21st AIAA Applied Aerodynamics Conference June 23 26, 2003/Orlando, FL

Transcription

1 AIAA Numerical Analysis and Design f Upwind Sails Sriram and Antny Jamesn Dept f Aernautics and Astrnautics Stanfrd University, Stanfrd, CA Margt G Gerritsen Department f Petrleum Engineering Stanfrd University, Stanfrd, CA 21st AIAA Applied Aerdynamics Cnference June 23 26, 23/Orland, FL Fr permissin t cpy r republish, cntact the American Institute f Aernautics and Astrnautics 181 Alexander Bell Drive, Suite 5, Restn, VA

2 Numerical Analysis and Design f Upwind Sails Sriram and Antny Jamesn Dept f Aernautics and Astrnautics Stanfrd University, Stanfrd, CA Margt G Gerritsen Department f Petrleum Engineering Stanfrd University, Stanfrd, CA The aim f this study is t develp and validate numerical methds that slve the inviscid field equatins (Euler) t simulate and design upwind sails. The three dimensinal Euler equatins fr cmpressible flw are mdified using the idea f artificial cmpressibility and discretized n unstructured tetrahedral grids t prvide estimates f lift and drag fr upwind sail cnfiguratins. Cnvergence acceleratin techniques like multigrid and residual averaging are used alng with parallel cmputing platfrms t enable these simulatins t be perfrmed in a few minutes. T accunt fr the elastic nature f the sail clth, this flw slver is cupled t NASTRAN t prvide estimates f the deflectins caused by the pressure lading. The results frm the aerelastic simulatins shw that the majr effect f the sail elasticity was in altering the pressure distributin arund the leading edge f the head and the main sail. Adjint based design methds have als been develped, and used t induce changes t the camber distributin f the main sail. The design prcess resulted in a camber distributin that allws smth entry f the flw thrugh the leading edge f the main sail thereby, reducing the leading edge suctin peaks that wuld be detrimental t the grwth f the bundary layer. 1. Intrductin Races like the Americas Cup have seen significant imprvements in the design f bth the hull and the sails ver the last tw decades. Cmpeting syndicates are cnstantly pushing the aerdynamic and structural limits f the designs as imprvements f less than.5 % in the speed f the bat results in savings f abut secnds which is typically greater than the margin f victry fr these races. Fr the windward leg f the race, a gd measure f the perfrmance f a design is the distance that the bat travels directly t windward in a given time. This perfrmance index (called the speed-made-gd by the bat) is dependent n bth the speed f the bat and her true sailing curse which in turn are dependent n the aerdynamic and hydrdynamic frces prduced by the sails and the hull. In general, the perfrmance f the bat can be imprved by reducing the resistance f the hull and the drag f the sails. Hwever changes in the aerdynamic and hydrdynamic frces alter the equilibrium f the sail bat which then has t adjust its speed and sailing curse t maintain equilibrium. Hence, the design f sails bats has t be carried ut in an envirnment where the analysis and design prcedures fr the sails and the hull are tightly cupled t realize a meaningful verall design. Traditinally, designers have used Velcity Predictin Prgrams (VPP)t design a bat with gd verall perfrmance. These prgrams slve fr the mtin f the bat using frce and mment balance relatins Prfessr, Stanfrd University Prfessr, Stanfrd University yright c 23 by authrs. Published by the American Institute f Aernautics and Astrnautics, Inc. with permissin. while ensuring that the resulting mtin is stable. These prgrams typically cmpute the aerdynamic and hydrdynamic frces using ptential flw slvers, which are cmputatinally inexpensive and easy t implement, enabling the designer t evaluate a wide array f designs. Hwever, there exists large regins f rtatinal flw and significant viscus interactin, where the assumptins f ptential flw are nt valid. Alternate descriptins f the flw field that accunt fr the rtatinal nature f the flw can prvide mre accurate estimates f the induced drag (which accunts fr 6 % f the ttal drag) fr the upwind leg f the race. Mdeling the viscus effects during the upwind and dwnwind leg f the race will help in the design f sails and hulls with reduced viscus drag. Simulatins f the evlutin f the free-surface f the sea and its interactin with the hull can help prvide a virtual tl that can supplement tw tank testing. The develpment f rbust and accurate tls t predict the abve mentined physical phenmena culd lead t imprved designs that can benefit the racing cmmunity in particular and the sailing wrld in general. Cmputatinal techniques that slve the Euler r the Navier-Stkes equatins are increasingly being used by cmpeting syndicates in races like the Americas Cup. Fr sail cnfiguratins, this desire stems frm a need t understand the influence f the mast n the bundary layer and pressure distributin n the main sail, the effect f camber and planfrm variatins f the sails n the driving and heeling frce prduced by them and the interactin f the bundary layer prfile f the air ver the surface f the water and the gap between the bm and the deck n the perfrmance 1 f 21 American Institute f Aernautics and Astrnautics Paper

