Research Note A 1D Model for Erosion Through Submerged, Prone Vegetation

Transcription

1 Rsarh Not A 1D Modl for Eroson Through Submrgd, Pron Vgtaton J. M.V. Saman Asso. Prof. of Agrultur Faulty, Tarbat Modarrs Unv. P.O.Box: saman_j@modars.a.r (Rvd: Sp. 3, Aptd: Ds. 4) Abstrat-Vgtaton an provd an nvronmntally sound lnng for hannls, floodways, low spllways and othr watr onvyan struturs. In ths papr, th fators afftng th roson rat n vgtatd hannls ar addrssd. A for-balan approah was dvlopd to stmat th shar strss atng on th sol undrlyng a vgtatv hannl lnng. Th roson rat was thn prdtd wth an aptd sdmnt transport rlatonshp basd on ths shar. Th dformaton of vgtaton ausd by th flow's momntum was aountd for by alulatng th amount of bndng of th vgtaton usng fors alulatd by th for-balan mthod. Th rsults ar n good agrmnt wth th xprmntal roson rat data for modld vgtatd hannls. Kywords: Vgtaton, Eroson, Shar strss, For qulbrum quaton. 1. Introduton Grass lnngs provd an attratv, ostfftv altrnatv for th protton of arth hannls xposd to ntrmttnt flow. Ths form of protton has long bn usd n agrultural dranag hannls [1] and, of ours, vgtaton ours naturally along th banks of many strams and rvrs. Grass-lnd mrgny spllways hav bn usd n both th USA and th UK as an altrnatv to ostly onrt laddng [-4]. Also, ground ovr n rlls gratly rdus th shar strss that ats on th undrlyng sol and hn rdus dtahmnt rats and sdmnt transport apaty of th flow vn n upland aras [5]. Today, th ntraton of th flow wth a vgtatd boundary s stll too omplatd to b analyzd n dtal. Baus of th omplx natur of th flow systm and th varty of ondtons that xst n th flow through vgtaton, t s dffult to dvlop an analytal flow modl basd ntrly on thortal drvatons. In th past, muh of th rsarh don on ths top has bn mpral n natur and wth almost no analytal bass. Ths papr dvlops a for-qulbrum modl that s apabl of dtrmnng th tratv strss at th sol-watr boundary ausd by th flow ovr a vgtatv hannl lnng, sn t s th man fator that affts roson. It s a physal and dtald approah whh mploys smpl turbuln onpts and numral alulatons. Of ours, mor omplatd turbuln modls would nd mor omplatd fforts to b appld. In ths papr, th roson rat data olltd at th Unvrsty of Watrloo n a hannl wth smpl modld vgtaton wr utlzd.. Bakground A numbr of mthods for assssng th sol loss hav bn dvlopd. Th most ommon rlatonshp s th Unvrsal Sol Loss Equaton (USLE) whh s an mpral modl dvlopd by th Untd Stats Dpartmnt of Agrultur, (USDA), [6].

