Numerical simulation of various flows over a square cavity

Transcription

1 Iteratoal Joural of Mechacal Egeerg ad Research, ISSN Vol. 5 No.1 (2015) Research Ida Publcatos; Numercal smulato of varous flows over a square cavty N.Sethl Kumar 1, M.Revath 2, P.Raju 3 Assstat Professor Abstract Numercal smulato of Euler equatos ad Naver-Stoke equato s carred out the preset work where for each smulato we took oe or more example problem say Quas-Oe- Dmesoal ozzle flow for Euler equatos ad vscous compressble flow over a square cavty, square cylder ad tall buldg for Naver-Stoke equato. The etre problems are solved here umercally ad vestgated. Here dfferet computatoal methods are appled for each problem ad show the results. Fte dfferece method ad Fte Volume method both are studed ad appled here for solvg the Euler ad Naver-Stoke Equatos respectvely for the respectve problem. All the problems are solved umercally the physcal plae tself wthout trasferrg t to the computatoal plae usg structured grd. I the preset vestgato, we got reasoably good agreemet betwee the umercal ad the aalytcal result. Keywords: Numercal smulato, Dmesoal ozzle flow, CFD. 1. INTRODCTION expesve, tme-cosumg or eve dagerous Numercal smulato usg computers expermets laboratores or o ste. or computatoal smulato has creasgly become a very mportat approach for solvg complex practcal problems egeerg ad Numercal smulato wth computers plays a valuable role provdg a valdato for theores offers sghts to the expermetal scece. Numercal smulato traslates results ad asssts the terpretato or eve mportat aspects of a physcal problem to a dscrete form of mathematcal descrpto, recreates ad solves the problem o a computer, ad reveals pheomea vrtually the dscovery of ew pheomea. It acts also as a brdge betwee the expermetal models ad the theoretcal predctos. The overall goal of the feld of accordg to the requremets of the aalysts. umercal smulato s the desg ad Rather tha adoptg the tradtoal aalyss of techques to gve approxmate but theoretcal practce of costructg layers of accurate solutos to hard problems. The assumptos ad approxmato, ths moder Netlb repostory cotas varous collectos umercal approach attacks the orgal of software routes for umercal problems, problem all ts detal wthout makg too may assumptos, wth the help of the mostly FORTRAN ad C. Commercal products mplemetg creasg computer power. Numercal may dfferet umercal algorthms clude smulato provdes a alteratve tool of scetfc vestgato, stead of carryg out the IMSL ad NAG lbrares; a free alteratve s the GN Scetfc Lbrary. There are 1 1

2 Iteratoal Joural of Mechacal Egeerg ad Research, ISSN Vol. 5 No.1 (2015) Research Ida Publcatos; several popular umercal computg applcatos such as MATLAB, S-PLS, Lab VIEW, ad IDL as well as free ad ope source alteratves such as Free Mat, Sclab, GN Octave (smlar to Matlab), IT++ (a C++ lbrary), R (smlar to S-PLS) ad certa varats of Pytho. Performace vares wdely: whle vector ad matrx operatos are usually fast, scalar loops may vary speed by more tha a order of magtude. May computer algebra systems such as Mathematc also beeft from the avalablty of arbtrary precso arthmetc whch ca provde more accurate results. 2. COMPTATIONAL METHODS The fte dfferece scheme for the umercal smulato of quas oe dmesoal ozzle flow problem cossts of followg methods. Here, both explct ad mplct methods are used for the same problem for uderstadg the dea ad usage. 1) Mac-Cormack s Predctor Corrector Method Mac Cormack s techque s a explct fte dfferece techque, whch s secod order accurate both space ad tme. The tme marchg soluto usg Mac Cormack s techque for the Euler equatos [1] proceeds as follows: The value of the soluto vector at tme +1 s obtaed as: t 1 t Where value of t u t av s a represetatve mea betwee tmes ad +1.Ths average tme dervatve s obtaed from a predctor corrector phlosophy as Follows: Predctor Step: I the goverg equato the spatal dervatves are replaced by forward dffereces to gve: F 1 F H t x I ths equato all the values at tme are kow.e. the RHS s kow. Now, a predcted value of s obtaed from the frst two terms of a Taylor seres as: t 1 t I eq. (3.3) s kow ad kow from eq. (3.2). Hece readly obtaed. However, 1 1 t s ca be s oly a predcted value ad s oly frst order accurate sce eq. (3.3) cotas oly frst two terms the Taylor seres. sg, H F F( ) H( ) ad ca be foud out. Corrector Step: I the corrector steps a predcted value of the tme dervatve at tme +1 s obtaed by substtutg the predcted value of the equato, replacg the spatal dervatves by rearward dffereces. F 1 F H t x... (3.17) 2 191

