plot of velocity-distribution in vertical

Transcription

1 620 TRANSACTIONS, AMERICAN GEOPHYSICAL UNION Fgure. Dmensnless plt f velcty-dstrbutn n vertcal Table l. The curves are very nearly parallel wthn the range shwn. The dsplacement s due t the tact that equatn () gves a value f F(z/d) f zer fr (z/d) f ne (at the surface) and the data n Table are such that F(z/d) s zer at a pnt n the vertcal whse velcty equals the mean [(z/d) = 0.33 by scale n Fg. ]. In applcatn t the evaluatn f n by the prcedure adpted heren, the dsplacement s f n sgnfcance, but snce the slpes f these tw lnes are nearly the same, It appears that n the average equatn () gves values whch, althugh perhaps nt perfectly suted t every channel-cndtn and In the absence f further nfrmatn, may be cnsdered t yeld acceptable results, lmted f curse by ther factrs pecular t ths methd f apprach. References [] M. P. O'Bren and B. D. RIndlaub, Dscussn n sand mvement n fluval mdels, Amer. Sc. Cv. Eng. Trans., v. Q0, pp , 935. [2] M. P. O'Bren, The vertcal dstrbutn f velcty n wde rvers, Trans. Amer. Gephys. Unn, pp , [3] F. C. Scbey, The flw f water In rrgatn channels, U. S. Dept. Agrc Prf. Paper, Bull. 94, 95. [4] Je w, Jhnsn, Dscussn f mechancs f flud turbulence, Trans. Amer. Sc. Cv. Eng., v. 02, pp , 937. [5] Fred C. Scbey, Flw f water n rrgatn and smlar canals, U. S. Dept. Agrc, Tech. Bull. 652, 939. U. S. Gelgcal Survey, Washngtn, D. C. CHANNEL-STORAGE AND UNIT-HYDROGRAPH STUDIES W. B. Langben (Publshed wth the apprval f the Drectr, U. S. Gelgcal Survey) Recent studes f the ranfal-and-runff relatn tend twards treatment f the subject n tw parts, namely, () the grund-phase, whch ncludes the study f such prcesses as nfltratn and evapratn, and (2) the channel-phase, whch cmprehends the study f the flw f water n the channel-system wth partcular reference t fld-wave mvement. Ths paper dscusses the channel-phase. Fld-wave mvement may be cnceved t be subject t tw prncpal knds f peratns, namely, unfrmly prgressve flw and reservr-actn. The frst peratn desgnates dwnstream mvement f a fld-wave wthut change f shape whch prbably nly ccurs under deal

2 REPORTS AND PAPERS, HYDROLOGY cndtns n a prsmatc channel n whch stage and dscharge are unquely defned at all places. Reservr-actn refers t transfrmatn f a fld-wave that takes place as a result f reservr-pndage. Fld-wave mvement n natural channel systems (that are unaffected by ders-mvements) s prbably ntermedate between the tw deal cndtns cted-- ne r the tner predmnatng n a partcular place. Hwever, the actual benavr s smetmes bscured because f the effects f trbutary r ther lcal nflw. The prcess f determnng prgressvely the tmng and shape f a fld-wave at dwnstream pnts s called fld-rutng. Several apprxmate methds f fld-rutng have been descred [see, 2, 3 f "References" at end f paper]. Thmas [] sets up equatns fr the exact slutn f fld-wave mtn but as he ndcates, practcal dffcultes make them vrtually napplcable t natural rver-channels. Rutter and thers [2] and McCarthy [3] descrbe methds whch are cnvenent fr general applcatn t rver-fld cndtns and, f prperly appled, can yeld clse results. In ref, McCarthy nas assumed that the vlume f strage n a rver-reach may be evaluated as fllws Strage t = K[xl t + ( - x) D t] () Where I t s nflw at tme t and D