AIAA 2002–0844 - Stanford Universityaero-comlab.stanford.edu/Papers/kim.aiaa.02-0844.pdf · 2019....

AIAA 2002–0844Design Optimization of High–LiftConfigurations Using a ViscousContinuous Adjoint MethodSangho Kim, Juan J. Alonso, and Antony JamesonStanford University, Stanford, CA 94305

40th AIAA Aerospace Sciences Meeting andExhibit

January 14–17, 2002/Reno, NV

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2002–0844

Design Optimization of High–LiftConfigurations Using a Viscous Continuous

Adjoint Method

Sangho Kim∗, Juan J. Alonso†, and Antony Jameson‡

Stanford University, Stanford, CA 94305

An adjoint-based Navier-Stokes design and optimization method for two-dimensionalmulti-element high-lift configurations is derived and presented. The compressibleReynolds-Averaged Navier-Stokes (RANS) equations are used as a flow model togetherwith the Spalart-Allmaras turbulence model to account for high Reynolds number effects.Using a viscous continuous adjoint formulation, the necessary aerodynamic gradient in-formation is obtained with large computational savings over traditional finite-differencemethods. A study of the accuracy of the gradient information provided by the adjointmethod in comparison with finite differences and an inverse design of a single-elementairfoil are also presented for validation of the present viscous adjoint method. The high-lift configuration design method uses a compressible RANS flow solver, FLO103-MB, apoint-to-point matched multi-block grid system and the Message Passing Interface (MPI)parallel solution methodology for both the flow and adjoint calculations. Airfoil shape,element positioning, and angle of attack are used as design variables. The prediction ofhigh-lift flows around a baseline three-element airfoil configuration, denoted as 30P30N,is validated by comparisons with experimental data. Finally, several design results thatverify the potential of the method for high-lift system design and optimization, are pre-sented. The design examples include a multi-element inverse design problem and thefollowing problems: Cl maximization, lift-to-drag ratio, L/D, maximization by minimiz-ing Cd at a given Cl or maximizing Cl at a given Cd (α is allowed to float to maintaineither Cl or Cd), and the maximum lift coefficient, Clmax , maximization problem for boththe RAE2822 single-element airfoil and the 30P30N multi-element airfoil.

Introduction

AERODYNAMIC shape design has long been achallenging objective in the study of fluid dy-

namics. Computational Fluid Dynamics (CFD) hasplayed an important role in the aerodynamic designprocess since its introduction for the study of fluidflow. However, CFD has mostly been used in theanalysis of aerodynamic configurations in order to aidin the design process rather than to serve as a directdesign tool in aerodynamic shape optimization. Al-though several attempts have been made in the pastto use CFD as a direct design tool,1–5 it has not beenuntil recently that the focus of CFD applications hasshifted to aerodynamic design.6–11 This shift has beenmainly motivated by the availability of high perfor-mance computing platforms and by the developmentof new and efficient analysis and design algorithms.In particular, automatic design procedures which useCFD combined with gradient-based optimization tech-

∗Doctoral Candidate, AIAA Member†Assistant Professor, Department of Aeronautics and Astro-

nautics , AIAA Member‡Thomas V. Jones Professor of Engineering, Department of

Aeronautics and Astronautics, AIAA FellowCopyright c© 2002 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

niques, have made it possible to remove difficulties inthe decision making process faced by the aerodynam-icist.

Typically, in gradient-based optimization tech-niques, a control function to be optimized (an airfoilshape, for example) is parameterized with a set ofdesign variables, and a suitable cost function to beminimized or maximized is defined (drag coefficient,lift/drag ratio, difference from a specified pressuredistribution, etc.) Then, a constraint, the governingequations in the present study, can be introduced inorder to express the dependence between the cost func-tion and the control function. The sensitivity deriva-tives of the cost function with respect to the designvariables are calculated in order to get a direction ofimprovement. Finally, a step is taken in this directionand the procedure is repeated until convergence to aminimum or maximum is achieved. Finding a fast andaccurate way of calculating the necessary gradient in-formation is essential to developing an effective designmethod since this can be the most time consumingportion of the design algorithm. Gradient informa-tion can be computed using a variety of approaches,such as the finite-difference method, the complex stepmethod12 and automatic differentiation.13 Unfortu-

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American Institute of Aeronautics and Astronautics Paper 2002–0844

nately, their computational cost is proportional to thenumber of design variables in the problem.

As an alternative choice, the control theory ap-proach has dramatic computational cost advantageswhen compared to any of these methods. The founda-tion of control theory for systems governed by partialdifferential equations was laid by J.L. Lions.14 Thecontrol theory approach is often called the adjointmethod, since the necessary gradients are obtained viathe solution of the adjoint equations of the governingequations of interest. The adjoint method is extremelyefficient since the computational expense incurred inthe calculation of the complete gradient is effectivelyindependent of the number of design variables. Theonly cost involved is the calculation of one flow so-lution and one adjoint solution whose complexity issimilar to that of the flow solution. Control theorywas applied in this way to shape design for ellipticequations by Pironneau15 and it was first used in tran-sonic flow by Jameson.6,7, 16 Since then this methodhas become a popular choice for design problems in-volving fluid flow .9,17–19 In fact, the method has evenbeen successfully used for the aerodynamic design ofcomplete aircraft configurations.8,20

Most of the early work in the formulation of theadjoint-based design framework used the potentialand Euler equations as models of the fluid flow.Aerodynamic design calculations using the Reynolds-Averaged Navier-Stokes equations as the flow modelhave only recently been tackled. The extension ofadjoint methods for optimal aerodynamic design ofviscous problems is necessary to provide the increasedlevel of modeling which is crucial for certain types offlows. This cannot only be considered an academicexercise. It is also a very important issue for the de-sign of viscous dominated applications such as the flowin high-lift systems. In 1997, a continuous adjointmethod for Aerodynamic Shape Optimization (ASO)using the compressible Navier-Stokes equations wasformulated and it has been implemented directly ina three dimensional wing problem .16,21 In 1998, animplementation of three-dimensional viscous adjointmethod was used with some success in the optimiza-tion of the Blended-Wing-Body configuration.22 Sincethese design calculations were carried out without thebenefit of a careful check on the accuracy of the result-ing gradient information, a series of numerical exper-iments in two dimensions that assessed the accuracyof the viscous adjoint gradient information were con-ducted by the authors.23

The research in this paper addresses the validity ofthis design methodology for the problem of high-liftdesign. Traditionally, high-lift designs have been real-ized by careful wind tunnel testing. This approach isboth expensive and challenging due to the extremelycomplex nature of the flow interactions that appear.CFD analyses have recently been incorporated to the

high-lift design process.24 Eyi, Lee, Rogers and Kwakhave performed design optimizations of a high-lift sys-tem configuration using a chimera overlaid grid sys-tem and the incompressible Navier-Stokes equations.25

Besnard, Schmitz, Boscher, Garcia and Cebeci per-formed optimizations of high-lift systems using anInteractive Boundary Layer (IBL) approach.26 Allof these earlier works on multi-element airfoil designobtained the necessary gradients by finite-differencemethods.

In this work, our viscous adjoint method is appliedto two-dimensional high-lift system designs, removingthe limitations on the dimensionality of the designspace by making use of the viscous adjoint designmethodology. The motivation for our study of high-liftsystem design is twofold. On the one hand, we wouldlike to improve the take-off and landing performance ofexisting high-lift systems using an adjoint formulation,while on the other hand, we would like to setup numer-ical optimization procedures that can be useful to theaerodynamicist in the rapid design and development ofhigh-lift system configurations. In addition to difficul-ties involved in the prediction of complex flow physics,multi-element airfoils provide an additional challengeto the adjoint method: the effect of the changes in theshape of one element must be felt by the other ele-ments in the system. While preliminary studies of theadjoint method in such a situation have already beencarried out,19,27 this research is designed to validatethe adjoint method for complex applications of thistype. Emphasis is placed on the validation and not onthe creation of realistic designs, which is beyond thescope of this work.

