Design Optimization of High Lift Configurations Using a Viscous Continuous Adjoint Method

Transcription

1 AIAA Design Optimizatin f High Lift Cnfiguratins Using a Viscus Cntinuus Adjint Methd Sangh Kim, Juan J. Alns, and Antny Jamesn Stanfrd University, Stanfrd, CA th AIAA Aerspace Sciences Meeting and Exhibit January 14 17, 2002/Ren, NV Fr permissin t cpy r republish, cntact the American Institute f Aernautics and Astrnautics 1801 Alexander Bell Drive, Suite 500, Restn, VA

2 AIAA Design Optimizatin f High Lift Cnfiguratins Using a Viscus Cntinuus Adjint Methd Sangh Kim, Juan J. Alns, and Antny Jamesn Stanfrd University, Stanfrd, CA An adjint-based Navier-Stkes design and ptimizatin methd fr tw-dimensinal multi-element high-lift cnfiguratins is derived and presented. The cmpressible Reynlds-Averaged Navier-Stkes (RANS) equatins are used as a flw mdel tgether with the Spalart-Allmaras turbulence mdel t accunt fr high Reynlds number effects. Using a viscus cntinuus adjint frmulatin, the necessary aerdynamic gradient infrmatin is btained with large cmputatinal savings ver traditinal finite-difference methds. A study f the accuracy f the gradient infrmatin prvided by the adjint methd in cmparisn with finite differences and an inverse design f a single-element airfil are als presented fr validatin f the present viscus adjint methd. The highlift cnfiguratin design methd uses a cmpressible RANS flw slver, FLO103-MB, a pint-t-pint matched multi-blck grid system and the Message Passing Interface (MPI) parallel slutin methdlgy fr bth the flw and adjint calculatins. Airfil shape, element psitining, and angle f attack are used as design variables. The predictin f high-lift flws arund a baseline three-element airfil cnfiguratin, dented as 30P30N, is validated by cmparisns with experimental data. Finally, several design results that verify the ptential f the methd fr high-lift system design and ptimizatin, are presented. The design examples include a multi-element inverse design prblem and the fllwing prblems: C l maximizatin, lift-t-drag rati, L/D, maximizatin by minimizing C d at a given C l r maximizing C l at a given C d (α is allwed t flat t maintain either C l r C d ), and the maximum lift cefficient, C lmax, maximizatin prblem fr bth the RAE2822 single-element airfil and the 30P30N multi-element airfil. Intrductin AERODYNAMIC shape design has lng been a challenging bjective in the study f fluid dynamics. Cmputatinal Fluid Dynamics (CFD) has played an imprtant rle in the aerdynamic design prcess since its intrductin fr the study f fluid flw. Hwever, CFD has mstly been used in the analysis f aerdynamic cnfiguratins in rder t aid in the design prcess rather than t serve as a direct design tl in aerdynamic shape ptimizatin. Althugh several attempts have been made in the past t use CFD as a direct design tl, 1 5 it has nt been until recently that the fcus f CFD applicatins has shifted t aerdynamic design This shift has been mainly mtivated by the availability f high perfrmance cmputing platfrms and by the develpment f new and efficient analysis and design algrithms. In particular, autmatic design prcedures which use CFD cmbined with gradient-based ptimizatin tech- Dctral Candidate, AIAA Member Assistant Prfessr, Department f Aernautics and Astrnautics, AIAA Member Thmas V. Jnes Prfessr f Engineering, Department f Aernautics and Astrnautics, AIAA Fellw yright c 2002 by the authrs. Published by the American Institute f Aernautics and Astrnautics, Inc. with permissin. niques, have made it pssible t remve difficulties in the decisin making prcess faced by the aerdynamicist. Typically, in gradient-based ptimizatin techniques, a cntrl functin t be ptimized (an airfil shape, fr example) is parameterized with a set f design variables, and a suitable cst functin t be minimized r maximized is defined (drag cefficient, lift/drag rati, difference frm a specified pressure distributin, etc.) Then, a cnstraint, the gverning equatins in the present study, 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 t a minimum r maximum 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 algrithm. Gradient infrmatin can be cmputed using a variety f appraches, such as the finite-difference methd, the cmplex step methd 12 and autmatic differentiatin. 13 Unfrtu- 1 f 20 American Institute f Aernautics and Astrnautics Paper

3 nately, their cmputatinal cst is prprtinal t the number f design variables in the prblem. As an alternative chice, the cntrl thery apprach has dramatic cmputatinal cst advantages when cmpared t any f these methds. The fundatin f cntrl thery fr systems gverned by partial differential equatins was laid by J.L. Lins. 14 The cntrl thery apprach is ften called the adjint methd, since the necessary gradients are btained via the slutin f the adjint 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. The nly cst invlved is the calculatin f ne flw slutin and ne adjint slutin whse cmplexity is similar t that f the flw slutin. Cntrl thery was applied in this way t shape design fr elliptic equatins by Pirnneau 15 and it was first used in transnic flw by Jamesn. Since then this methd 6, 7, 16 has becme a ppular chice fr design prblems invlving fluid flw. 9, In fact, the methd has even been successfully used fr the aerdynamic design f 8, 20 cmplete aircraft cnfiguratins. Mst f the early wrk in the frmulatin f the adjint-based design framewrk used the ptential and Euler equatins as mdels f the fluid flw. Aerdynamic design calculatins using the Reynlds- Averaged Navier-Stkes equatins as the flw mdel have nly recently been tackled. The extensin f adjint methds fr ptimal aerdynamic design f viscus prblems is necessary t prvide the increased level f mdeling which is crucial fr certain types f flws. This cannt nly be cnsidered an academic exercise. It is als a very imprtant issue fr the design f viscus dminated applicatins such as the flw in high-lift systems. In 1997, a cntinuus adjint methd fr Aerdynamic Shape Optimizatin (ASO) using the cmpressible Navier-Stkes equatins was frmulated and it has been implemented directly in 16, 21 a three dimensinal wing prblem. In 1998, an implementatin f three-dimensinal viscus adjint methd was used with sme success in the ptimizatin f the Blended-Wing-Bdy cnfiguratin. 