np+i E a f ( z ) - E b f"(£.-l) i = l ^ i = l ^ ^ (45)
i n w h i c h : z, i + 1 + z. i - l f o r i < np; a, = 1 z . + z , 1 1 i - l f o r i = np a. 1 2h 2h b. 1 f o r i < np+1; b. = 12h f o r i = np+1 I f f ( z ) i s a d e t e r m i n i s t i c f u n c t i o n o f z, i t s second d e r i v a t i v e w i t h r e s p e c t t o z can be e v a l u a t e d t h r o u g h o u t t h e v e r t i c a l , e i t h e r by a n a l y t i c a l d i f f e r e n -t i a -t i o n ( i f -t h e f u n c -t i o n a l r e l a -t i o n s h i p be-tween f and z i s known) or by numer-i c a l d numer-i f f e r e n t numer-i a t numer-i o n ( numer-i f f numer-i s o n l y known numer-i n d numer-i s c r e t e p o numer-i n t s and numer-i t s second de-r i v a t i v e shows no s t de-r o n g v a de-r i a t i o n s w i t h i n an i n t e de-r v a l ) . Then t h e l a s t t e de-r m o f (45) can be used t o e s t i m a t e t h e e r r o r i n f .
I f f ( z ) i s a measured q u a n t i t y , however, i n c l u d i n g random v a r i a t i o n s f r o m p o i n t t o p o i n t i n a v e r t i c a l , t h i s e s t i m a t i o n becomes f a r more c o m p l i c a t e d . As t h e measurements i n t h e v a r i o u s p o i n t s were t a k e n a t d i f f e r e n t t i m e s , t h e e r r o r s a t two d i f f e r e n t p o i n t s w i l l be u n c o r r e l a t e d . T h e r e f o r e t h e second de-r i v a t i v e o b t a i n e d f de-r o m t h e n u m e de-r i c a l d i f f e de-r e n t i a t i o n of t h e measude-red d a t a i n a v e r t i c a l g i v e s no r e l e v a n t i n f o r m a t i o n about t h e e r r o r i n f . I f t h e e r r o r s a r e s m a l l w i t h r e s p e c t t o t h e e x p e c t a t i o n s of t h e measured d a t a , however, t h e i n f l u e n c e o f t h e random v a r i a t i o n s on t h e e r r o r due t o t h e use o f t h e t r a p e -zium r u l e can be n e g l e c t e d .
Then t h e e r r o r can be e s t i m a t e d by t h e l a s t t e r m of ( 4 5 ) , t a k i n g f"(£^) as t h e second d e r i v a t i v e of t h e t h e o r e t i c a l d i s t r i b u t i o n of f e v a l u a t e d a t t h e p o i n t
6.4.2 E r r o r s i n t h e d e p t h - a v e r a g e d v e l o c i t y components
As some o f t h e e r r o r s t a k e n i n t o account are c o n s t a n t f o r a s i n g l e p o i n t , b u t v a r y randomly f r o m p o i n t t o p o i n t , e r r o r s t h a t are s y s t e m a t i c when c o n s i d e r i n g a s i n g l e p o i n t may become random when c o n s i d e r i n g a group o f p o i n t s . Hence, b e f o r e a p p l y i n g t h e t r a p e z i u m r u l e t o V. and V„ , t h e d i s t i n c t i o n between s y t e m a t i c and random e r r o r s must be r e c o n s i d e r e d . On c h e c k i n g whether t h e s y s t e m a t i c e r r o r s i n . and . ( ( 3 8 ) and ( 3 9 ) ) c o n t a i n e r r o r s o f t h e t y p e j u s t m e n t i o n e d , however, i t i s seen t h a t t h e y do n o t . As a consequence of
t a k i n g t h e mean o f two i n d e p e n d e n t o b s e r v a t i o n s o f N ( z ^ + dZ') and £' ( z ^ + dZ'), w h i c h can be c o n s i d e r e d as t a k i n g t h e mean o f two d i f f e r e n t p o i n t o b s e r v a t i o n s . h = i ( ^ i ^ ^ i l )
- -20-t h e e r r o r s o f -20-t h i s -20-t y p e were c o n s i d e r e d as b e i n g random i n -20-t h a -20-t s -20-t a g e a l r e a d y . A p p l y i n g ( 5 ) and ( 6 ) t o and ( i = 1,2,...,np) y i e l d s : i i V = Z a. - t , , 1 - t , - t 1=1 1 np np+1 8^v^ V , + E a. (V - v ^ ) + Z b. ( -) i = l ^ - ' ^ i " ' ' i i = l ^ ' i - l ( 4 6 ) V = E a. V — r . 1 1 — 1=1 1 np np+1 9^v V + E a. (V - V ) + E b. ( -) 1 — r . — r • , 1 — r . — r • • , i 2 1=1 1 1 1=1 9 z ? i - l (47) where: E { V ^ } - E { v ^ } np E a. i = l ^ E {VJ . } - E {v^ } i i np+! + E b, i = l ' 9 ^ ^ ( 1) 9 z 2 ' i - l , _ , dH , ( v ^ - v ^ ) np (48) Var { V ^ } np E a t Var {V^ } . , 1 - t . 1=1 1 (49) (dH d e n o t i n g t h e e r r o r i n t h e measured d e p t h o f f l o w ) and s i m i l a r e x p r e s s i o n s f o r t h e s y s t e m a t i c e r r o r and t h e v a r i a n c e o f t h e random e r r o r i n V^.
