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The flow of water in transition sections of rectangular open channels at supercritical velocities

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THE FLOW OF WATER & TR^ITION SECTIONS
OF RECTANGULAR OPEN CHANNELS AT
SUPERCRITICAL VELOCITIES
by
Warren E. W ilson
A t h e s i s su b m itte d i n p a r t i a l f u lf illm e n t o f th e req u irem en ts
f o r th e degree o f Doctor o f P hilosophy, i n th e Department
o f Mechanics and H y d ra u lic s, in th e G raduate C ollege
o f th e S ta te U n iv e rs ity o f Iowa
August 1940
ProQuest Number: 10583824
All rights reserved
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a n o te will indicate th e deletion.
uest.
ProQuest 10583824
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Dlxls t h e s i s I s hereby approved as a cred ­
i t a b l e r e p o r t on an e n g in e e rin g p r o j e c t o r re se a rc h
c a r r i e d o u t and p re s e n te d In a manner which w a rra n ts
I t s accep tan ce a s a p r e r e q u is ite f o r th e degree f o r
which i t i s su b m itted .
I t i s to be u n d ersto o d ,
how ever, t h a t n e i t h e r th e Department o f Mechanics
and H y d ra u lic s n o r th e t h e s i s a d v is e r I s re sp o n s­
i b l e f o r th e sta te m e n ts made o f f o r th e o p in io n s ex­
p ressed *
W e s i s M v is e r
A cii
naan o f Department
iii
AC&NQWLKDCMNTS
To P ro fe s s o r K« W. Lane, a s s o c ia te d i r e c to r i n
charge o f th e I n s t i t u t e o f H ydraulic R esearch and p r o f e s ­
s o r o f h y d ra u lic e n g in e e rin g o f th e s t a t e U n iv e rs ity o f
Iowa, th e a u th o r i s in d e b te d f o r th e su g g e stio n o f th e
stu d y u n d ertak en h e r e in , and th e a s s is ta n c e ren d ered in
c a rry in g i t o u t.
The a b le a s s is ta n c e ren d ered by P ro fe s s o r H u n ter
Rouse i n g u id in g th e t h e o r e t i c a l and ex p e rim e n ta l a n a ly s is
and c r i t i c i s i n g th e m anuscript i s d ee p ly a p p re c ia te d .
To P ro fe s s o r A. A. K alinske th e a u th o r i s in ­
d eb ted f o r h e lp f u l c o o p e ra tio n and c r i t i c i s m during th e
p ro g re ss o f th e s tu d y .
For th e c o n s tru c tio n o f th e a p p a ra tu s used a t
th e U n iv e rs ity o f Iowa th e au th o r e x p re sse s h i s th an k s to
Mr. Rnver M uratsade who had b u i l t and used th e equipm ent
i n a p re v io u s s tu d y .
TO th e o f f i c i a l s o f Wayne U n iv e r s ity th e a u th o r
i s in d e b te d f o r th e u se o f th e m echanical e n g in e e rin g
la b o ra to r y equipment and th e m a te ria ls used i n c o n s tru c t­
in g th e a p p a ratu s d u rin g th e w in te r o f 1959-40.
To th e I n s t i t u t e o f H y d rau lic Research th e au­
th o r w ish es to ex p ress h i s a p p re c ia tio n o f th e c o o p e ra tio n
o f th e e n t i r e s t a f f and h i s in d eb ted n ess f o r th e use o f
th e equipm ent.
iv
TABLE OF QQmWTS
Page
1
X*
In tro d u c tio n
II*
P ro p ag atio n o f D istu rb an ces i n Super­
c r i t i c a l Flow
4
(a) Wave v e lo c ity w ith h y d r o s ta tic p re s s u re
d is trib u tio n
(b) Wave a n g le
(o) Depth change on p a ss in g u n d er a wave
fro n t
(d) Wave v e lo c ity w ith n o n -h y d ro s ta tic
p re s s u re d i s t r i b u t i o n
(©) Wave a n g le
\ f ) Depth change on p a ssin g u n d er a wave
fro n t
g) I n t e r p r e t a t i o n o f e q u a tio n s
h) L im ita tio n s
10
11
15
i ) Summary
j ) E ffe c t o f f r i c t i o n
(k) D e sc rip tio n o f flow in t r a n s i t i o n s
17
18
19
!
4
4
5
7
9
I I I * L ab o rato ry I n v e s tig a tio n
87
IF* A n aly sis o f
55
E xperim ental Data
(a) T ra n s itio n s w ith r e v e r s a l o f w a ll
c u rv a tu re
(b) Flow a t d iv e rg in g curved w a ll
V.
summary and C onclusions
(a)
lb)
35
37
53
summary
C onclusions
53
54
B ib lio g rap h y
57
Appendix
58
y
▼
INDEX TO FIGURES
Page
F ig .
1
Wave p r o f i l e and v e c to r diagram (Normal)
6
2
Wave p r o f i l e and v e c to r diagram (Nonh y d r o s ta tlc p re s s u re )
8
3
Angular depth g ra d ie n t r a t i o
13
4
T y p ical t r a n s i t i o n s w ith nom enclature
20
5
Flow a t curved w a lls (sim ple curve)
22
6
Flow a t curved w alla (w ith r e v e r s a l o f
c u rv a tu re )
22
7
A pparatus a t th e U n iv e rs ity o f Iowa (Photo­
graph)
28
e
A pparatus a t Wayne U n iv e r s ity (Sfcotograph)
28
9
Sketch o f a p p a ra tu s a t th e U n iv e rs ity o f Iowa
29
10
Simple cu rv e in p la c e
31
11
J e t tube a t Wayne U n iv e r s ity
31
IB
Channel w ith slo p in g bottom
33
13
Maximum depth r a t i o f o r t r a n s i t i o n s w ith
r e v e r s a l o f w a ll c u rv a tu re
36
14
summary o f Huns 175-178
39
13
Summary o f m iscellan eo u s ru n s (Froude
number 10-25)
40
1©
Summary o f m iscellan eo u s runs (Froude
number 40-70)
41
17
P re s s u re diagram s f o r Huns 175-178. P lo ttin g
o f k a s a fu n c tio n o f Froude number
42
18
Hun 174
47
19
Hun 175
48
20
Run 176
49
ri
INDEX TO FIGURES C o a t'd .
Page
F ig .
£1
Run 177
50
£8
BUB 178
51
1
I . INTRODUCTION
In an e f f o r t to develop a method to be used In de­
s ig n in g t r a n s i t i o n s e c tio n s f o r r e c ta n g u la r open channels in
which th e flow would be a t s u p e r c r i t i c a l v e l o c i t i e s th e stu d y
d e sc rib e d h e r e in was made*
I t was r e a liz e d t h a t t h i s problem
i s c lo s e ly a l l i e d to t h a t o f s u p e r c r i t i c a l flow i n curved
open c h a n n e ls, hence due c o n s id e ra tio n was given to th e re*
s u i t s o b ta in e d by Ippen and Knapp a t th e C a lifo rn ia i n s t i t u t e
o f Technology in th e I n v e s tig a tio n o f such flo w s.
R eferences i n c u r re n t p e r io d ic a l l i t e r a t u r e to some
unusual and o b v io u sly In co m p letely u n d ersto o d phenomena as*
s o c ia te d w ith flow in t r a n s i t i o n s in open channels served as
an in c e n tiv e to c a rry on th e work.
I t was a t f i r s t b e lie v e d
t h a t in a d d itio n to an ex p erim en tal in v e s tig a tio n a com plete
t h e o r e ti c a l a n a ly s is m ight w ell be p o s s ib le .
However, as th e
work p ro g re sse d a number o f r e v is io n s , In c lu d in g a c o n sid e r­
a b le narrow ing o f th e scope o f th e I n v e s tig a tio n , were found
im p e ra tiv e .
The o r ig in a l p la n envisaged th e development o f de­
sig n p ro ced u res f o r v a rio u s ty p es o f t r a n s i t i o n s .
A tr a n ­
s i t i o n from a re c ta n g u la r channel o f a g iv en w idth to one o f
g r e a te r w id th , e ffe c te d by means o f w a lls form ing re v e rs e d
curves ta n g e n t a t e i t h e r end t o s t r a i g h t p a r a l l e l channel
w a lls , was th e f i r s t to be s tu d ie d .
I t was soon ap p aren t
t h a t t h i s ty p e o f t r a n s i t i o n in v o lv ed a flow p a t t e r n f a r
to o com plicated to p erm it complete a n a ly s is a t th e p re s e n t
£
time*
A t r a n s i t i o n in v o lv in g sim ple curved w a lls ta n g e n t
a t e i t h e r end to s t r a i g h t w a lls which a re i n i t i a l l y p a r a l­
l e l and which d iv erg e downstream from th e curved s e c tio n
was found t o he th e most c o u p le r s t r u c t u r e th e d esig n o f
which could he undertaken w ith some degree o f c o n fid e n c e .
The ex p e rim e n ta l work in v o lv ed a la r g e number o f
experim ents on t r a n s i t i o n s w ith r e v e r s a l o f w a ll c u rv a tu re
a s w ell as w ith th e sim p le r d iv e rg in g w a lls .
A wide range
o f p e r tin e n t r a t i o s o f le n g th s in v o lv ed In th e geom etry o f
th e t r a n s i t i o n s as w e ll a s a wide range o f Froude numbers,
th e l a t t e r b e in g lim ite d by th e v e l o c i t i e s o b ta in a b le in
th e m odels, were
used to f a c i l i t a t e th e fo rm u latio n o f an
e m p iric a l s o lu tio n i n th e event t h a t no a n a ly t ic a l one was
found.
A t h e o r e t i c a l a n a ly s is o f th e flow bounded by a
d iv e rg in g curved w a ll w ith n o n -h y d ro sta tic p re s s u re d i s t r i ­
b u tio n w ith p a r t i c u l a r re fe re n c e to p re s s u re s le s s th an
h y d r o s ta tic was made and an I n t e r p r e t a t i o n o f th e e x p e ri­
m ental d a ta i n th e l i g h t o f t h i s a n a ly s is i s g iv e n .
Equa­
t i o n s were d e riv e d which a id i n making a q u a l i t a t i v e stu d y
o f th e ex p erim en tal d a ta , b u t which a r e n o t as y e t s u f f i ­
c i e n t l y com plete to p r e d ic t q u a n ti ta t iv e l y th e elem ents o f
th e flow a t a d iv e rg in g curved w a ll o r in th e main body o f
th e liq u id i n th e ch an n el.