t s the dscharge r utflw frm the reach als at tme t, and tne factr k s the rat between strage and dscharge and has the dmensn f tme. The factr x, whch s dmensnless, defnes the manner n whch the strage-vlume s determned, relatve t the nflw and utflw dscharge-rates. The methd assumes a unfrm unbrken watersurface prfle (unfrm prgressn n dscharge) between the upstream and dwnstream pnts n tne reach, and assumes that stage and dscharge are unquely defned at these tw places, and that k and x are sensbly cnstant thrughut the range n stage experenced by the fld-wave. In general t mght be stated tnat the accuracy f a fld-rutng strage-equatn depends n the clseness wth whch the strage-vlume can be apprxmated. As a frst apprxmatn t has been stated that tne strage vares wth the utflw dscharge-rate as n a reservr, and later McCarthy ntrduced the factr x t gve expressn t the s-called "wedge-strage" and therefre affrded a secnd apprxmatn but whch may be n errr t the extent that the dscharge n the reach des nt vary unfrmly frm nlet t utlet. Rutter and thers [2] smlarly accunted fr the "wedge-strage" but by graphcal means, expressng strage based n rverprfles as a functn f utflw-rate usng the rate f nflw as a parameter. The factr x n McCarthy's equatn () s chsen such that the strage-vlume s the same fr a gven weghted dscharge, [xl + ( - x)d] r whether the rver s rsng r fallng. The factr k s the slpe f the strage-weghted dscharge-relatn, whch n mst fld-prblems appraches a straght lne. Fr spllway-dscharge frm a reservr, x may be shwn t be zer, snce the reservr-stage and hence the strage are unquely defned by utflw-dscharge, tne rate f nflw beng f neglgble nfluence as an ndex f strage n the reservr at any tme. Fr unfrmly prgressve flw x = 0.50, and bth nflw and utflw are f equal weght. In ths wave, n change n shape ccurs, the peak-dscharge remanng unaffected. In reservractn the hydrgraph s flattened. Analyses by the wrter f many fld-waves has demnstrated that tne tme-nterval etween the center f mass f vlume at the upstream pnt and at the dwnstream pnt s equal t the factr k. Fgure shws the effect f the value f x upn fld-waves. The Inflw-graph shwn was arbtrarly chsen fr llustratve purpses. The utflw-graphs were cmputed frm the gven nflw by the use f equatn (), usng a value f k = unty, and the centers f mass f each f them regardless f the value f x s cncdent, ccurrng ne unt f tme later than that f the gven nflw-graph. Althugh cncdent nsfar as ther centers f mass are cncerned, the three utflw-graphs dffer materally n shape-~that fr x» 0 (full strage-actn) gvng a flattened hydrgraph, that fr x * 0.50 beng a reprductn f the nflw-hydrgraph dsplaced by an nterval equal t k, and that fr x * 0.25 havng ntermedate characterstcs. Values f k are readly fund ether by cmputng the slpe f the strage-dscharge curve as descrbed by McCarthy r by cmputng the center f mass-lag nterval. Hwever, the hydraulc cndtns that determne the value f x that s applcable t a gven reach are dffcult t defne. In mst rver-reaches the value f x Is subject t varatn when cmputed by McCarthy's methd [3] dependng n the relatve amunt f lcal unmeasured Inflw. Lcal nflw s generally estmated, and the amunt and the tme-dstrbutn f the estmated nflw crrespndngly affect the cmputatns f the fld-wave mvement.

