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HYDRAULICS BRANCH

OFFICIAL FILE COPY

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UNITED STATES

DEP.AR'UlENT OF THE INTERIOR

BUFIBAU OF REC LALIJ,TION

HYDRAULIC LAEOR.ATORY REPORT NO. 60

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SiivIILITUDE OF HYDRONDYN.AMIC PHENCidE.NA IN *

WHICH -,J1u'ER IS CONSIDERABLY i.HXED \!ITH .A.IR *

BY J. Sl,iET.ANA *

TR.liJJSLATION FROU FREJ.'\JCH *

By

Donvor, Colorado

J"uly 31, 1939

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UNI'I'ED STA':.'ES

DEP.A.RTr,XNT OF THE UJTERIOR

BUREAU OF RECLAMATrmr

srmtTTGDE OF �-'YDRODY:''AI:IC P:IEN011Ei·!A rn WEICH

WATER IS COYSIDER?.?LY ?i::IXED 'NITJ! AIR-

A t r1::.nsl e. +,j or: of

SLILITUDE DES Fntl:OI·:t::ES FYDROEl'r.'A:HC':UF.s QT\

1 1 EAr EST co7:srDtRP.3LE!--;EFT ,:iu.::GEE AVEc DE L'.hIR

3:: J. S:·· eto.na

From Re;)ort on the First ),:eeting of the

Int0rna-tional Association for ::�··:fra,�lic Structures Resee.rch, 1938;

.Appendix 4; pP.�es 97-103.

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Transln-ted by

H. -G. DEV/EY, JR., A��ISTJJ'.'I E:mHTEER,

Under Di:r0.ction of

A. RliETT!'.:EF:S, SE:1IOR E�:O!:EER

Fl.nd

J. E. WLH11ff)CK, :RESEf,.JiCE EJ T(IViEER

DAnvQr, Cnlor�do, July 31, 1939.

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SIMILITUDE OF EYDRODY'NAMIC PHENOHEN'A

IN WHICH WATER IS CONSIDERABLY MIXED WITH AIR

For many hydrodynamic phenomena it is necessary to take into consideration, instead of a me.sr. of water, a mixture of ,;mter and air whose specific g;ravity is much less than unity.

For example, the hydraulic jurnp, either undro,me4 or drowned (figures 6a and 6b), is always accompanied on its Jurface by a roller which contains o. large or small amount of air� Sin�0 the hydraulic j't.unr serves as �he basis for determining the dimen­sions of an apron , the proof of the similitude of t!li s phenomencn, tho.t is to say, th.a possibility of extrapolation fror.i t.he red1.1ccd scale (model) to tho real scA.le (prototype), is vcr�.- important. This proof has not been ma.dl'l u::i to the present.

For this proof, it is neccstir,ry to proceed in u gent)rul manner ns follows:

(1) First stud�r to sec if th� quc.ntit:r of dr in t-hc mixt;uro of a.ir and wo.tcr is, constant or vc,ri.a:,10.

(2), study,.by tvn.:cn the dependent o.blc�

If the qu£;ntity of cir is vi:rinblo, it is nccossa.ry to experiments, to sc,c if there is· c. functionnl rclo.tion be,­, VC'.lw,.,s chc,.rc,ct0rizint, ,the phenomenon, takinr, thosq ,is in-Vt,.rir,blcs, Lnri the qu,·.nti t�· of :.:.ir us c. de:pc,ndcnt vt-.ri-

(3) After determining this function, it is [�guin nccossr.ry to r.ir.kc- the followin6 proof: Choose vc.lt1 cs for th0 charc,ctoristics of the ohonomcnon and set up the rhcnomonon for the chosc:1. vr;lucs for diff;rcnt reduced sc;.-.lcs (sec.le r�tios). The v�.luc of th,: ::.forc:­mcntioncd function must rcm:::..in const: nt for �,11 different reduced scales.

1 Ing. Dr. Jo.n Smote.no.: (1) Trr-.. vrux et Etudos des Inst.i tut;E

lfo.t:i,onr.ux Hydrologiquc ct H:•drotcchniquc T. G. 1ic..ear�.rk � Prnh:-.: No. 9 ., "Etudc cxpdrimcnt::..lc du rcss'.lut d I cxhr.usecincnt", ;ct No •.

·13, "Etudc cxpcrimcntule du r·,:ss�.ut no:0{." (2) XVIc Cong:r�s Intcrm: .. tionnl de Nc..vit;r.tion, Bruxollcs, 1935, H:rc Section, 2d Co!!'.r,tuni cation, Re.pport po.r J � Smctr.nc..