ProcedureIn this section we outline the overall design proce-

dure used for a variety of design calculations that willbe presented later. After the initial flowchart, each ofthe items of the procedure are explained in more detail.In practical implementations of the adjoint method, adesign code can be modularized into several compo-nents such as the flow solver, adjoint solver, geometryand mesh modification algorithms, and the optimiza-tion algorithm. After parameterizing the configurationof interest using a set of design variables and defining asuitable cost function, which is typically based on aero-dynamic performance, the design procedure can bedescribed as follows. First solve the flow equations forthe flow variables, then solve the adjoint equations forthe costate variables subject to appropriate boundaryconditions which will depend on the form of the costfunction. Next evaluate the gradients and update theaerodynamic shape based on the direction of steepestdescent. Finally repeat the process to attain an opti-mum configuration. A summary of the design processand a comparison with the finite-difference method areillustrated in Figure 1.

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Parameterize Configuration

Define Cost Function

Perturb Shape & Modify Grid

Solve Adjoint Equations

Perturb Shape & Modify Grid

Evaluate Gradient

Solve Flow Equations

Evaluate Gradient

Optimization

Optimum?

Yes

New Shape

Stop

No

Adjoint vs. Finite Difference in Cost, N = 100

0 �

20

40

60 �

80 �

100

120

c �

o s �

t �

Adjoint

Finite Difference

Solve Flow Equations

Finite Difference Adjoint

*note: process in the dashed box are repeated N times

�

where N is the number of design variables. �

Define Design Variables

Fig. 1 Flowchart of the Design Process

Fig. 2 Definitions of Gap, Overlap, and DeflectionAngles

Design Variables

The rigging quantities that describe the relative el-ement positioning of the slat, main element and flapare used as design variables. These variables includeflap and slat deflection angles, gaps, and overlaps. Themeaning of these variables can be easily seen in Fig-ure 2 for typical multi-element airfoil configurations.For the present study, gaps and overlaps are used inan indirect way since the rigging is controlled by trans-lation of the slat and flap leading edges in the x and ydirections. In this way, the element positioning vari-ables can be more easily changed independently of eachother. The actual values of overlaps and gaps can beeasily recovered from their leading and trailing edgelocations. The shapes of each of the elements are alsoused as design variables so as not to rule out the possi-bility that the optimum solution may be obtained witha combination of shape and position modifications.

In fact, for the drag minimization of a single-elementRAE2822 airfoil in transonic flow, the strong shockpresent at transonic flow conditions can only be elimi-nated using a small change in the shape of the airfoil.The coordinates of mesh nodes on the surface of theairfoil, Hick-Henne “bump” functions, patched poly-nomials and frequency-based decompositions can beused to represent each of the elements in the high-lift system. For example, a number of the followingHicks-Henne functions, which have been implementedand used for this study, may be added to the baselineairfoil to modify the shape:

b (x) = A[sin

(πx

log 5log t1

)]t2, 0 ≤ x ≤ 1.

Here, A is the maximum bump magnitude, t1 locatesthe maximum of the bump at x = t1, and t2 controlsthe width of the bump. Using this parameterization,two options are available for obtaining the optimumClmax . Firstly, Clmax can be predicted by maximiz-ing Cl at a given angle of attack, then predicting theClmax along a Cl vs. α line for that configuration,and repeating this procedure iteratively. Alternatively,Clmax may also be maximized directly by including an-gle of attack as a design variable in the optimizationprocess.

Baseline Configuration

Wind tunnel measurements have been performedon several three-element airfoil configurations at theNASA Langley Low Turbulence Pressure Tunnel(LTPT) at various Reynolds and Mach numbers 28–31

and the results of many CFD computations for thesegeometries have been reported using a variety of nu-merical schemes for the discretization of the Navier-Stokes equations and different turbulence models .32–35

One of these three-element configurations, denoted as30P30N, is used as a starting point for the present de-sign optimization process. It must be noted that the30P30N configuration had already been highly opti-mized for Clmax . The initial deflections of both theslat and the flap are set at 30◦, the flap gap and over-lap are 0.0127c/0.0025c whereas for the slat the gapand overlap are 0.0295c/0.025c, where c is the airfoilchord with the slat and flap retracted.

Grid Topology

A multi-block mesh is generated prior to the begin-ning of the iterative design loop so that the flow andadjoint equations can be suitably discretized. Figure 3shows a typical multi-block mesh generated around the30P30N configuration. As shown in close-up in Fig-ures 4 and 5, one-to-one point connectivity betweenblock faces is employed to ensure conservation acrossboundaries and to provide for continuity of the gridat block interfaces. Spacing at the wall is set to beless than 2 × 10−6c in order to obtain y+ = O(1)based on turbulent boundary layer thickness estimates

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Fig. 3 Grid around the 30P30N multi-elementconfiguration

Fig. 4 Grid around the 30P30N multi-elementconfiguration : Close-up between Slat and Mainelements

from a flat plate at the Reynolds number in question(Re = 9 × 106). The use of a grid with adequatelytight wall spacing (y+ = O(1)) has been reported tobe necessary in order to obtain accurate resolutionof the wall boundary layers, wakes, and shear lay-ers present in the problem.34,35 Grid lines are alsobunched along the expected wake trajectories of eachof the elements. Once the initial grid is generated,new grids corresponding to modified airfoil shapes areobtained automatically during the design process byusing a two-dimensional version of the automatic meshperturbation scheme (WARP-MB)20,36 that is essen-

Fig. 5 Grid around the 30P30N multi-elementconfiguration : Close-up around Flap element

tially equivalent to shifting grid points along coordi-nate lines depending on the modifications to the shapeof the boundary.

The modification to the grid has the form

xnew = xold +N (xnew

airfoil − xoldairfoil

)

ynew = yold +N (ynew

airfoil − yoldairfoil

).

Here,

N =lengthtotal − lengthj

lengthtotal.

The details of the procedure used have been pre-sented earlier and can be found in Ref.10,11

Multi-block Flow and Adjoint Solvers

The prediction of high-lift flows poses a particu-larly difficult challenge for both CFD and turbulencemodeling. Even in two-dimensions, the physics in-volved in the flow around a geometrically-complexhigh-lift device is quite sophisticated. In this studyFLO103-MB, a multi-block RANS solver derived fromthe work of Martinelli and Jameson37 and similar tothe three-dimensional version of Reuther and Alonso,9

is used for multi-element airfoil flow-field predictions.FLO103-MB satisfies the requirements of accuracy,convergence, and robustness that are necessary in thiswork. FLO103-MB solves the steady two-dimensionalRANS equations using a modified explicit multistageRunge-Kutta time-stepping scheme. A finite vol-ume technique and second-order central differencingin space are applied to the integral form of the Navier-Stokes equations. The Jameson-Schmidt-Turkel(JST)scheme with adaptive coefficients for artificial dissipa-tion is used to prevent odd-even oscillations and toallow for the clean capture of shock waves and con-tact discontinuities. In addition, local time stepping,

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implicit residual smoothing, and the multigrid methodare applied to accelerate convergence to steady-statesolutions. The Baldwin-Lomax algebraic model andthe Spalart-Allmaras one equation model are used tomodel the Reynolds stress. The adjoint gradient accu-racy study which was presented in a previous publica-tion23 was based on the Baldwin-Lomax model. Thismodel is also used for the single-element design casesin this paper. Although this algebraic model has someadvantages due to its implementational simplicity androbustness, the use of this model must be restricted todesign at lower angles of attack and to the design ofsimpler geometries such as single-element airfoils. Foractual high-lift designs such as Clmax maximization,the one-equation Spalart-Allmaras model gives betterpredictions of both the Clmax

and the flow physicsaround complex geometries.32,33,38 The turbulentequation is solved separately from the flow equationsusing an alternating direction implicit (ADI) method.The turbulence equation is updated at the start ofeach multistage Runge-Kutta time step on the finestgrid of the multigrid cycle only. The adjoint solutionis obtained with the exact same numerical techniquesused for the flow solution. The implementation exactlymirrors the flow solution modules inside FLO103-MB,except for the boundary conditions which are imposedon the co-state variables. The parallel implementationuses a domain decomposition approach, a SPMD (Sin-gle Program Multiple Data) structure, and the MPIstandard for message passing.

