Preliminary Study on Time-Spectral and Adjoint-Based Design

Optimization of Helicopter Rotors

Seongim Choi ∗

Department of Aeronautics and Astronautics

Stanford University, Stanford, CA 94305

Ki H. Lee †

Aerospace Operations Modeling Branch

NASA Ames Research Center

Moffett field, CA 94035

Juan J. Alonso ‡

Department of Aeronautics and Astronautics

Stanford University, Stanford, CA 94305

Anubhav Datta §

Eloret Corporation

Moffett field, CA 94035

We propose in this paper an adjoint-based design optimization methodology that isparticularly efficient for flow simulation of helicopter rotors in combination with a Time-Spectral (TS) method. The TS method is a fast and efficient algorithm to simulate theunsteady periodic flows with the narrow frequency spectrum which are often encounteredin rotorcraft, turbomachinery, and fixed-wing flutter analysis. It represents the time dis-cretization of the flow solver by Fourier-based modes/bases in which periodic steady-stateis assumed throughout the computation. The steady-state assumption reduces the com-putational cost significantly, which makes it possible to use the efficient adjoint methodapplicable to the rotor design problem. The adjoint solution method is widely acceptedas an inexpensive way to obtain the sensitivity information of flow solutions to a largenumber of design parameters. Integrated with gradient-based optimization technique,the adjoint method has been an essential module in aerodynamic/aero-structural shapeoptimizations. As a preliminary study before we progress towards a more complete andpractical desgin optimization of helicopter rotor, this study focuses on the accuracy andvalidity of our current design optmization tools. First of all, for the validation of the aero-dynamic tools, a flight 8534 test condition of the UH-60A configuration is simulated withtime-spectral computation, and the accuracy and efficieny of the time-spectral methodis demonstrated in comparison with the time accurate results and experimental data. Aloose coupling of the time-spectral method and the structural analysis via UMARC isalso demonstrated to prove the capability of the time-spectral method to simulate thehelicopter rotor problem in a comprehensive way. Finally, a simple design applicationproblem of the hover flight condition is considered and an analysis with single blade cou-pled with free wake model is employed. A baseline blade shape is modified to an optimalshape to minimize torque while the thrust is maintained or enhanced.

INTRODUCTION

Adjoint method1, 2 has been successfully employedas efficient and accurate technique in a wide range ofaerodynamic/aero-structural shape optimization.3–5

The ability to leverage an adjoint solution to inexpen-

∗Research Associate†Aerospace Engineer‡Associate Professor§Rotorcraft Dynamist

Presented at the AHS Specialist’s Conference on Aerody-namics, San Francisco, CA, January 23-25, 2008. Copyrightc©2008 by the American Helicopter Society International, Inc.All rights reserved.

sively obtain the sensitivity of a particular cost func-tion of interest with respect to a large number of designvariables has motivated the use of this tool in a varietyof design applications. This method becomes criticalin the gradient-based optimization process, since thenumber of design variables can be rather independentof the computational cost, and it provides higher ac-curacy as opposed to a finite-differencing or complexvariable method at a much reduced cost.3 Thereforea design problem, such as helicopter blade/planformshape optimization, which inevitably involves a largescale simulation with a huge number of design param-eters can greatly benefit from the adjoint method.

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However, until now the application of the adjointmethod has been focused mainly on steady-state prob-lems. Unsteady adjoint formulations require storingthe entire time history of the flow solutions, which en-tails massive storage/memory requirements and com-putational costs.6 With some efforts7 to amelioratethis situation, the robustness and efficiency of the trulyunsteady adjoint formulation have not yet been fullyinvestigated. Difficulties involved in the implementa-tion of the adjoint formulation for the unsteady flowsolvers make the problem more challenging. Thereforebecause of the difficulties associated with the inher-ent nature of the complicated unsteady flows aroundit, its direct application to the design optimization ofhelicopter rotors has been rather limited and less pop-ular

On the other hand, the Time-Spectral (TS)method11, 14, 15 has proved to be very fast and effi-cient in the simulation of the unsteady periodic flowsthat often occur in many flight configurations such aspitching airfoils, rotating blades, and flapping wings.A basic idea of the TS method is that a time deriva-tive term in the unsteady flow equation is representedby the inverse Fourier transform of its counterpartin the stationary nonlinear frequency-domain.8 Thepseudo-spectral approach in the frequency domain tosimulate the unsteady periodic flows8 is not specifi-cally new, but the TS method can further contributeto the computational time savings by eliminating theFourier transformation process back and forth betweenthe time and frequency domain, and also by removingthe transient time to reach periodic steady-state. Itsaccuracy has proved to be equivalent to the conven-tional time accurate computation.

One of the major advantages of the TS methodin our analysis is that it makes the adjoint methodapplicable to unsteady flow simulations. With the un-steady time response replaced by a Fourier-based rep-resentation, the entire computation marches throughpseudo-time as it would in a steady-state simula-tion. Therefore a steady adjoint formulation combinedwith the TS method becomes amenable to unsteadyperiodic flows, and any gradient-based optimizationmethod can be easily combined with the adjoint solu-tion method to locate the minimum of the objectivefunction.

Another distinctive characteristic of the flow simu-lation of helicopter rotors is the significant impacts ofthe structural loads and deformations on the aerody-namic performance and vice versa. The aerodynamicand structural disciplines are tightly coupled in thehelicopter rotor simulation, and the multi-disciplinaryaspects of both the flow analysis and the design op-timization are inevitable. A number of efforts tocouple CFD and computational structural dynamics(CSD) have recently been made with both tight andloose coupling,17–19 and the results have proved that

these approaches are successful. Thus the capabilityof the TS method to couple with the existing struc-tural/comprehensive analysis tools needs to be inves-tigated so that it can be used with confidence for theanalysis and design of helicopter rotors. The cost sav-ings of the TS method should also be able to make thecoupling process more efficient.

The main purpose of this study is to examine theaccuracy and the validity of our current optimizationmethodology, which efficiently combines time-spectraland adjoint-based methods, by applying it to real de-sign optimization problems.

A level flight condition of UH-60A configuration,flight 8534, is validated using the time-spectral com-putation. Flight conditions are simulated using boththe time-spectral and time accurate methods, and theresults are compared with the experimental data. Thetime savings using the TS method is shown by directlycomparing the wall clock CPU time for both compu-tations.