22 Since these design calculatins were carried ut withut the benefit f a careful check n the accuracy f the resulting gradient infrmatin, a series f numerical experiments in tw dimensins that assessed the accuracy f the viscus adjint gradient infrmatin were cnducted by the authrs. 23 The research in this paper addresses the validity f this design methdlgy fr the prblem f high-lift design. Traditinally, high-lift designs have been realized by careful wind tunnel testing. This apprach is bth expensive and challenging due t the extremely cmplex nature f the flw interactins that appear. CFD analyses have recently been incrprated t the high-lift design prcess. 24 Eyi, Lee, Rgers and Kwak have perfrmed design ptimizatins f a high-lift system cnfiguratin using a chimera verlaid grid system and the incmpressible Navier-Stkes equatins. 25 Besnard, Schmitz, Bscher, Garcia and Cebeci perfrmed ptimizatins f high-lift systems using an Interactive Bundary Layer (IBL) apprach. 26 All f these earlier wrks n multi-element airfil design btained the necessary gradients by finite-difference methds. In this wrk, ur viscus adjint methd is applied t tw-dimensinal high-lift system designs, remving the limitatins n the dimensinality f the design space by making use f the viscus adjint design methdlgy. The mtivatin fr ur study f high-lift system design is twfld. On the ne hand, we wuld like t imprve the take-ff and landing perfrmance f existing high-lift systems using an adjint frmulatin, while n the ther hand, we wuld like t setup numerical ptimizatin prcedures that can be useful t the aerdynamicist in the rapid design and develpment f high-lift system cnfiguratins. In additin t difficulties invlved in the predictin f cmplex flw physics, multi-element airfils prvide an additinal challenge t the adjint methd: the effect f the changes in the shape f ne element must be felt by the ther elements in the system. While preliminary studies f the adjint methd in such a situatin have already been carried ut, 19, 27 this research is designed t validate the adjint methd fr cmplex applicatins f this type. Emphasis is placed n the validatin and nt n the creatin f realistic designs, which is beynd the scpe f this wrk. Prcedure In this sectin we utline the verall design prcedure used fr a variety f design calculatins that will be presented later. After the initial flwchart, each f the items f the prcedure are explained in mre detail. In practical implementatins f the adjint methd, a design cde can be mdularized int several cmpnents such as the flw slver, adjint slver, gemetry and mesh mdificatin algrithms, and the ptimizatin algrithm. After parameterizing the cnfiguratin f interest using a set f design variables and defining a suitable cst functin, which is typically based n aerdynamic perfrmance, the design prcedure can be described as fllws. First slve the flw equatins fr the flw variables, then slve the adjint equatins fr the cstate variables subject t apprpriate bundary cnditins which will depend n the frm f the cst functin. Next evaluate the gradients and update the aerdynamic shape based n the directin f steepest descent. Finally repeat the prcess t attain an ptimum cnfiguratin. A summary f the design prcess and a cmparisn with the finite-difference methd are illustrated in Figure 1. 2 f 20 American Institute f Aernautics and Astrnautics Paper

4 *nte: prcess in the dashed bx are repeated N times where N is the number f design variables. Slve Adjint Equatins Perturb Shape & Mdify Grid Evaluate Gradient Define Design Variables Parameterize Cnfiguratin Define Cst Functin Slve Flw Equatins Adjint Finite Difference Perturb Shape & Mdify Grid Slve Flw Equatins Evaluate Gradient In fact, fr the drag minimizatin f a single-element RAE2822 airfil in transnic flw, the strng shck present at transnic flw cnditins can nly be eliminated using a small change in the shape f the airfil. The crdinates f mesh ndes n the surface f the airfil, Hick-Henne bump functins, patched plynmials and frequency-based decmpsitins can be used t represent each f the elements in the highlift system. Fr example, a number f the fllwing Hicks-Henne functins, which have been implemented and used fr this study, may be added t the baseline airfil t mdify the shape: [ )] b (x) = A sin (πx lg 5 t2 lg t 1, 0 x c s 40 t 20 0 Adjint vs. Finite Difference in Cst, Fig. 1 N = 100 Adjint Finite Difference Optimizatin Optimum? Yes Stp N New Shape Flwchart f the Design Prcess Fig. 2 Definitins f Gap, Overlap, and Deflectin Angles Design Variables The rigging quantities that describe the relative element psitining f the slat, main element and flap are used as design variables. These variables include flap and slat deflectin angles, gaps, and verlaps. The meaning f these variables can be easily seen in Figure 2 fr typical multi-element airfil cnfiguratins. Fr the present study, gaps and verlaps are used in an indirect way since the rigging is cntrlled by translatin f the slat and flap leading edges in the x and y directins. In this way, the element psitining variables can be mre easily changed independently f each ther. The actual values f verlaps and gaps can be easily recvered frm their leading and trailing edge lcatins. The shapes f each f the elements are als used as design variables s as nt t rule ut the pssibility that the ptimum slutin may be btained with a cmbinatin f shape and psitin mdificatins. Here, A is the maximum bump magnitude, t 1 lcates the maximum f the bump at x = t 1, and t 2 cntrls the width f the bump. Using this parameterizatin, tw ptins are available fr btaining the ptimum C lmax. Firstly, C lmax can be predicted by maximizing C l at a