UNSTEADY, LAMINAR FLOW SIMULATIONS USING THE NONLINEAR DISTURBANCE EQUATIONS

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1 UNSTEADY, LAMINAR LOW SIMULATIONS USING THE NONLINEAR DISTURBANCE EQUATIONS R. P. Hansen *, L. N. Lng, and P. J. Mrris Department f Aerspace Engineering The Pennslania State Uniersit Uniersit Park, PA 680 AIAA ABSTRACT The reslts f nmerical simlatins sing the Nnlinear Distrbance Eqatins (NLDE) fr iscs test cases in laminar flw are presented. The NLDE are implemented in a tw-dimensinal, finite lme cde t test the algrithms prperties fr iscs, wall-bnded flws. Testing fcsed n the accrac f nstead phenmena. An eriew f the methd, details n the implementatin f bndar cnditins, and reslts fr a benchmark scattering prblem and iscs flw er a circlar clinder are presented. The reslts shw this methd cnerges t a time accrate peridic cnditin sing a ariet f inpts fr the mean flw field, and nder sme circmstances ma cnerge sbstantiall faster t the peridic state cmpared t the same cde sing a standard Naier-Stkes frmlatin. Intrdctin This paper describes recent deelpments in the se f the Nnlinear Distrbance Eqatins (NLDE) fr the nmerical predictin f nstead flws. This wrk etends the applicabilit f the NLDE bend the preis wrks b Mrris et al.,,3, Li et al. 4, and Li and Lng 5. These earlier wrks sed the NLDE fr either an iniscid wall-bnded flw, r free shear flws. In this paper, the NLDE methd is applied t wallbnded, iscs flws. Reslts fr nstead iniscid and iscs, laminar test cases are reprted here. Preis sccess sing the NLDE inled the simlatin f jets and iniscid wall-bnded flws. Mrris et al. implemented the NLDE in the sltin f jet nise. In this wrk the distrbances were assmed iniscid. A finite difference methd n a strctred grid was sed and the cde was implemented n parallel hardware. This wrk was fllwed b frther research cnsidering bth circlar and rectanglar jets, as well as incrprating a sb-grid scale mdel. Cmparisns with eperimental data shwed the methd t accratel predict radiated nise leels t within 5 db. Mre recent calclatins 6 hae impred the abslte accrac f the predictins. In a lw-speed applicatin, the airwakes behind ships were stdied sing the NLDE 4. This simlatin was als cndcted n a strctred grid fr iniscid flw, and sccessfll predicted the wake strctre behind a landing helicpter aide (LHA) and a generic frigate fr aris wind cnditins. Applicatin f the NLDE t the aerdnamics and aeracstics f mlti-hle engine inlet liners 5 als demnstrated the ersatilit f this frmlatin. The mst crrent effrts sing NLDE inclde additinal jet nise stdies b Scheidegger et al. 6, se f large-edd simlatin with NLDE fr wall-bnded flws b Chczewski et al. 7, and a nn-cnseratie NLDE frmlatin b Lng 8. Sezer-Uzl and Lng 9 hae implemented a splitting methd related t NLDE fr calclating the rte strctres in wakes. Cmplete details f the NLDE are gien b Mrris et al.. An eriew is gien here. The methd is based n the traditinal splitting f the flw ariables int a mean cmpnent and a pertrbatin abt that mean. The mean cmpnent ma be btained frm an traditinal CD cde. In the case f the laminar tests cndcted here, the Naier-Stkes eqatins are sled sing the standard Jamesn, Schmidt, and Trkel finite lme frmlatin 0 t pride the mean. When trblence is inled, a cde priding the Renlds (time) aeraged reslts can be sed. The mean flw sltin is then inpt int the NLDE eqatin set t sle directl fr the pertrbatins. The NLDE sler sed here is a mdified ersin f the same Jamesn stle finite lme cde sed t btain the mean. Of particlar interest in this paper is the cmparisn f the NLDE sltins fr different tpes f mean flw inpt. There are seeral ther adantages f this decmpsitin methd. The magnitde f the pertrbatins sght that ield aeracstic infrmatin are ften seeral rders f magnitde smaller than the mean qantities. B sling fr the pertrbatins in a separate step, the nmerical rnd-ff errrs affecting * Dctral Candidate, LTC, United States Arm Prfessr, Assciate ellw, AIAA. Being / A.D. Wellier Prfessr, Assciate ellw, AIAA Cpright 000 b Hansen, Lng, and Mrris. Pblished b the American Institte f Aernatics and Astrnatics, Inc. with permissin American Institte f Aernatics and Astrnatics

