Journal of Appled Flud Mechancs, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. Avalable onlne at www.jafmonlne.net, ISSN 1735-3572, EISSN 1735-3645. Immersed Boundary Method for the Soluton of 2D Invscd Compressble Flow Usng Fnte Volume Approach on Movng Cartesan Grd S.M.H. Karman and M. Ardakan Center of Excellence n Computatonal Aerospace Engneerng Aerospace Engneerng Department, Amrkabr Unversty of Technology, Tehran, Iran Correspondng author Emal: m.ardakan@aut.ac.r (Receved Aprl 25, 2010; accepted March 13, 2011 ABSTRACT In ths study, two-dmensonal nvscd compressble flow s solved around a movng sold body usng Immersed Boundary Method (IBM on a Cartesan grd. Translatonal moton s handled wth a Cartesan grd generated around the body whch moves wth body on a background grd. In IBM, boundares are mmersed wthn the grd ponts. In ths paper soluton doman s dscretzed usng fnte volume approach. To mplement boundary condtons on mmersed boundares, a set of Ghost fnte volumes are defned along the wall boundares. Boundary condtons are used to assgn flow varables on these Ghost fnte volumes. Governng equatons are solved usng dual tme step method of Jameson. Fnally, numercal results obtaned from the present study are compared wth the other numercal results to evaluate the correct performance of the present algorthm and ts accuracy. Keywords: Immersed boundary method; Cartesan grd; Movng mesh; Unsteady Euler; Fnte volume; Jameson algorthm. NOMENCLATURE P E u v U V x y t t pressure energy velocty n x drecton velocty n y drecton x component of contravarant veloctes y component of contravarant veloctes x components of boundary velocty y components of boundary velocty 1. INTRODUCTION Immersed Boundary method (IBM s a new approach ntroduced n the last decade for the smulaton of flud flow around bodes wth complex statonary/movng boundares. In contrast to conventonal methods, IBM uses non-conformal Cartesan grd where surface of body do not necessarly pass through the grd ponts of the boundary. Therefore boundary condtons cannot be appled throughout the ordnary approaches. The technque by whch boundary condtons are appled n IBM s the key pont n these methods. Lookng back to the orgnal forms of IBM, t can be seen that boundary condtons have been appled by addng a forcng functon to the governng equatons. IBMs are categorzed nto two man groups. In the frst group, known as contnues forcng functon, the functon responsble for mplementng boundary condtons s added to the governng equatons before dscretzaton. R D t CFL Convectve flux force Numercal dsspaton real tme pseudo tme Densty Cell s area Currant number Works of Peskn (1972, La et al. (2000, Goldsten et al. (1993, Beyer (1992, Fauc et al. (1994 and Unverd et al. (1992 fall n ths category. In the second group, named dscrete forcng approach, forcng functon s added to the dscretzed form of the governng equatons. The sgnfcant advantage of ths method s that t would be ndependent of the dscretzaton method. Approaches n ths group are subdvded nto two categores drect approaches and ndrect approaches. In the ndrect approaches, forcng functon s well-selected to apply boundary condtons nto dscretzed governng equatons. Mohd-Yusof (1997 and Verzcco (2000 are good examples n ths group. In ther method, dscretzed form of the Naver-stokes equatons s frst solved and then correctons for boundary condtons are establshed. In ths method, forcng functon s determned n each tme step usng the latest velocty
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. feld calculated n the soluton doman. Ths method was used for modelng bodes wth complex boundares (Balaras 2004 and turbulence flow (Verzcco 2002. In some stuaton, mplementaton of forcng functons results n dffusng boundary effects over neghbor grd ponts. Followng ths, researchers ntroduced Drect approaches n whch boundary condtons are mplemented drectly on the dscrete boundary n the computatonal doman. Drect approaches are subdvded nto two groups, cutcell methods and ghost-node methods. In the cut-cell method Cartesan fnte volumes on the boundary are talored to conform to the boundary, as shown n Fg. 1a and Fg.1b. In ths case conservaton of conserved quanttes can be satsfed on the boundary fnte volumes (DeZeeuw et al. 1993; Qurk 1994; Yang et al. 1994; Udaykumar et al. 1996; Ye et al. 1999. In ths case we do not deal wth Cartesan grds on the boundary any more. In some