Nmeral Hydrals Blok 5 Transpor of soles n rvers Marks Holzner
Conens of he orse Blok 1 The eqaons Blok Compaon of pressre srges Blok 3 Open hannel flow flow n rvers Blok 4 Nmeral solon of open hannel flow Blok 5 Transpor of soles n rvers Blok 6 Hea ranspor n rvers
Relevane Assessmen of adens wh sable pollans n rvers Inerpreaon of raer epermens n rvers and sreams Bass for more omple models of rver waer qaly 3
Traer epermen Rhne 4
Traer epermen Rhne 5
Fl veor of a sole deomposon adveon molelar dffson rblen dffson dsperson J J D T me spae K Tme average over sohas varaons of Spae average over sysema varaons of oal fl 6
Proess of dsperson Cased by sysema veloy varaons n spae 7
Dsperson oeffen 1D rver model D b 0.011 h * h = waer deph, b hannel wdh wh 0 0 * ghi Applably of he 1D-model only afer flow dsane L wh L = average flow veloy over ross-seon, 1.8b r hy *
Inegral vew J = Mass fl [kg/s/m ] A Q Q q n, n, q 1/ o, 1/ AD 1/ AD 1/ Wh =1: onservaon of volme 9
Adveon-Dsperson eqaon o n o n n q q Q A q q AD Q A Sole mass and waer volme balane over an nfnesmal sle of he rver A q AD A n n 1 Transpor eqaon afer sbson 10
Seady sae flow who laeral nflows or oflows V Q Q AD AD 0 1/ 1/ 1/ 1/ Geomery a ross-seons ± 1/ gven: V = A -1/ + A +1/ / Prmary nknowns: onenraons n ell Dsperson: Whh graden a ± 1/? Adveon: Whh onenraon a ± 1/? Tme negraon? 11
Dsperson: deermnaon of graden Dfferene qoen nsead of dfferenal qoen: 1/ 1 Real onenraon Cell average + 1 1
Adveon: onenraon a ell nerfae Upwnd: +1/ = Downwnd: +1/ = +1 Cenral dfferenes: +1/ = + +1 / + 1 13
Cenral dfferenes +1/ = + +1 / Pro: More arae n he sense of an error analyss based on a Taylor seres epanson Conra: Osllaons negave onenraons 14
Osllaons n enral dfferenes I Transpor eqaon who dsperson or laeral n- /oflows D = 0, q n = 0 0 +1/ = + +1 / 0 - - 1 + 1 + 15
Osllaons n enral dfferenes II 0 0-1.5 0.5 1 0-1 + 1 + Corre! nreases when he fron moves! 0 aordng o he ranspor eqaon 16
Osllaons n enral dfferenes III 0 0.5 0.5-0 aordng o he ranspor eqaon Wrong! Shold derease when he fron moves! 0 1.5 0.5 1 0 Corre! - 1 + 1 + 0 17
Osllaons n enral dfferenes IV 0 Compleely wrong! Leads o negave onenraon! 0 1 0 1 0.5 1.5 - Wrong! Shold derease! 0.5 0.5 0-1 + 1 + 0 18 Smlaon of adveon wh enral dfferenes prodes al of osllaons 1.5 0.5 1 0 Corre!
Upwnd dfferenes +1/ = +1/ = +1 for posve veloy for negave veloy Pro: No osllaons Conra: Nmeral dsperson 19
Avodng osllaons by pwnd-dfferenes I +1/ = 0 0-1.5 0.5 0 1 0 Corre! - 1 + 1 + 0
0 - Avodng osllaons by pwnd-dfferenes II 0.5 0.5 0 0 0 Corre! 1.5 0.5 1 0 Corre! - 1 + 1 + 1
0 - Avodng osllaons by pwnd-dfferenes III 0 1 0 1 0.5 1.5 Corre! 0.5 0.5 0 Corre! 1.5 0.5 1 0 Corre! - 1 + 1 + 0
Nmeral dsperson from pwnd-dfferenes I 0 0 - - 1 + 1 + 3
Nmeral dsperson from pwnddfferenes 1 Move fron by ½ ell 0 - - 1 + 1 + 4
Nmeral dsperson from pwnddfferenes Averagng over ells leads o smearng o of Conenraon dsrbons looks lke dsperson Average n eah ell 0 - - 1 + 1 + 5
Nmeral errors n he smlaon Osllaons of adveon Negave onenraons are nphysal, Lead o nonsensal reaon behavor e.g. nrease nsead of derease Or o nsably e.g. nfne raes Nmeral dsperson Leads o eaggeraed mng of soles and herefore o eaggeraed reaon raes 6
The Easy Way O Appromaon error depends on dsresaon Fne resolon always helps Cenral dfferenes: Grd-Pele-nmber < Pe = Δ/D avods negave onenraons Inve: Mng lengh of nmers < *mng lengh of nare Upwnd dfferenes: Nmeral dsperson oeffen s proporonal o grd dsane Δ 7
There are oher ways o Godnov mehods: Slope lmers Mehod of haraerss. We wll se he smples mehod here, whh s feasble and effen as we say 1D Coran nmber 1 haraerss mehod Epl me negraon wh sably reqremens Basally sep mehod adveon-dsperson Or 3 sep mehod: reaon as hrd sep 8
Tme negraon: Epl Eler-mehod wh pwnd dfferenes Rgh hand sde onans only onenraons a me old me Every ell an be omped ndependenly 1 1/ 1/ 1 1/ 1/ 1/ 1 1/ V D A V D A V Q V Q 9
Tme sep lmaon neessary for adveon I Coran-Fredrh-Levy reron 1 =1 =0 =0-1 +1 30
Tme sep lmaon neessary for adveon II Coran-Fredrh-Levy reron 1 1 Cr 1 Coran nmber -1 0 =0 Move by one ell =1 +1 31
Opmal me sep wdh for epl negraon of adveon Cr = 1 Conenraons are moved ealy one ell downsream Ea solon for pre adveon One way of mplemenng mehod of haraerss Reqres non-nform grd dsanes for nonnform flow No feasble wh spaally fed grd f flow s me varyng 3
Lmaon of me sep neessary for dsperson n epl sheme I Nemann reron =0 =0 =1 +1-1 1 1 D D 33
Lmaon of me sep neessary for dsperson n epl sheme II Nemann reron +1-1 3 1 1 D D 3 1 D Ne =1/3 =1/3 =1/3 Nemann nmber 34 1 1 D D
Mamm me sep wdh for epl negraon of dsperson Ne < 1/3 Ereme vales are no nvered Ne < 1/ No negave onenraons appear General prnple: The smaller he me sep, he more ea he epl ompaon of dsperson 35
Comparson of mehods for adveon 36
Applaon o 1D ranspor n rvers Dsperson oeffen s omparavely large e.g. ompared o grondwaer Therefore enral dfferenes for adveon an be sed a sffenly fne spaal dsresaon For nform flow: se pwnd-dfferenes wh Cr = 1 and epl me negraon 37