2 AIAA American Institte f Aernatics and Astrnatics the acstic sltin are redced. When the mean flw is btained as described abe, it atmaticall satisfies all wall and farfield bndar cnditins. This allws the bndaries dring the NLDE sltin t be treated sing highl effectie radiatin and tflw cnditins deelped fr Cmptatinal Aeracstic (CAA) applicatins. The cntaminatin f aeracstic infrmatin frm reflectins at the cmptatinal bndaries is minimized as a reslt. The NLDE apprach als allws the se f efficient implicit r mltigrid CD schemes fr the mean flw, and a higherrder CAA algrithm fr the nstead flw. Gerning Eqatins The NLDE eqatins are btained frm the fll, time-dependent, Naier-Stkes eqatins. In tw dimensins the ma be written in the frm: S R G t q () The cnsered ectr ariable is gien b: e q () where is the flid densit, and are the elcit cmpnents, and e is the ttal energ per nit lme f flid defined as ( ) p e γ (3) where p and γ are the pressre and specific heat rati, respectiel. Eqatin 3 assmes the ideal gas law is alid. The iniscid fles, and G, and the iscs fles, R and S, are the same cnsered ariables defined b Mrris et al. Each ariable (densit, elcit cmpnents, energ, and pressre) is decmpsed int a mean and a pertrbatin and sbstitted int eqatin. The reslting set f eqatins are the same as a cnentinal Renlds decmpsitin, ecept the are nt time aeraged. The terms inling prel mean flw, btained frm Renlds aeraging, are instead placed n the right-hand side f the eqatins. All terms cntaining nstead cmpnents are placed n the lefthand side f the eqatins. In this frm the are called the Nnlinear Distrbance Eqatins (NLDE) and take the frm (in tw dimensins): ( ) ( ) ( ) ( ) n n n n Q S S R R G G t q (4) The terms nder the time deriatie and first spatial deriatie ma be written as: e q (5) ) ( ) ( p e p e p (6) ) ( p e n (7) The right-hand side f eqatin 4 cnsists f: S R G t q Q (8) The fl terms assciated with G, R, and S n the lefthand side can be similarl epanded. In this frm, q becmes a new ectr f cnsered ariables,, G, R, and S cntain linear pertrbatin terms, n,g n, R n, and S n cntain nnlinear terms. The right-hand side terms defined in Q (eqatin 8) are the same as the Naier-Stkes terms with the mean (sbscript "") ariable sbstitted fr the ttal ariable. Preis papers n the NLDE assmed the time deriatie f the mean qantities (q /t) was zer. These earlier references defined Q as S R G Q (9) The crrent effrt retains the time deriatie f the mean qantities (q /t) in the definitin f Q in an attempt t braden the range f mean flws that ma be stdied as inpts t the NLDE frmlatin. Specificall, this admitted the pssibilit f sing a "snapsht" f a time accrate, instantanes, flwfield as an NLDE mean