given angle f attack, then predicting the C lmax alng a C l vs. α line fr that cnfiguratin, and repeating this prcedure iteratively. Alternatively, C lmax may als be maximized directly by including angle f attack as a design variable in the ptimizatin prcess. Baseline Cnfiguratin Wind tunnel measurements have been perfrmed n several three-element airfil cnfiguratins at the NASA Langley Lw Turbulence Pressure Tunnel (LTPT) at varius Reynlds and Mach numbers and the results f many CFD cmputatins fr these gemetries have been reprted using a variety f numerical schemes fr the discretizatin f the Navier- Stkes equatins and different turbulence mdels One f these three-element cnfiguratins, dented as 30P30N, is used as a starting pint fr the present design ptimizatin prcess. It must be nted that the 30P30N cnfiguratin had already been highly ptimized fr C lmax. The initial deflectins f bth the slat and the flap are set at 30, the flap gap and verlap are c/0.0025c whereas fr the slat the gap and verlap are c/0.025c, where c is the airfil chrd with the slat and flap retracted. Grid Tplgy A multi-blck mesh is generated prir t the beginning f the iterative design lp s that the flw and adjint equatins can be suitably discretized. Figure 3 shws a typical multi-blck mesh generated arund the 30P30N cnfiguratin. As shwn in clse-up in Figures 4 and 5, ne-t-ne pint cnnectivity between blck faces is emplyed t ensure cnservatin acrss bundaries and t prvide fr cntinuity f the grid at blck interfaces. Spacing at the wall is set t be less than c in rder t btain y = O(1) based n turbulent bundary layer thickness estimates 3 f 20 American Institute f Aernautics and Astrnautics Paper

5 Fig. 3 Grid arund the 30P30N multi-element cnfiguratin Fig. 5 Grid arund the 30P30N multi-element cnfiguratin : Clse-up arund Flap element tially equivalent t shifting grid pints alng crdinate lines depending n the mdificatins t the shape f the bundary. The mdificatin t the grid has the frm ld xnew = xld N xnew airf il xairf il new ld y new = y ld N yairf il yairf il. Here, lengthttal lengthj. lengthttal The details f the prcedure used have been presented earlier and can be fund in Ref.10, 11 N = Multi-blck Flw and Adjint Slvers Fig. 4 Grid arund the 30P30N multi-element cnfiguratin : Clse-up between Slat and Main elements frm a flat plate at the Reynlds number in questin (Re = ). The use f a grid with adequately tight wall spacing (y = O(1)) has been reprted t be necessary in rder t btain accurate reslutin f the wall bundary layers, wakes, and shear layers present in the prblem.34, 35 Grid lines are als bunched alng the expected wake trajectries f each f the elements. Once the initial grid is generated, new grids crrespnding t mdified airfil shapes are btained autmatically during the design prcess by using a tw-dimensinal versin f the autmatic mesh perturbatin scheme (WARP-MB)20, 36 that is essen- The predictin f high-lift flws pses a particularly difficult challenge fr bth CFD and turbulence mdeling. Even in tw-dimensins, the physics invlved in the flw arund a gemetrically-cmplex high-lift device is quite sphisticated. In this study FLO103-MB, a multi-blck RANS slver derived frm the wrk f Martinelli and Jamesn37 and similar t the three-dimensinal versin f Reuther and Alns,9 is used fr multi-element airfil flw-field predictins. FLO103-MB satisfies the requirements f accuracy, cnvergence, and rbustness that are necessary in this wrk. FLO103-MB slves the steady tw-dimensinal RANS equatins using a mdified explicit multistage Runge-Kutta time-stepping scheme. A finite vlume technique and secnd-rder central differencing in space are applied t the integral frm f the NavierStkes equatins. The Jamesn-Schmidt-Turkel(JST) scheme with adaptive cefficients fr artificial dissipatin is used t prevent dd-even scillatins and t allw fr the clean capture f shck waves and cntact discntinuities. In additin, lcal time stepping, 4 f 20 American Institute f Aernautics and Astrnautics Paper

6 implicit residual smthing, and the multigrid methd are applied t accelerate cnvergence t steady-state slutins. The Baldwin-Lmax algebraic mdel and the Spalart-Allmaras ne equatin mdel are used t mdel the Reynlds stress. The adjint gradient accuracy study which was presented in a previus publicatin 23 was based n the Baldwin-Lmax mdel. This mdel is als used fr the single-element design cases in this paper. Althugh this algebraic mdel has sme advantages due t its implementatinal simplicity and rbustness, the use f this mdel must be restricted t design at lwer angles f attack and t the design f simpler gemetries such as single-element airfils. Fr actual high-lift designs such as C lmax maximizatin, the ne-equatin Spalart-Allmaras mdel gives better predictins f bth the C lmax and the flw physics 32, 33, 38 arund cmplex gemetries. The turbulent equatin is slved separately frm the flw equatins using an alternating directin implicit (ADI) methd. The turbulence equatin is updated at the start f each multistage Runge-Kutta time step n the finest grid f the multigrid cycle nly. The adjint slutin is btained with the exact same numerical techniques used fr the flw slutin. The implementatin exactly mirrrs the flw slutin mdules inside FLO103-MB, except fr the bundary cnditins which are impsed n the c-state variables. The parallel implementatin uses a dmain decmpsitin apprach, a SPMD (Single Prgram Multiple Data) structure, and the MPI standard fr message passing. Cntinuus Adjint Methd The prgress f a design prcedure is measured in terms f a cst functin I, which culd be, fr example the drag cefficient r the lift t drag rati. Fr the 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 [ ] I T δi = w I [ ] I T δw δf, (1) F II in the cst functin. Here, the subscripts I and II are used t distinguish the cntributins due t the variatin δw in the flw slutin frm the change assciated directly with the mdificatin δf in the shape. 