STEADY STATE AERODYNAMIC ANALYSIS OF AIRCRAFT WINGS WITH CONSTANT AND VARYING CROSS SECTION

Transcription

1 STEADY STATE AERODYNAMIC ANALYSIS OF AIRCRAFT WINGS WITH CONSTANT AND VARYING CROSS SECTION [1] Jimon Dennis Qudros, [] Dr D L Prbhkr Abstrct- The flight of n ircrft is minly generted by Lift force nd Drg force. These forces lter the reltive wind producing erodynmic forces tht ct on the wings of n ircrft. The present pper ims t computing the Co-efficients of Lift nd Drg for typicl wing plnform of Constnt nd Vrying cross section. Numericl studies re performed to investigte the erodynmic performnce of thin wing plnform hving Constnt nd Vrying Crosssection. The vortex lttice method (VLM) is utilized to simulte the Aerodynmic chrcteristics on the bsis of Computtionl Fluid Dynmics. The performnce chrcteristics tht minly govern the study re Co-efficient of Lift nd Co-efficient of Drg. The study lso dels with understnding the effect of vrious fctors such s Aspect Rtio, Sweep ngle, Vortex or Pnel formtion, Tper rtio, Ground Proximity nd Dihedrl Angle on Aerodynmic chrcteristics nd lso decide the wing dimensions for different wing configurtions under Stedy Subsonic flow. In the present work, the vlidity of the Vortex Lttice Method (VLM) for evluting the Lift nd Drg Coefficients is estblished by compring the results with the vilble experimentl dt. Keywords: Vortex Lttice Method, Co-efficient of Lift, Co-efficient of Drg. I. INTRODUCTION The erodynmic chrcteristics for subsonic flow bout n irfoil hve been widely discussed. Since the spn of n irfoil is infinite, the flow is identicl for ech spnwise sttion (i.e., the flow is two dimensionl). The lift is produced by the pressure differences between the lower surfce nd the upper surfce of the irfoil section [1] nd therefore the circultion (integrted long the chord length of the section), does not vry long the spn. Jimon Dennis Qudros [1], Dept. of Mechnicl Engineering, VTU/ S.C.E.M Mnglore, Indi, Mob No: Fig.1: Aerodynmic lod distribution for rectngulr wing in subsonic irstrem. Therefore determintion of the Lift nd Drg long the spn length of n ircrft wing becomes impertive s the two dimensionl results do not give n exct pproximtion of the lift nd Drg Co-efficient. Thus, n importnt difference in the three-dimensionl flow field round wing (s compred with the two-dimensionl flow round n irfoil) is the vrition in lift. Since the lift force cting on the wing section t given spnwise loction is relted to the strength of the circultion, there is corresponding spnwise vrition in circultion, such tht the circultion t the wing tip is zero. Procedures tht cn be used to determine the vortex-strength distribution produced by the flow field round three dimensionl lifting wing is presented in this pper. The Vortex Lttice Method (VLM) [] provides Three Dimensionl depiction for Lift nd Drg long the Whole spn of the Wing Plnform of n ircrft. Dr. D. L. Prbhkr [], Dept. of Civil Engineering, VTU/ S.C.E.M Mnglore, Indi, Mob No:

2 II. OBJECTIVE The present study ims t studying the vrition in Co- efficient of lift nd Co- efficient of Drg for Rectngulr Wing of constnt cross section due to vrition of different prmeters such s Aspect Rtio (AR), Sweep ngle nd ground proximity [3] nd lso compre the Theoreticl vlues of Co-Efficient of Lift with the vilble Experimentl Vlues. III. VORTEX LATTICE METHOD (VLM) The first step required to solve the eroelstic problem is the determintion of the erodynmic lods. In order to do so, one hs to develop computtionl progrm bsed on the Vortex Lttice Method (VLM).This section presents short review on the VLM method nd the fundmentl equtions used in the numericl implementtion. The VLM method is the simplest of the methods to solve incompressible flows round wings of finite spn. The method represents the wing s plnr surfce on which grids of horseshoe vortex re superimposed s shown in Fig. Ech of these Horseshoe vortices posses control point s shown in Fig 4. The computtion of the velocities induced by ech horseshoe vortex t ech specified control point is bsed on the Biot- Svrt lw [4]. n (dl r ) d V 3 4r (1) A summtion is performed for ll control points on the wing to produce set of liner system of equtions for the horseshoe vortex strengths tht stisfy the boundry conditions of no flow through the wing. The control points of ech element (or lttice) re locted t three-fourth of the element s chord nd the vortex strengths re relted to the wing circultion nd the pressure difference between the upper nd lower surfce of the wing. The pressure differentils re then integrted to yield the totl forces nd moments. In the pproch used here to solve the governing equtions, the continuous distribution of bound vorticity over the wing surfce is pproximted by finite number of discrete horseshoe vortices. The individul horseshoe vortices re plced in rectngulr (or trpezoidl) pnels lso clled finite elements or lttices. This procedure for obtining numericl solution for the flow is termed Vortex Lttice Method Fig. "Typicl" horseshoe vortex. The bsic expression for the clcultion of the induced velocity by the horseshoe vortices in the VLM is V = n 4 r 1 r r 1 r r r 1 r r1 r 0 () Where is the circultion round the pnel nd r 1 nd r re the vectors produced by Fce AB. To clculte the velocity tht is induced t generl point in spce (x, y, z) by the horseshoe vortex shown in Fig.. The horseshoe vortex mybe ssumed to represent tht for typicl wing pnel (e.g., the nth pnel). Segment AB represents the bound vortex portion of the horseshoe system. Then the influence of ech Horseshoe vortex on ech of the pnel is given by W m, n = Γ n /4π{((x m -x 1n )*( y m- y n )- (x m -x n )* (y m - y 1n )) -1 {[(x n - x 1n )* (x m -x 1n )+( y n - y 1n )* (y m - y 1n )/sqrt((x m -x 1n ) +(y m- y 1n ) )]-[(x n - x 1n )* (x m -x 1n )+( y n - y 1n )*(y m- y 1n )/sqrt((x m -x 1n ) +(y m- y 1n ) )] }+(1/ (y 1n - y m ) ) [1+((x m -x 1n )/ sqrt((x m -x 1n ) +(y m- y 1n ) )]- (1/ (y 1n - y m ) ) [1+((x m -x 1n )/ sqrt((x m -x 1n ) +(y m- y 1n ) )] (3) where the totl velocity is the sum of the contributions from the vortex segments shown in Fig 4. Assuming the point C(x,y,z) to be the control point of the mth pnel, with coordintes (x m, y m, z m ) locted t midspn of the element. 1553

3 The methodology implemented for Vortex Lttice Method is s follows [5] A. Choice of Singulrity Element The method by which thin wing plnform is divided into pnels is shown in Fig.4 nd some typicl pnel element. Fig.5: A spnwise Horseshoe Vortex element. Fig.4: Horseshoe Lifting Line Model for solving lifting line problem. In order to solve the problem by use of Horseshoe elements s shown in Fig.3, this element should consist of stright bound vortex segment BC tht models lifting properties nd two semi infinite triling vortices tht models the wke. The segment BC need not be prllel to the Y-xis, but t the element tips the vortex is shed into the flow where it must be prllel to the stremlines so tht no force will ct on the triling vortices. These vortex elements re used s the ner portions of vortex rings whose strting vortices extend fr bck so tht the effect of AD is negligible. The velocity induced by such n element t point P(x, y, z) s shown in Fig.5 cn be computed by velocity induced by horseshoe element. By use of Eqution 3 sub-routine is creted for simplicity of the Equtions. It is possible to include these computtions Equtions in subroutine such tht (u, v, w)=hshoe(x, y, z, z B, x, y,z,x, y, c c c D D, ) B. Discretiztion nd Grid geometry x A, y A, z A, x B, y B, z D (4) At this phse wing is divided into N spnwise elements s shown in Fig.4. For this exmple the spn is divided into 8 pnels nd lso the geometric pnel re like S, norml vector n nd the coordintes of the colloction point (x i, y i, z i ) re clculted t this phse. C. Influence co-efficients In order to fulfil the boundry conditions specified t ech colloction points. The velocity induced by horseshoe vortex element 1 t colloction point 1 cn be computed by using horseshoe routine developed before Fig.3: Horseshoe model of lifting wing. (u,v,w) 11 HSHOE ( x 1,y 1,z 1, A x, y, z,, y,z,x, y, B1 z 1, D B1 1.0) B1 xc1 c1 c1 D1 D1 x 1,y A 1,z A 1, The no norml flow cross the wing boundry condition t this point for the first colloction point cn be written s [( u, v, w) 111 (u, v, w) (u,v,w) 1N N (U,V,W )] n