2 Modars Thnal and Engnrng No. 3, Sprng 6 Th USLE s not ntndd for alulatng roson rats n vgtatd hannls. Most urrnt modls for roson rly on th gross haratrsts of flow, most ommonly thr hydraul shar strss or stram powr, to haratrz th apaty of th flow to dtah sol [7-9]. Myr t. al. [1] hav usd th rtal dsharg blow whh rll roson s nglgbl for stmatng th roson rat n th rlls ausd by runoff. Myr t. al. [1] and Fostr t. al. [11] hav suggstd that roson by flow ovr a tlld loos sdbd may b dsrbd by: 1 Er kk ( ). 5 (1) n whh kk = roson rat offnt; = avrag total shar strss on hannl bd (N/m ); = rtal shar strss at th solwatr ntrfa (N/m ); and Er = roson rat n volum pr unt ara pr unt tm (m/s). Othr rsarhs hav suggstd th sam form of Eq. (1), whr dtahmnt prosss ar onsdrd to domnat, and th roson rat s assumd to b proportonal to th fftv shar strss n xss of som rtal valu [4, 1-16]. Myr [1] has modfd th Duboys sdmnt transport quaton to stmat th dtahmnt of sol by flowng watr as: a Er kk ( ) () n whh: a = roson rat xponnt; and = fftv shar strss (N/m ). Both kk and ar assumd to b sol proprts. Th most mportant omponnt n Eq. () s, whh s alulatd by th for qulbrum mthod. Th for qulbrum quaton has bn usd by Ordway t al. [17] to modl th vloty dstrbuton of th flow nsd plants. Thompson and Robrson [18] hav attmptd to analytally rprodu vloty dstrbutons usng ths approah. Mru [19] and Shutzman [] hav modld th vloty profl of th flow ovr artfal flxbl roughnss usng th for qulbrum quaton. Jordanova and Jams [16] hav usd th for qulbrum quaton to stmat th bd load transport through mrgnt vgtaton by applyng mpral rlatonshp for alulatng th drag for. Th for balan for a layr of flud paralll to th bd an b wrttn as: FG FD d / dy (3) whr y s th dstan from bd hannl, F G s th for pr unt ara du to gravty n N, F D s th drag for pr unt ara n N, and s th shar strss n N/m. On th bass of th for qulbrum quaton, th drag for s gvn by: F N C A U D D s (4) whr, N = th numbr of grass unts pr unt ara, C D = drag offnt for a grass unt, = frontal projtd ara of a grass lmnt A s for an nrmntal dpth, dy, = dnsty of watr n kg/m 3. Th gravty for for on unt ara s gvn by: FG g dysn (5) Whr, g s th gravtatonal onstant, dy s th nrmntal dpth and s th hannl slop angl. Th shar strss an b wrttn n trms of th Boussnsq xprsson: du (6) dy whr, s th ddy vsosty. Usng th Prandtl mxng-lngth hypothss for b wrttn as: du dy, t an (7) whr, s th dynam vsosty n kg/m.s and s th mxng lngth n m. Tsujmoto t. al. [1] hav suggstd a paramtr,, as an addtonal for to mak t possbl to prdt th strutur of th flow abov and nsd th flxbl vgtaton du to th organzd wavng moton. Whn th flow s unform, th

3 A 1D Modl for Eroson Through... J. M.V.Saman for qulbrum quaton d boms: FG F D (8) d y Intgratng th abov quaton, on obtans th for qulbrum quaton wth a nw paramtr, M, whr, M M ( y ) dy y Tsujmoto t. al. [1] hav usd th followng xprsson for M : M whr,.. xp( max y ) (9) max = th dffrn btwn th shar strss at th grass tps n ts wavy moton and ts fxd ondton, and mpral paramtr. s also dfnd as th max dffrn btwn th shar strss at th grass tps n ts rt and ts wavy ondtons. Th latr dfnton s only vald whn th grass s just flxbl and has lmtd wavy moton lk 5% of th grass hght. Th addtonal trm proposd by Tsujmoto t al. (1993) for th wavng moton was not adaptd for ths study baus all th ass nvstgatd n ths study wr apron and out of th wavy moton rang. 3. Formulaton By wrtng th for qulbrum quaton (Eq.3) n a dffrntal form, w wll hav: B 1/ _ 1/ N CD A 1/ 1/ s 1/ 1 du dy U du dy (1) whr,, +1/ and +1 ndat th bottom, th mddl and th top lvls of th lmnt, rsptvly. Eq. (1) an b wrttn for ah y, th hght of th nrmntal lmnt, and a systm of non-lnar quatons rsults. Th boundary ondtons for th systm of N quatons ar N 1 = at watr surfa and vloty, U bd at th bd. In ths study t was assumd that thr s a shar strss at th bottom du to th frton: 1 1 C A U Df whr, C Df s th frton drag offnt assoatd wth th bd surfa; ths paramtr nluds th wak ffts ratd by th flow bhnd th grass lmnts, and A s th porton of th ara not oupd by th grass lmnts to th total ara, and U 1 s th flow vloty atd on th frst nrmntal lmnt. Provdng that N, C, C, and A, 1/ D 1/ Df s an b dfnd, Eq. (1) an b solvd as a st of N quatons wth N unknowns. Baus of th non-lnarty n th quaton, th systm of quatons must b solvd tratvly. Th systm of N quatons an b wrttn n a matrx form A U B, whr, A s th offnt matrx and U s th vtor of th unknown vlots. Th vtor B onssts of th known trms of Eq. (1), that s: B g y sn 1 A / sn (11) = Whr, A s th ross ston ara for a grass lmnt. Equaton (1) an yld th soluton for U by usng th Thomas Algorthm [] and th fftv shar strss on th bd an b alulatd by bak substtuton. To solv Eq.(1), two quantts must b spfd for ah layr: th mxng-lngth, haratrsts of th roughnss, C D. Th mxng-lngth an b xprssd as: FM, and th drag v y y 1 / for NB SF k y y k for NB (1)

