adjoint-based unsteady airfoil design optimization with...
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Adjoint-based Unsteady Airfoil Design Optimization withApplication to Dynamic Stall
Karthik ManiAssociate Research Scientist
Brian A. LockwoodGraduate Student
Dimitri J. MavriplisProfessor
University of Wyoming, Laramie, WY
Abstract
This paper presents the development and application of an adjoint-based optimization method to designing air-foils with the objective of alleviating dynamic stall effects in helicopter rotor blades. The unsteady flow problemis simulated using the NSU2D code, which is a two-dimensional unsteady, viscous, turbulent Reynolds averagedNavier-Stokes (RANS) finite-volume solver. The corresponding adjoint code computes sensitivities of the optimiza-tion objective functions with respect to a set of design inputs and is developed through discrete linearization of allcomponents building up the flow solver. Objective functions are defined such that their minimization results in allevi-ation of undesirable dynamic stall characteristics while the design inputs are parameters that control the shape of theairfoil. The method is applied to an example problem where the goal is to reduce the peak pitching moment during thehysteresis cycle of the SC1095 airfoil while maintaining the baseline time-dependent lift profile. To our knowledge,this work represents the first application of the unsteady adjoint method to optimizing airfoil geometry in dynamicstall problems.
IntroductionDynamic stall is an unsteady aerodynamic phenomenonon airfoils, wings, and rotor blades, which results in ad-verse lift, drag, and pitching moment characteristics, com-
Presented at the American Helicopter Society 68th Annual Forum, FortWorth, Texas, May 1-3, 2012. Copyright 2012 by the American Heli-copter Society International, Inc. All rights reserved.
pared with static stall, due to unsteady vortices and sep-aration patterns. Dynamic stall constitutes an importantproblem in the unsteady aerodynamics of rotorcraft. Theretreating blade of a rotorcraft in forward flight in a lowdynamic pressure, high angle of attack environment isparticularly susceptible, and can limit the lifting capabil-ity and forward flight speed of the helicopter [1]. Dy-
namic stall also adversely impacts the aeroelastics of therotor blade (flutter), rotor hub loads, and fatigue life.Fundamental research into dynamic stall has been ongo-ing for decades [2] in order to understand the underly-ing unsteady flow physics and airfoil shape dependen-cies. More recent advances have experimentally inves-tigated passive and active dynamic stall alleviation tech-niques [3, 4, 5, 6, 7, 8] with generally good success.These modifications include leading edge droop, vortexgenerators, airfoil recontouring, and active flow controlwith zero-mass jets. However, often the development ofthese modifications use trial and error, low-order aerody-namic models, and steady aerodynamic design methods.Thus the actual results are sometimes unpredictable whenwind tunnel tested. Overall understanding is still lack-ing as to the effects that airfoil shape modification haveon dynamic stall characteristics and the resultant detailedchanges to the flow field.
In order to successfully simulate dynamic stall, vis-cous and turbulent effects must be taken into account ina time-accurate manner, requiring, at a minimum, the useof an unsteady Reynolds averaged Navier-Stokes simula-tion capability with a suitable turbulence model. Com-putational fluid dynamics (CFD) analyses have achievedmixed success in predicting dynamic stall, mainly due todeficiencies in turbulence and transition modeling [4, 6, 9,10, 11, 12, 13]. Other deficiencies in experimental data,such as extreme unsteadiness, insufficient boundary con-dition information, and unknown test data error sources,also contribute to poor comparisons. Difficult cases, suchas smooth surface separation and trailing edge stall pro-gression (NACA0015), have been predicted successfullyusing advanced DES-like turbulence modeling [10, 11].However, in order to remain practical, design tools mustremain confined at most to simpler unsteady RANS meth-ods. The risk in relying on simple RANS CFD-baseddesign methods is that the calculations, particularly onthe initial geometry, will be grossly in error, and that fi-nal numerically generated designs may exploit incorrectphysics. However, it has been shown that some typesof dynamic stall progressions, for example leading edgeinitiation (e.g. OA209 and SC1095 airfoil) and milderstall, are often predictable using current RANS technol-ogy [4, 13]. These types of airfoils now dominate rotorblade development.
