The umercal smulato of compressble flow a Shub ozzle usg schemes of Bea-Warmg ad flux vector splttg Gh. Paygaeh a, A. Hadd b,*, M. Hallaj b ad N. Garjas b a Departmet of Mechacal Egeerg, Shahd Rajaee Teacher Trag Uversty, Lavza, Tehra, Ira b Departmet of Mechacal Egeerg, Tarbat Modares Uversty, Tehra, Ira. Artcle fo: Receved: 8/8/11 Accepted: 8/11/11 Ole: 3/3/1 Keywords: Compressble flow Shub ozzle Bea-Warmg scheme Flux vector splttg scheme Euler equato Abstract Over the last te years, robustess of schemes has rased a creasg terest amog the CFD commuty. The objectve of ths artcle s to solve the quasoe-dmesoal compressble flow sde a Shub ozzle ad to vestgate Bea-Warmg ad flux vector splttg methods for umercal soluto of compressble flows. Two dfferet codtos have bee cosdered: frst, there s a supersoc flow the etry ad a supersoc flow the outlet, wthout ay shock the ozzle. Secod, there s a supersoc flow the let ad a subsoc flow the outlet of the ozzle ad a shock occur sde the ozzle. The results show that the ru tme of the flux vector splttg scheme s more tha the Bea- Warmg scheme, ad, the flux vector splttg scheme s more accurate tha the Bea-Warmg scheme. However the flux vector splttg scheme s more complcated. 1. Itroducto Computatoal flud dyamcs (CFD) methods are based o the prcples of mass, mometum ad eergy coservato. The computed soluto provdes flow varables such as velocty, pressure, temperature, desty, cocetrato, etc. at thousads of locatos wth the doma. CFD methods ca be appled to exame dfferet equpmet desgs, or compare performace uder dfferet operatg codtos. Studes to exame the fluece of varous parameters o the flow behavor ca be coducted usg CFD methods. * Correspodg author: E-mal address: am.hadd@yahoo.com It also allows for varous cocepts to be examed a vrtual settg, wthout actually buldg a physcal model. I geeral CFD methods are appled to uderstad the overall flow ad heat trasfer behavor. I ths paper, two umercal methods are vestgated: (Bea-Warmg ad flux vector splttg schemes) for solvg Euler equatos Hgh Mach umber flows. The flux vector splttg scheme produces steady shock profles wth two teror zoes. Numercal solutos by frst ad secod-order schemes, cludg the above splt fluxes ca be foud Ref [1]. 111
JCARME Gh. Paygaeh et al. Vol. 1, No., March 1 The developmet of mplct fte-dfferece methods for the Euler ad Naver Stokes equatos s preseted by NASA Ames research scetsts [-7]. The major lmtg drawback to the explct methods s the applcato to vscous flows where fe grd spacgs are requred to capture boudary layers. Numercal stablty lmtatos of the explct methods led may to look at mplct schemes, wth ther heret ucodtoal stablty. Implct methods, though, were at frst hdered by ther vastly creased umercal work (maly due to the eed to vert large sparse matrces). The groudbreakg work by Bea ad Warmg led to effcet mplct approxmato schemes. Steger [8] made cotrbutos the umercal aalyss ad practcal applcato of mplct methods, alog wth Pullam ad Steger [9, 1] (oe of the frst three-dmesoal applcatos of mplct methods). Geerally, ths paper, focus s o Qus-Oe dmesoal Euler equatos ad methods whch apply for umercal soluto of Euler equatos such as Bea-Warmg methods ad Steger ad Warmg flux vector splttg.. Goverg equato The Euler equatos for a quas oedmesoal flow may be expressed as: E sq H t x (1) Where s s the cross-sectoal area assumed depedet of tme,.e. s=s(x) ad: u Q u, E s u p e t et p u ds ad H p () dx where ρ s desty, u s the velocty, p s the pressure, ad, e t s the total eergy: 1 et e u 3. Numercal ssues (3) Cosder a mplct algorthm for Eq. (1). The tme dervatve s approxmated by a frst-order backward dfferece approxmato to provde: Q s 1 where E Q t E x E Q 1 H 1 1 E Q O t (4) (5) ad 1 H H H Q O t Q (6) We wll assume a perfect gas ad therefore, p 1 e t u 1 (7) Wth defto of the speed of soud as a p/, we have: e t a 1 u 1 (1) (8) Hece, the Jacoba matrx E / Q wll be deoted by A ad the Jacoba matrx H / Q s deoted by B. The egevalues of A represet the characterstcs drecto of formato. Sce the flux Jacoba A possesses a complete set of egevalues ad egevectors, a smlarty trasformato exst such that: 1 A XDX where, u D u a u a (9) 11
