Guo, Jams C.Y. (006). Knmatc Wav Unt Hyrograph for Storm Watr Prctons, Vol 3, No. 4, ASCE J. of Irrgaton an Dranag Engnrng, July/August. Guo, Jams C.Y. (998). "Ovrlan Flow on a Prvous Surfac," IWRA Intrnatonal J. of Watr, Vol 3, No, Jun. INFLECION POIN ON RECESSION HYDROGRAPH Jams C.Y. Guo, PhD, P.E., Profssor Cvl Engnrng, U. of Colorao at Dnvr, Jams.Guo@cunvr.u Inflcton pont on th rcsson hyrograph has bn rcognz as on of th mportant tm paramtrs whn analyzng th watrsh rspons to ranfall. Convntonally, th nflcton pont on a rcsson hyrograph s fn by th cay of flow rat (Ponc 989). hs papr prsnts a rvaton to locat th nflcton pont by th normalz flow rat on th knmatc wavr unt hyrograph (KWGH). PARAMEERS A INFLECION POIN Ovrlan flow prouc by a slopng mprvous plan unr a unform ranfall xcss s knmatc n natur. h watr flow pth can b scrb as: y () t whr y flow pth, ranfall xcss, an ttm. At t0, th slopng plan s assum to b ry or y(t0)0 vrywhr. Unr ths ry b conton, Eq s ntgrat as: y t () h knmatc wav assocat wth Eq s propagat at th wav sp as: x t V αy α t w (3) α k (4) S 0 whr x stanc along watrway or slopng plan, V w knmatc wav sp, k roughnss of watrway, S o plan slop along watrway, α coffcnt trmn by plan slop an roughnss, an xponnt on ratng curv. In practc, 5/3 for Mannng s formula or 3/ for Chzy s formula, Intgratng Eq 3 yls αy x (5) Eq s an 5 fns th qulbrum watr profl across th watrway at t n whch s ranfall uraton. Eq 5 rprsnts th basc ratng curv rlatonshp n knmatc wav. Q x x αy (6)
Whr Q x unt-wth flow rat at Staton x. Aftr ran cass, t>, xcss ranfall ntnsty quals to zro (.. 0), thrfor, Eq bcoms: y t 0 (7) Intgratng Eq 7 wth rspct to t gvs yconstant. h rcng wav s propagat wth a constant pth by whch th knmatc wav sp s constant as: x t V αy w (yconstant aftr ) (8) As llustrat n Fgur, Eq 8 can b ntgrat for th rcng wav to travl from Staton x to Staton ovr th tm ntrval from to as: x α y t (9) x ( y )( ) x α (0) Substtutng Eq 4 nto Eq 0 yls ( αy ) ( αy ) () Eq pcts th rcng knmatc wav watr profl... par (, y) at Staton at tm. Fgur Propagaton of Rcng Wav A rcsson hyrograph s fn by th varaton of ts curvatur. h pont of nflcton s locat whr th curvatur vanshs. Mathmatcally, such a locaton on th rcsson hyrograph can b trmn by sttng th scon rvatv of Eq qual to zro.
