8/11/2019 Cap 1 Powell_1948

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.:i .

Eq4 E@ns, American Geophysical Union Voirerne 28, Nurnber 6 Becero'oer

IJ E tr] tr 11. N]KC]'V-, g C R i T'.8 JR NO }'] Fi O, F, U ], T.RA - I{a,.IJ IE FLO \i'/

Rail:h itri" l].o'r,ieii

A'bstract--.4 cI:iter'1on is given lorche stzbii'r't1r o, steady unifcrril flo" ir' c;;err crt?-r.-nels.flrne[1i11s number exeeeds one, 'the fi.ow is'.liir'a--rapid, ro11 waves form, :rnd tnsflo$, ca-nnot be stea-dy. Ti:is criterion rs conrL.a;red with several winiah il'ave beer. prc-pcseci, a,nd is ji't.dgeo tc i:e rncre corrii:rehert'sitre ;Llcl e:ract.

An em.pirrcai forn:uia ior Chez;z'srea"d\u pubilsheC by ihe author'.

trntroductioil--Iri tv/c ,,/ery interesirng pz..le::.s prr,lclisheol in Russian i:u'i a.vailahie in ling1ieii,\IEDEH,ffiOV liOef and 1946 lha-s elevelcped a r:r'ii:e::icr'r foi: tire s'ra.bility o-i stezi-v r.rnifor;t iior.ri:1e say5 that whei.i the cri';e:'ion i9 less tna.n oiie, vzi/e. terrd to ,lexr.pen out, btit t]lzi xrhen ir isactr1j:nl tc ci: ti,roie than one, ihe;r arrrpliiiy sl lnat ete:d.y iiow is ir:rpossii:ie and

i;rleharve

r''rLe-i.ireca].is i,r.itra-rapid flcw. Othe::.s have used tile expleEsicr:s:'cili wares ci' siug lc-'w.

Tlis cr1tericn, rv}licl; tr an calling ''"he XreCe;:r1ikorr lu.r*ber, -Irial' h: \,Yritten as

r"":here V is the mearu lreLociiy -c?urss 3-I1jexponents rn trre resistaoce eciu.ation

s =k.dFl6(t + B) ."."...t2)and h{ measures the effect of the shape of the cross section and is defined by

N6=1-Iqcltr1d.&

where S = slope of energy g::adient, R = hydratll.ic radius,

cross seciion. We aleohave the weltr known relationship

" - v =\m/eB

[3]

wlrere B = top widttr, a"{Ld. ais tire coeffiLcieiai whieh corr'ects for nonuniforniity it'r ths vs}sc)l-y dis-tribution.

Deyelqprneqf--In rough channels it is aow knovr-n-iirat for t::a.nquil fiow ehezy's e is eonsia.ntfcr z@Glative roughness, anel in (2), p = 2 a-nd S '= (}" nn srrocth e-hansretr fi,:rv oli the riherha;r.d, e is a funciion of Lhe P,cynolcls r:u:ttre::, whieh rco:"rircs that p + fi = ?, as csn he sscn b'i;writing s =kVp/R{l + ry) " \,?/CZR and (e 2k = v(2 - bllrepi. Since in Reynoids nurober'&- 4VR,/p,ihe veioeity anel,the hydraur.ic radius have the same expone.ut, this requires thai.2 - p = S.

^ I" tr*-prpers"Veccraikor iias ruuch orl the tin:.e lak-en p . 2, rvithout at thc same tiine iakjng

p= rr. lnsacr&Laflnrng s foysliiz r:rakesp -*?*nd fi= ti3 butitisbeginningtoirerealizeathatB{anning's fonmula is-a noi-too-suceessfiii attempt to rnake the sarse iaw eover-both smooth anorough cha-nnetr flow. Ira wha'e {oi.icv.rs we v,,iLl take p + S = 2, whieh leads t* anotirer forar of thecriterion

v'--{irF}tuuv,/tz fi\{stlatg '.'"" '(5

nf we define the Froud.e numbes, F'as th* ratia of the. imean veloeity of ilow to the vetaclt3l of agnavity wave iil still wa.ter. rve ltat'e frorn {4)