The li m ita tio n s upon th e use
o f th e d a ta , which were o b tain ed w ith r e l a t i v e l y sm all
s t r u c t u r e s , and s im ila r d a ta on sm all models f o r p r e d ic tin g
S
th e elem ents o f th e flow In a f u l l s c a le channel a re s e t
fo rth *
4
IX* PROPAGATION Of DISTURBANCES IN
SUPERCRITICAL FLOW
(a) Wav© v e lo c ity w ith h y d r o s ta tic
p re s s u r e d i s t r i b u ti o n
When w ater flow s i n an open channel a t a v e lo o ity
i n ex cess o f th e v e lo c ity o f p ro p a g a tio n o f a sm all wav© in
w ater o f th e g iv en d ep th , th e flow i s s a id to be a t super­
c r i t i c a l v e lo c ity , th e c r i t i c a l v e lo c ity b eing equal to th e
elem en tary wave v e lo c ity .
Small d istu rb a n c e s such a s in ­
crem ents o r decrem ents o f th e w ater depth a re propagated a t
wave v e lo c ity and cannot p ro ceed upstream i n s u p e r c r i t ic a l
flow*
The value o f th e elem en tary wave c e l e r i t y
c
re la ­
t i v e to th e f l u i d i s g iv en by th e e x p re ssio n ,
c
i n which
g
( 1)
i s th e a c c e le r a tio n due to g r a v ity and
th e depth o f th e w ater*
d
is
The d e r iv a tio n o f t h i s e q u a tio n
and a l l th o se d e sc rib e d h e re in may be found in th e appen­
dix*
I t i s assumed t h a t th e wave h e ig h t i s sm all and th e
wave le n g th g r e a t r e l a t i v e to depth and t h a t th e p re s s u r e
d i s t r i b u t i o n i s h y d r o s ta tic .
(b)
Wave an g le
I f a sm all d istu rb a n c e i s continuous i n n a tu re
such as would be th e r e s u l t o f continuous flow p a s t a
change in th e d ir e c tio n o f a channel w a ll, a wave f r o n t i s
form ed, s t a r t i n g a t th e d is tu rb a n c e p o in t and extending
5
in to th e flo w in g w ater a t an an g le w ith th e o r i g i n a l d ir e c ­
t i o n o f flow u s u a lly term ed /3 ,
th e wave a n g le .
I b is
a n g le i s d efin e d by th e e x p re ssio n
s i n /3 •
i n which
and
flow .
u
o
(2)
i s th e ware v e lo c ity d e fin e d by E quation (1)
i s th e mean v e lo c ity in th e o r ig i n a l d ir e o tio n o f
Upon s u b s t i tu t i o n o f th e e x p re ssio n f o r
s in
c
we have,
J
s in p »
or
where
~
f
(3)
(3a)
i s th e Froude number d efin e d b y th e e x p ressio n
(c) Depth change on p a ssin g under a ware f r o n t
As th e w ate r p a sse s under a ware f r o n t, th e com­
ponent o f momentum p e rp e n d ic u la r to th e ware fro n t under­
goes a change in m agnitude p ro p o rtio n a l to th e n e g a tir e
change i n depth o f th e w ate r, whereas th e component p a r a l ­
l e l to th e ware f r o n t rem ains unchanged, s in c e th e d e r a ­
t i o n o f th e su rfa c e alo n g th e ware f r o n t i s c o n s ta n t.
Ih©
changes in depth and components o f r e l o c i t y a re re p re se n te d
in F ig . 1.
A sim ple d e r iv a tio n on th e b a s is o f c o n tin u ity ,
geom etry o f tb e v e c to r diagram , and th e momentum p r in c ip le
le a d s to th e fo llo w in g e q u a tio n f o r th e increm ent o f depth
i n te rn s o f a n g u la r change i n th e d i r e c ti o n o f th e bound­
a ry , wave a n g le and v e lo c ity o f th e w a te r,
aa .
i p2 tan/3 • 4
e
(4)
I f th e p re s s u re d i s t r i b u ti o n i n th e body o f th e
l i q u i d may be co n sid ered e s s e n t i a l l y h y d r o s ta tic and i f
f r i c t i o n lo s s e s may b e ig n o re d , th e flow along a curved
boundary may be s u c c e s s f u lly an aly sed on th e b a s is o f th e
elem en tary p r in c ip le s s e t fo rth above*
A most s u c c e s s fu l
a n a ly s is o f t h i s ty p e was made by Xppen and Knapp in ”A
Study o f High V e lo c ity How in Curved S ectio n s o f Open
C hannels”, Pasadena, C a lif o r n ia , March £9, 1936*
In t h a t s tu d y an e q u a tio n d e riv e d on th e as sump-
7
tlo n o f c o n s ta n t v e lo c ity alo n g th e w a ll proved b e s t in
d e s c rib in g th e w a ll p r o f i l e o f th e w ater s u rfa c e along th e
curved wall*
I h i s e q u a tio n w il l be used h e r e in f o r p u r­
poses o f com parison and i s g iv en h e re ,
i n which th e s u b s c r ip t aero re fe r® , a s i t w i l l throughout
t h i s d is c u s s io n , to c o n d itio n s a t th e b eginning o f th e
curved wall#
The s ig n ific a n c e o f th e term s i n E quation
(5) i s a s fo llo w s;
d
« dep th o f th e w ater
Z3
m th e wave an g le
e
* th e t o t a l an g le betw een th e ta n g e n t
a t th e b eg in n in g o f curve and th e
ta n g e n t a t th e p o in t a t which th e
depth i s d .
F
» th e Froude number d e fin e d by th e ex­
p re s s io n
2
F m SL
gd
(d) Wave v e lo c ity w ith n o n -h y d ro s ta tic
p re s s u re d i s t r i b u ti o n
L et i t be assumed t h a t th e p re ss u re d i s t r i b u ti o n
a t th e w a ll and w ith in th e body o f th e liq u id i s not hy­
d r o s ta tic #
In F ig . 2 i s shown a s e c tio n s im ila r to sec­
ti o n A-A o f Fig# 1 tak en p e rp e n d ic u la r to a wave f r o n t,
which w i l l now be d efin ed as a l i n e along which th e re l a
no change o f momentum o f th e l i q u i d .
The force®
Vx
and
Pg p e r u n i t w idth a c tin g
on th e ends o f th e fre e body a re th e n o n -h y d ro s ta tic p re s ­
su re fo rc e s which w i l l be re p re se n te d th u s ,
( 6)
i n which /Q
I s th e u n i t d e n s ity o f th e liq u id and th e
c o e ffic ie n ts
and
kg
re p re s e n t th e f a c to r s by which
th e t o t a l h y d ro s ta tic p re s s u re co rresponding to th e depths
d^
and
p ressu re#
dg
must be m u ltip lie d to o b ta in th e a c tu a l t o t a l
I t w ill be assumed t h a t th e f r i c t i o n fo rc e s a c t ­
in g on th e elem ent o f li q u id a re n e g lig ib le In comparison
w ith th e p re s s u re fo rces*
We may now proceed to a d e r iv a tio n o f th e v e lo c i t y o f p ro p a g a tio n o f a sm all wave w ith c o n d itio n s as
d e sc rib e d above*
The v e lo c ity i s assumed t o be un ifo rm ly
d is t r i b u te d throughout a v e r t i c a l s e c tio n .
e q u a tio n i s w r itte n i n th e fo llo w in g form,
The momentum
9
*j. * P8
V
( • * H - ° l dl )
(7)
From th e c o n tin u ity r e la tio n s h ip we h ave,
°1
^
Assume now t h a t th e wave h e ig h t I s sm a ll, t h a t I s
I s a v e ry sm all q u a n tity compared to
•
d^ * dg
th e sim u ltan ­
eous s o lu tio n o f E quations (6 ), (7 ), and (8) le a d s to th e
fo llo w in g e q u a tio n f o r th e wave v e lo c ity
• ' f * h *
e,
(1 ' | ) f
m
For th e case o f h y d r o s ta tic p re s s u re d i s t r i b u t i o n
m kg
m l#
i n t h i s ease E quation (8) reduces to
E quation (1) which g iv e s th e v e lo c ity o f p ro p a g atio n o f th e
elem en tary wave w ith h y d ro s ta tic p re s s u re d i s t r i b u t i o n
(e)
Wave an g le
By d e f in i t i o n th e wave a n g le i s given a s th e
r a t i o o f th e wave v e lo c ity to th e mean v e lo c ity o f th e l i q ­
u id .
E quation (£) th u s g iv es th e s in e o f th e wave an g le
i n th e g e n e ra l form.
Equation (3) g iv e s th e v alu e f o r th e
p a r t i c u l a r case o f h y d ro s ta tic p re s s u re d is tr ib u tio n *
fo llo w in g eq u a tio n g iv e s th e s in e o f th e wave angle f o r
th e case o f n o n -h y d ro sta tlo p re s s u re d i s t r i b u t i o n ,
The
which In tu r n red u ces to th e e q u a tio n f o r th e ware angle
w ith h y d ro s ta tic p re s s u re d i s t r i b u t i o n when
k
*
!•
( f ) Depth change on p a s s in g under a ware f r o n t
Employing th e momentum e q u a tio n and w ith re g a rd
f o r th e geom etry o f th e r e c t o r diagram o f f i g . B, th e f o l low ing e q u a tio n f o r th e depth change ex p erien ced by th e
l i q u id p a ssin g under a ware f r o n t may be d e r ir e d 9
(U )
which may be reduced to th e more s i g n if i c a n t form
( 12)
through th e assum ption, which i s v a lid f o r la rg e fro u d e
numbers, t h a t
s in /? »
tan/7 •
s t a t i c p re ss u re d i s t r i b u t i o n
f o r the case o f hydro­
k « 1
and Ak/a& * 0
and
E quation (11) w ill red uce to E quation (4) which was de­
r iv e d s p e c i f i c a l l y f o r th e case o f h y d r o s ta tic p re s s u re
d is trib u tio n ,
xn u s in g
to r e p r e s e n t
ap p a ren t t h a t th e s ig n o f Ak /&3
s ig n o f
a&/ag
when
k
and
d
k^ - k 2
i t is
i s p o s itiv e as i s th e
r e s p e c tiv e ly in c re a s e in
th e d ir e c tio n o f p ro p a g a tio n o f th e wave in d ic a te d in
11
Fig* 2.
(g) I n t e r p r e ta t i o n o f equation®
E quation (12) may be employed to make a v e ry s ig ­
n i f i c a n t oo ap arieo n between th e r a t e o f change i n th e depth
AH/ao
which may be expected w ith n o n -h y d ro sta tlo p re ss u re
and th e corresp o n d in g r a t e o f change o f depth i f th e p re s ­
su re d i s t r i b u t i o n were h y d r o s ta tic ,
thro u g h o u t th e follow ­
in g d is c u s s io n , fo r th e sake o f b r e v ity th e t e r n "normal”
w i l l be used to r e f e r to elem ents o f flow w ith h y d ro s ta tlo
p re ss u re d i s t r i b u t i o n , and th e s u b s c r ip t
n
w i l l denote
th e q u a n titie s which a r e elem ents o f th e norm al flow .