The validation of the accuracy of time-spectral anddiscrete adjoint solution method is carried out by sim-ulating the pitching motion of NACA 0012 airfoil. Twoflight conditions, supersonic and transonic, are simu-lated, and the gradients of the average drag coefficientwith respect to the local shape changes on the airfoilsurface are calculated and compared with finite differ-ence results.

Finally, our design optimization method is appliedto a helicopter rotor design problem. Although ourmethodology has the potential to handle complicateddesign problems, and the TS method shows great ef-ficiency in highly unsteady problems, we have con-ducted a preliminary study before moving on to acomplete rotor design at the forward flight conditions.A straightforward hover flight condition of the UH-60Aconfiguration is chosen for a test optimization case.Although a hover flight is a steady state, it is treatedas periodic unsteady to test our method. To furthersimplify the problem, a single blade rotor is consid-ered, and a free wake model is added to the computa-tion to include the wake effects. The optimum bladeshape to minimize torque at a constant or improvedthrust is sought. The shape modifications are im-posed by applying Hicks-Henne bump functions withdifferent amplitudes around the airfoil surface at thevarious sections along the span. Sensitivity informa-tion with respect to the shape changes are calculatedby the adjoint solution method, and a gradient-basedoptimization algorithm, a nonlinear multi-dimensionalconjugate gradient method in our study, is employedto locate the optimum shape of the blade through thedesign iterations until the specified convergence is sat-isfied.

This paper is organized as follows. The details of thetime-spectral method and the discrete adjoint solutionmethod are described first, and the validation study is

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shown by the simulation of various flow conditions.Finally, results of the blade shape optimization arepresented and followed by the conclusion and futurework.

TIME-SPECTRAL METHOD

A simulation of helicopter flight inevitably entailsthe complete computation of three-dimensional rotor.The corresponding computational cost grows rapidlyin addition to the complexity related to the mesh con-struction. Therefore, the efficiency and accuracy of thenumerical scheme of the URANS solver becomes cru-cial. As the flows involved show an unsteady periodicnature, a second order implicit Backward DifferenceFormula (BDF) has been successfully used for its meritof A-stability, allowing larger time steps than thoseachieved using an explicit time-stepping method. A setof nonlinear equations for the new state are solved andadvanced in time, utilizing inner iterations involvingdual time-stepping. Combined with several conver-gence acceleration techniques,10 including multigridand implicit residual smoothing, its efficiency can beremarkably improved. However, the computationalcost of this methodology can still be considerably largewhen applied to unsteady periodic simulations whereat least two or three cycles (five or more for a pitchingmotion) should be preceded before it reaches periodicsteady state.

Fully taking the advantage of the periodic nature ofthe flow, and based on the idea of the Fourier seriesform of the periodic responses, Hall et.al.11 proposedHarmonic balance techniques to transform the un-steady equations in the physical domain into a steadyproblem in the frequency domain. This approachhas been extended as a non-linear frequency-domain(NLFD) method11, 12 to Euler and full Navier-Stokesequations and applied to a number of unsteady flowanalyses12 and aerodynamic/aero-structural shape op-timization.8 Compared to the time accurate compu-tations, the NLFD method can achieve a higher effi-ciency by eliminating the cost to compute initial tran-sient computation to reach the periodic steady-state.However, the solutions we are interested in are in thephysical time domain, and thus an inverse Fouriertransform back into the time domain is required ateach solution iteration, making the frequency-domainmethod less attractive.

On the other hand, in the same context of the NLFDmethod of Fourier representations in time, the time-spectral method has been proposed14 to further im-prove the efficiency. The main advantage of the TSapproach over the frequency-domain method is thatthe TS method represents the time derivative term inthe Navier-Stokes equations as Fourier series directlyin the time domain, and therefore eliminates the pro-cess required by the frequency-domain method, to inwhich the solutions must be transformed back and

forth to the time domain. The algorithm of the TSmethod is also more straightforward to implement inRANS solver. Details of the mathematical formulationand stability analysis are described in Reference,14 andonly a brief summary is shown here.

The Navier-Stokes equations in a semi-discrete formin the Cartesian coordinates can be written as

V∂ω

∂t+R(ω) = 0 , (1)

where ω is the vector of conservative variables,

ω =

ρρuρvρwρE

, (2)

and R(ω) is the residual of spatial discretizations ofviscous, inviscid, and numerical dissipation fluxes.

A discretization of Equation 1 using a pseudo-spectral formula13 renders equations

V Dtωn +R(ωn) = 0 (n = 0, 1, 2, ..., N − 1) , (3)

where N is the number of time intervals, and Dt isthe spectral time derivative operator. If a pseudo-timederivative term is directly added for a time integrationto steady state, then Equation 3 becomes,

V∂ωn

∂τ+ V Dtω

n +R(ωn) = 0 (n = 0, 1, 2, ..., N − 1) .

(4)The efficiency of the TS method arises from the treat-ment of the Dtω

n term in Equation 4. Instead oftransforming the entire equations into the frequencydomain, the inverse Fourier transform is performedonly to the time derivative term in Equation 3 as fol-lows,

Dtωn =

2π

T

k= N−1

2∑

k=−N−1

2

ikω̂keikn∆t (n = 0, 1, 2, ..., N − 1),

(5)where time period T is divided into N time inter-

vals, ∆t = TN

, and ω̂k is a Fourier mode. This can berewritten in the time domain as suggested,13

Dtωn =

2π

T

N−1

2∑

m=−N−1

2

dmωn+m (n = 0, 1, 2, ..., N − 1),

(6)A term dm can be rearranged as

dm =

{

12 (−1)m+1cosec(πm)

N) : m 6= 0

0 : m = 0 .(7)

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Thus a time derivative term in Equation 4 behaves asa matrix operator, and an additional cost in the TSmethod comes from the operation related to the mul-tiplication of the matrix with the elements dm and thevector ωn+m in Equation 7. The summation in Equa-tion 6 involves the solutions at all time levels, andthe solution at each time instance depends upon thesolutions at all other time instances. Thus the solu-tion at each time instance is updated simultaneously asthe computation advances in the physical time domainuntil the desired convergence is achieved. This does in-crease the memory requirement of the TS method, assolutions at all time levels need to be stored. How-ever, if the frequency contents of the problem we aresimulating do not span a wide range of the spectrum,this method can considerably contribute to improvingthe efficiency at an accuracy equivalent to that of thetime accurate computations.

