DESIGN OPTIMIZATION OF HIGH-LIFT CONFIGURATIONS USING A VISCOUS ADJOINT-BASED METHOD Sangh Kim Stanfrd University Juan J. Alns Stanfrd University Antny Jamesn Stanfrd University 40th AIAA Aerspace Sciences Meeting and Exhibit Ren, NV January, 2002
Outline Intrductin Objectives Descriptin f Cntinuus Adjint Methd Viscus Aerdynamic Sensitivity Accuracy Study High-Lift System Design Cnclusins and Future Wrk
High Lift System Cnfiguratin Secndary Flw Bundary Layer Viscus Shear Layer Wake Line Main Airfil Slat Flap Flap angle 40 deg Slats Extended Increase Flap Angle Lift Cefficient Lift Cefficient Slats Retracted Flap angle 0 deg Angle f Attack Angle f Attack
Intrductin High-Lift System Cnfiguratin Design Wind tunnel testing CFD as an Analysis Tl Difficulties in Decisin Making CFD as a Design Tl Autmatic Design Prcedure : CFD Gradient Based Optimizatin Gradient Based Optimizatin Cst functin t be Min/Maximized (Drag, L/D). Cntrl functin t be Optimized (Airfil Shape). Design Variables (Mesh pints, Sine Bump functin) Cnstraint (Euler eq. NS eq) Gradients (Finite Difference, Cmplex, Adjint) Optimizatin Algrithm (Steepest Descent) Cntrl Thery Apprach r Adjint Methd by Jamesn 1988. Cntinuus Adjint Design Methd using the Navier-Stkes equatins as a flw mdel and the Gradient Accuracy study. Applicatin t High-Lift System Design.
Objectives 1. Viscus Aerdynamic Sensitivity Accuracy Study 2D Implementatin f a Cntinuus Adjint Design Methd that uses the Navier-Stkes equatins as a flw mdel. Verificatin f the Accuracy and Efficiency f the present Cntinuus Adjint Methd by cmparisn with Gradients frm Finite Difference Methd. Demnstratin with Preliminary Examples. 2. High-Lift System Design Develp numerical ptimizatin tls fr design and develpment f high-lift system cnfiguratins. Imprve take-ff and landing perfrmance f high-lift systems using a Cntinuus Adjint Design Methd. Cl, Cd, L/D and Cl max.
Symblic Descriptin f Cntinuus Adjint Methd Let I be the cst (r bjective) functin where The first variatin f the cst functin is I = I(w, F) w = flw field variables F = design variables δi = I T δw I T δf w F The flw field equatin and its first variatin are R(w, F) = 0 δr = 0 = R δw R δf w F
Intrducing a Lagrange Multiplier, ψ, and using the flw field equatin as a cnstraint δi = = I T δw I T δf ψ T R δw R δf w F w F T I ψ T R T w w δw I ψ T R F F By chsing ψ such that it satisfies the adjint equatin we have R w T ψ = I w, T I δi = ψ T R F F = G T δf δf δf This reduces the gradient(g) calculatin fr an arbitrarily large number f design variables at a single design pint t One Flw Slutin One Adjint Slutin
Navier Stkes Inverse : Adjint Gradient 1.5 Cmparisn f Finite Difference and Adjint Gradiensts n 512x64 Mesh 1.5 Cntinuus Adjint Gradient fr Different Flw Slver Cnvergence 1 Adjint Finite Difference 1 7(4) rders 6(3) rders 5(2) rders 4(1) rders 3(0) rders 0.5 0.5 gradient 0 gradient 0 0.5 0.5 1 1 1.5 0 5 10 15 20 25 30 35 40 45 50 cntrl parameter 1a: Finite Difference vs. Adjint Gradient 1.5 0 5 10 15 20 25 30 35 40 45 50 cntrl parameter 1b: Cntinuus Adjint n Flw Cnvergence Levels
Flwchart f the Design Prcess Define Design Variables Parameterize Cnfiguratin Define Cst Functin Slve Flw Equatin Adjint Finite Difference 1 Lp Slve Adjint Equatins Perturb Shape & Mdify Grid Lp ver Gradients Perturb Shape & Mdify Grid Evaluate Gradient Slve Flw Equatins Evaluate Gradient Lp ver Gradients 120 100 Optimizatin 80 Adjint 60 40 20 Finite Difference New Shape 0 Adjint vs. Finite Difference Optimum? N in Cst, N=100 Yes Stp