I f E quation (4) i s solved f o r
tan/3 and
u
and
g
ndA®
and i f
a r e e v a lu a te d in term s o f th e Froude
number th e r e r e s u l t s ;
(13)
I f la rg e v a lu e s o f
F
a re to be co n sid ered th e s im p lif ic a ­
t i o n o f dropping th e f a c t o r u n ity i n th e denom inator may be
made and th e fo llo w in g sim ple e x p re ssio n r e s u l t s :
(14)
The r a t i o o f th e expected a n g u la r depth g ra d ie n t
ndAe to th e normal g ra d ie n t may now be ex p ressed i n th e
form,
Ad/AB
(A? K
d-fF
12
A o r ite r lo n may now be e sta b lish e d which w ill in d ic a te for
any given s e t o f co n d itio n s whether th e angular depth gra­
d ien t & d/ae
nay he expected to he g rea ter or l e s s than
th e normal v a lu e.
I f the r a tio expressed in Equation
(16)
has th e valu e u n ity th e angular depth grad ien t w ith nonh y d ro sta tlo pressu re d istr ib u tio n w ill he equal to th e
normal v a lu e.
S e ttin g th e r a tio equal to u n ity and reduc­
in g i t to sim p lest form one ob ta in s fo r the c r ite r io n the
ex p ressio n ,
£& -JL_
. g ji
1 - It
(16)
ne
I t I s now ap p aren t th a t f o r any given l o c a l v alu e
o f th e fro u d e number th e r e i s a c e r ta in com bination o f th e
f a c to r s d e s c rib in g th e c h a r a c t e r i s t i c s o f th e flow which
w ill r e s u l t i n an a n g u la r depth g ra d ie n t eq u al to th a t f o r
normal flow .
However, i f th e c h a r a c t e r i s t i c s o f th e flow
a r e such t h a t th e com bination o f term s on th e l e f t hand
s id e o f E quation (16) i s g r e a t e r o r le e s th an th e q u a n tity
2 F& th e s u rfa c e slo p e w i l l be l e s s o r g r e a t e r th a n normal
r e s p e c tiv e ly .
U iat such i s th e case i s a p p a re n t from f i g .
3 w herein a r e p lo tte d v alu es o f th e an g u lar depth g ra d ie n t
r a t i o f o r v a rio u s assumed com binations o f th e v a r ia b le s
Ff
k
and Akfae .
f o r any g iv e n v a lu e s o f
an in c re a s e o r d ecrease i n
Akfae
F
and
k
.w ill r e s u l t i n a de~
crease o r in c re a s e in th e a n g u la r dep th g ra d ie n t r a t i o
r e s p e c tiv e ly , which d em onstrates th e above sta te m e n t.
14
Uhe same c r i t e r i o n w i l l in d ic a te th e r e l a t i v e
v alu e o f th e wave angle*
However in t h i s case v alu es o f
th e q u a n tity on th e l e f t o f Equation (16) g r e a te r o r l e s s
th an
8 f¥ ~ in d ic a te
normal r e s p e c tiv e ly .
v a lu e s o f /& g r e a te r o r le s s th a n
O bviously th e same c r i t e r i o n a p p lie s
to th e wave v e lo c ity s in c e th e wave v e lo c ity divided by
th e average v e lo c ity o f th e flow i s equal to
s in p •
In
t h i s co n nection an i n t e r e s t i n g form o f th e eq u atio n fo r th e
an g le
i s o b tain ed by a tra n sfo rm a tio n s im ila r to t h a t
used i n o b ta in in g E quation (IE ),
-
t h i s eq u a tio n i s ,
<*'*I1*
,w l
th ese eq u a tio n s may be used to advantage i n d is ­
cu ssin g th e flow a t a curved boundary s in c e i t has been
e s ta b lis h e d e x p e rim e n ta lly th a t th e p re s s u re a t th e w all
i s i n c e r ta in cases n o n -h y d ro statlo *
p o s s ib le to in te g r a te th e eq u atio n f o r
I t i s a t p re s e n t im­
A&fae
to o b ta in
th e dep th d i r e c tl y as a fu n c tio n o f th e v a r ia b le s
k
F
and
s in c e th ese must f i r s t b e determ ined ex p e rim e n ta lly o r
a method must be dev ised to r e l a t e th e f a c to r
k
d ire c tly
to th e cu rv atu re o f th e s u rfa c e , th e Froude number and th e
depth*
ifoe Interdependence o f th ese q u a n titie s i s ap p a ren t
from th e f a c t th a t th e subnormal p re ss u re r e s u l t s from
v e r t i c a l a c c e le r a tio n , which i s a fu n c tio n o f the s u rfa c e
c u rv a tu re , which in t u r n depend® upon th e Froude number
15
and r e l a t i v e c u rv a tu re o f th e wall*
A method fo r th e de­
te rm in a tio n o f th e su rfa c e contour® i n s u p e r c r i t i c a l flow
based on th e analogy w ith su p erso n ic gas flow® ha® been
d ev ised and I s adm irably s e t fo rth by H rnst P reisw erk i n
a work which h a s been tr a n s la te d and p u b lish ed by th e
N atio n al A dvisory Committee f o r A eronautics i n T echnical
Memorandums #954 and 955*
In a s e r i e s o f experiment®
c a r r ie d on a t th e I n s t i t u t e fo r Aerodynamics o f th e
S id g en o essisch e Teohnisch© B ochsehule, Z urich, th e v a l i d i ty
o f t h i s method was demonstrated*
However, i t i s a p p lic a b le
o n ly when th e v e r t i c a l components o f the a c c e le ra tio n o f
th e f lu id a r e n e g lig ib le *
O bviously such a method cannot
be u sed s a t i s f a c t o r i l y f o r th e a n a ly s is o f flow s d iscu ssed
herein *
A s im ila r method o f a tta c k would be d e s ira b le fo r
t h a t type o f flow , b u t th e fo rm u latio n o f such an a n a ly s is
i s beyond th e scope o f th e p re s e n t work which seeks o n ly
to p o in t o ut th e e x is te n c e o f th e problem and th e g en eral
c h a r a c t e r i s t i c s o f th e flow*
(h)
L im ita tio n s
In o rd e r to i n t e r p r e t th e flow p ro p e rly i n term s
o f th e eq u atio n s s e t f o r th above, th e follow ing f a c to r s
must be k ep t in mind*
In d e riv in g th e eq u atio n s a wave
f r o n t was d efin ed a s a lin e along which th e re i s no change
in momentum o f th e liq u id *
A wave f r o n t i s th e re fo re no
lo n g e r f u l l y c h a ra c te riz e d by th e f e a tu r e th a t a l l p o in ts
on i t have th e same s u rfa c e e le v a tio n , as was th e case
16
f o r th e elementary* ware theory*
B ather th e wave f r o n ts
a r e l i n e s I n th e d ir e c tio n o f which th e re I s no n e t a c c e l­
e r a tin g fo rc e a o tin g on th e liq u id *
S urface e le v a tio n
co n to u rs w ill n o t serv e now a s wave f r o n ts sin c e th e tr u e
wave f r o n t i s a ls o determ ined in p a r t by contours o f con­
s ta n t mean p ressure*
However, th e c h a r a c t e r is t i c s o f th e
s u rfa c e co n to u rs do to some e x te n t in d ic a te th e shape o f
th e wave f r o n t s , and a re u s e f u l in stu d y in g c e r ta in fe a ­
tu r e s o f th e flow*
The wave f r o n ts a re determ ined by the
f a c t th a t th e q u a n tity
lin e s*
fed® must be a c o n sta n t along th e se
T his fo llo w s from th e f a c t th a t th e re i s no n e t
fo rc e in a given d ir e c tio n when F^ - Pg
th e d e f i n i t i o n o f
of
k
i s aero*
From
th e n e c e s s ity f o r th e c o n sta n t v alu e
kd® fo llo w s immediately*
I t i s to be n o ted th a t th e increm ents 4 d
nk
a re d e fin e d as th e increm ents in
d
t i v e l y p e rp e n d ic u la r to th e wave fro n t*
ao
in th e r a t i o s adyfoe
and
ak/ae
and
k
and
resp ec­
The increm ent
in d ic a te s the change
in d ir e c tio n o f th e v e lo c ity v e c to r upon p a ssin g under the
wave fro n t*
The d ir e c tio n a l c h a ra c te r o f th e se r a ti o s i s
im portant sin c e th e p re s s u re g ra d ie n t v a r ie s w ith the
d ir e c tio n i n which I t i s measured*
In o rd e r to e v a lu a te th e f a c to r
k
from th e ex­
p erim en tal d ata i t i s n e c e ssa ry to determ ine th e u n it
p re ssu re a t v a rio u s p o in ts on a v e r t i c a l l i n e extending
from s u rfa c e to bottom o f liq u id *
T his would make po ss-
17
lb l « a com putation o f th e t o t a l p re ss u re on a v e r t i c a l area*
By d e f in itio n
k
I s th e r a t i o o f t h i s t o t a l p re ss u re to
th e h y d r o s ta tic p re ss u re on an eq u al a re a s im ila r ly located*
(1) Summary
To summarise th e p r i n c ip l e fe a tu re s o f th e ex­
tended th e o ry o f th e p ro p a g a tio n o f sm all d istu rb a n c e s in
s u p e r c r i t ic a l flow th e follow ing statem en ts may be made*
1*
The wave v e lo c ity and wave angle as w ell as
th e an g u lar depth g ra d ie n t may be g r e a te r o r l e s s th an th e
normal v alu es depending upon th e r e l a t i v e l o c a l v alu es o f
th e q u a n titie s
k
and Ak/ae f o r a given Froude number*
I f th e angle /3
2.
