Leading Edge Vortex Flow Computations and Comparison with DNW-HST Wind Tunnel Data )

Transcription

1 (SYA) 14-1 Leadng Edge Vortex Flow Computatons and Comparson wth DNW-HST Wnd Tunnel Data ) F.J. Brandsma, H.S. Dol Natonal Aerospace Laboratory NLR Appled Aerodynamcs Dept. P.O. Box BM Amsterdam The Netherlands A. Elsenaar Natonal Aerospace Laboratory NLR Anthony Fokkerweg 1059 CM Amsterdam The Netherlands J.C. Kok Natonal Aerospace Laboratory NLR Mathematcal Methods and Models Dept. Anthony Fokkerweg 1059 CM Amsterdam The Netherlands Summary Computatons are presented for the vortcal flow around a sharp-edged cropped delta wng wth 65 o leadng edge sweep usng a computatonal method based on the Reynolds-averaged Naver-Stokes equatons. It s demonstrated that turbulence modellng plays a crucal role n the ablty to capture the vortcal structures. Standard one- and twoequaton turbulence models need correctons for vortcal flows n order to avod over-predcton of the levels of turbulent vscosty nsde vortex cores. In ths paper two types of modfcatons to the two-equaton k-omega turbulence model are nvestgated to overcome ths problem. One modfcaton conssts of lmtng the producton of turbulent knetc energy n the k-equaton, whereas the other modfcaton s amed at ncreasng the producton of dsspaton n the dsspaton equaton (omega equaton); omega represents the dsspaton of turbulent knetc energy. The computatonal results at the condtons M = 0. 85, α = 10o, and Re = 9 106, are compared wth detaled expermental surface and feld data obtaned from a seres of wnd tunnel tests n the DNW-HST at NLR. The comparsons show that the modfcaton whch ncreases the producton term for the dsspaton rate of turbulent knetc energy n the omega-equaton produces the best results when t comes to capturng the vortex core n a realstc way. The proposed modfcaton s n lne wth other approaches found n the lterature for one-equaton turbulence models. c R ) Part of ths nvestgaton has been carred out wthn the framework of WEAG-TA15 under a contract awarded by the Netherlands Agency for Aerospace programs (NIVR) for the Dutch Mnstry of Defence, contract number 07601N Paper presented at the RTO AVT Symposum on Advanced Flow Management: Part A Vortex Flows and Hgh Angle of Attack for Mltary Vehcles, held n Loen, Norway, 7-11 May 001, and publshed n RTO-MP-069(I).