(Engineer Dr. Jan Smcfa:.na.: (1) Works i:..nd Studies of the T. G. Mo.sr..ryk lfotionnl H�rdrolo[iC :::mcl. F.ydrotcchnic Institute o.t Prr,guo: No. 9, "Expurimcntul Study of the E:7drr...ulic Jump", c.nd lfo.13, 11 Expcrirr.c.ntc.1 Study of -. Drowned ::.I�·draul ic Jump. 11 ( 2) XVI Intc,r-m,tiom.l Cong:russ of N�·.vic:-,tion, Brussels, 1935� First Section, Scco::1c. Colll!nunic.:.tion, Report b1r J. Smct::,nn.

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a. Undrowned Hydraulic Jump

FIGURE 6

FIGURE 7

I II

r· t II

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I

I p_ , Y2- PYa

b. Drowned Hydraulic Jump

1

--- Velocity

FIGURE 8 -•

If /,;,'

..

The author conducted, in 1932 -1934 at the T . G a Masaryk National Hydrotechnical Institute of Pragu_e , a number of studies on the undrowned hfdraulic jump · (figure 6a) and on the drowned hydraul ic jump (figure 6b ) . T_he undrowned jump was formed f'or different values of M, the index of torrontuosity, from M = 1 to M -= 34 ; the index of torrentuos ity M being defined as the quotient of tho velocity of the i s suing jet v1 (f'iguros 6a and 6b ) and tho critical VC:Jlocity 'v'i'Yi', whero g is tho acc el eration due to gravity nnd y

1 is tho dopth·

of tho shootint flow.2 Drowned jumps were formed by inc roasine; the depth y2 downstream from tho undrowned jump to a dopth p y2 ., riving: p .,

· tho amount of' drowning ., valuos from p = 1. 1 to p = 2. 5 . This cho. s s i ­fication of phunomona indicated by M o.nd p, and tho proof that the vnluo cot e& = cot ap = 6 (figuros Su o.nd 6b ) for all tho vduos of p and M., ha.s mndc it possible for tho o.uthor to d orivo gon,,, rr..1 fonnu­lno for thoso two phonomona. . Tho author hns domonstro.tod thr t the undrownod jump is only o. po.rticula.r en.so o:f tho drowned j ump 3 • JLC ­c ording to tho c.uthor, the hydro.ul ic jump is., th erefore., ., cho.rn.ctor­izcd by two nwn.bors ., by the indox of torrontuos ity H n.nd tho dr.,r.rco of drowning p. It is nocos ,; c..ry, thon, to find the f'unct i on:-:.1 rola­tion bctwoon th r.a 1 u two ch:i. rnctoristics n.nc1 tho qur�nt i ty of ' c.ir in tho rol l er.

The qunntity of uir in tho rol ler is dot0rmin0d ty obtQin­ing tho spocific gra.vity of the wutcr-r.i r mb:turc. forr:linr the rol l er. This dctormin::-.t ion i s ,nr�do pos s ib l e only by tho r.i d of n i cz omc:tor·s . The bottom of tho c r.nei.l usod by the r.1.uthor in thoso cxnerimonts wns drilled with smo.1 1 holus to wt!ich \'1e3rc nttc.chc,d, 'by rubber tube�, vc,rtic:.1 pioz omotors of gla.fl s tubing open r.t the upper ond (figure 7 ) •

Dim ensions ns shown on figu res 6rt ., 6b , c.nd 7 ., fip.;urc 7 being uny cross section, �ro ns follows :

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y = depth of flow. y1 = spec ific grnvity oi' tho f'lo..-:

A3y = thicknoss of tho l ower z ori..o of tho roller, in which the diroction of fl ow is downstrc.::-.m.

y3 = s pecific grc.vity of zono t:,,3y A4,Y = thicknoss of top z one of roller, in which thc

0

diroction of fl ow is .upstro::un.

A3y + A4y = Ay = total thickness of roller.

Trnnslctor ' s not e : V1 Thr, inde;x cf' torrontuos i ty, M = hc.s ·vgy'i"'""

c.lso boon cnllad by many �-.uthors r, di s c h:i.rge: - ind1;x, flow indE..X , or veloc ity indox.