4 Modars Thnal and Engnrng No. 3, Sprng 6 whr, NB s th numbr of nrmntal ponts wthn th grass lmnt, k s th mxnglngth at th dfltd hght, k, y k s y at th hght k, v and ar Von Karmn's turbuln offnts for th rgons undr and ovr th grass, rsptvly. FM, SF, C D, and C Df ar offnts to b stmatd by optmzaton, whr, FM and SF ar dmnsonal f thy ar not qual to 1. It s also possbl to altr th xprsson for dmnsonlss varabls [3]. usng Th drag for atng on th grass lmnt by th flow was usd to alulat th dfltd hght of th grass lmnt usng th bam dflton thory. It was dvdd nto two omponnts for ah nrmntal lmnt of th grass: F D ( F D NC sn D A s ) U ( F D F os DN ) F DS 1 / 1 / (13) whr, F DN and F DS ar th normal and paralll omponnts of drag for, rsptvly. For mor dtals about th dflton analyss, s [4] and [3]. Ths approah has bn appld to th databas olltd at th Unvrsty of Watrloo [3]. 4. Exprmntaton Th xprmntaton rqurd to valdat th mathmatal modl ntrodud n th prvous stons whh nvolvs th masurmnt of dpth, rt and dfltd hght of th grass, flow rat, vloty profls and th roson rat. For ah run, th vloty profls wr obtand at th ntr of th hannl by at dffrnt lvls startng at th hannl bd. Th vlots wr masurd by a Ptot-stat tub onntd by two.9 m lngths of plast tub (damtr, 3 mm) to an ndutan typ dffrntal prssur transdur (Sanborn, Modl 68B). Th damtr of th mpat opnng at th Ptot tp was 1.1 mm and th prssur transdur was albratd wth stat watr olumns. 5. Rsults: Th vloty profls and roson rats masurd n ths nvstgaton wr usd to albrat th for qulbrum and th roson modls. Th vloty profls wr obtand by solvng Eq. 1 for th whol dpth. By applyng bak substtuton on Eq. 1, th fftv shar strsss (shar strss atng on th bd) wr alulatd and thn usd n Eq. to stmat th roson rat for ah run. Evaluaton of th dffrnt paramtrs nvolvd n Eq. (1) was aomplshd usng a pattrn sarh optmzaton shm [5] as programmd by Monro [6]. Th rsults of th optmzaton ar summarzd n Tabl 1. Tabl 1 Vloty profl paramtrs Varabl SF FM C D v C D F Pron Tabl 1 shows th optmzd valus of th dffrnt paramtrs for th pron as. Th pron as was th only as obsrvd n ths nvstgaton along wth th sdmnt roson. Equaton () was wrttn n dmnsonlss form as: a Er kk( ) (14) and part of th dmnsonlss masurd roson rats, ( prnt of th total data) was rgrssd aganst ( ) for th two gran

5 A 1D Modl for Eroson Through... J. M.V.Saman szs usd n th xprmnt to albrat th modls, (Equatons 15 and 16). Th dmnsonlss fftv shar strss s gvn by * ( D s w ) 5, Er s dmnsonlss roson rat gvn by Er/ u, and s th r dmnsonlss rtal shar strss. Th dmnsonlss fftv shar strss was alulatd by th vloty profl program by applyng bak substtuton of th alulatd vloty profls n th Eq. 1. In ths nvstgaton, th dmnsonlss rtal shar strss,, [7] was basd on th Shlds dagram and was st qual to.5. u s th r rtal shar vloty whh s qual to /. Th rsults for th two albratd modls ar as follows: Coars Sand: Er. 75 ( ) (15) Fn Sand: Er. 7 ( ) (16) Fgurs 1 and show th dmnsonlss masurd and alulatd roson rats for th whol data, whr th alulatd valus wr obtand from th albratd modls usng th alulatd shar strsss. Ths fgurs show that t s possbl to alulat th fftv shar strss atng on sand nsd a flxbl roughnss layr usng th for qulbrum approah wth rasonably good rsults provdd that th frontal ara and offnt of th drag for th rsstng lmnts an b spfd. masurd alulatd Fgur 1 Masurd and alulatd dmnsonlss roson rats for D 5 =.453 mm..1.1 masurd alulatd Fgur Masurd and alulatd dmnsonlss roson rats for D 5 =.44 mm

