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

Transcription

1 Design Optimizatin f Multi Element High Lift Cnfiguratins Using a Viscus Cntinuus Adjint Methd Sangh Kim, Juan J. Alns, and Antny Jamesn Stanfrd University, Stanfrd, CA ABSTRACT 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 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. The high-lift cnfiguratin parallel design methd uses a pint-t-pint matched multi-blck grid system and the Message Passing Interface standard fr cmmunicatin in 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 cmparisn with available experimental data. Finally, several design results that verify the ptential f the methd fr high- Pstdctral Schlar, Department f Aernautics and Astrnautics, Stanfrd University Assistant Prfessr, Department f Aernautics and Astrnautics, Stanfrd University, AIAA Member Thmas V. Jnes Prfessr f Engineering, Department f Aernautics and Astrnautics, Stanfrd University, AIAA Fellw 1

2 lift system design and ptimizatin, are presented. The design examples include a multi-element inverse design prblem and the fllwing ptimizatin prblems: lift cefficient maximizatin, liftt-drag rati maximizatin, and the maximum lift cefficient maximizatin prblem fr bth the RAE2822 single-element airfil and the 30P30N multi-element airfil. 2

3 NOMENCLATURE c = chrd length C d, C l = airfil cefficients f drag and lift C lmax = maximum lift cefficient C p = pressure cefficient C pcrt = critical pressure cefficient D = drag t/c = thickness-t-chrd rati F = bundary shape G = gradient vectr H = ttal enthalpy I = cst functin L = lift M = free stream Mach number p = static pressure p d = desired target static pressure R = gverning equatins r residual Re = Reynlds number S = surface area w = cnservative flw variables in Cartesian crdinates x,y = Cartesian crdinates y = dimensinless wall distance α = angle f attack α clmax = stall angle f attack δ = first variatin λ = step size in steepest descent methd 3 ψ = Lagrange multiplier, c-state r adjint variable

4 INTRODUCTION Aerdynamic shape design has lng been a challenging bjective in the study f fluid dynamics. Cmputatinal Fluid Dynamics (CFD) has played an imprtant analysis rle in the aerdynamic design prcess since its intrductin. Hwever, CFD has been mstly 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 1,2,3,4,5, it has nt been until recently that the fcus f CFD applicatins has shifted t aerdynamic design 6,7,8,9,10,11. 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 techniques have made it pssible t remve difficulties in the decisin making prcess faced by the applied aerdynamicist. In gradient-based ptimizatin methds, 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. Unfrtunately, except fr the case f the reverse mde f autmatic differentiatin, their cmputatinal cst is prprtinal t the number f design variables in the prblem, which can be rather large. An alternative chice, the cntrl thery apprach, has dramatic cmputatinal cst and strage advantages when cmpared t any f these methds. The fundatin f cntrl thery fr systems gverned by partial differential equatins was laid by 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 4

5 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 6,7,16. Since then, this methd has becme a ppular chice fr design prblems invlving fluid flw 9,17,18,19. 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. 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 a three-dimensinal wing prblem 16,20. Since these design calculatins were carried ut withut the benefit f a careful check f the accuracy f the resulting gradient infrmatin, a series f numerical experiments in tw dimensins were cnducted by the authrs 21 t determine the validity f the results. Existing appraches t the adjint methd can be classified int tw categries: the cntinuus adjint and discrete adjint methds. If the adjint equatins are directly derived frm the gverning equatins and then discretized, they are termed cntinuus, while if, instead, they are directly derived frm the discretized frm f the gverning equatins they are referred t as discrete. In thery, the discrete adjint methd shuld give gradients which are clser in value t exact finite-difference gradients. On the ther hand, the cntinuus adjint methd has the advantage that the adjint system has a unique frm independent f the scheme used t slve the flw-field system. Nadarajah and Jamesn have recently perfrmed a detailed gradient cmparisn study f the cntinuus and discrete adjint appraches using the Navier-Stkes equatins 22, and fund that in typical shape ptimizatin prblems in transnic flw the differences are small enugh that they have n significant effect n the final result. The wrk in ur grup has mainly been based 5