cases, ths talorng or reshapng may result n very small grd cells wth an adversely mpact on the numercal stablty. Ths can be overcome by a cell-mergng strategy shown n Fg. 1c and Fg. 1d (Clarke et al. 1985. volume methods, are sold nodes whch at least have one flud node n ther neghbors. Fg. 2. Grd-pont stencls for mposng boundary condtons (La et al. 2000 Sold and flud nodes are the nodes that are located wthn the sold and flud regons of doman, respectvely. Fg. 1. Treatment procedure of boundary cells n cutcell methods (Clarke et al. 1985 Ghost-node methods are ntroduced to rebuld the soluton at grd nodes n the vcnty of the mmersed boundary usng nterpolaton functons, to mplement the boundary condtons. The man ssue s that how the soluton s rebult near the boundary. The choce of nterpolaton functons makes the dfference between methods n ths category. One-dmensonal nterpolaton s used by Fadlun et al. (2000 along the grd lne ntersectng sold boundary, but the choce of ntersectng grd lne seems to be arbtrary; see Fg. 2. Later on, Balaras (2004 presented an approach where the soluton s reconstructed along a defned lne normal to the body; smlar approaches can be found n several exstng studes (Glmanov et al. 2003, 2005; La et al. 2000; Fadlun et al. 2000. Approaches usng nterpolaton functons are smple and straght forward. However, n cases where boundary passes from a dstance very close to the grd ponts, these approaches encounter some knd of nstablty. Grd-pont stencls whch are normally used n IBM, are shown n Fg. 2. Dependng on the locaton of boundary, some nodes n the sold secton of doman would become a part of soluton. These nodes are called ghost nodes. Ghost nodes or ghost cells n fnte Fg. 3. Image pont of a sample ghost node. G and I denote Ghost cell and Image pont (Petter et al. 2008 In IBM, varables at the ghost nodes are needed to close the governng equatons at the flud nodes. Boundary condtons along the sold boundary are mplemented throughout the determnaton of these ghost-node varables. IBMs are mostly recognzed by the method used to assgn ghost-node varables. These varables are normally extrapolated from ther values from the flud part of the soluton doman. The advantage of usng ghost nodes s that the same dscretzed form of equatons used nsde the soluton doman wll be appled for the nodes n the vcnty of the sold boundary. In words, there s no need to reformulate the numercal algorthm for the nodes near to the boundary. There are numerous ways for assgnng varables at ghost nodes, usng nterpolaton schemes (Tseng et al. 2003. Although hgher-order polynomals are more accurate, they are more senstve to numercal nstabltes. In ths category, frst order twodmensonal nterpolaton was used by Majumdar et al. (2001, and quadratc nterpolaton was used by Tseng et al. (2003, 2005. To resolve the nstablty problem, the concept of Image Pont (denoted by I was ntroduced and wdely used by Mttal et al. (2003. As 28
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. shown n Fg. 3, an mage pont, located wthn the flow doman, s the mrror pont of the ghost node wth respect to the sold boundary. Determnaton of mage ponts may rase dffcultes. Some of these dffcultes are shown n Fg. 4. a b Fg. 4. Specal cases n determnaton of ghost-node mage ponts. a Ghost node has no unque mage pont, b Ghost node les n flud part of doman. GC and BI denote Ghost cell and boundary ntercept [27] In Fg. 4a, due to the poston of ghost node wth respect to the sold boundary a unque mage pont cannot be defned for t. On the other hand as shown n Fg. 4b, sometmes one cannot fnd a ghost node n sold regon of doman. In ths case, duplcate of the correspondng flud node s consdered as a fcttous ghost node. Ths requres separate memory locatons for the fcttous ghost nodes. The correspondng mage pont s then determned. Works of Mttal cover dfferent Mach-number flows of vscd/nvscd condtons around statonary/movng bodes wth complex geometres (Mttal et al. 2002, 2003, 2004, 2005, 2008. Smulatons of movng boundares are carred out usng a qualfed grd. Ths qualty of grd must be preserved durng body moton. Ths can be done usng grd qualty mprovement procedure. Note that the moton of boundary causes contnuous change of flud, sold, and ghost nodes to each other. Therefore the algorthm should be able of handlng these changes. In the present paper, 2D nvscd compressble flow s smulated around movng bodes usng