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<br />614
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<br />HYDRAULIC ENGINEERING '94
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<br />small~r than roughly nA. Rurs (1990) data refers tn lorrenl~ in Austria, and ahmll
<br />half of lhe mcasur~mcnlS were made in partly regulated slrCum reaches. With bnlh
<br />data :-;e's. nnw velocity and discharge were obtained by the sail dilution method.
<br />Using the same mClhm.l, we measured now velocity ami discharge in lorrcnl~ of uur
<br />cXflCrimcnlill catchments in lhe Alptn) vallcy. a prcalJlinc region in Swil1.crland
<br />(Rickcnmann. in preJl.), These observations arc al:<<1 included in the ana'ysi~. h may
<br />be nOled that in the extensive data set uf Griffiths (1981) II part of lhe mcaSUI'cmcnls
<br />arc associulCd with bedload transport events (no. Hh in Tilhlc I).
<br />
<br />"nalysls and new velocity equallon
<br />
<br />According to Zeller (1<)91) there is a strong decrease in Ihe Strickler k value (Ill in-
<br />creasc in Manning's n) for slopes that are greater than a trunsiUon region of ahuut 0.6
<br />to 3%. Fm the data ~ts no. :3 to Ii (cL Tahle I) the value or Manning's n is .given ur
<br />an approximate value can he inferred from now tJcpth and vclocity. A plot ur n ver-
<br />sus Ihe slope shows a similar tendency liS observed by Zeller (1991). If the Man-
<br />ning's n values arc (dOlled against (he relative now depth, hld9U, a strong im;rca.'iC in
<br />n is noted for values h/d90 smaller than ahout 3 to 4. For the data used in this study,
<br />a transition region with stopes ranging from 0.6(Jf, tn 1.0 % can he defined. Must
<br />measuremenl~ with slopes in this nmge or above have relative now depths bcluw
<br />ahouI 4,
<br />
<br />For Ihe analysis, two dimensionless paramelers are delined: Y ~ V 1 (g d90)1/2, and
<br />X ~ Qtl3 1 (g 1/6 d905l6). A mulliple regression analysis was performed using X and
<br />the slope S a.~ independent, and Y as dependent paramcter, in order to determine a
<br />power form c.xflrcs.~inn for the mean now velocity. The analysis confirmed the
<br />presumption that a hclter correlation can be ohtaincd, if the S(l.-cp and low slope
<br />ranges are considercd separately. In order 10 have an overlapping region, the firM
<br />equation Wa."l determined for the measurements with S > 0.6% (N =. 211) umlthc
<br />second equation for all measurements with S < 1.0 % (N == 181); the oycrlnpping
<br />region inclodes 31 dala points,
<br />
<br />S > 0.6% : V ~ 0.37 go.n 00.34 50.20 1 d9(J0.35 (2M
<br />
<br />S <: 1.0% : V ~ 0.96 gO.36 00.29 S0.351 d900.2J (2b)
<br />
<br />In Ihe following analysis, equ. (2n) is applied 10 data poine, wilh S > 0.8%, and equ,
<br />(2h) to data with S < 0.8%. In terms of predicted versus me3."lured velocilics, the
<br />following. statistical parumctcrs were dClermincd: correlation coefficient r::::: 0,9) fllr
<br />bOlh equ. (2a) and equ, (2b), and slandard error or eslimale s. = 34% ror equ. (2a)
<br />and Sc ::;: 24 % for L"qU. (2b). Fig. I shows the ratio of PrcdictcJ to measured vchll:ity,
<br />against the slope for all data. There is no systematic hias. This statement also hnhla iI
<br />the ratio V rJV m is pltlUed against discharge, characteristic grain size or rchllive llllw
<br />depth; V p IS the predicted and V m is the mcasured velueity.
<br />
<br />For a comparison with the above equations, the Manning-Strickler formula is re.
<br />written using the conlinuity equalion and approximating Ihe hydraulic Radius by Ihe
<br />menn now deplh:
<br />
<br />V ~ (1/n)O.60 00.40 so.JO 1 WO.40 (3)
<br />
<br />To replace Ihe width, W. in equ, (3), a regime equalion may he used. Many regi",e
<br />equations havc been proposed. relating the width primarily tn the diSt:harge. In such
<br />a furm, dirfcrent relationships have been cstahlishcd, dcpending on whether .'oc
<br />variatiun al a .sHe or over a reach is considered. In this analysis, a neW regmlC
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<br />MEAN VELOCtTY-ALT. EQUATION
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<br />675
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<br />equation Wa.'i dctermined. As with the '1" ,.'.