y 3 α( ) [ ( )( ) y ( )] + y 0 α () y Y, Rarrangng Eq, th flow pth on th nflcton pont s obtan as: Y ( )( ) (3) A by Eq wth yy,, th corrsponng locaton, x, s rv to b: αy + (4) Eq 4 fns th curv of nflcton ponts along th watrway. As llustrat n Fgur, th nflcton pont on th outlt hyrograph can b trmn by Eq 4 wth xl. Fgur Illustraton of Rcng Watr Profl Applyng Eq 4 to th watrsh outlt, th flow pth at th nflcton pont s trmn as: αy L + (5) whr L lngth of watrsh or locaton of outlt an Y flow pth at nflcton pont. Accorng to th ratng curv rlatonshp us n th knmatc wav approach, w hav αy Q (6) Q E L (7) Whr Q unt-wth flow rat at nflcton pont, Q E qulbrum or unt-wth pak flow rat. A by Eq s 6 an 7, Eq 5 s rarrang as: Q Q Q (8) 3
whr Q mnsonlss flow rat at nflcton pont. Nxt, normalz Eq 3 by th tm of qulbrum, th tm to nflcton pont s foun to b: (9) Y Y ( ) ( ) Y E whr mnsonlss tm to nflcton pont aftr ran cass, tm to th nflcton pont aftr ran cass, ranfall uraton, an tm of qulbrum for watrway. A by th ratng curv n Eq 6 an th flow rato n Eq 8, Eq 9 s convrt nto Q ( ) QE ( ) aftr ran cass (0) Usng th bgnnng of th ranfall vnt as th bas, th tm to nflcton pont s calculat as: + aftr ran starts () whr o mnsonlss tm to nflcton pont aftr ran starts. Substtutng Eq 7 nto Eq 6 yls Y Y ( ) () Y E whr Y mnsonlss flow pth at nflcton pont. As scuss n th orgnal papr, th ranfall uraton for th KWUH s st to b th tm of qulbrum. Wth, th valus of ths mnsonlss varabls at th nflcton pont s summarz n abl usng ffrnt formulas: CLOSURE Formula Q Y Mannng s 5/3.04.04 0.67 0.34 Chzy s 3/ 0.658.658 0.333 0.439 abl Valus of Dmnsonlss Varabls at Inflcton Pont. As ncat n abl, th rcsson lmb of th KWUH cays fastr usng Mannng s formula than that usng Chzy s formula. Convrsly, Mannng s formula gvs a longr rcsson hyrograph.. Guo (006) rport that th tm to nflcton pont on th rcsson lmb of th KWUH was graphcally trmn to b.8 masur from th bgnnng of th vnt. As shown n abl, th tm to nflcton pont s.04 or.658, pnng on th mprcal formula. 3. Unr a unform ranfall xcss, th tm of concntraton s numrcally qual to th tm of qulbrum. Wthout rgorous mathmatcal proof, t has bn suggst that th tm of concntraton b th tm btwn th cntr of th mass of ranfall xcss an th nflcton pont on th rcsson lmb of th runoff hyrograph (McCun 998, Guo 00). hs papr has monstrat that th valu of s.04 or clos to unt. 4
.04 (3). 04 (4) As ncat n Eq 4, th tm of concntraton or qulbrum s closly qual to th tm to nflcton pont masur aftr th ran cass, nsta of th cntr of th ranfall xcss. APPENDI I Guo, Jams C.Y. (006). Knmatc Wav Unt Hyrograph for Storm Watr Prctons, Vol 3, No. 4, ASCE J. of Irrgaton an Dranag Engnrng, July/August. Guo, Jams C.Y. (00). "Ratonal Hyrograph Mtho for Small Urban Catchmnts," ASCE J. of Hyrologc Engnrng, Vol 6, No.4, July/August. Guo, Jams C.Y. (998). "Ovrlan Flow on a Prvous Surfac," IWRA Intrnatonal J. of Watr, Vol 3, No, Jun. McCun, R. H. (998) Hyrologc Analyss an Dsgn, Prntc Hall, Nw York. Ponc, V.M. (989) Engnrng Hyrology, Prntc Hall, Nw York. APPENDI II ranfall xcss L lngth of watrsh or locaton of outlt, Q flow rat at nflcton pont, Q E qulbrum or pak flow rat whn ran cass ranfall uraton, tm of qulbrum for th watrway tm to nflcton pont aftr ran starts mnsonlss tm to nflcton pont aftr ran cass. Y flow pth at nflcton pont, α coffcnt trmn by slop an roughnss of th plan, 5/3 for Mannng s formula or 3/ for Chzy s formula, mnsonlss varabl 5