Continuous Adjoint Method

The progress of a design procedure is measured interms of a cost function I, which could be, for examplethe drag coefficient or the lift to drag ratio. For theflow about an airfoil or wing, the aerodynamic proper-ties which define the cost function are functions of theflow-field variables (w) and the physical location of theboundary, which may be represented by the functionF , say. Then

I = I (w,F) ,

and a change in F results in a change

δI =[∂IT

∂w

]

I

δw +[∂IT

∂F]

II

δF , (1)

in the cost function. Here, the subscripts I and II areused to distinguish the contributions due to the varia-tion δw in the flow solution from the change associateddirectly with the modification δF in the shape.

Using control theory, the governing equations of theflow field are introduced as a constraint in such a waythat the final expression for the gradient does notrequire multiple flow solutions. This corresponds toeliminating δw from (1).

Suppose that the governing equation R which ex-presses the dependence of w and F within the flow

field domain D can be written as

R (w,F) = 0. (2)

Then δw is determined from the equation

δR =[∂R

∂w

]

I

δw +[

∂R

∂F]

II

δF = 0. (3)

Next, introducing a Lagrange Multiplier ψ, we have

δI =

[∂IT

∂w

]

I

δw +

[∂IT

∂F

]

II

δF

−ψT([

∂R

∂w

]Iδw +

[∂R

∂F]

IIδF

)

=

{∂IT

∂w− ψT

[∂R

∂w

]}

I

δw

+

{∂IT

∂F − ψT[

∂R

∂F]}

II

δF .

Choosing ψ to satisfy the adjoint equation[∂R

∂w

]T

ψ =∂I

∂w(4)

the first term is eliminated, and we find that

δI = GδF , (5)

where

G =∂IT

∂F − ψT

[∂R

∂F]

. (6)

The advantage is that (5) is independent of δw, withthe result that the gradient of I with respect to an ar-bitrary number of design variables can be determinedwithout the need for additional flow-field evaluations.In the case that (2) is a partial differential equation,the adjoint equation (4) is also a partial differentialequation and determination of the appropriate bound-ary conditions requires careful mathematical treat-ment. The computational cost of gradient calculationfor a single design cycle is roughly equivalent to thecost of two flow solutions since the adjoint problem hassimilar complexity. When the number of design vari-ables becomes large, the computational efficiency ofthe control theory approach over the traditional finite-differences, which require direct evaluation of the gra-dients by individually varying each design variable andrecomputing the flow field, becomes compelling. Theformulation of the adjoint equation and the boundaryconditions are described in greater detail in previouspublications22 and a detailed gradient accuracy studyfor the continuous adjoint method can be found inRef.23

Numerical Optimization Method

The search procedure used in this work is a simplesteepest descent method in which small steps are takenin the negative gradient direction.

δF = −λG,

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where λ is positive and small enough that the firstvariation is an accurate estimate of δI. Then

δI = −λGTG < 0.

After making such a modification, the gradient can berecalculated and the process repeated to follow a pathof steepest descent until a minimum is reached. Inorder to avoid violating constraints, such as a mini-mum acceptable airfoil thickness, the gradient may beprojected into an allowable subspace within which theconstraints are satisfied. In this way, procedures canbe devised which must necessarily converge at least toa local minimum.

ResultsValidation of the Adjoint Method for ViscousFlows

This section presents the results of a gradient accu-racy study for the RANS equations using the Baldwin-Lomax turbulence model, as well as a simple exampleof the use of the resulting gradient information in asingle-element airfoil inverse design case. Gradient ac-curacy is assessed by comparison with finite-differencegradients and by examination of the changes in themagnitude of the gradients for different levels of flowsolver convergence. For inverse design, the aerody-namic cost function chosen is given by:

I =12

∫

B(p− pd)

2dS, (7)

which is simply the Euclidean norm of the differencebetween the current pressure distribution and a de-sired target, pd, at a constant angle of attack, α. Thegradient of the above cost function is obtained withrespect to variations in 50 Hicks-Henne sine “bump”functions centered at various locations along the upperand lower surfaces of a baseline airfoil. The locations ofthese geometry perturbations are ordered sequentiallysuch that they start at the trailing edge on the lowersurface, proceed forward to the leading edge, and thenback around to the trailing edge on the upper surface.Figure 6 shows a comparison between the most ac-curate gradients obtained using both the adjoint andfinite-difference methods. There is a general agree-ment on all trends that validates the implementationof the present adjoint method. The fundamental ad-vantage is restated here that the gradient with respectto an arbitrary number of design variables, 50 for thisexample, is determined with the cost of only a singleflow field evaluation and a single adjoint evaluation forany given design cycle.

Figure 7 shows the computed adjoint gradients fordifferent levels of flow solver convergence. For Navier-Stokes calculations, the adjoint information is essen-tially unchanged if the level of convergence in the flowsolver is at least 4 orders of magnitude. This is an

additional advantage of using the adjoint method es-pecially for the design of high-lift configurations forwhich it is difficult to obtain levels of convergencemuch higher than 4 orders of magnitude. This is incontrast with the high levels of convergence requiredfor accurate sensitivity information when using the fi-nite difference method. In viscous flows it is typicalto require that the flow solver converge to about 6 or-ders of magnitude so that the gradient information issufficiently accurate.

An inverse design problem which starts with anRAE2822 airfoil geometry and tries to obtain theshape that generates the pressure distribution arounda NACA 64A410 airfoil at the same flow conditionsis presented here. The mesh used for this Navier-Stokes calculation is a C-mesh with 512×64 cells. Thetarget pressure specified is that of a NACA 64A410airfoil at M = 0.75 and α = 0.0. The Reynolds num-ber of this calculation was set at Re = 6.5 million.Figure 8 shows the progress of the inverse design calcu-lation. In 100 design iterations, the target pressure wasmatched almost exactly, including the correct strengthand position of the shock. The initial RAE2822 airfoilgeometry was altered to obtain a shape that is quiteclose to the NACA 64A410 airfoil that had producedthe target pressure distribution in the first place. TheRMS of the pressure error was reduced from 0.0504 to0.0029 in 100 design iterations.

FLO103-MB with SA Model Validation

Before using FLO103-MB with the Spalart-Allmaras(SA) turbulence model for the design of multi-elementairfoils we must demonstrate its ability to accuratelypredict the flow phenomena involved in this type ofproblem. In particular it is of primary importance tobe able to predict the values of Clmax , the elementpressure distributions, the lift curve slopes for eachelement and the details of the shear layers present inthe problem.