3 AIAA Mean Inpt Tpe Cnerged, Stead G R S q t Sm f les Q Ntes Time-Aeraged N! q q i N! i reestream! q q! n mean flw Instantanes!!! q q, t ( ) 3 American Institte f Aernatics and Astrnatics sltin reqired has nt prdced crrect sltins Q Cnerged, RANS 0 0 cntains 0 Renlds stresses Table. Interpretatin f Eqatin 8 fr Different Mean lw Chices inpt. While the q /t term is generall nn-zer in a time accrate simlatin, the sm f q /t with the fles mst ield Q 0. It was initiall assmed that this case wld be as effectie as sing a cnerged sltin fr the mean where bth Q and q /t are zer. In massiel separated flw cases, ne ften cannt btain a stead-state sltin becase f the inherent nsteadiness. In this eent, anther ptin fr btaining the NLDE mean inpt is t nmericall cmpte the time-aerage f the time accrate simlatin. Table smmarizes the tpes f mean flw cnsidered in this std, as well as a tpical case inling flw with trblence. The term "mean flw" is sed interchangeabl with mean inpt in this paper. Eer mean inpt t the NLDE is nt necessaril a tre mean flwfield, the freestream case being an etreme eample. In an case where Q des nt eqal zer, the right-hand side f eqatin 4 acts as a srce term fr the nsteadiness. The instantanes flwfield des nt appear t be an acceptable chice. N phsicall crrect NLDE sltins were btained sing this as a mean inpt. The eact reasns fr this are still being inestigated. The mean cmpnents are prided b an alidated CD cde. Using a time-aeraged mean reqires mre ttal (mean flw cde pls NLDE cde) cmptatin time, bt is inclded becase f the pssibilit that the mean flw sler will nt cnerge. It is als pssible that the mean flw field ma nl be partiall cnerged, sch as the smmetric flwfield abt a circlar clinder befre nsteadiness begins. Bth f these pssibilities, as well as sing a freestream flw as mean inpt, are addressed in this std. The last rw f table represents a case nt cnsidered in this std. The cnerged Renlds Aeraged Naier-Stkes (RANS) case assmes a steadstate sltin can be btained fr a flw inling bth mean nsteadiness and nsteadiness de t trblence. In cntrast t the laminar cnterpart in rw f a cnerged, stead sltin, Q des nt eqal zer fr the cnerged RANS case. This is becase the RANS eqatins inclde nt nl the terms in eqatin 8, bt als the Renlds stresses, which are nt accnted fr in the definitin f Q gien here. Nmerical Details Spatial and Tempral Discretizatin The NLDE eqatins are discretized sing the finite lme methd f Jamesn, Schmidt, and Trkel 0. This classic methd ses a blend f secnd and frth rder dissipatin terms fr stabilit. In the present wrk, the dissipatin peratr is applied in an analgs manner t the ectr f cnsered ariables in eqatin 6. The discretized dmain in each test case ses an "O grid

4 AIAA cmpsed f qadrilateral cells. A standard 4 th rder Rnge-Ktta integratin is sed t eplicitl adance the sltin in time. Bndar Cnditins The bndar cnditins b Tam and Dng are sed fr the farfield bndaries. Each cell n the bndar is ealated as either an inflw r tflw bndar. r sbsnic cnditins, an inflw bndar ma be classified as a radiatin bndar, where the interir f the cmptatinal dmain affects the bndar thrgh pressre pertrbatins nl. With a time inariant mean flw the radiatin bndar cnditins f Tam and Dng ma be written as V ( r, θ) t where V is defined b V! r r 0 p ( r θ) V eˆ ( a ( ) ) V eˆ /, r θ! (0) () and r is the distance frm the bndar cell center t the rigin f the crdinate sstem. In eqatin, a is the lcal speed f snd, V! is the mean elcit ectr, and e r and e θ are nit ectrs in the radial and tangential directins, respectiel. It is assmed that the rigin f the crdinate sstem is in the center f the dmain. Tam and Dng reprt the perfrmance f these bndar cnditins t be insensitie t the rigin as lng as the srce f the pertrbatins is nt lcated t clse t the bndar. Eqatin simpl accnts fr the speed f acstic distrbances (prpagatin speed pls mean flw) alng an θ directin. Otflw bndaries ma eperience an cmbinatin f pressre, rticit, r entrp distrbances. The nmerical cnditin frm Tam and Dng fr pressre n these bndaries is p 0 V ( r, θ ) t r r and fr densit and elcit pertrbatins are ()! p! V ( ) ( ) V p t a t! V ( ) ( p) t! V ( ) ( p) t (3) Nte that