~ ~,-

, ! "

. r

r .

, ' i

. . . . . . . . . . . ~-] .,.~.]o

~aeuumless Dam P~ofiles

by

A. 'Vtt o).s

* ~ m s ~ t e d by

Edward Y. ~Tilsey. AscOt, En~Ineo~

U.S.B.R.

" t

k .

r . • -

,]'~ i . ~ ; ~ , ~ ..

• . . . ~-.~ ~ii~ ;~

i n . .

. +

r .

2

i ~ ~ i~ , i i ~ ,~ ~ • ~

~ " ~ ~~:i~ "~ " ~~ ' ~ i ' ~ , ~ 7 ~ -

r~

~L

, " ~ ~ ~ ~ • ~ l ~ l ~ ~ ~ , ~ ~. . . . . ~ ~ ~ ~

~,~ ~ ~ , ~ 0~. ~ ~ , ~ 0 ~ , ~ : ~ 0 ~ ,

tb

L

0

J ~

! ,.,

i

k

"Th!:~ ~a~,~,~ , ~.~ c~." ~:~F:~i~,. ~ t ~ . ~ ' ~ , ~ t ~ j:,,~:~ ~o t h ~ f ~ ; ~ a ~ ¥ ~

k " ̧ . -

. - . o . ~ : o .

Ore~

- ~ t ¸ . , ~ , , . .

/-

i ~'~i ̧ r

~e ~mf~:vor~blo ~ctLo:~ of ~.~@,'~:t~v~ pressures on spillwa~s

• rreat~s t,he £oilowlzg'~"~i~ree eonaltions: ~

l~ A negative .~ressure renuces the b;:ck pressure on the spillway

so ~b:~•,t the resultant pri~s~ure is corres[~onalngly increasea~

~ , Y, nszatlve pressur~ causes, a~ a result of the vibrations

. . . . . . . ~..~-..~., . ; a . t ~ o c ~ l i . z e ~ _ ~i,~!nte~-atLon of the'surfeoe ,:m~terlal

of the ,,i~,~:~ for %~ n~p?~- fil,st aG_heros then s@rings fre~ fr~ml ~he

• ,,e,,~e~ ~urface of the spi!l~ay~

3,- Tb~ a'~tire d~m~ can acquire ~ state of considerable w.~w~.~on,

the ~:lspi~ce e~s of a sing).s point on the d~ ~re ,~indeed not large~

F . i. ̧ ". /~

the succeedln6 im~",lses, however, by increaJ£ug the accoleratiO~ and.

by inausuratin8 fO~c~$ ~ithin the boay of:the dam c~u give rise to

~.m~orta_ut ~eco~.aary force~ of varying ~ignlficance. This ~henom~on i

often increz:ses to: •such an ext~nt that i~ resembles ~ earthquake in

the vicinity of She power pl~t- One aBn. well imagine how destruotJve

such a pla~ ~- of force can be o~ tho ~hole structure~ In'this ~conneetien,.~.

~.~re ~re ~le~ of examples of pro jeers o~ 1%ulty ~,~nstrUc~iono

Th~ v~cuum prob!~a has arouse~ greet interest ~ In '~e~en~

~ :

ti~es, p,.~rt!cularly in Countries whe~ an aeZlve aemana for vJatpw

~o~er h~s arisen as~ for example, in :th~i:S~'vi@~ rh~lon) whore an ~t-~

tempt to solve ~hko problem has resulted in- a i repor) Of. the }lydro-

toclLuical Institute of Lenln?gradl~

grad, !927;. paper ~ui ~a~ian by, ~adame L. BO: tO 120. .~ " • ~~ ~ • " "

30 i ̧ ' .

'111/.. i~

.~.~.;~ ~,-~ ;-~ ,,, . .,~,~ ,,~.~ .,. ,~ , /, ;i J.., y~,.. . . . . . " ,•

: d i

4=•.. •

•

% . • . . . • ,

: 7 "

k

:~ :~,- . , ..