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3 (SYA) 14- Introducton Analyss of vortex domnated flows s of great mportance for the assessment of the aerodynamcs, the stablty and control, the aero-elastcs and the structural dynamcs of fghter arcraft. The mportance of vortex flow to fghter arcraft manfests tself for example as follows: - Aerodynamcs: manoeuvrng capabltes depend crtcally on vortex-nduced lft; maxmum vortex-nduced lft s affected by vortex stablty - Stablty and control: the roll stablty of complete fghter arcraft can heavly depend on asymmetrc vortex breakdown - Aero-elastcs: unsteady vortex flow can affect the flutter speed and the level of lmt cycle oscllatons - Structural dynamcs: fatgue lfe of tal surfaces and ventral fns depends sgnfcantly on the unsteady aerodynamc energy nput to the vbratons of these surfaces; ths energy nput can be due to vortces. These observatons motvate the nvestgaton of the ablty of CFD codes to capture the detals of vortcal flows around generc confguratons lke delta wngs. Prevous work (see [1], and []) shows that the accuracy of CFD predctons for ths type of flow and the ablty to arrve at a so-called grd-converged soluton rely heavly on the ablty to represent the turbulent structure of the vortces. Crucal for accurate vortcal flow predctons wth CFD codes based on the Reynolds-averaged Naver-Stokes (RaNS) equatons s the turbulence model used for the computatons (see [4]). One- or two-equaton turbulence models can potentally reach the requred mnmum level of modellng. In ths paper, new computatons are presented for the turbulent vortcal flow around a 65 o swept cropped delta wng-body confguraton wth sharp leadng edge usng a method based on the RaNS equatons [4] employng the Wlcox k ω two-equaton turbulence model [3]. It s well known that present-day one- and two-equaton turbulence models requre specal dampng n the vortex cores to represent the effects of flud rotaton (vortcty) as well as system rotaton on turbulence. See for example Spalart & Shur [5] and Dacles-Maran & Zllac et. al. [6] who proposed modfcatons to one-equaton models, and Hanalć & Launder [7] for modfcatons to the ε -equaton of the two-equaton k ε model. In ths paper two types of modfcatons of the k ω two-equaton turbulence model are nvestgated to mprove ts behavour for vortcal flow smulatons. Essentally, these modfcatons consst of ether lmtng the producton of turbulent knetc energy or ncreasng the dsspaton rate of turbulent knetc energy n vortex cores. The results of the computatons are compared wth detaled expermental data for the sharp-edged delta wng confguraton obtaned from a seres of wnd tunnel tests (see [9]). From ths comparson the effectveness of the proposed modfcatons to the turbulence model s assessed. The k-ω model and modfcatons for vortcal flows The Reynolds-averaged Naver-Stokes (RaNS) equatons are solved for the conservatve (Favre mass-averaged) varables densty, ρ, the momentum vector, ρ u, and the total energy, ρ E. In the present study the Wlcox k ω turbulence model s consdered [3], wth the addtonal cross-dffuson term that has been ntroduced by Wlcox to decrease the dependency of the solutons on the free-stream value of ω. The transport equatons for the turbulent

4 (SYA) 14-3 knetc energy, k, and the specfc turbulent dsspaton rate,ω, for the Wlcox model ncludng the so-called crossdffuson term, to be solved along wth the RaNS equatons, can be wrtten as (usng the summaton conventon) ( ρ k) ( ρ ku + t ( ρω) ( ρω u + t ) ) = τ R u ρ = α τ ω R * β ρ ω k + u * β ρ ω + * k [( µ + σ µ T ) ], (1a ) x ω [( µ + σµ T ) ] + σ d x ρ k max ω ω,0, (1b ) wth t the tme, x the poston vector and µ the molecular dynamc vscosty. The Reynolds-stress tensorτ R s modelled usng the Boussnesq hypothess τ R = 1 3 u ρ kδ, 3 k µ T S k () wth S the rate-of-stran tensor S = 1 ( u + u ). The turbulent (eddy) vscosty µ T s defned by * ρ k µ T = α. (3) ω The frst two terms on the rght hand sdes of both the k -equaton (1a) and the ω -equaton (1b) represent producton and dsspaton of k andω, respectvely. If we defne the rate-of-stran tensor wth zero trace ~ as S = S 1 Dδ 3, where the dlataton s defned by D = u x, and f we defne the magntude of ths rate-ofstran tensor as S = {S } 1 S, then the producton terms can be wrtten ~ ~ ~ as P P k ω u R = τ ω R = α τ k ~ = µ S u T ρ kd, 3 αω = P k k * ~ = αα ρ S αρω D. 3 (4) * The dsspaton term n the k -equaton s usually denoted by ρ ε = β ρ ω k. The last term on the rght hand sde of the ω -equaton n (1a) s the so-called cross-dffuson term mentoned above. The values of the coeffcents are taken as the hgh Reynolds number lmts for the k ω model as presented n [3]: α * = 1, α = 0.5, β * = 0.09, β = 0.075, σ * = 1, σ = 0.6, σ = 0.3. (5) It s shown by Kok (see [8]) that wth the cross-dffuson term ncluded the dependency on free-stream values can be completely removed wth the followng values for the coeffcents of the dffuson terms: σ * = / 3, σ = 0.5, σ d = 0.5, called the TNT set. However, for the computatons presented here, stll the Wlcox set of coeffcents (5) s used. In order to avod unphyscal producton of eddy vscosty n regons of stagnatng flow, the producton of turbulent knetc energy s lmted wth a commonly used standard lmter relatng the maxmum allowable producton to the dsspaton of turbulent knetc energy P = mn{ u, 0ρε}, (6) k P k d

