DENSIMETRIC EXCHANGE FLOW IN RECTANGULAR CHANNELS

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NOVEMBRE 1 9 6 3 - № 7 L A H O U I L L E B L A N C H E 7 3 9

I.—DEFINITIONS, REVIEW AND RELEVANCE TO MODEL DESIGN

BY D. I. H . BABE,

DEPARTMENT OF CIVIL ENGINEERING

THE ROYAL COLLEGE OF SCIENCE AND TECHNOLOGY, GLASGOW

This series of papers concerns investigations of the phenomenon of exchange flow in water; in parti

cular the situations where the two masses of water at different densities are contained in horizontal rectangular channels. It is considered (Barr 1958) that the correct scaling of hydraulic models involv

ing internal spread phenomena can only be determined in the knowledge of the characteristics and me

chanism of exchange flow. The type of problem particularly in mind is the prediction of the extent of recirculation of cooling water discharged from thermal power stations sited on estuaries or tidal rivers.

The continuing use of such models in Scotland (Barr, 1958; Smith, 1962) is considered to have justified the step by step approach made to the problem. First the overall pattern of advance of the fronts of exchange flow was studied. From these observations, and malting use of the extensive and valuable expe

rimental studies of Keulegan (1957, 1958) a design method for heat dissipation models has been devised, and has, in fact, been applied to three large model studies {Smith 1962, Barr, 1963 c).

In this paper {I) exchange flow is introduced and reviewed and some new experimental results given.

These include, it is thought for the first time, some details of "dam burst analogy" exchange flow. An attempt is made to explain the necessary compromising approach of those who practice the science—or art—of hydraulic modelling, and to apply this approach to the case of heat dissipation models.

1. L O C K A N D D A M B U R S T A N A L O G Y C A S E S

O F E X C H A N G E F L O W

I n t r o d u c t o r y .

A l o c k g a t e or o t h e r s u c h d i v i s i o n m a y s e p a r a t e b o d i e s of still w a t e r of t h e s a m e s u r f a c e level b u t w h i c h differ s l i g h t l y i n d e n s i t y . W h i l e t h e o p e n i n g of t h e g a t e m a y r e s u l t i n l o c a l d i s t u r b a n c e s , t h e p r e d o m i n a n t effect w i l l b e a c o n t i n u i n g e x c h a n g e p a t t e r n of flow w h i c h i s c a u s e d b y t h e d e n s i t y difference. T h e l e s s d e n s e w a t e r flows o v e r t h e m o r e d e n s e a n d a n e q u a l u n d e r f l o w of t h e m o r e d e n s e w a t e r m u s t p a s s t h e l o c a t i o n of t h e g a t e i n t h e o p p o s i t e d i r e c t i o n t o t h e s u r f a c e flow. F i e l d o b s e r v a t i o n s of t h e p h e n o m e n a h a v e b e e n m a d e o n v a r i o u s o c c a s i o n s , i n c l u d i n g t h o s e r e p o r t e d b y A l l e n a n d P r i c e (1959) a t t h e Q u e e n E l i z a b e t h I I D o c k o n t h e M e r s e y . T h e c i r c u m s t a n c e of a level b o t t o m

ed r e c t a n g u l a r o p e n c h a n n e l w i t h a n i n i t i a l l y v e r t i c a l d e n s i t y d i s c o n t i n u i t y , a s s h o w n i n F i g u r e la, h a s s e e m e d t h e s i m p l e s t c a s e f o r a t t e m p t s a t a n a l y s i s . A g o o d a p p r o x i m a t i o n t o t h i s m a y b e o b t a i n e d i n l a b o r a t o r y e x p e r i m e n t s b y fitting a t h i n v e r t i c a l c r o s s b a r r i e r i n t o s l i g h t g r o o v e s i n t h e s i d e s of a r e c t a n g u l a r f l u m e a n d t h u s i n i t i a l l y s e p a r a t i n g t h e d i s s i m i l a r b o d i e s of w a t e r . If t h e b a r r i e r is lifted s m o o t h l y , r e l a t i v e l y little local d i s t u r b a n c e is c a u s e d . At s m a l l d e p t h s ( < 0.75 ft, s a y ) t h e t i m e t a k e n for r e m o v a l is s l i g h t i n c o m p a r i s o n w i t h t h e t i m e t a k e n for e x c h a n g e m o v e m e n t s of t h e s a m e o r d e r of t r a v e l l e n g t h t o d e v e l o p , a n d l i t t l e d i s t o r t i o n of t h e flow p a t t e r n s is t h o u g h t t o r e s u l t f r o m t h e a p p r o x i m a t i o n . T h e i n i t i a l v e l o c i t i e s ( V⁰) of t h e t i p s o r f r o n t s of t h e u n d e r f l o w a n d overflow a p p e a r u n i f o r m for a g r e a t e r o r l e s s e r r e l a t i v e d i s t a n c e d e p e n d i n g o n t h e s c a l e of t h e e x p e r i m e n t , a n d a r e u n a f f a c t e d b y t h e c h a n n e l w i d t h t o d e p t h r a t i o ( B / H ) e x c e p t for e x t r e m e c a s e s w h e r e f B / H ) is w e l l u n d e r 0.5. L o c k e x c h a n g e flow is a c o n v e n i e n t d e s c r i p t i o n for t h i s t y p e of c i r -

Article published by SHF and available athttp://www.shf-lhb.orgorhttp://dx.doi.org/10.1051/lhb/1963052

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7 4 0 L A H O U I L L E B L A N C H E № 7 - NOVEMBRE 1 9 6 3

cu i n s t a n c e w h i c h h a s b e e n d e s c r i b e d b y H a r l e - m a n (1961) a s the classical problem i n unsteady, nonuniform flow in two layered systems. W h e  t h e r t h e l a t t e r p a r t of t h i s d e s c r i p t i o n is e n t i r e l y s u i t a b l e is d i s c u s s e d l a t e r i n t h i s s e r i e s of p a p e r s ( I I ) .

L o c k e x c h a n g e flow h a s a r o u s e d s p a s m o d i c i n t e r e s t d u r i n g t h e p a s t t h i r t y y e a r s or s o ; O ' B r i e n a n d C h e r n o (1934), Y i h (1957), K e u l e - g a n (1957) a n d B a r r (1959) h a v e d e s c r i b e d e x p e r i m e n t a l s t u d i e s , w h i l e K e u l e g a n (1946) a n d Schijf a n d S c h o n f e l d (1953) h a v e given a n a l y tical a p p r o a c h e s . It is o n e facet of t h e g r o u p of p h e n o m e n a v a r i o u s l y k n o w n a s d e n s i t y c u r r e n t s , s t r a t i f i e d flows, s u b - s u r f a c e flows o r i n t e r n a l flows. T h e e x i s t e n c e of p a r a l l e l s b e t w e e n s u b - s u r f a c e a n d free s u r f a c e h y d r a u l i c o c c u r e n c e s h a s b e e n s t r e s s e d b y v a r i o u s w r i t e r s , n o t a b l y K n a p p (1942), K e u l e g a n (1950) a n d m o r e r e c e n t l y , H a r l e m a n (1961). It is, h o w e v e r , i m p o r t a n t t o r e m e m b e r t h a t i n p r a c t i c a l cir

c u m s t a n c e s s m a l l d e n s i t y difference p h e n o m e n a a r e n o r m a l l y o b s e r v e d a s b e t w e e n m i s c i b l e fluids; f r e s h a n d s a l t o r w a r m e r a n d c o l d e r w a t e r , w a r m e r a n d c o l d e r air, o r b e t w e e n a i r a n d o t h e r g a s e s ( B a k k e , 1 9 5 9 ) . I n t h e l a b o r a t o r y m i s c i b l e l i q u i d s m a y b e u s e d , a s i n t h e e x c h a n g e flow⁷ s t u d i e s m e n t i o n e d a b o v e , or i m m i s c i b l e l i q u i d s a s in t h e e x p e r i m e n t s d e s c r i b e d b y L o n g (1953-1955) w h e r e t h e p r a c t i c a l c i r c u m s t a n c e s i n m i n d w e r e m e t e o r o l o g i c a l . T h e e x p e r i m e n t s of A b b o t (1961) r e l a t i n g to t h e s p r e a d of oil o r p e t r o l f r o m a h o l e d t a n k e r a n d of B a t a (1959), r e l a t i n g t o e x c h a n g e flow b e t w e e n p e t r o l a n d w a t e r in t h e p i p e s of a fuel s t o r a g e i n s t a l l a t i o n , a r e e x a m p l e s of t h e c o m p a r a t i v e l y f e w c a s e s w h e r e t h e l i q u i d s i n b o t h " m o d e l " a n d " p r o t o t y p e " a r e i m m i s c i b l e . I t is n o t u n l i k e l y t h a t t h e p a r a l l e l i s m b e t w e e n free s u r f a c e a n d s u b s u r f a c e p h e n o m e n a m a y b e r a t h e r m o r e m a r k ed i n t h e c a s e of i m m i s c i b l e l i q u i d s ; i t w o u l d c e r t a i n l y be difficult t o o b t a i n , s a y , s u c h a w e l l f o r m e d i n t e r n a l h y d r a u l i c j u m p i n m i s c i b l e li

q u i d s a s h a s b e e n o b s e r v e d i n i m m i s c i b l e l i q u i d s . O n t h e o t h e r h a n d v e r y m u c h of t h e i n t e r e s t of e x c h a n g e flow is r e l a t e d t o t h e m i x i n g p r o c e s s e s , a n d t h e r e is i n t h i s r e s p e c t , a s s h a l l b e s h o w n , l i t t l e s i m i l a r i t y b e t w e e n e x c h a n g e flows i n t h e m i s c i b l e a n d t h e i m m i s c i b l e c a s e s .