3 f the sail. Traditinally, experimental methds alng with ptential flw slvers have been widely used t quantify these effects. While these appraches are invaluable either fr validatin purpses r during the early stages f design, the ptential advantages f high fidelity cmputatinal methds makes them attractive candidates during the later stages f the design prcess. The aim f this study is t develp and validate numerical methds that slve the inviscid Euler field equatins t simulate and design upwind sails. The three dimensinal Euler equatins fr cmpressible flw are mdified using the idea f artificial cmpressibility, and discretized n unstructured tetrahedral grids t prvide estimates f lift and drag fr upwind sail cnfiguratins. Cnvergence acceleratin techniques like multigrid and residual averaging are used alng with parallel cmputing platfrms t enable these simulatins t be perfrmed in a few minutes. T accunt fr the elastic nature f the sail clth, this flw slver is cupled t NASTRAN t prvide estimates f the deflectins caused by the pressure lading. The results f this aerelastic simulatin, illustrate that the majr effect f the sail elasticity, is in altering the pressure distributin arund the leading edge f the head and the main sail. We als address the issue f hw t calculate imprved sail shapes using an ptimizatin prcedure based n the adjint methd. This is used t induce changes t the camber distributin f the main sail with the gal f reducing the leading edge suctin peaks that culd be detrimental t the grwth f the bundary layer. The design prcess results in an camber distributin that allws smth entry f the flw thrugh the leading edge f the main sail, thereby reducing the leading edge suctin peaks. 2. Analysis with CFD 2.1 Finite Vlume Discretizatin f the Flw Equatins A vast repertire f cmputatinal cdes have been develped by Antny Jamesn t analyze aerdynamic cnfiguratins in transnic flight. 1, 2 These cdes mdel the fluid as a cmpressible fluid and a variety f numerical techniques have been develped t efficiently slve the gverning equatins f a cmpressible fluid with embedded supersnic regins. In the limit f truly incmpressible flw, r zer Mach number, alternate methds must be used t cmpute the flw field t preserve the accuracy, rbustness and cnvergence prperties f the flw slutin. The fundamental prblem in transitining frm a cmpressible fluid mdel t an incmpressible ne is the lss f the evlutin equatin fr the density. Since the density is cnstant, a time dependent cnstraint must be impsed n the cntinuity equatins t ensure a divergence-free velcity field. In additin, the 2 f 21 eigenvalues resulting frm the system f cnventinal hyperblic Euler equatins fr cmpressible flws becme infinite in the limit f incmpressible flw. This is due t the fact that incmpressible flws exhibit infinite sund speeds. Hence, the use f cmpressible flw slvers in the incmpressible flw limit intrduces widely varying eigen-speeds, resulting in extremely stiff equatins. T vercme this difficulty, the present wrk uses the artificial cmpressibility methd, an apprach first prpsed by Chrin in as a methd t slve viscus flws. Artificial cmpressibility methds intrduce a psuedtempral equatin fr the pressure thrugh the cntinuity equatin. This apprach remves the trublesme sund waves assciated with cmpressible flw frmulatins as the Mach number appraches zer. The eigenvalues f the riginal system are nw replaced with an artificial set that renders the new set f equatins well-cnditined fr numerical cmputatin. When cmbined with multigrid acceleratin prcedures, artificial cmpressibility prves t be particularly effective 4. Cnverged slutins f incmpressible flws ver a main sail can be btained in abut 75-1 multigrid cycles. Using the idea f artificial cmpressibility, the equatins f mtin f an incmpressible, inviscid fluid can be cast in the fllwing frm. w t P { F x G y H dz } =. (1) Here, the dependent variables w, the inviscid flux vectrs f, g and h and the pre-cnditining matrix P are described by w = p u v w H =, F = w wu wv w 2 p u u 2 p uv uw, P =, G = Γ v vu v 2 p vw, (2) (3) This system f equatins has n physical meaning until a steady state is reached. At steady state, the time dependent pressure term drps frm the cntinuity equatin resulting in the true steady state equatins fr an incmpressible flw. Further, Γ can be selected t accelerate the time decay t steady state. Using the finite vlume apprach, the gverning equatins can be cast in the integral frm fr each cmputatinal vlume in the dmain as fllws, Cnservatin f Mass d dt V pdv American Institute f Aernautics and Astrnautics Paper S Γ 2 (u n) ds = (4)

4 Cnservatin f Mmentum d udv u(u n)ds = dt V S S pnds (5) Spatial discretizatin f equatin (4) and (5) leads t the fllwing equatin fr each pint f interest in the cmputatinal mesh. d dt V iw i k F k.n k S k = (6) where p is the pressure, u is the velcity vectr, n is the unit nrmal at the surface f the cntrl vlume, V and S are the vlume and surface area f the cntrl vlume respectively, F is the flux thrugh the cntrl vlume and the summatin f the fluxes is ver the cntrl vlume that surrunds each nde f the mesh. In the present study, a cell-vertex scheme is used fr the implementatin f the finite vlume scheme n unstructured tetrahedra. Dual meshes cnstructed frm planes bisecting each edge f the mesh are used t accumulate the fluxes at each nde. Bundary cnditins are then enfrced alng the triangular faces that lie n the bundary t accunt fr the ne-sided cntrl vlumes fr the ndes n the bundary. The rest f the discussin in this sectin utlines the details f the implementatin f the spatial discretizatin peratrs when used with artificial cmpressibility methds, the evaluatin f the numerical diffusin terms and the multigrid algrithm. In Equatin (3), Γ is called the artificial cmpressibility parameter in the light f the analgy that may be drawn between the abve equatins and the equatins f mtin fr a cmpressible fluid whse equatin f state is given by p = Γ 2 ρ. Thus, ρ is an artificial density and Γ may be referred t as an artificial speed f sund. When the tempral derivatives tend t zer, the set f equatins satisfy precisely the incmpressible Euler equatins, with the cnsequence that the crrect pressure may be established using the artificial cmpressibility frmulatin. The pre-cnditining matrix, P, may be viewed as a device t create a well psed system f hyperblic equatins that are t be integrated t steady state alng lines similar t well established cmpressible flw Finite Vlume frmulatins. In additin, the artificial cmpressibility parameter may be viewed as a relaxatin parameter fr the pressure iteratin. The eigenvalues f the system f equatins in equatin (1), are given by where, and λ 1 = U, λ 2 = U, λ 3 = U a, λ 4 = U a a 2 = U 2 Γ 2 (ψ 2 η 2 ξ 2 ) U = uψ vη wξ The terms ψ, η, ξ represent the slpes f the characteristic system f waves, are arbitrary and defined. The chice f Γ is crucial in determining the cnvergence and stability prperties f the numerical scheme. Typically, the cnvergence and stability rate f the scheme is dictated by the slwest waves and the stability f the scheme by the fastest. In the limit f large Γ, the difference in wave speeds can be large. Althugh this situatin wuld presumably lead t a mre accurate slutin thrugh the penalty effect in the pressure equatin, very small time steps wuld be required t ensure stability. Cnversely, fr small Γ, the difference in the maximum and minimum wave speeds may be significantly reduced, but at the expense f accuracy. Thus a cmprmise between the extremes is achieved by chsing Γ t be Γ 2 = C(u 2 v 2 w 2 ) where C is a cnstant f the rder f unity. In regins f high velcity and lw pressure where suctin ccurs, Γ is large t imprve accuracy, and in regins f lw velcity, Γ is crrespndingly reduced. Under these assumptins n the chice f the precnditiner, P, and the applicatin f the Finite Vlume methd fr a cell-vertex scheme results in a set f rdinary differential equatin fr each nde f the cmputatinal mesh, d dt (V iw) P Q i =. (7) where V i is the vlume arund each nde and Q i represents the flux thrugh the faces f the cntrl vlume. T prevent, dd-even decupling at adjacent ndes which may lead t scillatry slutins, a dissipatin term is added t the flux calculatin t mdify the abve equatin d dt (V iw) P [Q i D i ] =. (8) r d dt (V iw) R i =. (9) where R i is the residual at each nde in the cmputatinal mesh. The resulting system f equatins are integrated in time using an explicit multistage scheme with cefficients that maximize the stability regin f the timestepping scheme. T further accelerate cnvergence t steady state, lcal time-stepping and residual averaging techniques are intrduced. Detailed numerical analysis f the spatial discretizatin 6, 7 and the time stepping scheme 8 can be btained frm previus studies listed in the citatins. 2.2 Dissipatin Numerus research effrts during the eighties led t t the develpment f a mathematical frame-wrk t 3 f 21 American Institute f Aernautics and Astrnautics Paper