6 n the cntinuus adjint methd. In fact this methd has been successfully used fr the aerdynamic design f cmplete aircraft cnfiguratins 8,23 and has been extended t treat aer-structural design 24. 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 25,26,27. 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 28. Besnard, Schmitz, Bscher, Garcia and Cebeci perfrmed ptimizatins f high-lift systems using an Interactive Bundary Layer (IBL) apprach 29. All f these earlier wrks n multi-element airfil design btained the necessary gradients by finite-difference methds. Mre recently discrete adjint gradients have als been used fr the design f multi-element airfil cnfiguratins 30,31,32. In this wrk, the cntinuus 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 19,33,34, this research is designed t validate the adjint methd fr cmplex 6

7 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. 1 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 elements f the prcedure is 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 we slve the flw equatins fr the flw variables, fllwed by a slutin f the adjint equatins fr the cstate variables subject t apprpriate bundary cnditins which will depend n the frm f the cst functin. Next we evaluate the gradients and update the aerdynamic shape based n the directin f steepest descent. Finally we 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 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 bth the flw-field variables (w) and the physical lcatin f the bundary, which may be represented by the functin F. Then I = I (w, F), 7

8 and a change in F results in a change [ ] I T δi = w I δw [ ] I T F II δf, (1) 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 δr = [ ] R δw w I Next, intrducing a Lagrange Multiplier ψ, we have Chsing ψ t satisfy the adjint equatin [ ] [ ] I T I T δi = δw δf w I F II ([ ] [ ] ) R R ψ T δw δf w I F II { [ ]} I T R = w ψt δw w I { [ ]} I T R F ψt δf. F [ ] R δf = 0. (3) F II II [ ] R T ψ = I w w (4) the first term is eliminated, and we find that δi = GδF, (5) 8

9 where 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 flwfield evaluatins. In the case that (2) is a partial differential equatin, the adjint equatin (4) is als a partial differential equatin and the determinatin f the apprpriate bundary cnditins requires careful mathematical treatment. The frmulatin f the viscus adjint equatin and the bundary cnditins are described in greater detail in previus publicatins 35 and a detailed gradient accuracy study fr the cntinuus adjint methd has been carried ut under the assumptin that bth the viscsity (laminar plus turbulent) and heat cnductin cefficients are essentially independent f the flw, and that their variatins may be neglected 21. This simplificatin has been successfully used fr many aerdynamic prblems f interest. Hwever, if the flw variatins result in significant changes in the turbulent eddy viscsity, it may be necessary t accunt fr this effect in the calculatin. 36,37, 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 39,40,41,42 and the results f many CFD cmputatins fr this gemetry have been reprted using a variety f numerical schemes fr the discretizatin f the Navier-Stkes equatins and different turbulence mdels 26,43,44,45. The three-element cnfiguratin used in these studies, dented as 30P30N, is the 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 relative element psitining f the slat, main element and flap are described by the rigging quantities. These variables include flap and slat deflectin angles, gaps, and verlaps. The meaning f these variables can be 9

10 easily seen in Figure 2. 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. In these measurements, c is the airfil chrd with the slat and flap retracted. 1.3 Grid Tplgy Multi-blck viscus and inviscid meshes were generated using either a C- (viscus) r O-tplgy (inviscid) prir t the beginning f the iterative design lp s that the flw and adjint equatins culd be suitably discretized. Figure 3 shws a typical viscus multi-blck mesh generated arund the 30P30N cnfiguratin. One-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 1 based n turbulent flat plate bundary layer thickness estimates 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 44,45. 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) 10,11,23,46 that is essentially equivalent t shifting grid pints alng crdinate lines depending n the mdificatins t the shape f the bundary. Mesh perturbatins are applied nly t thse blcks that cntain a face that is part f the 30P30N baseline cnfiguratin surface. Since the blck interfaces are mdified nly by the mtin f their end pints, the initial ne-t-ne pint cnnectivity between blck faces remains valid thrughut the cmplete range f surface defrmatins. The authrs have previusly handled the cmplexity f multi-blck mesh perturbatin after large baseline gemetry changes