D e n s i m e t r i c F r o u d e a n d d e n s i m e t r i c F r o u d e - R e y n o l d s n u m b e r s , 9<^A a n d § >^A & l .

T h e effective g r a v i t a t i o n a l c o n s t a n t , g', for a n i m m e r s e d l i q u i d (or b o d y ) is g.(g^x — P 2 ) / p i w h e r e p⁵ a n d p² a r e t h e d e n s i t i e s of t h e i m m e r s ed a n d t h e s u r r o u n d i n g l i q u i d r e s p e c t i v e l y . If Pi — 9-> — Ap is s m a l l t h e a p p r o x i m a t i o n :

g' = g Ap/p

c a n be u s e d (i.e. p = 9i ' = Pa)- B y s u b s t i t u t i n g g' for g i n t h e s t a n d a r d F r o u d e n u m b e r , $ :

§i^V^JgL

w h e r e L a n d V a r e c h a r a c t e r i s t i c l e n g t h a n d A'elocity r e s p e c t i v e l y ) , t h e d e n s i m e t r i c F r o u d e n u m b e r $< A = N\/g^K, is o b t a i n e d . H i s t h e d e p t h of t h e d e n s i t y d i s c o n t i n u i t y a t t h e s t a r t of a n e x c h a n g e flow ( F i g . 1), a n d t h e t o t a l d e p t h i n lock e x c h a n g e flow: a s e x c h a n g e flow i s b e i n g c o n s i d e r e d it is b e s t t o h e r e u s e H , a s b e i n g t h e logical c h a r a c t e r i s t i c l e n g t h . T h e d e n s i m e t r i c F r o u d e n u m b e r i s , i n effect, t h e " o v e r a l l " R i c h a r d s o n n u m b e r ( E l l i s o n a n d T u r n e r 1959) (<f^A = — R i -¹/²) a l t h o u g h t h e l i n e s of r e a s o n i n g w^rhich led to t h e o r i g i n a l f o r m u l a t i o n of t h e s e p a r a m e t e r s a r e s o m e w h a t different ( B a r r , 1963), V A = t h e d e n s i m e t r i c v e l o c i t y , is a c o n  v e n i e n t c h a r a c t e r i s t i c velocity, p a r t i c u l a r l y for

lock e x c h a n g e flow s t u d i e s w h e r e t h e differing l i q u i d s c a n be r e g a r d e d a s m u t u a l l y i m m e r s e d .

Since it is d e s i r e d to find c r i t e r i a of s i m i l a r i t y b e t w e e n v a r i o u s c o m p a r e d s y s t e m s , i t i s c o n v e n i e n t t o u s e t h e t e r m s prototype a n d model a l t h o u g h n o t i n t e n d i n g t h a t a n a c t u a l h y d r a u l i c m o d e l n e c e s s a r i l y b e i m a g i n e d . I n free s u r f a c e h y d r a u l i c s a n d u s i n g w a t e r i n b o t h p r o t o t y p e a n d m o d e l , t h e s i m u l t a n e o u s u s e of b o t h t h e F r o u d e n u m b e r c r i t e r i a ( R e y n o l d s n u m b e r

== V H / v w h e r e v is t h e k i n e m a t i c v i s c o s i t y ) l e a d s to a m o d e l of t h e s a m e size a s t h e p r o t o t y p e . H o w e v e r t h e r e q u i r e m e n t s of b o t h d e n s i m e t r i c F r o u d e a n d R e y n o l d s n u m b e r c r i t e r i a c a n b e m e t t o g e t h e r o v e r a w i d e r a n g e of p r a c tical c i r c u m s t a n c e s still w i t h w a t e r a s t h e l i q u i d t h r o u g h o u t .

U s i n g s u b s c r i p t s p a n d m for p r o t o t y p e a n d m o d e l respectively—•

( D e n s i m e t r i c F r o u d e n u m b e r m o d e l c r i t e r i o n . )

( R e y n o l d s n u m b e r m o d e l c r i t e r i o n . ) T o s a t i s f y b o t h :

( s g ^ . H ^ ^ h^m. V „

WJ^1/2-K^m1/2 h „ . V „

o r :

¥^61= (g')!/². H3/²/v ^ V^A. H / v

is t h e d e n s i m e t r i c F r o u d e - R e y n o l d s n u m b e r . K e u l e g a n (1957, 1958) r e p o r t i n g o n h i s e x t e n -

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NOVEMBRE 1 9 6 3 - № 7 D . I . H B A R R 7 4 1

Removeoble barriers Barrières amovibles

I Barriers removed y Barrières enlevées

p p + àp

- ' . c

7—T—7—T—7-

-Í ' ' ' '

P + àP

H

-— p

/> + àp

P + A/=

T

^p

p + àp

FlG. 1

Diagramatic illustrations of various exchange flow configurations (pure density current cases)

Représentation schématique de divers profils d'écoulement d'échange (ces cas intéressent des courants de densité purs).

(a) Free surface lock exchange.

(b) Enclosed flume lock exchange.

a) Echange dans une écluse à surface libre.

b) Echange dans une écluse en canal couvert.

c) Ecoulement d'échange dans le cas représentant la rupture d'un barrage — écoulement « par en-dessus » (<:) Dam-burst analogy exchange flow—free surface overflow. à surface libre.

d) Ecoulement d'échange dans le cas représentant la rup- (d) Dam-burst analogy exchange flow—underflow. ture d'un barrage — écoulement « par en-dessous ».

sive a n d v a l u a b l e s t u d i e s of e x c h a n g e flow, first d e m o n s t r a t e d t h e g r e a t u s e f u l n e s s of t h e c o n - c e p t s of d e n s i m e t r i c v e l o c i t y a n d d e n s i m e t r i c F r o u d e - R e y n o l d s n u m b e r . H e u s e d t h e n o m e n - c l a t u r e " d e n s i m e t r i c R e y n o l d s n u m b e r ( d i^A) "

for t h e a b o v e d i m e n s i o n l e s s n u m b e r w h e r e t h e d e n s i m e t r i c F r o u d e - R e y n o l d s n u m b e r (£F^A 01) h a s b e e n u s e d h e r e : t h e r e a s o n s for t h e c h a n g e a r e g i v e n i n S e c t i o n 3 w h e r e r e f e r e n c e is m a d e t o a m o r e c o m p l e t e d e r i v a t i o n of t h e $^A Oi n u m b e r w h i c h is g i v e n a s a n a p p e n d i x ( 2 ) ( B a r r ,

1 9 6 3 b).

I n i t i a l v e l o c i t i e s .