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 multiple flw slutins. This crrespnds t eliminating δw frm (1). Suppse that the gverning equatin R which expresses the dependence f w and F within the flw field dmain D can be written as R (w, F) = 0. (2) Then δw is determined frm the equatin [ ] [ ] R R δr = δw δf = 0. (3) w F I Next, intrducing a Lagrange Multiplier ψ, we have [ ] [ ] I T I T δi = δw δf w F I II ([ ] [ ] ) ψ T R R δw δf w I F II { I T [ ] } R = w ψt δw w I { I T [ ] } R F ψt δf. F Chsing ψ t satisfy the adjint equatin [ ] T R ψ = I w w the first term is eliminated, and we find that where II II (4) δi = GδF, (5) G = IT F ψt [ ] R. (6) F The advantage is that (5) is independent f δw, with the result that the gradient f I with respect t an arbitrary number f design variables can be determined withut the need fr additinal flw-field evaluatins. In the case that (2) is a partial differential equatin, the adjint equatin (4) is als a partial differential equatin and determinatin f the apprpriate bundary cnditins requires careful mathematical treatment. The cmputatinal cst f gradient calculatin fr a single design cycle is rughly equivalent t the cst f tw flw slutins since the adjint prblem has similar cmplexity. When the number f design variables becmes large, the cmputatinal efficiency f the cntrl thery apprach ver the traditinal finitedifferences, which require direct evaluatin f the gradients by individually varying each design variable and recmputing the flw field, becmes cmpelling. The frmulatin f the adjint equatin and the bundary cnditins are described in greater detail in previus publicatins 22 and a detailed gradient accuracy study fr the cntinuus adjint methd can be fund in Ref. 23 Numerical Optimizatin Methd The search prcedure used in this wrk is a simple steepest descent methd in which small steps are taken in the negative gradient directin. δf = λg, 5 f 20 American Institute f Aernautics and Astrnautics Paper

7 where λ is psitive and small enugh that the first variatin is an accurate estimate f δi. Then δi = λg T G < 0. After making such a mdificatin, the gradient can be recalculated and the prcess repeated t fllw a path f steepest descent until a minimum is reached. In rder t avid vilating cnstraints, such as a minimum acceptable airfil thickness, the gradient may be prjected int an allwable subspace within which the cnstraints are satisfied. In this way, prcedures can be devised which must necessarily cnverge at least t a lcal minimum. Results Validatin f the Adjint Methd fr Viscus Flws This sectin presents the results f a gradient accuracy study fr the RANS equatins using the Baldwin- Lmax turbulence mdel, as well as a simple example f the use f the resulting gradient infrmatin in a single-element airfil inverse design case. Gradient accuracy is assessed by cmparisn with finite-difference gradients and by examinatin f the changes in the magnitude f the gradients fr different levels f flw slver cnvergence. Fr inverse design, the aerdynamic cst functin chsen is given by: I = 1 (p p d ) 2 ds, (7) 2 B which is simply the Euclidean nrm f the difference between the current pressure distributin and a desired target, p d, at a cnstant angle f attack, α. The gradient f the abve cst functin is btained with respect t variatins in 50 Hicks-Henne sine bump functins centered at varius lcatins alng the upper and lwer surfaces f a baseline airfil. The lcatins f these gemetry perturbatins are rdered sequentially such that they start at the trailing edge n the lwer surface, prceed frward t the leading edge, and then back arund t the trailing edge n the upper surface. Figure 6 shws a cmparisn between the mst accurate gradients btained using bth the adjint and finite-difference methds. There is a general agreement n all trends that validates the implementatin f the present adjint methd. The fundamental advantage is restated here that the gradient with respect t an arbitrary number f design variables, 50 fr this example, is determined with the cst f nly a single flw field evaluatin and a single adjint evaluatin fr any given design cycle. Figure 7 shws the cmputed adjint gradients fr different levels f flw slver cnvergence. Fr Navier- Stkes calculatins, the adjint infrmatin is essentially unchanged if the level f cnvergence in the flw slver is at least 4 rders f magnitude. This is an additinal advantage f using the adjint methd especially fr the design f high-lift cnfiguratins fr which it is difficult t btain levels f cnvergence much higher than 4 rders f magnitude. This is in cntrast with the high levels f cnvergence required fr accurate sensitivity infrmatin when using the finite difference methd. In viscus flws it is typical t require that the flw slver cnverge t abut 6 rders f magnitude s that the gradient infrmatin is sufficiently accurate. An inverse design prblem which starts with an RAE2822 airfil gemetry and tries t btain the shape that generates the pressure distributin arund a NACA 64A410 airfil at the same flw cnditins is presented here. The mesh used fr this Navier- Stkes calculatin is a C-mesh with cells. The target pressure specified is that f a NACA 64A410 airfil at M = 0.75 and α = 0.0. The Reynlds number f this calculatin was set at Re = 6.5 millin. Figure 8 shws the prgress f the inverse design calculatin. In 100 design iteratins, the target pressure was matched almst exactly, including the crrect strength and psitin f the shck. The initial RAE2822 airfil gemetry was altered t btain a shape that is quite clse t the NACA 64A410 airfil that had prduced the target pressure distributin in the first place. The RMS f the pressure errr was reduced frm t in 100 design iteratins. FLO103-MB with SA Mdel Validatin Befre using FLO103-MB with the Spalart-Allmaras (SA) turbulence mdel fr the design f multi-element airfils we must demnstrate its ability t accurately predict the flw phenmena invlved in this type f prblem. In particular it is f primary imprtance t be able t predict the values f C lmax, the element