I n a d d i t i o n , t h e c o v a r i a n c e s between V , V , V and #. can be e v a l u a t e d by — t ' — r —tot£ — 1
u s i n g ( 1 7 ) .
6.5 Magnitude and d i r e c t i o n o f t h e depth-averaged v e l o c i t y v e c t o r
The magnitude v^^^. and t h e d i r e c t i o n a o f t h e d e p t h - a v e r a g e d v e l o c i t y v e c t o r ( i . e . , t h e v e c t o r o b t a i n e d by composing v^. and v ^ ) a r e i m p o r t a n t q u a n t i t i e s f o r t e s t i n g t h e m a t h e m a t i c a l model o f c u r v e d s h a l l o w channel f l o w (see P a r t I ) . T h e r e f o r e t h e s y s t e m a t i c e r r o r s and t h e v a r i a n c e s o f t h e random e r r o r s i n t h e s e q u a n t i t i e s a r e g i v e n f u r t h e r a t t e n t i o n . Making use o f (15) t h r o u g h ( 1 7 ) , t h e s e e r r o r s can be expressed i n terms o f t h e e r r o r s i n t h e measured d a t a . The r e s u l t s are g i v e n i n Appendix A, t o g e t h e r w i t h t h e a p p r o x i m a t i o n s f o r s m a l l v a l u e s o f (*. - a ) , and -^-^ .
6.6 N o r m a l i z e d main and h e l i c a l v e l o c i t y components
I n o r d e r t o v e r i f y t h e t h e o r y as t o t h e v e r t i c a l d i s t r i b u t i o n o f t h e main and h e l i c a l v e l o c i t y components (shape o f t h e c u r v e s , s i m i l a r i t y between v a r i o u s v e r t i c a l s ) , t h e n o r m a l i z e d main and h e l i c a l v e l o c i t y components r e s u l t i n g f r o m
the measurements have been c o n s i d e r e d . T h e r e f o r e t h e r e s u l t s o f t h e e r r o r ana-l y s i s f o r these q u a n t i t i e s a r e g i v e n i n A p p e n d i x A f o r s m a ana-l ana-l v a ana-l u e s o f t h e f l o w d e v i a t i o n a n g l e and i t s f i r s t and second v e r t i c a l d e r i v a t i v e s .