3 622 TRANSACTIONS. AMERICAN GEOPHYSICAL UNION Inflw ^y Center f mss f nflw 5 Fg. Sc hematc cr parsn f cmputed utflw-hy drgraphs wth gv en nflw a nd value f K, as sumng dff erent values f X lw X = 0.50 jr«out If? \ jtflw X=0.25 N >'l\\ / \ V ^ Outflw X-0.00 / / / O I Tme The value f x tends t apprach zer n rver-reaches that apprach reservr-cndtns, that s, n thse n whch much f the rver-prfle s cntrlled by the dscharge at a pnt dwnstream rather than by the dscharge at the several pnts n the reach. At places wthn such reaches, the stage-dscharge relatns tend t becme lps, wth greater dscharge n the rsng sde than n the fallng sde. The lp s prduced ether by varable slpes r by backwater frm a dwnstream cntrl. On the ther hand, n prsmdal channels where stage and dscharge are unquely defned at all places, the value f x prbably tends twards It s, f curse, generally understd that n rutng a fld thrugh a rver-system, It s necessary t break, t up nt reaches such that the rver-surface can be safely assumed as unfrm and unbrken. A seres f pls and rffles r cataracts r excessvely lng reaches wuld nt cmply wth ths crtern, althugh specal treatment n evaluatng strage may remve sme f the dffcultes n usng lng channel-reaches. An example f a unfrmly prgressve wave n a natural channel s shwn n Fgure 2. The.wave was generated by the abrupt release f a vlume f water at Guernsey Reservr n the Nrth Platte Rver. By the tme t reached Brdgeprt, Nebraska, 0 mles belw, the wave had becme stablzed as shwn. The wave passed Lsc, Nebraska, 28 mles belw Brdgeprt wth substantally the same shape. There was neglgble nflw frm trbutares, but there were small dversns fr rrgatn. The dscharge at Lsc as shwn n Fgure 2 ncludes the estmated vlume f such dversn between Brdgeprt and Lsc. The lag-nterval between ther centers f mass s 7 hurs. The peaks were 5 hurs apart. Fgure 3 shws the strage n the reach pltted aganst the dscharge.at Lsc, Nebraska, gvng a lp shwng a larger vlume f strage fr a gven dscharge durng rsng stages than durng fallng stages. It shuld be nted that the rver-surface between Brdgeprt and Lsc was nt unfrm at all tmes. Thus at nn f June 2, the dscharge at Brdgeprt was,300 cfs, whereas dwnstream at Lsc the rver had nt yet begun t rse, the wave-frnt beng smewhere between these tw places. At ths tme the strage In the reach was 58,000 secnd-ft hurs. The abrupt change n dscharge ndcates that ths reach wuld have been t lng fr drect applcatn f McCarthy's methds. Hwever, the dscharges at ntermedate pnts culd be estmated frm the hydrgraphs themselves, that Is, the dscharge half-way between s prbably the average f that n the utflw-hydrgraph 7/2 hurs earler wth that n the nflw-hydrgraph 7/2 fllwng. The frst gves 3,600 secnd-feet, and the secnd 3,200 secnd-feet. Cnsequently McCarthy's frmula culd be mprved fr sme applcatns by usng the average dscharge durng an nterval nstead f nstantaneus dscharges. The equatn may be mdfed as fllws S t - UXL ( t _g x k ) + ( - ) D ( T + 2 X K )3 (2)

4 REPORTS AND PAPERS, HYDROLOGY where S t s strage at tme t, x and k are as befre but I(t-2xk) s t h e average nflw durng an nterval 2xk precedng t and D^t+2xj-j s the average dscharge durng an nterval 2xk fllwng t. When x * 0.50 the frmula becmes s t * k[0.5 l^^) + * 5 D (t+k) ^ m ^ snce I(t-k) * D(t+k) t h e n s t = ^(t+k)* w e n x * 0 equatn (2) becmes S+. * kd t. Therefre an apprxmatn that s useful n sme prblems s suggested as fllws where x and k have the same sgnfcance and values as befre. S t " ki >(t+2xk) (3) 4,000 2,000 -Q j 5 h urs j! j / \ : \! / \ «/ 57 V fl ' 2* ml -J / a / J \J j I I I Fg. 2 Hydrgrap h f dscharge f Nr th Platte Rver at Br dgeprt, Nebraska. and at Lsc, Ne braska, June 935 June, 935 2,000 K=I7 hurs X=0,47 0 G + N G E X P L A N A T I O N O Strage pltted aganst dscharge at Lsc 0 Strage pltted aganst weghted average dscharge at Brdgeprt and Lsc Fg. 3 Stra ge-dscha rge relat n ISO Strage n thusands at secnd-ft hurs