g

in uli;ra-:rzpid flcw is derl.ved frorn daea. ai--

, u is the veiocity of the wave, and S and p are

For(*E"/dy)(Ei, fov

t eaRor.IsE

Tkreproposedsh*ws h

Oneanl Eaultra-rapthe artic

Thefollews\r= 1dF,cz = [v

1n*

Yaluesho diffchannelsrectangu

lri

ssg

i6i

and subs

and the

Cnmbini

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UJ-Y,fr,A-R,APID FI-OW

(orn'i:ining (?) ancl (8), .-ve have

Decernher l.g4g

w in open ehan-orrn, and thee been pro-

from data a1-

labie in Englistr,ady unifoy'm flow.t that r,/hen it iswe have whai heow.

as

...".....{1)

and p and p are

19\.."".,.."\ l

and substit,.rting in (5) we have

1r=(xrp)\trE/Q_F)." ..."tx)and the critical value of E {or vr}rich Y = i ;.s

1/ - & / "': zt=(

For a rectangular channel, l' = 8y, tr = E + 2y, eodFi = By/iB + 2v)" Aiso dP.'/dt\" = tdPidYi/(#../d,y) =2,/8, sttharfrorn (3), 1"{=n - Zyi(E +2v)=]Eltts + 2y) - \/(.1 +Tv/w)" T.'herefore, frorn(8), for a rectangular channel

Ec =[(? - P]/(\ -'E)l (1 + zy/a\ " ."' . . (i0)

e omparison witti oth itefia--Xri a very na.nrcw chanine)., Eo uio'utld'oe vez'y large and ulir"arapid@l;cc;r.0ntheotherlrand,inaveryieidech.anne1ttre(1+2y/g)in.(1ti)approachesone,, andE" =Q -B)/$ +6)" Forroughchannelf1ow,6-0andIc =2. This isihe value given by IEF FREYS [1925] and hy THOn{"&S [1939]"

For smooth channel flcw with 6 = 0.20, the wide channel gives 3- critical Froude number of1,8/1"2 = 1.5, which is the result obtained bl, KEUn-,EGAN and F/*TTERSOI{ [i940] by quite dif-ferent reasoning.

I-ea's fonnaula for s:-aooth channels gives S= 0"25, and Ec = 1'40 as given hy R'otsE&TsOi{ a"nciRousE [1941].

The advantage of Veelernikov's criterion is that ii is muchr more general than those hei:etoforeproposed, applying to channels of any shape, and io troth sn:ooth and rough channetr flow, and that itshows how to ccrreet for the effect of non-uniforrn velocity rnistrihution.

One interesting result is that for laminar flow in very wide channels (sheet fiow), (J = 1, I"( - x,

(e)

.......,.i:i) andEc =0.50" Thj,s seemstoindicatethai-withinereasingveloeiiy, tranq,uiiflowwlllchangetoultra:-rapici directly, without passing ttrrougtr the rapid stage. This resrrlt -"eems to be impl-ied in

ter, and A = area of the article by ROBERTSON and ROUSE [1941]"

Thre question as to l.vhat vaiue of 6 should, be used in any giverr case rnay be answered_asfoilolvs" I'or non-turbulent f1ow, 6 = 1 anc g = 2ndE. Fo:'turbuien't rougtr-chrannel flow, S= 0 anclv =brIF/2. For fuqt:uleni^sn-looth-channel fiow, (2) rnay be corn'oinaci uith Chezy's fornrula 13 gi\.'.CZ = CVmlpA = nErS/+SX" Djfferentiating rhis and rnakirg farrobs suLs;itLltions w3 havc

s={usie)(de/uel " ..."iil}An as yet unpubl.ished study by ihe writetr inCicates that for rapici flow in srnooth ehannels

e =41"2iog16 (&/e) + l?.e6p ........(i2)where,6k is the shape correction used b5r KEUI,EGA'N 1938, trq. (53)"""]"

Regarding 6y as a constani anci differer-,iiating (. ,2) glves (clelei$)li -:- 2i1"2 icg16(e)/e] = rr.i.iioglEe,/&, whicir cornbiued rvith (1x) gives