11 1.4 Multi-blck Flw and Adjint Slvers 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 are quite sphisticated. In this study FLO103-MB, a multi-blck RANS slver derived frm the wrk f Martinelli and Jamesn 48 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 Navier-Stkes 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, 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 accunt fr the Reynlds stresses. The adjint gradient accuracy study which was presented in a previus publicatin 21 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 arund cmplex gemetries 26,43,49. The slutin f the Spalart-Allmaras turbulence mdel is an adaptatin f Dr. Creigh Mc- Neil s implementatin embedded in the three-dimensinal versin f ur flw slver 50. The trip terms are nt included in the turbulence mdel and fully-turbulent flw is assumed. The turbu- 11

12 lence equatin is slved separately frm the flw equatins using an alternating directin implicit (ADI) methd and it 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 laminar flw slutin with the assumptin f frzen eddy viscsity. 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) prgram structure, and the MPI standard fr message passing. 1.5 Design Variables Fr the present study, gaps and verlaps are used in an indirect way since the rigging is cntrlled by the 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 the leading and trailing-edge lcatins f the varius elements in the airfil. 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. In fact, fr the drag minimizatin f a single-element RAE2822 airfil in transnic flw, the strng shck which appears 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 high-lift 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 1. (7) Here, A is the maximum bump magnitude, t 1 lcates the maximum f the bump at x = t 1, 12

13 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 C lmax alng a C l vs. α line fr that cnfiguratin, and repeating this prcedure iteratively. In this way, the angle f attack, α, is nly cnsidered as a design variable nce the intermediate shape f the airfil has been frzen within each design cycle. Alternatively, C lmax may als be maximized directly by including angle f attack as a design variable in the ptimizatin prcess. 1.6 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, 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. 2 Results 2.1 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 13

14 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, (8) 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 numbered sequentially such that they start at the lwer surface f the trailing edge, prceed frward t the leading edge, and then back arund t the upper surface f the trailing edge. Figure 4 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. Sme discrepancies exist which are attributed t several surces. First, there is the issue that the finite-difference gradients never achieved step size independence, and, therefre, a step size had t be selected fr this plt which may nt be the crrect ne. Secnd, there is the issue f mesh reslutin 21. Third, the present adjint equatin is btained withut taking int accunt the dependence f the dissipatin cefficients n the flw variables. A detailed study n these issues was presented by Nadarajah and Jamesn 22. The surce f the discrepancies is believed t be a cmbinatin f these influences. Figure 5 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 ver finite-differencing 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. Fr 14

15 viscus flws and finite-differencing 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 next. The mesh used is a Navier-Stkes C-mesh with cells. The target pressure specified is that f the NACA 64A410 airfil at M = 0.75 and α = 0. The Reynlds number f this calculatin was set at Re = Figure 6 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 areas arund the trailingedge are very sensitive t small changes in the gemetry. The surce f the C p discrepancy in the neighbrhd f the trailing edge can be fund in the inability f ur parameterizatin t exactly recver the shape f the NACA 64A010 airfil. Mre gemetry cntrl may be required t prduce mre exactly matched slutins in the areas arund the trailing-edge surface. Fllwing ur design prcess, 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. Cnvergence f the design prcedure is simply measured by the clseness f the resulting pressure distributin t the specified target. N attempt t satisfy the Kuhn-Tucker cnditins was made. The nrm f the pressure errr decreased ver tw rders f magnitude frm t in 100 design iteratins. 2.2 FLO103-MB with SA Mdel Befre using FLO103-MB with the Spalart-Allmaras (SA) turbulence mdel fr the design f multielement airfils we must demnstrate that the mdel has been crrectly implemented cnfirming that the slver can 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 15

16 distributins, the lift curve slpes fr each element and the details f the shear layers present in the prblem. Flw cnvergence Figure 7 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 sufficient 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 purpses. The cde, FLO103-MB, and the related turbmachinery cde TFLO 51, have been extensively validated fr a variety f test cases, ranging frm flat plates and transnic axisymmetric bumps, t full three-dimensinal cnfiguratins. Figure 8, 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 9. 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 16