eqatin is the same as the radiatin cnditin fr pressre in eqatin 0. Eqatins 0-3 are applied t each farfield bndar ghst cell at each time step. The bndar eqatins are discretized sing a first rder backward difference in time, and a secnd rder backward difference fr the spatial deriaties. The reslting finite difference frmla is an eplicit, algebraic epressin fr the pertrbatin ale in the ghst cell. This ghst cell ale is then sed t calclate the apprpriate ales f the cnsered ariables in eqatin 6. Nte that nl the bndar cnditin fr the pertrbatins is applied. The mean flw will atmaticall satisf the bndar cnditins. The wall bndar cnditins enfrce zer elcit and zer pressre gradient at the wall. These cnditins are applied nl t the pertrbatins, becase the mean flw alread satisfies the wall bndar cnditins. Mean lw The mean flw is btained frm the baseline finite lme cde (LOW). A separate mean flw sltin is nt needed fr the scattering test becase the flid is qiescent. The clinder test cases all se the same grid fr bth the mean flw sltin and the NLDE sltin. In general, the grids need nt be the same, and shld nt when er fine iscs grids are needed. The mean flw ma be determined n a carse grid sing as little cmptatin time as pssible, and then interplated t the finer mesh sed in NLDE. Since nne f the phsics f the Naier-Stkes eqatins is remed r apprimated in frming the NLDE eqatins, it was hpthesized that an flw field cld be sed as the mean and the NLDE pertrbatins shld adjst t pride the crrect ttal sltin (mean pls pertrbatin). Using the freestream ales f all flw ariables as the mean flw field is tested t inestigate this hpthesis. Since a nifrm freestream bisl des nt satisf wall bndar cnditins, a simple adjstment t their applicatin is reqired. Instead f appling the wall cnditins t jst the pertrbatins, the are enfrced n the ttal flw. The apprpriate pertrbatins at the wall can then be calclated based n the identit that the ttal flw eqals the freestream ale pls the pertrbatin ale. 4 American Institte f Aernatics and Astrnatics

5 AIAA Reslts and Discssin Scattering frm a Circlar Clinder The first test case is the scattering f a Gassian plse b a circlar clinder, selected frm the nd Cmptatinal Aeracstics Wrkshp n Benchmark Prblems (Prblem, Categr ). The analtical sltin is prided b Krbatskii 3. This case is chsen t ealate the abilit f the NLDE algrithm t accratel prpagate waes, ensre bth the farfield and wall bndar cnditins are effectiel implemented, and fr direct cmparisn f the NLDE sltin with the baseline finite lme cde. The prblem descriptin frm reference is smmarized here. igre diagrams the prblem. igre. Scattering Benchmark Prblem All length references are dimensinless, based n clinder diameter. The center f the tw-dimensinal Gassian plse is lcated fr nits frm the clinder center. The bserer lcatins (pints A, B, and C) are fie nits frm the clinder center as shwn. The anglar psitins f pints A, B, and C are at θ 90, 45, and 80 degrees, respectiel. The initial spatial distribtin f the pertrbatin pressre is gien b at the farfield. The first frame is the pressre field after Pressre Time igre. Scattering Prblem Sltin NLDE: Pint A; NLDE Pint B; O NLDE : Pint C ) ( Eact ; 000 time steps. The peak f the plse has jst reaches the clinder srface and part f the eterir bndar. The nn-dimensinal time is After 4000 time steps (t 7.64), the plse has passes the clinder, generating a scattered wae. The farfield bndar ehibits negligible reflectins as the plse eits the dmain. t3.84 t 7.64 ( 4) p ep ln. (4) 0. The maimm amplitde f the pertrbatin is ne. There is zer mean flw fr this prblem. The grid sed fr this case is 000 pints in the circmferential directin and 600 pints