DEPTH-AVERAGED SHALLOW WATER EQUATIONS

Transcription

1 Lecre maeral Enrnmenal Hdralc Smlan Page DEPTH-AVERAGED SHALLOW WATER EQUATIONS Te Renlds eqans a were dered n e las caper descrbe e mn prcesses f a flw n all ree dmensns. Ts elabrae apprac s apprprae fr eample fr e maemacal calclan f flws clse cnsrcs r n srngl meanderng rers, s bascall eerwere were e flw s dmnaed b ree-dmensnal effecs. Snce e nmercal ree-dmensnal calclan s sll er csl, makes sense redce e Renlds eqans fr calclans w smpler flw cndns. Te dep-aeraged w-dmensnal flw eqans, als called sallw waer eqans, prde an eample. Te sallw waer eqans are baned, as e name sggess, b aeragng e Renlds eqans er e dep. Te fllwng cndns ae be me n rder fr e sallw waer eqans be applcable: - e ercal mmenm ecange s neglgble and e ercal elc cmpnen w s a l smaller an e rnal cmpnens and : w << and w <<. - e pressre gan s lnear w e dep (parallel flw lnes drsac pressre dsrbn): p() γ w beng e dep measred frm e waer srface and γ g. Tese assmpns make pssble redce e basc ssem f eqans nl ree eqans: e cnn eqan (as sal) and e mn eqans n drecn f e - and -as, as fllws: -cmpnen: w F γ ( ) Eq. -7 -cmpnen: w F γ ( ) Eq. -7 Prr negrang e mmenm eqans er e dep, we wll wrk n e cnn eqan, as a small praccal eample warm p. An mpran l fr e dep-negran are e s-called knemacal bndar cndns a ge nfrman ab e cange f e waer srface er me.

2 Lecre maeral Enrnmenal Hdralc Smlan Page 76 Knemacal bndar cndn a e free srface: An epressn fr e srface mn s dered. Sle Wasserefe Srface ( ) D w, were f,, D ( ) ( ) ( ) w w, can rle frm rer bed w Eq. -7 Knemacal bndar cndn a e rer bed: Grnd s mpermeable n mass fl perpendclar bed. n, n nrmal ekr f e rer bed s s w Eq. -73 w

3 Lecre maeral Enrnmenal Hdralc Smlan Page 77 Oer ls are e Lebn erem and e fndamenal erem f negran, a b sall be resaed n s place (s make sre): Lebn erem : d d Inegran erem : d Eq. -74 W ese nsrmens we ackle e negran f e cnn eqan ab e ercal as, a means beween e bed and e free srface (w beng e waer dep): w d w d d d d frs erm d secnd erm rd erm w Eq. -75

4 Lecre maeral Enrnmenal Hdralc Smlan Page 78 In a lle dfferen frm and sbsng e knemacal bndar cndns elds: d d ( ) ( ) w (empral cange f e srface) w (n mass fl perpendcla e rer bed) d d Eq. -76 Inrdcn f dscarge as an negral f e flw elc er e dep, and e dep-aeraged flw elces and (n be cnfsed w e smbls fr e me-aeraged felds menned befre) we ge: q d nd q d Eq. -77 Te dep-negraed cnn eqan can s fnall be saed as: ( ) ( ) Eq. -78 Te dep-negraed cnn eqan sws a e dfference beween e flw n and f a lme f waer cmes w a cange f e waer dep. Te deran f Eq. -78 as been dne w smplfng assmpns. Te eqan represens e cndns eacl. Cnseqenl we wll d e dep-negran f e mmenm eqans. Te prcedre s a lle leng, b we wll g rg n deal anwa eemplar fr e -cmpnen. w g ( ) F Eq. -79

5 Lecre maeral Enrnmenal Hdralc Smlan Page 79 And dep-negraed: ( ) w d g d empral deran pressre erm adece erm rnal scse erm ( ), ercal scs erm (3) d F d lme frces Eq. -8 Fr beer erew, e dep-negran wll be dne separael fr eac erm. Te frs erm ncldes e derae and e adecn par. Ten e Lebn erem wll be appled b e derae and e w rnal adecn erms, and e fndamenal erem f negran wll be appled e rd adecn erm: w d d d d ( ) ( ) ( ) ( ) ( w) Eq. -8 ( ) ( w) Te cmpnens f s eqan can be ransfrmed s a e knemacal bndar cndn can be appled: w d d d d ( ) ( ) w, becase e rer bed d esn` cange w Eq. -8

6 Lecre maeral Enrnmenal Hdralc Smlan Page 8 Te erms n e pareneses are er becase f e knemacal bndar cndns and we ban e fllwng fr e frs erm: d d d w Eq. -83 Te secnd erm, n e fllwng referred as pressre erm, wll reman ncanged. Te rd erm, cnanng e scs erms, wll parl be ransfrmed w e Lebn erem and parl w e fndamenal erem f negran. Frs e rnal scs pars: d d Eq. -84 d d Te ercal scs erm wll be negraed w elp f e fndamenal erem: d Eq. -85 Dep-negrang e lme frces n drecn f e -as w reference Eq. -3 elds: θ ω θ ω sn d sn d F Eq. -86