5 (SYA) 14-4 wth P the unlmted producton defned by (4). In the dscusson that wll follow the model as descrbed above wll u k be referred to as standard k ω model n order to dstngush t from the modfed versons dscussed below. It s well known that most one- and two-equaton models produce too much eddy vscosty n vortces causng a far too strong dffuson of vortcty. For the Baldwn-Barth and the Spalart-Allmaras one-equaton models, modfcatons were proposed by Dacles-Maran et al. (see [6]) and by Spalart & Shur (see [5]) to cure ths problem. Both modfcatons use the rato r of the magntude of the rate-of-stran tensor and the magntude of vortcty Ω = Ω } 1 Ω, wth the vortcty tensor 1 ~ Ω = ( u / u / ), r = S / Ω. In shear layers, the velocty { gradent s domnated by the gradent n the normal drecton, so that r 1. In the core of a vortex, the flow approaches pure rotaton, mplyng that r << 1. In the one-equaton models, a PDE for the eddy vscosty s defned. In the modfcatons of Dacles-Maran et al. [6] and of Spalart & Shur [5] the producton of eddy vscosty n ths equaton s essentally modfed by multplcaton wth a functon f(r). Ths functon has the propertes f ( 1) = 1, so that the producton s unchanged n boundary layers, and f ( r) < 1 for r < 1, so that the producton s reduced n vortex cores. The modfed producton term s gven by Pν = C ρν T Ω f ( ) wth C1 a (postve) constant and ν T = µ T / ρ. T 1 r In order to mprove the behavour of the k ω model for vortex domnated flows we have consdered two modfcatons of the source terms dependng on the varable r. The frst modfcaton s an extenson of the lmtng of the k-producton as done n equaton (6) usng the dsspaton term as a lmter but now wth the coeffcent beng a lnear functon of r, Modfcaton 1: { u Pk = mn P,( Ck1 + Ck mn{0, r 1}) ρε}, (7) wth C k1 > 1 and C k > 0 k. In ths way the k-producton s reduced or even turned nto a dsspaton term nsde vortex cores. For boundary layers, however, takng a value of C k1 close to 1 may also result n a reducton of the producton, f the boundary layer s not n equlbrum (balance between producton and dsspaton). For ths type of modfcaton we have nvestgated two choces for the parameters. In both cases C k1 =. 0 s taken whereas the value of the coeffcent of the r -dependent part s set to C k =. 0 and C k = 8. 0, respectvely. In the second modfcaton of the k ω model, we have left the producton term of the k-equaton unchanged (n whch, n fact, the only modellng assumpton s the Boussnesq hypothess), and modfed the producton term of the ω -equaton as follows, ~ Modfcaton : P = αα * ρ max{ Ω, S }, (8) ω whch s equvalent to dvdng the producton term n the ω -equaton by mn{ r,1}, and where a non-zero dlataton has been neglected. In ths way, the producton of ω s ncreased n vortex cores, thus ncreasng the dsspaton of turbulent knetc energy and, as a consequence, a reducton of the producton of eddy vscosty s obtaned. From the k-equaton and the modfed ω -equaton, an equaton for the eddy vscosty can be derved for whch the producton s now of the form P ( 1 α) α * ρ ν = Ω f ( r) ω, wth f ( r) = [ r α max{ r,1}]/(1 α ), ν T / T whch shows some smlartes wth the expressons used by Spalart & Shur [5] and Dacles-Maran et al. [6] for oneequaton turbulence models mentoned above. Furthermore, the second type of modfcaton s qute smlar to the