DISCRETE ADJOINT METHOD

A design problem in our study can be posed as a gen-eral form of the optimization problem which amountsto finding the minimum of the objective/cost func-tion, I, with respect to a number of design variables,x, while satisfying a set of constraints, C. A costfunction, I, is also dependent upon the state vector,ω, which can be obtained by solving the governingequations, R. This design problem can be written inmathematical formulation as follows,

Minimize I(x, ω(x))

w.r.t x ,

subject to R∗(x, ω(x)) = 0

Ci(x, ω(x)) = 0 (i = 1, ...,m) ,

where the governing equation R∗ in our problem rep-resents the time-spectral form of the Euler or Navier-Stokes equations in Equation 4, and m additional con-straints are satisfied by the equations Ci(x, ω(x)). Thesensitivities of the cost function with respect to the de-sign variables are obtained by applying the chain rule,and a variation of I can be represented, to the firstorder, as

δI =∂IT

∂ωδω +

∂IT

∂xδx (8)

Since the derivative of the cost function is dependenton both the design variables, x, and the state vector,ω, a new flow solution is required for each parameterwith respect to which we are seeking a derivative. Asan alternative to the direct solution of Equation 8,we introduce the governing equations as a constraintusing the method of Lagrange Multipliers, ψ, whichmakes it possible to obtain an expression for δI thatis independent of δω. The gradient of I with respect

to n arbitrary number of design parameters can becalculated without re-evaluating the flow.

In other words, since the governing equations oftime-spectral formulation of the Euler/Navier-Stokesequations are given by R∗(x, /omega) = 0, we can in-fer from the fact that the governing equations shouldalways be satisfied, that the variation of the residualmust be zero and can be derived as,

δR∗ =

[

∂R∗

∂ω

]

δω +

[

∂R∗

∂x

]

δx = 0 (9)

As δR∗ is identically zero, it can be added or sub-tracted to Equation 8 to yield,

δI =∂IT

∂ωδω +

∂IT

∂xδx −

ψT

([

∂R∗

∂ω

]

δω +

[

∂R∗

∂x

]

δx

)

=

{

∂IT

∂ω− ψT

[

∂R∗

∂ω

]}

δω + (10)

{

∂IT

∂x− ψT

[

∂RT

∂x

]}

δx

To get rid of the dependence of δI on δω, adjoint vari-able, ψ can be chosen for the first part of the righthand side of the Equation 10 to be zero. Thus as longas ψ satisfies the adjoint equation,

[

∂R∗

∂ω

]T

ψ =∂I

∂ω, (11)

then the sensitivity Equation 8 becomes independentof δω as follows,

δI =

{

∂IT

∂x− ψT

[

∂R∗

∂x

]}

δx (12)

The adjoint matrix[

∂R∗

∂ω

]

in Equation 11 is a sparse

matrix of constant coefficients (which depend on theperiodic flow solution ω) and can be set up by deriv-ing or computing the dependence of the residual in onecell of the mesh on the flow solution at every cell inthe domain. Since the residual evaluation has a com-pact stencil (mostly composed of nearest neighbors)the number of non-zero entries in each column of thematrix is small. For the current flow solver, SUmb,and the discretization used in this work, the residualat one point is influenced by the flow solution at 33neighboring cells in Navier-Stokes solutions and 9 cellsin Euler solutions. In addition, through the couplingof the time-spectral derivative term, the solution at apoint also depends on the solution at the same point,but at all the time instances considered in the solutionof the periodic problem. The influence of all othertime instances on one is included in an adjoint matrixin a fully coupled manner rather than separated in theright-hand side of the adjoint equation

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The computation of all non-zero entries of the ma-trix and storage during the iterative convergence pro-cess requires a considerable amount of memory. In-stead, we recompute these terms at every iterationof the adjoint solution, reducing the memory penalty.Furthermore, we have chosen a solution procedure thatemploys an ILU-preconditioned GMRES algorithm (inthe form of the PETSc toolkit) that has proved to beboth efficient and robust in the solution of the rela-tively small problems treated in this work. For largerproblems, the convergence rates of this solution ap-proach deteriorate, but the additional multigrid as apreconditioner for the GMRES iteration is able to re-store the convergence rates to more reasonable values.

The final sensitivity information in Equation 12 be-comes very straightforward to obtain, as ∂I

∂xand ∂R∗

∂x

are easy to compute. Once we know the objective func-tion values and the gradient information with respectto the design variables, any gradient-based optimiza-tion algorithm can be applied to locate the optimumvalue of the objective function. A conjugate gradientmethod was chosen in our study. The summary ofthe entire design optimization procedure is shown atFigure 1.

Fig. 1 Procedure for design optimization

VALIDATION OF AERODYNAMIC

ANALYSIS TOOL

The accuracy of the analysis tools is crucial to deter-mine the credibility of the design optimization results,and a validation study was performed using our cur-rent analysis tools, the time-spectral and adjoint solu-tion method. Simulations of several level flight condi-tions of a UH-60A configuration were carried out usingboth the time-spectral and time accurate computa-tion, and the results were compared with experiments.A loose coupling of time-spectral computation with acomprehensive analysis code was also carried out toprove its efficiency for an aerodynamic and structuralcoupling process.

An adjoint solution method implemented in thetime-spectral form of an inviscid flow solver was used

to simulate the pitching motion of the NACA 0012airfoil, and the gradients of a time-averaged drag co-efficient with respect to the airfoil shape changes werecompared with the finite difference results. An expen-sive computational cost of the finite difference methodhas limited the choice of the validation problem to thesimple NACA 0012 airfoil, but the accuracy of the cur-rent discrete adjoint solution method has been shownto be independent of the size of the problem.