IBM approach. Boundary condtons are mplemented by drect method usng ghost nodes. Soluton doman s dscretzed nto Cartesan fnte volumes. To mplement boundary condtons Ghost fnte volumes (GFV are establshed along the boundary. The man ntenton of ths paper s to combne fnte volume technque wth IBM. As a result conservaton laws wll be satsfed wthn the doman. As wll be dscussed later flow varables at GFVs wll be assgned based on the boundary condtons. For the smulaton of flow around movng bodes there exst many approaches; see Mttal s works (Mttal et al. 2002, 2003, 2004, 2005, 2008. These nclude approaches n whch the grd s regenerated after each tme step of body moton, or approaches n whch the grd s contnuously adapted to the movng boundares. Each of these methods has ts own advantage and drawbacks. Methods n whch the grd s generated repeatedly wll be computatonally costly. In addton to ths flow varables should be nterpolated from one grd to the other one. Ths wll cause sgnfcant numercal error. Fully dynamc grds are also very complcated, and ther codng wll be cumbersome. Snce Cartesan grd s used here, smulaton of flud flow around bodes wth translatonal moton can be smply modeled. As mentoned before, Mttal et al. (2008 used fnte dfference to descretze governng equatons. Boundary condtons are mplemented n Mttal s work usng ghost nodes for whch correspondng mage ponts n flud doman have been defned, as mentoned before. Obvously wth boundary moton ghost nodes and ther related mage ponts wll be changed. Therefore, search would be needed for new ghost nodes and ther related mage ponts n each tme step. To handle movng boundares wth translatonal moton n ths paper, hybrd grd approach of Mrsajed et al. (2006 s employed. Soluton doman s dvded nto two zones. As shown n Fg. 5, frst zone ncludes a Cartesan grd n whch the movng body s located. Ths zone wll move and the body s statonary wth respect to t. Second zone s a background Cartesan grd n whch the frst zone can be moved. Any grd refnement can be performed n frst zone to provde soluton accuracy. Snce the frst zone has rectangular boundares any translatonal moton can be smply modeled. Fg. 5. Grd confguraton; two zone approach The advantage of ths approach s that the number of node deleton/nserton process s mnmzed, and therefore very few data nterpolaton s requred. On the other hand, snce the movng body s statonary n the frst zone, flud, sold and ghost fnte volumes reman unchanged durng the soluton procedure. 29
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. 2. GRID GENERATION In ths study, movng body smulaton s handled usng two-grd zone approach of Mrsajed et al. (2006. As shown n Fg. 5, mmersed boundary of a movng body s located n the frst zone whch tself moves wthn the second zone, called back ground. Grds of both zones are Cartesan. As mentoned n the prevous secton ths grd confguraton allows smple handlng of translatonal moton of a movng boundary. Although not employed n ths paper, rotatonal moton of a movng body can be also modeled by ths approach f one more zone s added to ths grd confguraton; see paper of Mrsajed et al. (2006 for more detals. As mentoned earler, sold boundares are defned as mmersed boundares n a Cartesan grd. The key task n IBM s to accurately mpose boundary condtons on these mmersed boundares. For ths purpose, hgh qualty grds are requred n the vcnty of mmersed boundares. Ths s carred out n ths paper usng a mult-layer refnement algorthm.there s other works such as the work of Mttal et al. (2007 n whch the grd s refned around the body locally. However as shown n Fg. 6, these refnements spread out n x and y drectons of the doman as banded refned zones, whch s not desrable. Refnement procedure s started from outer layer to the nner layer. One level of refnement s appled to Cartesan grds wthn all layers, as shown n Fg. 8b. In the next step one more level of refnement s appled to Cartesan grds wthn all layers except the most outer one, as shown n Fg. 8c. Havng excluded the two most outer layers grd refnement s appled once agan to the rest of Cartesan grds wthn the nner layers. Dependng on the number of layers ths procedure s contnued up to the layer neghbor to the body, as shown n Fg. 8d. Fg. 8. grd refnement procedure n frst zone around body surface Fg. 6. Grd refnement around body n x and y drectons (Ghas et al. 2007 Grd generaton procedure ncludes, 1 Cartesan grd generaton