i s g r e a te r than normal th e
an g u lar depth g ra d ie n t w ill be l e s s and i f th e angle
I s I ess th a n normal th e an g u lar depth g ra d ie n t w i l l be
g r e a te r than normal*
3*
In o rd e r th a t no d istu rb a n c e may be propa­
g ated I t i s not o n ly n e c e ssa ry t h a t th e lo c a l v alu e o f
be aero b u t a ls o t h a t
A k/^e
be zero*
k
This i s apparent
from a stu d y o f E quation (9) f o r th e wave v e lo c ity *
4*
Wave f r o n ts do n o t co in c id e w ith th e su rfa c e
e le v a tio n co n to u rs o f the body o f th e liq u id In th e channel*
The d eterm in atio n o f th e lo c a tio n and form o f th e wave
f r o n ts re q u ire s a knowledge o f th e p re ssu re d is tr ib u tio n
w ith in th e flu id *
3*
P re d ic tio n o f th e elem ents o f the flow in ­
c lu d in g th e s u rfa c e p r o f i l e a t th e curved w a ll in any b u t
18
an e m p iric a l manner must aw ait f u r th e r a n a ly s is o f th e me­
ch a n ic s ©f th e flow w ith s p e c ia l re fe re n c e to th e e v a lu a tio n
o f th e p re s s u re i n term s o f th e s u rfa c e s lo p e , th e r e la tio n ­
s h ip b ein g dependent upon th e components o f th e a c c e le ra ­
t i o n w ith in th e body o f th e liq u id *
(J) E ffe c t o f f r i c t i o n
I f a v isc o u s liq u id flow s a t s u p e r c r it ic a l ve­
l o c i t y on a h o riz o n ta l su rfa c e in a channel o f uniform
w id th th e lo s s In m echanical energy due to conversion in to
h e a t by th e v isc o u s fo rc e s w ill r e s u l t in an in c re a s e in
depth o f th e liq u id *
This may be v e r if ie d e i t h e r by a d i­
r e c t c o n s id e ra tio n o f th e s p e c if ic energy diagram o r by an
In s p e c tio n o f th e d i f f e r e n t i a l e q u a tio n o f v a rie d flow*
This e q u a tio n i n g e n e ra l form i s ,
,,
dd
dr-
where
s lo p e ,
,
u
vSo~cx/e
j~
J s<*
L
B i s th e h y d ra u lic ra d iu s ,
c
s0
i s th e Chezy c o e f f ic i e n t ,
i s th e bottom
x
i s th e co o rd in a te
i n th e d ir e c tio n o f which th e v e lo c ity
u
th e v e lo c ity i s s u p e r c r it ic a l th e team
u ^ /s ^ which i s th e
Eroude number i s la rg e compared to unity*
case o f flow on a h o riz o n ta l su rfa c e
s0
i s taken*
When
Also fo r th e
i s zero*
The
eq u atio n th e n red u ces t o ,
dd
—cV?
d*
/~ ja "
T his q u a n tity i s o b v io u sly p o s itiv e in d ic a tin g an In crease
19
l a d ep th .
T his c o n tr a s ts w ith th e d e c reasin g depth o f sub-
o r i t i o a l flow*
Xt stay he concluded th e re fo re th a t th e
e f f e c t o f f r i c t i o n on th e depth would he to in c re a s e i t
above th e v alue which would he expected f o r a p e r f e c t f l u i d .
(k) D e sc rip tio n o f flow i n t r a n s it io n s
S ince t r a n s i t i o n s a r e com binations o f curved and
s t r a i g i t channel w a lls , a good id e a may he fo raed o f th e
n a tu re o f th e flow which may he e j e c t e d on th e b a s is o f
th e p r in c ip le s s e t f o r th above*
sim ple ty p e s o f t r a n s i t i o n s .
In F igure 4 a r e shown two
C onsider f i r s t , to e lim in a te
co m p licatin g f a c t o r s , a t r a n s i t i o n o f th e type shown i n
Fig* 4 a, o f very g r e a t width*
As th e w ater e n te rs th e
d iv e rg in g s e c tio n , th e su rfa c e a t th e w all w ill f a l l in
accordance w ith th e elem entary th e o ry and Equation (5)
w ill ad eq u ately d e sc rib e th e s u rfa c e p r o f i l e a t th e w a ll
p ro v id in g th e assum ptions made in th e d e riv a tio n a re sub­
s t a n t i a l l y f u l f i l l e d , i . e * , v e r t i c a l a c c e le r a tio n I s neg­
l i g i b l e and f r i c t i o n lo s s e s may be ignored*
In o rd e r th a t
such be th e c a se , th e cu rv a tu re must be slow enough to in*
su re h y d ro s ta tic p re s s u re d i s t r i b u ti o n and y e t not so slow
th a t th e f r i c t i o n e f f e c t s are appreciable*
She su rfa c e
co n to u rs w ill th en be s t r a ig h t l i n e s making th e a n g le /?
w ith th e w a ll a t a l l p o in ts*
t h i s an g le p
w ill decrease
i n m agnitude along th e w a ll a s th e depth decreases s in c e
o b s e rv a tio n s in d ic a te t h a t th e v e lo c ity rem ains e s s e n t i a l l y
co n stan t*
51253 ?
ai
In fig # 5 i s shown a sk e tc h o f th e flow along a
d iv e rg in g boundary which f o r th e sake o f o l a r l t y in de­
s c r i p t i o n i s shown a s a s e r ie s o f s h o rt s t r a i g h t segments
each making th e an g le
a©
w ith the p reced in g segment*
She wave f r o n ts th e n appear a s l i n e s w ith f i n i t e spacing*
Ih e stre a m lin e s w i l l be a s in d ic a te d in th e sketch*
me
sp ac in g o f th e stre a m lin e s g ra d u a lly becomes g r e a te r a s
th e flow proceeds alo ng th e boundary*
As th e w ate r p asses
u n d er each s u c c e ssiv e wave fro n t i t changes i n d ir e c tio n
so t h a t th e flow i s alw ays p a r a l l e l to th e corresponding
segment o f th e w all*
In th e case o f a smooth curve th e
flow would be m o d ified to th e e x te n t t h a t th e wave f r o n ts
would be spaced a t in f in ite s im a l d is ta n c e s , and th e stream ­
li n e s would be smooth cu rv es.
In Fig* 6 i s shown a t r a n s i t i o n o f a s l i g h t l y
more com plicated type*
Ihe flow up to th e p o in t o f in ­
f l e c t i o n o f th e curve w ill be a s p re v io u s ly described*
A fte r p a ssin g th e p o in t o f in f le c tio n o f th e curve th e
e f f e c t w ill be s im ila r to th a t i n th e i n i t i a l p a r t o f the
channel but w ith In c re a s in g r a th e r th a n d ecreasing depth
and w ith th e stre a m lin e s becoming c lo s e r spaced r a th e r than
w ider spaced as before*
Ifce u ltim a te depth a tta in e d a t
th e s t r a i g h t waU w ill be i d e n t i c a l w ith th e I n i t i a l depth
and th e sp acin g o f th e stre a m lin e s w ill assume th e o r ig in a l
value*
I h l s r e s t o r a t io n o f i n i t i a l c o n d itio n s w il l occur
p ro v id in g th e re i s ai d:
boundar y in t h i s
FIGURE
22
re g io n , th e boundary drag i s n e g lig ib le and th e waves a re
o f th e same type fo r both convex and conoave c u rv atu re o f
th e w a ll,
i f th e s e c o n d itio n s a re f u l f i l l e d the liq u id
w ill be moving i n th e same d ir e c tio n w ith undim inished
v e lo c ity a f t e r p a s s in g th e curved w a ll.
There i s th e re ­
fo re no n e t change i n momentum, and consequently no change
in d ep th .
I f a channel o f th e f i r s t type i s o f f i n i t e w idth,
waves c ro ss in g from th e o p p o site s id e s o f the channel w ill
augment th e depth change a t a l l p o in ts o f in te r s e c tio n , in
acco rd w ith th e p r in c ip le s o f wave in te r f e r e n c e .
The s u r­
face c o n to u rs , m oreover, w i n then be curved lin e s sin c e
in te r f e r e n c e o f waves augmenting th e depth change w ill p ro ­
duce l e s s e r depths i n th e c e n tr a l re g io n o f th e flow in
th e channel along a given wave f ro n t than e x i s t a t th e w a ll.
I t i s th e re fo r e n ec essary to go upstream to fin d a p o in t
w ith a depth eq u al to t h a t a t th e w all on a given wave
f r o n t.
The an g le between contours and w all w i ll then equal
th e wave an g le o n ly a t p o in ts u n a ffe c te d by waves c ro ssin g
th e ch an n el.
S im ila rly i n a
tr a n s it i o n o f th e second type
th e n e g a tiv e waves c ro s s in g th e channel w ill reduce the
e f f e c t o f th e p o s itiv e wave.
The su rfa c e contours w ill
ag a in be curved l i n e s .
I f th e f r i c t i o n lo s s e s may not be n eg lec te d ,
t h e i r e f f e c t w ill be evidenoed a t th e w all by an in c re a se
i n th e e le v a tio n o f th e w ater su rfa c e and a corresponding
34
d i s t o r t i o n o f th e wave fro n ts*
I f th e p re ss u re d i s t r i b u ti o n i s not h y d ro s ta tic ,
th e e f f e c t s in d ic a te d i n th e development o f th e equ atio n s
o f wave v e lo c ity , wave a n g le , and change in depth f o r nonh y d ro s ta tic p re s s u re d is tr ib u tio n may be expected*
Pro­
ceed in g ag a in from th e sim ple to th e complex, th e flow a t
th e d iv erg in g curved w all may f i r s t be discussed*
As th e
l iq u id e n te r s th e d iv erg in g s e c tio n th e d e v ia tio n o f th e
w a ll from a s t r a i g h t lin e produces a re d u c tio n in th e w all
p r e s s u r e , which in tu r n produces a v e r t i c a l a c c e le ra tio n o f
th e liq u id *
t h i s p re ss u re re d u c tio n has an immediate e f f e c t
upon th e wave v e lo c ity , wave a n g le , and slope of th e liq u id
s u r fa c e , i n t h a t th e s e q u a n titie s w ill be e i th e r la r g e r o r
s m a lle r th an th e normal v alu es depending upon th e r e la tiv e
v alu es o f th e f a c t o r
k
and th e r a t i o Ak/A&
to th e l o c a l Froude number*
k
m r e la tio n
A q u a n tita tiv e e v a lu a tio n o f
i n term s o f th e su rfa c e slo p e end Fronde number has not
y e t been made, hence only tre n d s may be in d icated *
T&q r e l a t i v e ra d iu s and Fronde number to g e th e r
determ ine th e m agnitude o f th e p re ss u re reduction*
For
s h o r t r e l a t i v e r a d i i , o r la rg e Froude number, th e v e r tic a l
a c c e le ra tio n s o f th e p a r t i c l e s a re la rg e ; consequently a
c o n sid e ra b le re d u c tio n i n p re ssu re i s to be expected*
Fig*
3 i s a g ra p h ic a l re p re s e n ta tio n o f th e eq u atio n f o r th e
an g u lar depth g ra d ie n t from which may be o b tain ed d e f in ite
in fo rm atio n on th e g e n e ra l foim o f th e su rfa ce curve*
If
25
th e l i q u id i s assumed to o u te r th e channel w ith uniform
T e lo c ity throughout th e cro ss s e c tio n , h y d ro s ta tic p re ssu re
d i s t r i b u ti o n ! and w ith n e g lig ib le boundary drag th e v alu es
of
k
and th e r a t i o
s p e c tiv e ly #
Ak/&&
would b© u n ity and zero r e ­
Th© p o in t i n F ig . 5 in d ic a tiv e o f t h i s i n i t i a l
c o n d itio n would H e on th e h o riz o n ta l l in e in d ic a tin g th e
v alu e o f r a t i o
^ /<$£)*
equal to unity#
on t h i s l in e would be determ ined by
I t s p o s itio n
th e i n i t i a l Froude
number#
Th® us© o f th e diagram to p r e d ic t su rfa c e pro­
f i l e s I s n o t f e a s ib le sin c e th e re i s one degree o f in determ inancy in v o lv ed due to th e f a c t t h a t k
y e t been expressed a s a fu n c tio n o f
rad iu s#
F
and
has not
th e r e la ti v e
The diagram w ill be used i n th e a n a ly sis o f flow s
o f t h i s type f o r which d ata a re a v a ila b le #
The strea m lin e
diagram and su rfa c e contour diagram f o r a flow o f th® type
u nder d is c u s s io n would c o n s titu te a m o d ificatio n o f those
shown in Fig# 5 f o r th e case o f h y d ro s ta tic p re ssu re d is ­
tr ib u tio n #
The c h ie f d iffe re n c e s would be curved r a th e r
th a n s t r a i g h t wave f r o n t s , and d i f f e r e n t an g les between
co n to u rs and w all and wave fro n ts and wall#
Th© e f f e c t s o f the n o n -h y d ro sta tic p re ssu re
d i s t r i b u t i o n on th e flow i n a t r a n s i t i o n o f th© second
ty p e would c e n te r p rim a rily in th e upstream p o rtio n o f th e
tra n s itio n .