Flow convergenceFigure 9 shows the convergence history of the av-

eraged density residual for the calculation of the flowfield around the 30P30N high-lift configuration usingthe flow solver, FLO103-MB. The Spalart-Allmarasone-equation turbulence model is used for this calcu-lation. The solution converges down to somewhere inthe range of 10−4 and 10−5 in about 2000 iterations.Although small oscillations in the residual remain after2000 iterations, the Cl, one of the cost functions usedfor subsequent designs, has converged without oscilla-tions. As mentioned earlier, this level of convergenceis also good enough to obtain accurate sensitivity in-formation using the adjoint method.

Comparisons with Experimental DataComparisons between computational results and ex-

perimental data are presented below for validation

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purposes. The code, FLO103-MB, and the relatedturbomachinery code TFLO,39 have been extensivelyvalidated for a variety of test cases, ranging fromflat plates and transonic axisymmetric bumps, to fullthree-dimensional configurations.

Figure 10, shows the comparison of the computa-tional and experimental Cp distributions around the30P30N configuration at M∞ = 0.2, α = 8◦, andRe= 9 × 106. The agreement between experimentaland computational distributions is very encouraging.Integrated force coefficients also agree quite well.

In order to validate the ability of the flow solver topredict stall using the SA turbulence model, a compar-ison of Cl versus angle of attack is shown in Figure 11.The total coefficient of lift, together with the individ-ual lift from the three components, is plotted in therange of −5◦ < α < 25◦. The computed results agreequite well with experiment with slightly higher predic-tions of Clmax

and angle of attack at Clmax. Although

the results do not agree with experiment exactly, ithas been observed that the choice of turbulence modelcan have a substantial impact on the numerical valuesof some of these parameters. The stall prediction ca-pability can be a critical factor for actual design cases,such as Clmax maximization. With other turbulencemodels, these quantities can be over-predicted sub-stantially.

Finally, for further validation, velocity profiles fromboth computation and experiment are compared attwo different locations on the main element and theflap. Streamwise velocity components are plottedagainst normal distance to the wall at x/c = 0.45 onthe main element and at x/c = 0.89817 on the flap.The computed velocity profiles predict well both thevelocity gradient in the boundary layer and the gradi-ents due to the wakes of the slat and main element.

Single-Element Airfoil Design

Before embarking on multi-element airfoil design, astudy of the use of optimization for single-element air-foils was performed to gain insight into the possibilitiesfor improvements and the behavior of the method. Cd

minimization at a fixed Cl and Cl maximization ata fixed Cd were tested in order to guarantee the im-provement of the lift over drag ratio, L/D, which is ameasure of the aerodynamic efficiency. The design ex-amples presented in this subsection were all carried outusing a 4 block multi-block mesh around an RAE2822airfoil with a total number of cells equal to 512 × 64.The two designs presented had as a starting point theRAE2822 airfoil and computations were carried out ata Reynolds number of 6.5 million. The surface of theairfoil was parameterized using 50 Hicks-Henne bumpfunctions, 25 of which are distributed evenly along theupper surface of the airfoil, while the remaining 25 areplaced in a similar fashion along the lower surface.

Cd Minimization at a Fixed Cl

Figure 13 shows the result of a typical viscous de-sign calculation where the total coefficient of drag ofthe airfoil is minimized using the parameterization de-scribed above. The free stream Mach number is 0.73and the optimization procedure is forced to achieve anear constant Cl = 0.83. This constraint is achievedby periodically adjusting the angle of attack duringthe flow solution portion of the design procedure. Fig-ure 13 shows the result of 50 design iterations for thistest case. The optimizer is able to eliminate the strongshock wave that existed in the initial design by us-ing the values of the same 50 design variables. Oncethe design process is completed, the total coefficient ofdrag has been reduced from 0.0167 to 0.0109, while theCl has increased very slightly from 0.8243 to 0.8305.The L/D has improved significantly, leading to moreefficient designs. In the case of Cd minimization at afixed Cl, the L/D improved by 54.36% from 49.36 to76.19. This test case also provides a validation of themultiblock design procedure, since a similar test casehad previously been run using the single-block designcode.

Cl Maximization at a Fixed Cd

In this test case, we attempt to maximize the Cl

of the RAE2822 airfoil by altering its shape usingthe same 50 Hicks-Henne design functions while con-straining the coefficient of drag to be constant (Cd =0.0153). Figure 14 shows the result of this type ofdesign optimization. The front portion of the uppersurface of the configuration is modified considerablyto produce a very different pressure distribution thatallows for the existence of a shock wave on the uppersurface that considerably increases the amount of liftcarried by the airfoil. In addition, since the Cd is con-strained to be constant (this is imposed by allowingthe angle of attack to float), the resulting angle of at-tack is also higher, again leading to the creation of ahigher lift coefficient. The numerical results presentedshow significant improvements in L/D. The resultingL/D increased 21.70% from 52.22 to 63.55.

Multi-Element Airfoil Design

Except for the inviscid test case presented in thefirst subsection below, all of the results in this sec-tion were computed using multi-block viscous meshesconstructed using a C-topology. The C-topology meshhas 26 blocks of varying sizes and a total of 204,800cells. All calculations were carried out at a free streamMach number, M∞ = 0.20 and a Reynolds number,Re= 9× 106. The computation of the Reynolds stresswas carried out using the Spalart-Allmaras turbulencemodel.

Apart from the first inviscid test case, the results inthis section mimic those in the single-element airfoilsection as far as the design procedure is concerned.

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Inverse Design Using the Euler Equations

In order to verify the implementation of our de-sign procedure, we present a simple test case whichis aimed at verifying that the multi-block flow andadjoint solvers are capable of producing correct sen-sitivities to both shape modifications and rigging vari-ables in a multi-element airfoil design environment.For this purpose, a multi-block inviscid grid aroundthe 30P30N configuration was constructed. A per-turbed geometry was created by activating a singlebump on the upper surface of the main element andby deflecting the flap by an increment of 2◦. The pres-sure distribution around the original geometry is usedas a target pressure distribution (seen as a solid linein Figure 15) for the perturbed geometry to arrive atthrough an inverse design process. Because this targetpressure distribution is achievable, we can indirectlymeasure the correctness of the sensitivity informationby observing whether the design evolves towards theknown specified target. Notice that the modification ofthe geometry described above influences the pressuredistribution in all three elements: slat, main element,and flap.

A total of 156 design variables were used to param-eterize the complete configuration. 50 bump functionsare used in each of the three elements. In addition,both the slat and the flap were allowed to translatein the x and y directions and to rotate about theirleading-edges. After 100 design iterations where sen-sitivities with respect to all design variables were cal-culated, the target pressure distribution was recoveredas expected. The original geometry was also recoveredas shown in Figure 15. The results of this inviscid testcase provide the necessary confidence to tackle someof the more complex viscous cases presented below.

Cd Minimization at a Fixed Cl

In this test case, we attempt to minimize the totaldrag coefficient of the configuration without changingthe lift coefficient. This design task is one of the mostinteresting problems since decreasing Cd without lossin Cl is the most effective way of increasing the lift overdrag ratio of airfoil at low Mach number (no shockwaves) and at high angle of attack flight conditions(high initial lift coefficient). However, this case is alsothe most difficult problem, since a decrease in Cd usu-ally comes at the expense of a decrease in Cl for thehigh-lift system configuration design. Notice that, asopposed to the single-element test case, the Mach num-ber of the flow is subsonic throughout (Cpcrt = −16.3for M∞ = 0.2) and, therefore, no shock waves arepresent. 9 design iterations were carried out and, asexpected, with a slight increase in Cl, a small decrease(10 drag counts) in Cd was achieved as shown in Fig-ure 16. The resulting L/D increases by 1.73% from62.17 (Baseline L/D at α = 16.02◦) to 63.25. Noticethat the resulting α has increased slightly while trying

to maintain Cl unchanged.