5 add numerical diffusin t the discretized equatins. While the fcus f these studies were n deriving stable and accurate techniques t reslve shcks in the flwfield, the mathematical frame-wrk can be inherited fr incmpressible flws that use the artificial cmpressibility methd with sme mdificatins that limit the amunt f numerical diffusin. Lcal Extremum Diminishing (LED) schemes that guarantee that new extrema are nt generated during the evlutin f the slutin have prven t be rbust and efficient. These schemes limit the recnstructed slutin and fluxes at cell interfaces by using limiters that can be cnstructed frm gradient infrmatin frm a stencil f pints arund each cmputatinal pint. A variety f chices exist fr the frm f the limiters, and the JST scheme 8 can be refrmulated in the class f SLIP schemes fr a particular chice f the limiter. Over the years the JST scheme has prven t be rbust and accurate fr a variety f aerdynamic flws. It can be represented as d j 1 = ɛ2 2 j w 1 2 j 1 ɛ4 2 j 1 2 ( w j w j 1 2 w j 1 2 When used fr incmpressible flws, the first term n the R.H.S. can be drpped and the remainder used as the numerical diffusin term that prevents dd-even cupling. All the cmputatins in this study has been perfrmed using the JST scheme as the basis fr numerical dissipatin. 2.3 Multigrid Multigrid techniques are widely used t accelerate the cnvergence f a system f equatins t steady state. A general framewrk fr the develpment f full-apprximatin multigrid methds fr nn-linear equatins can be utlined as fllws. Cnsider, Lu = F discretized n a mesh with spacing h as This can be rewritten as L h v h = F h L h (v h δv h ) = F h where δv represents crrectin t the present estimate v h r L h δv h R h = (1) where R h is the residual. n the carse grid, the abve equatin can be replaced by L 2h δv 2h I h 2h R h = (11) 4 f 21 where I2h h represents the aggregatin r restrictin peratr. The crrectin t the present fine grid slutin can be represented as Then ) v new h = v h Ih 2h δv h where I 2h h represents an interplatin peratr. We can add and subtract the fllwing frm equatin (11) t get L 2h v 2h F 2h = R 2h L 2h (v 2h δv 2h ) F 2h I h 2h R h = This leads t the full apprximatin scheme (FAS) L 2h (v 2h ) F 2h I h 2hR h R 2h = v h = v h I h 2h (v 2h v 2h) In the present wrk a series f nn-nested meshes are used fr the multigrid cycle and the slutin and residual frm each mesh are aggregated t the carser mesh while interplating the crrectin frm the carser t the fine mesh. Detailed descriptins f the multigrid scheme can be btained frm 9. Fr unstructured grids, the nature f the grids t be used in the multigrid cycle is a questin f nging debate. In the present study a series f nn-nested meshes was generated by a grid generatr (MESH- PLANE). The initial slutin frm a particular mesh is advanced in pseud-time t btain new estimates f the flw variables. On transfer t the next level f the multigrid, the slutin fr the carse grid mesh pints is interplated frm the fur ndes f the fine mesh tetrahedrn that cntains this nde. Further, the accumulated residual at each fine mesh pint is distributed t each nde f the tetrahedrn in the carse mesh that that enclses the fine mesh nde. The interplating factrs fr each nde are cmputed frm weights which are based n the vlume included by a given nde and ppsite face f the tetrahedrn (figure 1). This reduces t a secnd rder interplatin scheme n equilateral tetrahedra and has been fund t be sufficient fr the present calculatins. The estimate f the residuals frm the fine mesh is used t advance the slutin n the carse mesh where the larger scale errrs can be mre efficiently accmplished n the carse mesh. Further levels in the multigrid cycle invlve the same peratins are befre, thereby using grids that are carser and carser t cnvect the errr terms ut f the cmputatinal dmain faster. The ascending side f the multigrid cycle estimates a crrectin frm each grid which is then interplated t the next finer mesh in the sequence (figure 2). The crrectins frm the carser mesh are transferred using American Institute f Aernautics and Astrnautics Paper

6 n a1 p a2 n a3 y n1 x Fig. 1 Interplatin cefficients fr use in the multigrid cycle Fig. 3 Dmain decmpsitin f a rectangular regin using a mdified bisectin methd inter-prcessr bundary edges that are shared between prcessrs fine grid ndes carse grid ndes Fig. 2 Transfer f slutin, residuals and crrectins between the fine and carse mesh similar interplating factrs as fr the aggregatin peratins. Multigrid cycles which prgress in the shape f a W have generally been fund t prvide faster cnvergence t steady state than the simple V cycle. 3. Parallel Implementatin f the Unstructured Multigrid Flw Slver T explit the availability f mdern parallel cmputing platfrms, a cmbinatin f the AIRPLANE 1 and FLO77 1 cmputatinal prgrams, was parallelized. Due t the unstructured nature f the cmputatinal grid, a wide variety f pssible data structures t implement the underlying numerical algrithms exist. The fllwing sectins utline the chice f data structures and algrithms that were made t parallelize the flw slver. 3.1 Dmain Decmpsitin, Lad Balancing A mdified crdinate bisectin methd was used t recursively divide a given cmputatinal mesh int sub-dmains (figure 3). The sub-dmains were created such that they cntained apprximately the same number f cmputatinal ndes. N effrt was made t ptimize the dmain decmpsitin prcess s as t minimize the number f edges that are shared between Fig. 4 Hal ndes and the distributin f edges alng prcessr bundaries the sub-dmains. The sub-dmains are distributed amng the available prcessrs in a manner that minimizes a cmbinatin f the cmputatinal cst assciated with the dmains in each prcessr and the cst f cmmunicatin amng the prcessrs fr a given distributin. This methdlgy was fund t result in well balanced lad distributins. As the flw slver uses an edge-based data structure t accumulate the fluxes at each vertex, the edges surrunding the ndes that lie within a partitin are accumulated. If an edge cnnects a pair f ndes that lie acrss prcessr bundaries, this edge is duplicated in the tw prcessrs and hal ndes are cnstructed fr bth prcessrs. This idea is illustrated further in figure Parallel Implementatin f the Multigrid Algrithm The use f multigrid techniques fr flw analysis necessitates the need t exchange infrmatin between fine and carse grid pints and vice-versa. While the 5 f 21 American Institute f Aernautics and Astrnautics Paper