6 (SYA) 14-5 modfcaton of Hanalć & Launder [7] who propose to ntroduce a term proportonal to k Ω n the ε -equaton of the k ε two-equaton model. However, Hanalć s term has a sgn opposte to the present proposal. Test case As a test case for the present study the flow around a 65 o cropped delta wng wth sharp leadng edge s used. The wng s the same as the one that has been the subect of the Internatonal Vortex Flow expermental study reported n [9]. The balance mounted delta wng model was measured n dfferent wnd tunnels and the expermental data were amed to set up an expermental data base for the valdaton of Euler codes to be used for the predcton of vortcal flow characterstcs n the subsonc and transonc speed regme. In 1988 NLR manufactured a new model wth the same geometry (the sharp edge varant) wth a very dense matrx of pressure taps. Snce then ths model, whch wll be referred to as the WB1-SLE model, has been the subect of fve test programs n the DNW-HST faclty at NLR throughout the years. All fve tests were amed at gettng detaled expermental data to be used for valdaton purposes for CFD codes capturng the characterstcs of varous aspects of vortcal flows. The frst test durng whch detaled surface pressure measurements and surface flow vsualsatons for symmetrc (no sde-slp) subsonc and transonc condtons have been obtaned are reported n [10]. The tests that followed ncluded a flow feld nvestgaton usng a 5-hole pressure probe at a subsonc and a transonc condton, where a number of crossflow planes have been surveyed, and nvestgatons nto asymmetrc flow (sde-slp angle sweeps). Also a flow feld nvestgaton wth the Partcle Image Velocmetry (PIV) technque at a subsonc and a transonc condton, focussng on the angle of ncdence / slp-angle combnatons where transton to vortex breakdown takes place, has been carred out. An overvew of the tests and an analyss of all expermental data obtaned n the dfferent tests are gven n [11]. The basc chordwse wng sectons of the sharp-edged delta wng are defned by the NACA64a005 profle. Between the leadng edge and the 40% chord lne the geometry s changed nto two crcular arcs, defnng the sharp leadng edge. Between the 75% chord lne and the tralng edge the geometry s replaced by a straght-lne blend towards the tralng edge (see Fgure 1). The wng s mounted on an underwng fuselage that has served n the wnd tunnel experments as the support for the nstrumentaton of the model. An mpresson of the complete model s shown n Fgure 1. The flow case consdered n the present study s the transonc flow around the sharp edged delta wng (WB1-SLE) for a Mach number of M = 0. 85, an angle of ncdence of α = 10o, and a Reynolds number based on the root chord, c R, of Re = At these condtons detaled surface pressure measurements are c R avalable from the DNW-HST experments as well as flow-feld data n three cross-flow planes obtaned wth the 5- hole pressure probe as mentoned above. CFD method and computatonal grd Computatons have been performed usng the flow solver ENSOLV, that s part of NLR s ENFLOW system for flow smulatons based on the Euler- or the Naver-Stokes equatons (see [1]). A cell-centred, central dfference, fnte volume scheme s used to dscretze the RaNS equatons n space, where hgh-aspect-raton scalng of the artfcal dsspaton, and a matrx dsspaton formulaton used n surface normal drecton are appled. The turbulence equatons are dscretzed n the same way as for the basc flow equatons, where t should be noted that for the turbulence model equatons a TVD swtch s used n the formulaton of the artfcal dsspaton leadng to a second order TVD scheme for these equatons. It should be mentoned that for the mplementaton of the k ω model the