However, secondary e f f e c ts would b© ev id en t
i n such a t r a n s i t i o n o f f i n i t e w idth sin ce the waves c ro ss -
26
lu g th e channel would have d i f f e r e n t angles th an those of
th e normal flow .
She r e s u ltin g re d u c tio n in su p erelev a­
t i o n would d i f f e r th en in m agnitude from th a t o f th e nor­
mal ca se.
27
I I I . LABORATORY INVESTIGATION
The n a tu re o f th e ap p aratu s used In th e e x p e ri­
m ental work may h e a t be understood by re fe re n c e to th e
photographs in F ig s. 7 and 8 and th e sk e tc h in F ig. 9,
These show th e g e n e ra l arrangem ent o f th e apparatus a t both
th e U n iv e rs ity o f Iowa and Wayne U n iv e rs ity .
The two s e ts
o f equipment were e s s e n t i a l l y th e same in p r in c ip le , p ro ­
v id in g f o r th e in tr o d u c tio n o f a j e t o f w ater re c ta n g u la r
in c r o s s - s e c tio n to a channel w ith w a lls which formed a
t r a n s i t i o n o f th e d e s ire d type*
I t was assumed th a t o n ly
o n e -h a lf o f th e channel need be u sed , th e c e n te r - lin e be­
in g sim u la te d by a p l a t e o f g la s s form ing a s tr a i g h t
v e r t i c a l w a ll, and th e curved w all being sim ply a p iece o f
p y r a lln so c o n stru c te d th a t i t could be formed to th e de­
s ir e d p la n and h e ld in p la c e .
As may be seen in F ig s. 7 and 9 , th e source o f
w ater a t th e U n iv e rs ity o f Iowa was a te n inch d is tr ib u ­
tio n p ip e coming from th e c o n sta n t head ta n k s .
from t h i s p ip e was c o n tro lle d by a s i x
The flow
in ch valve lo c a te d
i n a p ip e le a d in g from a te e i n th e main l i n e .
The w ater
passed through a w e ir box equipped w ith a 90° v-notch
w eir and hook gage to a channel i n which th e flow was
q u ie te d by means o f b a f f l e s , thence through a device which
pro v id ed a means f o r v ary in g th e s iz e o f th e j e t o f w ater
in tro d u c e d to th e ch an n el.
The j e t Issu e d from a tu b e ,
r e c ta n g u la r in c r o s s - s e c tio n w ith c o n tra c tio n s suppressed
28
vie. e
H ook
Gage and 'S+////n<y We//
PLAN
W e / r
IV/df/t Oep-tfi Control
7502
>’Pyra/fn Wo/I
//T o
KSump
S F C T /O N
P R /N C
/P A L
APPARATUS
AT
U N ! A T P S / T P O P I OH/A
Figure
F
30
on two s id e s by moans o f c y l i n d r i c a l s u rfa c e s o f one fo o t
r a d iu s , th e o th e r two s id e s b ein g th e bottom and one s id e
o f th e approach ch an n el.
T his equipment made p o s s ib le th e
p ro d u c tio n o f a J e t w ith minimum dim ensions o f 1" x 1" and
maximum dim ensions o f ©" x 6 ".
A ll com binations o f th e
in te rm e d ia te d ep th s and w id th s were p o s s ib le b u t l im ita ­
tio n s on th e q u a n tity o f w ate r and T e lo c ity a v a ila b le fix e d
th e maximum s in e a t about tw en ty -fo u r sq u are Inches in
c r o s s - s e c tio n a l a r e a .
The t r a n s i t i o n s e c tio n was con­
s tr u c te d on a h o r iz o n ta l p la n e su rfa c e which c o n s iste d o f
com position board mounted on a s u ita b le s t r u c t u r a l fram e,
p a in te d and marked o f f in a re c ta n g u la r c o o rd in a te system .
D e ta ils o f a
t r a n s i t i o n s e c tio n may be seen in f i g , 10,
The ap p aratu s u sed a t Wayne U n iv e rs ity d iffe re d
i n one e s s e n t i a l re s p e c t from t h a t d e sc rib e d above — no
ad ju stm en t o f th e e ls e o f J e t was p o s s ib le .
The J e t was
formed by means o f a g a lv a n ise d iro n t r a n s i t i o n p ie c e from
a th re e in c h c i r c u la r p ip e to a one and o n e -h a lf inch
square tu b e .
The J e t Issu e d d i r e c tl y upon a h o riz o n ta l
s u rfa c e formed by a tw o-inch plan k mounted on a s tr u c tu r a l
fram e.
The s u rfa c e was p a in te d and marked in a re c ta n g u la r
c o o rd in a te system as was th e one a t th e U n iv e rsity o f Iowa.
The d isch arg e was determ ined by means piezom eters in th e
t r a n s i t i o n s e c tio n o f th e tube which was c a lib ra te d in
p la c e .
This tu b e i s shown in g r e a te r d e t a i l in Big, 11,
The w ate r was pumped d i r e c t l y to t h i s tu b e by a c e n tr if u g a l
SL
Fig* XO
F ig . XX
32
pump from a sump w ith no p ro v is io n f o r c o n sta n t head main­
te n a n c e .
Hfc a p p re c ia b le head v a r ia ti o n was n o ted .
At b o th th e U n iv e rs ity o f Iowa and Wayne Uni­
v e r s i t y th e dep th o f th e w a te r was determ ined by means o f
a p o in t gage mounted on a tr a v e lin g b a r which gave f u l l
freedom o f m otion In a h o r iz o n ta l p la n e .
The lo c a tio n o f
th e gage was determ ined In term s o f two c o o rd in a te s from
th e o r i g i n o f th e c o o rd in a te system marked on th e channel
bottom by means o f g rad u ated ta p e s and In d ic a to r s a tta c h e d
to th e t r a v e li n g b a r and gage I t s e l f .
The e le v a tio n o f
th e w a te r s u rfa c e was determ ined by p o in t gage read in g s
on th e s u rfa c e and channel bottom.
With th e a p p a ra tu s d esc rib ed above i t was p o ss­
i b l e to d eterm in e th e q u a n tity o f w ater flow ing i n th e
channel and th e depth a t any p o in t w ith in th e boundaries
o f th e t r a n s i t i o n .
By d i r e c t com putation th e lo c a tio n o f
s u rfa c e c o n to u rs and th e shape o f su rfa c e p r o f i l e s could
th e n be d eterm in ed .
A d d itio n a l equipment was used i n s e v e ra l s tu d ie s
to se c u re c e r t a i n in fo rm a tio n in a d d itio n to th a t which
could be o b ta in e d w ith th e ap p a ratu s d e sc rib e d above.
3 h ls equipm ent c o n s is te d o f a channel w ith slo p in g bottom
I l l u s t r a t e d i n T ig . I S , piezom eters In channel bottom and
w a lls and th re a d s a tta c h e d to th e channel bottom , th e
l a t t e r p ro v id in g both v is u a l and photographic in d ic a tio n s
o f th e d ir e c tio n o f th e v e lo c ity v e c to rs a t s p e c if ie d
Rubber
Seal
f
x4$r/ngo
a d j u s t a b l e
OHA N N E
Fl GURL
3 LO PE
BOTTOM
/td /o s fo b /e
Pipe
34
p o in ts i n th e c h a n n el.
In a m a jo rity o f th e ru n s , n o ta b ly th o se made a t
th e U n iv e r s ity o f Iowa during th e summer o f 1939, th e ob­
j e c t was sim ply to determ ine th e e le v a tio n o f th e li q u id
s u rfa c e w ith in and beyond th e t r a n s i t i o n s e c tio n .
Ihe pro­
cedure under th e s e circum stances was sim ply to e s ta b lis h
th e c o n d itio n s d e s ire d by forming th e w all to th e r e q u is i t e
form , p ro v id in g a j e t w ith th e n e c e ssa ry cro ss s e c tio n and
e s ta b lis h in g a flow w ith th e s p e c if ie d Froude number, and
th e n to make a s many o b se rv a tio n s w ith th e p o in t gage on
th e w ate r s u rfa c e a s was deemed a d v is a b le .
In l a t e r r u n s , when th e im portance o f th e p re s ­
s u re d i s t r i b u ti o n had been r e a l i s e d , more numerous observa­
tio n s on th e e le v a tio n o f th e s u rfa c e o f th e liq u id were
made and th e p re s s u r e s a t c e r ta in s p e c if ie d p o in ts were
determ ined by means o f p iezo m eters.
55
IT . ANALYSIS OF EXPERIMENTAL DATA
(a j T ra n s itio n s w ith r e v e r s a l o f w all c u rv a tu re
The e s s e n t i a l fin d in g s o f t h i s stu d y e re summa­
r iz e d b r i e f l y l a F ig . 13 w herein i s shown th e r a t i o o f th e
mailTOm depth alo n g th e curved w all
depth ex p ressed by th e r a t i o
to th e i n i t i a l
a s a fu n c tio n o f th e
i n i t i a l Froude number, w ith th e p e r tin e n t g eo m etrica l r a t i o s
In d ic a te d i n th e le g e n d .