F i g u r e s 1 a n d 2 i n d i c a t e t h e p r e s e n t s t a t e of k n o w l e d g e of i n i t i a l velocities, V⁰, ( V⁰ = K . V^A) of v a r i o u s c a s e s of e x c h a n g e flow i n a r e c t a n g u - l a r c h a n n e l ; F i g u r e 1 a is t h e s i m p l e free s u r f a c e c a s e so f a r d i s c u s s e d ( K e u l e g a n 1 9 5 7 , B a r r , 1 9 5 9 ) . T h e i n i t i a l v e l o c i t i e s of u n d e r f l o w a n d overflow differ b y 1 2 % o v e r t h e r a n g e s t u d i e d . If s i m i l a r t e s t s a r e c a r r i e d o u t i n a r e c t a n g u l a r p i p e ( F i g . 1 b) t h e coefficient o b t a i n e d for b o t h u n d e r f l o w a n d overflow a r e s i m i l a r t o t h o s e for t h e n o r m a l u n d e r f l o w ( B a r r , 1 9 6 1 ) . T h i s l a t t e r

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DAM BURST ANALOGY

. CAS REPRÉSENTANT LA RUPTURE D UN BARRAGE

o Overflow - Écoulement "par en dessus"

• • Underflow- - d° - "par en dessous"

FREE SURFACE LOCK EXCHANGE

ECHANGE DANS UNE ÉCLUSE À SURFACE LIBRE e Thermal overflow (surface tension bolanced) - Flume A

Écoulement "par en dessus" thermique (tension superficielle équilibrée) - Canal A q Thermol underflow - Flume A

Écoulement "par en dessous'' thermique - Canal A B Saline overflow - F(ume A

Écoulement "par en dessus " salin - Canal A Soline overflow - Flume B

Écoulement "par en dessus" satin - Canal B A Saline underflow - Flume B

Écoulement "paren dessous" salin - Canal B

ENCLOSED FLUME

CANAL COUVERT A Saline overflow in tilting flume (B)

Écoulement "paren dessus"salin en canal à pente variable (B)

y Saline underflow in Hiring flume (B) Écoulement "par en dessous " salin (B)

• Saline underflow with normal barrier arrangement flumes B + C - Écoulement "par en dessous " salin avec barrière disposée normalement (canaux B et Ci

200 300 400 600 1,000 FIG. 2

Coefficient of proportionality for initial velocities of exchange flow configurations shown in Figure 1.

Coefficients de proportionalitê correspondant aux vitesses initiales des schémas d'écoulement d'échange de la figure 1.

FIG. 3

Diagramatie illustrations of two further cases of exchange flow.

Représentation schématique de deux autres cas correspondant à des écoulements d'échange.

(a) Coupled "hackwater" curves at interface. a) «Courbes de remous» couplées à Vinterface.

(&) Intermediate case. fc) Cas intermédiaire.

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FIG. 4. — Views of lock exchange flow experiments (depth 4 in., breadth 4 1/8 in. A Q /⁰ = 0.011.).

Photographies des essais sur les écoulements d'échange en écluse (profondeur V, largeur 4 1/8", &Q/Q — 0,011).

(a) Overflow, less dense water coloured. «) Ecoulement «par en-dessus». Eau moins dense colorée.

(b) Overflow, more dense water coloured. b) Ecoulement «par en-dessus». Eau plus dense colorée.

(c) Underflow, more dense water coloured. c) Ecoulement «par en-dessous ». Eau plus dense colorée, id) Underflow, less dense water coloured. d) Ecoulement «par en-desous». Eau moins dense

colorée.

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(M

FIG. 5

Views of enclosed flume lock exchange flow experiments (depth and breadth both 4 1 / 8 in.)

Photographies des essais sur les écoulements d'échange en écluse en canal ouvert (profondeur il/S'", largeur il/8").

(a) Mow initiated by tilting flume (Ag/o = 0 , 0 0 6 0 ) . a) Ecoulement provoqué par l'inclinaison du canal (AQ/Q = 0,0060).

(b) Flow initiated by removal of barrier (AQ/Q = 0 . 0 1 4 5 ) . V) Ecoulement provoqué par l'enlèvement d'une barrière ( A⁰/ p = 0 , 0 * 4 5 ) .

c i r c u m s t a n c e will b e t e r m e d t h e e n c l o s e d f l u m e c a s e of l o c k e x c h a n g e . Also s h o w n a r e r e s u l t s r e c e n t l y o b t a i n e d d u r i n g o b s e r v a t i o n s of d a m - b u r s t a n a l o g y e x c h a n g e flow ( F i g u r e 1 c a n d F i - g u r e 1 d) so called b e c a u s e of t h e s i m i l a r i t y t o t h e t r u e d a m b u r s t p h e n o m e n a . H is a g a i n t a k e n a s t h e d e p t h of t h e i n i t i a l l y v e r t i c a l i n t e r - face. V⁰ for t h e d a m b u r s t a n a l o g y c a s e s is n o t , of c o u r s e , i n d é p e n d a n t of t h e H^e/ H r a t i o , w h e r e H^e is t h e t o t a l d e p t h , b u t t h i s s e e m s t o h a v e l i t t l e i n f l u e n c e b e y o n d v a l u e s of t h i s r a t i o of t h e o r d e r of 2 . It will be seen t h a t a s i m i l a r i n c r e a s e i n K w a s f o u n d b e t w e e n t h e free s u r f a c e d a m b u r s t overflow a n d t h e d a m b u r s t u n d e r f l o w a s e x i s t s b e t w e e n t h e overflow a n d t h e u n d e r f l o w of

n o r m a l l o c k e x c h a n g e flow. I n t h e l a t t e r c a s e b o t h overflow a n d u n d e r f l o w a r e o b t a i n e d i n t h e s a m e e x p e r i m e n t ( F i g . 1 a) w h i l e t h e d a m b u r s t a n a l o g y overflow ( F i g . 1 c) a n d u n d e r f l o w (Fig. 1 d) a r e b o t h c o u p l e d w i t h a t y p e of b a c k - w a t e r c u r v e , t h e l i m i t of e x t e n s i o n of w h i c h i s difficult t o o b s e r v e . D o u b t l e s s e x a m p l e s of d a m b u r s t a n a l o g y overflow c o u l d b e o b s e r v e d i n a n e n c l o s e d f l u m e a n d t h e n c e w o u l d b e o b t a i n e d t h e s a m e v a l u e s of K a s for e q u i v a l e n t u n d e r - flows. A g a i n , b y a r r a n g i n g t h a t t h e v e r t i c a l i n t e r f a c e e x t e n d s t o n e i t h e r s u r f a c e n o r b o t t o m of t h e f l u m e , i t i s p o s s i b l e t o c o u p l e t h e i n t e r - facial s h i f t s as i n d i c a t e d i n F i g u r e 3 a, o r t o o b t a i n a n i n t e r m e d i a t e c a s e ( F i g . 3 b).

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NOVEMBRE 1 9 6 3 - № 7 D . I . H BARR 7 4 5

F i g u r e 4 s h o w s p h o t o g r a p h s of free s u r f a c e l o c k e x c h a n g e flow, a l l t a k e n w h e n t h e e x t e n s i o n r a t i o L / H of t h e o v e r f l o w w a s 1 8 i n a 4 i n c h d e e p c h a n n e l . T h e t e c h n i q u e of c o l o u r i n g first t h e l e s s d e n s e a n d t h e n t h e m o r e d e n s e w a t e r t o s h o w t h e l i m i t s of m i x i n g w a s p r e v i o u s l y u s e d b y K e u l e g a n ( 1 9 5 7 ) . F i g u r e 5 s h o w s p h o t o g r a p h s of e n c l o s e d flume l o c k e x c h a n g e flow. I n t h e e x p e r i m e n t of F i g u r e 5 a flow w a s i n i t i a t e d b y t i l t i n g t h e flume t h r o u g h 9 0 ° , t h u s o b v i a t i n g t h e n e e d for a b a r r i e r ( B a r r , 1 9 6 1 ) w h i l e i n t h a t of F i g u r e 5 6 a n o r m a l b a r r i e r w a s u s e d . T h e a p p a r e n t difference b e t w e e n t h e c o m p l e t e l y s i m i l a r overflow a n d u n d e r f l o w is c a u s e d b y t h e d o m i n a n c e of t h e r e d d y e , a n d t h i s w i l l b e c o n s i d e r e d i n p a p e r I I of t h e s e r i e s .

T h e l o c k e x c h a n g e u n d e r f l o w l i n e o n F i g u r e 2 w a s first p r o p o s e d t e n t a t i v e l y b y K e u l e g a n ( 1 9 5 7 ) a n d f u r t h e r r e s u l t s o b t a i n e d b y t h e a u t h o r ( B a r r , 1 9 5 9 ) t e n d e d t o c o n f i r m t h e g e n e r a l t r e n d of

i n c r e a s i n g v a l u e s of K w i t h i n c r e a s i n g fll.

T h e l a r g e n u m b e r s of t e s t p o i n t s g i v e n p r e v i o u s l y a r e n o t a g a i n s h o w n , b u t t h e o p p o r t u n i t y h a s b e e n t a k e n t o i n c l u d e s o m e m o r e r e c e n t e v a l u a t i o n s .

D i m i n u t i o n o f f r o n t v e l o c i t y .