Trcms lc. tor ' s note : Prosu.'llr,b ly in some oth er s tudy rot :ncr:t i oL c d , lJt·::; s oc .:-ou ·: .�,o ;.,o l a

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The author hns cal culat ed the di s c har1_::e q r-.nrl the rl i s ­chnrg e 13 and 14 of the lower and upper z ones of the rol l c r by n p i tot tube. The di schnrg e 14 of' the upper z or_ie i s pruc t i c r•. l lr equc.l . as the o.uthor hns p rovad3 , to the tl i s c hn�ge 13 , of th e, l o·:1 !.:r zone of turbulence . As shown on figure a. the vr, luo _ of A4y i s c. s ­tl'.'.bl-i shod di roctly. By toking tho nrec:. 3 -4-5 equr;l t o the n ru. 1 -2 -3 , the vo.luos · of A3y and y a.r'o als o dotonnino d .

Expres s ing stutic equi librium o.s shown o n figure 7 , it i s p o s si b l e to write tho equnt ion :

Tho a.uthor C'.dds to the right s i do of' thi s oquo.tion tho vnluo yz , which mus t express tho prcs suru L'..gu.i l:st tho botto111 of tho c ,· .. m.l duo to tho curvcturo of tho s tro��l inos . z b eing the hoight ropre ::; 0nt -1ng thi s prcs �ure; (pros ::;u r ,:; -hc i::.d ) . The we.to r in th o p i cz om<.;tvr hu; c. spec ific g r: .vi ty y a 1 .

vrri to : By l o_tting Y3_ = y4 o.nd d ividi ng equnti"on ( n ) by y, wcJ m:·.y

p = y + .!!_ Ay + z •· • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . ( b )

I n thi s oqunti on T4 r-.nd z nro unknovm. Lotting z a 0 , it i s p o s s ib l e to detc nnino tho othor unknown, y4 :

= y (P-y) . , { ) V 4 Ay • • • • • • • • • •. . . . . . . . . . . . . . . . . . . . . . . . . . . . C

3 . ::. · Tho c.uthor hns p rovod thr.t. for n dcgroo of rl r owninr.

p a 1 .4 , the turbulenc e is nonrly fro(> of nir. thr.t is to s ny Y4. i s unity. For thi s cu.c o it i s pos n iblo to dotormino the; •:>r t:;s s uro hL nd duo to tho curvcturo of tho st romnl ines ncco rdingly :

z = P - {y + Ay) • .

For n hydraul ic jump ,rith H = G . 79 , p = 1.4. ::.nd q = o .0866 m3/ s/m ( 0 . 931 c .f ..a ./rt . ) , for oxru11�> l c , v�luus of' z dctonninod from thi s rolnt i on � re· in socti on I • ( fi �uro 6b ) in matQrs , 0 . 02 7 ( 0 . 089 ' ) ; in s oction I" , o .018 { 0 .• 069 ' ) • in s ootion I " ' • 0 . 01 (0 .033 1 ) ; nnd in s octi on -I ' v , 0.001 {0.003 • ) ; c o rresponding vcluos of' (y + Ay) b eing in motors . 0 . 2 29 (0 . 7 5 ' ) , o .243 (o. ao • ) , o .2 77 ( 0 . 91 ' ) , r.md o . 304 ( 0 . 99 1 ) • ,. It is ovidont thnt tho pros suro-hcn.d z duo to tho curv6.turo · of tho stroo.mlino s . i s relo.tivoly vory sma.11 nnd therof'oro Iieiglie;iblo; and consequently th,:. vnluos of y4 detorminod by equntion ( c ) n.ro clos e _ onough and only very little grontor tho.n nctunl vnlucs .

In f iguro 9n tho vuluos of y4 dot� rminod by equat i on { c ) i:: r-0 ropros entod by o. hydrn.ulic jump p = 1 , nnd M = 1 to M = rn . Sf,.7 .

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· . r f.. f e r Ir. i' ::. . :uro 9b t !� : , v · lu •.;; s :: i ' Y1; d t.;. t ,'.· rminc d b y 1., qu : ·.-:: L :, ( c )

. , � ;,; = 6 . 79 , but t :·. i:, v0. lt.o of p v· · . :ri 0 ::: f' rcu p = 1 t -:-· p = 1 . -1 .