7 SpeedUp Actual SpeedUp Ideal SpeedUp Number f Prcessrs Fig. 5 Speedup frm the parallel implementatin residuals frm the fine grid pints need t be accumulated at the carse grid pints, the crrectins frm the carse grid pints need t be transferred back t the fine grid pints. Hence, efficient pint lcatin methds are used t determine the interplatin cefficients assciated with each cmputatinal nde in the multigrid cycle. The required search methdlgy has been implemented using an ctree-based search rutine. Each grid in the multigrid cycle is recursively divided int ctants that cntain a certain number f pints. Using this data structure, a given pint is identified within an ctant and the clsest nde clsest in this ctant is determined. The tetrahedra cnnected t this nde are checked t see if they cntain the search pint. Once such a tetrahedrn is identified, interplatin cefficients t aggregate the residuals and t interplate the crrectins are cnstructed. Further, t reduce the cmmunicatin cst amng prcessrs during the transfer f infrmatin between the fine and carse grids, sub-dmains n the carser grids are cnstructed and distributed s as t cnfrm t the divisin and distributin f the fine mesh. Typical cmputatinal times t build the ctree and t build the interplatin tables are in the range f a few minutes fr a sequence f meshes cntaining a millin ndes. The efficiency f the parallel implementatin is shwn in figure Aerelastic Cmputatins t Predict the Flying Shape f Upwind Sails The incmpressible flw slver described in the previus sectins has been used t btain the characteristics f sail cnfiguratins f high perfrmance yachts fr races like the Americas Cup. Mdificatins t the inlet bundary cnditin were incrprated s as t simulate the bundary layer prfile ver the sea. Figure 7 shws a representative head and main sail cmbinatin used in the Americas Cup. Due t the fast turn-arund times f the flw slver it was pssible t use this cmputatinal package t btain aerdynamic characteristics f the sail cnfiguratins fr a variety f wind cnditins, and als study the effect f varying twist, camber and sail trim. Further, in rder t predict the flying shape f the sail gemetries, this flw slver was cupled t NASTRAN. The aerelastic package uses an iterative algrithm that transfers the pressure lading btained frm the flw slver t the structural mdel, and uses the deflectins frm the structural analysis t mdify the cmputatinal mesh fr the fluid. This prcess is iteratively carried ut until the deflectins are belw a particular threshld. 4.1 Structural Mdel In the structural mdel f the sail the sail clth is discretized int quadrilateral membrane finite elements with fur ndes (after neglecting the presence f batten pckets). These elements withstand all external frces thrugh tensin and are incapable f resisting bending mments. The translatinal and rtatinal degrees f freedm alng the ft f the main sail are suppressed. Alng the mast, the translatinal degrees f freedm are inhibited while allwing fr rtatinal mtin. Fr the head sail, the pint f attachment f the ft t the rig is cnstrained. The leech f the main and head sail are allwed t mve freely t induce a gemetric twist due t the aerdynamic lading. The mast is assumed t be rigid during the structural and aerelastic calculatins. The presence f battens and tensin cables and ther structural elements f the sail rig is neglected frm this analysis. The linear system f equatins relating the displacements t the frce field is advanced t a steady state by an iterative prcess that incrementally adds the lad while btaining a cnverged displacement field fr each step. This prcedure was intrduced in rder t allw fr large deflectins f the sail gemetry. Wrinkling f the structure, which is an imprtant cnsideratin especially arund the leading edge (luff) and at the sail tip, is nt anticipated by this mdel, but the use f a numerical algrithm t track large defrmatins allws fr wrinkling mdels t be included at a later stage. Figures 13 and 14 pictrially depict the bundary cnditins used fr the structural mdel. 4.2 Aerelastic Cupling Prcedure The pressure lading frm the flw slver is fed t the structural analysis t estimate the deflected shape f the sail. T enable the transfer f lads and displacements t be cnservative, the fluid mesh n the surface and the structural mesh are identical, eliminating the need fr interplatin. The deflected shape f the sail is used t defrm the cmputatinal mesh. The ppular spring-analgy methd has been used t track the mesh defrmatins. While this methd restricts the allwable deflectins and may impair the quality f the defrmed mesh, it prvides a simple tl t track 6 f 21 American Institute f Aernautics and Astrnautics Paper