The moat, 9 o p u l a r ~o.lu~ic~'.i. of t h e I .... U,~,~ ~?,~2.,~,.:~:s ~tO be ; ith~t : - / '.:,: 0

C " ~ - .-Of the American en~inser~ rea~er-~. '~'~ a~ ~il t~a o t h o r .invssti~a-

tloas, is based on the weir experim~uts of Bazln ~ conc~rnlng .the shasta- ::-.:i

:gAmnal~s des Ponts et .Dhaussees, ~Slo .5, no, 19~ i890~ fat semoster, ' ; .i! ~. 56; ,Also a~German :tmmmslation by Ko ..E~ller, Zei~s, dos V~DJI~ / vol. ~., .1.~...90, p . : ~ B . ,~"~

..of .an aemat~d ' f r e e na~pe at .the upper portion .of its trajsclisr~, -:

Brlefly,. Q~ea~er's 9z~ceaure is as follows: A particle of ~vatsr ~:~.~ .... a. "

di~tauce of about one-thlra of the thickness of the nappy, imeasured

~ lts .u~der surface '(fi~ ~.) is conceived as a free.solla~p~r..

%icle ~Ich moves in a vacumm~ that is, without friction, The-ini.,,

tiai location of this .particle corresponds ~%o %h~ polntof.maximum

presauro in the nappe.:~(flgure !). and ~h~ Init2al velocity of .the~par-

tlcle is the average :over :the-~hole. seoti@m, :iThe ~trajeetory .of the'

" k:

p~!cle la ' the p ~ a :

D •

'ox :Y . -

g

where ,70X i s ~ h e h o r i z ~ a , t e ~ c o ~ e n t of the :~Iti~ ,veloclty.

:..):'/.:,:

. ,- :> :11;

If ~ t h l s , c u r v e : i s p l o t t e d on O r e ~ t e r c o o r d i n a t e p ~ p e r .and '

if ~he, orlg~ is eblf%e~..the ,.,~o1~o~ng .Ore~er-, equatloa ~u~ts: ~.:-i ,::

. .= Z. ¢~I~ :(y + b) g

-~ const~m~s a , b~ ena Vo~ are :~eterminea by:exl)eri

h~a~o This equation re~resents the.kern 'be~entthe

The th~e~ ~ °• ' : - • .('::" .:L:! J:. •_. i ¸ ' " " ) -̧

k 7

L" " ".

,-)i

a •

,.-.~

~ :.': ~.

. o

t-~JeO~O~3 r i~ de%er=!nsd., a~. foLLow:

d

[

The tblc~ of the Je~ I~ found from,

J

.The kern .llos .-.a~ a di~anoe of. D/B m~aaur~omChoconally. ::¢~om ~he

~ d e r ~ u r f a o e o f - ~ h e nepp~o .Pz'oooe~Llng , in t h i s fa~hion~ bo~;h ~:th~

~ p e z " end l o w ~ pzof~J~es m~ .'be p~o~ted~..--,~e .G~eage~ table :o~" ,.¢o-

o z ~ u a ~ t e a t~ ~l,(~,Ll, a~ed f o r h , .1; f o r o t h e r :heads , the ~ x % t i n a g e s

a~ ~ucz~a~ed i n pzopoz~;ion $o the :.he~Clo " . . . .

: ~ e ~ s ~ a n ~ s a ~ .b b y - e ~ e ~ t , ~ o b l e ~ s : ~ o t ~ e ~ . ~ : h o e ~ -..:

b~,~resg~ as-the Zzee. po~i: ' " .... ' : ~.:~' ....... ,~. " • i

Y~U her ~ual~sls ~ .~h~,ha~ : O h o s ~ '~he e~'er :of ~ ,:O£::(t~he "

.:a~.jppe, ~. t h e ..free po~l ; , i

~ e ~ e o m e t r l ~ l : t m ~ r e t a t i o m of' ~ l s p ~ o ~ o ~ l :re.Chat: :. "

. feun~ . i t ~to "-be a t . G . 4 a 4 o f ,~he ~:~hlekne~ ~o~t '!: ! .:': "~L. :

?

: :i~ ~ •

i ':i: ~

. . . . : ;,:--: . .., , : .

" "i " I

the t:~aJ e c ~ z ~ :

i " The consists n, m, 8a~ ~ a~e detezm~Ined i~m Baz~e= experlmen~e by. i,:

• applying :~he %heol~ o. ~ least squares t o t h e observes coordinates.

The coox~tes of the napp~ ~rofiles are computed in ~he

same w~y as by Creager B that is, ~rom. the thickness of the Je~ and

the path of the ~ree pal~cle. "

T~e progoe~l o~ the RusSian, Puslrevsky, shoui~ also b e . ' ' '

mentioned, ~n~ a~in is concerned ~rlth the motion of a, i~ee paFtlcle..