Cl Maximization at a Fixed Angle of AttackA similar calculation to the one presented earlier is

discussed in this section. Instead of minimizing Cd,however, we maximize the Cl of the configuration us-ing all 156 design variables in the problem. In thistest case, the angle of attack of the whole configura-tion remains constant, α = 16.02◦. The optimizer isable to make improvements in Cl after 19 design it-erations: the lift coefficient has increased from 4.0412to 4.1881 as shown in Figure 17. Large changes areobserved in both the flap and slat rigging parameters.The flap deflection angle has increased in order to al-low for larger camber and the slat deflection angle hasdecreased achieving a higher effective angle of attack,and therefore carrying more lift. Figure 18 shows thesame design attempt using only the setting parame-ters as the design variables. After 19 design iterations,the lift coefficient has increased from 4.0412 to 4.1698.From these results it is evident that a large portion(about 85%) of the increase in Cl is due to the modifi-cation of the rigging parameters of each airfoil element.However, notice that very small geometry changes ineach element through the bump functions still deliv-ered more than 100 counts (15% of the increase) in liftcoefficient.

Cl Maximization at a Fixed Cd

We now allow the angle of attack of the configu-ration to float by fixing the value of Cd to that ofthe baseline design point at α = 16.02◦. As we cansee in Figure 19, in 10 design iterations, the optimizerhas increased the lift by an amount of 815 lift countsfrom 4.0412 to 4.1227 while reducing the angle of at-tack from 16.02◦ to 15.127◦ with small changes in thetotal coefficient of drag from 0.0650 to 0.0661. Thisresult appears counterintuitive at first but highlightsthe power of both the adjoint methodology and thecareful parameterization of the surface, since the pro-cedure still yields a higher Cl, while the angle of attackis forced down to match the prescribed Cd = 0.0650.

Maximum Lift Maximization

Single-Element Airfoil ResultsThe present adjoint method was also applied to

the optimization of an airfoil shape that maximizesthe maximum lift coefficient (Clmax). The RAE2822single-element airfoil and the same 4 block mesh for theprevious single-element airfoil design cases were usedfor calculations at a design condition of M∞ = 0.2and Re = 6.5 × 106. Bump functions were used asbefore, and the angle of attack (α) was included asan additional design variable for the maximization ofmaximum lift. Two different approaches were tried.In the first approach three steps were taken as follows:firstly using α alone as a design variable, Clmax andα at Clmax(αclmax) were predicted along the Cl vs.

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α curve for the baseline configuration and then, us-ing 50 bump functions, a new airfoil configuration wasobtained that maximized Cl with α = αclmax fixed.Finally, starting from this value of α, the first stepwas repeated to obtain a new Clmax

and new αclmax

for the updated configuration. Figure 20 shows thedesign results from this approach. In the first step,Clmax

was predicted to be 1.6430 at αclmax = 14.633◦

(O – A in Figure 20 ). Next, as shown in Figure 21,Clmax

increased to 1.8270 while the initial airfoil wasupdated to have more thickness near the leading-edgeand more camber (A – B in Figure 20). The final Clmax

obtained was 1.8732 (Point C in Figure 20) and theoverall Clmax improved 14% from 1.6430 while αclmax

increased 11.1% from 14.633◦ to 16.250◦. This designexample verifies the design capabilities of the adjointgradients using α. Although this example represents asingle iteration of the overall procedure (note that 31iterations were used during the A – B step), this proce-dure could be repeated iteratively to attain even higherlevels of Clmax

. Figure 22 also shows how accuratelythe adjoint method using α can predict Clmax . Themodification of angle attack, α, based on the adjointgradient and the corresponding changes in Cl agreevery well. Both the Cl and α have converged to Clmax

and the αclmax respectively after some small oscilla-tions.

Based on the information gathered from the first ap-proach, both the bumps and the angle of attack weresimultaneously used for the design in the second ap-proach (D – E in Figure 20 ). As shown in Figure 23,using the baseline RAE2822 at α = 11.85◦, Clmax im-proved by 13.3% to 1.8617 and αclmax changed by 6.8%to 15.623◦ from the Clmax and αclmax of the baselineRAE2822 configuration.

Multi-Element Airfoil Results

This design example is the culmination of the ef-forts in this paper. Using the newly developed viscousadjoint procedure, the multi-block flow and adjointsolvers, and the lessons learned in the previous designexamples, we can now attempt to redesign the 30P30Nmulti-element airfoil to optimize its value of Clmax . Inthis case, a total of 157 design variables are used, in-cluding 50 Hicks-Henne bump functions on each of thethree elements, 3 rigging variables for each the slatand flap components, and the angle of attack (α) ofthe complete configuration.

As shown in Figure 24, the design started at α = 22◦

which is near the αclmax of the baseline 30P30Nconfiguration and the baseline was modified in thedirection of Cl improvement using all of the designvariables. As shown in Figure 25 and Figure 25, Clmax

improved by 1.12%, 501 counts, increasing from 4.4596to 4.5097, with a slight change (0.43%) in αclmax.

ConclusionsMaking use of the large computational savings pro-

vided by the adjoint method when large numbers ofdesign variables are involved, we have been able toexplore high-dimensional design spaces that are nec-essary for high-lift system design. In this study, the30P30N multi-element airfoil has been used becauseexperimental data was available for validation pur-poses. We have shown that high-lift system designwith up to a total of 157 design variables is feasiblewith our method. These design variables can includethe parameterization of the element shapes, the riggingvariables, and the angle of attack of the configuration.

Results for L/D maximization, and Clmaxmax-

imization for the RAE2822 single-element airfoilshowed significant improvements. For the multi-element design cases, the relative improvements weresmaller than those of the single-element design cases.The lift increments, however, are of comparable magni-tude although the baseline values for the multi-elementcases are much higher. Of course, one of the mainreasons for this was the fact that the baseline high-liftconfiguration was already a highly optimized one. Theresults obtained are encouraging and point out thatthe adjoint method can have great potential for thedesign of high-lift systems. The design cases in thisportion of the work are purely academic and meantto validate the sensitivity calculation procedure only.Future work will focus on expanding the results of thecurrent paper and on utilizing the method describedabove to perform realistic two-dimensional high-liftsystem designs.

AcknowledgmentsThis research has been made possible by the gener-

ous support of the David and Lucille Packard Foun-dation in the form of a Stanford University School ofEngineering Terman fellowship. The solution of theSpalart-Allmaras turbulence model is an adaptation ofDr. Creigh McNeil’s implementation embedded in thethree-dimensional version of our flow solver. Thanksalso go to Dr. Kuo-Cheng of the Boeing Company forproviding us with the multi-element airfoil experimen-tal data.

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percritical Wing Sections II. Springer Verlag, New York, 1975.2P. R. Garabedian and D. G. Korn. Numerical design

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4J. Fay. On the Design of Airfoils in Transonic Flow Usingthe Euler Equations. Ph.D. Dissertation, Princeton University1683-T, 1985.

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6A. Jameson. Aerodynamic design via control theory. Jour-nal of Scientific Computing, 3:233–260, 1988.

7A. Jameson. Optimum aerodynamic design using CFD andcontrol theory. AIAA paper 95-1729, AIAA 12th ComputationalFluid Dynamics Conference, San Diego, CA, June 1995.

8J. Reuther, A. Jameson, J. Farmer, L. Martinelli, andD. Saunders. Aerodynamic shape optimization of complexaircraft configurations via an adjoint formulation. AIAA pa-per 96-0094, 34th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1996.