pressure distributins, the lift curve slpes fr each element and the details f the shear layers present in the prblem. Flw cnvergence Figure 9 shws the cnvergence histry f the averaged density residual fr the calculatin f the flw field arund the 30P30N high-lift cnfiguratin using the flw slver, FLO103-MB. The Spalart-Allmaras ne-equatin turbulence mdel is used fr this calculatin. The slutin cnverges dwn t smewhere in the range f 10 4 and 10 5 in abut 2000 iteratins. Althugh small scillatins in the residual remain after 2000 iteratins, the C l, ne f the cst functins used fr subsequent designs, has cnverged withut scillatins. As mentined earlier, this level f cnvergence is als gd enugh t btain accurate sensitivity infrmatin using the adjint methd. Cmparisns with Experimental Data Cmparisns between cmputatinal results and experimental data are presented belw fr validatin 6 f 20 American Institute f Aernautics and Astrnautics Paper

8 purpses. The cde, FLO103-MB, and the related turbmachinery cde TFLO, 39 have been extensively validated fr a variety f test cases, ranging frm flat plates and transnic axisymmetric bumps, t full three-dimensinal cnfiguratins. Figure 10, shws the cmparisn f the cmputatinal and experimental C p distributins arund the 30P30N cnfiguratin at M = 0.2, α = 8, and Re= The agreement between experimental and cmputatinal distributins is very encuraging. Integrated frce cefficients als agree quite well. In rder t validate the ability f the flw slver t predict stall using the SA turbulence mdel, a cmparisn f C l versus angle f attack is shwn in Figure 11. The ttal cefficient f lift, tgether with the individual lift frm the three cmpnents, is pltted in the range f 5 < α < 25. The cmputed results agree quite well with experiment with slightly higher predictins f C lmax and angle f attack at C lmax. Althugh the results d nt agree with experiment exactly, it has been bserved that the chice f turbulence mdel can have a substantial impact n the numerical values f sme f these parameters. The stall predictin capability can be a critical factr fr actual design cases, such as C lmax maximizatin. With ther turbulence mdels, these quantities can be ver-predicted substantially. Finally, fr further validatin, velcity prfiles frm bth cmputatin and experiment are cmpared at tw different lcatins n the main element and the flap. Streamwise velcity cmpnents are pltted against nrmal distance t the wall at x/c = 0.45 n the main element and at x/c = n the flap. The cmputed velcity prfiles predict well bth the velcity gradient in the bundary layer and the gradients due t the wakes f the slat and main element. Single-Element Airfil Design Befre embarking n multi-element airfil design, a study f the use f ptimizatin fr single-element airfils was perfrmed t gain insight int the pssibilities fr imprvements and the behavir f the methd. C d minimizatin at a fixed C l and C l maximizatin at a fixed C d were tested in rder t guarantee the imprvement f the lift ver drag rati, L/D, which is a measure f the aerdynamic efficiency. The design examples presented in this subsectin were all carried ut using a 4 blck multi-blck mesh arund an RAE2822 airfil with a ttal number f cells equal t The tw designs presented had as a starting pint the RAE2822 airfil and cmputatins were carried ut at a Reynlds number f 6.5 millin. The surface f the airfil was parameterized using 50 Hicks-Henne bump functins, 25 f which are distributed evenly alng the upper surface f the airfil, while the remaining 25 are placed in a similar fashin alng the lwer surface. C d Minimizatin at a Fixed C l Figure 13 shws the result f a typical viscus design calculatin where the ttal cefficient f drag f the airfil is minimized using the parameterizatin described abve. The free stream Mach number is 0.73 and the ptimizatin prcedure is frced t achieve a near cnstant C l = This cnstraint is achieved by peridically adjusting the angle f attack during the flw slutin prtin f the design prcedure. Figure 13 shws the result f 50 design iteratins fr this test case. The ptimizer is able t eliminate the strng shck wave that existed in the initial design by using the values f the same 50 design variables. Once the design prcess is cmpleted, the ttal cefficient f drag has been reduced frm t , while the C l has increased very slightly frm t The L/D has imprved significantly, leading t mre efficient designs. In the case f C d minimizatin at a fixed C l, the L/D imprved by 54.36% frm t This test case als prvides a validatin f the multiblck design prcedure, since a similar test case had previusly been run using the single-blck design cde. C l Maximizatin at a Fixed C d In this test case, we attempt t maximize the C l f the RAE2822 airfil by altering its shape using the same 50 Hicks-Henne design functins while cnstraining the cefficient f drag t be cnstant (C d = ). Figure 14 shws the result f this type f design ptimizatin. The frnt prtin f the upper surface f the cnfiguratin is mdified cnsiderably t prduce a very different pressure distributin that allws fr the existence f a shck wave n the upper surface that cnsiderably increases the amunt f lift carried by the airfil. In additin, since the C d is cnstrained t be cnstant (this is impsed by allwing the angle f attack t flat), the resulting angle f attack is als higher, again leading t the creatin f a higher lift cefficient. The numerical results presented shw significant imprvements in L/D. The resulting L/D increased 21.70% frm t Multi-Element Airfil Design Except fr the inviscid test case presented in the first subsectin belw, all f the results in this sectin were cmputed using multi-blck viscus meshes cnstructed using a C-tplgy. The C-tplgy mesh has 26 blcks f varying sizes and a ttal f 204,800 cells. All calculatins were carried ut at a free stream Mach number, M = 0.20 and a Reynlds number, Re= The cmputatin f the Reynlds stress was carried ut using the Spalart-Allmaras turbulence mdel. Apart frm the first inviscid test case, the results in this sectin mimic thse in the single-element airfil sectin as far as the design prcedure is cncerned. 7 f 20 American Institute f Aernautics and Astrnautics Paper