It C '.".n ·o e z e en i' ron the z e tv10 r:'. i .:r, rr.:::s thc.t t· e'..:·::e t' ,., t. 't: � · V '." l , ; e s of :.: , p , '.:n 1 t:1c di s t r.nc e x ( f �g;urcs fa·, c·.r.d 6b ) f Y-01�. tLc u : , ­s t re :_:..r., l i::1i t of -cr. e rol l e r t o the s e c t i on i n que sti on o ::: o�· e h, n r:' , · r. d tr,e s p e c if i c . g :rcv i ty -r

4 on_ t:-, c other hr,nd , th,.,t the r0 d�� i st s

t h'ci 1' ol lc·:dr.[; func t i o:1e.l r d r·.t i o n : The spec i f i c �r :·.vi ty "(4.

of t r.e ro l l (.. r is srr.c..1 1 : ( 1 ) ·i� : ,en the de[r<.;o of" d rO',min,:; p is sn�.1 1 ; ( 2 ) when -:h0 ir.d(;;X o f torrent\;.os i ty ; ; i s h.rr e ; '. .nd ( 3 ) y:hor. t:r: c s 1:;c -c i on i n qui;, s ;. i on i s !, ,.;; ' rer the. Ui) s t roc.::i l i:rai t of tr."- ro l h, r . :' lms i;:,c S (;c onrJ cc,ndi t i on o f the; s i:-:,il i tudo ns st:·.tcd �·tt. ovw i s s �·� ti s f' i <) d e

Tho s c.t i sfyir:r.; o f tLo thi rc c on.di t i on i s s ho':111 by th0 d i ",l!. l'"'. ..r., on f i ,· un., 10 . Th.:. s di r, ;; 1"..r.. r- 0p r c s cnt ::; the r0 s ul t r of' · s t1.: d ­

. i . , s ':::-. ich hr.v,) b r, ,;;::1 · "1ulu f .::. r :� h:rd 1 · � ·.ul i c j U:!lfJ ·11i th :.: = 6 . 7 ':J :· nd p = 1 .0 , :-.nd ·:rh i c h hr� s b ocn tc s tc n for the s c :· . l c s 1 : 1 , 1 : 2 , 1 , 4. , 1 : 6 , 1 : 8 , 1_ .nc. 1 : 10 ; thc.t i s to c · y , ,·; ith :·. di :; c hr.rgo q = c . O 1 i 1i !:i.3/s/m ( 0 . 12 3 c . f . s ./ft . ) ; 0 .c ;:; 1 1 ( O . 0 3 ti ) ; o . o e r, 5 ( O . 931 ) ; o . F,71 ( 1 . 690 ) ; C . 242C ( 2 . 6Qb ) ; : n� t- . 3 � G 6 ( � . 619 ) . !

' For r.1 1 ,' th e s e s c r.l ,_; :o '.; !-. c. s p ,;c i f'i c s:r .vi ty y 4: hr. c tOi rc,,,r . in

the s s.i:ie f. :_, r r. pive� c :rc s �· s ecti or. � . T b e l ira i t s i n wh:i. c � y4 v,a.-ricc : -. - as-,z

Ul"Od Oel t/m" v1hich i s quite '.'cc c eptr .bl e c or.d derinr; the rr,etl:od .� nd e r rors of �ens ure�ent .

iienc e , on fi rure 1 0 , th �, func t i o m1 l rel s.ticr. r·. l s o -r: !'.l . . r· i :·. R

c ons tant for d i fferent s c al es .

For p > 1 . 4 r.nd y4 = 1 t/;;:3 ( unity ) the rol l e r c o?·,tr: b s p r�c t ic �l ly n o �i r .

From the sp ec i fi c g rr� ity e s t�bfi shed, i t i s e�sy t o de­t0m.ine the vohune of G.ir c ontc.ined in the rol l e r . If, for O:V.:'.:.mp l e , y

4 ':' 0 . 4 t/m3 ( 0 . 4 ) the volume of' d r i s , nq;lucting tht: s p eci ti c

g r; ·.rity of :J.i r , 0 . 6 t/-..3 ( O . G ) for 1 ?Tl3 of ·volumo ( unit 1rol ume ) of

:rol l or .

Th0 two firurcs 9 n. : ,nd 9 b c l e: .rly show, then, th, t th(·Jru ex i s t s £'. functi onnl rol:-..ti on b (;twc0n th 0 ch1 �rnc t0ri st i r; s U r..nd p of � hydrnul i c jump , r·.n-:l th r; voltunt.; o f' r. i :r c ontninod in tho r o l l 0 r .

I n thi;; f'o r0p;oing; di scu s s i on the cutho:r b <;l i r;v u s to h1·. 1J c domonstr�.tcd oxr:.ctly th o simil itude of r_.._ hydraul i c ,ju.'llp unr1 r·(;)vmv1 r_nd drowned within thr.:J l imits of nc ·,-_l c from 1 : 1 to 1 : 10 . 'l' i H,, r o is no runson to doubt th �t th0 s imi l itudo doosn 1 t hol d b eyon d tho l imits s tudi od.

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