n both zones as was shown n Fg. 5, 2 mplementaton of mult-layer refnement algorthm n the frst zone to mprove grd qualty around the body. The followng refnement procedure s only appled to the frst zone. Refnement layers are shown n Fg. 7. Fg. 7. Mult layers n refnement procedure Snce the sold body under consderaton s fxed n the frst zone, ths grd zone wll move wth t. As shown n Fg. 9a, background grd lnes become close to each other n front of the frst zone, and become far from each other at the back of the frst zone. To preserve grd qualty, deformatons are lnearly dstrbuted wthn the three or four rows of the background grd; ths shown n Fg. 9b. In ths algorthm, whenever the 1st grd lne of background grd n the front of the frst zone reaches to the old locaton of the 2nd grd lne of background grd, these two grd lnes wll be merged wth each other. In ths case, the two grd lnes of background grd n the rear of the frst zone wll be splt nto three lnes; ths s shown n Fg. 9b. In ths procedure the total number of grd ponts n the doman wll be constant. As shown n Fg. 10, dscretzaton of the soluton doman produces three types of fnte volumes when usng IBM. A fnte volume wth ts center located wthn the body s known as sold fnte volume (SFV. These are the fnte volumes on whch flow equatons are not solved. In contrast to ths, f the center of fnte volume s located n the flud part of the soluton doman the fnte volume s named flud fnte volume (FFV. These are the fnte volumes on whch flow equatons wll be solved. In addton to these two types of fnte volumes flow varables should be determned on a set of fnte volumes called Ghost fnte volume (GFV. GFVs are sold fnte volumes whch at least has one flud fnte volume n ther neghbors. Flow varables on GFVs are needed for calculaton of fluxes on the surface of FFVs. As wll be dscussed later, ths 30
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. s where the boundary condtons wll be appled on the sold boundares. Note that snce sold boundary s statonary wth respect to frst zone, sold, flud, and ghost fnte volumes reman unchanged durng the body moton. where W, F, and G are defned as, u W, v E U V uu p uv F, G vu vv p E pu xtp E pv yt p (2, puv,, and E are densty, pressure, velocty components, and total energy, respectvely. Contravarant veloctes U and V are defned as Fg. 9. Frst zone movng to left; A Before lne deleton/nserton, B After lne deleton/nserton a b Where U ux t V v yt xt and yt are velocty components of fntevolume boundary. In addton to these, equaton of state for a perfect gas s used to complete the set of equatons. Equaton (1 s appled to each fnte volume wth area of and boundary of. Ths results n the followng equaton d R D 0 dt (3 where R (w s the convectve flux over the surfaces of th fnte volume, and D (w s the numercal dsspaton term whch s ntroduced to prevent odd and even pont oscllatons, and oscllatons n the vcnty of shock waves (Jameson et al. 1986. Equaton (3 s mplctly dscretzed n tme;.e. Jameson et al. (1996 d n1 n1 n1 n1 R D 0 (4 dt Usng second order accurate backward dfferencng (Jahangran et al. 2004 Eq. (4 can be wrtten as 3 n1 n1 2 n n 2t t (5 1 n1 n1 n1 n1 R D 0 2t The above equaton s solved usng dual tme steppng scheme. Ths s done by solvng the followng equaton at each tme step. Fg. 10. Three types of fnte volumes created by the mmersed boundary; S: sold fnte volume, F: Flud fnte volume, and G: Ghost fnte volume 3. GOVERNING EQUATIONS Integral form of the two dmensonal unsteady Euler equatons for compressble flow n the Cartesan coordnate are gven by d WdA Fdy Gdx 0 (1 dt w R * n 0 (6 where s pseudo-tme n each tme step, and * n R s the unsteady resdual, defned as * 3 1 1 2 R n n n n n 2t t 1 n 1 n 1 R n 1 D n 1 2t (7 Equaton (6 s a modfed steady state problem n pseudo-tme. Ths problem can be solved usng explct Runge-Kutta multstage scheme or any tme marchng method desgned to solve steady state problems. To 31