4 Estblishing the sme procedure for ech of the colloction points in the discretized form of the boundry condition NN Q n NN Q n NN Q n N1 1 N N NN N Q Where the Influence co-efficient re defined s i (u, v, w) n (5) i i n N on itself is lrgest, the mtrix will hve dominnt digonl nd the solution is stble. F. Secondry computtions: pressures, lods, velocities The solution for the bove set of equtions results in vector (Γ 1, Γ Γ n ). The lift of ech bound vortex segment is obtined by kutt-owkoski condition: L Q y (7) Where Δy is the pnel bound vortex proection norml to the free strem. The induced drg computtion is somewht more complex. The totl erodynmic lods re the sum of contribution of individul pnels. Following lifting line results Equtions The norml velocity components of the free strem flow re known nd moved to the R.H.S of the eqution RHS (U,V, W ) n i (6) These set of N lgebric equtions with N unknown T tht cn be solved by stndrd solution mtrix techniques. As n exmple for the cse of plnr wing with constnt ngle of ttck α, this results in the following set of equtions D w y (8) ind where the induced downwsh w ind t ech colloction point is computed by summing up the velocity induced by ll triling vortex segments s shown in fig 6. This cn be done during the phse of the influence co-efficient computtion or even lter by using HSHOE routine with the influence of bound vortex segment turned off N N... 1N 3N NN N D. Estblish R.H.S vector. 1 3 N = - Q 1 1 sin 1. 1 The right hnd side vector Eqution (6) is ctully the norml component of the free strem which cn be computed within the influence co-efficient computtions. However if n upgrde is plnned to include unstedy effects or simultion of norml trnspirtion flows, then its recommended to do this clcultion seprtely. Fig.6: Arry of Vortex segments responsible for induced downwsh on the three-dimensionl wing. E. Solve liner set of Equtions The solution for the bove described problem cn be obtined by stndrd mtrix methods. Furthermore since the influence of such n element G. Ground Effect - Clssicl Theory Wieselsberger presents prmeter tht he refers to s n influence coefficient (σ).since Wieselsberger ttributes the formul to Prndtl, we will refer to the formul s Prndtl s formul. 1555

5 1 0.66([ h / b]) (9) ( [ h / b]) Murice Le Sueur clrified the usge of the bove eqution. At given Lift Co-efficient nd the drg due to Lift Co-efficient CL S CD (1 ) (10) b Since the Aspect Rtio (AR) is b AR= S Therefore the Drg due to Lift becomes CL C (1 ) (11) D AR This eqution cn be used to study the effect of Ground proximity [6] nd Dihedrl ngle on wings of constnt nd vrying cross-section. V. RESULTS Fig.8: Vrition of lift with respect to ngle of ttck for different spect rtios. Fig. 7: Theoreticl nd Experimentl vlues of coefficient of lift with respect to Angle of Attck for 45 swept wing. Fig. 9: Vrition of Co-efficient of lift with respect to Aspect Rtio for different Sweep Bck ngles. 1556

6 Fig.10: Vrition of Co-efficient of Drg with respect to spect rtio for different sweep ngles. Fig.1: Vrition of Co-efficient of lift with respect to Dihedrl ngle for different vlues of Ground Proximity (h/b). Fig. 11: Vrition of Co-efficient of lift with respect to Ground proximity for different Aspect Rtios (AR). Fig. 13: Vrition of Lift with respect to Drg. 1557