Validation of the Time-Spectral Method

Time-Spectral Computation with Loose Coupling ofCFD/CSD

Time-Spectral analysis is coupled to a rotorcraftcomprehensive analysis, UMARC (University of Mary-land Advanced Rotorcraft Code).20 The temporalcoupling method is unique to rotorcraft and is referredto as ‘delta coupling’. The terminology ‘loose coupling’is also used by rotary wing researchers to describe thismethod. However, note that the method bears no re-semblance to ‘loose coupling’ as used by fixed wingresearchers. Unlike the ‘loose coupling’ of fixed wingresearch, the ‘delta coupling’ ensures strict time ac-curacy of the response harmonics. The original ‘deltacoupling’, or ‘loose coupling’ was proposed by Tunget al.21 in 1986. In its present form, it has served asthe cornerstone of the recent advances in trimmed ro-torcraft CFD/CSD loads prediction in the U.S.17, 22, 23

and Europe.24–26 The original method is described inReference,21 and its current adaptations in any of theabove references (see for example References.17, 22) Fora comprehensive review of the contemporary rotarywing CFD/CSD efforts, see Datta et al.27

During the Time-Spectral computa-tion/Comprehensive analysis coupling, the lattersupplies the structural dynamic model, the trimmodel, and the airload sensitivities as required by thedelta coupling method. The trim model is a validatedfull aircraft, six-axes, free flight trim model.20, 22 Inthe present study, the time-spectral computation isused in its single-blade form. Only the near-field iscalculated by RANS. The far-field, which includesthe effect of all blades, is accounted for via theinflow generated by a free wake model. The inflowis incorporated within the near-blade CFD domainusing the field velocity approach of Sitaraman andBaeder.28 The wake model is coupled iterativelyalong with the vehicle trim iterations.

The UH-60A counter 8534 is a vibration critical highspeed (155 kts) flight. It is characterized by a speedratio of µ = 0.368 and the vehicle weight coefficient tosolidity ratio CW /σ = 0.0783. The calculated thrustcoefficient to solidity ratio is CT /σ = 0.084. The pri-mary mechanisms of rotor vibratory loads at this flightwere identified by Datta and Chopra29 as: (1) largeelastic twist deformations near the blade tip, and (2)inboard wake interactions on the advancing side. The

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elastic twist deformations are caused by 3-D unsteadytransonic pitching moments near the blade tip, whichcan only be predicted using CFD. The inboard wakeinteractions can only be predicted in presence of thecorrect elastic twist deformations. In the case of single-blade CFD, the inboard interactions can be predictedusing a dual peak or a moving vortex type roll-upmodel. In the present simulation, a moving vortexmodel was used.

The evolution of predicted pitching moments (ap-proximately a quarter-chord) near the blade tip withcoupling iterations is shown in Figure 3. Usually, 6 to 8iterations are necessary for a converged time accuratesolution. The present results correspond to the fourthiteration. The peak pitching moments are correctlypredicted near the tip (96.5% R). Thus the primarymechanism of the 3-D unsteady transonic shock reliefappears to be in place. The unsteady waveform in thefirst quadrant, however, is not satisfactory, which canbe explained by the fact that this high frequency wakeintegration can not be captured with TS method un-less more time instances are used. The discrepancy ismore pronounced at the 92% R.

The pitching moments determine the elastic twistdeformations. The twist deformations are shown inFigure 2. The waveforms from the baseline UMARC(unsteady lifting-line based analysis) and CFD arecompared with that obtained using measured airloadson the UMARC structural model. In absence of mea-sured deformations, the latter, obtained from a vali-dated structural model,29 provides a reasonable basisfor comparison. The elastic twist deformation showsthe same trends as the pitching moments. The gen-eral trend is improved (i.e, closer to that obtainedusing measured airloads). However, the waveform inthe first quadrant (between 0 and 90◦) shows a sig-nificant discrepancy. Capturing the waveform here isimportant for the accurate prediction of the vibratorylift harmonics (3-20/rev). Even though not accurate,the twist deformation is adequate enough to providethe correct vibratory lift phase near the tip (see Fig-ure 5.) The result of the correct vibratory lift phaseis seen in Figure 4, where the phase of the predictedadvancing blade lift moves closer, gradually, to that ofthe test data.

Comparison with Time Accurate Computation

A time-spectral computation has been used to sim-ulate three level flight conditions of the UH-60A con-figuration as a test cases, flight 8534, 8515, and 9017,and the corresponding sectional force results have beencompared with the time accurate results and experi-ments. Although good agreements for all the test caseshave been found and demonstrated in Reference,16

only the results from the flight 8534 test case are shownin this paper.

Two separate approaches to flow simulation were

Fig. 2 Changes of twist distribution during cou-pled runs.

rotating speed (rad/sec) 27.025

Mtip 0.6415M∞ 0.2359

advance ratio 0.368Reynolds number(chord based) 3.22 × 107

shaft angle (o) -7.31

Table 1 Flight conditions of 8534 test case

considered: single blade analysis with free wake cou-pling, and complete four blade analysis. A meshtopology for each approach is demonstrated in Fig-ure 6, and the flight condition is summarized at Ta-ble 1. A free wake model was added to the singleblade analysis to account for the wake effects andother blades. Prescribed deformation obtained fromOVERFLOW/CAMRAD analysis was imposed for anaero-elasticity effects, and CFD/CSD coupling is notconsidered. Time instances of as many as fifteen wereused for the time-spectral computation to enhance theaccuracy. A previous study16 indicates that the fewernumbers of time instances were sufficient for the ac-curate simulation of the flight 8534 case, as the bladeand wake interactions were less severe than in the otherflight conditions. However the accuracy of the solutionappears to be improved for the cases of flight 8515 and9017 where wake interactions and dynamic stall cycleswere not moderate and can not be computed withoutsome higher frequency content.

Figures 17, 18, and 19 show the results of the sec-tional normal force, chord force, and pitching momentfrom the single blade analysis. The same flight condi-tion was simulated using four blade analysis and theresults are plotted at Figure 20, 21 and 22. Goodagreements between single and four blade analysis in-dicated that the wake interference in flight 8534 wasnot critical and free wake coupling was a reasonablealternative to the complete wake capturing method ata better computational cost.