17 angle f attack at C lmax. Althugh the results d nt agree with the 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. 2.3 Single-Element Airfil Design Befre embarking n multi-element airfil design, a study f the use f ptimizatin fr singleelement 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 10 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 in what is smetimes called a reduced gradient apprach. Figure 10 shws the result f 50 design iteratins fr this test case. The ptimizer is able t eliminate the strng 17

18 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 by 54.36% frm t This test case als prvides a validatin f the multi-blck 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 11 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. 18

19 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 12) 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, including 50 bump functins 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 12. Figure 13 shws the cnvergence histry f the bjective functin. 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 19

20 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 ne, 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. A ttal f 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 14. 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 We nw 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 15. Large changes are bserved in bth the flap and slat deflectin angles. The flap deflectin angle has increased frm 30 t 32.6 in rder t allw fr larger camber and the slat deflectin angle has decreased frm 30 t 26.7 achieving a higher effective angle f attack, and therefre carrying mre lift. Changes in gaps and verlaps are small and the resulting flap gap and verlap are c/0.0020c whereas fr the slat the gap and verlap are c/0.0258c. Figure 16 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 20

21 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 17, in 10 design iteratins, the ptimizer has increased the lift by 815 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 used 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. 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 18 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 19, C lmax increased 21

22 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. 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 21, 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 22, 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 22 and Figure 23, C lmax imprved by 1.12%, 501 cunts, increasing frm t , with a slight change (0.43%) in α clmax. Even fr highly ptimized cnfiguratin such as the 30P30N, further small imprvements can be achieved by using the viscus 22

23 adjint sensitivities that are the central prtin f this paper. 2.6 Results f Design Optimizatin In the present design study all f the ptimizatins emplyed the methd f steepest descent. Cnstraints are applied in the ptimizatin by simply prjecting the gradient vectr nt the subspace f feasible designs, eventually leading t a cnstrained ptimum which is arrived at by mving alng cnstraint bundaries. N attempt t use an augmented Lagrangian was made and therefre, the actual ptimum design is nt measured by the satisfactin f the Kuhn-Tucker cnditins, but, instead, by mnitring the prgress f the cst functin f interest. Obviusly, a nn-linear cnstrained ptimizer culd have been used instead (as we have dne in sme f ur previus wrk 10,11.) Hwever, we have ften fund that, in engineering terms, slutins very clse t the ptimum can be fund with the steepest descent methd. Althugh fr gradient-based ptimizatin, cnvergence t the exact ptimum cannt be expected in fewer iteratins than the actual number f design variables in the prblem, in typical adjint-based design parameterizatins, it is nt unusual t have design variables whse gradients are rather small and, therefre, have very little effect n the value f the cst functin at the ptimum pint. Fr this reasn, sme f the design calculatins were stpped after a given number f design iteratins, nce the functin f interest shwed n evidence f cntinued decrease. Figure 24 shws the cnvergence histry fr the C l maximizatin f the RAE2822 airfil using the angle f attack as the nly design variable. As expected by the thery, C l cnverged t the lcal maximum and the gradient value crrespnding t the lcal ptimum became f its initial value, 6.506, in 44 design iteratins. Similarly, fr the prblem f drag minimizatin f an RAE2822 airfil at a fixed angle f attack, the lcal minimum fr the specified parameterizatin was achieved as shwn in Figure 25. Hwever, this minimum may be further imprved if a different shape parameterizatin were t be emplyed; 23