I n F ig . 4 I s shown th e nomen­
c l a t u r e used h e re in on a sk e tc h o f t y p ic a l tr a n s itio n s e
u se o f th e r a t i o s
m/bc
and
b0/ b o
was based on a stu d y o f
th e In flu e n c e o f th e fundam ental r a t i o s
m/d0 ,
b0/ d 0
b0/ d o
which in d ic a te d th a t th e use o f th e p ro d u ct
d$/bo
*
xn/b0
The
and
m/d0 x
was j u s t i f i e d sin c e a l l p o in ts re p re se n tin g
d ata based on t r a n s i t i o n s w ith equal v alu es o f t h i s produot
o f le n g th r a t i o s la y on th e same s t r a i g h t lin e *
s i t u a t i o n e x is te d w ith re s p e c t to th e r a ti o
A s im ila r
b0/b o •
th e p r in c ip le f e a tu r e o f t h i s diagram i s th e f a c t
t h a t i t conforms to e x p e c ta tio n s on th e b a s is o f th e e le ­
m entary wave th eo ry in a g e n e ra l manner*
The r a t i o
d^/do
exceeds th e valu e u n ity f o r o n ly a few cases which may be
ex p la in e d on th e b a s i s o f high l o c a l v e r t i c a l a c c e le ra tio n s
a t th e downstream p o in t o f tangency between curve and
s t r a i g h t w all*
At low Froude numbers th e value o f
ddp
I s low , s in c e th e n e g a tiv e waves, making la rg e angles w ith
th e o r ig in a l d ir e c tio n o f flow , cro ss th e ohannel and r e ­
duce th e s u p e re le v a tio n o f th e w ater surface*
c u rv e s, which a re in d ic a te d by la rg e v alu es o f
In long
m/b0 ,
a
37
s im ila r e f f e c t I s n o tic e d , s in c e th e wave f r o n t s , even
though making sm all a n g le s w ith th e w a ll, cro ss th e channel
w ith in th e l i m i t s o f th e t r a n s i t i o n and red u ce th e super*
e le v a tio n *
The flow in th e upstream p o rtio n o f t h i s type o f
t r a n s i t i o n i s id e n t i c a l w ith th a t i n a t r a n s i t i o n w ithout
r e v e r s a l o f w all curvature*
The a n a ly s is o f th e d a ta f o r
t h i s p a r t o f th e channel w i ll he c a rrie d o u t in th e next
s e c tio n which r e l a t e s o n ly t o flow in a channel hounded hy
d iv e rg in g curved w a lls and s t r a i ^ i t tangents*
A q u a n tita tiv e a n a ly s is o f th e re d u c tio n i n super­
e le v a tio n hy th e waves c ro ss in g th e channel i s n o t f e a s ib le ,
s in c e th e flow w ith subnormal p re ssu re i s n o t y e t s u b je c t
to com plete a n a ly s is even when th e waves do n o t cro ss w ith­
i n th e l i m i t s o f th e t r a n s i t i o n .
(b) Flow a t d iv e rg in g curved w all
The th e o ry o f th e p ro p ag atio n o f sm all d is tu rb ­
ances in a li q u id flow ing a t s u p e r c r it ic a l v e lo c ity was
extended to in c lu d e th e s i t u a t i o n wherein subnormal p re s ­
s u re s e x i s t w ith in th e body o f th e liq u id in an e f f o r t to
e x p la in d iffe re n c e s which became ev id en t during th e in v e s t­
ig a tio n betw een a c tu a l s u rfa c e p r o f i l e s and th o se p re d ic te d
by th e th e o ry based on norm al p re s s u re d i s t r i b u ti o n .
In
o rd e r to determ ine th e e x p la n a tio n f o r th e s e d iffe re n c e s
between th e p r e d ic tio n s o f th e elem entary th e o ry and a c tu a l
o b s e rv a tio n i t i s n e c e s s a ry to e s ta b lis h c e r ta in fa cts*
38
S in ce th e elem entary th e o ry assumed th a t th e
f r i c t i o n e f f e c t s were n e g lig ib le and th e p re s s u re d is tr ib u ­
t i o n was h y d r o s ta tic , la c k o f conform ity w ith e i t h e r o f
th e s e assumed c o n d itio n s would m e rit c o n s id e ra tio n .
I t has
been shown p re v io u s ly th a t th e f r i c t i o n a t th e boundaries
would tend to in c re a s e th e s u rfa c e e le v a tio n .
She e x is t­
en ce o f a no n -h y d ro statl© p re ss u re d is tr i b u tio n could have
th e e f f e c t o f e i t h e r r a is in g o r low ering th e e le v a tio n
above th e norm al, depending upon th e r e la ti v e ra d iu s and
th e Fronde number*
I t rem ains th e re fo re to e i t h e r e s ta b lis h th e
f r i c t i o n a l d ra g a s th e e x p la n a tio n f o r th e d iffe re n c e be­
tween normal and a c tu a l p r o f i l e s o r to e lim in a te i t from
c o n s id e ra tio n and to e s ta b lis h o r disprove th e e x iste n c e
o f subnormal p re s s u r e s and d iscu ss th e p o s s ib le e f f e c t s i f
th e e x is te n c e o f such p re s s u re s i s dem onstrated, , In o rd e r
t o c a rry o u t th e n e c e ssa ry a n a ly s is o f th e experim ental
d a ta th e p l o t t i n g s shown i n F ig s. 14, 15, 16, and 17 were
p re p a re d .
In stu d y in g th e e f f e c t o f f r i c t i o n a l drag i t i s
f i r s t n e c e s s a ry to e s t a b l i s h , i f p o s s ib le , th e probable
m agnitude o f th e s u rfa c e slo p e s due to t h i s f a c to r a lo n e .
Secondly, th e r e l a t i v e m agnitude o f the e f f e c t o f f r i c t i o n
i n v ario u s ca ses should be e v a lu a te d .
Bie f i r s t o f these
p o in ts i s sim p ly handled by eosnm tlng th e m agnitude o f the
s u rfa c e slo p e f o r a g e n e ra l case by means o f th e v a rie d
iH B H rainsam
M IM W IHHIll
B H H H
g i m n s i B a s s s s s
M
a a a B
H B a m
V^^WHinsnilliHllllllBAIll
;MaBM3maMRBSMBRM««gfUMMaRl
!C»nR«iEsaBaaBHBRaBnRaBB2aBB»»HHn
HSaiilSIMHS
D
w e m s io n l e s s
Curved
h/ALL
JD/MENSIONLESS FOKM
BGURE
D
B o jk o b h
k ’ XT i T i T
f* R E s s u r b 5
fw R tfiE c r r o N
H
r I j i Li.i U
\ Li i 1 IT. f ' M 1 l.j 1 1 .i. i..U J J
43
flow equation*
As h as been shown p re v io u s ly , t h i s e q u a tio n
re d u c e s to th e fo llo w in g form f o r la r g e Froude numbers w ith
flow ta k in g p la e e on a h o riz o n ta l su rface*
M
E
-
-tt2/ ***
1 - u 8/ gd
Tbia e q u a tio n may a ls o be w ritte n more sim ply l a th e f o l ­
low ing form ,
g
■
This e q u a tio n may be used to determ ine approxim ately th e
s u rfa c e slo p e due to f r i c t i o n a l drag a lo n e .
The approxima­
t i o n l i e s c h i e f l y in th e v alu e o f Chezy*s c o e ffic ie n t* ta b ­
u la te d v a lu e s o f which have been e v a lu a te d fo r uniform flow.
The second p o in t l a concerned c h ie f ly w ith th e f a c t th a t
th e lo n g e r th e curved su rfa c e along which th e liq u id I s in
c o n ta c t th e g r e a t e r should be th e t o t a l e f f e c t o f th e
f r i c t i o n a l d ra g .
TO s tu d y th e s e two p o in ts F ig s. 14, 15, and 16
w ill be u s e f u l.
In Huns 175-178 th e i n i t i a l Froude number
was 10, th e i n i t i a l depth was approxim ately 0.33 f t . , and
th e i n i t i a l v e lo c ity 10 f t . p e r sec.
The value o f Chezy's
C was n o t determ ined b u t f o r rough c a lc u la tio n s th e value
100 w i l l be found to be a f a i r l y good average fig u re .
The
h y d ra u lic ra d iu s would b e i n i t i a l l y approxim ately 0 .1 f t .
and would n o t d e c re a se m a te r ia lly f o r a c o n sid erab le
4-4
d is ta n c e downstream*
The s u rfa c e slo p e due to f r i c t i o n
would he th e n ap p ro x im ately
0*01
*
We should ex p ect th e n In th e d is ta n c e 0*25 f t . from th e be­
g in n in g o f curve to a p o in t downstream t h a t th e r e would be
accum ulated due to f r i c t i o n a lo n e an increm ent in e le v a tio n
above th e normal o f approxim ately 0.QQ3 f t .
A c tu a lly we
observe th e fo llo w in g ,
R
ft.
A ctual d
ft.
Normal d
ft.
Increm ent
ft.
0.7 1
1 .0
2 .0
3 .5
0 .1 3
0 .2 4
0 .3 3
0 .3 4
0.004
0.123
0.250
0.290
0.116
0.112
0.080
0.050
These in crem en ts a re n o t o n ly much l a r g e r th an th o se i n d i­
c a te d f o r f r i c t i o n alo n e b u t t h e i r v a r ia tio n w ith r e l a t i v e
ra d iu s i s in th e d ir e c tio n o p p o site to th a t which would
o b ta in were th e se in crem ents due to f r i c t i o n a lo n e .
The
l a t t e r f e a tu r e becomes even more marked i f we co n sid er
g r e a t e r d is ta n c e s downstream.
E v en tu ally th e lo n g e st r a ­
d iu s curve shows a s u rfa c e p r o f i l e dipping below th e normal
p ro file .
A d d itio n a l evidence on th e se two p o in ts may be
o b ta in e d from th e d ata i n F ig s. 15 and 16.
In th e case o f
th e v e ry la rg e Froude numbers one might expect a more p ro ­
nounced f r i c t i o n e f f e c t .
C onsider fo r example a Froude
number o f 70, a depth o f 0.1 3 f t . , an h y d ra u lic ra d iu s o f
45
0*04 f t . and a v e l o c i t y o f 17 f t . p e r se c .
The slo p e due
to f r i c t i o n i n t h i s case would h© ag ain 0 .0 1 .
I t i s ob­
v io u s w ith o u t f u r t h e r c a lc u la tio n th a t the observed d i f ­
fe re n c e s betw een normal and a c tu a l p r o f i le s exceed th o se
which would be caused by f r i c t i o n .
Ho d e f in ite tre n d s
w ith r e s p e c t t o th e r e l a t i o n betw een th e r e l a t i v e ra d iu s
and th e s u p e re le v a tio n in th e case o f th e high Froude
numbers i s o b s e rv a b le .
The o th e r p lo ttin g s f o r low er Froude
numbers shown i n F ig s. 15 and 16 p re se n t s im ila r evidence.
I t i s c l e a r from th e a v a ila b le d a ta th a t a f r i c t i o n a l drag
i s I n s u f f i c i e n t to e x p la in th e observed d iffe re n c e s between
normal and a c tu a l d ep th s.
The e x is te n c e o f subnormal p re ssu re s in th e flows
in v e s tig a te d i s e s ta b lis h e d by th e d ata shown in th e p lo t­
tin g s o f th e in d iv id u a l Buns 175-178 a s w ell a s by th e
summary o f th ese d a ta in F ig . 17.
of
k
ranged from
*0.7
to
-0 .4 7 .
tio n s o f l e s s e r accu racy v alu es o f
were o b serv ed .