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. accelerate convergence, local pseudo tme steppng and mplct resdual averagng are used. The four-stage Runge-Kutta scheme used n ths paper s ntroduced by w w w w w (0 (1 (2 (3 (4 w w w w (0 (0 (0 (0 n1 m1 n1 m 1 4 1 R 3 1 2 1 4 * R (4 * (0 (1 * R (2 * R (3 (8 GFV, mage pont, and sold boundary. For the case of statonary boundares, normal velocty component at the GFV wll be equal to the negatve value of normal velocty component at the mage pont. For the velocty component tangent to the boundary, ts value at the GFV s set equal to tangental component of velocty at the mage pont; these are shown n Fg. 12. Where * ( l 3 ( l 1 2 R n ( n n 2 w t t 1 1 1 ( l n n R D n 1 2 t (9 To ncrease computatonal effcency, numercal dsspatve term s only calculated n the frst stage of Eq. (8. The allowable pseudo-tme step for each cell s restrcted by stablty consderatons and s gven by mn CFL N J 1 2t, 3 Where j denotes edge of the correspondng cell, and (10 = u y v x (11 4. BOUNDARY CONDITIONS In the far feld, non-reflectng boundary condtons are used based on the characterstc analyss. Sold boundares are mmersed wthn the grd. Therefore, boundary condtons cannot be appled by conventonal methods. Instead, boundary condtons are mplemented through the determnaton of the flow varables on the ghost fnte volumes, whch are defned along the sold boundares. To determne flow varables on ghost fnte volumes, an mage pont (I should be defned for each ghost fnte volume. Image pont whch s located wthn the flow doman s the mrror pont of the center of ghost fnte volume wth respect to the sold boundary. Dfferent mage ponts are shown n Fg. 11. Flow varables on an mage pont are known snce t s wthn the soluton doman; ths wll be dscussed later. Assumng zero normal-pressure gradent on the sold boundary, pressure of a ghost fnte volume would be equal to ts mage-pont pressure. For an adabatc boundary, agan temperature of a ghost fnte volume would be equal to ts mage-pont temperature. Flud flow should have a velocty whch ts component normal to the sold boundary s equal to the normal velocty component of boundary. Ths s mplemented usng lnear nterpolaton between normal veloctes of Fg. 11. Dfferent mage ponts and ther surrounded fnte volumes At ths stage determnaton of flow varables at the mage ponts should be dscussed. Although, t seems that flow varables can be easly determned on mage pont, n some cases ths would be hard to do. As shown n Fg. 11a f the mage pont are ghost fnte volume, varables on mage pont are calculated usng blnear nterpolaton of the most recently updated varables at the GFVs; see Fgs. 11b-d. In ths case calculaton of varables of the ghost fnte volume s repeated untl the converged values of varables at all of the GFVs are reached. Fg. 12. Veloctes on the Ghost Fnte Volume 5. RESULT AND DISCUSSION To valdate present algorthm the followng test cases are solved. 5.1 Statonary Arfol Subsonc, transonc and supersonc flows are smulated around NACA0012 arfol at dfferent angles of attack. Frst test case ncludes subsonc flow of Mach 0.5 passng the arfol at 0 degree angle of attack. Soluton 32
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. s carred out on three dfferent grds coarse, mddle and fne, whose specfcatons are gven n Table 1. For each grd number of flud fnte volumes (FFV, ghost fnte volumes (GFV, and sold fnte volume (SFV are gven. Ghost fnte volumes play the same role that boundary nodes play n conventonal algorthms for the mplementaton of boundary condtons on body ftted grds. More accurate boundary condtons can be mplemented wth hgher number of GFVs. As seen, the number of GFVs are doubled from coarse to mddle, and from mddle to fne grds. Smlarly, transonc flow of Mach 0.85 over a NACA0012 arfol at 1 degree angle of attack s numercally solved by the present algorthm. Cp dstrbutons of dfferent grds are shown n Fg.15. Agan results of fne grd have excellently matched the results of AGARD (1985. As seen, the captured shocks are well postoned and ther strengths are correctly calculated. Table 1 Specfcatons of coarse, mddle and fne grds Number of Coarse grd Wth 5 layers Mddle grd wth 6 layers Fne grd wth 7 layers FFV 2221 4591 12094 GFV 48 104 216 SFV 30 200 998 In Fg. 13, Cp dstrbutons of ths subsonc flow on three grds are compared wth each other and wth the expermental results of AGARD (1985. As s obvous, results of fne grd are very close to the results of AGARD (1985. Convergence hstory of the fne grd soluton s presented n Fg. 14. As seen after 130 real tme steps error s decreased to the order of 1e -9. Fg. 15.Comparson between Cp dstrbutons of NACA0012 arfol on dfferent grds wth result of AGARD (1985; M=0.85, Incdence angle =1 As mentoned earler, flow varables of ghost fnte volumes are determned n each tme step based on the boundary condtons on the sold boundary. Therefore, convergence