A computational time saving for time-spectral com-putation was estimated by a direct comparison with

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0 90 180 270 360−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015r/R=0.920

Cm

*M2

0 90 180 270 360−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02r/R=0.965

Expiter 0iter 1iter 4

Fig. 3 Changes of sectional pitching moment during coupled runs.

0 90 180 270 360−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3r/R=0.920

Cn*M

2

0 90 180 270 360−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25r/R=0.965

Expiter 0iter 1iter 4

Fig. 4 Changes of sectional normal forces (0/rev∼20/re) during coupled runs.

0 90 180 270 360−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08r/R=0.920

Cn*M

2

0 90 180 270 360−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1r/R=0.965

Expiter 0iter 1iter 4

Fig. 5 Changes of sectional normal forces (3/rev∼20/rev) during coupled runs.

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a) C-O-type mesh around singleblade (single-block with 568,816nodes)

b) mesh around 4 blades (536blocks with 2,401,776 nodes)

Fig. 6 Grid topology for UH60

computation method Time (sec)/per one MGcycle

time accurate 4.47time spectral 49.611

(9 time instances)time spectral 92

(15 time instances)

Table 2 Comparison of wall clock CPU time be-tween time-spectral and time accurate method.

a time accurate computation of wall clock CPU timefor the flight 8534 simulation using single blade anal-ysis with the results shown in Table 2. Scaling factorsof five or more were observed for both computations,to be converged up to 10−4 order (from the L2 norm-based residual of the density). A conventional secondorder BDF scheme was used with a time accuratescheme, 30 multigrid inner iterations, 0.2 ∼ 0.3o timestep and 2 ∼ 3 revolutions. Nine to fifteen time in-stances were used in time-spectral computation. Wealso employed a free wake coupling method for a hoverflight simulation, which is considered in the design op-timization application. The computational time sav-ing is critical for the optimization procedure to shortenits turnaround time for function evaluations.

Validation of Time-Spectral and Adjoint

Solution Method

A good way to validate the accuracy of the proposedmethodology is to compare the sensitivity informationwith other gradient computation methods that have

equivalent accuracy. Such methods as finite difference,complex variable and automatic differentiation meth-ods have different characteristics in their accuracy andcomputational cost. A finite difference method waschosen for the comparison as its implementation is easyand straightforward; however the step sizes are care-fully taken to minimize the dependence on the stepsizes. On the other hand, the Automatic Differen-tiation (AD) method31 uses numerical algorithm toevaluate the adjoint solutions by a computer program.It achieves numerical accuracy higher than that of themanual derivation as it is not associated with round-of errors due to democratization or cancellation errorsdue to finite number precision. Its application to thesolution of discrete adjoint method is currently beingpursued.

A pitching motion of a NACA 0012 airfoil was sim-ulated at two flow conditions, both transonic (M∞ =0.8) and supersonic (M∞ = 2.0) inviscid flows. Theairfoils were pitched about the quarter chord with areduced frequency of k = 0.2 and with a pitching mo-tion amplitude of ±5o. The corresponding angularfrequencies were ω = 35.43, 88.5HZ for the transonicand supersonic flow condition respectively. The testcases presented here are quasi-three-dimensional inthat they consist of a two-dimensional airfoil extrudedin the third direction. A total of six time instances fora period of the pitching motion were used for both theflow and discrete adjoint solution. A body-fitted O-mesh with 81× 17× 9 grid points were used for all thecalculations. Figure 7 shows Mach number contoursfor four snapshots of the time-spectral solution for thetransonic test case, and the supersonic case is shownin Figure 8. A comparison of the time-spectral results,with a time accurate computation using second-orderbackwards difference formula and a dual time-steppingapproach, has shown good agreements, although thecomparison is not included in this paper. A similaragreement is found for the supersonic flow simulation.

The corresponding snapshots for the discrete, time-spectral, and adjoint solutions for both the transonicand supersonic cases (for the first adjoint variableonly) can be found in Figures 9 and 10. To assessthe accuracy of the resulting discrete adjoint solution,a finite difference method was used to compare thesensitivity of the time-averaged coefficient of drag

CD =1

T

∫ T

0

CD(t)dt (13)

with respect to the amplitude of Hicks-Henne bumpfunctions centered at mesh points on the upper surfaceof the airfoil. The Hicks-Henne functions are smoothfunctions that can be used to modify the shape of theNACA 0012 airfoil. Their effect (appropriately non-dimensionalized) can be seen in Figure 12.

The values of the sensitivities of the average dragcoefficient (over a pitching cycle), CD (shown in Fig-

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a) α = 0.0o, downstroke b) α = −5.0o

c) α = 0.0o, upstroke d) α = 5.0o,

Fig. 7 Four snapshots of the local Mach numbercontours for a pitching NACA 0012 airfoil in tran-sonic flow, M∞ = 0.8, k = 0.2, ∆α = ±5

o

a) α = 0.0o,downstroke b) α = −5.0o

c) α = 0.0o,upstroke d) α = 5.0o,

Fig. 8 Four snapshots of the local Mach numbercontours for a pitching NACA 0012 airfoil in su-personic flow, M∞ = 2.0, k = 0.2, ∆α = ±5

o

ure 11) computed using both the time-spectral adjointsolutions and the method of finite differences (withcarefully chosen step sizes to yield accurate deriva-tives). The agreement between these two approachesgives us confidence that the results of the time-spectraladjoint calculations are correct. Small differences be-tween the two sets of results still exist, and are at-tributed to the fact that the time-spectral adjointoperator we have implemented has neglected somevariations of the residual R∗(w) involving the spec-tral radius computation and the artificial dissipationcoefficients.