in the radial directin. The farfield bndar is placed eight diameters frm the clinder srface. A CL nmber f ne is sed in the time stepping. igre plts the pressre-time histr (nndimensinal) sltins fr the benchmark scattering prblem sing the NLDE cde. The LOW cde prdced the same reslts as the NLDE cde. The diminished amplitdes cmpared t the analtical respnse were anticipated since the baseline algrithm is nl secnd rder accrate. igre 3 illstrates the effectieness f the Tam and Dng bndar cnditins 5 igre 3. Gassian Plse Impinging n arfield Bndar Circlar Clinder lw The secnd series f tests inle flw past a circlar clinder. Of ke interest is the accrac f the nstead rte shedding behair and the time reqired t generate an NLDE sltin cmpared t btaining the sltin with the baseline finite lme cde. An O- grid sing 30 eqall spaced pints in the circmferential directin and 65 pints in the radial directin is sed. The grid is stretched in the radial directin. The distance f the first cell center nrmal t the clinder srface is /500 f the clinder diameter. The largest cell aspect rati is apprimatel three. American Institte f Aernatics and Astrnatics

6 AIAA Cnditins f Renlds nmber 000, and Mach nmber f 0.3 are sed. While the bndar laer n the clinder srface is still laminar at this Renlds nmber, the wake has alread transitined t trblence in the real, threedimensinal, flw. Zer secnd rder artificial iscsit is sed. Where standard frth rder artificial iscsit is referred t in sbseqent sectins, the cefficient ale f /56 riginall recmmended b Jamesn et al. 0 is sed. All cmptatins are carried t n a single nde f a PC clster with a 400 Mhz prcessr speed. Additinal details f the PC clster sed are dcmented b Mdi and Lng 4. Using the standard ale fr frth rder artificial dissipatin, initial tests f the clinder demnstrated a resistance t the nset f peridic rte shedding when a CL nmber f ne was sed. Tw methds were sed t assist the initiatin f nsteadiness. The first was the intrdctin f freestream trblence at the inflw bndar. This methd was nt sccessfl. Up t 5 % freestream pertrbatins were sed at the inflw bndaries with n impact tward initiating peridic rte shedding. Breer 5 btained similar reslts when intrdcing freestream trblence fr sqare clinder cmptatins. As Breer states, this is de in part t the larger cells near the bndaries that tend t damp t the distrbances that are intrdced. The distrbances d reach the clinder, bt case lift pertrbatins nl n the rder f 0-5. The secnd methd was t intrdce an asmmetric distrbance clse t the clinder. After the mean flw had sfficientl deelped er the clinder, the streamwise elcit in the pper half-plane is mltiplied b.05 fr ne time step. This is similar t the apprach b Hardin and Lamkin 6 wh sed a rticit pertrbatin t initiate nsteadiness. In the case f the LOW cde, the clinder frces begin respnding immediatel after this pertrbatin is intrdced, slwl bilding t their peridic states. r the NLDE cases, the frth rder artificial iscsit cefficient was redced t 0% f its standard ale. This ale was determined thrgh trial and errr. This ale seems reasnable gien that the mean flw can nl generate distrbances thrgh interactins with the pertrbatins, which are mch smaller. This techniqe f "starting" the nsteadiness made it mre difficlt t assess the actal time needed t reach peridicit. Eperimentatin with higher CL nmbers, clser t the limits f stabilit fr the frth-rder Rnge-Ktta time integratin, reealed that self-starting cld be achieed witht affecting the final sltin. A CL nmber f.7 is sbseqentl sed in all cases reprted belw. Reslts fr the peridic state f rte shedding are gien in Table. These reslts are btained er ten perids f shedding. The clmn headings