6 A 1D Modl for Eroson Through... J. M.V.Saman szs usd n th xprmnt to albrat th modls, (Equatons 15 and 16). Th dmnsonlss fftv shar strss s gvn by * ( D s w ) 5, Er s dmnsonlss roson rat gvn by Er/ u, and s th r dmnsonlss rtal shar strss. Th dmnsonlss fftv shar strss was alulatd by th vloty profl program by applyng bak substtuton of th alulatd vloty profls n th Eq. 1. In ths nvstgaton, th dmnsonlss rtal shar strss,, [7] was basd on th Shlds dagram and was st qual to.5. u s th r rtal shar vloty whh s qual to /. Th rsults for th two albratd modls ar as follows: Coars Sand: Er. 75 ( ) (15) Fn Sand: Er. 7 ( ) (16) Fgurs 1 and show th dmnsonlss masurd and alulatd roson rats for th whol data, whr th alulatd valus wr obtand from th albratd modls usng th alulatd shar strsss. Ths fgurs show that t s possbl to alulat th fftv shar strss atng on sand nsd a flxbl roughnss layr usng th for qulbrum approah wth rasonably good rsults provdd that th frontal ara and offnt of th drag for th rsstng lmnts an b spfd. masurd alulatd Fgur 1 Masurd and alulatd dmnsonlss roson rats for D 5 =.453 mm..1.1 masurd alulatd Fgur Masurd and alulatd dmnsonlss roson rats for D 5 =.44 mm

7 Modars Thnal and Engnrng No. 3, Sprng 6 [16] Jordanova, A. A., and Jams, A. S. (3) "Exprmntal Study of Bd Load Transport through Emrgnt Vgtaton". Journal of Hydraul Engnrng, ASCE, 18(1): [17] Ordway, D. E., A. Rttr, D. A. Spn, and H. S. Tan (1963). "Effts of Turbuln and Photosynthss on CO Profls n th Lowr Atmosphr". In th Enrgy Budgt at th Earth's Surfa, Part, d. E. R. Lmon, Produton Rsarh Rport No. 7, Agrultural Rsarh Srv, US. Dpartmnt of Agrultur, Washngton, D.C., 3-6. [18] Thompson, G. T., and J. a. Robrson (1976). "A Thory of Flow Rsstan for Vgtatd Channls". Transatons, ASAE, 19(): [19] Mru, T. J. (1981). "A Physal Basd Modl for Dtrmnng Vloty Profls n Vgtaton". M.A.S Thss, Unvrsty of Watrloo, Watrloo, Ontaro. [] Shutzman, W. C. (1981). "Partl Sttlng n Vgtatd Watrways". M.A.S Thss, U. of Watrloo, Watrloo, Ontaro. [1] Tsujmoto, T., T. Okada, and K. Kontan (1993). "Turbulnt Strutur of Opn-Channl Flow Ovr Flxbl Vgtaton". KHL Progrssv Rport, Hydrauls laboratory, Kanazwa Unvrsty, pp [] Wang, H. F.,and M. P. Andrson (198). Introduton to Groundwatr Modlng. Fnt Dffrn and Fnt Elmnt mthods. W. H. Frman and Compand, Nw York. [3] Saman, J. M. V. (1995). "Stablty and Eroson of Vgtatv Channl Lnngs". Ph.D. Thss, Unvrsty of Watrloo, Watrloo, Ontaro. [4] Saowapon, C. M., and N. Kouwn (1989). "A Physally Basd Modl for Dtrmnng Flow Rsstan and Vloty Profl n Vgtatd Channls". Prodng of th Intrnatonal Confrn on Channl Flow and Cathmnt Runoff, Charlotsvll, Vrgna 99, USA. [5] Hook, R., and T. H. Jvs (1961) "Drt Sarh Soluton of Numral and Statstal Problms". Journal Asso. of Comput. Mah., 8(), 1-9. [6] onro, J. C. (1971). "Drt Sarh Optmzaton n Mathmatal Modlng and a Watrshd Applaton". NWS HYDRO 1, Natonal Wathr Srv, Slvr Sprngs, MD. [7] Shlds, A. (1936). "Applaton of Smlarty Prnpls and Turbuln Rsarh to Bdload Movmnt", Hydrodynam Lab. Rp. 167, Calforna Insttut of Thnology, Pasadna, Calf. [8] Saman, J. M. V., N. Kouwn (1998). " Eroson n Vgtatv Channl Lnngs". Journal of Hydraul Engnrng, ASCE, 18(1): [9] Moffat, R. J. (1988). "Dsrbng th Unrtants n Exprmntal Rsults". Exprmntal Thrmal and Flud Sn, 1:3-17.