9 Inverse Design Using the Euler Equatins In rder t verify the implementatin f ur design prcedure, we present a simple test case which is aimed at verifying that the multi-blck flw and adjint slvers are capable f prducing crrect sensitivities t bth shape mdificatins and rigging variables in a multi-element airfil design envirnment. Fr this purpse, a multi-blck inviscid grid arund the 30P30N cnfiguratin was cnstructed. A perturbed gemetry was created by activating a single bump n the upper surface f the main element and by deflecting the flap by an increment f 2. The pressure distributin arund the riginal gemetry is used as a target pressure distributin (seen as a slid line in Figure 15) fr the perturbed gemetry t arrive at thrugh an inverse design prcess. Because this target pressure distributin is achievable, we can indirectly measure the crrectness f the sensitivity infrmatin by bserving whether the design evlves twards the knwn specified target. Ntice that the mdificatin f the gemetry described abve influences the pressure distributin in all three elements: slat, main element, and flap. A ttal f 156 design variables were used t parameterize the cmplete cnfiguratin. 50 bump functins are used in each f the three elements. In additin, bth the slat and the flap were allwed t translate in the x and y directins and t rtate abut their leading-edges. After 100 design iteratins where sensitivities with respect t all design variables were calculated, the target pressure distributin was recvered as expected. The riginal gemetry was als recvered as shwn in Figure 15. The results f this inviscid test case prvide the necessary cnfidence t tackle sme f the mre cmplex viscus cases presented belw. C d Minimizatin at a Fixed C l In this test case, we attempt t minimize the ttal drag cefficient f the cnfiguratin withut changing the lift cefficient. This design task is ne f the mst interesting prblems since decreasing C d withut lss in C l is the mst effective way f increasing the lift ver drag rati f airfil at lw Mach number (n shck waves) and at high angle f attack flight cnditins (high initial lift cefficient). Hwever, this case is als the mst difficult prblem, since a decrease in C d usually cmes at the expense f a decrease in C l fr the high-lift system cnfiguratin design. Ntice that, as ppsed t the single-element test case, the Mach number f the flw is subsnic thrughut (C pcrt = 16.3 fr M = 0.2) and, therefre, n shck waves are present. 9 design iteratins were carried ut and, as expected, with a slight increase in C l, a small decrease (10 drag cunts) in C d was achieved as shwn in Figure 16. The resulting L/D increases by 1.73% frm (Baseline L/D at α = ) t Ntice that the resulting α has increased slightly while trying t maintain C l unchanged. C l Maximizatin at a Fixed Angle f Attack A similar calculatin t the ne presented earlier is discussed in this sectin. Instead f minimizing C d, hwever, we maximize the C l f the cnfiguratin using all 156 design variables in the prblem. In this test case, the angle f attack f the whle cnfiguratin remains cnstant, α = The ptimizer is able t make imprvements in C l after 19 design iteratins: the lift cefficient has increased frm t as shwn in Figure 17. Large changes are bserved in bth the flap and slat rigging parameters. The flap deflectin angle has increased in rder t allw fr larger camber and the slat deflectin angle has decreased achieving a higher effective angle f attack, and therefre carrying mre lift. Figure 18 shws the same design attempt using nly the setting parameters as the design variables. After 19 design iteratins, the lift cefficient has increased frm t Frm these results it is evident that a large prtin (abut 85%) f the increase in C l is due t the mdificatin f the rigging parameters f each airfil element. Hwever, ntice that very small gemetry changes in each element thrugh the bump functins still delivered mre than 100 cunts (15% f the increase) in lift cefficient. C l Maximizatin at a Fixed C d We nw allw the angle f attack f the cnfiguratin t flat by fixing the value f C d t that f the baseline design pint at α = As we can see in Figure 19, in 10 design iteratins, the ptimizer has increased the lift by an amunt f 815 lift cunts frm t while reducing the angle f attack frm t with small changes in the ttal cefficient f drag frm t This result appears cunterintuitive at first but highlights the pwer f bth the adjint methdlgy and the careful parameterizatin f the surface, since the prcedure still yields a higher C l, while the angle f attack is frced dwn t match the prescribed C d = Maximum Lift Maximizatin Single-Element Airfil Results The present adjint methd was als applied t the ptimizatin f an airfil shape that maximizes the maximum lift cefficient (C lmax ). The RAE2822 single-element airfil and the same 4 blck mesh fr the previus single-element airfil design cases were used fr calculatins at a design cnditin f M = 0.2 and Re = Bump functins were used as befre, and the angle f attack (α) was included as an additinal design variable fr the maximizatin f maximum lift. Tw different appraches were tried. In the first apprach three steps were taken as fllws: firstly using α alne as a design variable, C lmax and α at C lmax (α clmax ) were predicted alng the C l vs. 8 f 20 American Institute f Aernautics and Astrnautics Paper

