Block 4 Numerical solution of open channel flow

Transcription

1 Numeral Hydrauls Blok 4 Numeral soluon o open hannel low Markus Holzner 1

2 onens o he ourse Blok 1 The equaons Blok 2 ompuaon o pressure surges Blok 3 Open hannel low (low n rers) Blok 4 Numeral soluon o open hannel low Blok 5 Transpor o solues n rers Blok 6 Hea ranspor n rers 2

3 Numeral soluon o open hannel low - Fne derenes mehod - haraerss mehod - Fne Volume dsrezaon 3

4 Bas equaons o open hannel low n arables h and or reangular hannel onnuy ( Flu-onserae orm ) Momenum equaon h ( h ) 0 () h g gis gi hb Ie rhy k r 2h b 2 4/3 s hy 4

5 Bas equaons o open hannel low n arables h and q or reangular hannel onnuy h Momenum equaon 2 4/3 2 s hy q 0 2 q ( q / h) h gh gh( IS I ) qq hb Ie rhy k r h 2h b 5

6 Bas equaons o open hannel low or general ross-seon n arables A and Q onnuy A Q 0 Momenum equaon 2 1 Q 1 (Q / A) ( A/ bwsp ) g gi0 gi A A QQ A Ie rhy k r A L 2 4/3 2 s hy u e 6

7 Boundary ondons A nlow boundary usually he nlow hydrograph should be gen A he oulow boundary we an use waer leel (also me arable e.g. or de) waer leel-low rae relaon (e.g. wer ormula) slope o waer leel or energy In superral low wo boundary ondons are neessary or one boundary (or boh and h) 7

8 Boundary ondons Number o boundary ondons rom number o haraerss In 1D: subral low: IB: 1, OB: 1 superral low: IB: 2, OB: 0 IB = Inlow boundary, OB = Oulow boundary 8

9 Dsrezed bas equaons n arables h and or reangular hannels pl mehod onnuy ( Flu-onserae orm ) = 2,, N I h h ( h h ) / new old old old old old 1 1 Momenum equaon ( h h ) = 2,, N old 2 old 2 old old new old 1 1 g gi0 gie 2 old old old h 1 e r 2 4/3 hy old s hy 2 1 k r h b b hh derenes o hose n order o be able o buld n lower and upper boundary 9 ondons?

10 Dsrezed bas equaons n arables h and or reangular hannels pl mehod Boundary ondons ( = 1) eample h h ( h h ) / q /( b h ) new old old old old old new new new n 1 Boundary ondons ( = N+1) eample h = h ( wer) new N1 0 new N1 new ompuaon as 1,...,N 1 or wer ormula q ( h). e. ( h) h 10

11 Dsrezed bas equaons n arables h and or reangular hannels pl mehod pl mehod requres sably ondon ouran-fredrhs-ley (FL) rerum mus be ullled: /( ) s he relae wae eloy wh respe o aerage low gh ga( h) / b( h) 11

12 Non-onserae orm: h h h h 0 g I S I g onserae orm: h h 0 2 q = h q 1 q g IS I g h h 2 h h q q q gh ghis h 2 I 12

13 Mar ormulaon o he las orm o he equaons 13

14 Assgnmen Deermne he wae propagaon (waer surae prole, mamum waer deph, oulow hydrograph) or a reangular hannel wh he ollowng daa: wdh b = 10 m, k sr = 20 m -1/3 /s lengh L = m, boom slope I S =0.002 Inlow beore wae, base low Q 0 = 20 m 3 /s Boundary ondon downsream: er wh waer deph 2.2 m Boundary ondon upsream: Inlow hydrograph Inlow hydrograph Q (s added o base low Q 0 ): Tme (h) Q (m 3 /s)

15 Inlow/Oulow hydrographs Q (m 3 /s) abou 4 h me seps n 10s 15

16 1D Shallow waer equaons h h h 0 h g I S I g - The oal derenal or =(,) and h=h(,) s: D D D h h D 16

17 haraers equaons e mulply he rs o he orgnal equaons wh a mulpler l and add he wo equaons up: h g h lh l g IS I l To oban oal derenals n he brakes we hae o hoose g l1,2 h gb A 17

18 Thus we oban he haraers equaons: D D g Dh g IS D I along d d D D g Dh g IS D I along d d 18

19 Types o haraerss Pose and negae haraerss or sub-ral, ral and super-ral low: P P P Termnology: P, (es) und (as) nsead o, -1, +1 19

20 Inegraon o he haraers equaons Mulplaon wh d and negraon along haraers lne P d dh g P g and along haraers lne P yelds: P I S I d dh g P g P I S I d d d d d d 20

21 P S P P I I g h h g P S P P I I g h h g P p P h P n P h P S p I I g h g P S n I I g h g ( / ) ( / ) g g or Ths mples a lnearsaon. The wae eloy beomes onsan n he elemen. 21 Inegraon o he haraers equaons