of varables at GFVs would be a good ndcaton of soluton convergence as well. Consder prevous test case. For ths steady state case, convergence hstory of pressure at GFV whch s located at leadng edge of arfol, s shown n Fg. 16. As seen, after about 1000 teratons ts value approaches to ts converged value of 1.5. Fg. 13. Comparson between Cp dstrbutons of NACA0012 arfol on dfferent grds wth result of AGARD (1985; M=0.5, Incdence angle =0 Fg. 16. Convergence hstory of GFV pressure n the steady state case of M=0.85, Incdence angle =1 Fg. 14. Convergence hstory of the fne grd n the flow smulaton around NACA 0012 arfol at M=0.5 and Incdence angle =0 In the next case, Cp dstrbuton of supersonc flow of Mach 1.2 over NACA0012 arfol at 7 degrees angle of attack s plotted n Fg. 17 for the fne grd. Results of the present algorthm and that of AGARD (1985 match wth each other very well. Note that the unsmooth result n the front of arfol s due to hgh curvature of body geometry n ths area that nfluence on determnng GFVs flow varables. 33
S.M.H. Karman, and M. Ardakan / JAFM, Vol. 4, No. 2, Specal Issue, pp. 27-36, 2011. ncdence angle of 3 degrees wll be smulated n the followng three dfferent setups. Fg. 17.Comparson between Cp dstrbutons of NACA0012 arfol wth result of AGARD (1985; =1.2, Incdence angle =7 5.2 Arfol wth horzontal moton Here we demonstrate some test cases to valdate the capablty of the present algorthm n solvng unsteady flow felds wth movng boundares. Frst test case ncludes unsteady flow around NACA 0012 arfol whch moves wth Mach 0.5 n a statonary flud. After an ntal transton, the flow feld around the movng arfol should become steady wth respect to the arfol. For comparson purpose, ths problem was also solved n the steady mode,.e. the arfol s statonary and the ar flows wth speed of Mach 0.5. Comparson of the predcted Cp dstrbutons wth the expermental data (AGARD 1985 s shown n Fg. 18. As seen, calculated Cp dstrbutons match wth each other and wth the data of (AGARD 1985. Note that the lttle dfference between statonary and movng results s due to movng algorthm and determnng boundary velocty on ntersect ponts n order to determnng GFVs normal velocty that take effect on GFV s ndrectly. In the frst setup, arfol s postoned parallel to the x- axes of the soluton doman. Flow wth Mach 0.5 and 3 degrees angle of attack then passes over the arfol. In the second setup, an arfol wth ncdent angle of 3 degrees wth respect to the x-axes s translated wth speed of Mach 0.5 along the x axs n statonary flud. Fnally, n the thrd setup, a NACA0012 arfol parallel to the x-axes, s translated wth the speed of Mach 0.5 n a drecton wth 3 degrees ncdence n a statonary flud. The predcted Cp dstrbutons obtaned from these three cases are compared wth each other n Fg. 19. All of the results match wth each other. Wth ths excellent test one can make sure that the method s perfectly mplemented and s ndependent of grd. The lttle dfference between statonary and movng results s same as prevous test case that menton earler. Fg. 19. Comparson of Cp dstrbutons of flow of Mach 0.5 over NACA0012 arfol at 3 degrees angle of attack for three dfferent setups 6. CONCLUSIONS In ths paper, a movng-mesh algorthm s presented for the soluton of two-dmensonal compressble nvscd flow on Cartesan grd usng Immersed Boundary Method (IBM. Soluton doman s dscretzed to a number of fnte volumes. Boundary condtons on sold boundares are mplemented throughout the determnaton of ghost fnte volume varables n the soluton doman. Grd refnement s performed n dfferent layers around the body to prevent extra producton of grd ponts. Flow equatons are solved usng dual tme step method of Jameson. Numercal results obtaned from the present study are compared very well wth other numercal results. REFERENCES Fg. 18.Comparson of Cp dstrbuton of NACA0012 arfol wth expermental data (AGARD 1985 n a horzontal translatonal movement 5.3 Arfol wth oblque moton Ths test case s desgned to examne the correct performance of the present algorthm n all drectons. Ar flow of Mach 0.5 passng NACA 0012 arfol wth AGARD Flud Dynamc Panel (1985. Test cases for Invscd flow feld methods. AGARD Advsory Report, AR-211. Balaras, E. (2004. Modelng complex boundares usng an external force feld on fxed Cartesan grds n large-eddy smulatons. Journal of computatonal fluds 33, 375 404. 34
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