indicate the cde sed. The NLDE cases differ b hw the mean is btained. NLDE ses a mean that is btained frm the LOW cde b aeraging er ten cmplete ccles f shedding. NLDE ses a mean btained b psed-time marching the LOW cde ntil large, stable, recirclating znes were btained in the clinder wake. The NLDE3 case ses a freestream flwfield as the mean. 6 American Institte f Aernatics and Astrnatics NLDE NLDE NLDE LOW 3 C l min C l ma C l ag C d min C d ma C d ag θ s St Table. Time-Aeraged Reslts fr Circlar Clinder Test sing LOW cde and NLDE cde with Different Mean Inpts. igre 4 plts the streamlines fr the mean flw btained b time aeraging. The lack f recirclating znes in the clinder wake is tpical f tw-dimensinal sltins where three-dimensinal phsics plas a rle in the real flw. Breer 5 demnstrated this phenmenn when sling the flw er a circlar clinder sing bth a D and 3D simlatin, bth witht trblence mdeling. Een when trblence mdeling is applied, a time-aerage sltin similar t figre 4 can reslt de t weaknesses f the trblence mdel. Sch a case is discssed b Rdi 7. NLDE ses a mean btained b psed-time marching the LOW cde with a CL.0. The LOW cde was rn ntil the cnergence was dwn fie rders f magnitde. igre 5 shws the mean flw streamlines btained sing a psed-time marching sltin. The mean flw fr this case has C l , C d 0.77, and θ s The same streamline ales are sed in bth figres 4 and 5. The NLDE3 case ses freestream cnditins as the mean flw. This reqires the wall bndar cnditins t be applied t the ttal sltin since a freestream flw des nt satisf the wall bndar cnditins. arfield bndar cnditins are satisfied b a freestream mean. In table, the angle f separatin is θ s and is determined b the pint where the time-aeraged cefficient f frictin becmes zer. The Strhal

7 AIAA better than the LOW cde. NLDE3 prides impred estimates in each f the measred parameters, with the imprement in Strhal nmber being the mst significant. Y X igre 4. Time-aeraged streamlines fr the mean flw sed in Case NLDE. C l nmber is based n the perid f the lift cefficient ccle. The time step, reqired b stabilit cnsideratins, is The generall accepted eperimental ales 8 fr Re000 are Strhal nmber , C d.0, and θ s The drag cefficient was epected t be in errr becase f the relatiel carse discretizatin sed net t the srface. Despite this, the data cmpares well with a nmerical eperiment cndcted b C et al. 9 The lift Y X igre 5. Streamlines fr the mean flw btained b psed-time marching sed in Case NLDE. and drag etremes are apprimatel the same as reference 9 and the Strhal predictin fr the NLDE and NLDE3 cases are significantl better. C et al. 9 sed the cde CL3D and the distance t the first pint ff the clinder srface was an rder f magnitde smaller than the present std. The reslts shwn here are, therefre, cnsidered a alid basis fr cmparisn between the baseline LOW cde and the NLDE ersin. NLDE prides slightl impred drag estimates, a slightl less accrate separatin angle, bt a Strhal nmber that is within the eperimental range. NLDE prides drag estimates that are less accrate, a mean lift that is clser t zer, a separatin angle that is er accrate, and a Strhal nmber that is nl marginall igre 6. Lift Cefficient Histr fr NLDE and NLDE ( NLDE; NLDE ) 7 American Institte f Aernatics and Astrnatics Iteratin NLDE represents the case where cnergence is nattainable, s a time aerage is sed t represent the mean. In this eent, the Q term behaes as a srce term and mst be retained in the eqatin set. If a well cnerged mean sltin can be btained as in NLDE, time aeraging is nt reqired and Q 0. igre 6 plts the lift cefficient histr fr these tw cases. The NLDE cnerges t a peridic state in fewer iteratins, bt as