Aerodynamic shape optimization can be used to
achieve specific performance objectives through the appli-cation of suitable geometric shape modifications. Shapeoptimization has been used extensively for steady-stateproblems, such as aircraft cruise drag reduction. Two-dimensional airfoil optimization techniques date backdecades and have seen widespread use. The developmentof adjoint techniques has proven to be an enabling tech-nology for steady-state aerodynamic optimization, espe-cially in 3D, since the full sensitivity vector of a singleobjective with respect to any number of design variablescan be computed with a single adjoint solution, at a costroughly equivalent to a single flow solution [14]. How-ever, because of the flow physics complexity and compu-tational cost, optimization of unsteady and moving bodyproblems has received significantly less attention [15, 16,17, 18, 19, 20]. However, recent efforts have lead to thedevelopment of both two-dimensional [17, 19, 21] andthree-dimensional [18, 20] unsteady adjoint capabilities.
Given these developments, it seems that the technologyis in place for the development of a useful dynamic stallalleviation design methodology. The success of currenttwo and three-dimensional practical RANS simulationsfor realistic configurations, coupled with new develop-ments in unsteady adjoint optimization techniques, can beleveraged to produce effective two and three-dimensionalhelicopter rotor design tools targeting dynamic stall al-leviation. Even though current capabilities for the pre-diction of dynamic stall effects are still under develop-ment, advancement in state-of-the-art optimization cannotpossibly wait until dynamic stall analysis is a completelysolved problem. Rather, advances in dynamic stall anal-ysis and optimization capabilities must be sought concur-rently.
This paper presents the development of the adjoint lin-earization of the unsteady RANS equations in two dimen-sions for computing the gradient vector for use in an op-timization environment. All components building up theflow solver including the turbulence model are linearizedand the gradient vector computed is the discretely exactsensitivity of the objective function with respect to the setof design inputs that control the shape of the airfoil. Theoptimization example problem presented, where the peakpitching moment is reduced without compromising lift,shows great promise for the method within the limits ofthe solver’s ability to capture the correct physics in theflow.
Analysis Problem Formulation
Governing equations and discretizationThe conservative form of the Navier-Stokes equations isused in solving the flow problem. In vectorial form theconservative form of the Navier-Stokes equations may bewritten as:
∂U(x, t)∂t
+∇ · (F(U)+G(U)) = 0 (1)
where U represents the vector of conserved variables(mass, momentum, and energy), F(U) represents the con-vective fluxes and G(U) represents the viscous fluxes.Applying the divergence theorem and integrating over amoving control volume A(t) bounded by the control sur-face B(t) yields:∫
A(t)
∂U∂t
dA+∫
B(t)F(U) ·ndB+
∫B(t)
G(U) ·ndB = 0
(2)Using the differential identity:
∂
∂t
∫A(t)
UdA =∫
A(t)
∂U∂t
dA+∫
B(t)U(x ·n)dB (3)
equation (2) is rewritten as:
∂
∂t
∫A(t)
UdA+∫
B(t)[F(U)− xU] ·ndB
+∫
B(t)G(U) ·ndB = 0
(4)
Considering cell-averaged values for the state U and amoving but non-deforming control volume A, the spatiallydiscrete version is then:
A∂U∂t
+S(U,x(t), x(t),n(t)) = 0 (5)
where S represents the spatially integrated discrete con-vective and viscous fluxes. A is the control volume, x(t)the time varying mesh coordinates, x(t) the time vary-ing grid speeds, and n(t) the time varying normals of thecontrol volume boundary faces which are computed fromx(t).
The time derivative term is discretized using ei-ther a first-order accurate backward-difference-formula
(BDF1), or a second-order accurate BDF2 scheme. Thesediscretizations are shown in equations (6) and (7) respec-tively. The index n is used to indicate the current time-level as the convention throughout the paper.