24 the present design parameterizatin did nt allw fr mdificatins f the leading and trailing edge lcatins and all f the Hicks-Henne sine bump functins had fixed a, t 1, and t 2 parameters (in Equatin 7) and therefre, the numerical ptimizatin prcedure was unable t span the cmplete design space f all pssible airfil shapes. Althugh n direct cnstraints were intrduced, the use f the present bump functins (with bunds n their pssible values) and design parameterizatin can be cnsidered an indirect way f applying cnstraints. In the single-element lift maximizatin cases using Hicks-Henne bump functins, the designs were run fr a pre-specified number f design iteratins and were nt necessarily allwed t cnverge t the exact lcal maximum, althugh inspectin f the results reveals that the difference was small. Further imprvements can be btained by reducing the airfil thickness and therefre, a mre cmplete treatment f thickness cnstraints is suggested fr mre detailed design wrk. Fr the multi-element 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. One f the main reasns fr this was the fact that the baseline high-lift cnfiguratin was already a highly ptimized ne. Tw ther pssible reasns can als be mentined. Firstly, the additinal design variables such as the flap and slat deflectin angles, gaps, and verlaps, made the multi-element airfil design space much mre cmplex than that f the single-element airfil design. Fr the scalings used in bth the bump amplitudes and the flap and slat angle deflectins, the magnitudes f the lift and drag sensitivities differ cnsiderably. Typically, fr the units chsen in this wrk, the sensitivities t the bump amplitudes are much larger then thse due t angle f attack variatins, althugh the cst functin imprvements that may be derived frm angle f attack changes may be mre substantial after a large number f steepest descent design iteratins. Since, due t the cst f the ptimizatin prcedure fr viscus multi-element design, we were limited in the ttal number f design iteratins that we culd cmpute, a user-defined scaling was used in the angle f attack variables t arrive at 24

25 the lcal minimum in a smaller number f iteratins. Secndly, ne can expect further imprvements by using alternative cst functins and adjint bundary cnditins. Interestingly, fr the drag minimizatin f the multi-element cnfiguratin, a significant imprvement was bserved using pressure drag as the cst functin instead f ttal drag, thugh the actual numerical result was nt included in this paper since a thrugh study has nt been cmpleted. This may be due t the fact that mst f the drag f the multi-element airfil is the pressure drag. Althugh the flw in high-lift systems is dminated by viscus phenmena, the typical rati f pressure drag t skin frictin drag is f rder f 10. The high-lift system ptimizatins f the 30P30N cnfiguratin were carried ut using a multiblck mesh with 26 blcks and a ttal f 204,800 mesh pints. Each design iteratin invlved the slutin f the flw using the Spalart-Allmaras turbulence mdel and it required abut 5, 000 iteratins. The adjint slutin is cmputed in the same mesh and, since it des nt need t be cnverged as far as the flw slutin in rder t btain accurate gradient infrmatin, nly 700 iteratins were required. Each design iteratin, cnsisting f ne flw and ne adjint slutin and all the necessary mesh perturbatins and gradient calculatins tk apprximately 1.5 hr n 26 prcessrs f an SGI Origin 2000 cmputer with the 195MHz R10000 prcessr. 3 Cnclusins A numerical ptimizatin prcedure using the adjint methd fr high-lift system design has been develped and presented. The prcedure is based n a multi-blck RANS flw slver, FLO103-MB, that uses the Spalart-Allmaras turbulence mdel fr high Reynlds number flws. FLO103-MB has been implemented in parallel s that the turnarund fr design calculatins can be even faster. Multi-element airfils are parameterized using the well-knwn Hicks-Henne bump functins and additinal design variables that allw the gaps and verlaps, as well as the angle f attack f the slat and flap elements t be represented. Making use f the large cmputatinal savings prvided by the 25

26 adjint methd when large numbers f design variables are invlved, we are able t explre highdimensinal design spaces that are necessary fr high-lift system design. In this study, the 30P30N multi-element airfil is used because experimental data is available fr validatin purpses. A ttal f 157 design variables crrespnding t 50 Hicks-Henne bump functins n each f the elements f the cnfiguratin, the x and y lcatins and angles f attack f the slat and flap elements, and the verall angle f attack f the cnfiguratin are used fr C d minimizatin and C l maximizatin subject t several kinds f cnstraints. The angle f attack f the high-lift system was used as a design variable fr C lmax maximizatin nly. Results fr L/D maximizatin, and C lmax maximizatin fr the RAE2822 single-element airfil shwed significant imprvements. Fr the multi-element 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 wrk are purely academic and meant t validate the sensitivity calculatin prcedure nly. Future wrk will fcus n expanding current results and n utilizing the methd described abve t perfrm realistic tw-dimensinal high-lift system designs. 4 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 authrs acknwledge Dr. Creigh MaNeil fr assistance with his implementatin f the Spalart- Allmaras turbulence mdel 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 26

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