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<br />AN ALTERNATIVE EQUATION FOR THE MEAN VELOCITY IN GRAVEL-
<br />BED RIVERS AND MOUNTAIN TORRENTS'
<br />
<br />Dieter Rickcnmano2
<br />
<br />Abstract
<br />
<br />Induding new licld data in the slupe range of lorre~ls ~p III 6~%. ,..n ?n~piric:~1 ~qur
<br />,,'uo is develuped It) I~XPrcSS the mean Jlow YCluclly In terms ul ,the ,UuW rale. ! lie
<br />. . . The ..1 "HuDlinn is apphcah c over a WI( C
<br />'Inpc and a c.:haraclcrislic grain SI1.c. . pmposcu -, .
<br />:angc of now condit inns. including R\lUllI,aiR slrcams and Immols Wll~. a slope ~a~~
<br />. . bout l'~ 10 63'''. and relative fll1w depths a.'\ small as ahn~l n,4. . 'H~ grave -. t
<br />(~I ~ . with ~luflCs' hcluw allllut.. Pt, ami f1nw rules up In 24()(l !n-/S. 11 SlUlI,lar cqual!ulI
<br />n~~rs. m '~hal different cX[luncnls is presented. The ahcrnauvc vclul,;lly cllualllln
<br />WI '''~)l C 'f rmcd lnw a Curm similar In the Manning.Slridlcr formula; the pafil-
<br />ean.~ runs 0 ,I',n' II> Mannl'ng's n ,'s shown 10 be a funcliun (If Ihe llnw ralC, the
<br />meler corrcspnnu g . ' :.
<br />slupe and a char.lclcri...lil.: grain Size 01100 bcd.
<br />
<br />Introduction
<br />
<br />In SIeCp muuntain lurrcnL" wilh large relaliye l11ughncss, il is ,difficult ~l' dl.'~l'n:.~inl.'
<br />: 'un Ill)W depth. linwcver. must f1uw resislallce ct.Iualums n.~qUl~~ l. e t uw
<br />~c t~~ in UI parameler. Fur hed slupes ahov~ ahuul I(~I, ~mar,t.all~ ~ag.g~ ,t I ~X1.:
<br />. ,~ Balhur~1 (19K5) have presentetl currcspundlOg clluauuns whu.:h arc hasc~ I,n j
<br />~~rlically h~garilhmic vl'lucily distribution. In Sleep lorrents, mea,n ~lnw v~locn~ a~
<br />. .. ,")f 'n mcasured by the salt dilution methud. anti nn IOlonnaUtlll lIlay
<br />dlsc.:h, "hrgl C IS tlhl~ w"l.hh and deplh uf the Ilnw in the very irregular slremuhcll !!l~ume.
<br />aval a e un c , 11 d II' 'npr" 'liC'l1 in steep
<br />try. Therefnrc, a lest uf equaliuns using the relative lIW ep 1 IS II . {;. .
<br />streams wilh a mugh hed.
<br />
<br />Using field mellSllfl.~menlS including.slt)(lC~ from .9 (~I,tvlI ~KI'~i R(()uf~~XK~(~~.vf~;~~
<br />. , . I' r lh' meun lluw velocity, V In Imlsl. as - .-. '. 7U . '. '.
<br />~l~l ~l~U~,I,t;~. u,r, 1~13/sl S is the hed ~du(lC. and dx is the l'hamcte~'isllc gml~1 Sl~.l~ It;'.
<br />:~icch ~~~~~~l by w~ighl Ult.: liner. Bused un lahuraltll'Y ex.rcnmcnls wult sh1lll.S
<br />
<br />1 Paper lor the ASCE 1994 NaIMmal Conference on Hydraulic Eftljne8ling~ HydIauic& 01 Moullt.llR Rivers
<br />2 R ~..,- fA.....a1Ins~IuI8.)1 folt$l, Snow and landscape Research. CH-8003 Bimlensdort. Swilzetlaod
<br />8i.8Il!J..~-
<br />
<br />672
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<br />
<br />MEAN VELOCITY-ALT. EQUATION
<br />
<br />673
<br />
<br />fwm 3 % to 2U % ami intense bedluad transporl. Rickcnmann (1990, 1991) proposed
<br />an empirical hut dimensionally currect Clfuation for Ihe mean Ilow velocity:
<br />
<br />V = 1.3 gll.2 411.6 SO,2/ U901l.4 ( I )
<br />
<br />where g is lhe accclcmlinn due to gravity, and q is the Iluw rate per unit width. In-
<br />lerestingly, the same equaliun has been lheoretically eSlablishet! to estimate the ve.