9J. Reuther, J. J. Alonso, J. C. Vassberg, A. Jameson, andL. Martinelli. An efficient multiblock method for aerodynamicanalysis and design on distributed memory systems. AIAA pa-per 97-1893, June 1997.

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11J. J. Reuther, A. Jameson, J. J. Alonso, M. Rimlinger, andD. Saunders. Constrained multipoint aerodynamic shape opti-mization using an adjoint formulation and parallel computers:Part II. Journal of Aircraft, 36(1):61–74, 1999.

12J. R. R. A. Martins, I. M. Kroo, and J. J. Alonso. Anautomated method for sensitivity analysis using complex vari-ables. AIAA paper 2000-0689, 38th Aerospace Sciences Meeting,Reno, Nevada, January 2000.

13C. Bischof, A. Carle, G. Corliss, A. Griewank, and P. Hov-land. Generating derivative codes from Fortran programs. In-ternal report MCS-P263-0991, Computer Science Division, Ar-gonne National Laboratory and Center of Research on ParallelComputation, Rice University, 1991.

14J.L. Lions. Optimal Control of Systems Governed by Par-tial Differential Equations. Springer-Verlag, New York, 1971.Translated by S.K. Mitter.

15O. Pironneau. Optimal Shape Design for Elliptic Systems.Springer-Verlag, New York, 1984.

16A. Jameson. Re-engineering the design process throughcomputation. AIAA paper 97-0641, 35th Aerospace SciencesMeeting and Exhibit, Reno, Nevada, January 1997.

17J. Reuther, J.J. Alonso, M.J. Rimlinger, and A. Jameson.Aerodynamic shape optimization of supersonic aircraft configu-rations via an adjoint formulation on parallel computers. AIAApaper 96-4045, 6th AIAA/NASA/ISSMO Symposium on Multi-disciplinary Analysis and Optimization, Bellevue, WA, Septem-ber 1996.

18O. Baysal and M. E. Eleshaky. Aerodynamic design op-timization using sensitivity analysis and computational fluiddynamics. AIAA paper 91-0471, 29th Aerospace Sciences Meet-ing, Reno, Nevada, January 1991.

19W. K. Anderson and V. Venkatakrishnan. Aerodynamicdesign optimization on unstructured grids with a continuousadjoint formulation. AIAA paper 97-0643, 35th Aerospace Sci-ences Meeting and Exhibit, Reno, Nevada, January 1997.

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21A. Jameson, N. Pierce, and L. Martinelli. Optimum aero-dynamic design using the Navier-Stokes equations. AIAA pa-per 97-0101, 35th Aerospace Sciences Meeting and Exhibit,Reno, Nevada, January 1997.

22A. Jameson, L. Martinelli, and N. A. Pierce. Optimumaerodynamic design using the Navier-Stokes equations. Theo-retical and Computational Fluid Dynamics, 10:213–237, 1998.

23S. Kim, J. J. Alonso, and A. Jameson. A gradient accu-racy study for the adjoint-based navier-stokes design method.AIAA paper 99-0299, AIAA 37th Aerospace Sciences Meeting& Exhibit, Reno, NV, January 1999.

24S. X. Ying. High lift chanllenges and directions for cfd.Technical report, AIAA/NPU AFM Conference Proceedings,China, June 1996.

25S. Eyi, K. D. Lee, S. E. Rogers, and D. Kwak. High-liftdesign optimization using navier-stokes equations. Journal ofAircraft, 33:499–504, 1996.

26Eric Besnard, Adeline Schmitz, Erwan Boscher, NicolasGarcia, and Tuncer Cebeci. Two-dimensional aircraft high liftsystem design and optimization. AIAA paper 98-0123, 1998.

27S. Kim, J. J. Alonso, and A. Jameson. Two-dimensional high-lift aerodynamic optimization using thecontinuous adjoint method. AIAA paper 2000-4741, 8thAIAA/USAF/NASA/ISSMO Symposium on MultidisciplinaryAnalysis and Optimization, Long Beach, CA, September 2000.

28W. O. Valarezo. High lift testing at high reynolds numbers.AIAA paper 92-3986, AIAA 9th Aerospace Ground TestingConference, Nashville, TN, July 1992.

29W. O. Valarezo, C. J. Dominik, R. J. McGhee, W. L. Good-man, and K. B. Paschal. Multi-element airfoil optimization formaximum lift at high reynolds numbers. In Proceedings of theAIAA 9th Applied Aerodynamics Conference, pages 969–976,Washington, DC, 1991.

30V. D. Chin, D. W. Peter, F. W. Spaid, and R. J. McGhee.Flowfield measurements about a multi-element airfoil at highreynolds numbers. AIAA paper 93-3137, July 1993.

31F. W. Spaid and F. T. Lynch. High reynolds numbers,multi-element airfoil flowfield measurements. AIAA paper 96-0682, AIAA 34th Aerospace Sciences Meeting & Exhibit, Reno,NV, Jan 1996.

32Christopher L. Rumsey, Thomas B. Gatski, Susan X. Ying,and Arild Bertelrud. Prediction of high-lift flows using turbulentclosure models. AIAA Journal, 36:765–774, 1998.

33S. E. Rogers, F. R. Menter, and P. A. Durbin Nagi N.Mansour. A comparison of turbulence models in computingmulti-element airfoil flows. AIAA paper 94-0291, AIAA 32ndAerospace Sciences Meeting & Exhibit, Reno, NV, January1994.

34S. E. Rogers. Progress in high-lift aerodynamic calcula-tions. AIAA paper 93-0194, AIAA 31st Aerospace SciencesMeeting & Exhibit, Reno, NV, January 1993.

35W. Kyle Anderson and Dary L. Bonhaus. Navier-stokescomputations and experimental comparisons for multielementairfoil configurations. Journal of Aircraft, 32:1246–1253, 1993.

36J. Reuther, A. Jameson, J. Farmer, L. Martinelli, andD. Saunders. Aerodynamic shape optimization of complex air-craft configurations via an adjoint formulation. AIAA paper 96-0094, AIAA 34th Aerospace Sciences Meeting and Exhibit,Reno, NV, January 1996.

37L. Martinelli and A. Jameson. Validation of a multigridmethod for the Reynolds averaged equations. AIAA paper 88-0414, 1988.

38P. R. Spalart and S. R. Allmaras. A one-equation turbu-lence model for aerodynamic flows. AIAA paper 92-0439, AIAA30nd Aerospace Sciences Meeting & Exhibit, Reno, NV, January1992.

39J.J. Alonso J. Yao, RogerL. Davis and A. Jameson. Un-steady flow investigations in an axial turbine using massivelyparallel flow solver tflo. AIAA paper 2001-0529, 39th AIAAAerospace Sciences Meeting, Reno, NV, January 2001.

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0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5Comparison of Finite Difference and Adjoint Gradiensts on 512x64 Mesh

control parameter

grad

ient

Adjoint Finite Difference

Fig. 6 Navier-Stokes Inverse Design: Comparison of Finite-Difference and Adjoint Gradients for a 512x64Mesh.

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5Continuous Adjoint Gradient for Different Flow Solver Convergence

control parameter

grad

ient

7(4) orders 6(3) orders 5(2) orders 4(1) orders 3(0) orders

Fig. 7 Navier-Stokes Inverse Design: Adjoint Gradients for Varying Levels of Flow Solver Convergence.