RESULTS

The main purpose of the current study is to as-sess the validity and accuracy of the current design

a) α = 0.0o,downstroke b) α = −5.0o

c) α = 0.0o,upstroke d) α = 5.0o,

Fig. 9 Four snapshots of the local Mach numbercontours of the first adjoint variable for a pitchingNACA 0012 airfoil in transonic flow, M∞ = 0.8, k =

0.2, ∆α = ±5o

a) α = 0.0o,downstroke b) α = −5.0o

c) α = 0.0o,upstroke d) α = 5.0o,

Fig. 10 Four snapshots of the local Mach numbercontours of the first adjoint variable for a pitchingNACA 0012 airfoil in supersonic flow, M∞ = 2.0, k =

0.2, ∆α = ±5o

optimization tool and to investigate its capability forfuture applications to more viable and complete de-sign problems. A simple yet realistic rotorcraft prob-lem is considered in this paper. The choice of thehover flight condition made the optimization problemmore straightforward, as the trim condition was notviolated exceedingly. An optimum blade shape thatminimized torque was sought while a total thrust pro-duced by the baseline configuration was maintainedconstant or enhanced. A blade of UH-60A configu-ration was chosen for the baseline, and single bladeanalysis described from the earlier section was carriedout with the mesh topology shown at Figure 6(a). AnMtip was 0.65, and fixed collective pitch angle was set

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0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

Chord

Gra

dien

t

Mach 2.0 NACA 0012 Average Drag Coefficient Unsteady Gradient

FDADJ

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1

0

1

2

3

Chord

Gra

dien

t

Mach 0.8 NACA 0012 Average Drag Coefficient Unsteady Gradient

FDADJ

Fig. 11 Comparison of sensitivity derivatives ∂CD

∂bi

using discrete adjoint solution and finite differencesfor the supersonic (top) and transonic (bottom)test cases.

11.8◦. A total of three time instances were used fortime-spectral computation and the aero-structural de-formation was prescribed at each time instance fromthe previous analysis using OVERFLOW/CAMRADcomputation.17

Fig. 12 Non-dimensional effect of Hicks-Henneperturbation on the surface of the airfoil.

Another advantage of optimizing the hover condi-tion is that constraint handling became more straight-forward as the number of constraints was reducedby excluding the trim related parameters. We em-ployed a nonlinear multi-dimensional conjugate gra-dient method for a gradient-based optimization al-gorithm. Since the constraint handling in conjugategradient method was not straightforward, a constraintof constant thrust was integrated into the objectivefunction as an implicit form via the properly weightedpenalty.Also a second form of an objective functionwithout a constraint of constant thrust was consideredas the ratio of torque to thrust, if improved thrust isallowed.

Using the nondimensionalized coefficients of torqueand thrust, mathematical formulations of the objectivefunctions are shown at Equation 14,

I = CQ/CT , (14)

I = CQ + α ((CTo− CT ))

2,

where CQ and CT are the coefficient of torque andthrust, respectively, after the blade shape is modified,and CTo

is the coefficient of the initial thrust of thebaseline. Blade shape modifications were obtained byimposing smooth Hicks-Henne bump functions on thesurface of the airfoils along the various span sections,and the corresponding new meshes were generated us-ing the mesh warping routine without the need for theregenerating the complete meshes. As the gradient-based optimization algorithm performs well with thesmooth design space, the shape changes, thus the am-plitudes of the bump functions are limited to smallamounts, with the assumption that few variations inthe shape modifications produces smooth design spaceand less deviation in structural deformation from theinitial values. A total of nine sections along the spanwere selected, with even distribution on the blademidboard sections and close distribution on the bladeinboard sections and tip sweep region. Perturbationswere applied at each section on ten locations of theairfoil surfaces, and the maximum amplitudes of thebumps were scaled to the local airfoil thickness. Atotal of 90 design variables (the amplitudes of bumpfunctions) were introduced, and it has been tested thatthe inclusion of more design variables do not signifi-cantly affect the results.

Prescribed aero-elastic deformation was set as con-stant, and maintained as the initial value throughoutthe design iterations where the blade shape changed,although this is not practically true. If the blade shapewere to experience significant variation, such as thatrequired for planform shape optimization, structuraldeformation must be properly updated at each designiteration through CFD/CSD coupling. However, whenwe introduced a small changes only in design variables,we assumed that aero-elastic deformation would notdeviate much from the initial values.

Optimized results were obtained after 19 design it-erations and a total of 12% torque reduction at the7% thrust increase. These values came at a surprise,considering the small variations in the airfoil shapes(mainly camber changes), but this fact implies thataerodynamic loads are very sensitive to even smallchanges in the airfoil shapes. The corresponding shapeof the optimized blade is shown at Figures 13 and 14.As the shape changes are limited to the small vari-ations, the direct comparison of shapes between thebaseline and the optimized blade is not obvious.

Changes in the pressure distribution on the bladeare demonstrated in Figures 15 and 16 on the upperand lower surfaces respectively. It can be noted thatpressure around the tip area becomes lower and con-tributes to reducing the pressure drag.

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CONCLUSIONS AND FUTURE WORK

A new optimization methodology was proposedbased on the combined time-spectral and adjoint-based approaches for the design problem involving un-steady periodic flows. The accuracy and efficiency ofthe time-spectral solution method was validated by thesimulation of the flight 8534 test case of the UH-60Aconfiguration, and the results showed good agreementswith time accurate results and the experimental data.The adjoint solution method with the time-spectralimplementation of the flow solver was applied to thesimulation of the pitching motion of a NACA 0012

a) overlaid blade shapes (up-per surface), red:baseline, opti-mized:blue

b) overlaid blade shapes (lowersurface), red:baseline, opti-mized:blue

Fig. 13 Comparison of blade shapes between base-line(red) and optimized(blue)

XY

Z

Fig. 14 Shape changes of the airfoils along thespan (baseline(red) and optimized(blue)).

X

YZ

Pressure

1.060531.007890.9552630.9026320.85

a) baseline

X

YZ

Pressure

1.060531.007890.9552630.9026320.85

b) optimized

Fig. 15 Comparison of the surface pressure con-tours between the baseline and the optimized (onthe upper surface).

Z

X

Y

Pressure

1.060531.007890.9552630.9026320.85

a) baseline

Z

X

Y

Pressure

1.060531.007890.9552630.9026320.85

b) optimized

Fig. 16 Comparison of the surface pressure con-tours between the baseline and the optimized (onthe lower surface).