alread nted is nt as accrate in either drag cefficient r freqenc f shedding. NLDE begins nsteadiness frm the start f the NLDE sltin, bt des s slwl er abt 40,000 iteratins befre reaching a peridic state. The NLDE3 case ses the same mean as the initial cnditin in the LOW cde, and a direct cmparisn f the nmber f ccles needed t btain peridicit can be made. Cmparisns f lift and drag cefficient cnergence are shwn in figres 7 and 8. The ariable "is4" in the legend refers t the fractin f standard frth rder artificial iscsit sed. The NLDE3 rn sing is4 0. is bsered t cnerge t the peridic state mch mre qickl than the LOW case sing the standard frth rder iscsit (is4.0). T make sre this is nt a damping effect f artificial iscsit, the LOW cde was als rn at 60% and 40% f the standard artificial dissipatin. The LOW cde becmes nstable when 0% iscsit is attempted. The cmparisns in figres 7 and 8 shw that cnergence t the peridic state is delaed in the LOW cde when smaller ales f artificial iscsit are sed. This reslt was nepected. Once the peridicit des begin fr the lwer ales f artificial iscsit, the peaks are mch

8 AIAA mre erratic, which is epected becase f the smaller srface. The rticit strctre is er cmple near the clinder srface C l igre 9. Abslte Vrticit Cntrs Dring Peridic Shedding Iteratin igre 7. Lift Cefficient Histr Cmparisn ( NLDE3; LOW(is4.0); LOW(is40.6); LOW (is40.4) ) damping. The NLDE3 case btains a fll peridic state b iteratin 05,000 while the peaks f all the LOW cases still flctate past 50,000 iteratins. The LOW case sing standard iscsit reaches cnsistent peridicit at iteratin 80,000, while the 60% and 40% cases fail t cnerge t a peridic state een at 00,000 iteratins when the calclatin is terminated. C d Iteratin igre 8. Drag Cefficient Histr Cmparisn ( NLDE3; LOW(is4.0); LOW(is40.6); LOW (is40.4) ) igre 9 is a cntr plt f the abslte ales f rticit near the clinder btained frm an NLDE3 sltin. The shed rte detaches between tw and three clinder diameters dwnstream f the clinder Table 3 smmarizes the differences between the aris mean flw inpts and the crrespnding NLDE sltin as a percentage errr relatie t the eperimental ales. In the case f the freestream (NLDE3), the aerage C d data is listed as nt applicable (NA) becase the freestream mean des nt satisf the clinder wall bndar cnditins. Psitie nmbers in the drag, separatin angle, and Strhal nmber rws indicate the nmerical ale is abe the eperimental ale. Negatie nmbers indicate the nmerical ale is belw the eperimental ale. The NLDE case has the strngest crrelatin between accrac f the mean flw and the accrac f the crrespnding NLDE sltin. The time-aeraged mean sed in NLDE cntains mre infrmatin abt the flw than the ther tw cases. The NLDE mean cntains n infrmatin abt the freqenc f nsteadiness, s NLDE is epected t ield a mre accrate Strhal nmber. The NLDE mean cntains a mch mre accrate separatin angle than the ther cases, and prides the best NLDE sltin fr this parameter. The NLDE3 case appears t ilate the hpthesis that the mre accrate the mean, the mre accrate the NLDE sltin. The NLDE3 mean cntains n infrmatin abt the tre mean flwfield, et the NLDE sltin gies a Strhal nmber within the spread f eperimental data, a drag predictin that is better than NLDE, a separatin angle almst as gd as NLDE, and the mst accrate lift sltin. This sggests that priding a mean flw with "netral" infrmatin as in NLDE3 ma be better than sing a mean flw that cntains inaccrate infrmatin. 8 American Institte f Aernatics and Astrnatics Cnclsins The first test f the NLDE algrithm that incldes the effects f iscsit with a wall-bnded flw has been cndcted here. These tests are als the first effrt t inclde the iscs pertrbatin terms in an NLDE simlatin. The reslts are specific t the test cases eplained and ma nt be generalized ntil frther testing is cmpleted. r the nstead, iniscid test case,