8 mesh defrmatins. The defrmed mesh is then used t cmpute a new pressure lading fr the sail. This iterative prcess ffers n frmal guarantee f cnvergence, but it usually predicts the deflected shape t reasnable accuracy in a few steps (typically 5 fr sail gemetries). 4.3 Results f Aerelastic Simulatins The deflected shapes f the head and the main sail are shwn in figures 19 and 2. It can be seen frm these plts that the lwer sectins f the head and the main sail d nt underg appreciable defrmatin. The largest deflectins ccur in the mid-sectins f the main sail. As the pint f attachment f the main sail t the mast and the leading edge f the head sail were nt allwed t mve, the aerdynamic lading changes the twist f the sail gemetry. This has a favrable influence n the pressure distributin, especially n the head sail (figures 17 and 18). The pressure distributin ver the head and sail after the aerelastic simulatin highlights the need t perfrm aerelastic analysis t btain the flying shape f these sail gemetries. While the lift and the drag distributin f the defrmed shape is nt significantly different frm the undefrmed shape, the pressure distributin ver the sail sectins shw that the twist and the camber distributin f the head and the main sail have been altered t prvide a smth entry fr the flw ver the leading edge f bth cmpnents. 5. Aerdynamic Shape Optimizatin With the availability f high perfrmance cmputing platfrms and rbust numerical methds t simulate fluid flws, it is pssible t shift attentin t autmated design prcedures which cmbine CFD with gradient-based ptimizatin techniques. Typically, in gradient-based ptimizatin techniques, a cntrl functin t be ptimized (the sail shape, fr example) is parameterized with a set f design variables and a suitable cst functin t be minimized is defined. Fr aerdynamic prblems, the cst functin is typically lift, drag r a specified target pressure distributin. Then, a cnstraint, the gverning equatins can be intrduced in rder t express the dependence between the cst functin and the cntrl functin. The sensitivity derivatives f the cst functin with respect t the design variables are calculated in rder t get a directin f imprvement. Finally, a step is taken in this directin and the prcedure is repeated until cnvergence is achieved. Finding a fast and accurate way f calculating the necessary gradient infrmatin is essential t develping an effective design methd since this can be the mst time cnsuming prtin f the design prcess. This is particularly true in prblems which invlve a very large number f design variables as is the case in typical three dimensinal shape ptimizatin. The cntrl thery apprach has dramatic cmputatinal cst advantages ver the finite-difference methd f calculating gradients. With this apprach the necessary gradients are btained thrugh the slutin f an adjint system f equatins f the gverning equatins f interest. The adjint methd is extremely efficient since the cmputatinal expense incurred in the calculatin f the cmplete gradient is effectively independent f the number f design variables. In this study, a cntinuus adjint frmulatin has been used t derive the adjint system f equatins. Hence, the adjint equatins are derived directly frm the gverning equatins and then discretized. Hence, this apprach has the advantage ver the discrete adjint frmulatin in that the resulting adjint equatins are independent f the frm f discretized flw equatins. The adjint system f equatins have a similar frm t the gverning equatins f the flw and hence the numerical methds develped fr the flw equatins 1, 8, 14 can be reused fr the adjint equatins. The gradient is derived directly frm the adjint slutin and the surface mtin independent f the mesh mdificatin. This is critical fr this design methdlgy t wrk n unstructured meshes. If the gradient depends n the frm f the mesh mdificatin, then the field integral in the gradient calculatin has t be recmputed fr mesh mdificatins crrespnding t each design variable. T reduce the cmputatinal cst with this apprach, the number f design variables wuld have t reduced by parameterizing the gemetry. Hwever, this reduced set f design variables wuld be incapable f recvering all pssible shape variatins. Using the gradients cmputed with this new frmulatin, a steepest descent methd is used t imprve an existing design. 5.1 The General Frmulatin f the Adjint Apprach t Optimal Design Fr flw abut an airfil, r wing, the aerdynamic prperties which define the cst functin are functins f the flw-field variables, w, and the physical lcatin f the bundary, which may be represented by the functin, F, say. Then I = I(w, F), and a change in F results in a change δi = IT IT δw δf, (12) w F in the cst functin. Using cntrl thery, the gverning equatins f the flw field are intrduced as a cnstraint in such a way that the final expressin fr the gradient des nt require re-evaluatin f the flwfield. In rder t achieve this, δw must be eliminated frm equatin (12). Suppse that the gverning equatin R which expresses the dependence f w and F within the flw field dmain D can be written as 7 f 21 American Institute f Aernautics and Astrnautics Paper

9 R(w, F) = (13) Then δw is determined frm the equatin i = 1 t 3 is implied by a repeated index i. Then, the three-dimensinal Euler equatins may be written as w t f i x i = in D, (18) δr = [ ] R δw w [ ] R δf = (14) F Next, intrducing a Lagrange Multiplier ψ, we have ([ ] [ ] ) δi = IT IT R R δw δf ψt δw δf w F w F ( I T δi = w ψt [ ] ) ( R I T w δw w F ψt Chsing ψ t satisfy the adjint equatin [ ] R δf F where w = ρ ρu 1 ρu 2 ρu 3 ρe, f i = ρu i ρu i u 1 pδ i1 ρu i u 2 pδ i2 ρu i u 3 pδ i3 ρu i H ) and δ ij is the Krnecker delta functin. Als, (19) { p = (γ 1) ρ E 1 ( ) } u 2 2 i, (2) [ ] T R ψ = I w w the first term is eliminated and we find that where (15) δi = GδF (16) G = IT F ψt [ ] R F (17) This prcess allws fr eliminatin f the terms that depend n the flw slutin with the result that the gradient with respect with an arbitrary number f design variables can be determined withut the need fr additinal flw field evaluatins. After taking a step in the negative gradient directin, the gradient is recalculated and the prcess repeated t fllw the path f steepest descent until a minimum is reached. In rder t avid vilating cnstraints, such as the minimum acceptable wing thickness, the gradient can be prjected int an allwable subspace within which the cnstraints are satisfied. In this way ne can devise prcedures which must necessarily cnverge at least t a lcal minimum and which can be accelerated by the use f mre sphisticated descent methds such as cnjugate gradient r quasi- Newtn algrithms. There is a pssibility f mre than ne lcal minimum, but in any case this methd will lead t an imprvement ver the riginal design. 5.2 Design Using the Euler Equatins The applicatin f cntrl thery t aerdynamic design prblems is illustrated in this sectin fr the case f three-dimensinal wing design using the cmpressible Euler equatins as the mathematical mdel. It prves cnvenient t dente the Cartesian crdinates and velcity cmpnents by x 1, x 2, x 3 and u 1, u 2, u 3, and t use the cnventin that summatin ver and ρh = ρe p (21) where γ is the rati f the specific heats. Cnsider a transfrmatin t crdinates ξ 1, ξ 2, ξ 3 where [ ] [ ] xi K ij =, J = det (K), K 1 ξi ij =, ξ j x j and S = JK 1. The elements f S are the cfactrs f K, and in a finite vlume discretizatin they are just the face areas f the cmputatinal cells prjected in the x 1, x 2, and x 3 directins. Using the permutatin tensr ɛ ijk we can express the elements f S as Then S ij = 1 2 ɛ jpqɛ irs x p ξ r x q ξ s. (22) S ij = 1 ( 2 ξ i 2 ɛ x p x q jpqɛ irs x p 2 ) x q ξ r ξ i ξ s ξ r ξ s ξ i =. (23) Nw, multiplying equatin (18) by J and applying the chain rule, where J w t R (w) = (24) f j R (w) = S ij = (S ij f j ), (25) ξ i ξ i using (23). We can write the transfrmed fluxes in terms f the scaled cntravariant velcity cmpnents U i = S ij u j 8 f 21 American Institute f Aernautics and Astrnautics Paper