6.7 N o r m a l i z a t i o n o f t h e e r r o r s L e t i t be assumed t h e v e r t i c a l d i s t r i b u t i o n s o f t h e e x p e c t a t i o n s and t h e v a -r i a n c e s o f t h e main v e l o c i t y and t h e f l o w d e v i a t i o n a n g l e a -r e s i m i l a -r i n a l l v e r t i c a l s , i . e . : ^ o t = \ o t ^main (5°) ^dev (51) Var {N} = Var {N} f ^ (?) ( 5 2 )
Var { $ ' } = Var { $ ' } f ^ , ( ? ) + Var { d $ ^ } (53)
i n w h i c h t h e f u n c t i o n s f . , f , , f., and f . , o n l y depend on t h e r e l a t i v e main' dev' N $' ^ ^ h e i g h t ? f o l l o w s : h e i g h t ? = T- . The e r r o r s i n a c e r t a i n v e r t i c a l can t h e n be n o r m a l i z e d as E {V } - E { v } dZ c^ dCj t £ L , t o t _ ^ p 1 ^ _ ^ ^ _ 1) ^ + F (54) v^ ^ ^ h v^ c, mam ^ h 2 ^ ' t o t t o t 1 np
Var {V } Var { d Z } c^ Var { N }
^ f + ~ (55)
-2 3 2 r„9 -9
v f ^ h ^ T^ V^ ^
-22-E { A } - -22-E { a } „ dZ, dC, .„
i - 7 i " " ? + E, + E, — + f . f , i f + ( 5 6 )
h/R h o 5 h 6 c, mam dev h 7
1 np np
Var { A } Var { d Z } Var { $ ' } Var { d $ ' } c^ Var {N }
^ + F„ + F,„ ^ + F , , - i ( 5 7 ) h V R ' ' h^ ' h V R ' h V R ' v^^ ^ t o t E x p r e s s i o n s f o r t h e c o n s t a n t F^ t h r o u g h F^^ a r e g i v e n i n A p p e n d i x B. dZ c, dC, E { V . } - E { v ' , } - f , ( C ) - r i + f , ( ? . ) ~ + f ( ? . ) = ^ + f (C.) -mam, - m a i n . i ^ ^ i ' h 2 ^ i h 3 ^ ^ i ^ v^ ^ c, i 1 1 t o t 1 (58) Var { d Z } c^ Var { N } Var {r^,J - f , ( ? , ) : + f , (C,) ( 5 9 ) ^ t o t ^ ^ a i n ^ ^ 5 - i ^ ^ 2 - 6 ^ 2 _ 2 7. 1 t o t 1 (60)
Var { d Z } Var { $ ' } Var { d $ ' } Var {V' } - f (? ) — — + f (? ) + f . , ( ? , ) — + - h e l . 1 1 1 ^ 2 1 2 1 h V R ^ " h V R ' c^ Var {N } + f - . (?.) " 1 T 2 V 2 14 t o t (61)
The c o n s t a n t s F^ t h r o u g h F^^ and t h e f u n c t i o n s f ^ ( ? ^ ) t h r o u g h F^^^ ( ? ^ ) can be c o n s i d e r e d as i n d i c a t i n g t h e i n f l u e n c e o f t h e v a r i o u s e r r o r s i n t h e measured d a t a on t h e e r r o r s i n t h e e l a b o r a t e d r e s u l t s .
As a consequence o f t h e assumptions u n d e r l y i n g t h e t h e o r y (see S e c t i o n 6 . 1 ) , t h e s y s t e m a t i c e r r o r s and t h e v a r i a n c e s o f t h e random e r r o r s i n t h e e l a b o r a t e d r e s u l t s a r e seen t o have a l i n e a r r e l a t i o n t o t h e s y s t e m a t i c e r r o r s and t h e v a r i a n c e s o f t h e random e r r o r s i n t h e measured d a t a .
6.8 G e n e r a l i z a t i o n t o any v e r t i c a l
So f a r t h e a n a l y s i s has been l i m i t e d t o one v e r t i c a l , c o n s i d e r i n g t h e e r r o r s t h a t a r e c o n s t a n t i n a v e r t i c a l as b e i n g s y s t e m a t i c , even though t h e y may v a r y randomly f r o m v e r t i c a l t o v e r t i c a l . When g e n e r a l i z i n g t h e r e s u l t s o f t h e e r r o r a n a l y s i s t o any v e r t i c a l , t h i s random c h a r a c t e r must be t a k e n i n t o a c c o u n t . As a l l g e n e r a l s y s t e m a t i c e r r o r s i n t h e measured d a t a a r e n e g l e c t e d (see
Chap-t e r 3 ) , o n l y t h e s y s t e m a t i c e r r o r s due t o t h e use o f t h e t r a p e z o i d a l i n t e g r a -t i o n r u l e and -t h o s e caused by n o n - z e r o e x p e c -t a -t i o n s o f dZ^ , dC^^, dH and r e m a i n : 5, , ' i h * <' T—> S * "main t o t t o t 1 np E {dH} — h — ^ (62) E { A } - E { a } E { $ ' } E {dZ } E {dC } ^ .^^^_ _ ^ ^ p ,. "f" F " f f h/R h/R 5 h 6 c main dev 1 np np E {dH} — h —
'
^ (63) 2 E {dZ } E {dH} ^ ^ ^ a i n > - ^ < ^ a i n > ^ ^ ' h — R - ^a = t o t E {dC } c ' + f 4 ( q ) (64) E {dZ } c, dC, ^ ^ ^ h e l > - ^ ^ ^ h e l > ^ ^ ^ ^^a ' ^3 ^ t o t 1 E {dH} + f 3 ( q ) ( q > (65)
-24-Th e i n f l u e n c e o f t h e random v a r i a t i o n o f dZ^, dC^, dH and #J must be a c c o u n t e d f o r i n t h e v a r i a n c e s o f t h e random e r r o r s :
Var { v . .} Var { d Z } c, Var {dC } Var { d H }
= ^ ^ ( 1 „ ^ ) 2 =zL_4. = _ ( f . - 1) 2 v ^ , , ' h^ ^ o t S h^ '"^^"np
Var { d Z } c\ V a r {N }
, F 3 — — — + F ^ - — (66)
^ T ^ L t
Var { A } Var { $ ' } Var {dZ } Var {dC } Var {dH}
. + F2 z z i _ + p 2 z z ^ - ^ f 2 _ f 2 Z H .