He also introduces e presSUre foroe ~h!ch~InTluenees the .motion of the

I t Is Zo be n o ~ e d that in this case 'the o ~ , ~ n a z y p a r a b o l a ~ " :

~eeczlbed by a f~ee fall is dla~r~el. Be-gives no fixed method,-.fOX" .

dete~inlnE this foree, end, ~herefore, ~here is ~anger of dAst~z~

the pe~abcle either too m~ch o~ too little. The de1~lls of this pz~-;

e e ~ u m e c a ~ o ~ be .~eprodu~e~[ h a z e ; t h e reacLer I s r e : ~ e r r e d : t o .~he o x ' i ~ . -

h a l p , ~ e r . " "

c o n c e p t of a "free iS~iCle", ~h WhiCh hy~raul$ce h~e-little ox" : ~..,

nothlns to ~. A p p ~ r c n t l y ~ ~ a soXution can be aehleve~ on the bas£s " i~."i:

o f orc~Lmezy hyC~r~u!Ice; o~he~Lee, all propoel~i~:Of

z~ise above %he level of wank emplrlcellsm e~d t h u ~ ~ ~'en~e~'.;~£ab~

• . .., / ,." ~ -~ ~:~.~ ~

. - , , . .','<,7"" : ',- -

: .~e u n r ~ i a b l l l t y .of a n e m p i r i c a l 9z'ocedure i s incz , ezzsgd

~henevez o n e ~eparts from the conaltion~ o f .the .experiments. .~.,:the. :

fc t l~oedL~, an at temp~ 'is made t o .solve the vacuum 9rob lem on the b a s i s r "

of, the sveryaay hyd~auli.cs assisted by Bazin~s ex~erlments,. -"

' i . , ~ -~

~i ~i 7:" .

. : j

• he px~b~kmn ,is reduoed to "~b,e speete]L case o:f a d ~ t i t ih ' :e ": . :' ::: . ,<] :i - !

! , 7

I~ ~ l ~ o t l c e . . N a l : ~ . ~ h i s . a a ~ e l e oan be elten~ea to o t h e r 'cases.

The 'ueli, i , - l l i lOim ~ ~ i equa¢ion is expendefi :toinc~uae-an -:: "

a d d i t i o n ~ t e ~ f o r t h e p u ~ o s e o f . i n t r o d u c i n g . ' into th~ dlscusetox~ .,the 1 " ' ' - -

ee~ii~ i~Iz~e ~li~ aeeo~ganies ou~vi~ine~ motion e~c which re--,. ;::- : - , }

~eH rthe .p~essux~ between ~he nappe an~ .the s~rrace o f the spillway. '

.The exBend-ed equa¢ion is as toZ~ows:

+ i4=bt U

l u ~ £ e l i l

T : : Che.evex~ge veZee£ty o f the nsppe;-- "

Y o a I ~ s tm~Lt te l . , ~ tue ;

. _L

~/o=L.~he i n i t i a l ' . . l ~ l u e .of t h i s . . c o e f f i c i e n t ; - .-

T . J . ~ ~ r ! a ~ao'te, ~ i B USUBL~ denotea b y j~ez~c~:~i~e~.~as: - ' - ' . a u ~ £ e de, Iliad 'by the fo.l,,].o'w'Lo~ e~u~tlon:- ,~" ~ I .

e ) " ~ ; 1 " * . . . .

+ " J * . • + , - > . : 4 ' - , ~ ' ~ - "

:be c e . ~ e a :::~ i.J'~'": -;~.,~i ¸- :~.: .:

: ~ ":?

, .:.:;

4"

!ii

. . ! .

i ..

r . .

t h e o u ~ p w ~ ~:Lch , i s d ~ w n o ~ t h o g o n ~ - l ~ y t o ~he ! . ~ l ; ~ n . i l lne~; .,:: : "

r ' a t h e r ~ l ~ u s o~ ~ s ' t u x , e of ~ ~tre~'~JJ~e,~' the_~o~i~on.o~ ' , ~he

.9o~j~t ~n.~he ~ p p e ~ u ~ t h e 2.0es o ~ - . h e ~ ,Is~

The ~eo~l~a~The ~I~bO~Sof,the.~ ere ~o~Z, g e ~ ~ . .. ,, .~: :,. •..i ,.- ....... ~,,,:::~ :!;~i! I

. . . . . . , : j .