10 as F i = S ij f j = ρu i ρu i u 1 S i1 p ρu i u 2 S i2 p ρu i u 3 S i3 p ρu i H. Assume nw that the new cmputatinal crdinate system cnfrms t the wing in such a way that the wing surface B W is represented by ξ 2 =. Then the flw is determined as the steady state slutin f equatin (24) subject t the flw tangency cnditin U 2 = n B W. (26) At the far field bundary B F, cnditins are specified fr incming waves, as in the tw-dimensinal case, while utging waves are determined by the slutin. The weak frm f the Euler equatins fr steady flw can be written as D φ T ξ i F i dd = B n i φ T F i db, (27) where the test vectr φ is an arbitrary differentiable functin and n i is the utward nrmal at the bundary. If a differentiable slutin w is btained t this equatin, it can be integrated by parts t give φ T F i dd = ξ i D and since this is true fr any φ, the differential frm can be recvered. If the slutin is discntinuus (27) may be integrated by parts separately n either side f the discntinuity t recver the shck jump cnditins. Suppse nw that it is desired t cntrl the surface pressure by varying the wing shape. Fr this purpse, it is cnvenient t retain a fixed cmputatinal dmain. Then variatins in the shape result in crrespnding variatins in the mapping derivatives defined by K. As an example, cnsider the case f an inverse prblem, where we intrduce the cst functin I = 1 (p p d ) 2 dξ 1 dξ 3, 2 B W where p d is the desired pressure. The design prblem is nw treated as a cntrl prblem where the cntrl functin is the wing shape, which is t be chsen t minimize I subject t the cnstraints defined by the flw equatins (24). A variatin in the shape will cause a variatin δp in the pressure and cnsequently a variatin in the cst functin δi = (p p d ) δp dξ 1 dξ 3. (28) B W Since p depends n w thrugh the equatin f state (2 21), the variatin δp can be determined frm the variatin δw. Define the Jacbian matrices A i = f i w, C i = S ij A j. (29) 9 f 21 The weak frm f the equatin fr δw in the steady state becmes φ T δf i dd = (n i φ T δf i )db, ξ i where D B δf i = C i δw δs ij f j, which shuld hld fr any differential test functin φ. This equatin may be added t the variatin in the cst functin, which may nw be written as δi = (p p d ) δp dξ 1 dξ 3 B W ( ) φ T δf i dd D ξ i ( ni φ T ) δf i db. (3) B On the wing surface B W, n 1 = n 3 =. Thus, it fllws frm equatin (26) that S 21 δp δs 21 p δf 2 = S 22 δp δs 22 p. (31) S 23 δp δs 23 p Since the weak equatin fr δw shuld hld fr an arbitrary chice f the test vectr φ, we are free t chse φ t simplify the resulting expressins. Therefre we set φ = ψ, where the c-state vectr ψ is the slutin f the adjint equatin ψ t CT i ψ ξ i = in D. (32) At the uter bundary incming characteristics fr ψ crrespnd t utging characteristics fr δw. Cnsequently ne can chse bundary cnditins fr ψ such that n i ψ T C i δw =. Then, if the crdinate transfrmatin is such that δs is negligible in the far field, the nly remaining bundary term is ψ T δf 2 dξ 1 dξ 3. B W Thus, by letting ψ satisfy the bundary cnditin, S 21ψ 2 S 22ψ 3 S 23ψ 4 = (p p d ) n B W, (33) we find finally that ψ T δi = δs ijf jdd ξ D i American Institute f Aernautics and Astrnautics Paper (δs 21ψ 2 δs 22ψ 3 δs 23ψ 4) p dξ 1dξ 3. (34) B W

11 Here the expressin fr the cst variatin depends n the mesh variatins thrughut the dmain which appear in the field integral. Hwever, the true gradient fr a shape variatin shuld nt depend n the way in which the mesh is defrmed, but nly n the true flw slutin. In the next sectin, we shw hw the field integral can be eliminated t prduce a reduced gradient frmula which depends nly n the bundary mvement. 5.3 Adjint Equatins fr the Euler Equatins Mdified by the Artificial Cmpressibility Methd Althugh the adjint equatin represents a linear set f partial differential equatins fr the adjint variables, they are f the same frm f the flw equatins. The numerical slutin prcedures develped fr the flw equatins are applied t the adjint system with the apprpriate bundary cnditins. The adjint c-state flux terms are mdified t accunt fr the intrductin f the artificial cmpressibility terms in the gverning flw equatins. The methdlgy fllwed here is derived frm the wrk f Luigi Martinelli and G. Cwles. 2 The adjint field equatins can be expressed as a time dependent system f the frm, where ψ t [A i] T ψ = (35) x i ψ = p φ 1 φ 2 φ 3 (36) Hence, this system can be integrated t steady state using a pre-cnditiner similar t that used in the methd f artificial cmpressibility. The adjint cntinuity equatin is augmented by a time derivative f the adjint pressure p. p t φ Γ2 i = (37) x i The frm f Γ is identical t that used fr the flw equatins since the magnitude f the eigenvalues f the flux Jacbians fr the tw systems are identical. Tgether with equatin (37), the adjint system is discretized and slved in a manner that is cnsistent with that used fr the flw equatin. 5.4 Reduced Gradient Frmulatins Cntinuus adjint frmulatins have generally used a frm f the gradient that depends n the manner in which the mesh is mdified fr perturbatins in each design variable. T represent all pssible shapes the cntrl surface shuld be regarded as a free surface. If the surface mesh pints are used t define the surface, this leaves the designer with a thusands f design variables. On an unstructured mesh evaluating the 1 f 21 gradient by perturbing each design variable in turn, wuld be prhibitively expensive because f the need t determine crrespnding perturbatins f the entire mesh. This wuld inhibit the use f this design tl in any meaningful design prcess. In rder t avid this difficulty an alternate frmulatin t the gradient calculatin is fllwed in this study. This idea was develped by Jamesn and Sangh Kim 18 and was validated fr tw and three dimensinal prblems with structured grids. Hwever, as it is pssible t devise mesh mdificatin rutines that are cmputatinally cheap n structured grids, the majr benefit f this alternate gradient frmulatin is fr general three dimensinal unstructured grids. T cmplete the frmulatin f the cntrl thery apprach t shape ptimizatin, the gradient frmulatins are utlined next. The frmulatin fr the reduced gradients in the cntinuus limit is presented in the cntext f transfrmatin between the physical dmain and the cmputatinal dmain, and is easily extended t unstructured grid methds where these transfrmatins are nt explicitly used. The evaluatin f the field integral in equatin (34) requires the evaluatin f the metric variatins δs ij thrughut the dmain. Hwever, the true gradient shuld nt depend n the way the mesh is mdified. Cnsider the case f a mesh variatin with a fixed bundary. Then, δi = but there is a variatin in the transfrmed flux, δf i = C i δw δs ij f j. Here the true slutin is unchanged. Thus, the variatin δw is due t the mesh mvement δx at fixed bundary cnfiguratin. Therefre and since it fllws that δw = w δx = w x j δx j (= δw ) ξ i δf i =, ξ i (δs ij f j ) = ξ i (C i δw ). (38) It is verified by Jamesn and Sangh Kim 18 that this relatin hlds in the general case with bundary mvement. Nw D ψ T δrdd = = ψ T D American Institute f Aernautics and Astrnautics Paper B D ξ i C i (δw δw ) dd ψ T C i (δw δw ) db ψ T ξ i C i (δw δw ) dd. (39)