h V R ' h V R ' ' h^ cf ^ ^ % p h^'
Var { d Z } Var { $ ' } Var { d $ ; } c^ Var { N }
+ F + F + F L - ^ _J_ ' ' h V R ^ h V R ^ T^ v^
t o t
Var { d Z } Var {dH} c^ Var {dC }
{ ^ a i n , > " 'I < q ) — ^ - ( q ) — ^ - < q ) I T ^ i n h V c t o t 1 Var { d Z } c f Var { N } ^ 1 1,2 6 1 2 - 2 t o t v a r { V - ^ ^ ^ } ^ f ? ( q ) Var {dZ } c, Var {dC } . { f ^ (,.) . (,.) p i -t o -t
Var {dH} Var { d Z } Var {<!>'}
+ f ( r ) + f ( r . ) ^ ^ 1 , 2 12 1 . 0 , _ 0 h2/R2 Var { d $ ; } . f , 3 ( ? i ) „ „ + f , u (C.) -c^ Var {N } h V R ' ' 1 T2 ^2 t o t (69)
6.9 Q u a n t i f i c a t i o n o f i n f l u e n c e f a c t o r s and f u n c t i o n s
The i n f l u e n c e f a c t o r s and f u n c t i o n s g i v e n i n A p p e n d i x B have been e v a l u a t e d u s i n g t h e m a t h e m a t i c a l e x p r e s s i o n s f o r f . ( ? ) and f , ( ? ) d e r i v e d i n [sl: ° mam dev '—' ^main = ^ ^ É É ^ ' ' ^ W ( ? ) = - ^ (1 - # M 2 / 1B£ a? . ^ / ^ ^ ^ > / ^ ^ ^ f m a i n (^)> ? t ( 7 1 ) i n w h i c h : g = a c c e l e r a t i o n due t o g r a v i t y K = Von Karman's c o n s t a n t C = Chezy's f a c t o r ? * = exp ( 1
-I n o r d e r t o a t t a i n b e t t e r agreement w i t h t h e measured d a t a , f ( ? ) has been m u l t i p l i e d by 1.5, y i e l d i n g t h e same v e r t i c a l d i s t r i b u t i o n o f t h e d e v i a t i o n
a n g l e as was used i n t h e n u m e r i c a l a n a l y s i s .
The v e r t i c a l d i s t r i b u t i o n f u n c t i o n s f ^ ( ? ) and f ^ , ( ? ) o f t h e v a r i a n c e s o f N and $' have a l s o been t a k e n t h e same as i n t h e n u m e r i c a l a n a l y s i s :
f ^ ( ? ) - 1.8 f o r ? i 0.3 - 3.6 - 7.2 ? + 3.6 ?2 f o r 0.3 < ? j< 0.7 ( 7 2 ) - 0.3 f o r ? > 0.7 f ^ , ( ? ) = 1.6 - 1.5 ? + 0.3 ? 2 ( 7 3 ) U s i n g (70) t h r o u g h ( 7 3 ) , t h e i n f l u e n c e f a c t o r s F, t h r o u g h F, and t h e i n f l u e n c e 1
f u n c t i o n s f ( ? ) t h r o u g h f , . ( ? ) have been e l a b o r a t e d f o r C = 70 m V s and C = 50 m V s , b o t h f o r t h e 1 0 - p o i n t case ( ? ^ - 0 . 1 , 0.2 1.0) and f o r t h e 6 - p o i n t case ( ? . = 0 . 1 , 0 . 2 , 0.4, 0.6, 0.8, 1.0) .