In Runs 175-178 v alu es
In previous observa­
k
as low as
-4 .0
In Runs 175-178 th e o b se rv a tio n s f o r
k
were made a t a v e r t i c a l s e c tio n a s c lo s e a s was p o s s ib le
to th e b eg inning o f th e cu rv e.
Subnormal p re s s u re s o f such a magnitude th a t th ey
were a c t u a l l y su b -atm o spheric were f i r s t observed in ex­
p erim en ts perform ed w ith th e ap p aratu s a t Wayne U n iv e rsity .
There was a t f i r s t some doubt concerning th e accuracy o f
th e p iezo m eter re a d in g s which In d ic a te d the sub-atm ospheric
46
p re ssu re s*
The p o s s i b i l i t y o f i r r e g u l a r i t i e s in th e appa­
r a tu s was co n sid ered and th o ro u g h ly in v e s tig a te d and th e
o b s e rv a tio n s repeated*
th e same*
The r e s u l t s were s u b s t a n ti a lly
s e v e r a l months l a t e r th e runs numbered 174-178
were made a t th e U n iv e rs ity o f Iowa w ith th e r e s u l t s shown
i n th e p l o t t i n g s o f Figs* 18, 19, 20 , 21 , and 88*
The
su b -atm o sp h eric p re s s u re s were ag ain observed although th ey
were o f a l e s s e r magnitude*
The l e s s e r m agnitude was to
be ex p ected s in c e th e Froude number was o n ly 10 in t h a t
s e r i e s o f ru n e whereas i t had been a s h ig h as 70 i n th e
ru n s made a t Wayne U n iv e rs ity .
An e x p la n a tio n o f th e e x is te n c e o f a sub-atm os­
p h e ric p r e s s u r e c lo se to th e f r e e s u rfa c e o f a body o f
l i q u id h a s n o t been u n d ertak en a t t h i s time*
I t may w ell
be e x p la in e d by th e f a c t t h a t th e p re s s u re a s y m p to tic a lly
approaches atm ospheric a s th e d ista n c e from th e f r e e s u r­
fa ce decreases*
However, t h i s has n o t y e t been f u l l y
e s ta b lis h e d and se rv e s o n ly a s a p o s s ib le working hypoth­
e sis*
The e f f e c t o f subnormal p re s s u re s may b e s t be
u n d ersto o d by re fe re n c e to Fig* 3*
th a t fo r
Assume f o r example
F0
*
100
and a c e r ta in r e l a t i v e ra d iu s th e
re d u c tio n o f
k
from i t s i n i t i a l value o f u n ity to a
v alu e o f 0*8 o ccu rs w ith
n k /a o
■
ond to r a s h o rte r
r e l a t i v e r a d iu s and i d e n t i c a l Froude number th e re d u c tio n
i s to
k
•
0*6
w ith
Akfoe
»
4*0 .
I t i s apparent
!KV53EI
SM IllbilMlill
laftBisaisaBiii
licSsaciaaw
ESSSSSSSSSSSEBSS&SBSSSSSSBiiiS£BSp"™3 iiiH
■ m i i « l i l MWlMM«a!MWWPesgiga5gi8Sia a a 5 5
a W B M B l B t f t t i l d B I
BSSSSSSBSfiSSSSSSBSSSSSSSBSSSSiSSE”**!"**""
liisssBBsssssassssssssssSassssssisiisiiiii
a s 8 iia a iB s a g a M 8 3 8 a ^ B B
ISgFSaBBBBaBliaHB!
iB
B
R
B
H
B
B
B
B
IS
K
H
H
i
laBBBgiBBBBBlBHl
li&HsiaiBHaiiMaa
ssssasssssssss
ig£3S!3£sii££i!3
!£ £ £ £ £ £ £ £ £ £ £ £ £ £ £
m
a m
iS8SSSSSIS1SSS8|
SSSBSBSSBf
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lliliigiS
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l|W
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iBBBBBBBaBBBBflaa
BBBBBBBBBBBBRBB
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missus
IttBBBMBBBBB
1881811181
aiH SSSBS
iBiaaBBBBBi
mbmbm
iissssssiS
InS S B pP rE sS B B B B S S sSS
iMaSHajaBNsaBaBnBBJBBRsaa!
M
rBBBi*aa
M H S B B BBS
InfcHSsI
a a m
a a c
eBBH
jSSkSSS!
SSffigSBRBSI
■ h u iiu i
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n
E
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■m 8S88u Sb83S
58
from th e diagram t h a t the r e s u lt in g d e p ressio n o f th e s u r fa e e w ill he g r e a t e r fo r th e s h a rp e r curvature* f o r a g iv en
AG assum ing th e change In
s i m il a r s e t of v alu es fo r
F to he small*
k
and
Ak/ne
However a
would y ie ld an
e n t i r e l y d i f f e r e n t r e s u l t f o r an i n i t i a l Froude number of
10*
In t h i s ca se th e s itu a tio n would he re v e rse d and th e
l e s s e r cu rv atu re would give th e g r e a te r depression*
l a t t e r s i tu a tio n i s ap parent in Runs 175-178*
The
I t fo llo w s
th e r e f o r e th a t th e extended th e o ry p rovides a q u a l i t a t i v e
b a s i s fo r th e e x p la n a tio n fo r a p o rtio n of th e observed
effe c t® o f subnormal pressure*
The v a r ia ti o n o f
w ith
F
k w ith th e r e l a t i v e r a d iu s and
1® in d ic a te d from the a v a ila b le d a ta as shown in
Figs* 14 and 17.
S h o rt r e la tiv e r a d i i give sm all v alu es
of
k and h i # Froude numbers
of
k
g iv e sm all
k Ts •
The value
approaches u n ity as th e r e l a ti v e ra d iu s in c re a s e s
and a s th e Froude number approaches unity*
A compariaon o f th e su rfa c e contours o f Buns 174
and 175 in d ic a te s t h a t th e re d u c tio n o f e le v a tio n o f th e
s u rfa c e i s e f fe c te d more ra p id ly w ith a sh o rt ra d iu s w all
i n p la c e th an in th e fre e jet*
53
V. SUlkSMARY AND CONCLUSIONS
(a) Summary
The o r ig in a l o b je c tiv e o f th e in v e s tig a tio n which
was th e e s ta b lis h in g o f p r in c ip le s and th e fo rm u latio n o f
a p ro ced u re to be used in th e desig n o f t r a n s i t i o n s in
r e c ta n g u la r open channels i n which th e flow would be a t
s u p e r c r i t i c a l v e l o c i t i e s , was m odified to th e e x te n t de­
s c rib e d below*
The g en e ral c h a r a c te r is ti c s o f th e flow in
a t r a n s i t i o n w ith r e v e r s a l o f w all cu rv a tu re were in v e s t i­
g a te d th o ro u g h ly and on th e b a s is o f th a t stu d y i t was con­
clu d ed t h a t alth o u g h th e g e n e ra l b ehavior was in accord w ith
th e th e o ry a q u a n tita tiv e p re d ic tio n o f the elem ents of th e
flow was n o t f e a s ib le a t t h i s tim e fo r two reasons*
firs t
o f th e se was t h a t the flow was a c tu a lly very complex and
could n o t always be broken down in to sim ple flow s th e e le ­
m ents o f which could be p red icted *
The second reaso n was
t h a t th e assum ption o f h y d ro s ta tic p re ss u re d i s t r i b u ti o n
used i n th e elem en tary th eo ry and equations developed by
Ippen and Knapp was n o t f u lf ille d *
As a consequence o f
th e l a t t e r f a c t n e ith e r th e elem ents o f th e flow in th e up­
stream p o rtio n o f a t r a n s i t i o n w ith re v e rs a l o f w all curva­
t u r e nor th o se o f th e ensuing downstream flow , which was
d i r e c t l y a f f e c te d th ere b y , were su b je c t to q u a n tita tiv e
p re d ic tio n *
An a n a ly s is , both th e o r e tic a l and experim ental in
c h a r a c te r , was made o f th e flow along a d iv e rg in g curved
54
w a ll assum ing n a n -h y d ro s ta tic p re ssu re d is trib u tio n , to de­
term in e th e c h a r a c t e r i s t i c s o f th e su rfa c e p ro file *
The
r e s u l t la g e q u a tio n s p ro v id e an e x p la n a tio n o f th e d i f f e r ­
ences between th e observed su rface p r o f i l e s and th o se p re ­
d ic te d by th e th e o ry based upon the assum ption o f hydro­
s t a t i c p re s s u re d is trib u tio n *
The e x te n sio n o f the theory o f th e p ro p ag atio n o f
sm all waves i n a liq u id flow ing a t s u p e r c r i t ic a l v e lo c ity
to in c lu d e th e s i t u a t i o n involving non-hydros t a t io p re s ­
s u re d i s t r i b u t i o n and th e dem onstration t h a t p re s s u re s
l e s s th a n h y d ro s ta tic and indeed sub-atm ospherle may e x i s t
i n a flow bounded by d iv erg in g curved w a lls a re th e p r in c ­
i p l e f e a tu r e s o f th e in v e s tig a tio n *
(b)
Conclusions
The summary o f th e fin d in g s r e la te d to t r a n s i ­
tio n s w ith r e v e r s a l o f w all cu rv atu re and th e co n clu sio n
t h a t in g e n e ra l th e flow w ith th a t type o f boundary i s too
complex f o r complete a n a ly s is a t p re se n t a re s e t f o r th
above*
The fo llo w in g conclusions summarize th e fin d in g s
r e l a t e d to th e flow bounded by sim ple d iv erg in g curved w alls*
The word normal i s used to r e f e r to th e elem ents o f th e
flow p re d ic te d by th e elem entary th eo ry based upon th e
assum ption o f h y d ro s ta tic p re ss u re d is tr ib u tio n and n e g lig ­
i b l e f r i c t i o n a l e ffe c ts *
1*
Experimental observations e sta b lish the fa c t
55
t h a t th e r e i® a d iffe re n c e between th e a c tu a l p r o f il e s o f
th e l i q u i d s u rfa c e a t the curved boundary and those p re ­
d ic te d by th e th e o ry based upon th e assum ption o f n e g lig ­
i b l e f r i c t i o n a l e f f e c t s and h y d ro s ta tic p re ss u re d is tr ib u tio n *
8.
The experim ental d ata in d ic a te th a t th e d i f ­
ference® between th e normal and a c tu a l su rfa c e e le v a tio n
a r e o f a m agnitude which cannot be ex p lain ed by the f r i c ­
t i o n a l drag*
3.