10 α curve fr the baseline cnfiguratin and then, using 50 bump functins, a new airfil cnfiguratin was btained that maximized C l with α = α clmax fixed. Finally, starting frm this value f α, the first step was repeated t btain a new C lmax and new α clmax fr the updated cnfiguratin. Figure 20 shws the design results frm this apprach. In the first step, C lmax was predicted t be at α clmax = (O A in Figure 20 ). Next, as shwn in Figure 21, C lmax increased t while the initial airfil was updated t have mre thickness near the leading-edge and mre camber (A B in Figure 20). The final C lmax btained was (Pint C in Figure 20) and the verall C lmax imprved 14% frm while α clmax increased 11.1% frm t This design example verifies the design capabilities f the adjint gradients using α. Althugh this example represents a single iteratin f the verall prcedure (nte that 31 iteratins were used during the A B step), this prcedure culd be repeated iteratively t attain even higher levels f C lmax. Figure 22 als shws hw accurately the adjint methd using α can predict C lmax. The mdificatin f angle attack, α, based n the adjint gradient and the crrespnding changes in C l agree very well. Bth the C l and α have cnverged t C lmax and the α clmax respectively after sme small scillatins. Based n the infrmatin gathered frm the first apprach, bth the bumps and the angle f attack were simultaneusly used fr the design in the secnd apprach (D E in Figure 20 ). As shwn in Figure 23, using the baseline RAE2822 at α = 11.85, C lmax imprved by 13.3% t and α clmax changed by 6.8% t frm the C lmax and α clmax f the baseline RAE2822 cnfiguratin. Multi-Element Airfil Results This design example is the culminatin f the effrts in this paper. Using the newly develped viscus adjint prcedure, the multi-blck flw and adjint slvers, and the lessns learned in the previus design examples, we can nw attempt t redesign the 30P 30N multi-element airfil t ptimize its value f C lmax. In this case, a ttal f 157 design variables are used, including 50 Hicks-Henne bump functins n each f the three elements, 3 rigging variables fr each the slat and flap cmpnents, and the angle f attack (α) f the cmplete cnfiguratin. As shwn in Figure 24, the design started at α = 22 which is near the α clmax f the baseline 30P 30N cnfiguratin and the baseline was mdified in the directin f C l imprvement using all f the design variables. As shwn in Figure 25 and Figure 25, C lmax imprved by 1.12%, 501 cunts, increasing frm t , with a slight change (0.43%) in α clmax. Cnclusins Making use f the large cmputatinal savings prvided by the adjint methd when large numbers f design variables are invlved, we have been able t explre high-dimensinal design spaces that are necessary fr high-lift system design. In this study, the 30P30N multi-element airfil has been used because experimental data was available fr validatin purpses. We have shwn that high-lift system design with up t a ttal f 157 design variables is feasible with ur methd. These design variables can include the parameterizatin f the element shapes, the rigging variables, and the angle f attack f the cnfiguratin. Results fr L/D maximizatin, and C lmax maximizatin fr the RAE2822 single-element airfil shwed significant imprvements. Fr the multielement design cases, the relative imprvements were smaller than thse f the single-element design cases. The lift increments, hwever, are f cmparable magnitude althugh the baseline values fr the multi-element cases are much higher. Of curse, ne f the main reasns fr this was the fact that the baseline high-lift cnfiguratin was already a highly ptimized ne. The results btained are encuraging and pint ut that the adjint methd can have great ptential fr the design f high-lift systems. The design cases in this prtin f the wrk are purely academic and meant t validate the sensitivity calculatin prcedure nly. Future wrk will fcus n expanding the results f the current paper and n utilizing the methd described abve t perfrm realistic tw-dimensinal high-lift system designs. Acknwledgments This research has been made pssible by the generus supprt f the David and Lucille Packard Fundatin in the frm f a Stanfrd University Schl f Engineering Terman fellwship. The slutin f the Spalart-Allmaras turbulence mdel is an adaptatin f Dr. Creigh McNeil s implementatin embedded in the three-dimensinal versin f ur flw slver. Thanks als g t Dr. Ku-Cheng f the Being Cmpany fr prviding us with the multi-element airfil experimental data. References 1 F. Bauer, P. Garabedian, D. Krn, and A. Jamesn. Supercritical Wing Sectins II. Springer Verlag, New Yrk, P. R. Garabedian and D. G. Krn. Numerical design f transnic airfils. In B. Hubbard, editr, Prceedings f SYNSPADE 1970, pages , Academic Press, New Yrk, R. M. Hicks and P. A. Henne. Wing design by numerical ptimizatin. Jurnal f Aircraft, 15: , J. Fay. On the Design f Airfils in Transnic Flw Using the Euler Equatins. Ph.D. Dissertatin, Princetn University 1683-T, R. Campbell. An apprach t cnstrained aerdynamic design with applicatin t airfils. NASA Technical Paper 3260, Langley Research Center, Nvember f 20 American Institute f Aernautics and Astrnautics Paper

11 6 A. Jamesn. Aerdynamic design via cntrl thery. Jurnal f Scientific Cmputing, 3: , A. Jamesn. Optimum aerdynamic design using CFD and cntrl thery. AIAA paper , AIAA 12th Cmputatinal Fluid Dynamics Cnference, San Dieg, CA, June J. Reuther, A. Jamesn, J. Farmer, L. Martinelli, and D. Saunders. Aerdynamic shape ptimizatin f cmplex aircraft cnfiguratins via an adjint frmulatin. AIAA paper , 34th Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January J. Reuther, J. J. Alns, J. C. Vassberg, A. Jamesn, and L. Martinelli. An efficient multiblck methd fr aerdynamic analysis and design n distributed memry systems. AIAA paper , June J. J. Reuther, A. Jamesn, J. J. Alns, M. Rimlinger, and D. Saunders. Cnstrained multipint aerdynamic shape ptimizatin using an adjint frmulatin and parallel cmputers: Part I. Jurnal f Aircraft, 36(1):51 60, J. J. Reuther, A. Jamesn, J. J. Alns, M. Rimlinger, and D. Saunders. Cnstrained multipint aerdynamic shape ptimizatin using an adjint frmulatin and parallel cmputers: Part II. Jurnal f Aircraft, 36(1):61 74, J. R. R. A. Martins, I. M. Kr, and J. J. Alns. An autmated methd fr sensitivity analysis using cmplex variables. AIAA paper , 38th Aerspace Sciences Meeting, Ren, Nevada, January C. Bischf, A. Carle, G. Crliss, A. Griewank, and P. Hvland. Generating derivative cdes frm Frtran prgrams. Internal reprt MCS-P , Cmputer Science Divisin, Argnne Natinal Labratry and Center f Research n Parallel Cmputatin, Rice University, J.L. Lins. Optimal Cntrl f Systems Gverned by Partial Differential Equatins. Springer-Verlag, New Yrk, Translated by S.K. Mitter. 