11 of 19

airfoil, and the gradients of the time-averaged dragcoefficient with respect to the airfoil shape changeswere compared with the finite difference results. Goodagreements were found for both the supersonic andtransonic flow conditions. Finally a simple optimiza-tion problem was solved for a helicopter blade shapedesign at the hover flight condition. A minimization oftorque at a constant or improved thrust was pursuedby a multi-dimensional conjugate gradient method. Asmuch as 12% reduction of torque was observed, whilethe total thrust was enhanced by 7%. This fact in-dicates that our design optimization methodology issuited for helicopter rotor design. However, to fullyemploy our methodology to realistic design cases ofhelicopter rotors, such as at the forward flights, a num-ber of issues should be resolved. First, a study onthe proper choice of design variables should proceedfor a realistic helicopter rotor design. More designvariables, in addition to the design variables not onlylimited to the camber changes in our study, will be ableto produce further improvement in aerodynamic per-formance. However, when adjoint-based optimizationuses gradient information to locate optimum values,special care must be taken to ensure smooth designspace.

A more viable design test for helicopter flight is tooptimize the rotor shapes during the forward flightcondition, where our design approach using time-spectral solution method is best suited. Howeverhighly unsteady motion during the forward flight posesnumerous problems. A more rigorous form of con-straints to satisfy the trim condition needs be includedin the optimization process, and a more efficient androbust constrained optimization algorithm is neces-sary. Our design methodology is currently tested withNPSOL SQP method and ready to be applied to therealistic design optimization of helicopter rotors, andoptimization results with a better optimization algo-rithm will be shown in the future. Finally the currentadjoint solver implementation in the Euler flow solverwill be extended to the full Navier-Stokes flow solver.Introducing an automatic differentiation algorithm31

would enable the adjoint solver development to be eas-ier and more straightforward, and this is also part ofour on-going research.

Acknowledgment

This work has been carried out under the supportof the AFDD at Ames Research Center, under con-tract NNA06CB11G. We gratefully acknowledge thesupport of our points of contact, Mark Potsdam andTom Maier, in supplying us with the experimental datanecessary to complete this study.

References1J.J.Reuther, A. Jameson, J.J.Alons, M.Rimlinger, and

D.Saunders, ” Constrained multipoint aerodynamic shape op-timization using an adjoint forumlation and parallel computers:Part I & II,” Journal of Aircraft, 36(1):51-74,1999

2A. Jameson, ”Aerodynamic design via control theory ,”Journal of Scientific Computing, 3:233-260,1988

3J.R.R.A.Martins, ”A Coupled-Adjoint Method for High-Fidelity Aero-Structural Optimization,” ph.D dissertation, De-partment of Aeronautics and Astrounautics, Stanford Univer-sity, Stanford, CA, Oct. 2002.

4S. Kim, J.J.Alonso, and A.Jameson, ”A gradient accu-racy study for the adjoint-based Navier-Stokes design method,”AIAA Paper 99-0299,Reno, Jan. 1999.

5K.Leoviriyakit and A.Jameson, ”Aero-structural wing plan-form optimization,” AIAA 42nd Aerospace Science Meetings &Exhibit,Reno, NV, Jan. 2004.

6K. Mani, D. J. Mavriplis, ”An Unsteady Discrete AdjointFormulation for Two-Dimensional Flow Problems with Deform-ing Meshes,” AIAA 45th Aerospace Science Meetings & Ex-hibit,Reno, NV, Jan. 2007.

7K.Palaniappan, P.Sahu, J.J.Alonso, and A.Jameson, ”Ac-tive flutter control using an adjoint method,” AIAA Paper2006-0844, Reno, NV, Jan. 2006.

8S.Nadarajah, M.McMullen, and A.Jameson, ”Optimal con-trol of unsteady flows using time accurate and non-linear fre-quency domain methods,” AIAA Paper 2003-3875,Orlando, FL,Jun. 2003.

9S.Nadarajah and A.Jameson, ”A comparison of the contin-uous and discrete adjoint approach to automatic aerodynamicoptimization,” AIAA Paper 2000-0677,Reno, NV, Jan. 2000.

10A. Jameson, “Time dependent calculations using multi-grid, with application to unsteady flows past airfoils and wings,”AIAA Journal, 91-1956, June 1998.

11J.P.Thomas, E.H.Dowell, and K.C.Hall, ”Modeling three-dimensional inviscid transonic limit cycle oscillation using aharmonic balance approach,” ASME International MechanicalEnginnering Conference and Exposition,New Orleans, 2002.

12M. S. McMullen, “The Application of Non-Linear Fre-quency Domain Methods To the Euler and Navier-Stokes Equa-tions ,” Ph.D thesis, Stanford University, March 2003.

13C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A., Jr. Zang“Spectral Methods in Fluid Dynamics,” Springer, 1998.

14A. Gopinath and A. Jameson, ”Time spectral method forperiodic unsteady computations over two- and three- dimen-sional bodies,” AIAA Paper 05-1220, Reno, NV, Jan. 2005.

15E.van der Weide, A.Gopinath and A.Jameson, ”Turboma-chinery applications with the time spectral method,” AIAAPaper 05-4905, Toronto, Canada, Jun. 2005.

16S. Choi, J. J. Alonso, E. van der Weide, and J. Sitaraman,”Validation Study of Aerodynamic Analysis Tools for DesignOptimization of Helicopter Rotors,” AIAA Paper 07-3929, Mi-ami FL., June 2007.

17M.Potsdam, H.Yeo, and W.Johnson, “Rotor Airloads Pre-diction Using Loose Aerodynamic/Structural Coupling,” AHS60th Annual Forum, Baltimore MD, June 7-10 2004.

18A. Datta, and I. Chopra, “Prediction of UH-60A DynamicStall Loads in High Altitude Level Flight using CFD/CSD Cou-pling,” AHS 61st Annual Forum, Grapevine, TX, June 1-32005.

19M. J. Bhagwat, R. A. Ormiston, H. A. Saberi, and H. Xin,“Application of CFD/CSD Coupling for Analysis of RotorcraftAirloads and Blade Loads in Maneuvering Flight,” AHS 63rdAnnual Forum, Virginia Beach VA, May 1-3 2007.

20Datta, A., Chopra, I., “Validation and Understanding ofUH-60A Vibratory Loads in Steady Level Flight,” Journal ofthe American Helicopter Society, Vol. 49, No. 3, July 2004, pp271-287.

21Tung, C., Cardonna, F.X., and Johnson, W.R., “The Pre-diction of Transonic Flows on an Advancing Rotor,” Journalof the American Helicopter Society, Vol. 31, (3), July 1986, pp.4–9.