A∂U∂t
=A(Un−Un−1
)∆t
(6)
A∂U∂t
=A( 3
2 Un−2Un−1 + 12 Un−2
)∆t
(7)
The solver uses a spatially second-order accurate node-centered finite-volume full Navier-Stokes discretizationwith matrix dissipation for the convective fluxes. The oneequation Spalart-Allmaras [24] turbulence model is usedto simulate the effect of turbulence in the flow. Furtherdetails of the discretization used in the flow solver can befound in references [25, 26].
Solution procedure
For an implicit time-step the nonlinear flow residual is de-fined as:
Rn = A∂U∂t
+S(xn, xn,Un) = 0 (8)
where the discretization of the time derivative is based onthe chosen time-integration scheme. The second term isthe spatial residual of the flow and is a function of the gridvelocities x, the current mesh coordinates x and the flowsolution at the current time-level Un. The mesh orienta-tion at the current time-level is obtained through rigid pre-scribed rotations as a function of time and the correspond-ing grid speeds are computed using the analytic deriva-tive of the prescribed motion equations. The non-linearresidual is linearized and solved using Newton’s methodat each time-level as:[
∂R(Uk)
∂Uk
]δUk =−R(Uk) (9)
Uk+1 = Uk +δUk
where the linear system is solved using either agglomer-ation multigrid [27] or preconditioned GMRES [28] withmultigrid as the preconditioner. Jacobi iterations are usedas the smoother in the multigrid algorithm.
Geometry parametrizationModification of the baseline geometry is achieved throughdisplacements of the surface nodes defining the geometry.In order to ensure smooth geometry shapes, any displace-ment of a surface node is controlled by a bump functionwhich influences neighboring nodes with an effect thatdiminishes moving away from the displaced node. Thebump function used for the work presented in this paperis the Hicks-Henne Sine bump function [29]. The de-sign variables or inputs for the optimization examples pre-sented form a vector of weights controlling the magnitudeof bump functions placed at various chordwise locations.Modifications to the surface of the airfoil are propagatedto the interior mesh using the tension spring analogy al-gorithm for mesh deformation. Details of the implemen-tation of the mesh deformation algorithm can be found inreferences [21, 30].
Sensitivity Formulation for theUnsteady Flow Problem
The time-integrated objective function Lg that is to beminimized can be written with intermediate dependenciesas a function of the design variables D in order to assistwith the chain rule differentiation process. For the prob-lem of the unsteady pitching airfoil, the design variablesform a vector of weights controlling the magnitudes ofbumps placed at various chordwise locations. With inter-mediate dependencies the objective function can be writ-ten as:
Lg =nsteps
∑n=0
Ln (Un(D),xn(x0(D)))
(10)
where Ln,Ln−1 · · · denote local objective function valuesat each time-step. Differentiating equation (10) results ina general expression for the final sensitivity derivative dLg
dDfor the unsteady flow problem as:
dLg
dD=
nsteps
∑n=0
∂Lg
∂Ln
[∂Ln
∂Un∂Un
∂D+
∂Ln
∂xn∂xn
∂D
](11)
where intermediate derivatives are not shown for simplic-ity. For typical airfoil shape design problems there aremultiple design variables and only a single objective func-tion. This requires the transpose of equation (11) in order
to obtain the adjoint or reverse linearization of the sensi-tivity as shown below.