. h . ~ ~ . : . $~e ~ p e o l ' ~ - - . e i ~ . "/: " ~ .o. ::.if:i: i i ':~' :'~' :if:: i

! d...1~ d.~ ~ ~ ..::..

. 0 . . . . . " . : : . . . . . - . - -'" . . . .

, ,, ,,

~ I I * b

,XS,

I ,

2 ' / ~:

°

o

(3 "

" <Y.~o., v;,~ ,,

! ,,w ~..¢

• . • . "

- i!

, il

!

- - d

- %: ~ :'/7 , S ~ " '

- ~ , L i

/ ~ . : : . , : " / ., ~': . v

~ ~ j ~ I

Ey

a~oz, e ; ~ : o ~ o n a L cmose e ~ l o n o t : , ~ h e ~ n & g p o .

/T /

' q t ' - : ~ ' ~ : ~ % " ~ r : , " ' : ~ . ,:,~i: . . . . . ! :L,;" ~.',~ C~" : ' ~ • ~ • ' :' trl> "" < ' • i : . /i~ ~ q l '~'~ ;r~p'i~'<~ . . . . . . "VbVq£m!:: . . . . . %--b"~'--~':~ ," ' ~ ' ' ' ~i~ ~J/~' '~ ~ X ~ : ~ ' * ~ t " : ' ~ ' ~ " ~ ~:'~'~;'~:'~

p~' -- ~" . .~ " J'rX. :~ : ~ - , " , . ~* .

~ - ~ ! : ; ~ : !, i : ~ ~,,b -L , ~ ,

,.. >" :/'!

n~

" " " , L . , ' . . . . J , . . . . . . r r " ~ 9 : :

; : ~ ~ay : fo r - :¢he abovo co~d~Y, t o n , ~b.e ~re.~l~blos : ~ ,¢he~s a ~ ¢ ~ o ~ /i:: ~ ~ ....

: - . - [

- i

! : : : i - ~ R ' ~ " • < . . . . . " . . . . . ~ ' ; P }b ' - - ' ~ : . ' r , " M " ." ' [ ' d* ' ~ I : ' . " ~ ~:.': [ : ; ' : " ' p . . . . ~ ' ,

£ / i

• °

~!:~i::!. "

C; , , '/

: -

. .~.~J ' ~,, ,, ~ . .

::2>,:." : :

I

C' , ',-

tlve~, thu~: ,.,; ~I_ ; ,..,

j .

0~

..,._.. cC',~ • ~ / ~..,.~'~, ~ 'W ,I:: .~ ~.:c.~,:~

2. /

. . , ' !

.T-. e ~ £ o ~ :

2

\' <' 'W j - ' - ~ t

..... i

i

T

. ,o

1 1 " "

L

. : < •

~r ~

to. , i '-.'.LL'2 r . s - ,',

( /

Equstlon (4) iS the differeatlal ~ ~ 1 ef t~e V a C ~ ~ S~I~ ~

profile for the limttln~ ca~e ( n o res~wve ~ r ~sltlve pres~ure).

Equation (4) eu~nlt3 a s~Lm~l~ficat$on yhtch throws light on the 7ti.h.m~ll~,,,

enon as o b s e r v e d b y Ba~tn . This t s accompl i shed by i n t r o d u c l n ~ d i -

m e ~ s i o n ~ e e s c o o r ~ i n a t e s b y meens o f l ~ t c h t h e phenomenon m e y b e b e t t e r

Investi~ate~. Theseooo~Ina~esa~e:

"J;t ~,..,,'~" t,4D~., < i ,

~e now.have:

~md . /

d'2,,./!~,t - t~

/ ..

H r, ' "o ,. r_./,t;,,. :~:. o L ct,. ~ . )

/ /

d

t

~S,, ~ ~ ~ "~=~ "

i . . - ! ~ - " . . . . ~ YL

Illar~wlth the followLug method 0£ obt~InLug the aecon~-ae~va~iTez

. . . . . . . • ~ ; : i i ~ - - -

l l "

+ .

1 .

r q l

. ' , o

but s~nce

~ ~ ' _ _ _ _1_

~, ~ ~

..J.._ t '- Io

~ l q, . . . .