5 624 TRANSACTIONS, AMERICAN GEOPHYSICAL UNION Fgure 3 als shws a well-defned relatn between the vlumes f strage and the weghted dscharge as defned abve by equatn (2), usng a value f x fund by tral equal t 0.47, ndcatng nearly deal unfrmly prgressve flw as mght be cnfrmed by nspectn f Fgure 2. The relatn clsely apprxmates a straght lne f slpe equal t 7 hurs and culd have defned by equatn (3) as well, usng a tme-nterval f 2 X 0.47 x 7» 6 hurs. In cmparng ranfall wth the runff frm a dranage-basn, It s frequently f assstance t adjust the hydrgraph f utflw frm the basn fr strage. Hrtn [4] and the wrter [5j have descred metnds fr dng ths. The frst-named reference made n allwance fr accuntng fr greater strage n the rsng sde and the secnd nly treated f the prblem n whch upstream gagng-statns were avalable. A methd wll be descrbed fr evaluatng strage n tne Dasn abve the mst upstream gage durng a fld by the use f equatn (3). The prblem essentally s ne f the selectn f the prper values f k and x fr the gven basn, partcularly the latter. Ether equatn () r equatn (3) may be used. Althugh the use f equatn () may be mre accurate, ts slutn fr the nflw requres mre wrk than des equatn (3). If equatn (3) s used, mprved results may frequently be btaned by evaluatng D(t+2xk) as tne gemetrc mean ver the nterval 2xk nstead f usng the arthmetc mean. When the range n dscharge durng the nterval s small, the dfference s nt great. In rder t fnd a reasnable value f x applcable t entre basns, several basns were examned n whch there were a number f small gaged upstream-areas. The strage n these basns at apprxmate ntervals durng tw r three dfferent fld-rses was cmputed by subtractng the ttal channel-nflw as ndcated by the small gagng statns frm the ttal utflw as gven by the gagng statn at the utlet. Fgure 4 shws a strage-lp cmputed by ths metnd fr Hsc Rver near Eagle Brdge, New Yrk (50 square mles), durng the fld f September, 938. The upstream statns n ths prblem were the fllwng: Hsc Rver at Adams, Massachusetts (46.3 square mles; Nrth Branch Hsc Rver at Nrth Adams, Massachusetts (39.0 square mles); and Wallmsac Rver near Nrth Bennngtn, Vermnt ( square mles). Fgure 4 als shws the strage-lp clsed by the use f equatn (3) by plttng the strage at a gven tme aganst the mean dscharge durng an nterval f egnt hurs fllwng the gven tme. The pnts shwn apply t the fld f March, 936, as well as that f September, 938. The slpe f the average lne shwn s 9 hurs and therefre x «[8/(2 X 9)]» 0.2. Cmparsn f tne tme-lag between center f mass f effectve ranfall durng the strm f September, 938, and center f mass f drect runff was cmputed as 2.5 hurs. The dfference f 2.5 hurs may be accunted fr n tw ways: (a) The slpe f the dscharge-strage relatn des nt nclude the "nlet" tme r ntal lag-nterval but ncludes nly the tme f transt n the defned stream-channels; (b) nadequaces f the selected methd f evaluatng ttal strage n Hsc Rver Basn; analyss by use f the recessn-curve as explaned belw gave a value f k 23 hurs. Fld-waves n ther basns that have been examned have ndcated that as an average a value f x * 0.25 s apprxmately crrect fr evaluatng the strage-effects and that the lag between ranfall and drect runff s sensbly the same as the slpe f the weghted dschargestrage curve. The value f x * 0.25 seems t Indcate that n the average fld-flws n natural rver-basns seem t be affected as much by reservr-actn as by unfrmly prgressve flw. There s flattenng f tne fld-crest due t reservr-actn and desynchrnzed tmng f the flw frm dfferent parts f the basn, but nt as much as wuld be ndcated by the same vlume f strage perated as a strage-reservr. The amunt f strage n the rver-channel system abve a sngle gage can mst readly be apprxmated by analyss f the recessn-lmb as suggested by Hrtn [4] and the wrter [5]. In the prblem f translatng utflw nt nflw t s desrable t knw nt nly the ttal vlume f strage but als Its relatn t the rsng sde as well as t the recessn-lmb f the hydrgraph. Fr applcatn t the rsng sde, as well as t the fallng sde f a hydrgraph, cgnzance must be taken f the x-factr, and accrdng t the studes descrbed abve ths factr fr mst basns may be taken as 0.25 and the recessn dscharge-strage relatnshp develped accrdngly. Thus If equatn (3) s used, such a dagram wuld shw the relatn between the dscharge ver an nterval equal t 0.50k (termed ndex-nterval) and the strage at the begnnng f that nterval. Snce k s nt generally knwn, ts value must frst be apprxmated, befre the 0.50k ndex-nterval can be derved. Table, fllwng, ndcates the methd f cmputatn emplyed. An Interval f eght hurs was used nstead f exactly 0.50k n ths analyss n the bass f the precedng studes f Hsc Rver Basn abve Eagle Brdge, New Yrk; the tme-nterval chsen can be runded ff t btan a cnvenent tabulatng nterval.