F u r t h e r v a r i a t i o n s of coefficient of p r o p o r t i o n a l i t y w o u l d be o b t a i n e d i n n o n - r e c t a n g u l a r c h a n n e l s , o r if a v a r i a t i o n i n c h a n n e l s h a p e w e r e i n t r o d u c e d a t t h e b a r r i e r ( K e u l e g a n , 1 9 5 8 ) . I n e v e r y c a s e t h e v e l o c i t y of t h e f r o n t s falls a w a y a f t e r s o m e a d v a n c e , t h e r e l a t i v e d i s t a n c e b e f o r e t h i s c a n b e d i s t i n g u i s h e d i n c r e a s i n g w i t h i n c r e a s i n g v a l u e s of $«^A (R.. T h i s d i m i n u t i o n of v e l o c i t y h a s b e e n s t u d i e d e x p e r i m e n t a l l y i n c o n s i d e r a b l e d e t a i l b y K e u l e g a n ( 1 9 5 7 , 1 9 5 8 ) f o r t w o r e s t r i c t e d c a s e s . I t h a s b e e n f o u n d t o b e

FIG. 6

View of dam-burst analogy overflow experiment in progress.

Essai en cours

avec écoulement « par en-dessus », pour le cas représentant

la rupture d'un barrage.

FIG. 7 View of dam-burst analogy underflow experiment in progress.

Essai en cours avec écoulement

«par en-dessous », pour le cas représentant la rupture d'un barrage.

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7 4 6 L A H O U I L L E B L A N C H E № 7 - NOVEMBRE 1 9 6 3

100 80

These results from Keulegan (1957) Ces résultats proviennent de Keulegan (1957}-

100 200 300 400

FIG. 8

Gongruency diagram for underflow of lock exchange flow in " w i d e " flume w i t h no reflection.

Schéma de congruence correspondant à l'écoulement «par en-dessous» d'un écoulement d'échange dans un canal

« de grande largeur » sans réflexions.

NOTES (on Fig. S):

(i) The practice of plotting both this diagram and the diagram showing variation of coefficient of proportionality for the initial velocities (Fig. 2) against K.S»^A<3v was initiated by Dr. Keulegan.

In the case of Figure 2 there are no real advantages and it is, in fact, somewhat irrational to plot the values of K for an underflow and an overflow, which can be obtained in the same experiment, against slightly different base values. However, the intention has been to compare with past work, and as K varies rather slowly with • tH.

and thus with K . S ^ d l little harm results. If desired, it is simple to convert the "best" lines as the author found expedient on one occasion (Barr, 1962).

With regard to Figure 8, there have been advantages in having arrangement whereby the slow variation of K is allowed for in proportioning the diminution of velocity ratios. Again the diagram could be transferred to a ff'^tfL base but only by returning to the original plots of travel against time.

Hi) The results at very low values of K.$>^&o% were obtained in a %/i inch deep enclosed flume (i.e. two "underflows" occurred) instead of in the normal open flume. However, the experiments is inevitably rather imperfect at this scale and this approximation is quite suitable to obtain the trend. It was necessary to assume that initial velocities as obtained from the extension of the lock exchange underflow line on Fig. 2 momentarilv existed.

Note au sujet de ta figure 8 :

a) La pratique suivant laquelle on trace, à la fois, le présent abaque, et celui exprimant la variation du coefficient de proportion- nalité correspondant aux vitesses initiales en fonction de K. S'^ÛX, a été employée pour la première fois par le Dr. Keulegan. Le procédé né présente aucun avantage réel dans le cas de la figure 2; en effet, il ne serait pas très rationnel de porter les valeurs de K correspondant à des écoulements « par en-dessus i et t par en-dessous », qui peu- vent être déterminées au cours d'une même expérience, en fonction de valeurs de base un tant soit peu différentes. Par contre, l'intention de l'auteur a été de confronter ses résultats avec ceux obtenus' au;

cours d'études antérieures, et puisque la valeur de K varie assez lentement en fonction de S'^OX, et donc de K.ff<^hûX, les conséquences ne sont pas trop graves. On peut, si on Je désire, aisément effectuer la conversion des courbes «optimales*, ainsi que l'auteur a trouvé utile de le faire dans un cas (Barr, 1962).

En ce qui concerne la figure 8, une disposition, dans laquelle on a tenu compte de la lente variation de K en proportionnant les rapports de diminution de vitesse, a présenté certains avantages.

On pourrait également reporter l'abaque sur une base en gï^A ûX, mais à condition de revenir aux graphiques initiaux du déplacement en fonction du temps.

b) Les résultats correspondant aux très faibles valeurs de K.$i^Aô\

ont été obtenus dans un canal couvert profond de (c'est-à-dire il s'est produit deux écoulements <c par en-dessous »), à. la place du canal découvert normal. Mais l'expérience reste inévitablement assez imparfaite à cette échelle, et cette approximation suffit pour déter- miner l'allure (la tendance) du phénomène. Il a fallu admettre l'exis- tence momentanée des vitesses initiales obtenues par l'extension de la courbe de ta figure 2, correspondant à l'écoulement « par en- dessous » d'un échange en écluse.

affected b y t h e v a r i a t i o n s i n c o n f i g u r a t i o n m e n - t i o n e d a b o v e , b y c h a n n e l w i d t h t o d e p t h r a t i o , a n d m a r k e d l y hy t h e p r e s e n c e of a reflection if

e i t h e r c h a n n e l l e n g t h is s h o r t ( K e u l e g a n , 1 9 5 7 ; B a r r , 1961). T h e s i m p l e s t c a s e for e i t h e r l o c k or d a m b u r s t a n a l o g y e x c h a n g e flow is f o r t h e r e

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NOVEMBRE 1 9 6 3 - № 7 D . I . H B A R R 7 4 7

t o b e sufficient b r e a d t h for t h e d i m i n u t i o n of v e l o c i t y t o b e u n a f f e c t e d b y t h e s i d e s ; a b r e a d t h t o d e p t h r a t i o of a b o u t s i x b e i n g n e c e s s a r y ( K e u - l e g a n , 1957), a n d for t h e l e n g t h t o d e p t h r a t i o o n b o t h s i d e s of t h e b a r r i e r t o b e sufficiently g r e a t f o r n o r e f l e c t i o n t o o c c u r w i t h i n t h e p e r i o d of t h e e x p e r i m e n t . T h e s e w i l l b e r e f e r r e d t o a s t h e s t a n d a r d c a s e s .

K e u l e g a n u s e d a p l o t of L / H a g a i n s t K . g <Â (R., w h e r e L is t h e d i s t a n c e f r o m t h e b a r r i e r b y w h i c h t h e f r o n t v e l o c i t y V h a s f a l l e n t o s o m e p r o p o r t i o n of V⁰, t o f u r t h e r i l l u s t r a t e t h e o v e r a l l c h a r a c t e r i s t i c s of e x c h a n g e flow, a n d c a l l e d t h i s a c o n g r u e n c y d i a g r a m . F i g u r e 8 s h o w s s u c h a d i a g r a m , o r r a t h e r t h e f r a g m e n t so f a r a v a i l a b l e of t h a t f o r t h e s t a n d a r d c a s e of l o c k e x c h a n g e u n d e r f l o w a s s h a l l b e e x p l a i n e d . F i r s t it s h o u l d b e s a i d t h a t if §<Â (R, is a sufficient c r i t e r i o n of s i m i l a r i t y i n e x c h a n g e flow, it s h o u l d be p o s s i b l e t o o b t a i n d a t a for t h e f o r m u l a t i o n of a c o n g r u e n c y d i a g r a m for a n y s e l e c t e d c o n f i g u r a t i o n . T h a t u s e d p r i n c i p a l l y b y K e u l e g a n w a s for t h e m o r e d e n s e w a t e r to b e i n i t i a l l y i n a s h o r t l e n g t h ( L⁰) of c h a n n e l w h e r e L⁰ = 7.2 H, a n d w i t h t h e c h a n n e l w i d t h B = H / 2 . H e c a r r i e d o u t sufficient e x p e r i m e n t s a t v a l u e s of K . g <Â (R. r a n g i n g f r o m 600 t o 200,000 t o p r o v i d e a v e r y c o m p l e t e c o n g r u e n c y d i a g r a m ( w i t h i n t h e a f o r e s a i d l i m i t s of K . 5<Â (R) f o r t h i s c o n f i g u r a t i o n , a n d t o d e m o n s t r a t e t h e s o u n d b a s i s of h i s a p p r o a c h . H o w e v e r t h e a c t u a l d i a g r a m i s d o m i n a t e d b y t h e side effects a c c r u i n g f r o m t h e s m a l l w i d t h t o d e p t h r a t i o a n d , m o r e e s p e c i a l l y , b y t h e i n e v i t a b l e p r e s e n c e of a r e f l e c t i o n . B e c a u s e of t h e w i d t h a n d , p a r t i c u l a r l y , t h e l e n g t h of f l u m e r e q u i r e d t o a l l o w for s t a n d a r d c a s e o b s e r v a t i o n s e x c e p t a t s m a l l d e p t h s t h e a u t h o r h a s so f a r b e e n able t o p r o v i d e o n l y t h e f r a g m e n t a r y c o n g r u e n c y d i a g r a m s h o w n i n F i g u r e 8.