11 of 20


RAE2822 TO NACA64A410 MACH 0.750 ALPHA 0.000

CL 0.3196 CD 0.0029 CM -0.0952

GRID 513X65 NCYC 10 RES0.220E+00

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++

++++++++++

++++++++

++++++

+++++

++++

+++++

+++++++++++

++++++

++

++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++

+++++++

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++

++++++++

++++++

++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++

++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial, Perror = 0.0504RAE2822 TO NACA64A410 MACH 0.750 ALPHA 0.000

CL 0.5650 CD 0.0111 CM -0.1445

GRID 513X65 NCYC 10 RES0.129E+00

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++

++++++++

++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

++++

+

+

+

+

+

+

+

+

+

+

+

+

++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++

++++

++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 100 Design Iterations, Perror = 0.0029

Fig. 8 Typical Navier-Stokes Inverse Design Calculation, RAE 2822 airfoil to NACA 64A410, M = 0.75,α = 0.0, Re = 6.5 million.

0 200 400 600 800 1000 1200 1400 1600 1800 2000−2

0

2

4

6

8

Cl

Cl Convergence History

0 200 400 600 800 1000 1200 1400 1600 1800 200010

−2

100

102

104

Multigrid Cycles

Res

idua

l

FLO103MB Convergence History

Fig. 9 Convergence History of Cl and Density Residual for a Multi-Element Airfoil using the Spalart-Allmaras Turbulence Model.

12 of 20


30P30N TEST MACH 0.200 ALPHA 8.010

CL 3.1514 CD 0.0651 CM -1.3693

GRID 118784 NCYC 2000 RES 0.950E+00

2.0

00

1.0

00

0.0

00

-1.0

00

-2.0

00

-3.0

00

-4.0

00

-5.0

00

-6.0

00

-7.0

00

-8.0

00

Cp

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++

+++

++

+++

++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+++++++++++++++++++++++++++++

+++++

++++

++++

++

++

+

+

++++++++++++++++++++++++++++++++++++++++++++++

+

+

++++

+

++

++

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++

++

++

++

++

++

+

++++

+++++

+

++++++++++++++++++++++++++++++++++++++++++

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+++++++++++++++++++++++++++++++

+

oooooooooooooo

oo

oo

o

o

oooo

oo

ooooo

oooo

o

o

o

o

oo o o o o o o o o ooooooooo

ooooooooo

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o

o

o

oooooooo

ooooooo

oo

o

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ooo

oo

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o

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Fig. 10 Comparison of Experimental and Computational Pressure Coeffient Distributions for the 30P-30N Multi-element Airfoil. o - Experiment, + - Computation.

−5 0 5 10 15 20 25−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

angle of attack

Cl

plot Cl vs. alpha

Total

Main

Slat

Flap

−*− : Comp. o : Exp.

Fig. 11 Experimental and Computational Lift Versus Angle of Attack Comparison.

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a) Location of Experimental Velocity Profiles.

0 0.1 0.2 0.3 0.40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

u/ainf

d/c

velocity profile at x=0.45

0 0.1 0.2 0.3 0.4 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

u/ainf

d/c

velocity profile at x=0.89

Comp 9MExp 9M

b) Velocity profiles at x/c=0.45 and x/c=0.89817

Fig. 12 Comparison Between Computational and Experimental Velocity Profiles, M = 0.20, α = 8.0,Re = 9 million.

RAE2822 DRAG MIN MACH 0.730 ALPHA 2.944

CL 0.7991 CD 0.0174 CM -0.2947

GRID 32768 NCYC 200 RES 0.192E-01

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++

+++++++++

++++++++

++++++

++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++

+++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cd = 0.0167, Cl = 0.8243, M∞ = 0.73, α = 2.977◦,Re = 6.5× 106.

RAE2822 DRAG MIN MACH 0.730 ALPHA 3.172

CL 0.8305 CD 0.0109 CM -0.2858

GRID 32768 NCYC 200 RES 0.155E-01

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++

+++++++++

++++++++

++++++++

++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

++

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 50 Design Iterations, Cd = 0.0109, Cl = 0.8305, M∞ =

0.73, α◦=3.172, Re = 6.5× 106.

Fig. 13 Typical Navier-Stokes Drag Minimization Calculation at Fixed Cl = 0.83, RAE 2822 Airfoil. −−−Initial Airfoil, Current Airfoil.

14 of 20


RAE2822 TEST MACH 0.730 ALPHA 2.794

CL 0.7989 CD 0.0153 CM -0.2946

GRID 32768 NCYC 400 RES 0.484E-03

0.1E

+01

0.8E

+00

0.4E

+00

-.2E

-15

-.4E

+00

-.8E

+00

-.1E

+01

-.2E

+01

-.2E

+01

Cp

+++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++

+++++++++

++++++++

++++++

++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++

+++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cl = 0.7989, Cd = 0.0153, M∞ = 0.73, α =

2.794◦, Re = 6.5× 106.


CL 0.9723 CD 0.0153 CM -0.3472

GRID 32768 NCYC 400 RES 0.187E-02 0.

1E+

010.

8E+

000.

4E+

00-.

2E-1

5-.

4E+

00-.

8E+

00-.

1E+

01-.

2E+

01-.

2E+

01

Cp

++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++

++++++++++

++++++++

+++++++

+++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++

++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 51 Design Iterations, Cl = 0.9723, Cd = 0.0153, M∞ =

0.73, α = 3.355◦, Re = 6.5× 106.

Fig. 14 Typical Navier-Stokes Lift Maximization Calculation at Fixed Cd = 0.0153, RAE 2822 Airfoil.−−− Initial Airfoil, Current Airfoil.


CL 3.4606 CD 0.0775 CM -1.5246

GRID 28160 NCYC 200 RES 0.877E-03

0.2E

+01

0.1E

+01

0.0E

+00

-.1E

+01

-.2E

+01

-.3E

+01

-.4E

+01

-.5E

+01

-.6E

+01

-.7E

+01

-.8E

+01

Cp

++++++++++++++++++++++++

+++++++++++++++++++

+++++++

++

+

++++++++++++++++++++++++++++

+

+

+

+

+

+++

+

+++++++++++++++++++++++++++++++++

+++++++

+

+

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++

++++++++++++++++++++++++++++++++++++++

+++++

+++

+

+

+

+++++++++++++++++++++

+

+

+

+

+

+++++++++++++++++++

++

+

++++++

+

++++++++++++++++++++++++++++++++++++++++++++++++

+

+

++++++++++++

++++++++++++++++++++++++++++++++

++

a) Initial Geometry with One Bump on the Main Element anda 2◦ Deflection on the Flap


CL 3.5643 CD 0.0833 CM -1.5790

GRID 28160 NCYC 200 RES 0.104E-02

0.2E

+01

0.1E

+01

0.0E

+00

-.1E

+01

-.2E

+01

-.3E

+01

-.4E

+01

-.5E

+01

-.6E

+01

-.7E

+01

-.8E

+01

Cp

++++++++++++++++++++++++

+++++++++++++++++++

+++++++

++

+

++++++++++++++++++++++++++++

+

+

+

+

+

++

+

+

+

+++++++++++++++++++++++++

++++++++++++

++

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

+++++++++++++++++++++++++++++++++++++++++++

+++

+

+

+

+++++++++++++++++++++

+

+

+

+

+

+++++++++++++++++++

++

+

++++++

+

+++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+++++++++++++++++++

+++++++++++++++++++++++++++

b) 100 Design Iterations Using All Bumps and Rigging Vari-ables

Fig. 15 Example of the 30P30N Multi-Element Euler Inverse Design. M∞ = 0.2, α = 8.0◦. + Actual Cp,Target Cp. −−− Initial Airfoil, Target Airfoil.

15 of 20



CL 4.0414 CD 0.0650 CM -1.4089

GRID 204800 NCYC 5000 RES 0.285E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

000

-16.

000

-18.

000

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cd = 0.0650, Cl = 4.0412, M∞ = 0.2, α = 16.02◦,Re = 9× 106.