12 Here n the wall bundary C 2 δw = δf 2 δs 2j f j. (4) Thus, by chsing ψ t satisfy the adjint equatin and the adjint bundary cnditin, we have finally the reduced gradient frmulatin that δi = ψ T (δs 2j f j C 2 δw ) dξ 1 dξ 3 B W (δs 21 ψ 2 δs 22 ψ 3 δs 23 ψ 4 ) p dξ 1 dξ 3.(41) B W 5.5 The need fr a Sblev inner prduct in the definitin f the gradient Anther key issue fr successful implementatin f the cntinuus adjint methd is the chice f an apprpriate inner prduct fr the definitin f the gradient. It turns ut that there is an enrmus benefit frm the use f a mdified Sblev gradient, which enables the generatin f a sequence f smth shapes. When the metric perturbatins, δs ij, are related t the surface mtin, equatin (41) finally yields the cst variatin in the frm f an inner prduct defined ver the surface δi = (G, δf) = GδFdξ B where δf dentes the surface displacement. Then the update B F n1 = F n λg n wuld result in an imprvement 6. Mesh Defrmatin The mdificatins t the shape f the bundary are transferred t the vlume mesh using the spring methd. This apprach has been fund t be adequate fr the cmputatins perfrmed in this study. The spring methd can be mathematically cnceptualized as slving the fllwing equatin x i t N K ij ( x i x j ) = j=1 where the K ij is the stiffness f the edge cnnecting nde i t nde j and its value is inversely prprtinal t the length f this edge, x i is the displacement f nde i and x j is the displacement f nde j, the ppsite end f the edge. The psitin f static equilibrium f the mesh is cmputed using a Jacbi iteratin with knwn initial values fr the surface displacements. 7. Results The verall design prcess is illustrated in figure 6. Generate sequence f meshes Pre prcessr δi = λ (G, G)) It turns ut, hwever, that the gradient is generally a functin in a lwer smthness class than the initial shape, with the result that there is a prgressive lss f smthness with each iteratin. This can be crrected by intrducing a mdified gradient which crrespnds t a weighted Sblev inner prduct, f the frm u, v = (uv ɛu v )dξ This is equivalent t replacing G by G where in ne dimensin Flw Slver (Parallel) Adjint Slver (Parallel) Gradient evaluatin, camber line changes G ξ ɛ G ξ = G, with G = at the end pints and making a shape change δf = λg This bth preserves the smthness f the redesigned shape, and acts as a precnditiner which reduces the number f design steps needed t reach an ptimum slutin. Fig. 6 Mesh defrmatin Repeat until design criterin is satisfied Flw chart f the verall design prcess 11 f 21 American Institute f Aernautics and Astrnautics Paper

13 7.1 Shape Optimizatin f Airfils in Incmpressible Flw In rder t validate the design prcedure, tw dimensinal prblems were studied first. An inverse design prblem that recvers the pressure distributin ver the Onera M6 airfil was used t validate the gradient frmulatins and the adjint slutin. A three level multigrid cycle was used fr btain steady state slutins fr the flw and the adjint equatins. The grids were generated using a cnfrmal mapping technique. The initial airfil shape had a NACA 12 prfile and the initial pressure distributin is shwn in figure 21. Arund 4 design cycles were required t recver the target pressure and shape by the design prcess (figure 21). 7.2 Inverse Design f Wings in Incmpressible Flw T validate the design prcess fr three dimensinal flws, a test prblem similar t the tw dimensinal case was used. The initial wing had the planfrm f the Onera M6 but had NACA 12 airfil sectins. The target pressure distributin crrespnded t the steady state pressure distributin ver the Onera M6 wing. Three levels f multigrid were used t btain steady state flw and adjint slutins. The meshes were generated using an autmated grid generatr and interplatin cefficients were accumulated in a preprcessing step. The parallel implementatin f the flw and adjint slvers were used t reduce the cmputatinal time f the design prcess. Mdificatins t the shape f the wing were transmitted t the interir mesh using the spring defrmatin methd which wrked well fr this prblem. Figures 22,23,24,25 shw that the target pressure distributin has been recvered in abut 5 design cycles. These cmputatins tk under 3 minutes requiring 8 prcessrs f an SGI Origin Inverse Design fr Sail Gemetries The results f the flw and aerelastic simulatins, shw that the interactin f the head sail with the main reduces the develpment f sharp pressure gradients arund the leading edge. This interactin is crucial t the perfrmance f the main sail as it allws the main sail t be set at a higher angle t the center-line f the bat. These results als shw that the regin abve the head sail has large suctin peaks which is a cause f cncern. The aerdynamic shape ptimizatin prcedure validated in the previus sectins was used t redesign the main sail, with an aim f reducing the pressure gradient arund the luff f the main sail. An inverse design prcedure was emplyed and the target pressure distributin was btained by smthing the pressure distributin n the main sail btained frm the aerelastic analysis. Figures 26,27,28 and 29 shw that a significant prtin f the leading edge f the main sail has been redesigned t allw fr smth entry f the flw. The assciated reductin in sharp suctin peaks shuld have a favrable affect n the grwth f the bundary layer ver the upper surface. The change t the sectins are shwn in figure 8. Cnclusins Rtatinal inviscid flw slutins t sail cnfiguratins can nw btained in a cheap and efficient manner. These can be used t substitute ptential flw mdels t btain better estimates f the induced drag. Adjint based design methds can be used t redesign sail shapes t smth ut suctin peaks and imprve the sail perfrmance. The use f gradient frmulatins that depend nly f the surface mesh allws adjint based methds t be used fr unstructured grids in a cmputatinally efficient manner. Hence, it is nw pssible t devise cmpletely autmated shape ptimizatin prcedures fr sail cnfiguratins. Further expliting the flexibility f unstructured grids, it is pssible t devise a hierarchy f ptimizatin prcedures that can be used t tackle planfrm ptimizatin, sail settings, hull/keel shapes and ther cmpnents f the verall design prcess. This hlds great prmise fr the design f imprved sails. 9. Onging and future directins f Research Our current effrts are fcussed n develping cmputatinal methds fr viscus simulatins. Preliminary results frm a tw equatin turbulence mdel have prvided gd estimates f the frictin drag fr airplane cnfiguratins. Bundary layer methds cupled t inviscid flw slutins are als being investigated t prvide a cmputatinally cheap methd t estimate the viscus effects. Redesign f hull and keel shapes are within sight, and it is hped that in the near future, a cmbined analysis and design tl fr high perfrmance bats will be realized. References 1 A. Jamesn and T.J. Baker, Imprvements t the Aircraft Euler Methd, AIAA Paper , 25 th AIAA Aerspace Sciences Meeting, Ren, January, J. Reuther, A. Jamesn, J. Farmer, L. Martinelli and D. Saunders, Aerdynamic Shape ptimizatin f cmplex aircraft cnfiguratins via an adjint methd, AIAA Paper 96-94, 34 th AIAA Aerspace Sciences Meeting, Ren, January, A. Chrin, A Numerical Methd fr Slving the Incmpressible Viscus Flw prblem, Jurnal f Cmputatinal Physics Vl. 2, pp 12-26, J. Farmer, L. Martinelli and A. Jamesn, A Fast Multigrid Methd fr Slving Incmpressible Hydrdynamic Prblems with Free Surfaces, AIAA Paper , 31 st AIAA Aerspace Sciences Meeting, Ren, January, A. Rizzi and L. Erikssn, Cmputatin f inviscid incmrpessible flw with rtatin, Jurnal f Fluid Mechanics Vl. 153, pp , A. Jamesn, Analysis and Design f numerical schemes fr gas dynamics II, Artificial diffusin and discrete shck structure, Internatinal Jurnal f Cmputatinal Fluid Dynamics, Vl. 5, pp. 1-38, f 21 American Institute f Aernautics and Astrnautics Paper