<br />!cICily of a dchris llows frunl.
<br />
<br />Here un attempt is presented lu express the mean nnw velucity us a funclion uf the
<br />flow rate, Ihe bet! slupe and a characteristic grain si".c. In nrder tu include a wide
<br />mnge uf Ilow cundilinns, several dala sclS on field measuremcnls 1)1' mean nuw
<br />velucity were used in the development of a regression equatiun.
<br />
<br />Data preparation
<br />
<br />OVerall, eight d;'ll.j 't'ls were used, fur which informalion is given nn the mean flow
<br />velocity, the lluw ralc. the bed ur wuler surface slupe (Sb ur Sw)' ami a chnraclerislic
<br />gmin size. The lutal numher of meusurements (N) used in Ihe regression analysis in-
<br />dudct! 373 dala puints. The data suurces and infurmation related In the following
<br />analysis are Iisled in Table I. In the regrcssinn analysis, a characteristic grain si",e
<br />dl)U was used, Fur sume Ill' the data sets, Ihis inrormalion is nul directly availahle. Tn
<br />lIetcrmine d90. empirical relationship... ~Iven hy Zeller (199 I) Were lIsed.
<br />
<br />No Data source N Range, PLIO! 5 doo W Rh
<br /> slo";'s orh orn
<br />I Hodel/19931 94 0.9-63 Sh d - -
<br />2 AlDtal fAickenmann In oreD.\ 7 15 - 45 Sh d - -
<br />3 Auf 11990\ 11 1.7 - 20.5 Sh d W- n
<br />4 Jarrell 119841 69 0.2 -3.9 S d W.h 'c'
<br />5 Bathurst 11985\ 41 0.5-3.8 S~ 'a' W.h 'c'
<br />6 Thomoson & Camnbe1l11979\ 5 5.2 S 'a' W.h 'c'
<br />7 Thorne & Zevenberoen 11985\ 12 1.4-2.0 S 'a' W.h A.
<br />8a G"lIilhs (i U81l: riaid bed 82 0.0085 -1.1 S~ 'b' W A.
<br />8b G"lIillls 19811: mobile bed 52 0.083 - 0.71 S 'b' W Ah
<br />
<br />Tub/e I: Data SUUl't.'Cs. including some rclevanl information. N refers In the number
<br />tlf dilla used in the anillysis. Sb Sw and Stt indicate water surface slope and hed
<br />slupe, respectively. Fur the cases marked with 'a' tl9U is tlctcrmined by dt){l = 1.1 K
<br />ull4 ' anu fnr 'he eases marked wilh 'h' u911 is UcICrmincu hy u9l) = 2.1 d511 lafter
<br />Zeller, 191.) I). W ur h indicate whether the width andlor the fluw deplh arc given.
<br />Rh ur n illl.Jicale whcther Ihe hydraulic nrdius andlor the Ml.lOning's rcsisll.lllCc
<br />coefficient ure given; fur lhe cases marked wilh 'c' an appmxinulle value uf II WilS
<br />delcrmined by using Manning's cf.Jualion with Ihe mean Ilnw tIerlh.
<br />
<br />11...1.:1 (1993) performed II Inial of almust 400 mcmmrcnwnL... on llnw velocity ami
<br />lll,\thilrge in l11uunlain torrents in Switi".crland wilh hed slupes ranging frnm I IU
<br />UN)%, He fuund, that (he reluliun helween now velocily and jl parumcter including
<br />discharge nnd slope is different for Inrrenl reaches with a distinct step-punl murphn.
<br />lugy nnd and those wilh a more even hed slupc. In (he present analysis, 94 uuln
<br />ptlints uf the I,,"er category were used with slopes from (),9'~, (0 63% and flow rales
<br />larger than tUn m3/s. Fur data wilh smaller flow ruleS, a preliminary analysis
<br />showed a diffcren( behaviour which is prubably due tCl relalive flow depths heing
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