-26-6.9.1 S y s t e m a t i c e r r o r s
E l a b o r a t i n g ( r e p r e s e n t i n g t h e s y s t e m a t i c e r r o r due t o t h e use o f t h e t r a p e z o i d a l r u l e when computing V^^^) f o r C = 70 m^/s and ? i _ j = 5 ^ ^ i ^ ^ i - i ^
( i S np) y i e l d s a v a l u e o f - 4.5 x 10"^ f o r t h e 1 0 - p o i n t case and - 5.6 x 10~ f o r t h e 6 - p o i n t case. A p p l y i n g t h e t r a p e z o i d a l r u l e t o ( 7 0 ) , however, y i e l d s a d e p t h - a v e r a g e d v a l u e o f 0.97 b o t h f o r t h e 1 0 - p o i n t case and f o r t h e 6 - p o i n t case, w h i c h i m p l i e s a s y s t e m a t i c e r r o r o f - 3 x 10~^. The l a r g e d i f f e r e n c e be tv^een t h i s e r r o r and t h e computed v a l u e s o f F^ must be a t t r i b u t e d t o t h e
f a i l u r e o f t h e e r r o r e s t i m a t i o n on t h e i n t e r v a l ( 0 , C j ) , where t h e second de-r i v a t i v e o f f . v a de-r i e s between - 10 and - <». Then t a k i n g t h e v a l u e o f t h e
mam ° second d e r i v a t i v e a t = | i n e s t i m a t i n g t h e e r r o r becomes r a t h e r a r b i t r a r y . T h i s can be i l l u s t r a t e d as f o l l o w s : t h e e x a c t i n t e g r a t i o n o f f . over ^ * mam t h e i n t e r v a l ( 0 , 5^) y i e l d s a v a l u e o f 0.074 and t h e c o n t r i b u t i o n o f t h i s i n -t e r v a l -t o -t h e r e s u l -t s o f -t h e -t r a p e z o i d a l r u l e i s 0.043, so -t h e e x a c -t e r r o r i s 0.031. A c c o r d i n g t o ( 9 5 ) w i t h Cg = 5 C^, t h e e s t i m a t e d e r r o r on t h i s i n t e r -v a l i s - 0.004, i . e . , 0.027 t o o l a r g e . T h i s e x p l a i n s t h e g r e a t e r p a r t o f t h e d i f f e r e n c e between F^ and t h e e r r o r i n t h e depthaverage o f f ^ ^ ^ ^ when a p p l y -i n g t h e t r a p e z o -i d a l r u l e . T h e r e f o r e t h e e x a c t v a l u e - 0.031 o f t h e e r r o r on t h e i n t e r v a l ( 0 , ?^) has been used, r a t h e r t h a n e s t i m a t i n g t h i s e r r o r by
d ^ f . 0 main S i m i l a r r e a s o n i n g c o n c e r n i n g t h e c o n t r i b u t i o n o f t h e i n t e r v a l ( 0 , ?^) t o F, ( i n d i c a t i n g t h e i n f l u e n c e o f t h e s y s t e m a t i c e r r o r due t o t h e use o f t h e t r a p e z o i d a l r u l e when computing A) l e a d s t o a v a l u e o f + 0.502 f o r t h e c o n t r i b u -t i o n o f -t h i s i n -t e r v a l . O b v i o u s l y , -t h e v a l u e s o f A based on -t h e -t r a p e z o i d a l r u l e a r e t o o l a r g e . T a b l e V I g i v e s a r e v i e w o f t h e c o n s t a n t s F^ and F^ computed w i t h t h e c o n t r i -b u t i o n s o f t h e i n t e r v a l ( 0 , C^) as suggested a-bove, t h e v a l u e s o f these c o n t r i b u t i o n s , and t h e v a l u e s o f t h e e q u i v a l e n t e x a c t e r r o r s i n t r o d u c e d by t h e use o f t h e t r a p e z o i d a l r u l e .
O b v i o u s l y , t h e i n t e r v a l ( 0 , ?^) s u p p l i e s by f a r t h e most i m p o r t a n t c o n t r i b u -t i o n -t o -t h i s -t y p e o f s y s -t e m a -t i c e r r o r , i n a l l cases c o n s i d e r e d b o -t h i n