22 Grd or subral low (1) haraerss sar on grd pons Ze j+1 P Problem: haraerss nerse beween grd pons n pons P a me leels whh do no onde wh he me leels o he grd. Resuls hae o be nerpolaed. j Termnology (ener), (es) und (as) nsead o, -1, +1 22

23 Ze Grd or subral low (2) haraers lnes end a pon P, sarng pons do no onde wh grd pons. Values a sarng pons are obaned by nerpolaon rom grd pon alues j+1 P e hoose hs aran! j Termnology (ener), (es) und (as) nsead o, -1, +1 23

24 Inerpolaon (le) 24 L L L P L L L L L

25 Inerpolaon (rgh) R R P R R R R R R 25

26 Sarng pon L Soluon or L and L yelds: Inrerpolang analogously or h: 26 L 1 L L 1 L L L h h h h

27 Sarng pon R or subral low In analogy o pon L, arables or pon R R and R R 1 R 1 R h h h h R R R The mehod s an epl mehod. The FL-rerum s auomaally ullled. 27

28 Sarng pon or superral low Sarng pon o haraers beween and Usng eloy - we oban 28 R 1 R R 1 R R R h h h h

29 Fnal epl workng equaons Inegraon rom L o P and rom R o P P p Lh h P P n R P wh and p n L R g g L R h h L R g g I I S L I I ( g / ) ( g / ) L L R R S R 2 equaons wh 2 unknowns Boundary ondons are requred as dsussed n FD mehod 29

30 lassal dam break problem: Soluon wh mehod o haraerss Propagaon eloy o rons slghly oo hgh 30

31 Mar orm o he S. Venan equaons (1D) 31

32 Fne olume mehod ell boundares Ԧ Ԧ For smply we onsder a sysem whou soure erm. Inegrang n spae we oban: d d න 1 2 u, d = Ԧ u 1/2, Ԧ u +1/2, 32

33 Addonally, we negrae n me beween n and n+1 : න 1 2 u, n+1 d = න 1 2 u, n+1 d + n+1 න Ԧ u 1/2, n n+1 d න Ԧ u +1/2, n d Ths s he negral orm o he equaons. 33

34 Denng: u n = 1 න u, n d and 1 2 Ԧ ±1/2 = 1 n+1 න Ԧ u ±1/2, n d we an wre: u n+1 = u n + ( Ԧ 1/2 Ԧ +1/2 ) 34

35 Numeral shemes: Deren shemes an be desed aordng o he apporah o epress lues a he ell boundares Ԧ ±1/2 e dsngush: - enered shemes, whh ge equal wegh o neghborng ells and do no need normaon on dreon o wae propagaon. They are easy o mplemen bu lead o srong numeral duson; - Upwnd shemes, whh use wae propagaon o epress lues. More dul o mplemen bu muh more aurae. 35

36 enered sheme: La-Fredrhs Flues are epressed as Ԧ +1/2 = 1 2 (u Ԧ n +1 + Ԧ(u n 1 2 (u +1 n u n ) and Ԧ 1/2 = 1 2 (u Ԧ n n + Ԧ(u (u n u n 1 ) Mehod s rs order aurae and ery smple o mplemen. Howeer, leads o numeral duson ha s oo srong or praal applaons. FL reron: /( ) 36

37 Smples upwnd sheme: epl upwndng +1/2 beomes and -1/2 beomes -1 he wae propagaes n pose -dreon Ԧ +1/2 = Ԧ(u n ) and Ԧ 1/2 = Ԧ(u n 1 ) Mehod s more robus n he presene o shoks bu numeral duson s sll onsderable. FL reron: /( ) 37

38 Smples upwnd sheme: epl upwndng (2)

39 Sae o he ar upwnd sheme: Goduno Idea: sole loal Remann problems orward n me Mehod requres he soluon o he Remann problem a eery ell boundary and on eah me leel. Ths amouns o alulang he soluon n he regons ha orm behnd he non-lnear waes deelopng n he Remann problem as well as he wae speeds neessary or derng he omplee wae sruure o he soluon. 39

40 Remann problem aes are eher shoks (soluon s dsonnuous) or rareaons (soluon s onnuous) To nd he ea soluon n he sar regon, we need o esablsh approprae jump ondons 40

41 ample: HHL (*) soler I s assumed ha wo waes propagae n oppose dreons wh eloes S L and S R, generang a sngle sae n beween hem: How o ompue U HLL? e sar rom he negral orm o he equaons (slde 33). (*) Mehod s an appromae Remann soler deeloped by Haren, La and an Leer (1983) and mproed by neld (1988) 41

42 HHL (2) Inegral orm o equaons: onrol olume or alulaon o HLL lu whh ges on he rgh hand sde: (*) where and The le hand sde an be spl no 3 negrals: (**) 42