15 O. Pirnneau. Optimal Shape Design fr Elliptic Systems. Springer-Verlag, New Yrk, A. Jamesn. Re-engineering the design prcess thrugh cmputatin. AIAA paper , 35th Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January J. Reuther, J.J. Alns, M.J. Rimlinger, and A. Jamesn. Aerdynamic shape ptimizatin f supersnic aircraft cnfiguratins via an adjint frmulatin n parallel cmputers. AIAA paper , 6th AIAA/NASA/ISSMO Sympsium n Multidisciplinary Analysis and Optimizatin, Bellevue, WA, September O. Baysal and M. E. Eleshaky. Aerdynamic design ptimizatin using sensitivity analysis and cmputatinal fluid dynamics. AIAA paper , 29th Aerspace Sciences Meeting, Ren, Nevada, January W. K. Andersn and V. Venkatakrishnan. Aerdynamic design ptimizatin n unstructured grids with a cntinuus adjint frmulatin. AIAA paper , 35th Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January J. Reuther, A. Jamesn, J. J. Alns, M. J. Rimlinger, and D. Saunders. Cnstrained multipint aerdynamic shape ptimizatin using an adjint frmulatin and parallel cmputers. AIAA paper , 35th Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January A. Jamesn, N. Pierce, and L. Martinelli. Optimum aerdynamic design using the Navier-Stkes equatins. AIAA paper , 35th Aerspace Sciences Meeting and Exhibit, Ren, Nevada, January A. Jamesn, L. Martinelli, and N. A. Pierce. Optimum aerdynamic design using the Navier-Stkes equatins. Theretical and Cmputatinal Fluid Dynamics, 10: , S. Kim, J. J. Alns, and A. Jamesn. A gradient accuracy study fr the adjint-based navier-stkes design methd. AIAA paper , AIAA 37th Aerspace Sciences Meeting & Exhibit, Ren, NV, January S. X. Ying. High lift chanllenges and directins fr cfd. Technical reprt, AIAA/NPU AFM Cnference Prceedings, China, June S. Eyi, K. D. Lee, S. E. Rgers, and D. Kwak. High-lift design ptimizatin using navier-stkes equatins. Jurnal f Aircraft, 33: , Eric Besnard, Adeline Schmitz, Erwan Bscher, Niclas Garcia, and Tuncer Cebeci. Tw-dimensinal aircraft high lift system design and ptimizatin. AIAA paper , S. Kim, J. J. Alns, and A. Jamesn. Twdimensinal high-lift aerdynamic ptimizatin using the cntinuus adjint methd. AIAA paper , 8th AIAA/USAF/NASA/ISSMO Sympsium n Multidisciplinary Analysis and Optimizatin, Lng Beach, CA, September W. O. Valarez. High lift testing at high reynlds numbers. AIAA paper , AIAA 9th Aerspace Grund Testing Cnference, Nashville, TN, July W. O. Valarez, C. J. Dminik, R. J. McGhee, W. L. Gdman, and K. B. Paschal. Multi-element airfil ptimizatin fr maximum lift at high reynlds numbers. In Prceedings f the AIAA 9th Applied Aerdynamics Cnference, pages , Washingtn, DC, V. D. Chin, D. W. Peter, F. W. Spaid, and R. J. McGhee. Flwfield measurements abut a multi-element airfil at high reynlds numbers. AIAA paper , July F. W. Spaid and F. T. Lynch. High reynlds numbers, multi-element airfil flwfield measurements. AIAA paper , AIAA 34th Aerspace Sciences Meeting & Exhibit, Ren, NV, Jan Christpher L. Rumsey, Thmas B. Gatski, Susan X. Ying, and Arild Bertelrud. Predictin f high-lift flws using turbulent clsure mdels. AIAA Jurnal, 36: , S. E. Rgers, F. R. Menter, and P. A. Durbin Nagi N. Mansur. A cmparisn f turbulence mdels in cmputing multi-element airfil flws. AIAA paper , AIAA 32nd Aerspace Sciences Meeting & Exhibit, Ren, NV, January S. E. Rgers. Prgress in high-lift aerdynamic calculatins. AIAA paper , AIAA 31st Aerspace Sciences Meeting & Exhibit, Ren, NV, January W. Kyle Andersn and Dary L. Bnhaus. Navier-stkes cmputatins and experimental cmparisns fr multielement airfil cnfiguratins. Jurnal f Aircraft, 32: , J. Reuther, A. Jamesn, J. Farmer, L. Martinelli, and D. Saunders. Aerdynamic shape ptimizatin f cmplex aircraft cnfiguratins via an adjint frmulatin. AIAA paper , AIAA 34th Aerspace Sciences Meeting and Exhibit, Ren, NV, January L. Martinelli and A. Jamesn. Validatin f a multigrid methd fr the Reynlds averaged equatins. AIAA paper , P. R. Spalart and S. R. Allmaras. A ne-equatin turbulence mdel fr aerdynamic flws. AIAA paper , AIAA 30nd Aerspace Sciences Meeting & Exhibit, Ren, NV, January J.J. Alns J. Ya, RgerL. Davis and A. Jamesn. Unsteady flw investigatins in an axial turbine using massively parallel flw slver tfl. AIAA paper , 39th AIAA Aerspace Sciences Meeting, Ren, NV, January f 20 American Institute f Aernautics and Astrnautics Paper

12 1.5 1 Cmparisn f Finite Difference and Adjint Gradiensts n 512x64 Mesh Adjint Finite Difference 0.5 gradient cntrl parameter Fig. 6 Mesh. Navier-Stkes Inverse Design: Cmparisn f Finite-Difference and Adjint Gradients fr a 512x Cntinuus Adjint Gradient fr Different Flw Slver Cnvergence 7(4) rders 6(3) rders 5(2) rders 4(1) rders 3(0) rders 0.5 gradient cntrl parameter Fig. 7 Navier-Stkes Inverse Design: Adjint Gradients fr Varying Levels f Flw Slver Cnvergence. 11 f 20 American Institute f Aernautics and Astrnautics Paper

13 0.1E01 0.8E00 0.4E00 -.2E E00 -.8E00 -.1E01 -.2E01 -.2E01 0.1E01 0.8E00 0.4E00 -.2E E00 -.8E00 -.1E01 -.2E01 -.2E01 RAE2822 TO NACA64A410 a) Initial, P errr = MACH ALPHA RAE2822 TO NACA64A410 b) 100 Design Iteratins, P errr = MACH ALPHA CL CD CM CL CD CM Fig. 8 Typical Navier-Stkes Inverse Design Calculatin, RAE 2822 airfil t NACA 64A410, M = 0.75, GRID 513X65 NCYC 10 RES0.220E00 GRID 513X65 NCYC 10 RES0.129E00 α = 0.0, Re = 6.5 millin. 8 Cl Cnvergence Histry 6 4 Cl FLO103MB Cnvergence Histry Residual Multigrid Cycles Fig. 9 Cnvergence Histry f C l and Density Residual fr a Multi-Element Airfil using the Spalart- Allmaras Turbulence Mdel. 12 f 20 American Institute f Aernautics and Astrnautics Paper