I | _ _ D J I

1< ('Ho.- Ho ~,~ n,'"- I - ~ ( ~°+ ' ~ ) ~ "

- i m w

~ ~ + n;'~ ~'~" - - 0

lsl-

~ :, Thls is ~he final 61ffe~ntial e~uatlon of the vaeu~esa

. ~ spili~ ~zo~A~e oxpreased in dlmensionless coordAnage8. The c o o l -

~,~ ~::i~i .

flcl~t K Is ~t a t~e ~ns~t;. for the ti~e belnll it ll;i£ be con- k~ ........ :i::! p-

/ #

1 " 0

r ,

.

7 f , : ~ ! "

.77~ <-

,+,"!{:7, > •

'" ' [ ' " I

i n ~ h i ~ form t h ~ e i s s m a l l hope o~ i n t e ~ r a t i n g t h e I q u n l t o n , - 1 ~ - "

T l s l o n ~ t l y E w i l l be assumed ~ n a t a n t o I t l l l l . 1 be s h o ~ u l a t e ~ ' that

t h i s a s s u m p t i o n d o e s n o t i n w ~ l d a t e t h e f i n a l : e m l l l s , 1~o~ b y . ~ l v i i ~ !

K Tarloits Talu~s~ t~ whole family o! ~ ~cuzwe8 ilill be found lihich llill

+I . :J

be. i n t e r s e c t a ~ . r by t h e p l ~ o f i l e o f t h e spillwa~.Q 1 t

This f i r s t inteEralion Of e q u a t i o n {51 is e f f e ~ t e d in,.~he

f O l l o w ~ n g manner:

, , V~. ~'t,. "= - - - ' = t ,.::L ~- -

I

-

w t ~ l l o e i

l - , , -

• m

. . . . ~ / . . . . . . . . . . . . . • . . . . . ~ , - . ..... -

B'u, i ;

~D an~ the i n t e z m l ~ l s :

• ,.,.,,. ' " " < " I

e o

7

~ " • , " . ' ' ~ = . / / - ' " < ",' . - ~ . , 4 ' 7 , ~ ~ ' ~ . • • ~i ~ ' L " ~ " ~ • , " .~ ~ . : ~ ~ , ~ ; ~ x . ~ > t < ~ , ~ L "~ ," .'~ , " . ~ . '~ . ~ "

!:~i I and t h e , ~ u t o g r a l , : b o ~ n o 8

0 1 "

~ : .

k - !~-.

/.i.7

~ h i o h may b e ~ t t t e n ; ~ . I~

H~moe:

/

,~i. ~

:

~ 2 Z .

and

- I (6}

• ~ e B e Q a s o s a ~ 8 ~

':i I. ~./.. , !.,~./ ,,~

:,~ :.,. 0~' *L

.i, ,,:~ : ̧ "

m~ich ~e~uoos theequatlon.tOi

cC,~

" • " L

m

k q _ _

'r

(b).K,,..l o~,"= : = ~l : then

' W : '

,... ]

:! ~::E

:i!~i; .:L ~k.

- ~ L .... '~ But .hen ' " . .

7

n b

' , . ,

C , .

. : 2 , ","

L ' ' C

L

°

: -. . . . . . . . i ~ , - , ¸ , i:::i,~:,:,+! ~

"T'

%

21t ' -

f .

: ~tll

2 , ' ° < : - 7 D ~

D { ' L

,.,~7~;7:b.,:~ <- ,. :, ' :,: 7 ,.< ~:. .......... .. t<:':"::,, t~

. , r ~

' ? .

. . " "

I~ wo li~.ah I ;o cozlsl~ai' -.the l~a')~h ,oT :a . llo~ pe~Tticle, ,i,m jgi'o¢ell, i ~li:i

"~ho ~'olloar~,nL; im~unur: The ho.-~l.zontel ~ccelez-ation ilil

" ° " cL ~''j< ._ ..cA. :'~7>, J~ '-" x --" c'i

a~,~. lhe ..l~l, tl~l,, ~ccalol-allon in . tho . y - ~ I e ~ l ; l o ~ .i~ :~hf:t oi" t t l ~ t y , o r ; ,

o x ..,,~.-,0 ~ ~=o:

'i~en

I s ) .

: i ' 7

<,C/i.~.