dLg
dD
T=
nsteps
∑n=0
[∂Un
∂D
T∂Ln
∂Un
T
+∂xn
∂D
T∂Ln
∂xn
T]
∂Lg
∂Ln
T
(12)
Expanding the summation backward:
dLg
dD
T=
(∂Un
∂D
T∂Ln
∂Un
T
+∂xn
∂D
T∂Ln
∂xn
T)
∂Lg
∂Ln
T
︸ ︷︷ ︸(X)
+
(∂Un−1
∂D
T∂Ln−1
∂Un−1
T
+∂xn−1
∂D
T∂Ln−1
∂xn−1
T)
∂Lg
∂Ln−1
T
︸ ︷︷ ︸(Y )
· · · · ·
(13)
Here ∂Lg/∂Ln is a scalar quantity, while ∂Ln/∂Un and∂Ln/∂xn are vectors that are easily computable since Ln
is a scalar quantity. These are merely linearizations withrespect to the flow variables and the mesh coordinates.However, the sensitivity of the flow state to the designvariables ∂U
∂D is a matrix and cannot be directly obtained.In order to obtain a suitable expression for this term werevisit equation (8) which forms the constraint to be sat-isfied at each time-step in the sensitivity problem. For aBDF1 time discretization the flow constraint equation (i.e.the non-linear flow residual at a given time-level n) can bewritten as:
Rn (xn(x0(D)),Un(D),Un−1(D))= 0 (14)
Differentiating this with respect to the design variables Dyields:
dRn
dD=
∂Rn
∂xn∂xn
∂D+
∂Rn
∂Un∂Un
∂D+
∂Rn
∂Un−1∂Un−1
∂D= 0 (15)
which, when transposed and rearranged, results in an ex-pression for the flow variable sensitivity as:
∂Un
∂D
T
=−
(∂xn
∂D
T∂Rn
∂xn
T
+∂Un−1
∂D
T∂Rn
∂Un−1
T)[
∂Rn
∂Un
]−T
(16)
Considering only term (X) in equation (13) and substitut-ing the above expression we obtain:
(X) =−
(∂xn
∂D
T∂Rn
∂xn
T
+∂Un−1
∂D
T∂Rn
∂Un−1
T)
×[
∂Rn
∂Un
]−T∂Lg
∂Un
T
+∂xn
∂D
T∂Lg
∂xn
T(17)
Introducing an adjoint variable ΛnU at time-level n defined
as:
ΛnU =−
[∂Rn
∂Un
]−T [∂Lg
∂Un
]T
(18)
we obtain a linear flow adjoint system to be solved itera-tively as shown below.[
∂Rn
∂Un
]T
ΛnU =−
[∂Lg
∂Un
]T
(19)
This is nearly identical to the linear systems that arisein the Newton solver for the nonlinear analysis problemwith the exception of the transposed flow Jacobian ma-trix. Convergence of both systems are similar since theeigenvalues of the linear operator in both cases are identi-cal. The same methods employed to solve the linear sys-tems in the nonlinear problem may be used to solve forthe adjoint variable vector.
Once the adjoint variable vector has been obtained iter-atively, term (X) now becomes:
(X) =∂xn
∂D
T∂Rn
∂xn
T
ΛnU︸ ︷︷ ︸
(A)n
+∂Un−1
∂D
T∂Rn
∂Un−1
T
ΛnU︸ ︷︷ ︸
(B)n
+∂xn
∂D
T∂Lg
∂xn
T
︸ ︷︷ ︸(C)n
(20)
The sum of terms (A)n and (C)n forms the contribution tothe total gradient vector [dL/dD]T from time-level n as:
(A)n +(C)n =∂x0
∂D
T∂xn
∂x0
T{
∂Rn
∂xn
T
ΛnU +
∂Lg
∂xn
T}
(21)
where the second term is simply the transpose of the gridrotation matrix:
[β]nT =∂xn
∂x0
T
(22)
The backward recurrence relation is now clear since term(B)n can be combined with the appropriate flow sensitiv-ity term at time-level n− 1 that appears in (Y ) and theprocess can and be repeated. Assembling the completesensitivity vector [dL/dD]T involves starting at the finaltime-level and sweeping backward in time while solvingfor a vector of flow adjoint variables at each time-level.The general form of the flow adjoint equation to be solvedat an arbitrary time-level n for BDF1 and BDF2 basedtime discretizations are given as:
BDF1 :[
∂Rn
∂Un
]T
ΛnU =− ∂Lg
∂Un
T
+∂Rn+1
∂Un
T
Λn+1U (23)
BDF2 :[
∂Rn
∂Un
]T
ΛnU =− ∂Lg
∂Un
T
+∂Rn+1
∂Un
T
Λn+1U
+∂Rn+2
∂Un
T
Λn+2U
(24)
where ΛnU are the variables to be solved for and all terms
on the right-hand-side are known from previous (future)steps during the backward sweep in time.