Navier Stkes Inverse : RAE2822 NACA64A410 Cp 0.1E01 0.8E00 0.4E00 -.2E-15 -.4E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.2E-15 -.4E00 -.8E00 -.1E01 -.2E01 -.2E01 RAE2822 TO NACA64A410 2a: MACH Initial, 0.750 ALPHA 0.000 P errr = 0.0504 CL 0.3196 CD 0.0029 CM -0.0952 GRID 513X65 NCYC 10 RES0.220E00 RAE2822 TO NACA64A410 2b: 100 Design MACH 0.750 ALPHA Iteratins, 0.000 P errr = 0.0029 CL 0.5650 CD 0.0111 CM -0.1445 GRID 513X65 NCYC 10 RES0.129E00
Navier Stkes C d Min. : RAE2822 at Fixed C l =.84 Cp 0.1E01 0.8E00 0.4E00 -.2E-15 -.4E00 -.8E00 -.1E01 -.2E01 -.2E01 Cp 0.1E01 0.8E00 0.4E00 -.2E-15 -.4E00 -.8E00 -.1E01 -.2E01 -.2E01 3a: RAE DRAG Initial, CD t =0.0169 MACH 0.730 ALPHA 2.739 M = CL 0.73, 0.8411 CD α 0.0169 = 2.739, CM -0.0975 C GRID 513X65 NCYC 400 RES0.624E-03 l = 0.8411 3b: 40RAE Design DRAG Iteratins, C MACH 0.730 ALPHA 2.949 d =0.0097 M = CL 0.73, 0.8406 CD α 0.0097 = 2.949, CM -0.0824 C GRID 513X65 NCYC 100 RES0.346E-02 l = 0.8406
Advantages f Viscus Adjint Methd Over Finite Differencing Cmputatinal Cst was Drastically Reduced. Required Level f Flw Slver Cnvergence is Lwer than fr Finite Differencing. Step Insensitive. (In Adjint Methd, Step is nly used fr Gemetry Perturbatin.) Required Level f Cnvergence f Adjint is Minimal. = Mre Efficient, Mre Rbust Methd fr Sensitivity Analysis in Viscus Flws.
Design Variables Sine Bumps : Shape design fr each element. 0.1E01 0.8E00 0.4E00 High-Lift System Design : SYN103MB RiggingRAE Variables AIRFOIL : Gaps, TO KORN Overlaps and Deflectins MACH 0.730 ALPHA 2.790 CL 0.6465 CD 0.0415 CM -0.0884 GRID 513X65 NCYC 8 RES0.547E01 Angle f Attack (α) : Cl max maximizatin
' $ Grid Tplgy : Multi-blck structured, Overall C- and O- mesh tplgies. Figure 4: Viscus Mesh fr 30P30N Multielement Airfil Mesh Perturbatin New grids are generated by shifting the grid pints alng the radial crdinate lines depending n the shape bundary mdificatin. & %
FLO103-MB PV3 CLIENT Z COORDINATE @ 1.5000 MACH NUMBER frm 0.0000 t 0.3825
High-Lift System Design : SYN103MB (cnt.) FLO103-MB Methdlgy Runge-Kutta Explicit Time Stepping Cell Centered Spatial Discretizatin Jamesn-Schmidt-Turkel(JST) Scheme Lcal Time Stepping Implicit Residual Smthing Multigridding Turbulence Spalart-Allmaras One Equatin Mdel slved by ADI Scheme fr Multi-Element Airfil Designs. ( Dr. Creigh McNeil ) ADJ103-MB : The same methdlgy. MPI (Message Passing Interface) parallel slutin methdlgy.
FLO103-MB CONVERGENCE HISTORY (SA Mdel) 8 Cl Cnvergence Histry 6 4 Cl 2 0 2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 10 4 FLO103MB Cnvergence Histry Residual 10 2 10 0 10 2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Multigrid Cycles 4a: Cnvergence histry fr the Spalart-Allmaras mdel
Cp Cmparisn : M=0.2, α = 8, Re = 9 10 6, 30P30N Cp 2.000 1.000 0.000-1.000-2.000-3.000-4.000-5.000-6.000-7.000-8.000 4b: Experimental 30P30N and TEST cmputatinal Cp : - Exp.,x - Cmp. MACH 0.200 ALPHA 8.010 CL 3.1514 CD 0.0651 CM -1.3693 GRID 118784 NCYC 2000 RES 0.950E00
Stall Predictin : M = 0.2, Re = 9 10 6 4.5 plt Cl vs. alpha 4 * : Cmp. : Exp. Ttal 3.5 3 Main 2.5 Cl 2 1.5 1 Slat 0.5 Flap 0 0.5 5 0 5 10 15 20 25 angle f attack 4c: Experimental and cmputatinal Cl vs. α