7 (SYA) 14-6 equatons are reformulated such that nstead ofω a newly ntroduced quantty τ = /( ω + ω ) s used as the second 1 0 turbulence varable, removng the sngular behavour of the soluton at sold walls (see [1]). The turbulence varables k and τ are both set to zero at sold walls. At the nflow parts of the far feld boundares the free-stream values for the turbulence varables are computed from specfed values of the free-stream turbulence Reynolds 6 number and the free-stream dmensonless turbulent knetc energy (set to 0.01 and 10, respectvely). A sutable computatonal grd has been defned usng the doman modeller ENDOMO and the grd generaton program ENGRID, both programs beng part of NLR s ENFLOW flow smulaton system (see [1]). A CO-type topology has been used wth a sngular lne runnng from the apex to the upstream far-feld boundary, where an O-type sngularty occurs at the edges of the wake-cut downstream of the tralng edge towards the downstream far-feld boundary. It has been demonstrated before (see for example [1] and []) that ths type of topology allows for grds that are very well suted for the flow under consderaton, whch s expected to be concal over large part of the delta wng. The computatons have been carred out for the half model due to the symmetrc flow condtons. The farfeld boundary s placed at approxmately 3 root chords from the geometry. The grd conssts of = 1,796,985 grd ponts. A rather fne mesh spacng has been used n spanwse drecton (15 cells dstrbuted over upper wng, lower wng and fuselage) to be able to capture the vortex structures n an accurate way. For the 81 ponts n normal drecton t s checked wth the computed results that about 30 to 35 ponts are located n the boundary layer. Furthermore t has been checked that for almost the entre wng- and fuselage surface the dmensonless (based on the frcton velocty) heght of the frst cell normal to the wall s n the range of 0.5 < y + < An mpresson of the computatonal grd s gven n Fgure. Dscusson of results The pressure coeffcents on the upper wng surface computed wth the NLR mplementaton of the k ω model and ts modfcatons are compared wth expermental data n Fgure 3. The expermental data ndcate a prmary vortex of whch the footprnt s vsble n the regon of hgh sucton and a pressure plateau between the footprnt and the leadng edge. The standard NLR k ω model results obvously do not represent ths stuaton. Only a small regon ndcatng vortex formaton close to the wng apex s vsble (probably caused by a separaton of the locally lamnar flow) after whch the prmary separaton covers the entre regon between the locaton of the sucton rse and the leadng edge. In Fgure 4 the upper surface lmtng streamlnes for the dfferent computatons are compared. All computatonal results show a prmary separaton over the entre leadng edge attachng at smlar postons ( A1 n Fgure 4). The results obtaned wth the modfcatons of the k ω model show a secondary separaton ( S ) underneath the prmary vortex that attaches agan outboard of the secondary separaton lne. Ths secondary separaton, also observed n the experments, s mssng n the results obtaned wth the standard k ω model. Ths s due to the fact that wth the standard NLR k ω model, a large amount of turbulent knetc energy s produced nsde vortces. Ths hgh level of turbulence strongly dffuses the vortcty and dsspates some of the knetc energy assocated wth the swrlng flow component of the vortex. The results obtaned wth the subsequent modfcatons of the turbulence model dffer n the locaton of the secondary separaton, S, and the locaton of the secondary reattachment, A. However, the locaton of the prmary reattachment, A 1, s close to the expermentally observed poston as can be determned from the sectonal pressure dstrbutons presented n Fgure 5 and Fgure 6. It s