&-1l(0.02?se+0.5) t13)with

for

t4)

y in the veiocity dis-

ezy's C is colstantflow on the other

as can be seen bYs number R= 4VR/u,that2-P=8.e same time taking

E to be realized thatier both smooth andother form o{ the

(5)

w to the velocity of a

Yaiues t-rom ihis formuia zre pioi:ted in Figrrre 1. Ttie rraiues of -R tlrere giveii are ;iroar(i-2)

two different values of S5' The r""ppc'r lir-: is for 6q = 0' vri-iictr is the vahre for in:flnieel-rz -u:ricechannels. The lc-r,re.t iinJis fcr Su = O.ll:fn and 1T.$ Eir = 3.4C, ti4ri.clt is the r+axii-r.u:.n valu-e

rectangu.lar ehannels and occurrs v,hen the dc'pth is hali the lvidtli"EEBgffmenq?iES--As a test oi this crii:ei:icil, sorre da'ea oi:taineci iiJi i.lit' \,'].-iLel IFOi"E1L,ig46fi; fior&, in a Ffra*gnlar flur:re were used" trt. shcuici, periraps ho lloi:ed that v/-hile by titie ihe

studii dealt with rough chinneJ.s, d"ata we::e a-t ibe sarne tiiae obtained on sm.oottr1 channeLs" It qp"s(6)

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.,N lRA I*Fli-'r;l.,PGV,/ f, LI

AG-{-I, V

stance for

(1.1i

..,....(ls,

Bber issEP-

0 C, the Yaluegobservedformula. TbP

(able i--Da4- i ObsRuni^iu

11

11C

r- r 1q- ) 1,- 3 1i- + 12- 5 1z- 6 12- 7 1,- B \2i- s 1o-rn 1,-i1 ll2-12 12-73 13- L 1D- n 13- 3 13- 4 13- "4 13- 6 13- X 13- I 13- I 18- 1 1B- 2 18- 3 18- 4 18- 5 tB- 68- 7 L8- 88" I 18-108-1iB-izB-13B-14B-158-168-1?

23- 123- 223- 323- r.,9?- ;

23- n1a- e1a* ?

14- 414- 5

t-e

\V ,ro.tol

i- 1t-Z1-3t- L1- 5

,lT

Fo.asl

,rr[

tlrt,,.[ ',l l t1L.l tol

. Fig. 1--Reiationsh:

-tll r -.-L---t Lrfto5

L/ALU Of Re|HOLOS NAMACR 4Atlp between 6, C, and Bfor smooth recta

rlto6

/vngula

The method of computing Table 1 was as follows: For the rough channels, p was taken aszero, and for the smooih channels it was computed from the observed Chezy coefficient by (13),Using these values of , the critical Froude number, Icr was computed by (10), the channel widthB being 0.682 ft.and the values of the depth y being taken from Table l^ of POVfELL [1946]. TIg_FroudeDumber f wascomputedfrom(6),whichforrectangularchannelsreducestoF=VVq1gy.Yalues of V and y were talen from Tabie 1of POWXLL [1946j, and g was taken as 32.16 ft/sesz.Following KEULEGAN [1942], cr was defined as the ratio of the mean of the squares of the velcci'ties to the square of the mean velocity. Its actual value was unknown and probably varied fromruntto run, but for want of a better value it was taken as 1.02. The resulting values of Fare thlsabout one per cent larger than those given by POWELL [1946] where crwas omitted from thedefinition of . The Vedernikov number, Y, was then computed by (9). To avoid computationalerrors from accumulating, the computations were made on a machine to four figures, aithough itis realized that the data are not nearlv that accurate,

For ryrla fo u llalap 41 g - -Il.ra-rapid flow would include the Ve checked

L The

wasrsed erm qinno *he M in lhe Vpde

rDleinst

${i

c,

o

5{,

c = 41.2}ogls (E1c) + 42"3{(Ic - 0.515) - 113.?. . . . .