6 REPORTS AND PAPERS, HYDROLOGY Table Recessn-curve analyss Hsc Rver near Eagle Brdge, New Yrk, fld f September 938 Date Tme Dscharge, Base, Drect runff, Strage, 8-hr unts 8-hr mean dscharge, Sep a.m. 29,000,200 27,800 59,990 20,900 4 P m. 6,000,200 4,800 38,690 2 P m. 9,500,200 8,300 27,40 Sep a m. 6,650,300 5,350 20,35 4 P m. 5,260,300 3,960 5,660 2 P m. 4,450,400 3,050 2,55 Sep. 24 8" a m. 3,900,400 2,500 9,380 4 P m. 3,490,500,990 7,35 2 P m. 3,50,500,650 5,35 Sep a m. 2,880,550,330 3,825 4 P m. 2,630,600,030 2,645 2 P m. 2,40,600 80,725 Sep a m. 2,270, ,00 4 P 2,30, ra. P m.,970, AS =59,505 AQ = 20,550 K = (AS/AQ) = (59,505/20,550) = 2.9 eght-hur unts = 23 hurs In adjustng a hydrgraph fr strage usng equatn (3), the strage at a gven tme s cmputed as k tmes the gemetrc mean dscharge, q-^, q 2 Q n v er an nterval equal t 0.50 k (r whatever value f x was used n cmputatn f k as n Table ) fllwng the gven tme. Or a dagram may be prepared fr the slutn f equatn () frm knwn values f k and assumng x» 0.25 as shwn n Fgure 5, n whch strage has been expressed n terms f the utflw-rate and the nflw-rate. The graphcal methd has an advantage f flexblty n that the

7 626 TRANSACTIONS, AMERICAN GEOPHYSICAL UNION slpe f the lnes can be vared t allw fr changes In k wth dscharge. In general the usefulness f equatn (3) s lmted t cmputng nflw frm a graph f dscharge. After adjustment fr strage by ths methd, the center f mass f the cmputed nflw wll precede the center f mass f the bserved drect runff by an nterval equal t k. If k equals the lag between effectve ranfall and runff, the adjusted hydrgraph wll cmpare favrably n a tme-bass wth the effectve ranfall. Hyt [6] referred t the lag-nterval as a pssble means f crrelatng unt-hydrgraphs, and Snyder [7] has descrbed methds f crrelatng the unt-hydrgraph fr a gven basn wth ts lag-interval. Snyder has used the lag-nterval between center f mass f effectve ranfall and peak-dscharge f the unt-hydrgraph as the crrelatng factr. The wrter suggests that the lag between center f mass f effectve ranfall and center f mass f drect runff s f greater flexblty, and mrever has ther physcal sgnfcances as descrbed befre. Mrever, determnatn f the lag-nterval as thus defned Is nt lmted t unt-strms. Its use as a measure f the channel-strage r unt-hydrgraph characterstcs f a basn seems t be prmsng.

