MATEMATIČKO MODELISANJE 1D NEUSTALJENOG TEČENJA U OTVORENIM TOKOVIMA
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1 MATEMATIČKO MODELISANJE 1D NEUSTALJENOG TEČENJA U OTVORENIM TOKOVIMA Mirjana Isić 1, Miodrag Spasojević 2 UDK: :167.7 Rezie: U radu su prikazani razvoj i testiranje odela za linijsko neustaljeno tečenje u granatoj reži kanala sa unutrašnji granični uslovia. Von-Neuann-ova analiza stabilnosti Preissann-ove šee je sprovedena za linearizovane jednačine tečenja. Nueričko rešenje de St. Venant-ovih jednačina je testirano na osnovu serije nueričkih eksperienata, pri čeu su, izeďu ostalog, korišćeni i standardni ASCE testovi. U odel granate reže kanala su uključeni i testirani unutrašnji granični uslovi. Unutrašnji granični uslovi su testirani tako što je proveravano da li svaki od njih zadovoljava odgovarajuće hidrauličke zavisnosti za niz situacija neustaljenog tečenja. Ključne reči: linijski odel, neustaljeno tečenje, granata reža tokova, unutrašnji granični uslovi. 1. UVOD U radu se razatra odel linijskog neustaljenog teĉenja u granatoj reţi kanala gde su od unutrašnjih graniĉnih uslova obuhvaćeni retenzija, prag i ustava. Model je napravljen prieno Preissann-ove šee koja je prethodno testirana Von Neuannovo analizo stabilnosti na pojednostavljeni jednaĉinaa. Sa odel je testiran prieno standardnog ASCE testa i pooću odgovarajućih hidrauliĉkih zavisnosti. 2. VON NEUMANN OVA ANALIZA STABILNOSTI ZA LINEARIZOVANE JEDNAČINE TEČENJA Prieno de St. Venant-ovih pretpostavki na jednaĉine teĉenja dobijene su jednaĉine koje opisuju jednodienzionalno neustaljeno strujanje u otvoreni tokovia, jednaĉina kontinuiteta i dinaiĉka jednaĉina (1). A Q 0, t x 2 ga S0 gas f 0 Q Q h t x A x (1) 1 Mirjana Isić, dipl. graċ. inţ. aster, GraĊevinski fakultet Subotica, Kozaraĉka 2a, tel: , e ail: isic@gf.uns.ac.rs 2 Dr. Miodrag Spasojević, dipl. graċ. inţ., GraĊevinski fakultet Subotica, Kozaraĉka 2a, tel: , e ail: spasojevic@gf.uns.ac.rs ZBORNIK RADOVA 18 (2009) 25
2 Von Neuann-ova analiza stabilnosti vaţi za linearne parcijalne diferencijalne jednaĉine zbog ĉega je prikazane jednaĉine potrebno linearizovati. Linearizacijo se dobija h h t u 0 0 x u h, g 0 t x (2) Dobijen je linearan siste od dve parcijalne diferencijalne jednaĉine (2). Kako je dobijeni siste linearan, rešenje se oţe napisati kao linearna kobinacija eksponencijalnih funkcija, tj. dovoljno je posatrati jednu koponentu rešenja. Posatra se -ta koponenta rešenja u vreenu i prostoru, ;, j x jt j x jt h x t H e e u x t U e e, (3) U dato sisteu prva eksponencijalna funkcija opisuje raspored rešenja po prostoru, a druga raspored po vreenu. Uvrštavanje ovih izraza u polazne jednaĉine se dobija hoogen siste od dve jednaĉine, gde je uslov postojanja netrivijalnog rešenja da je deterinanta sistea nula. Prieno ovog uslova i uvoċenje brzine propagacije talasa, c gh0, odreċuje se veza konstanti i. Traţena veza je c. Dobijena su dva rešenja, a ukupno rešenje se dobija njihovo linearno kobinacijo, g g h e A e Be ; u e A e B e c c j x jt jt j x jt jt (4) U nastavku se razatra faktor aplifikacije j t e j t odnosno e, j 2 j 2 j e e R e I e cos sin 1. (5) Apsolutna vrednost faktora aplifikacije je jedan što znaĉi da se proizvoljna koponenta toko vreena ne aplificira tj. poĉetna vrednost putuje nizvodno brzino c gh 0 bez proene oblika i bez aplifikacije tj. rasplinjavanja. 