7 Lecre maeral Enrnmenal Hdralc Smlan Page 8 Cnsderng eqans Eq. -8 Eq. -83, we ge e prelmnar dep-negraed mmenm eqan n drecn f e -as: d d g d rer bed Eq. -87 ( ) ( ) waer leel ω sn θ Te fllwng epressn wll be nrdced fr e wnd and bed sear sresses: ( ) ( ) wnd, and Eq. -88 s, In analg e me-aeragng f e NS eqans, a dsn f e mmenar ercal feld n a dep-negraed mean ale par and a dean f e mean ale s dne. Fr e elc cmpnen n drecn f e -as s dsn s: () Eq. -89 w beng e mean elc er e ercal as (n be cnfsed w e me-aeraged mean ale a as e same smbl) and beng e dean f e mean elc. Te fllwng rle f negran lds fr s dsn: ɶ ɶ d d ɶ ɶ d ɶ d w ɶ d, ɶ Eq. -9

8 Lecre maeral Enrnmenal Hdralc Smlan Page 8 Sbsng e epressns fr e wnd and bed sear sresses as well as e pars f e dsn frm Eq. -87 n Eq. -85 elds e fllwng dep-aeraged mmenm eqan n drecn f e -as: ( ) g ( ) ( ) ( ) ( ) ωsn θ d d wnd, s, Eq. -9 As ne f e las seps, e dep-negraed frm f e cnn eqan s slaed frm e frs ree erms n e lef sde f e eqan w elp f paral dfferenan. ( ) ( ) ( ) ( ) ( ) dep negraed cnn eqan Eq. -9 Sbseqenl, we dde b e waer dep and grp e flcang erms w e scs erms. We nw ge e mmenm eqan n drecn f e -as: g ( ) wnd, s, ωsn θ Eq. -93 Fr a beer erew, e general frm f e dep-aeraged cnn eqan and e depaeraged mmenm eqans, e s-called sallw waer eqans: ( ) g 3 ( ) wnd, s, a 4 5 were a F acceleran cmpnen f e lme frces. 6 F Eq. -94

9 Lecre maeral Enrnmenal Hdralc Smlan Page 83 Te nddal erms wll nw be eplaned srl. Te frs erm n e lef sde s e rae f cange (er me) and e secnd erm e cnece mmenm ranspr. Te rg sde ncldes e graan frce (rd erm) and e fr erm s e dffse mmenm ranspr w: µ ν k δ 3,(m ) T Eq. -95 Te ff erm s e dsperse mmenm ranspr, wc s a maemacal resl smlar e Renlds sresses nl b dep-negran. Fr nfrm and mgenes flws s erm can be negleced. I descrbes e ecange prcesses de ercal nn-nfrmes. If ere s a srng secndar flw, fr eample becase f srng meanderng, e dsperse erm becmes mpran. Te s erm cmbnes e frces frm e sde, a s bed sear sress and wnd sear sress. Te wnd sear sress can be negleced ms f e mes, weer. Te bed sear sress s bnd e dep-aeraged elc b a qadrac elc law. s s c f c f (al lg emen) Eq. -96 Here, c f s e frcn ceffcen, wc can be deermned accrdng e flw laws f Gackler- Mannng-Srckler r Darc-Wesbac. Darc-Wesbac s flw law s defnel preferred. λ λ cf Eq were drag ceffcen λ can be deermned w elp f Clebrk & We s frmla as a fncn f e felds: f crss secn f frm drag ceffcen, Re Renlds nmber, k s eqalen sand rgness, r dralc rads. Wen sng e Gackler-Mannng-Srckler flw law, s f mprance a e emprc k S -ale depends n Re and r,.e. e dscarge sae. Te k S -ale s s n parameer f general, glbal ald, b as be adaped e dscarge sae.

10 Lecre maeral Enrnmenal Hdralc Smlan Page 84 If we lk a e dered sallw waer eqans Eq. -9, s cnspcs a we ae a clsre prblem agan. In rder sle e prblem, e same prcedre as fr e me-aeragng can be appled. We smplf and smmare e fr and ff erms f Eq. -9 : ν ν ν ν ν g, D T m g,g ),(m,d Eq. -98 We see a e al scs s cmpsed f a mleclar par, a rblen par and a dsperse par. Frer ransfrmans eld: ε ν ε ν g, g, Eq. -99 Tese smplfcans make pssble wre e sallw waer eqans n e fllwng frm: θ ω θ ω sn g sn g wnd, s, wnd, s, Eq. - And n ensr frm: F wnd, s, a g Eq. - In s eqan we ae med e dspersn par w e dffsn, s sa, b nrdcng a al scs. Hweer, s wa des n prde a pre rblence mdellng anmre, ndependen frm e rblence mdel sed, snce s negraed apprac als mdels e dspersn.

11 Lecre maeral Enrnmenal Hdralc Smlan Page 85 In Eq. -98 and Eq. -99 all felds are me- and dep-aeraged. Bars acrss, waes r er means f smbl declaran fr e me- and dep-aeragng ae been lef awa and wll n be sed n e fllwng capers eer.