8 REPORTS AND PAPERS, HYDROLOGY The summatn-hydrgraph (S-hydr graph) Is defned by Mrgan and HulHnghrst [8] as the hydrgraph f runff frm a basn wth cntnuus generatn f runff. It affrds a cmpact means fr cmparng the hydrgraph-characterstcs fr strms f dfferent duratn. As descrbed by Mrgan and HulHnghrst, unt-hydrgraphs fr strms f any effectve length can readly be derved frm the summatn-hydrgraph. Ths s dne by placng tw Identcal S-hydrgraphs alng ne anther, dsplacng them by a tme-interval equal t the length f the desred unt-hydrgraph strm, and subtractng the rdnates f the ne frm the ther at crrespndng tmes. Examnatn f a number f S-hydrgraphs fr dfferent basns suggests that lag-interval may be emplyed In cmparng them. The results are shwn n Fgure 6. The rdnates f the summatn-graphs have been expressed as a percentage f the ultmate dscharge (vlume f runff dvded by perd f effectve ranfall) and the abscssas as tme frm begnnng f effectve ranfall cnsdered as a per cent f the lag-interval fr the dfferent graphs. The pnts shwn apply t abut ten dfferent summatn-graphs fr areas rangng frm 30 t 4,000 square mles n sze n the Nrtheastern States. When equlbrum s reached, that s, when the rate f utflw equals Inflw, the vlume f strage In the basn s equal t the area abve the summatn-hydrgraph and belw the 00 per cent dscharge-lne. Under these stablzed cndtns, the rat between vlume and rate f dscharge equals the lag-nterval r n the case shwn, unty. The dscharge f a unt-hydrgraph based n the summatn-hydrgraph s q = s[(vlume f runff)/(lag-interval)] (4) where s s the slpe f a chrd f the summatn-hydrgraph (per cent f ultmate dscharge dvded by per cent f lag) ver an nterval equal t the duratn f the perd f effectve ranfall. Fr a unt-hydrgraph the vlume f runff s ne Inch and f the lag-tme s expressed In hurs, equatn (4) becmes s[646/(lg-nterval)] where q s rate n secnd-feet per square mle and s s dmensnless. The maxmum slpe f the summatn-hydrgraph,.20, ccurs at 0.5 lag. Ths slpe Is the measure f the maxmum dscharge fllwng an ne-nch ranstrm f neglgble duratn (generally less than ne-tenth lag). The crrespndng dscharge n secnd-feet per square mle s as fllws Peak-dscharge =.20 [646/(lag-nterval)] = [770/(lag-nterval)] (5) Snyder [7] gves an average dscharge f [400/(lag t peak)] fr a strm f duratn equal t the lag t the peak dvded by 5.5. The tw are nt drectly cmparable. The peak-dscharge gven by equatn (5) r dscharges defned by Fgure 6 are averages defned by a number f hydrgraphs and may nt ft any partcular basn. N lmts can be set fr varatns frm the mean at ths tme. It s beleved, hwever, that dfferences n shape f basn and arrangement f trbutares, slpe f the several basns, and nn-unfrmty f ranfall wuld accunt fr such varatns. The wrter acknwledges the valuable assstance f L. C Gelgcal Survey n the preparatn f ths paper. References Crawfrd f the Unted States [] H. A. Thmas, The hydraulcs f fld mvements In rvers, Carnege Inst. Tech., 934. [2] E. J. Rutter, Q. B. Grves, and F. F. Snyder, Fld rutng, Trans. Amer. Sc. Cv. Eng., V. 04, pp , 939. [3] G. T. McCarthy, The unt hydrgraph and fld rutng, unpublshed manuscrpt presented at cnference f Nrth Atlantc Dvsn, Crps f Eng., U. S. Army, June 24, 938. (4] R. E. Hrtn, Natural stream channel-strage, Trans. Amer. Gephys. Unn, pp , 936. [5] W. B. Langben, Sme channel-strage studes and ther applcatn t the determnatn f Infltratn, Trans. Amer. Gephys. Unn, pp , 938. [6] w. G. Hyt, Studes f relatns f ranfall and runff n the Unted States, U. S. Gel. Surv. W.-S. Paper 772, p. 209, 936. [7] F. F. Snyder, Synthetc unt-graphs, Trans. Amer. Gephys. Unn, pp , 938. [8] R. Mrgan and D. W. HulHnghrst, Unt hydrgraphs fr gaged and ungaged watersheds, unpublshed manuscrpt, U. S. Eng. Offce, Bnghamtn, N.Y., July 939. U. S. Gelgcal Survey, Washngtn, D. C.