Velcity Prfiles : M = 0.2 α = 8 Re = 9 10 6 0.1 velcity prfile at x=0.45 0.14 velcity prfile at x=0.89 0.09 0.12 0.08 0.07 0.1 Cmp 9M Exp 9M 0.06 0.08 d/c 0.05 d/c 0.04 0.06 0.03 0.04 0.02 0.01 0.02 0 0 0.1 0.2 0.3 0.4 u/a inf 0 0 0.1 0.2 0.3 0.4 0.5 u/a inf 4d: High-Lift Cnfiguratin 4e: Cmparisn f velcity prfiles
High Lift System Design Philsphy Neccessary in take-ff and landing cnditins Minimize runway length and Maximize paylad. Typical Cnfiguratins Take Off Cnfiguratin Landing Cnfiguratin Appraches High Maximum Lift Cefficient High L/D (Climb) Take ff : Maximize Cl while maintaining high L/D. Fix Cd and Maximize Cl (shape and α) Fix Cl and Minimize Cd (shape and α) Landing : Maximize Cl withut wrrying abut Cd r L/D. Cl max maximizatin. High Maximum Lift Cefficient Less Strict Requirements n L/D
Cp 2.000 0.000-2.000-4.000-6.000-8.000-10.000-12.000-14.000-16.000-18.000 Multi-Element Cl Max. : 30P30N with Fixed Cd=0.065 Cp 2.000 0.000-2.000-4.000-6.000-8.000-10.000-12.000-14.000-16.000-18.000 5a: Initial, Cl=4.0412 30P30N TEST MACH 0.200 ALPHA 16.020 M = 0.2, CL 4.0414 α CD = 0.0650 CM -1.4089, Re = 9 10 6, GRID 204800 NCYC 5000 RES 0.285E-02 Cd = 0.0650 5b: 1030P30N Design TEST Iteratin, Cl=4.1227 MACH 0.200 ALPHA M = 0.2, CL 4.1227 α = CD 0.0661 15.127 CM -1.4610, Re = 9 10 6, GRID 204800 NCYC 5000 RES 0.185E-01 Cd = 0.0661
RAE2822 Cl max Maximizatin. Cl max Maximizatin fr RAE2822 Airfil (Apprah I) 2 1.8 B C 1.6 Cl = 2π α (radian) A 1.4 1.2 Cl 1 0.8 α at Cl max : 11.1 %, Cl max : 14 % imprved. 0.6 0.4 O 0.2 0 2 4 6 8 10 12 14 16 18 angle f attack(α), degrees
RAE2822 Cl max Maximizatin. Cp 2.000 0.000-2.000-4.000-6.000-8.000-10.000-12.000-14.000-16.000-18.000 Cp 2.000 0.000-2.000-4.000-6.000-8.000-10.000-12.000-14.000-16.000-18.000 RAE2822 TEST 5c: Initial, Cl=1.6430 MACH 0.200 ALPHA 14.633 CL 1.6430 CD 0.0359 CM -0.4346 GRID 32768 NCYC 1000 RES 0.150E-02 M = 0.2, α = 14.633, Re = 6.5M, Cd = 0.0359 RAE2822 TEST 5d: 31 Design Iteratins, Cl = 1.8270 MACH 0.200 ALPHA 14.633 CL 1.8270 CD 0.0326 CM -0.4901 GRID 32768 NCYC 1000 RES 0.168E-03 M = 0.2, α = 14.633, Re = 9 10 6, Cd = 0.0326.
RAE2822 Cl max Maximizatin Using α and Bump. Cl max Maximizatin fr RAE2822 Airfil (Apprach II) 2 1.8 Apprach II Design Curve B E C 1.6 A 1.4 1.2 Cl = 2πα(radian) D Apprach I Design Curve Cl 1 0.8 α at Cl max : 6.8 %, Cl max : 13.3 % imprved. 0.6 0.4 0.2 O 0 2 4 6 8 10 12 14 16 18 angle f attack(α), degrees
30P30N Cl max Maximizatin. 4.55 Maximum Cl Maximizatin fr 30P30N E designed 4.5 4.45 Design Curve E baseline 4.4 S Cl Baseline Cl vs. α Curve 4.35 4.3 α at Cl max : 501 Cl cunts, α clmax : 0.1 increased. 4.25 4.2 19 20 21 22 23 24 25 26 angle f attack(α), degrees
Cnclusins and Future Wrk Making use f the Large Cmputatinal Savings prvided by the Adjint Methd, a Numerical Optimizatin Prcedure fr Designs f High-Dimensinal Design Space has been develped. High-Lift System Cnfiguratin Design Examples have been perfrmed in rder t validate the Sensitivity Calculatin Prcedure using ur Cntinuus Viscus Adjint Methd. Further Study f Adjint Slver and Adjint Bundary Cnditins. Enhancement f Design Methds by Survey f Optimizatin Algrithm and Design Parameterizatin. Extensin t Mre Realistic Prblems inculding 3D Design Prblems and Multi Pint Designs