CL 4.0478 CD 0.0640 CM -1.3988

GRID 204800 NCYC 7000 RES 0.944E-03 2

.000

0.0

00 -

2.00

0 -

4.00

0 -

6.00

0 -

8.00

0-1

0.00

0-1

2.00

0-1

4.00

0-1

6.00

0-1

8.00

0

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++++++++++++++

+

+

+++++++++++++++++++++++++++++++++++++++++++++

+++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 9 Design Iterations, Cd = 0.0640, Cl = 4.0478, M∞ =

0.2, α = 16.246◦, Re = 9× 106.

Fig. 16 Multi-Element Airfoil Drag Minimization Calculation at Fixed Cl=4.04, 30P30N. − − − InitialAirfoil, Current Airfoil.


CL 4.0414 CD 0.0650 CM -1.4089

GRID 204800 NCYC 5000 RES 0.285E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

000

-16.

000

-18.

000

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cl = 4.0412, Cd = 0.0650, M∞ = 0.2, α = 16.02◦,Re = 9× 106.


CL 4.1881 CD 0.0697 CM -1.4501

GRID 204800 NCYC 5000 RES 0.517E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

000

-16.

000

-18.

000

Cp

+++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+

++++++

++

++++++

+++++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


0.2, α = 16.02◦, Re = 9× 106.

Fig. 17 Multi-Element Airfoil Lift Maximization Calculation at Fixed α = 16.02◦, 30P30N. + Current Cp,Initial Cp. Current Airfoil, −−− Initial Airfoil.

16 of 20



CL 4.0414 CD 0.0650 CM -1.4089

GRID 204800 NCYC 5000 RES 0.285E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

000

-16.

000

-18.

000

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++



CL 4.1698 CD 0.0689 CM -1.4439

GRID 204800 NCYC 5000 RES 0.334E-01 2

.000

0.0

00 -

2.00

0 -

4.00

0 -

6.00

0 -

8.00

0-1

0.00

0-1

2.00

0-1

4.00

0-1

6.00

0-1

8.00

0

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

++++++

++

+++++

+

+++++++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


0.2, α = 16.02◦, Re = 9× 106.

Fig. 18 Multi-Element Airfoil Lift Maximization Using Settings Only at Fixed α = 16.02◦, 30P30N. +Current Cp, Initial Cp. Current Airfoil, −−− Initial Airfoil.


CL 4.0414 CD 0.0650 CM -1.4089

GRID 204800 NCYC 5000 RES 0.285E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

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000

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000

-12.

000

-14.

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-16.

000

-18.

000

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++

++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++



CL 4.1227 CD 0.0661 CM -1.4610

GRID 204800 NCYC 5000 RES 0.185E-01

2.0

00 0

.000

-2.

000

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000

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Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++

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+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++

++

++++++++++++++

+

+

++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


0.2, α = 15.127◦, Re = 9× 106.

Fig. 19 Multi-Element Airfoil Lift Maximization Calculation at Fixed Cd = 0.0650, 30P30N. + CurrentCp, Initial Cp. Current Airfoil, −−− Initial Airfoil.

17 of 20


0 2 4 6 8 10 12 14 16 180.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

angle of attack(α), degrees

Cl

Clmax

Maximization for RAE2822 Airfoil

Approach II Design Curve

D

A

B E C

O

Approach I Design Curve

Cl = 2 π α (radian)

Fig. 20 Design Curve of Approach I (O-A-B-C) and II (D-E) for the RAE2822 Clmax Maximization.M∞ = 0.20, Re = 6.5× 106.


CL 1.6430 CD 0.0359 CM -0.4346

GRID 32768 NCYC 1000 RES 0.150E-02

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

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-18.

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Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

+

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cl = 1.6430, Cd = 0.0359, M∞ = 0.2, α =

14.633◦, Re = 6.5× 106.


CL 1.8270 CD 0.0326 CM -0.4901

GRID 32768 NCYC 1000 RES 0.168E-03

2.0

00 0

.000

-2.

000

-4.

000

-6.

000

-8.

000

-10.

000

-12.

000

-14.

000

-16.

000

-18.

000

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

+++++

+

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 31 Design Iterations, Cl = 1.8270, Cl = 0.0326, M∞ =

0.2, α = 14.633◦, Re = 6.5× 106.

Fig. 21 RAE2822 Clmax Maximization Calculation at a Fixed α = 14.633◦. + Current Cp, Initial Cp.Current Airfoil, −−− Initial Airfoil.

18 of 20


37 38 39 40 41 42 43 4414.6315

14.632

14.6325

14.633

14.6335

14.634

α clm

ax

design iteration

αclmax

and Clmax

Convergence History for RAE2822 airfoil

37 38 39 40 41 42 43 441.6429

1.6429

1.643

1.643

1.643

1.643

Cl m

ax

design iteration

Fig. 22 Accuracy of Adjoint Gradient Using α Near Clmax for the RAE2822. M∞ = 0.20, Re = 6.5× 106.


CL 1.4667 CD 0.0215 CM -0.4120

GRID 32768 NCYC 1000 RES 0.681E-04

2.0

00 0

.000

-2.

000

-4.

000

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000

-8.

000

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000

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-14.

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000

-18.

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Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Initial,Cl = 1.4470, Cd = 0.0215, M∞ = 0.2, α = 11.850◦,Re = 6.5× 106.


CL 1.8617 CD 0.0417 CM -0.4966

GRID 32768 NCYC 1000 RES 0.980E-02

2.0

00 0

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+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

+

+

+

+

+

+

+++

+

+

+

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 35 Design Iterations, Cl = 1.8617, Cl = 0.0417, M∞ = 0.2,

α = 15.623◦, Re = 6.5× 106.

Fig. 23 RAE2822 Clmax Maximization Calculation Including α as a Design Variable. + Current Cp,Initial Cp. Current Airfoil, −−− Initial Airfoil.

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19 20 21 22 23 24 25 264.2

4.25

4.3

4.35

4.4

4.45

4.5

4.55Maximum Cl Maximization for 30P30N

angle of attack(α), degrees

Cl

Design Curve

Baseline Cl vs. α Curve

α at Clmax

: 501 Cl counts,α

clmax : 0.1o increased.

S

Edesigned

Ebaseline

Fig. 24 Design Curve of the 30P30N Clmax Maximization. M∞ = 0.20, Re = 9× 106.


CL 4.4596 CD 0.1062 CM -1.2460

GRID 204800 NCYC 7000 RES 0.200E-02

2.5

00 0

.000

-2.

500

-5.

000

-7.

500

-10.

000

-12.

500

-15.

000

-17.

500

-20.

000

-22.

500

Cp

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

++

+

++

++++

++

+

+

+

++

+

++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

a) Baseline, Cl = 4.4596, Cd = 0.1062, M∞ = 0.2, α = 23.5◦,Re = 9× 106.


CL 4.5097 CD 0.1236 CM -1.2607

GRID 204800 NCYC 7000 RES 0.149E-02

2.5

00 0

.000

-2.

500

-5.

000

-7.

500

-10.

000

-12.

500

-15.

000

-17.

500

-20.

000

-22.

500

Cp

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++

+

+

+

+

+

++

+

++

++++

+

+

+

++

+

++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

b) 6 Design Iterations, Cl = 4.5097, Cd = 0.1236, M∞ = 0.2,

α = 23.601◦, Re = 9× 106.

Fig. 25 Multi-Element Airfoil Clmax Maximization Calculation for the 30P30N. + Current Cp, InitialCp. Current Airfoil, −−− Initial Airfoil.

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AIAA 2002–0844 - Stanford Universityaero-comlab.stanford.edu/Papers/kim.aiaa.02-0844.pdf · 2019....

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