8 A 1D Modl for Eroson Through... J. M.V.Saman 3. Owng to thr nflun on th vloty profls, von Karman s mxng-lngth offnt,, and FM for th flow rgon v nsd th grass zon, hav a sgnfant fft on th outom of th for qulbrum modl, ndatng that th modl s snstv to th valu of th mxng lngth n ths zon. Thrfor, t s nssary to hav a larg numbr of vloty masurmnts n ths zon for th optmzaton pross. Othr vloty profl paramtrs hav shown lss nflun, allowng thm to b onsdrd as onstants. 4. Th rsults of ths study just an b takn as a gud study to othr rsarhrs to ondut xprmnts wth ral vgtaton through valuatng th drag for. 8. Rfrns [1] Tmpl, D. M. (1983). "Stablty of Grass- Lnd Opn Channls". Transatons, ASAE, 6(4): [] Bow, A. J. (198). "Invstgaton on Vgtaton Erodng Stram Banks". Transaton of th ASAE, pp [3] Brhall, J., and Pnyan C. (1986). "Grass Roots Approah to Spllways". Engr. Nws-Rord, 17(9): [4] Tmpl, D. M. (1987). "Vgtatd Protton of Embankmnts and Spllways". Prodngs of th 1987 Natonal Confrn on Hydrauls Engnrng, d. R. M. Ragan, ASCE, [5] Narng M. A., Lan L. J., Lafln J. M. (199b)."Prdton Thnology for Sol Eroson by Watr: Status and Rsarh Nds". Sol S. So. Am. J. 54: [6] Wshmr, W. H., Smth D. D. (1965). "Prdtng Ranfall Eroson Losss from Mountans". Agrultur Handbook No. 8, US. Dpartmnt of Agrultur, Washngton, D. C. [7] Fostr, G. R. (198). "Modlng th Eroson Pross": In Hydrolog Modlng of Small watrshds, ds. C. T. Haan, H. P. Johnson and D. L. Braknsk, St. Josph, MS: ASAE. [8] Ros, C. W. (1985). "Dvlopmnts n Eroson and Dposton Modls". In Advans n Sol Sn, Vol., d. B. A. Stwart, Nw York Sprngr-Vrlag. [9] Harsn, P. B. (1988). "A Physally basd Modl of Th Eroson of Cohsv Sols." Ph.D. Dss., Grffth U., Brsban, Australa. [1] Myr, L. D., G. R. Fostr, and S. Nkolov (1975). "Efft of Flow and Canopy on Rll Eroson". Transatons of th ASAE, 18(5): [11] Fostr G. R., L. J. Lan, and J. D. Nowln (198). "A Modl to Estmat Sdmnt Yld from Fld Szd Aras". Vol. 11: Usr Manual. Consrvaton Rsarh Rport No. 6. USDA. S. and Edu. Admn. Chapt.. pp [1] Myr, L. D. (1964). "Mhans of Sol Eroson by Ranfall and Runoff as Inflund by Slop Lngth, Slop Stpnss, and Partl Sz". Ph.D. Thss. Purdu Unv., W. Lafaytt, IN. [13] Narng M. A., L. D. Asough, and J. M. Lafln (199a). "Snstvty Analyss of WEPP Hllslop Profl Eroson Modl". Transaton, ASCE, 33 (3): [14] Patronsky, R. J. and D. M. Tmpl (1983). "Dsgn of Grassd Watrways Usng Sols Data". ASAE papr No , ASAE, St. Josph, MI [15] Wks, J. M., J. C. Bathurst, and C. W. Johnson (199). "Calbratng SHE Sol- Eroson Modl for Dffrnt Land Covrs". J. Irr. and Dra. Eng. 118(5):