T h e p r e s u m p t i o n , i n d i c a t e d i n F i g u r e 8, t h a t t h e L / H r a t i o s for t h e v a r i o u s V / V⁰ r a t i o s b e c o m e s i n d é p e n d a n t of K . p ^A (R, is b a s e d o n t h e r e s u l t s of i d e a l i s e d c h a n n e l a n d s e a e x c h a n g e flow e x p e r i m e n t s b y K e u l e g a n ( 1 9 5 8 ) . F i g u r e 9 s h o w s w h a t is m e a n t b y t h e c h a n n e l a n d s e a c o n f i g u r a t i o n t h e m o r e d e n s e w^ra t e r b e i n g i n i t i a l l y i n t h e s e a . K e u l e g a n a g a i n u s e d a r e l a t i v e l y n a r r o w a r r a n g e m e n t s (B = H a n d H / 2 w h e r e B is t h e b r e a d t h a n d H t h e t o t a l d e p t h ) a n d o b t a i n e d for t h e u n d e r f l o w i n i t i a l v e l o c i t i e s s u c h t h a t K a v e r a g e d 0.57. I t m u s t b e p r e s u m e d t h a t t h i s coefficient w o u l d ~vary s l o w l y w i t h

&>Â (R for a n y g i v e n H / B r a t i o , a n d d e p e n d s c o n s i d e r a b l y o n t h e H / B r a t i o . If t h e c h a n n e l w e r e v e r y w i d e , t h e v a l u e s for K w o u l d b e close to t h e v a l u e s a p p r o p r i a t e t o t h e n o r m a l lock e x c h a n g e u n d e r f l o w ( u p t o a b o u t 0.49). H o w e v e r t h e m o s t i m p o r t a n t a s p e c t of t h e r e s u l t s w a s t h a t a s '§iÂ(R. (g'Â(R = 1,75 K . $ >Â CR i n

SECTIONAL ELEVATION A A COUPE VERTICALE EN A A

FIG. 9

Channel and sea configuration.

Disposition « mer et canal »

t h i s c a s e ) a p p r o a c h e d 45,000 t h e d i m i n u t i o n of v e l o c i t y p a t t e r n b e c o m e s i n d é p e n d a n t of ~&^iA~dt a s is s h o w n i n F i g u r e 10.

T h e e v i d e n c e so f a r a v a i l a b l e i n n o w a y c o n t r a d i c t s t h e s u p p o s i t i o n t h a t t h e g e n e r a l p a t t e r n of d i m i n u t i o n of v e l o c i t y for t h e l o c k e x c h a n g e overflow is s i m i l a r t o t h a t for t h e u n d e r f l o w . F i g u r e 11 show's t y p i c a l r e s u l t s o b t a i n e d i n t h e s a m e e x p e r i m e n t . Of c o u r s e t h e s c a l i n g m e t h o d m e n t i o n e d i n t h e s y n o p s i s , a n d e x p l a i n e d i n t h e n e x t s e c t i o n , d e p e n d s o n t h e f o r e g o i n g b e i n g t r u e — h e a t d i s s i p a t i o n m o d e l s b e i n g c o n c e r n e d w i t h w a r m overflows.

0 1 1 1 1 1 1 1 1 1

0 40 80 120 160 200 240 280 L/H

FIG. 10

Typical results obtained by Keulegan (1958) for intrusion of saline underflow from " s e a "

into "freshwater" channel.

Résultats typiques obtenus par Keulegan (1958) pour la pénétration « par en-dessous » d'une langue salée,

de la mer dans le canal «d'eau douce».

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7 4 8 L A H O U I L L E B L A N C H E № 7 - NOVEMBRE 1963

M '

f^a

1

i i

I I

1

i 1

^!_!

r Ì i

10 20 30 40 50 SO 70 80 Time in seconds from lifting of barrier

Temps en secondes, compté à partir du souf&vement de la barrière

FLG. 1 1

Comparison of pattern of diminution of velocity of underflow and overflow obtained

in typical lock exchange flow experiment.

Comparaison entre la diminution,, d'une part, de la vitesse de l'écoulement « par en-dessous », et d'autre part, de celle de l'écoulement «par en-dessus », correspondant (t une expérience-type sur l'écoulement d'échange en écluse.

2 . R E L E V A N C E

O F E X C H A N G E F L O W S T U D I E S T O H Y D R A U L I C M O D E L D E S I G N

N a t u r a l m o d e l s i n v o l v i n g e x c h a n g e flow o n l y . T h e p r e s e n t s t a t e of k n o w l e d g e of i n i t i a l v e l o - cities ( F i g s . 1 a n d 2) a n d of t h e p a t t e r n of d i m i - n u t i o n of f r o n t v e l o c i t y a s given b y K e u l e g a n for r e l a t i v e l y n a r r o w c h a n n e l s a n d s h o w n o n F i g u r e 8 for t h e s t a n d a r d w i d e c h a n n e l case of t h e l o c k e x c h a n g e u n d e r f l o w a l l o w s t h e follow- i n g g e n e r a l a s s e s s m e n t of t h e m o d e l a p p r o a c h t o be m a d e . F o r n a t u r a l m o d e l s (i.e. t h o s e w i t h full g e o m e t r i c s i m i l a r i t y ) t h e r e a r e t w o m a i n p o s s i b i l i t i e s .

(£) I n f o r m a t i o n o n i n i t i a l velocities c o u l d b e o b t a i n e d f r o m a m o d e l o n t h e b a s i s of t h e d e n s i - m e t r i c F r o u d e c r i t e r i o n , a n d u s i n g t h e d e n s i - m e t r i c F r o u d e - R e y n o l d s c r i t e r i o n to c h e c k t h a t little d i m i n u t i o n of v e l o c i t y i s l i k e l y w i t h i n t h e p e r i o d of o b s e r v a t i o n . T h e r e w o u l d of c o u r s e b e little p o i n t i n b u i l d i n g a s m a l l s c a l e m o d e l t o e x a m i n e t h e i n i t i a l s t a g e s of lock e x c h a n g e flow w h e r e t h e p r o t o t y p e is a r e c t a n g u l a r c h a n - nel b e c a u s e t h i s c a s e h a s b e e n f a i r l y w e l l cover- ed. H o w e v e r i n i t i a l velocities i n a l a r g e n o n - r e c t a n g u l a r o r i n a n o n - p r i s m a t i c c h a n n e l c o u l d be f a i r l y c l o s e l y p r e d i c t e d f r o m s m a l l scale m o d e l t e s t s .

(z7) O n c e t h e p a t t e r n of d i m i n u t i o n of v e l o c i t y b e c o m e s of i m p o r t a n c e it w o u l d b e b e s t t o a d h e r e t o t h e s a m e d e n s i m e t r i c F r o u d e - R e y n o l d s n u m b e r i n m o d e l a s i n p r o t o t y p e , a n d e s s e n t i a l

t o m a k e t h e m o d e l K.g>Âdl v a l u e of t h e o r d e r of 1 0⁵ or g r e a t e r . A s s u m i n g w a t e r of t h e s a m e a v e r a g e v i s c o s i t y b e u s e d i n t h e m o d e l a s o b t a i n s i n t h e p r o t o t y p e , it c a n b e s e e n t h a t , b e c a u s e g ?Â 61 is p r o p o r t i o n a l t o t h e 1.5 p o w e r of t h e d e p t h a n d t h e 0.5 p o w e r of t h e d e n s i t y differ- e n c e , a s e v e r e l i m i t a t i o n i s i m p o s e d o n t h e s c o p e of t h i s a p p r o a c h . T o m a k e a r e a s o n a b l y s m a l l m o d e l ( s a y 6 i n c h e s d e e p ) of a l o c k a n d c a n a l s y s t e m w h e r e t h e p r o t o t y p e d e p t h i s 30 ft. a n d Ap/p = 0.027 ( f r e s h a n d s a l t w a t e r ) w o u l d r e - q u i r e a n i m p o s s i b l e d e n s i t y difference i n t h e m o d e l e v e n t o o b t a i n a K.îôX v a l u e of 1 0⁵, let a l o n e t h a t of t h e p r o t o t y p e . T h e p o i n t w h e r e t h e a p p r o x i m a t i o n g'= g. Ap/p i s n o l o n g e r a d m i s s a b l e h a s n o t b e e n d e t e r m i n e d a n d w o u l d d o u b t l e s s v a r y f r o m c a s e t o c a s e ( B a r r , 1 9 6 3 b).