UPPR LtN tS FOR ile= o(tHF WtrtLY Vtoe catttttt)

LOUCR LIH t.t FOD tr,g przr.a6(DprH HALF rct *rora)

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ULT'RA.R,"&PID FLOW

Tnbt" l--885

1- tr 103.4L* 2 111.31- 3 115.6"t'4 118.?1- 5 123.5

0.295 1.75? 2.7250.277 1.903 2.8050.268

1.963 2.8420.262 2.282 2.6430.253 2.445 2.5342-1a- ,Z.J2-42-62-72-8,- o2-LA2-LX2-L22 -i3c- 1J- Z

a- A

3- 6J_ Oq- o

8-L8-38-48-58-68- '.?B-88-9

8-10B -118-12

O-IJ8-14I -15

8-16g-1ry

116. i115 .6118.3120.1114.5129.1L27.4727.1132.9116.3116 "2116.2

110.3L 2.4

111.6116"?1 18.9118. g127 "2lta o126 "9

102.9106.7108.0

112.1113.811^ A

Ll'.t.2120.5121 .6

115"21 10.511i"5

L22"fr125.E\22.8

123.8125.6

130.6137"?

13?.0

145.0

rJ" {80"?70 Q

78. S80.4

.1.JI_t3 .0813.0403.1083.1543.0213.0603.A443.04?a.o Lo2.97 43.0383 .021

2"9422.935

2.9462.8852.89 L2 .914Z.I LLZ^GdJ2.705

2.7552.'.t662. ?69

a. t I I2.X66

2.7392"?50L" IAH

2.6272.54 .2.52A

Z "OZ,)2 "6412.565

2 .607d"otc

1.7111..692x.625

x "$84I"DD"11.464

1 .9702"1231.986

1".8s 11..820

1.55Xr.47 4

L .4481.1581.477

1.X 44L "6'.t 41.6611.?051.6931.6621.3 ?31.3671.3661.1031.5741.5 651.563

1.6931.5 71

1"5581.44n1.3741.3461.1201 .085

i."5 x4n "4?3

1 .42 i-1.3?5t.J,){

1.2961"2XA1.23 6

1.171 "x221. .092

1. 102:L.0931"049

1"040n.023

0.90'/0.82I0.69,1

0.0300.5540.464

0. ?290"?49G.63r..

0.55$0"484

100.9110.0

1.13.3Lt?.7Lzr.1

L24.5115.01 13.3114.511?.9t12.4129.5t29.2129.6130.01 15.0119.6118.6

106.5113.5

LL',2L[1 .2121.3L24.5L24.t123.9L27.L

L03.9107. ?110.L

L13"0114.81 15.9

L1,7 "21 19.0L20.4

11B.0L 16.21 16.6

i.21.0L22,Lt20.4

123.3L24.6

2"3x"3

1.0

i.rd"o

2.29l

ls

b) of Table I oIatter slopesVless than

classifieation(or Rapid)

and Run 28,epest slopee runs ln(n) of the table00" A few of

s taken asent-ny (18).channe] width19461" Theo s. = VN a/Cy."16 ft/sec?.

ofthe veloci-arieci from

of X" are thisrom the

putationalalthough it

ance forchecked

r/. fire

(14)

...(15)

ber is suP-C, the Yaluesbservedormula' The

0.256 1.8990.267 L.84L0.268 1.8300.263 1.8230.260 1.863o.27L 1.8180.244 2.2284.247 2.2270.247 2.23L0.238 2.5540.267 1.8890.26't 1.9410.267 1.9330.280 1.?380.2?5 1.8680.271 1.8910.266 1.9960.262 2.LO40.262 2.1650.247 2.4200.248 2.4634.24',t 2.5560.291 L.1520.288 1.8270.285 1.8800.276 1.9540.272 2.0110.271 2.0580.265 2.1130.259 2.1660.257 2"2L6

0.269 2.244a.m9 2.2650.277 2.30.?0.255 2.3814.249 2.4t70.255 2.4460.253 2.5060.250 2.5580.245 1.8870.241 2.0400.230 2.3420.231 2.5160.223 2.8040.220 3.158