3 Iteratoal Joural of Mechacal Egeerg ad Research, ISSN Vol. 5 No.1 (2015) Research Ida Publcatos; The average value of the tme dervatve of s ow obtaed from the arthmetc mea of 3 u a ad t 1 (3.4) respectvely. obtaed from eq. (3.2) ad eq. 1 1 t av 2 t t... (3.18) Thus the fal corrected value of ca be obtaed from eq. (3.1), repeated below: t 1 t I Mac Cormack s techque the use of forward dffereces o the predctor step ad rearward dffereces o the corrector step s ot sacrosact; the same order of accuracy s obtaed by usg rearward dffereces o the predctor ad forward dffereces o the corrector. Ideed, a tme marchg soluto ca be carred out by alteratg betwee these two sequeces at every other tme step. 2) Crak Ncolso method wth Beam ad Warmg learzato techque Quas Lear formulato: As F=F () by the cha rule of F F dfferetato: x x F Deotg A, A s called the Jacoba of the flux vector. It s obtaed by dfferetatg each of the elemets of flux vector F oe by oe by each of the elemets of vector. Equato show above s ow wrtte as: A H 0 t x Case 1: Coverget-Dverget ozzle (Fg : Nozzle -1 show above) doma wth Purely subsoc flow Nozzle cross secto area A(x) =1+ (2.2*(x- 1.5)*(x-1.5)); 0 x 3 ft. No. of grd pots: 31.e. step sze x = 0.1 Courat No. = a) usg Coservato form of goverg equato (after 1400 tme steps) b) usg o coservato form of goverg equato (after 1400 tme steps) Case 2 : Coverget-Dverget ozzle (fg: Nozzle-2 show above) doma wth subsoc-supersoc setropc flow Nozzle cross secto area a) sg Coservato form of goverg equato (after 5000 tme steps) 192

4 Iteratoal Joural of Mechacal Egeerg ad Research, ISSN Vol. 5 No.1 (2015) Research Ida Publcatos; Case 3a: Coverget-Dverget ozzle (fg: Nozzle-1 show above) doma wth subsoc-supersoc flow wth ormal shock sde (wthout Artfcal vscosty) The umercal results were obtaed usg Naver- Stokes equato based o Cosstet Flux Recostructo cocept as descrbed chapter 4 for uformly dstrbuted grd. The covergece crtero of was appled. Fgure 2a). Pressure cotours ad Streamles patter for Reyolds umber 75 Valdato of the results for Re 75,150 &150 ad blockage rato of 11% Reyolds Grd Drag Co Lft Co Number Sze effcets effcets Numercal calculatos of lamar flow past a square cylder for varous Reyolds umber (say 75,150,500) are doe by solvg the Naver-Stoke equato ad the results are obtaed. The drag ad lft coeffcets are calculated for all the flow cases ad tabulated above. If compared wth the results of some the research papers for the approprate blockage rato, Reyolds umber, grd sze ad where the square cylder was located, some results get matched. Fgure 2b). Pressure cotours ad Streamles patter for Reyolds umber 150 Refereces A.K.Saha, G.Bswas, K.Muraldhar S. Turk, H. Abbass, Reyolds umber Grd Sze 178 X X 77 Drag Co effce t Fgure 2c). Pressure cotours ad Streamles patter for Reyolds umber 500 R.Frake, W.Rod ad B.Schoug X

5 Iteratoal Joural of Mechacal Egeerg ad Research, ISSN Vol. 5 No.1 (2015) Research Ida Publcatos; 3. CONCLSION derstadg of the mplemetato of varous CFD techques to varous problems ad got the umercal results matchg wth aalytcal results. I Quas oe dmesoal ozzle flow problem, the Euler equatos wll be solved umercally usg fte dfferece scheme. For uderstadg of the mplemetato of fte volume method (CFR Scheme) to develop a compressble Naver- Stokes solver for two dmesoal geometres usg structured grd, flow over a square cavty problem was studed ad matched the umercal result wth joural paper results. REFERENCES 1. ABDALLAH S Numercal soluto for the pressure Posso equato wth Neuma boudary codto usg a o staggered grd, I. Joural of Computatoal Physcs; 70: ABDALLAH S.1987 Numercal solutos for the compressble Naver Stokes equato prmtve varables usg a o-staggered grd, Joural of Computatoal Physcs;70: A.B.HARICHANDRAN et al, CFR: A fte volume approach for computg compressble vscous flow, Joural of Appled flud mechacs. 4. AHMAD SOHANKAR, C. NORBERG, AND L. DAVIDSON 1999, Smulato of three-dmesoal flow aroud a square cylder at moderate Reyolds umber Physcs of fluds 11, AKHILESH K.SAH, R.P.CHHABRA, V.ESWARAN Two-dmesoal usteady lamar flow of a power law flud across a square cylder. Joural of No-Newtoa Flud Mechacs 160, 6. A. ROY et al., 2006 A fte volume method for vscous compressble flows usg a cosstet flux recostructo scheme, Iteratoal joural for umercal methods fluds, Volume 52, Issue 3, pages B.GALLETTI, C.H.BRNEA, L.ZANNETTI, A.IOLLO, 2004 Loworder modelg of lamar flow regmes past a cofed square cylder, Joural of Flud Mech, Vol.503, pages CHENG M. TAN S. H. HNG K.C, 2005, Lear shear flow over a square cylder at low Reyolds umber Physcs of fluds, Volume 17. Pages DENG GB, PIQET J, QETEY P, VISONNEA M Icompressble flow calculatos wth a cosstet physcal terpolato fte volume approach. Computers ad Fluds; 23(8): FRANKE.R. RODI.W. SCHNNG.B. 19, Numercal calculato of lamar vortex sheddg Flow past cylders, Joural of Wd Egeerg ad Idustral Aerodyamcs, Volume 35. Pages GERA.B.PAVAN K. SHARMA,SINGH.R.K 2010,CFD aalyss of 2D usteady flow aroud a square cylder Iteratoal Joural of Appled Egeerg Research, Ddgul Volume1, 12. GRESHO PM Some curret CFD ssues relevat to the compressble Naver Stokes equatos. Computer Methods Appled Mechacs ad Egeerg; 87: HARLOW FH, WELCH JE. 1965, Numercal calculato of tme-depedet vscous compressble flow of flud wth free surface. Physcs of Fluds; 8(12):