4"

' 4

. e a . ~ =

l i u t i h e n

' i h e n I

) , s "m'; <7 - ° D

" ? . '

. . - . .

. " . ] . ' i <

i4<,r

" , ' , Z

) _ ' - " , . : . .

" ~ ' , , , ' 4 < '< ) - . . . .

' L "

, t [ "

. ? /

t .

; - j

i _

9 /o b u t , ~ h e n t ~ o ; . y ~ 3"o; t h e r e f o r e :C~ ~ o , .

e n d

/~..~. ---~. ,© d

'b :i,,'i ' .

Then

,dG~ t'~x

L?" 'i " -

, %- . . . . . .

.' ~i ̧,'~/ :

• ,L " '-"

T h e r e f o r e : ~ "~-

{'~i .~.!

X

a n ~ ~ - t ~ e r e ~ e

. . .., •

o

.~ ~.,- ~ ~ ~i~ ~ ....

_ ;'!:~i,>i ̧-,

-! !i

.,. .

• ,ii ~, ~

~!~ - ,

H

7

:i-:.i

il. i ~:.

:~. -

k k

- • • ..

: ~,j

:i~!i: :::/::~:.;: ........... " -! ?~:.:?": i :::::: ......

%

. k ̧ ~

~ h e r e f o r e equation (7) I n d i c a t e s . t h a t - t h e promiSe ' :o f the

!i

/;

sp~way el~h no negative press~ures:Is a p~ral~ ~Ich Is'~tx~Ae.pro-

vldi~ ~e ~ater particAes ~.~ the nappe ~ve as a sys~e~ offree :.s~lid

particles in. a vacuum. ~t . aince tilts :is not the easa~ it :.£a~Aca~es ~.

t h a t t h e p a r ~ l ~ l ~ ~ t c h . ~ y e ~ t n e e : ~ s , u s e ~ .has t ~ o ~ch ~ e e a ~ v e

a g a i n s t t h e ~ o ~ n a t t o n o f a v a c u u ~ and s u c h , a pro~J.~e i t s . t h e r e f o r e

uneconomAoa] . , Thus : t h e p a ~ b o ~ ;nay ~ be , c c n o e l v ~ a s t h e o u t e r ~tt

of alA. vae~eas profiles.

• he observ~ ~a ~ BaR 0~ ~l~h i~ves~Iga~rs u s ~

"'base ~helr 'des1~ of.vacuu~ess spilA~y .pr~fl~es.~ not fit elixir

mLrv~, which im~Ioates tha~ di~fe~lal ~e~£on~(~} mus't;~be ln '~o-

grated, ~l~h n ~ ~. ~Is-opex~t~on:~t be~c~om~ishe~.:glx~c.i~.

We tha~fo1~e ~xpand the ~tunoi~io~

as an ~fin~t~ serle~hus:

...... , L. ~

/~.'~ ~ (~ .,~"V°4 ' ~ --.-

L., \

- • = k

/ "

~.,~ : _

",L k

tre~e~i~Anves~ete~ ~bF ]~sIIA~, ~ran :f.n~pe~to.u o f 2a~,e ..clat~ •

ahows that the mexlm~m ~ ~ (eee ~81~e ~, ~azAa:s dete),m~1~ :.~o~-

k .

• The te~ms:of,,i~e se~ries b e ~ ~ ~r~h the-i~: ~b~Is ~si~ITe,

: d . ~ " . ~ I . . . . . .

. • r "~ii ~I~ ~-i' i~ ~ii'.~: ,.~

• . . . i ,

f : . :

i.,i

~11%h t h £ s approxlnmte venue :the .d t~ ' fe~-~t£~ l . ' , "@~US'ttOD.':r b(~oOI08:

- L

i.<.

• • " k V

, oi ' w,b.leh i;~e l n t , e g z ~ . ; t 8 ; % r < ,

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FIGURE l -D ISTRIBUTION OF VELOCITY A N D

PRESSURE WITHIN A FREE NAPPE,

i

F IGURE 2 - T H E CREAGER PARABOLA.