90

. ..d2.5

3.42.4

i.i3"60.5,A

::0.i

4.Ocn(1

't6

2.4

0.5ln4=ffi 45=

2.4

;o1.3

3"8

1.01.0., .t

0"91.01"6

0"0 0.0L"2

23- 123- 223- 323, 46d - 323- 61d.- l"14,- z14:3'lt - ,J.14- 5

a,704z"clz3" 1S8

3.4023.760

00s

00

EunObserved

Up F r = 0"1?81v1 y g= ry/Ec e omputed

C

Discrepancy+

=uzo

R-'::1

tI

, i

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886 R,AI,PE W. POWELL [Trans" ,tGU, V" Z* _ 6]im.prol'ement is hardly enough to be significant, but it m.ay tre nol-ed that in J4 oi rhe 44 rune .i.- ..is improved. Equations (14) and (15) rnust be regarded ai tentatlve untitr a iarger **l.r.t .,....'a-ccu::ate measurements are available" trt is entireiy possible that an equation of oiffur"r,t ]:ni'would fit the data better. But it seerns quite certain that the resistance 1aw

isrjlifferentr,

ii, r,:irl-rapid flow than in rapicl, and that in the former, e depends upon ttre Vede:"nikov number.

. ,{plqqrytg4gtlo9pt-q--The author tiranks V. V. Yedernikov and Garkris I{. Keulegan for- ie;tei:swhich have been of assistance to him, hu.t he is alone responsiirle for anJr errcrs"f irt.rt rr*:]o,.

Eete{eqq_eE

JEFFIt,EYS, }IAROLD, The florv of vrater in an inclined channel of rectangular sectlon, Fh.ii_. i,...ta_g.ser. 6, v. 49, p" ?93, 1928.KtrutrEGAiq, G" H., Laws of turbulent flovr in open channels, triesearch Faper 1151, [. F.esea;:ci:,National Eureau of Standards, v" 21, December j.gBB.KEULEG.AN, G" H', Equation of motion for steady mean flow of water in open channels, Resea..l:,Paper lo{o" 1488, J. Res., National Bureau of Standards, v. 29, pp. g?_i11, }uiy 194i.KEULEG"&Id, G. Fi., andIATTEIISON, G. W., A criterion for instabihty of flow in steep ch::,n:.ie1sI'rans. Amer. Geophys. Union, pt" Z, pp. Bg4-S96, 1940.POII/ELL, R." W., FIov" in a chrannel of definite rou.ghness, Trans. Arner" Soc. eirr" Engrs., i7" .p1:.531-566,1946.ROBERTS0I{, J. M., andEO{JSE, I{{JNTER, On the four regim,es of operr channel f}ow, e i".," X;.ig."v. 1i., pp. 169-1?1, 1941.trHOMAS, H. -&', The propagation of waves in steep prisnxatic conduits, Froc. Hyclraulic e ollfe::e:ice.Univ. of lowa, pp. 214-229, 1939"VEDEENIKOV, V. V., Conditions at the front of a translation \rave disturbing a stead5T n:o,uic4 oi rreal fluid, C.-R. (Doklady), Acad. Sci., URSS, v. 4g, no. 4, pp. ZB9_242,1g4:b.VEDERNIKOV, V. lf", e haracteristic features of a liquid flow-in nn open cliannel, el .*F-,(iEokird;r),

Acad. Sci., URSS, v. 52, pp" 207-210, 1.946.

Ohio State [Iniversity,eolumhus, Ohio

(Man':seript receivedApril 19, 1948; p;ese;ited at the Annual h,{ee'iing, Roeky Miountzin i,ijrd:ia_i:ti::tr-aboratory, Allenpark, eolorado, August 1.6, L}AX; open fo;: forrnal Cllcussion until iV.ia.v il, 1 :i .i

fransact

-

widef9ermeiaydowThmiingablcul

n"oa suitaipropertsoil. Tis, rn ii:ercentait rvitrlditional

Tn

Gecphydiscusseferredious m

gravitaid-eas.

L{to {ir dcapacit

Ma soilsoil rnwatexmeasr.irsoil tutably thwiliingby BRIstant rwilted,1928]plieatisoitr at

pe;L{Emilbepe: