Numerical simulation of hydraulic jump using ENO scheme

Transcription

1 Avalable ole Joual of Chemcal ad Phamaceutcal eseach, 4, 6(6):63-67 eseach Atcle ISSN : CODEN(SA) : JCPC5 Numecal smulato of hydaulc jump usg ENO scheme Y.. u*, Y. X. Ca, W.. We, K. Yag ad Zh. Ma State Key aboatoy of Eco-Hydaulc Egeeg Shaax, X a vesty of Techology, X a, Shaax, Cha ABSTACT Ths pape s coceed wth a hgh-esoluto mathematcal model fo oe-dmesoal (D) hydaulc jump flows. The D shallow wate equatos of Sat-Veat wee solved wth the essetally o-oscllatoy (ENO) scheme, the fte volume method (FVM) ad the total vaato dmshg (TVD) uge-kutta-type method. Ths model was used to pedct the hydaulc jump flows, ad the umecal esolutos ae ageemet wth the expemetal oes, whch dcate that ths model s faly effectve ad accuate fo smulatg hydaulc jump flows. Keywods: ENO scheme; FVM; hydaulc jump, umecal smulato, wate equatos of Sat-Veat INTODCTION Hydaulc jump s fomed whe the state of flow chages fom supectcal to subctcal oe. The hydaulc jump s usually accompaed by a aeated ecculatg olle o the top, whee tesve tubulece esults eegy dsspato ad the mxg of a also takes place. The hydaulc jump s teated as a macoscopcally steady pheomeo fo egeeg puposes, ad tempoally aveaged quattes ae examed. Mathematcally, the hydaulc jump s commoly descbed by the shallow wate equatos (also amed the Sat Veat equatos fo the D case). Oe featue of hypebolc equatos of ths type s the fomato of boes (.e., the apdly vayg dscotuous flow). It s a mpotat bass fo valdatg the umecal method whethe the scheme ca captue the hydaulc jump waves accuately o ot. Ths gves se to a ceasg teest solvg such a poblem. Seveal fte-dffeece schemes that hadle dscotutes effectvely wee used to compute opechael flows, such as the appoxmate ema solve [-4]. I ecet yeas, moe ad moe hgh-esoluto schemes such as MacComack, TVD ad ENO schemes [5-7] ae appled to umecal smulato of dscotuous poblem such as the shock wave. Although TVD scheme ca keep the total vaato dmshg, t causes the accuacy dop to oe-ode at the local exteme pot of smooth aea. To ovecome ths weakess, ENO (Essetally o-oscllatoy) scheme [7] s poposed, whch amplfes the estcto of total vaato dmshg ad also allows a ty cease of total vaato, the the scheme s kept ufomly hgh esoluto ad essetally o-oscllatoy. The chaacte of ENO scheme s that t elmates the mootocty lmt, ad ca adaptvely choose the best tepolatg pot the tepolatg fuctos ecostuctos. Based o the above eseach esults, the goal of the cuet wok s to develop a mathematcal model capable of dealg wth hydaulc dscotutes such as steep fots, hydaulc jump, etc. The wate goveg equatos has bee solved by the ENO scheme. GOVENING EQATIONS The D equatos of Sat Veat ae as follows [7]: t F x G () 63

2 Y.. u et al J. Chem. Pham. es., 4, 6(6):63-67 A, Q ; whch the vaables,f, ad Q ae defed matx foms as follows: ( ) T ( ) T G, gas f ; P P( z) ( z η) B( η) dη z z ; S f Q Q B A,whee ttme, xdstace, Asteam coss-secto aea, Qdschage, Ggavty, Phydostatc pessue o the coss-secto, z wate level, z S s vaable fcto slope, s the lowest elevato of coss-secto, ( η) B s the wdth whe the wate level s η, f s the Mag coeffcet, gue shows the defto sketch fo geeal flows chaels. g. Defto sketch fo geeal flows chaels NMEICA SCHEME The umecal schemes clude the umecal dscetzato of wate equatos, umecal flux splttg, ENO schemes ecostucto ad tme dscetzato [7,8]. Numecal dscetzato of wate equatos: The equato () ca be dscetzed as [8]: ( ) tg λ () whee pot; λ t x ; t tme step; ad x spatal step; epesets the tme laye; epesets the spatal ae the umecal fluxes cosstet wth flux F of equato (). Numecal flux splttg: The fte-dffeece method (FDM) was used to solve the umecal flux, such as : { F( ) F( ) a~ ( )} (3) whee a ~ s the oe aveage value of a F o ad,ad s the Jacoba matx fo a F. ad ae the values of o the left o ght had of pot x espectvely, whch ca be computed by the stecl opto of ENO.The oe velocty s as follows: a~ ( ) F( ) F (4) Smlaly, ca be computed wth same method ad the values of stecl opto of ENO schemes. - ad - ca also be obtaed by the ENO schemes ecostucto: ENO schemes adaptvely choose the smoothe stecl by compag the sze of dvded dffeece. To acheve k - ode esoluto schemes, k cell s eeded whe choosg the stecl. Total cells ae added to k. Assume the k stecl as follows: ( ) { x, x } S,,, k, (5), k 64

3 Y.. u et al J. Chem. Pham. es., 4, 6(6):63-67 So k dffeet ecostucto method to calculate o ca be acheved: k ( ) j C,,, k (6), j j whee s the aveage value of at pot x,whch ca be wtte as: k, w (7) ( ) et the coeffcet of the decded stecl whose magtude of dvded dffeece s mmal, the the stecl opto s fshed. If w ad w k,the the equatos was chaged to WENO(Weghted ENO) schemes. w, the othes Tme dscetzato: The uge-kutta TVD method was used to dscetze the equato (). The equato ca be wtte as [9]: w t ( ) whee ( ) s appoached by F( ) G. x Accodg to the uge-kutta TVD schemes, ( ) t( ) ( ) 3 t t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (8) (9a,b,c) ESTS AND DISCSSION The test poblem elates to a hydaulc jump a wde hozotal ectagula chael of costat wdth. A hydaulc jump s fomed wheeve the flow chages fom supe-ctcal flow (F>) to sub-ctcal flow (F<), whee F epesets the Foude umbe. The tal flow codtos the hozotal chael ae a depth of.4 m ad a velocty of.65 m/s. These flow codtos gve a upsteam Foude umbe of 4.3. A Couat umbe of.8, a oughess coeffcet of.3, ad a fe gd spacg of.3 m wee used. As supectcal flow has a upsteam cotol, the flow vaables at the upsteam ode wee kept equal to the tal codtos. At the dowsteam ed a costat depth of. m was specfed ad held costat fo all tme levels. gue s the taset depth pofle. The computed esults ad the expemetal oes ae a close ageemet. Theoetcally, a hydaulc jump wll occu whe the upsteam Foude umbe F, depth h ad dowsteam depth h F satsfy the Bélage fomula h / h ( 8 ). Ths ca be used to check the esults by the model. Fom the umecal esults, the dowsteam wate depth h.3m. Substtuto of the upsteam Foude umbe ad depth (F 4.3, h.4m) to the Bélage fomula esults the theoetcal dowsteam depth equed fo the jump as ĥ.m. h s thus vey close to the theoetcal dowsteam depth ĥ. The elatve eo s %. 65

4 Y.. u et al J. Chem. Pham. es., 4, 6(6):63-67 The close ageemet betwee the computed esults ad the expemetal oes shows that the poposed method s compaatvely accuate..5.4 deepth(m) dstace(m) g. Hydaulc jump steady state pofles fo F 4.3( computed, expemetal) gue 3 shows the effect of gd spacg o shock educto. de the same codtos that F 4.3, h.4m,the compasos of the computed esults wth x of.3m ad.m show that vayg the gd spacg ad Couat umbe dd ot alte the locato of the shock fot, whch says that gd spacg of.3m has coveged to steady state wth eough accuate soluto..5.4 deepth(m) dstace(m) g.3.effect of gd spacg o shock educto, F 4.3 ( x.3, x., expemetal) CONCSION The hgh-esoluto mathematcal model has bee developed fo D shallow wate equatos (Sat-Veat equatos) fo the computato of hydaulc jump flows. The D shallow wate equatos wee solved wth the ENO scheme, the fte volume method (FVM) ad TVD uge-kutta-type method. Numecal smulato fo the hydaulc jump flows dcates that ths model s faly effectve ad accuate fo smulatg hydaulc jump flows. The hgh-esoluto mathematcal model ca be futhe modfed ad exteded to mult-dmesoal hypebolc cosevato laws to effectvely smulate D ad 3D hydaulc jump flows. Whe utlzg the techque of bouday teatmet such as a body-ftted coodate system, the poposed model ca also effectvely smulate the hydaulc jump flows wth complex boudaes. The esults wll be gve a futue wok.. Ackowledgmets Patal facal suppot of ths study fom the Natoal Natual Scece Foudato of Cha (Gat No.57839), the scetfc eseach poject of Shaax Povce (4K5-3-5)ad the specal fuds fo the developmet of chaactestc key dscples the local uvesty suppoted by the cetal facal (Gat No: 6-X, 6-5X5) ad the Shax Povce key subject costucto fuds. EFEENCES []K Hu, C G Mgham, D M Causo, It. J.Nume. Methods Fluds, 998,8,pp4~6. 66

5 Y.. u et al J. Chem. Pham. es., 4, 6(6):63-67 []K Aastasous, C T Cha, It. J. Nume. Methods fluds, 997, 4,pp5-45. [3]C G Mgham, D M Causo, J. Hyd. Egg., ASCE, 998, 4, pp [4]P Glaste, It. J. Nume. Methods fluds, 993, 6,pp [5]A. Hate, S. Egqust, Oshe ad S. Chakavathy, Joual of Computatoal Physcs, 987, 7, pp3-33. [6]A. Hate, J Comp Phys, 983,49, pp [7]Seka Vukovc, uka Sopta, Joual of Computatoal Physcs,,79, pp [8]W.. We, Computatoal hydomechacs theoy wth applcatos, st Edto,X a,shax Scece ad Techology Publshg House,,9-93. [9]Y.. u, YF Hog, W.. We, Joual of Chemcal ad Phamaceutcal eseach, 4, 6(4),