Solver Validation
Dynamic stall predictionThe NSU2D code was used to simulate a dynamic stallevent for the SC1095 airfoil at a Mach number of 0.302and a Reynolds number of 3.86 million. The mean angle-of-attack for this case was 9.92o and a sinusoidal pitch-ing motion of amplitude 9.9o with a reduced frequencyof 0.148 was prescribed. This case is documented exper-imentally as Frame 37109 in the NASA Technical Memo84245 [2]. The computations using NSU2D were per-formed on a grid of approximately 80,000 points, usinga total of 10,000 time-steps of size ∆t = 0.02, with 25multigrid (5 grid levels) per time-step. The simulationcovered just over 3 periods of motion, with the results dis-playing periodic behavior after the first period. The entiresimulation required a total of 7 hours wall-clock time us-ing OpenMP parallelization on 8 cores. The flow was as-sumed to be fully turbulent for this case and no transitionwas specified. Figure 1 shows the time-dependent loadscomputed by NSU2D and compares them to experimen-tal data. Reasonable agreement is seen in the histories of
all three load distributions, lift, drag and moment. Fig-ure 2 shows the typical convergence of the flow equationsand the adjoint equations at an arbitrary time-step with 25multigrid cycles. A parameter study was performed wherethe number of time-steps and number of multigrid cyclesat each time-step were varied to determine optimal valuesand the results are shown in Figure 3.
Gradient verification
Verification of the gradient vector obtained from the ad-joint solver was done using a three-step process. Theforward or tangent linearization gradients were first ver-ified against gradient components obtained from a com-plex version of the analysis solver, after which the prin-ciple of duality [17] between transpose operations wasused to verify the adjoint gradients against the forwardlinearization values. The complex version of the analysissolver was developed be redefining all floating point num-bers as complex and overloading certain operators suchMIN,MAX,ABS etc. to handle complex arguments. Thecomplex solver takes advantage of the fact that functionsevaluated using complex arguments can be used to deter-mine the derivative of the function with respect to the ar-guments by introducing a complex perturbation in the ar-gument. The derivative f ′(x) of any real function f (x) canbe computed as:
f ′(x) = Im[ f (x+ ih)]/h (25)
where h is the complex perturbation introduced into theargument x and f (x+ ih) is the complex equivalent of thereal function f (x). The derivative is then simply the imag-inary part of the complex function divided by the pertur-bation. Complex perturbations are introduced into the de-sign variables one at a time and the solver is run in com-plex mode to determine the sensitivity of the objectivefunction to that design variable. While the complex stepmethod much like the finite-difference method requires Ncomplex solutions for N design variables, it is not subjectto subtraction errors that affect traditional finite differenc-ing. This allows the perturbation to be significantly smallwith values generally being in the 10−100 range for thecurrent work. In fact it is a requirement that the complexperturbation be small enough such that operations do notresult in the contamination of the real part, i.e. less than
Angle of attack (degrees)
Lift
coef
ficie
nt
0 5 10 15 200
0.5
1
1.5
2
ExperimentTURNSNSU2D
(a) Time variation of lift coefficient
Angle of attack (degrees)
Dra
gco
effic
ien
t
0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ExperimentTURNSNSU2D
(b) Time variation of drag coefficient
Angle of attack (degrees)
Mo
men
tcoe
ffic
ient
0 5 10 15 20-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
ExperimentTURNSNSU2D
(c) Time variation of moment coefficient
Figure 1: Predicted time-dependent loads for theSC1095 airfoil as computed by NSU2D compared withTURNS [31] and experimental data.
Multigrid Cycles
Res
idua
l
0 5 10 15 20 25
10-8
10-7
10-6
10-5
10-4
10-3
FlowAdjoint
Figure 2: Typical convergence of the flow and adjointequations at an arbitrary time-step for the SC1095 testcase with 25 multigrid cycles.
Figure 3: Time-step and multigrid cycles sensitivity studyfor the SC1095 test case.