A n y t r u e m o d e l s t u d y i s a c o m p r o m i s e , t h e e x t e n t of t h e a p p r o x i m a t i o n s a l l o w a b l e b e i n g i n i t i a l l y a m a t t e r of j u d g e m e n t a n d l a t e r a f a c t o r i n a s s e s s i n g t h e r e s u l t s . B u t t h e v a l u e s of Ap/p = 0.1 a n d r a t h e r g r e a t e r r e a c h e d b y K e u l e - g a n ( F i g . 10) i n h i s b a s i c s t u d i e s of t h e c h a n n e l a n d s e a c o n f i g u r a t i o n w o u l d a p p e a r t o b e a r e a s o n a b l e l i m i t f r o m b o t h t h e o r e t i c a l a n d p r a c - t i c a l s t a n d p o i n t s .

T h e d e v i c e o f e x a g g e r a t i o n o f t h e v e r t i c a l s c a l e i n h y d r a u l i c m o d e l s g e n e r a l l y .

T h e p o w e r of t h e m e t h o d of m o d e l s i n solv- i n g p r o b l e m s i n free b o u n d a r y h y d r a u l i c s h a s b e e n d e m o n s t r a t e d i n t h o u s a n d s of s u c c e s s f u l s t u d i e s . M o d e l i s a g a i n u s e d i n t h e r e s t r i c t e d s e n s e of a n a t t e m p t a t a s m a l l e r s c a l e s i m u l a t i o n of a l a r g e , a n d n e a r l y a l w a y s u n i q u e , n a t u r a l o r artificial h y d r a u l i c s y s t e m : f o r i n s t a n c e , p a r t of a n e s t u a r y , o r a l a r g e s t r u c t u r e s u c h a s t h e overflow s p i l l w a y of a d a m . If t h e p o s i t i o n i s e x a m i n e d i n m o r e d e t a i l w e find t h a t w h e r e g r a v i t a t i o n a l forces p r e d o m i n a t e a n d a fixed c o n t a i n i n g b o u n d a r y o b t a i n s , c o n s i s t a n t a g r e e - m e n t is f o u n d b e t w e e n m o d e l a n d p r o t o t y p e , s o l o n g as v a r i o u s w e l l e s t a b l i s h e d p r e c a u t i o n s a r e t a k e n i n t h e d e s i g n a n d u s e of t h e m o d e l . W h e r e f r i c t i o n a l forces h a v e t h e s a m e o r d e r of i n f l u - e n c e a s t h e g r a v i t a t i o n a l f o r c e s , s u c c e s s h a s a l s o b e e n m e t w i t h , b u t t o a l e s s e r e x t e n t ; i n m a n y c a s e s i n v o l v i n g l a r g e b u t r e l a t i v e l y s h a l - low p r o t o t y p e s s u c c e s s h a s b e e n d e p e n d e n t o n t h e device of e x a g g e r a t i o n of t h e v e r t i c a l s c a l e i n c o m p a r i s o n w i t h t h e h o r i z o n t a l s c a l e of t h e m o d e l . T h i s is c e r t a i n l y t r u e f o r m o s t m o d e l s of t i d a l w a t e r s : e x a m p l e s of s u c h s t u d i e s w e r e , of c o u r s e , t h e first t r u e h y d r a u l i c m o d e l s , a n d b e i n g c o n c e r n e d w i t h b e d m o v e m e n t w e r e a t t e m p t s on w h a t is still o n e of t h e m o s t c o m - p l e x of m o d e l p r o b l e m s . W h e r e t h e c o n t a i n i n g b o u n d a r y is w h o l l y o r l a r g e l y m o v e a b l e r e s u l t s

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NOVEMBRE 1 9 6 3 - № 7 - D . L H B A R R 7 4 9

S I

Non- exaggerated model (flow laminar) Modèle non-distordu ( régime laminaire 1

Probable diffusion limit Limite de diffusion probable

Exaggerated model (flow turbulent) Modèle distordu (régime turbulent)

Diffusion limit required for similorify Limite de diffusion nécessaire pour la similitude

V^s indicates surface velocities - désigne des vitesses superficielles Vm indicates mean velocities - désigne des vitesses moyennes

FIG. 12

Diagrammatic illustrations of the effect of exaggeration on velocity profiles and on the diffusion limits of a "tracer".

Représentation schématique de l'influence de la distorsion sur les profils de vitesse et les limites de diffusion d'un élément traceur.

still t e n d t o b e q u a l i t a t i v e r a t h e r t h a n q u a n t i t a - t i v e , g i v i n g , s a y , a n i n d i c a t i o n of s h o a l i n g a t s o m e p o i n t a f t e r s o m e p r o p o s e d m o d i f i c a t i o n r a t h e r t h a n a close i n d i c a t i o n of t h e e x t e n t of t h e s h o a l i n g o v e r a definite t i m e i n t e r v a l . T u r n - i n g t o t h e free s u r f a c e b o u n d a r y w e find t h a t v e r y close s i m u l a t i o n of t h e s e q u e n c e of c h a n g e of w a t e r l e v e l s c a n b e o b t a i n e d t h r o u g h o u t t h e l e n g t h of a n e s t u a r y ; t h e c o n t a i n i n g b o u n d a r y d u r i n g s u c h t e s t s b e i n g e i t h e r fixed, w i t h i t s r o u g h n e s s d e t e r m i n e d f r o m c a l i b r a t i o n r u n s , or, p e r h a p s , m o b i l e w i t h a g r e a t e r o r l e s s e r d e g r e e of s i m i l a r i t y t o w h a t w o u l d o b t a i n i n t h e p r o t o t y p e , u n d e r c o n d i t i o n s c o r r e s p o n d i n g t o t h o s e b e i n g i m p o s e d o n t h e m o d e l . T h u s v a r i o u s a s p e c t s of t h e p h e n o m e n a o c c u r i n g i n a p r o t o - t y p e m a y b e s i m u l t a n e o u s l y s i m u l a t e d i n a m o d e l w i t h v a r y i n g d e g r e e of s u c c e s s .

T h e R e y n o l d s n u m b e r c r i t e r i o n is n o r m a l l y a p p l i e d t o a s e l e c t e d l e n g t h a n d v e l o c i t y i n a p r o p o s e d m o d e l t o c h e c k t h a t t u r b u l e n t flow w i l l o b t a i n . F o r i n s t a n c e , t h e m i d s t r e a m d e p t h a n d v e l o c i t y a t t h e s h a l l o w e s t p a r t of a m o d e l e s t u a r y m i g h t b e t a k e n . I n d e s i g n i n g a m o d e l w i t h s o m e m i n i m u m v a l u e of R i n m i n d it w i l l b e f o u n d t h a t a s t h e h o r i z o n t a l scale, l/.x, is i n c r e a s e d , t h e m i n i m u m e x a g g e r a t i o n , x/ij ( w h e r e

1/y is t h e v e r t i c a l s c a l e ) , f o r t u r b u l e n t flow w i t h d e c r e a s e , e v e n t u a l l y r e a c h i n g t h e l i m i t of o n e

(i.e. n o e x a g g e r a t i o n ) . T h i s is c o n v e n i e n t i l l u s - r a t i o n of t h e difference b e t w e e n e x a g g e r a t i o n a n d d i s t o r t i o n . C o n s i d e r a flow o v e r a m o r e or less level b e d i n a p r o t o t y p e . B y e x a g g e r a t i n g t h e v e r t i c a l s c a l e of a m o d e l w e m a y e l i m i n a t e d i s - t o r t i o n of t h e v e l o c i t y d i s t r i b u t i o n , w h i c h w o u l d o c c u r if t h e flow w a s l a m i n a r , a s i l l u s t r a t e d i n F i g u r e 12. B u t w e c a n o n l y a l t e r a n d n o t eli- m i n a t e t h e d i s t o r t i o n of t h e s t r i k e p o i n t o n t h e b o t t o m of t h e l i m i t of diffusion of a t r a c e r i n - t r o d u c e d o n t h e s u r f a c e i n p r o t o t y p e a n d m o d e l (also s h o w n i n F i g u r e 12). T h e g r e a t e r t h e e x a g g e r a t i o n t h e g r e a t e r t h e e r r o r of t h e p r e d i c - t i o n i n t h i s l a t t e r r e s p e c t . I n f a c t w e w i l l a l w a y s r e q u i r e t u r b u l e n t v e l o c i t y d i s t r i b u t i o n — g i v i n g , for i n s t a n c e , t h e p r o b a b i l i t y of c o r r e c t s i m u l - a t i o n of t h e t r a v e l of s u r f a c e floats if w e r a t i o n - a l l y s c a l e t h e flows i.e. t h e m e a n v e l o c i t i e s — b u t o n l y i n n o n - h o m o g e n e i t y m o d e l s , w h e r e p a r t s of t h e w a t e r m a s s is i d e n t i f y a b l e , a r e w^re l i k e l y t o b e i n t e r e s t e d i n t h e diffusion of a t r a c e r .