43 HHL (3) ombnng (*) and (**) ges: Ddng by he lengh we nally oban: Usng hs epresson wh he Rankne Hugono ondons (see e.g. Guno p. 150) we ge he epresson or he HHL lu o be used n he Goduno sheme: 43

44 2D Shallow waer equaons The 2D shallow waer equaons an be wren n mar orms as: 44

45 Derene o 1D Addonal arable (spe. low n y-dreon) and addonal momenum equaon. Flues n -dreon are ormulaed equal o he lues n 1D. Flues n y-dreon are ormulaed n analgoy o lues n -dreon. In addon o e(es) and w(wes) he ndes n(norh) und s(souh) are nrodued. The dreon o upwndng s deermned ndependenly rom he upwnd dreon n he -oordnae. The onseraon s oer he whole elemen. I.e. soure erms and lues oer eas/wes and norh/souh boundares ener he same balane. 45

46 Speales o 2D modellng The 2D ompuaonal grd s always a projeon on he horzonal plane. Deren poson o nodes n z-dreon nluene he soure erm (gray). 2D elemens are more general,.e. only n he ase o reangular elemens he 2D problem an be dded no wo 1D problems. In he ase o general elemens (e.g. rangular elemens) he lues orhogonal o he elemen sdes mus be used. They are deomposed no omponens along he orhogonal /ydreons. Appromaely reangular elemens mproe ompuaonal auray. I elemen szes ary srongly he numeral error nreases as. In ha ase hgher order shemes hae o be used. 46

47 Flows wh ree waer surae (Naer-Sokes approah, erally 2D or 3D) For he soluon o he paral derenal equaons he doman has o be dsrezed. Ths s no mmedaely possble as he poson o he surae s no a pror known. An erae proedures s neessary. There are deren ways o akle he problem: Surae Trakng: he grd ollows he ree surae. Soluon o an addonal adeon equaon.e. on a ed grd normaon s ranspored wh he adee low. The normaon s he waer onens o a ell (Fraon o olume - FOV) or he dsane rom a daum o he surae (LS). On a ed grd parles are moed oneely wh he low (Marker n ell - MA). 47

48 48

49 ommens on he Naer-Sokes approah Addonal non-lneary means addonal ompuaonal eor and he danger o non-onergene. The dsresaon s onsderably more dul and requres een grd generaors. Inapproprae dsresaon may lead o wrong soluons. Numeral duson an smooh ou he poson o he surae. The soluon o he low s onsderable beer erally ured sreamlnes es. The Naer-Sokes approah s more or loal phenomena, shallow equaon approah or global low phenomena. 49

50 H-RAS 50

51 onepual model o H 1D bu sll akng no aoun hannel and loodplans hannel Sorage (whou low) loodplan 51

52 How o say 1D n energy, momenum, and pezomer head? z = eleaon o waer surae s he same or hannel and lood plans = h p n our nomenlaure 52

53 Some denons I here s no oodplan, he model desrbes wha we dd so ar I here s a loodplan, he hannel s subdded no seeral seons wh he same waer leel bu possbly deren ron oeens subdson oneyane K or eah subdson: Toal low: Q N 1 Q Q K I wh K 1 n A r 2/3 hy, 53

54 quaons or hannel and loodplan l q q S A Q q A Q Indes: = loodplan, = hannel M I z ga Q Q M I z ga Q Q,, ) ( ) ( onnuy equaons: Momenum equaons: z = eleaon o he waer surae, q l =laeral nlow, S area o non-oneyng ross-se. q = low rom loodplan o hannel (per lengh), q = low rom hannel o loodplan (per lengh), M, M orrespondng momenum lues (per lengh) The rgh hand sdes are elmnaed by addng he equaons or hannel and loodplan 54

55 quaons ombned n 1D Fnal 1D equaons The unknowns are Q and z (= h p ) A, A and S are known unons o z I, and I, are known unons o z and Q 55 0 ) / ) ((1 ) / ( 0 ) ) ((1 ) (,, I z ga I z ga A Q A Q Q Q Q A ) /( ) (1 K K K wh Q Q Q Q Q S A A A ( work) preous our n h I z h z z S boom

56 quaons n derene orm Momenum equaon wh onnuy equaon eloy oeen equalen -oordnae 56 Q Q Q A A A I A A I AI,,. 0 I z ga Q Q Q 0 l Q S A A Q ) ( ) )( ( , , j j j j spae j j j j me F F F F F F F F F F

57 Soluon mehod Impl, lnearzed Fne Derene sheme Impl sheme: All spaal deraes are aken as a weghed aerage beween old and new me, wegh θ All me deraes are aken n he mddle o The spaal dsrezaon neral j+1 θ j Lnearzaon mehod: ample: erm 2, a more omplaed erm would be Q 2 /A j1 j j1 2 j 2 j 2 j 2 j j j1 2 2 Soluon o bg equaon sysem or 2N unknowns per me sep, where N s he number o nodes As he equaon sysem s sparse, ehnques or sparse mares are used 57