JCARME The umercal smulato... Vol. 1, No., March 1 Also, X s the egevector matrx ad X -1 s the verse of the egevector matrx. Moreover, recall that the flux vector E possesses the homogeous property; therefore, t may be spltted to sub-vectors such that each subvectors s assocated wth postve or egatve egevalues of the flux matrx Jacoba. Thus, the egevalues may be grouped as postve or egatve. For a subsoc flow, two of the egevalues, amely u ad u+a, are postve, whereas the thrd egevalue, u-a, s egatve. Therefore, the Jacoba matrx A s spltted accordg to: 1 A A A, A XD X, 1 A XD X (1) The elemets of the dagoal matrces D + ad D - are the postve ad egatve egevalues,.e. u D s u s u a u a u a s u a (11) Now, the flux vector E may be spltted accordg to: E ad, E A A Q Q (1) (13) Note that for a supersoc flow, all three egevalues are postve ad, therefore, A A A (14) The flux Jacoba matrces A + ad A - (for the subsoc flow) are easly determed by Maple. At ths pot, pause a momet to determe the reaso for all the mathematcal mapulatos cosdered so far. Recall that the objectve s to develop effcet ad stable umercal schemes to solve a system of hyperbolc PDEs, for the tme beg the model Eq. (1). To vestgate the stablty requremet of the equato, a lear stablty aalyss s employed. The results dcate that f oe-sded dfferecg s used for the spatal dervatves, t must be a forward dfferecg for the terms assocated wth the egatve egevalues ad a backward dfferecg for the terms assocated wth the postve egevalues. Ths requremet s used for the FDEs whch oe-sded dffereces are used. A secod cosderato, a very mportat oe, s the specfcato of the flow ad outflow boudary codtos based o the egevalues. Ths pot wll be explored after the examato of the FDEs. 4. Implct formulatos The mplct formulato for the oedmesoal Euler equato s gve by Eq. (4). Substtuto of Eq. (5) ad Eq. (6) to Eq. (4) yelds: Q s E t x H E Q Q H Q Q (15) Ths equato may be expressed terms of the Jacoba matrces A ad B as: A si t x Bt Q E t H (16) x where I s the detty matrx ad A/ xq mples AQ / x. 5. Steger ad Warmg Flux Vector Splttg 113
JCARME Gh. Paygaeh et al. Vol. 1, No., March 1 I the flux vector splttg scheme, E ad the flux Jacoba matrx A are spltted accordg to the prevous dscusso to provde: si t A A B x Q t E E H (17) x Hece, whe frst-order approxmatos are used, the followg fte dfferece equato s obtaed: t si A A 1 A 1 A tb x 1 Q t E E 1 E 1 E H x 6. Bea-Warmg (18) I the Bea-Warmg scheme, the mplct formulato for the oe-dmesoal Euler equato s gve by: sq E t 1 1 E x 1 1 H 1 (19) Substtuto of Eq. (5) ad Eq. (6) to Eq. (19) ad smplfyg yelds: ra ra 1 1 Q Q 1 SI tb where R r 1 Q R F F 1 th De D () 1 (1) For stablty of umercal soluto must add D e must be added to R. For creasg of speed D must be added to the left- had sde of Eq. (). De ad D are gve by: D D e 4 4 x 4 Q e x Q x () x 7. Results ad dscussos Results of the umercal soluto usg the Steger ad Warmg flux vector splttg scheme are preseted, ad the that of the Bea-Warmg method are preseted. The cross-sectoal area of the Shub ozzle s defed by: S x 1.398.347 tah.8x 4 x xmax The quas-oe-dmesoal Euler equatos are varat uder the followg scalg: u u u a t t l x S p e S p e * S p a The subscrpt o deotes the reservor codto. The flow codto for ths ozzle s as follow:.5861 p p.719 supersoc flow u u 1.99184 ( M 1.614 ) Ad outflow codto whe the flow s subsoc the outlet s: p x p. 5156 max out The results of mesh depedecy Fg. 1 show that results do ot chage f the umber of mesh s more tha 5. Hece, ths work, we cosder 11 grds x-drecto. For usg the CFL umber, the maxmum value of (u+a) must be kow. Therefore, varato of x l 114
JCARME The umercal smulato... Vol. 1, No., March 1 (U+a) alog the ozzle are plotted Fgs. ad 3. whereas the desty ad the pressure are decreasg. As s obvous from the Fg. 4, the Mach umber across the ozzle s more tha oe. Hece, there s ot ay shock across the ozzle. I Fg. 5(a) ad (b), results are show for the state whch (a) Dagram of M versus x the flow wth Supersoc flow ad supersoc outflow. (b) Dagram of M versus x the flow wth Supersoc flow ad subsoc outflow. Fg. 1. Mesh depedecy the supersoc flow ad, (a) supersoc, ad (b) subsoc out outflow. Fg.. (U+a) to x for supersoc outflow (wthout shock). Fg. 3. (U+a) versus x for subsoc outflow (wth shock). I Fg. 4, dstrbuto of pressure, Mach umber, desty ad velocty across the ozzle for state whch there are ot ay shock are show. As s clear, the velocty ad the Mach umber are creasg across the ozzle a shock occurs the ozzle ad the flow s subsoc the outlet. They show that the shock occurs exactly the mddle of the ozzle. Ths fact s cofrmed by expermets ad aalytcal data [14]. 115