A s r e g a r d s t h e s u c c e s s f u l s i m u l a t i o n of a n y p a r t i c u l a r a s p e c t of t h e p r o t o t y p e c o n d i t i o n s t h e r e a r e , t h e n , s e v e r a l p o s s i b i l i t i e s w h i c h t h e m o d e l d e s i g n e r s h o u l d c o n s i d e r i n c l u d i n g :

(i) Useful r e s u l t s c a n be o b t a i n e d o v e r a w i d e r a n g e of e x a g g e r a t i o n , a n d t h u s a s m a l l h o r i - z o n t a l s c a l e is f e a s i b l e .

(ii) U s e f u l r e s u l t s c a n b e o b t a i n e d w i t h i n a

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P R O T O T Y P E NATURE

b a ,

' 7

V E R T I C A L L Y E X A G G E R A T E D M O D E L MODÈLE OtSTOROU VERTICALEMENT

M O D E L R E S U L T S IN T E R M S OF P R O T O T Y P E

COMPARAISON DES RESULTATS DU MODÈLE.AVEC L'EVOLUTION DANS LA NATURE

(The edge of the sand bank moves from A to B in the prototype and from a to b in the vertically exaggerated model ) {Le bord du banc de sable se déplace de A en 8 dans lo nature et de a en b sur le modèle distordu verticalement )

FIG. 1 3

Diagrammatic illustration of satisfactory simulation of movement of sandbank in an exaggerated model.

Schéma d'une représentation correcte du mouvement d'un banc de sable sur un modèle distordu.

r e s t r i c t e d r a n g e of e x a g g e r a t i o n ; p e r h a p s a n u p p e r l i m i t m i g h t o b t a i n o r t h e r e m a y b e a b e s t e x a g g e r a t i o n , w i t h a d e c r e a s i n g d e g r e e of s i m i - l a r i t y a t g r e a t e r or l e s s e r e x a g g e r a t i o n s . F u r - t h e r t h i s b e s t e x a g g e r a t i o n may b e r e l a t e d t o t h e h o r i z o n t a l s c a l e c h o s e n .

(Hi) Useful r e s u l t s c a n n o t b e o b t a i n e d — e x c e p t i n a n a t u r a l m o d e l — t h u s g r e a t l y i n c r e a s i n g t h e h o r i z o n t a l s c a l e if t u r b u l e n t flow is t o be o b t a i n - e d . T h i s m a y w e l l b e q u i t e u n e c o n o m i c .

(iv) Useful r e s u l t s c a n n o t be o b t a i n e d . I t i s , b r o a d l y s p e a k i n g , t h e r e l a t i v e i m p o r - t a n c e of v e r t i c a l m o t i o n s a n d effects a s c o m p a r - ed w i t h h o r i z o n t a l m o t i o n s a n d effects t h a t d e t e r m i n e s t h e c a t e g o r y i n t o w i i i c h s o m e p a r t i - c u l a r c a s e is l i k e l y t o fall. T h e a n g l e of r e p o s e of t h e b e d m a t e r i a l i n a loose b o u n d a r y m o d e l s t u d y m a y i m p o s e a l i m i t o n e x a g g e r a t i o n . B u t w e w o u l d r e g a r d a s s u c c e s s f u l a m o d e l s t u d y if it c o r r e c t l y p r e d i c t e d t h e o v e r a l l m o v e m e n t of a flat t o p p e d s t a n d b a n k w i t h s t e e p l y s l o p i n g s i d e s d e s p i t e t h a t t h e s t r i c t s c a l i n g u p of t h e m o d e l b e d profiles w^Tould i n d i c a t e u n d u l y g e n t l e s l o p e s a t t h e edge of t h e p r o t o t y p e b a n k . T h i s p o i n t is i l l u s t r a t e d i n F i g u r e 1 3 .

A p p l i c a t i o n of k n o w l e d g e of e x c h a n g e flow t o h e a t d i s s i p a t i o n m o d e l s — a n d m o d e l s i n v o l v - i n g s m a l l d e n s i t y d i f f e r e n c e s p r e a d in g e n e r a l .

T h e p r a c t i c e of u s i n g s m a l l d e n s i t j ' differences, s a l i n e o r t h e r m a l , i n a m o d e l is c o m p a r a t i v - ely r e c e n t , t h e first i n s t a n c e s k n o w n t o t h e

a u t h o r o c c u r r i n g s o m e t w e n t y y e a r s ago (Tiffany, 1942; A p p e n d i x 1). D e m o n s t r a t i o n s of t h e t y p e of flow w h i c h m a y o c c u r w h e r e d e n s i t y differences a r e p r e s e n t h a d b e e n m a d e s o m e t w e n t y y e a r s p r e v i o u s l y ( F r e e m a n , 1 9 2 9 ) . As r e g a r d s m o d e l s t u d i e s it h a s n e a r l y a l w a y s b e e n a s s u m e d t h a t b o t h t h e F r o u d e a n d t h e d e n s i - m e t r i c F r o u d e c r i t e r i a s h o u l d b e a d o p t e d ;

Vj,

V

^{H p}

( s t a n d a r d F r o u d e n u m b e r m o d e l c r i t e r i o n ) :

for a n e x a g g e r a t e d m o d e l :

V n •g-n»t

_ I M , rfflr

'^V» .g.H,

( d e n s i m e t r i c F r o u d e n u m b e r c r i t e r i o n ) . .'. T o s a t i s f y b o t h ^ =

9m Pp

Since pm = pp w a t e r b e i n g u s e d i n t h e m o d e l , Ap^m = Ap^p.

F o r , s a y , a full t i d a l m o d e l of a n e s t u a r y t h e u s e of t h e F r o u d e c r i t e r i o n is b a s e d o n b o t h t h e o r y — t h e t i d e s b e i n g a w a v e e f f e c t — a n d

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NOVEMBRE 1 9 6 3 - № 7 D . I . H B À R R 7 5 1

k n o w l e d g e of a n u m b e r of s u c c e s s f u l i n v e s t i g - a t i o n s .

S u p p o s e t h a t w e a c c e p t t h a t A p^m = Ap^p a n d c o n s i d e r w h a t b e s t m i g h t b e d o n e t o give s i m i - l a r i t y a s r e g a r d s t h e p a t t e r n of d i m i n u t i o n of f r o n t v e l o c i t i e s of a l a r g e p r o t o t y p e e x a m p l e of l o c k e x c h a n g e flow i n a s m a l l s c a l e m o d e l w h e r e t h e h o r i z o n t a l s c a l e i s t o b e 1/x. W e k n o w f r o m F i g u r e 8 t h a t a n a t u r a l m o d e l w o u l d fail c o m - p l e t e l y i n t h i s c a s e . T h e d e v i c e of e x a g g e r a t i o n of t h e v e r t i c a l s c a l e , o f f e r s a v e r y definite i m p r o - v e m e n t ; so l o n g a s t h e b r e a d t h t o d e p t h r a t i o i n t h e e x a g g e r a t e d m o d e l i s still of t h e o r d e r of six or g r e a t e r a n d w e c a n t h u s i g n o r e s i d e effects i n b o t h m o d e l a n d p r o t o t y p e . W e c h o s e a v a l u e of V / V⁰ a n d find f r o m t h e c o n g r u e n c y d i a g r a m ( F i g . 8) t h e c o r r e s p o n d i n g v a l u e of L / H a t t h e g<^A 61 v a l u e of t h e p r o t o t y p e . U n t i l t h e d i a g r a m is e x t e n d e d — o r p o s s i b l y r e p l a c e d b y a s i m i l a r d i a g r a m f o r a d i f f e r e n t e x c h a n g e flow c a s e — w e m u s t c h o o s e V / V⁰ = 0.9. W e t h e n find a v a l u e of y (1 /y is t h e v e r t i c a l scale) so t h a t :