Y FIGURE 3- s.LENGTH OF CURVE ORTHOGANAL TO STREAM LINES; h-THE THICKNESS OF THE NAPPE NORMALTO THE SPILLWAY PROFILE; s 'AND h'. THE~.R VERTICAL PROJECTIONS,

TABLE 1' BAZIN 'S C O O R D I N A T E S FOR THE UNDER SURFACE OF T H E ' N A P P E r . . . Y ~ . . . . . . . . . . . .~--~,~ • AND COMPUTED-VALUES OF n I I | ~ ~

- - H i Ho . E , a . T O F C * . T O F * E , . ' , ' 3 ~ , L ~ o , ~ - - . . x

• . 'HIEt, D ONWEIR,h.,O.IS"O.iSm r . . . . . . i " , , ~ , . ~ . i l " '~,, .

Y . " %

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0 r~ 0 " : O. , ";' ,' " ~0S~ : O 0 0 , , ~ . . . . j " = ;: F ' O ~ w h i c h i f fOl lOWS:

. . . . . . . '~ . : . , , _ . - h H o ~ ~ -

. . . . ~ " ' t . . . . . ; ; . . . . ~ ' ' " ~ " 0 , 8 8 8 ' - O ~ 5 : 0.005 I O22~ ;0,0304 i : S m l l o r l y :

0 5 5 :o .o54 i 0.358 '0o653 2 2 4 ~ '~ . ~r~ 0 . 8 8 8 , i

080 o.o55 ~ o s g ~ : o o B , l z.=o o . 8 5 ~ 0 . o ~ 5 { o . 4 ; 0 ' o , , , S z . , o ' .,,: F O R M U L A S : . F O R

~ ..... t " ; . . , T R A N S F O R M A T I O N o . ~ o o . o o ~ o . 5 o 8 ~ ( J ~ , 2 ) ' n ' .~ ....... : _ F ~ A X E-.

" T A B L E ' l ' r

C O M P U T E D C O O R D I N A T E S [

A C C O R D I N G 1"0 A C C O R D I N G ~ O r ' " ro~Mu,~s {S)~NG(~) rO~MUL~(X)(~R~SO~A)

' Hol n ' ' ~o~oaee~ )o .88n ; ' Hol ~ .olo..~;to8=.~ 00583 OOOtZ3 C.O50,0, OOt I O,2OOO,2OO O.Ot O 17) '6 'O08B8

o o ~ z s o o o B ~ o ~ o o , o o o 6 1 o ~ o o 0 0 4 o a , ' , s z o o 5 5 ~

~)~'~v,0o=Bs'0tso'o.os5~10.8000.800 o o ~ : o 5 : 5 2 . ~ 0 . o " t 9 8

0 '2~ " 0 0 ~ 0 " " ~r Z O O ' 0 0 ~ 7 J 0 . 8 0 0 ' 0 ' 6 0 ~ ' 0 4 : 0 ' ' ~'; I

0 2 @ 2 " O 0 4 6 2 ; O Z " o : o e 4 t I .O00 O ~ 5 : 0 8dBO O22ZO

O.538 . 0 0 6 5 3 • O 3 0 0 0 0 5 8 [ .200 0 3 6 0065610,3ZOO . : ~ ';~.-X

0.395 OOEIIF~?:O ~,50 "O077 I 4OO 0 4 9 1.2432 0 4 3 5 0 f ' * ~ . ~ t ~

0 ~ 5 0 0115 0 4 0 0 ~0 09'~ I 800 0 6 4 1.4ZO8 O58~5

I 175 0 500 i L 0 4 5 . 0 .444 1,800 O81 L5984 0 7ZOO

7 ; ' O LOO i l , 57Z : 0 8 8 8 2 0 0 0 tOO . 7 7 6 O O B 8 8 0

2 8 1 7 2.00 • 2 3 2 4 1,7T6 2,~f28 2 0 0 13.51Z0 I 7~'f~

3 2 4 0 , 5OO ! 2 8 # ? :2.~6<.

7Z! 4.OO : 3 . 5 2 0 1 3 , 5 5 2

4 . ~ } ' 1 5 0 0 ~5,~OO t4 .440

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! . . . .

k ; ' i ;

F I G U R E 4 - GEOMETRICAL. EXPLANATION OF S Y M B O L S .

P o r o [ o l o ~ " ~ ~ ~

V i f o l s ' - -

Creoger 's recommP'nocd pqOhle--

C f .aogc r - -

PuZirCvsh~ - - -

Ko£o n lko ] to . . . .

F I G U R E 5 - VAGUUMLESS 'PROFILES ACCORDING TO VARIOUS AUTHORS.

x - 0 - 2 i 8 5