3. VON NEUMANN-OVA ANALIZA STABILNOSTI PRIBLIŽNOG REŠENJA LINEARIZOVANIH JEDNAČINA Pribliţno rešenje se dobija prieno Preissann-ovih aproksiacija (6) na linearizovane jednaĉine (2). n 1 n 1 n n n 1 n n 1 n i 1 i i 1 i f i i i1 i1 f f f f f f f f f 1 ; ; x x x t 2t 2t n1 n1 n n i1 i i1 i f f f f f 1 (6) 26 ZBORNIK RADOVA 18 (2009)
3 Razvijanje pribliţnog rešenja u Fourier-ov red se dobija siste od dve jednaĉine, ĉije rešenje je dato sa (7), gde je Cr c t x gh0 t x, Courant-ov broj. 2 2 sin ( x) sin( ) 1 4 (1 ) x Cr Cr j 1 cos( ) 1 cos( ) t x x 1,2 j sin ( x) sin ( x) 14 Cr 14 Cr 1 cos( x) 1 cos( x) e A B (7) Apsolutna vrednost faktora aplifikacije je j t. Pošto se kod pribliţnog e A B rešenja ne zna da li je realan ili kopleksan broj, posatra se brzina propagacije definisana kao Re( ). Nakon niza ateatiĉkih transforacija, brzina propagacije ˆ ˆ ˆ t. Definišu se aplitudna, R 1, i fazna, R 2 greška. 1 talasa je Re( ) tan B A apsolutna vrednost faktora aplifikacije približnog rešenja R1 A B apsolutna vrednost faktora aplifikacije taĉnog rešenja brzina propagacije -te koponente približnog rešenja tan BA R2 brzina propagacije -te koponente taĉnog rešenja 2 Cr ( x) U nastavku se daje grafiĉki prikaz fazne i aplitudne greške, gde je M=ζ/ x. 1 Slika 1 Grafički prikaz aplitudne greške ZBORNIK RADOVA 18 (2009) 27
4 Slika 2 Fazna greška za =0.5 UporeĊivanje grešaka R 1 i R 2 se zakljuĉuje da Preissann-ova šea daje taĉno rešenje linearizovanih jednaĉina za kobinaciju Cr=1 i = TEST DUGOTRAJNOG OČUVANJA ZAPREMINE MODELA ZA JEDNODIMENZIONALNO NEUSTALJENO TEČENJE (ASCE TEST) Model se prienjuje na horizontalan, prizatiĉan kanal duţine 10k. Popreĉni presek kanala je trapezni sa osnovo 10 i nagibo strana 1 : 2. Manningov koeficijent je /3 s. Raĉunski korak po prostoru je 1k, a raĉunski korak po vreenu je 6in. Za poĉetni uslov se uzia Q 0 i konstantna dubina vode od 7. Nizvodni graniĉni uslov je zatvoren kanal, a uzvodni graniĉni uslov je opisan sinusoido (8) toko perioda od 6 ĉasova nakon ĉega do kraja proraĉuna nea dotoka. U tabeli 1 su prikazane relativne greške zapreine za razliĉite kobinacije i. Isti test se ponavlja za neprizatiĉan kanal sa osnovo 10 na uzvodno kraju, do 20 na nizvodno kraju. Relativna greška zapreine je za ovaj sluĉaj data u tabeli 2. Koef. pond. Br. Iter. Zapr. greška (%) Tabela 1 Relativna greška zapreine * zapreinska greška anja od 10-3 % se zanearuje < >1 1 Nestabilno >1 1 > * * * >1 28 ZBORNIK RADOVA 18 (2009)