14 7 A. Jamesn, Multigrid algrithms fr cmpressible flw calculatins, In W. Hackbusch and U. Trttenberg, editrs, Lecture Ntes in Mathematics, Vl. 1228, pages Prceedings f the 2nd Eurpean Cnference n Multigrid Methds, Clgne, 1985, Springer-Verlag, A. Jamesn, W. Schmidt and E. Turkel, Numerical Slutin f the Euler equatins by finite vlume methds using Runge- Kutta time stepping schemes, AIAA Paper , June, A. Jamesn, Multigrid algrithms fr cmpressible flw calculatins, Prceedings f the 2nd Eurpean Cnference n Multigrid Methds, Clgne, A. Jamesn, Unpublished ntes and cmputatinal techniques fr the slutin f the unsteady Euler equatins n unstructured grids using multigrid methds. 11 J.L. Lins, Optimal Cntrl f Systems Gverned by Partial Differential Equatins, Springer-Verlag, New Yrk, Translated by S.K. Mitter. 12 O. Pirnneau, Optimal Shape Design fr Elliptic Systems, Springer-Verlag, New Yrk, A. Jamesn, Optimum Aerdynamic Design Using Cntrl Thery, Cmputatinal Fluid Dynamics Review 1995 Wiley, T.J. Barth, Apects f unstructured grids and finite vlume slvers fr the Euler and Navier-Stkes equatins, AIAA Paper , 29 th AIAA Aerspace Sciences Meeting, Ren, January, K. Andersn and V. Venkatakrishnan, Aerdynamic Design Optimizatin n Unstructured grids using a cntinuus adjint frmulatin, AIAA Paper , 34 th AIAA Aerspace Sciences Meeting, Ren, January, J. Ellit and J. Peraire, Aerdynamic design using unstructured meshes, AIAA Paper , 33 rd AIAA Aerspace Sciences Meeting, Ren January, S. E. Cliff, S.D. Thmas, T. J. Baker, A. Jamesn and R. M. Hicks, Aerdynamic Shape ptimizatin using unstructured grid methd, AIAA Paper 2-555, 9 th AIAA Sympsium n Multidisciplinary Analysis and Optimizatin, Atlanta, September, A. Jamesn and Sangh Kim, Reductin f the Adjint Gradient Frmula in the Cntinuus Limit, AIAA Paper, 41 st AIAA Aerspace Sciences Meeting, Ren January, A. Jamesn, L. Martinelli and J. Vassberg, Using CFD fr Aerdynamics - A critical Assesment, Prceedings f ICAS 22, September 8-13, 22, Trnt, Canada. 2 G. Cwles and Luigi Martinelli, A Cntrl-Thery Based Methd fr Shape Design in Incmpressible Viscus Flw using RANS, AIAA Fluids , June 19-22, Denver, CO. 13 f 21 American Institute f Aernautics and Astrnautics Paper

15 Twisted inflw with bundary layer prfile 1 m AIRPLANE CP frm -1. t -.5 Main Sail 24 m Jib 2.3 m 1 m Fig. 7 Sail gemetry Fig. 9 Pressure cnturs n the leeward side AIRPLANE 1 2 Cnvergence Histry 1 1 grid 2 grid 3 grid CP frm -.6 t Lg(Errr) Number f iteratins Fig. 8 Cnvergence histry fr sail gemetries with artificial cmpressibility Fig. 1 Pressure cnturs n the windward side 14 f 21 American Institute f Aernautics and Astrnautics Paper

16 1.2 Lift and Drag distributin alng the height f the head sail Cl Cd Translatinal degrees f freedm 1 suppressed at the mast.8.6 Leech is allwed t mve freely Mast is assumed t be rigid Height alng the span Translatinal and rtatinal degrees f freedm suppressed at the bm Fig. 11 sail Spanwise frce distributins n the head Fig. 13 Bundary cnditins fr the main sail 1.4 Lift and Drag distributin alng the height f the head sail mast head.8.6 Cl Cd leech stay Translatinal degrees f freedm supressed alng the stay.4 luff.2 clew ft Height alng the span All degrees f freedm suppressed Fig. 12 sail Spanwise frce distributins n the main Fig. 14 Bundary cnditins fr the head sail 15 f 21 American Institute f Aernautics and Astrnautics Paper