L / H ( m o d e l f o r V / V „ = 0.9)^v x_ . L / H ( p r o t o t y p e f o r V / V⁰ = 0.9)^X y

L / H f o r t h e m o d e l d e p e n d s o n &>^A 61 f o r t h e m o d e l a n d t h u s u p o n y, so t h e p r o c e s s i s o n e of a t r i a l a n d e r r o r . T o e x p r e s s t h i s i n w o r d s it is c o n v e n i e n t t o t h i n k of t h e d e v i c e of e x a g g e r - a t i o n a s r a t h e r a r e d u c t i o n of t h e h o r i z o n t a l s c a l e from, a c h o s e n v e r t i c a l s c a l e . W e k n o w t h e a c t u a l d i s t a n c e s a t w h i c h t h e v e l o c i t i e s of t h e p r o t o t y p e a n d m o d e l w i l l h a v e d e c r e a s e d t o t h e c h o s e n p r o p o r t i o n of t h e r e s p e c t i v e i n i t i a l v e l o c i t i e s . T h e r a t i o of t h e m o d e l d i s t a n c e t o t h e p r o t o t y p e d i s t a n c e is t h e r e q u i r e d h o r i z o n t a l s c a l e t o give s i m i l a r i t y .

T h e m e t h o d c a n be e x t e n d e d t o t r u e m o d e l s b y i m a g i n i n g t h a t a l o n g s i d e t h e e s t u a r y , or o t h e r s u c h l a r g e b o d y of w a t e r w i t h w h i c h t h e m o d e l s t u d y is c o n c e r n e d lies a c h a n n e l i n w h i c h o c c u r s a l o c k e x c h a n g e flow of t h e s a m e o r d e r of m a g n i t u d e a s t h e o c c u r r e n c e of i n t e r e s t i n t h e t r u e p r o t o t y p e . T h i s is a v i t a l p o i n t ; t h e o r d e r of m a g n i t u d e of a s a l i n e u n d e r f l o w e n t e r - i n g a r i v e r a f t e r flood c l e a r i n g — a s h a s b e e n o b s e r v e d b o t h i n n a t u r e a n d i n a m o d e l i n t h e c a s e of t h e M i s s i s s i p p i (Tiffany, 1942) is m u c h g r e a t e r t h a n w o u l d Be t h a t of a local s u r f a c e s p r e a d of h e a t e d w a t e r f r o m a p o w e r s t a t i o n d i s c h a r g i n g c o n d e n s i n g w a t e r t o t h e s a m e r i v e r . T o a s s e s s t h e s c a l e of t h e p h e n o m e n o n of i n t e - r e s t a r e p r e s e n t a t i v e d e p t h a n d d e n s i t y differ- e n c e m u s t b e e s t i m a t e d a n d a £F^A 61 n u m b e r f o r m e d .

H a v i n g d e c i d e d o n t h e h o r i z o n t a l scale of t h e r e a l m o d e l ( a n d h e n c e of t h e i m a g i n a r y m o d e l c h a n n e l a l o n g s i d e ) w e find t h e e x a g g e r a t i o n

r e q u i r e d i n t h e i m a g i n a r y m o d e l c h a n n e l so a s t o give s i m i l a r i t y w i t h t h e i m a g i n a r y p r o t o t y p e c h a n n e l i n r e s p e c t of t h e e x c h a n g e flow d i m i - n u t i o n of v e l o c i t y p a t t e r n . T h i s a p p r o a c h g a v e v e r y f a v o u r a b l e r e s u l t s w h e n a p p l i e d to t h e s p r e a d of h e a t e d w a t e r f r o m a s i m p l i f i e d outfall s y s t e m ( B a r r , 1 9 5 9 ) . B o t h m o d e l a n d p r o t o t y p e w e r e of l a b o r a t o r y size a n d t h e h o r i z o n t a l s c a l e of t h e m o d e l w a s 1 / 5 .

It m a y be o b j e c t e d t h a t t h i s m e t h o d is c o n - c e r n e d o n l y w i t h t h e l i m i t s of s p r e a d . B u t t o k n o w t h a t t h e l i m i t s of t h e s p r e a d s of a b u o y a n t field c a n b e r e a s o n a b l y w e l l s i m u l a t e d i n a m o d e l is a c o n s i d e r a b l e s t e p f o r w a r d . T h e logical n e x t s t e p w o u l d b e t o e x a m i n e t h e d e g r e e of s i m i l - a r i t y o b t a i n e d i n oLher a s p e c t s of t h e e x c h a n g e flow o r t h e s p r e a d .

M o d e l s t u d i e s of t h e o v e r a l l p a t t e r n of e x t e n - s i o n of t h e f r o n t s i n e x c h a n g e f l o w — o r t h e s e q u e n c e of e n l a r g e m e n t of t h e p e r i p h e r y of a b u o y a n t field h a s b e e n p l a c e d i n c a l a g o r y ( a ) of t h e list of p o s s i b l e effects of e x a g g e r a t i o n g i v e n i n 2 b. T h e r e is a b e s t e x a g g e r a t i o n for a n y p a r t i c u l a r c a s e a n d a n y r a d i c a l d e p a r t u r e f r o m t h a t e x a g g e r a t i o n will h a v e a m o s t d e t r i - m e n t a l effect o n t h e v a l u e of a m o d e l s t u d y . T h i s is n o t t o s a y t h a t t h e " b e s t " e x a g g e r a t i o n for a p a r - t i c u l a r h o r i z o n t a l s c a l e w i l l n e c e s s a r i l y give good r e s u l t s i n a t h r e e d i m e n s i o n a l m o d e l . T h e r e m u s t be a f u r t h e r a s s e s s m e n t of t h e d i s t o r t i o n effects of t h e e x a g g e r a t i o n . T h e s e a r e r e a l l y o u t s i d e of t h e s c o p e of t h i s p a p e r a l t h o u g h i t is p e r h a p s w o r t h m e n t i o n i n g o n e o r t w o p o i n t s . T h e r e is p o s s i b l y a p a r t i a l a l t e r n a t i v e t o t h e m o s t o b v i o u s , a n d costly, r e m e d y : to i n c r e a s e t h e h o r i z o n t a l s c a l e . S o m e e v i d e n c e h a s b e e n g i v e n t h a t i n m o d e l s of s h o r t l e n g t h s of e s t u a r i e s — t h e c a s e of m a n y h e a t d i s s i p a t i o n m o d e l s — t h e v e l o c i t i e s c o u l d be i n c r e a s e d t o s e v e r a l t i m e s t h o s e g i v e n b y t h e F r o u d e n u m b e r c r i t e r i o n . T h e d e n s i t y differ- e n c e s m u s t b e i n c r e a s e d c o r r e s p o n d i n g ; i.e. t h e s q u a r e of t h e i n c r e a s e i n v e l o c i t y ( B a r r , 1 9 6 3 ) . C o n s i d e r t h e c a s e of a d i s c h a r g e of b u o y a n t w a t e r p r o j e c t e d i n t o a s t a g n a n t p o o l a l o n g a c h a n n e l w h i c h is s h a l l o w i n c o m p a r i s o n w i t h t h e pool, a s s h o w n i n F i g u r e 14. T h i s is t h e c o n f i g u r a t i o n of t h e simplified o u t f a l l s t u d i e s ( B a r r , 1959) r e f e r r e d t o p r e v i o u s l y . S u p p o s e w e h a v e a m o d e l of t h e s i t u a t i o n , h a v e c h o s e n t h e v e r t i c a l s c a l e a n d a d o p t e i t h e r v e l o c i t i e s given b y t h e F r o u d e n u m b e r c r i t e r i o n o r s o m e definite s m a l l m u l t i p l i e r w i t h Àp/p i n a d j u s t e d a c c o r d - i n g l y . T h e flow p e r u n i t w i d t h of t h e c h a n n e l is i n d é p e n d a n t of t h e h o r i z o n t a l s c a l e a n d if t h e b r e a d t h t o d e p t h r a t i o of t h e p r o t o t y p e is v e r y l a r g e ( s a y 30-50), t h e v e r t i c a l m i x i n g p a t t e r n on a s e c t i o n a l o n g t h e c e n t r e l i n e of t h e c h a n n e l w i l l a l s o b e m o r e or l e s s i n d e p e n d e n t of t h e e x a g g e r a t i o n so l o n g a s t h e b r e a d t h t o d e p t h r a t i o i n t h e m o d e l i s still f a i r l y l a r g e — s a y 6