8 (SYA) 14-7 observed there that the locaton of the prmary reattachment n the result obtaned wth the standard NLR k ω model s too far nboard as compared wth the expermental data. The locatons for prmary reattachment and secondary separaton shown for the experment are derved from ol flow vsualsatons. From the pressure dstrbutons t can be seen that all the modfcatons consttute a strong mprovement over the standard model. For the modfcaton usng the lmter for the k - producton term wth a coeffcent of C k = 8, however, the reducton of turbulence producton has been exaggerated, resultng n a large over-predcton of the sucton peak. The other two results are close to each other and to the expermental results. In partcular, the wdth of the sucton peak and the pressure plateau between the peak and the leadng edge are predcted well. The man dfference between the Pk - lmter and the Pω - modfcaton approach s the way the pressure dstrbuton changes n downstream drecton. Wth the Pk -lmter approach (wth C k = ), the heght of the sucton peak gradually drops compared to the Pω - modfcaton approach and the expermental data. In order to nvestgate the behavour of the two most promsng modfcatons models further, we have looked at the downstream development of the dstrbuton of the totalpressure-loss Fgure 7 and of the turbulence Reynolds number Re T = ρ k /{ ω µ } Fgure 8. For both modfcatons, the total pressure loss shows a farly strong prmary vortex wth mld secondary separaton. Movng n downstream drecton, the prmary vortex obtaned wth the Pk -lmter approach appears to become more dffuse than the results obtaned wth the Pω - modfcaton approach. Ths s related to the dstrbuton of the turbulence Reynolds number, where we see that for the Pk -lmter modfcaton, the turbulence Reynolds number ncreases strongly n downstream drecton, whle ths ncrease s lower for the P ω - modfcaton. Furthermore, the dstrbuton of the turbulence Reynolds number obtaned wth the Pω - modfcaton shows for each secton a local mnmum n the vortex core. In partcular ths last observaton s mportant, snce t s known from theory that a turbulent vortex can have a lamnar core f the flud rotaton becomes strong enough. In Fgure 9computed total-pressure-losses are compared n detal wth expermental feld data for a cross-flow plane (normal to the free stream drecton) at 90% root chord poston. Agan t s clear that the standard NLR mplementaton of the k ω model generates a far more dffuse vortex. Both modfcatons presented for the k ω model show qualtatvely the same vortex structure as n the experment. Dfferences between the modfcatons of the k ω model become most clear when lookng at the total-pressureloss dstrbuton along a horzontal traverse and a vertcal traverse through the prmary vortex core n these planes. The comparson s made as a functon of the dmensonless dstance from the prmary vortex core for each soluton. In ths way a dfference n locaton of the vortex core n the solutons does not show up and only the dfference n total-pressure-loss dstrbutons wthn the vortex s udged upon. It can be seen that compared wth the experment all modfcatons over-predct the level of total-pressure-loss n the vortex core. However, t s concluded that wth respect to the downstream development of the sze of the vortex as well as for the levels of total-pressure-losses n the regon ust outsde the core the Pω - modfcaton gves the best results as compared wth the experment. Comparsons of n-plane velocty components at a cross-flow plane normal to the free stream drecton at 97% chordwse poston n Fgure 10 supports ths observaton. Especally the dstrbuton of the vertcal velocty component along a horzontal traverse through the vortex core as obtaned wth the Pω - modfcaton s n good agreement wth the expermental data at ths staton.

9 (SYA) 14-8 Concludng remarks Computatons have been performed for the turbulent vortcal flow over a sharp-edged cropped delta-wng / underwng-fuselage confguraton wth the ENFLOW Naver-Stokes method. The computatons have been carred out on a computatonal grd of relatvely hgh resoluton consstng of 1,796,985 grd ponts. The computatons have been carred out at a transonc free-stream Mach number of 0.85, an angle of ncdence of 10 degrees and a Reynolds number of 9 mllon. The purpose of the computatons s to nvestgate the capablty of dfferent modfcatons of a two-equaton turbulence model to mprove predctons of turbulent vortcal flow. Results obtaned wth the NLR mplementaton of the Wlcox k ω turbulence model (ncludng the cross-dffuson term, [1]), and two modfcatons for vortcal flow to ths k ω model, have been presented. One of these modfcatons s amed at reducng the unphyscal hgh producton of turbulent knetc energy n the vortex core predcted wth the standard k ω model and has been tested for two combnaton of the ts parameters. The second modfcaton bascally has the same effect but accomplshes ths effect by ncreasng the producton of ω n vortex cores. Based on detaled comparson wth expermental data for ths case the followng conclusons are drawn: Standard k ω models produce unphyscal hgh levels of turbulent vscosty nsde vortex cores, resultng n vortcty dffuson that s larger than found n experments. The modfcaton based on ncreasng the ω -producton term s demonstrated to produce the best agreement wth expermental surface pressure and flow-feld data. Ths modfcaton also s the only one that mantans a local mnmum of the turbulence Reynolds number at the vortex centre throughout the flow, whch agrees wth the theoretcal observaton that turbulent vortces can have a lamnar sub-core. The approach to modfy the ω -equaton seems consstent wth approaches adopted by other authors to modfy one-equaton turbulence models for vortcal flow smulatons. It s recognsed that there s stll a need for a better theoretcal foundaton of modfcatons to the ω -equaton to properly account for hgh levels of vortcty. Although detaled expermental data have been used n the present paper more complete nformaton on the turbulence n vortex cores s requred, generated ether by DNS or LES smulatons of vortex cores or new dedcated experments (see for example [13]). References [1] Brandsma, F.J., and Sytsma, H.A., Naver-Stokes Computatons for a Round-Edged Delta Wng; Influence of underwng body, grd densty, and lamnar turbulent transton locaton, NLR CR L, [] Arthur, M.T., Kordulla W., Brandsma, F.J., and Ceresola, N., Grd adaptaton n vortcal flow smulatons, AIAA 15 th Appled Aerodynamcs Conference, Atlanta GA, USA, June [3] Wlcox, D.C., A Two-Equaton Turbulence Model for Wall-Bounded and Free-Shear Flows, AIAA , AIAA 4 th Flud Dynamcs Conference, Orlando, July 6-9, [4] Kok, J.C., Brandsma, F.J., Turbulence Model based Vortcal Flow Computatons for a Sharp Edged Delta Wng n Transonc Flow usng the full Naver-Stokes Equatons, NLR-CR , 000. [5] Spalart, P.R. and Shur, M., On the Senstzaton of Turbulence Models to Rotaton and Curvature, Aerospace Scence and Technology, No. 5, pp.97-30, 1997 [6] Dacles-Maran, J. and Zllac, G.G., Numercal/Expermental Study of a Wngtp Vortex n the Near Feld, AIAA Journal, Vol. 33, No. 9, pp , 1995.