7 V. CONCLUSION The present work dels with the study of Lift nd Drg vritions of n ircrft wing. The formultion is bsed on finding out Lift nd Drg of wing by use of 3-dimensionl method clled s Vortex Lttice Method or Pnel method. The theoreticl method is employed by dividing the ircrft wing into finite number of pnels nd determining the required prmeters. Results re presented bsed on the nlyticl study of vrition of Lift nd Drg of n ircrft wing with respect to different prmeters such s Aspect Rtio, Bckwrd Sweep ngle, Effect of ground proximity nd Dihedrl ngle. The most importnt result of Vortex Lttice Method is to estblish the Lift nd Drg of wing of prticulr Spn length nd chord length. [4] John Bertin nd Russel Cummings, Aerodynmics for Engineers, Fifth Edition, Prentice Hll Publishers, US, 008. [5] Joseph Ktz, Low Speed Aerodynmics, Second Edition, Cmbridge Univ. Press, Cmbridge, Englnd, U.K, 005. [6] Sng-Hwn, Juhee Lee, Aerodynmic nlysis nd multi-obective optimiztion of wings in ground effect, Ocen Engineering, Vol. 68, Aug 013, pp [7] G. G. Brebner, J. Weber, Low-Speed Tests on 45-deg Swept-bck Wings, Aeronuticl Reserch Council Reports nd Memornd, Reports nd Memornd No. 88*, London, My The Lift obtined from VLM is in good greement with experimentl vlue of lift obtined for Brebner nd weber [7] therefore vlidting the result. The lift of wing decreses s the Aspect Rtio becomes smller for wing of generl spnwise circultion. The induced Drg of wing increses s the wing Aspect Rtio decreses for wing with generl spnwise circultion. This theory lso provides vluble informtion bout the wing s spnwise loding nd bout the existence of the triling vortex wke. The theory is limited to smll disturbnces nd lrge Aspect Rtio since ngle of ttck is considered to be very smll upto level of10. There re possible modifictions to this theory such s ddition of flpped wings. However the study of the wings with more complex geometry is difficult with this method. Using the results from this method we re ble to study the vrition of Lift nd drg with respect to different prmeters. REFERENCES [1] Prbhu Rmchndrn, S. C. Rn, M. Rmkrishn. An Accurte Two-Dimensionl Pnel Method, Deprtment of Aerospce Engineering, IIT- Mdrs, Chenni. [] D. Levin, J. Ktz, Vortex Lttice Method for Clcultion of Nonstedy Seprted Flow over Delt Wings, Journl of Aircrft, Vol. 18, No., 1981, pp [3] Kyoungwoo Prk, Juhee Lee, Optiml design of two-dimensionl wings in ground effect using multiobective genetic lgorithm, Ocen Engineering, Vol. 37, No. 10, Jul 010, pp AUTHOR S PROFILE Jimon Dennis Qudros is working in Shydri College of Engineering nd Mngement s n Assistnt Professor in the deprtment of Mechnicl Engineering. He received the M.Tech degree in Mchine Design from Shydri college of Engineering nd Mngement, Mnglore ffilited to Visvesvry Technologicl University (VTU) nd B.E degree from P.A College of Engineering in 011. He hs presented ntionl conference ppers in the field of Aerospce Engineering. His res of interest re Mechnicl Vibrtions, Strength of Mterils nd Computtionl Fluid Dynmics. 1558

8 Dr. D. L. Prbhkr, n eminent scholr nd renowned cdemicin. Dr Prbhkr did his M.Tech (Applied Mechnics) t IIT Mdrs nd his Ph.D (Aerospce Engg.) t IIT Khrghpur. He hs put in 37 yers of service in the teching profession nd hs published bout 60reserch ppers in Ntionl nd Interntionl ournls nd conferences. Dr Prbhkr is reviewer of reputed ntionl nd interntionl ournls. Currently 5 reserch scholrs re pursuing Ph.D progrm under his guidnce. He is lso n uthor of the book titled 'Grphic Sttics'. He hs served in lmost ll the cdemic nd sttutory bords of lmost ll the universities including VTU. 1559