JCARME Gh. Paygaeh et al. Vol. 1, No., March 1 Fgure 5(c) shows that CPU tme depeds o the CFL umber. It s clear that wth crease of the CFL umber, the CPU tme wll decrease, but t caot be creased more tha 1.4, because the the soluto wll dverge. The best amout for CFL umber s 1.3 ths case. (a) (b) Fg. 4. Dagrams of mach ad desty (a), velocty ad pressure (b) versus x by flux vector splttg method. (a) (b) (c) 116
JCARME The umercal smulato... Vol. 1, No., March 1 Fg. 5. Dagrams of Mach, desty, velocty ad pressure versus x (a ad b) ad dagram of CPU tme versus CFL usg the flux vector splttg method (c). Geerally, the Bea-Warmg scheme for the codto that the flow s supersoc the outlet, does ot have ay dfferece from the flux vector splttg scheme. But for the secod codto, subsoc outlet the results are dfferet. Fgure 6(a) ad (b) show the varato of pressure, velocty, desty ad Mach umber alog the ozzle, usg the Bea-Warmg scheme, whe there s a shock the ozzle. It ca be see that there are fluctuatos ear the locato of the shock. These fluctuatos deped o D e. Dagram of CPU tme versus CFL umber Fg. 6(c) shows that the best CFL umber s.3. Comparg the CPU tme of the two methods ther optmum CFL umbers, we otce that the Bea-Warmg method s faster tha the flux vector splttg. The results also show that the results from the flux vector splttg are more accurate, but the algorthm s more complcated also show that the results from the flux vector splttg are more accurate, but the algorthm s more complcated. (a) 8. Coclusos I ths paper, we the Qus-Oe dmesoal Euler equato s solved for the flow a Shub ozzle, usg methods of Steger- Warmg flux vector splttg ad Bea- Warmg. Numercal soluto s doe for two dfferet codtos: wthout ay shock the ozzle ad wth a shock the ozzle. From the CPU tme ad the accuracy pot of vew, t was foud that both methods have the same results whe there s ot a shock the ozzle. However, whe there s a shock, the flux vector splttg method s more accurate tha the Bea-Warmg method. However, the flux vector splttg s slower ad more complcated tha the Bea-Warmg method. All all, the umercal results have a full complace wth aalytcal solutos. Choosg of a sutable CFL umber s mportat for havg a fast covergece to the soluto. 131
JCARME Gh. Paygaeh et al. Vol. 1, No., March 1 (b) (c) Fg. 6. Dagrams of mach, desty, velocty ad pressure versus x (ab) ad dagram of CPU tme versus CFL (c). [6] R. Warmg ad R. B., "Upwd secodorder dfferece schemes ad applcatos aerodyamc flows", Amerca Ist of Aeroautcs ad Astroautcs,Vol. 14, pp 141-149, (1976). [7] J. Qu, "Developmet ad comparso of umercal fluxes for LWDG methods", Numercal Mathematcs, Theory, Methods ad Applcato, Vol. 1, pp. 1-3, (8). [8] J. Steger, "fte dfferece smulato of flow about arbtrary geometres wth applcato to arfols", Amerca Ist of Aeroautcs ad Astroautcs, Vol. 16, pp. 679-686, (1978). [9] D. W. Zgg, S. De Rago, M. Nemec ad T. H. Pullam, "Comparso of several spatal dscretzatos for the Naver-Stokes Equatos", Joural of computatoal Physcs, Vol. 16, pp683-74, (). [1] T. H. Pullam, "Early developmet of mplct methods for Computatoal Flud Dyamcs at NASA Ames", Computers ad flud, Vol. 38, pp. 491-495, (9). Refereces [1] B. VaLeer, "Flux-vector splttg for the Euler equatos", Eghth Iteratoal Cofereceo Numercal Methods Flud Dyamcs, Germay, (198). [] R. Bea ad R. Warmg, "A mplct fte-dfferece algorthm for hyperbolc systems coservato law form", Joural of ComputatoalPhyscs, Vol., pp. 87-11,(1976). [3] R. Bea ad R. Warmg, "A mplct factored scheme for the compressble Naver-Stokes equatos", Amerca Istof Aeroautcsad Astroautcs, Vol. 16, pp. 393-4, (1977). [4] G. May ad A. Jameso, "A spectral dfferece method for the Euler ad Naver-Stokes equatos o ustructured meshes", AIAA 44 th Aerospace Scece Meetg ad Exhbt, (6). [5] H. Bucker, B. Pollul ad A. Rasch, "O CFL evoluto strateges for mplct upwd methods learzed Euler equatos", Iteratoal Joural for Numercal Methods Flud, Vol. 59, pp1-18, (9). 118
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