10 (SYA) 14-9 [7] Hanalć, K., Launder, B.E., Senstzng the Dsspaton Equaton to Irrotatonal Strans, ASME J. Fluds Eng., Vol. 10, pp , [8] Kok, J.C., Resolvng the dependence on free-stream values for the k-ω turbulence model. NLR-TP-9995, 1999 (also AIAA Journal, July 000). [9] Elsenaar, A., et.al., The Internatonal Vortex Flow Experment, AGARD CP 437, Valdaton of Computatonal Flud Dynamcs, May [10] Elsenaar A., Hoemakers, H.W.H.M., An expermental study of the flow over a sharp edged delta wng at subsonc and transonc speeds, AGARD CP-494, Vortex Flow Aerodynamcs, Paper 15, [11] Boersen, S.J., Analyss of surface pressure and flow feld measurements on the WB1-SLE confguraton (65 o sharp-edged cropped delta wng) at subsonc and transonc speeds n the DNW-HST, NLR-CR-9955, [1] Kok, J.C., Spekrese, S.P., Effcent and accurate mplementaton of the k-ω turbulence model n the NLR multblock Naver-Stokes system, presented at the ECCOMAS 000 Conference, Barcelona, Span, September 000, NLR-TP [13] Rley, A.J., Lowson, M.J., Development of a three-dmensonal free shear layer, J. of Fl. Mech., Vol. 369, pp.48-89, 1998.

11 (SYA) Fgures Fgure 1 Defnton of the 65o cropped sharp-edged delta wng and mpresson of the wndtunnel model geometry Fgure Impresson of the computatonal grd (surface, symmetry plane and cross-flow grd planes) Fgure 3 Comparson of upper-surface pressure dstrbutons (half wng)

12 (SYA) Fgure 4 Upper surface lmtng streamlnes and skn frcton dstrbutons Fgure 5 Comparson of spanwse pressure dstrbutons

13 (SYA) 14-1 Fgure 6 Fgure 7 Fgure 8 Comparson of spanwse dstrbutons Total pressure loss dstrbuton n cross-flow planes Turbulence Reynolds number dstrbuton n cross-flow planes

14 (SYA) Fgure 9 Fgure 10 Comparson of of total pressure losses n a cross-flow plane at x / cr = 0.9 wth expermental data Comparson of n-plane velocty components n a cross-flow plane at x / cr = 0.97 wth expermental data

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