9 AIAA NLDE NLDE NLDE3 Ep. Mean Inpt NLDE Sltin Mean Inpt NLDE Sltin Mean Inpt NLDE Sltin C d ag NA 55.6 θ s St Table 3. Percent Differences Relatie t Eperimental Vales n adantages f the NLDE algrithm are discernable er the baseline cde. r the nstead, iscs case the ale f frth rder iscsit is fnd t be an imprtant factr in btaining the nset f peridic shedding as well as achieing a cnerged peridic state. The NLDE were fnd t perate sitabl sing nl 0% f the standard frth rder artificial dissipatin. The se f three different tpes f mean flw were stdied as inpts t the NLDE frmlatin. Mied reslts were btained when sing either a time-aeraged r a psed-time marched stead-state sltin fr the mean flw. The benefits f increased accrac btained in ne parameter were generall ffset b redced accrac in ne r mre ther parameters. Using a mean eqal t the freestream prides the mst intriging reslts, priding gd accrac in all measred parameters, as well as cnerging t a peridic state sing nl 58% f the iteratins needed fr the LOW cde. References Mrris, P.J., Lng, L.N., Bangalre, A., Wang, Q., A Parallel Three-Dimensinal Cmptatinal Aeracstics Methd Using Nnlinear Distrbance Eqatins, J. f Cmptatinal Phsics, Vl. 33, 997. Mrris, P.J., Lng, L.N., Lckard, D.P., Wang, Q., Nmerical Predictin f High Speed Jet Nise, AIAA Paper , Mrris, P.J., Lng, L.N., Scheidegger, T.E., Wang, Q., Piln, A.R., High Speed Jet Nise Simlatins, AIAA Paper 98-90, Li, J., Lng, L.N., Mdi, A., Higher Order Accrate Sltins f Ship Airwake lw ields sing Parallel Cmpters, AGARD Cnference, Amsterdam, Li, J., Lng, L.N., Direct Aeracstic and Aerdnamic Simlatin f Mlti-Hle Engine Liners, AIAA Paper , Scheidegger, T.E., Mrris, P.J.,, and Lng, L.N., Jet Nise Simlatins fr Circlar and Rectanglar Nzzles, AIAA Paper , Chczewski, T, Mrris, P.J., and Lng, L.N., Large- Edd Simlatin f Wall Bnded Shear lw Using the NLDE, AIAA Paper , Lng, L.N., A Nn-Cnseratie Nnlinear lwfield Splitting Methd, AIAA Paper , Sezer-Uzl, N. and Lng, L.N., High Accrac Wake and Vrte Simlatin Using a Hbrid CD/DV Methd, AIAA Paper , Jamesn, A., Schmidt, W., and Trkel, E., "Nmerical Sltin f the Eler Eqatins b inite Vlme Methds Using Rnge-Ktta Time Stepping Schemes", AIAA Paper 8-59, 4 th lid and Plasma Dnamics Cnference, 98. Tam, C. K. W., and Dng, Z., "Radiatin and Otflw Bndar Cnditins fr Direct Cmptatin f Acstic and lw Distrbances in a Nnnifrm Mean lw", CEAS/AIAA Paper ,995. NASA Cnference Pblicatin 335, nd Cmptatinal Aeracstics Wrkshp n Benchmark Prblems, Langle Research Center, Hamptn, VA, Jne Krbatskii, K., "Analtical Sltins f the Categr, Benchmark Prblems and ", NASA Cnference Pblicatin 335, nd Cmptatinal Aeracstics Wrkshp n Benchmark Prblems, Langle Research Center, Hamptn, VA, Jne American Institte f Aernatics and Astrnatics

10 AIAA Mdi, A. and Lng, L.N., Unstead Separated lw Simlatins Using a Clster f Wrkstatins, AIAA Paper , Breer, M., "Large Edd Simlatin f the Sb-Critical lw Past a Circlar Clinder: Nmerical and Mdeling Aspects", Internatinal Jrnal fr Nmerical Methds in lids, Vl. 8, N., Dec. 998, pp Hardin, J. and Lamkin, S., "Aeracstic Cmptatin f Clinder Wake lw", AIAA Jrnal, Vl., N., Janar Rdi, W. "On the Simlatin f Trblent lw Past Blff Bdies", Prceedings f the st Internatinal Smpsim n Cmptatinal Wind Engineering, Agst Zdrakich, M.M., lw Arnd Circlar Clinders, Ofrd Uniersit Press, Ofrd, C, J. Rmse, C., Brentner, K., and Ynis, B. "Cmptatin f Snd Generated b Viscs lw Oer a Circlar Clinder", NASA TM 0339, March, American Institte f Aernatics and Astrnatics