<br />same two dimensinnless imJcpcndcn( pa~grc;~~lOn a~~'ysls for the now VClnrity. lhe
<br />and the slnpe S. Wilh Y' = WId a~ de a.me~ers ~~rc ?~d: X ~ ()1I.1. f (gllh d90~/()),
<br />cxprcs:;ion was tlmcrmined: 90 renden( parameter lhc lnllnwmg pllwer form
<br />
<br />. W ~ 5.111 00.32 d!lOo.2tl (gO.t6 SO.2-') 4
<br />Despite the (,t,ct thilt SHme n( 'he data fist. f' T . ( J
<br />m~nts at a she, there is It rca."itlnahlc Cf)~( 'JI~, .ul>~ I r~'cr.n ~ series III' me~'~Urc-
<br />Widths: r ::: n.H2. Sc ::: 44 %. Cumbjnin t n: a .I(~n . 1W(..-cn predlctcd ill1d mcasured
<br />exprc.~"i()m; fur Manning's n arc nhtaincd~ cquatrons (21, (3) and (4), the fullnwing
<br />
<br />~ > 11.6% : Un ~ 056 gO.44(j'.tt1 (So.3J d9fJ'1.45) (Sa)
<br />S <: 1.0% : I/n ~ 2.73 gO.49 (jUlJ 1 (SIUll! d9f,".24) (5b'
<br />II .blluld he "Oled lba' ellu (Sa) and (5b) d' .
<br />value in metric unjt.~ ls/m.1/:lf If th t- aI' rc l1!1cnsltmally turrecl .ilmJ give the n
<br />. - e ra 10 n prcdlclcd 10 m " lJ I
<br />1.1i plotted against fin' slope for the dnla 'u 3 cas,urc n. va ucs. np/"m.
<br />uverprediclinn of n hJr lhe ..necpcr ~Iupc m'~.c (c to ~' thThe~ .I~,.l shght systcmalil:
<br />t~ now depth inslC1ld of (he h dra r. g., quo . ~). . l.s I~ ~anly due In using
<br />higher slope dala with smaller Jdrh ~(:c.:d :dt~u~, .W~ICThh IS s,lgmflcanl unly for the
<br />the rutio n.Jo is ploucd a ai d" ep rUIIOs... ere IS nn systematic hias u-
<br />depth, If Jqu'Sa) is applic~ I~Sl,~SC~::;f:' ~~~~CtC~I~11C grain Si1.c or relative nuw
<br />s <; ~.H%. and in terms or predicted yc~f . ' > : (~. and cqu. (5b) 10 da(it wi.h
<br />.oUalJ.'ttK:al parameters arc determined' r _ 0 ;;:5 ~easurcd n ~aluc.~. the fulluwing
<br />0.57 and "e ~ 24% "Jrcqu. (5b). '-, an "e ~ 36% lor equ. (5a), and r ~
<br />
<br />VpfVm
<br />2-00
<br /> . equ. (2a)
<br /> 0
<br />1.50 o squ. (~b) 0 0
<br /> 0
<br />1-00
<br />0.50
<br />0.00
<br />0.00001 0.0001 O.Oot 0.01
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<br />
<br />0.1 S
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<br />Fil:. I: Ratio ofprcdkteu to measured velocity, VplYrn. again.'illhc slope, S.
<br />
<br />~~t~rn:a~.;,~w~y ~~:a .hllllCd equ. (I) is combined with Ihe widlh relationship cqn
<br />and s..) as'(2a) gwhcC ,Ht:rIY'~ed~ua'lUhn ~hows a similar gmKlnes,s of 11, (cxpr~ssl'li hy r'
<br />"", , I appl' '01 eheld "t 'd' h- ..
<br />henL~ for'l San" d . (.l a use 10 t IS study. Uuwcvcr the exnn.
<br />'<, , , u 90 arc somewhar dIfferent The .,- '''d' ' ,.
<br />Inca.liun: of (he rcfalivc Onw depth hid S..,. ,ra. III '";' ~O I.... If) sOInc.cKlc.o' a
<br />depths are interrelated it is (to~~iblc' thal~~rl~ mee the slope ~ and lhe rclauvc 'h,w
<br />Sand d n" . " I erenl ex-Jlonents 1m lhe parcmclcrs g Q
<br />.I~, . d 90. may rc eet " Similar trend. Furthcrl11ure the 'hc.~l' e~punc I.. '.' ,
<br />U\:pcn In snme eXlent un the sinN' rc . . J '. n..1O .l;<..'Cnt (0
<br />lWn equations (la) and (2b) " If:-.. .J:~IU,n. Us '~ a. so .Jlu.'itrClfctl by introdudllg the
<br />. n cresting y, a Similar Slalcmenl hnlds fur llccJlnad
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