Functional Cl CmAdjoint -0.02578964199504 -2.42401026265272Forward -0.02578964199504 -2.42401026265272Complex -0.02578964199505 -2.42401026265272
Table 1: Comparison of adjoint sensitivity against for-ward sensitivity and those obtained by the complex-stepmethod for a single design variable, i.e. single bumpfunction located on the upper surface at midchord. (Non-matching digits in bold font)
machine precision for the real part. Table 1 compares thegradients obtained from different methods and indicatesnear perfect agreement.
Optimization ExampleThe goal of the optimization example presented here wasto reduce the peak pitching moment during the dynamicstall cycle without compromising the time variation of thelift coefficient. The dynamic stall case is identical to theone presented in the validation section, i.e. SC1095 air-foil undergoing periodic pitching at a reduced frequencyof kc = 0.148 in freestream conditions of Mach num-ber 0.302 and Reynolds number 3.86× 106. The meanangle-of-attack is α0 = 9.92o and the pitch amplitude isαm = 9.9o. The time variation of the angle-of-attack ofthe airfoil is given by:
α = α0 +αm sin(ωt−π/2) (26)
where a phase shift of 90o is included to assure that theairfoil begins (t = 0) the pitch cycle with the lowest angle-of-attack possible. This is necessary to preserve code sta-bility during startup since free-stream flow parameters areused as the initial condition at all nodes in the mesh. Thetime-integration used 750 time-steps of size ∆t = 0.08which translates into a approximately 1 period of pitch.25 multigrid cycles were used at each time-step for boththe flow and adjoint solutions.
The objective function for this case was defined as:
Lg =
[nsteps
∑n=1
(Cnm)
100
] 1100
+Wnsteps
∑n=1
(Cnl −Cl
ntarget)
2 (27)
X
Y
0 0.2 0.4 0.6 0.8 1-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4: Computational mesh of the SC1095 airfoil con-sisting of 56268 nodes used in the optimization example.
where the first term corresponds to a high order norm thattargets the peak moment during the pitch cycle and thesecond term acts as a constraint on the time variation ofthe lift. W is a weight on the constraint which may bemodified during the course of the optimization if neces-sary. The target lift coefficients at each time-step werethose of the baseline SC1095 airfoil.
The geometry parameterization consisted of 10 designvariables acting as weights on bump functions, half ofwhich were placed on the upper surface and the other halfplaced on the lower surface of airfoil. The bump func-tions were equally spaced in the chordwise direction be-tween a non-dimensional chord of c = 0 and c = 0.8. Thiswas done to prevent the optimization from making drasticmodifications to the trailing edge turning angle. The opti-mization algorithm used in this case was the L-BFGS re-duced hessian gradient-based method [23]. The gradientvector is computed by running the flow solver followed bythe adjoint code and then passed into the L-BFGS routineto obtain a modified design. The case was run on an sin-gle cluster node with 16 cores and utilized OpenMP par-allelization. A single function/gradient call consisting ofa forward analysis run and a backward adjoint solve tookapproximately 20 minutes of wall clock time. The com-putational mesh consisted of 56298 nodes and is shown inFigure 4.
The optimization was started with the baseline SC1095airfoil using a loose lift constraint weight of W = 0.001and run until a limit of 50 function/gradient requestsby the optimizer were reached. L-BFGS being a localgradient-based optimization algorithm has a tendency tonot move from the initial airfoil configuration when astrong weight for the constraint is used initially since thatis where the constraint is satisfied exactly. Loosening theweight during startup allows the optimizer to focus onminimizing the peak moment during the pitch cycle andmove out of the starting area in the design space as op-posed to focusing on satisfying the constraint. Once asignificant reduction in the peak moment was achieved,the lift constraint weight was tightened to W = 0.01. Theoptimization was then run again until 50 requests for func-tion/gradient values were made by the optimizer. At thispoint the constraint weight was further tightened to its fi-nal value of W = 0.1. This final stage of the optimiza-tion primarily focused on satisfying the constraint and ter-minated when 122 requests for function/gradient valueswere made by the optimizer. The L-BFGS optimizationalgorithm uses multiple function/gradient values within asingle design iteration as it involves line searches duringthe process. As a consequence the number of design itera-tions is typically less than the number of calls to the func-tion/gradient code. However, it is important to track andprovide termination criteria based on the number of func-tion/gradient requests rather than the number of designiterations since the cost of the optimization is directly as-sociated with the evaluation of the function and gradient.17 design iterations were performed in the third and finalstage of the optimization by the L-BFGS algorithm usinginformation from 122 function and gradient values. Over-all approximately 200 function/gradient runs were madeduring the design process resulting in a wall clock time ofabout 70 hours.