The e f f e c t o f a r e l a t i v e l y long ra d iu s i n p ro ­
ducing a la rg e r e l a t i v e In c rease in th e e le v a tio n due to
f r i c t i o n a l drag was n o t observed*
O ccasio n ally long r e l a ­
t i v e r a d i i were accompanied by lee® su p e re le v a tio n above
nonaal e le v a tio n th an was found f o r s h o rt r e l a t i v e ra d ii*
4*
Subnom al pressure® were observed a t the w all
and in th e body o f a l i q u id flow ing a d ja c e n t to a curved
wall*
5*
The extended th eo ry o f th e p ro p ag atio n o f
sm all disturbance® i n a liq u id flow ing a t s u p e r c r itic a l ve­
l o c i t i e s in d ic a te s th e same type o f d iffe re n c e s between
a c tu a l and normal p r o f i l e s as were observed*
6*
The p re se n ce o f a sh a rp ly curved waU may
e f f e c t a more ra p id re d u c tio n o f the su rfa c e e le v a tio n o f
a l i q u i d flow ing on a h o riz o n ta l su rfa c e than could be
accom plished in a fr e e j e t flowing on th e same su rfa ce a t
th e same i n i t i a l Froude number*
7*
The e x is te n c e o f subnormal p re s s u re s , which
56
may fee a c tu a ll y sub-atm ospheric, a t th e curved w all o f an
open channel t r a n s i t i o n c re a te s a s tr u c tu r a l problem whioh
should fee co n sid ered i n th e desig n o f such s tr u c t u r e s .
B7
BIBLIOGRAPHY
1.
Knapp, R. T ., and A. T. Ipp©nt "C u rv ilin e a r Flow o f
L iq u id s w ith Free Surface a t V e lo c itie s above t h a t
o f Wave P ro p ag a tio n ," Proceedings o f th e F if th
I n te r n a tio n a l Congress f o r A pplied M echanics,
page 531. John F. Wiley and Sons. 1938.
2.
P re isw e rk , E rn s t, "A pplication o f th e Methods o f Gas
Dynamics to Water Flows w ith F ree S u rfa c e ," Tech­
n i c a l Memorandums o f th e n a tio n a l Advisory Com- *
m ltte e f o r A ero n au tics, #934 and 933, March 1940.
3.
Rouse, H unter, "F lu id Mechanics f o r H ydraulic Engi­
n e e r s ," McGraw-Hill Book Company In c . 1938.
4.
von K am an, Theodor, "Eine p ra k tia e h e Anwendung d er
A nologie zwischen U eberschallstroem ung in Gasen
und u e b e r k r itls c h e r Stroemung in o ffe n en G erlnnen,"
Z e it. f . Aug. Math. u . Meoh., Band 18, H o f f l.,
pages 49-56.
58
AEPENDIX
(a) D e riv a tio n o f Equations
1 . Wav© v e lo c ity w ith h y d ro s ta tic p re ssu re
d is trib u tio n
f o r th e sake o f s im p lic ity th e a r t i f i c e o f su p e rim posing upon th e liq u id a uniform motion having th e ve­
l o c i t y o f p ro p a g a tio n o f th e sm all wav© w ill be u sed .
S his
tra n sfo rm s a problem o f unsteady flow to one o f ste a d y flow .
We s h a ll c o n sid e r a n e g a tiv e wave, th a t i s one a r is i n g from
a d e p ressio n o f th e w ater s u rfa c e .
to th e l e f t in f i g . £•
This wave would move
The e n t ir e body o f f lu id w ill then
be im agined to be moved to th e r ig h t a t the v e lo c ity
th u s h o ld in g th e wave form m o tio n less.
o
The liq u id w il l th en
appear to flow through th e wave form towards th e r i g h t .
I t i s assumed th a t th e v e lo c ity i s uniform
throughout th e d ep th , th e p re s s u re d i s tr ib u tio n i s hydro­
s t a t i c , and th e f r i c t i o n a l lo s s e s a r e n e g lig ib le .
The momentum equation i s w ritte n fo r s e c tio n s
1 and £ o f f i g . 1 in t h i s manner:
The c o n tin u ity eq u atio n gives th e follow ing in fo rm atio n ,
•"■"V' 'lr
^
S ince th e p re s s u r e d is t r i b u ti o n i s h y d ro s ta tic i t follow s
th a t
,0
12
... - , " /
59
S u b s titu tio n o f t h i s value fo r
and upon e lim in a tio n o f
(P^ * Pg)
above g iv e s
Og by means o f th e c o n tin u ity
e q u a tio n th ese re s u lt® ,
which may be w r itte n
£d - <J'
'fc Li j il-i d
d - d. / -
d77
J d 1 *-~u / — c /,z-j/
D iv id in g b o th s id e s o f t h i s eq u atio n by
d^ - dg
we haves
wkibh may be s i s ^ l i f i e d b y tak in g n o te a t the assum ption
o f a sm all wave in which case
u n i t y and
(d^ ♦ dg)
d^/dg
i s approxim ately
i s approxim ately
Sd.
The sim pli~
fie d fo ra i s ,
o r i n a more f a m ilia r form
d
-
/ ci d
2 . Depth change upon p a ssin g under a wave fro n t
P roa th e momentum eq u atio n given above i n th e
d e r iv a tio n o f E quation (1) we have upon s u b s titu tio n o f
&d
f o r dj, - dg,
60
2
i
{c/ ~ 6'J? J -
(C+AcfCd-ad) ~ c zd z
whioh upon expansion and e lim in a tio n o f t e r n s o f h ig h e r
o r d e r feecome a;
I
f d <z~ d Lr 2 d £ d l - - C Ld rC & r J i c i s c c i - C Z/AG
J
*-
and s im p lif ie s to
Cj d A d
X C A cd -
-
C 7" £ d
a d ( ij d + c l) ~ 2 c * c d
or
We have shown t h a t
c
■
£ ^ c/
^
0S?
-
fgd, th e re fo re
c
/j c c /
c *c
Now from th e geometry a t th e v e c to r diagram
n
~ c c o //? /$ '
and
/ d f ' l i d jjd :, f t (/3 -h A S ) ~~ 61 C O 5 / ?
or
6c
C —
S u b s titu tin g f o r
o
and no
/i a-
or
3.
bution*
we o b ta in :
u^Snl/3
-r.. - f-r, !crjjrw~ r “”
/
&a
<3 #
■j
,
ud‘
/
r-
£&
(4)
Wave v e lo o ity w ith non-hyflroatatlo p re ss u re d i s t r i ­
61
The sim ultaneous s o lu tio n o f E quations (6), (7),
and (6) p ro c eed s as fo llo w s
■ps.fc
*i
1i
F - k , *»
«[.
Z
"Pf — /“?
£
(6)
?
- //O (\ c £Lci I ~ C
«“ ( lO
, tt J)
ft t.-*v v7
(7)
/*> p
^ 2- '~4t
tJpon s u b s t i tu t i n g In (7) th e v alu es o f
P1 and P2
from
E quation (6) we o b ta in :
+*L.
But from th e c o n tin u ity Equation (3) we have
Uz «.
.
0>t
which p e rm its th e fo llo w in g s im p lif ic a tio n :
I h l s may be s im p lifie d co n sid erab ly in th e follow ing
manner*
F a c to rin g , d iv id in g
and m u ltip ly in g by k ,
o b ta in
q k . f v/ 'L
if
z
1 c!t ! " */
^ r- I~
-J
^ J
or
and upon d iv id in g by
d^ - dg,
; h ->•-/
L a(
^
j
we
Assuming
dx - dg v e ry sm all,
ri & f z d + .™SL~ / / - J i t \ / 2~'L
(
which g iv e s a s th e v alu e f o r
a
^
c
l/ycf I k
YJ
-
e i
f
h
, // - a . )
k,t
4 . Wave an g le
Since
s i n /?»
§>
we h av e,
J /* /?
-
- / ^ C./ r .
/V
Z' t /
/
. .,
M , +iTf e ) U ~ t ) J
5. Depth change on p a ssin g under a wave fro n t
from momentum c o n s id e ra tio n s we have:
/ /f c / L- A, {‘J-'&d] j
- /& c
^C
or
J ^ ^J - ^
d
'j s
Cd <
26
which becomes
(sf/1 »■ /
L
2. /r.
1
„./
,■
~ t- J “ cc "c
Upon re a rra n g in g term s we have,
63
© l a may be w r itte n
0
-—
ad
*i.1
+ f
•"'/“
A)
*1}
From th e geometry o f th e r e c to r diagram :
c •+ &C.
^
to n
^
^
or
,
^
£j C,
(■<. O
■
“
™
How s u b s t i tu t i n g t h i s value f o r -a o
and u s ln /£ f o r
c
we have:
f.-ff “
j
^ u
.-"v ,
i, 1 '
*
'O
A, ^
which may be s im p lifie d to :
^ j
How l e t
kx -
™ j~ -
kg
* a
,-V
/J Of
or
f- 6' - ^
^ ■"O
, JL
.. ~
ko
si ,C)
-. O a
—
/'V//
-fJO •
, i.
^
J i5
A
( 11)
and we have
k
,
a d
A /?/? ^
&^
*■ -—a
iil
J.
i.
-/-«„■ <3 -
'
4
^
4 4
For la r g e v alu es of the Froude number one may co n sid er
ta n ,9m s in /?
Then e v a lu a tin g
to
s in p from Equation (10) and eq u a tin g i t
ta n /? we have*
th e re fo re ,
ad
j
~2To
-\j* + M
)2
A
*
-
4
<ac *
-4
**
^ a n a p o s e th e second ta n a on th e r i g h t and fa o to r o u t
^ d$ b
,
7~k I k *h SL ~ } ss l~f rj ']//' y~ 2*
^ ^
z t dj
/-
r
Ad
__
K
yr
^.J
_li //\r ./-lh 21- a
yx
ds</ z
th e n
Ad
&&
Square b o th s id e s ,
/j] S i
and
„ ...^
/ a d )
,
K2i &/
*
+
S o lv in g th e q u a d ra tic i n
/{ + 4c/ £~
a
2
-
Ak
/J&
/L i
C '
T A§
<'.s
o r more sim ply
t,/z
Zi <3
we have,
-j- > / / t (
f<? ‘^ & 1
-f— /
/ "
/ rj
2.
L
65
6. C r ite r io n fo r normal a n g u la r depth g ra d ie n t
From E quation (15) which I s
/ 0 i
we may o b ta in , upon eq u a tin g
to u n ity , th e
fo llo w in g ,
which becomes, a f t e r tra n sp o sin g term s and squaring both
s id e s ,
or
C o lle c tin g term s and sim p lify in g we o b ta in ,
&“
/
1 -k
a;
7
I
^ *
(b) P lo ttin g s o f Experim ental Data
P lo ttin g s o f the type shown in P ig s. 18*22 have
been mad©, in a form s u ita b le fo r re p ro d u c tio n , of th e d a ta
f o r a l l th e runs deemed usable*
Ih ese drawings a re on f i l e
i n th e o f f ic e o f th e departm ent o f mechanics and h y d ra u lic s
o f th e S ta te U n iv e rs ity o f Iowa w ith th e m aster copy o f the
m anuscript*
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