5 2 t Q 200sin Koef. pond. Br. iteracija Zapr. greška (%) ; (t je u sekundaa) za 0 t 6 h (8) Tabela 2 Relativna greška zapreine < Nestabilno >1 1 >1 1 1 > * * * * 5. TESTIRANJE MODELA GRANATE MREŽE KANALA SA UNUTRAŠNJIM GRANIČNIM USLOVIMA Razatra se neprizatiĉan kanal pravougaonog popreĉnog preseka, duţine 24k i konstantnog nagiba od Širina kanala linearno raste od 23.6 na uzvodno do 26 na nizvodno kraju. Manning-ov koeficijent je 1/15-1/3 s. Na 3k od uzvodnog kraja se nalazi boĉna retenzija konstantne površine od 10ha. U kanal se, na 16k od uzvodnog kraja, uliva pritoka sledećih karakteristika: prizatiĉan kanal pravougaonog popreĉnog preseka širine 20, duţine 10k, Manning-ovi koeficijento /3 s i nagibo dna Na 2k uzvodno od ušća je prag širine 20, sa koeficijento prelivanja 1. Uzvodni graniĉni uslovi su poznati hidrograi. Slika 3 Q(x) glavnog toka ZBORNIK RADOVA 18 (2009) 29
6 U prvo sluĉaju je nizvodni graniĉni uslov poznata veza Q(Z) koja vaţi za jednoliko teĉenje Q K S0. Na slici 3 su prikazane skokovite proene protoka na estu ušća pritoke u glavni tok. Punjenje, odnosno praţnjenje retenzije, je prikazano na slici 4 u vidu hidrograa za presek neposredno uzvodno i nizvodno od retenzije. Rezultati proraĉuna pritoke su dati na slici 5 u vidu hidrograa za presek sa prago. Slika 4 Hidroga za presek neposredno uzvodno i nizvodno od retenzije Slika 5 Hidrogra za presek sa prago 30 ZBORNIK RADOVA 18 (2009)
7 U drugo sluĉaju je nizvodni graniĉni uslov veza Q(Z) usled isticanja ispod tablaste ustave koje je ograniĉeno na sluĉaj slobodnog isticanja za alu otvorenost ustave. Rezultati su prikazani na slici 6. Slika 6 Linija nivoa glavnog toka 6. ANALIZA REZULTATA Slika 7 Uticaj koeficijenta ponderacije na nelinearizovane jednačine tečenja ZBORNIK RADOVA 18 (2009) 31
8 Na slici 7 su prikazani rezultati proraĉuna, za deonicu kanala u okolini ušća pritoke u glavni tok. Za sve proraĉune je Cr>1, što ukazuje na postojanje fazne greške. Sa porasto koeficijenta ponderacije raste i nueriĉka difuzija što dovodi do rasplinjavanja nastale fazne greške, kao što je i prikazano na slici 7. LITERATURA [1] J.A. Cunge, F.M. Holly, A. Verwey, Practical Aspects of Coputational River Hydraulics, Pitan publishing liited, London, [2] M. Spasojević, Nuerička hidraulika, Pisana predavanja, Subotica, [3] M. Stipić, Ustava kao erni objekat pri slobodno isticanju, Seinarski rad iz predeta Eksperientalna hidraulika, Subotica, MATHEMATICAL MODELING OF 1D UNSTEADY FLOW IN OPEN CHANNELS Suary: The paper presents the developent and testing of one-diensional unsteady flow in a branched network of open channels with internal boundary conditions. The Von-Neuann s stability analisis of the Preissann s schee was done for the siplified flow equations. The nuerical solution of de St. Venant s equations was tested using a series of nuerical experients, including standard ASCE tests. The odel of the branched channel network includes internal boundary conditions. Internal boundary conditions were checked by akig sure that each condition satisfies related hydraulic laws for a series of unsteady flow situations. Key words: One-diensional odel, unsteady flow, branched channel network, internal boundary conditions. 32 ZBORNIK RADOVA 18 (2009)
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