The convergence of the last stage of the optimization isshown in Figures 5 and 6. The reduction in the objectivefunction at each design iteration and the number of func-tion/gradient requests made at each iteration is shown inFigure 5. Figure 6 shows the convergence of the objec-tive function and the constraint separated out of the ob-jective during the course of the optimization. Figure 7shows the final optimized design for the airfoil and com-pares it to the baseline SC1095 airfoil. Figure 8 shows thetime-variation of the loads for the final optimized design
Figure 5: Convergence of objective for the optimizationexample with L-BFGS design iterations and number offunction/gradient evaluations.
Figure 6: Convergence of objective function and con-straint separated from objective function for the optimiza-tion example against number of function/gradient evalua-tions.
(a) Comparison of the baseline SC1095 airfoil with the finaloptimized airfoil.
(b) Comparison using an exaggerated scale.
Figure 7: Comparison of optimized and baseline airfoils.
(a) Time variation of lift coefficient
(b) Time variation of drag coefficient
(c) Time variation of moment coefficient
Figure 8: Comparison of initial and final time-dependentloads for the optimization example.
of the airfoil. A large reduction of nearly 60% is seen inthe peak value for the moment coefficient during the cy-cle, while the time variation of the lift is not significantlychanged particularly during the upstroke. Although thedrag coefficient was not included in the objective formu-lation, a significant reduction in the peak drag value isalso observed in the final design. It was noticed from theflowfield animations of the final design that leading edgeinitiation of separation had been eliminated and separa-tion with a weaker vortex closer to midchord had beenintroduced. This explains a mitigated peak moment as aweaker vortex rolls past the trailing edge and a shallowerstall in lift distribution as less circulation is lost in the pro-cess. Figure 9 compares the density contours between thebaseline SC1095 airfoil and the final optimized design ofthe airfoil at the highest angle-of-attack during the pitchcycle. The plot indicates a significantly weaker vortex be-ing shed from the optimized airfoil.
Conclusions
This work has demonstrated the possibility of performingairfoil shape optimization for dynamic stall alleviation.The use of a time dependent adjoint approach enables thecalculation of the sensitivities with respect to any num-ber of design variables with a single adjoint backwardstime-integration sweep which is equivalent to the cost of asingle forward time-dependent flow simulation. Thus, fortwo-dimensional simulations, optimization cases can berun on a multicore workstation within several days. Al-though the example shown herein was successful in re-ducing pitching moment magnitudes for the prescribedtest case, the performance of the resulting thicker airfoilshape will likely degrade at other operating conditions.Thus, future work will focus on extending the processto include multi-point and multi-objective optimization.Additionally, the bump functions used to define the air-foil shapes are ill-suited for producing leading-edge droopconfigurations, which are well known to be beneficial forstall alleviation. Thus, the incorporation of more sophis-ticated and flexible shape design parameters will also beinvestigated.
x
y
0 0.5 1
-0.5
0
0.5
RHO
1.041.024291.008570.9928570.9771430.9614290.9457140.930.9142860.8985710.8828570.8671430.8514290.8357140.82
(a) Baseline SC1095 airfoil.
X
Y
0 0.2 0.4 0.6 0.8 1-0.6
-0.4
-0.2
0
0.2
0.4
RHO
1.0400001.0242861.0085710.9928570.9771430.9614290.9457140.9300000.9142860.8985710.8828570.8671430.8514290.8357140.820000
(b) Final optimized design.
Figure 9: Comparison of the density contours between thebaseline SC1095 airfoil and the final optimized design air-foil close to the highest angle of attack in the pitch cycleof the optimization example presented.
AcknowledgementsThis work was supported by ARMY SBIR Phase 1 con-tract W911W6-11-C-0037.
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