22Datta, A., Sitaraman, J., Chopra., I, and Baeder, J.,“CFD/CSD prediction of rotor vibratory loads in high speed

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flight,” in press, Journal of Aircraft, Vol. 43, (6), November–December 2006, pp. 1698–1709.

23Datta, A. and Chopra, I., “Prediction of UH-60 MainRotor Structural Loads using CFD/Comprehensive AnalysisCoupling,” 32nd European Rotorcraft Forum, Maastricht, TheNetherlands, September 12–14, 2006.

24Servera, G., Beaumier, P., and Costes, M. “A Weak Cou-pling Method between the Dynamics Code Host and the 3DUnsteady Euler Code Waves,” 26th European Rotorcraft Fo-rum, The Hague, Netherlands, September 14–16, 2000.

25Altmikus, A. R. M., Wagner, S., Beaumier, P., and Servera,G., “A Comparison : Weak versus Strong Modular Coupling forTrimmed Aeroelastic Rotor Simulation,” American HelicopterSociety 58th Annual Forum, Montreal, Quebec, June 2002.

26Pahlke, K. and Van Der Wall, B., “Chimera simulations ofmultibladed rotors in high-speed forward flight with weak fluid-structure-coupling,” Aerospace Science and Technology, Vol. 9,(5), 2005, pp. 379–389.

27Datta, A., Nixon, M. and Chopra, I., “Review of RotorLoads Prediction with the Emergence of Rotorcraft CFD,” Jour-nal of the American Helicopter Society, Vol. 52, (4), October2007, pp. 287–217.

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0 90 180 270 360−0.1

−0.05

0

0.05

0.1

0.15r/R=0.225

Cn*M

2

0 90 180 270 360−0.05

0

0.05

0.1

0.15

0.2r/R=0.400

0 90 180 270 3600

0.05

0.1

0.15

0.2

0.25r/R=0.550

ψ

0 90 180 270 3600

0.05

0.1

0.15

0.2

0.25r/R=0.675

ψ

Cn*M

2

0 90 180 270 360−0.1

0

0.1

0.2

0.3r/R=0.775

ψ0 90 180 270 360

−0.2

−0.1

0

0.1

0.2

0.3r/R=0.865

ψ

0 90 180 270 360−0.2

−0.1

0

0.1

0.2

0.3r/R=0.920

ψ

Cn*M

2

0 90 180 270 360−0.2

−0.1

0

0.1

0.2r/R=0.965

ψ0 90 180 270 360

−0.1

0

0.1

0.2

0.3r/R=0.990

ψ

expTATS (15T)

Normal force comparison for a single blade with wake coupling

Fig. 17 Section normal force of flight 8534 (one blade case with wake coupling (red-TA; green-TS)).

14 of 19

0 90 180 270 360−0.01

0

0.01

0.02r/R=0.225

Cc*M

2

0 90 180 270 360−0.01

0

0.01

0.02

0.03r/R=0.400

0 90 180 270 360−0.02

−0.01

0

0.01

0.02

0.03r/R=0.550

0 90 180 270 360−0.01

−0.005

0

0.005

0.01

0.015r/R=0.675

ψ

Cc*M

2

0 90 180 270 360−0.015

−0.01

−0.005

0

0.005

0.01r/R=0.775

ψ0 90 180 270 360

−0.02

−0.01

0

0.01

0.02r/R=0.865

ψ

0 90 180 270 360−0.02

−0.01

0

0.01

0.02r/R=0.920

ψ

Cc*M

2

0 90 180 270 360−0.01

−0.005

0

0.005

0.01

0.015r/R=0.965

ψ0 90 180 270 360

−0.02

−0.01

0

0.01

0.02r/R=0.990

ψ

expTATS (15)

Chord force comparison for a single blade with wake coupling

Fig. 18 Section chord force of flight 8534 (one blade case with wake coupling (red-TA; green-TS)).

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0 90 180 270 360−0.01

0

0.01

0.02

0.03r/R=0.225

Cm

*M2

0 90 180 270 360−4

−2

0

2

4x 10

−3 r/R=0.400

0 90 180 270 360−0.01

−0.005

0

0.005

0.01r/R=0.550

ψ

0 90 180 270 360−0.01

−0.005

0

0.005

0.01r/R=0.675

ψ

Cm

*M2

0 90 180 270 360−0.02

−0.01

0

0.01r/R=0.775

ψ0 90 180 270 360

−0.02

−0.01

0

0.01

0.02r/R=0.865

ψ

0 90 180 270 360−0.03

−0.02

−0.01

0

0.01

0.02r/R=0.920

ψ

Cm

*M2

0 90 180 270 360−0.03

−0.02

−0.01

0

0.01

0.02r/R=0.965

ψ0 90 180 270 360

−0.02

−0.01

0

0.01

0.02r/R=0.990

ψ

expTATS (15)

Pitching moment comparison for a single blade with wake coupling

Fig. 19 Section pitching moment of flight 8534 (one blade case with wake coupling (red-TA; green-TS)).

16 of 19

0 90 180 270 360−0.2

0

0.2r/R=0.225

Cn*M

2

0 90 180 270 360−0.2

0

0.2r/R=0.400

0 90 180 270 3600

0.2

0.4r/R=0.550

ψ

0 90 180 270 3600

0.2

0.4r/R=0.675

ψ

Cn*M

2

0 90 180 270 360−0.5

0

0.5r/R=0.775

ψ0 90 180 270 360

−0.5

0

0.5r/R=0.865

ψ

0 90 180 270 360−0.5

0

0.5r/R=0.920

ψ

Cn*M

2

0 90 180 270 360−0.2

0

0.2r/R=0.965

ψ0 90 180 270 360

−0.2

0

0.2r/R=0.990

ψ

expTATS(15)

Fig. 20 Section normal force of flight 8534 (four blade case (red-TA; green-TS)).

17 of 19

0 90 180 270 360−0.01

0

0.01

0.02r/R=0.225

Cc*M

2

0 90 180 270 360−0.02

0

0.02

0.04r/R=0.400

0 90 180 270 360−0.02

0

0.02

0.04r/R=0.550

0 90 180 270 360−0.01

0

0.01

0.02r/R=0.675

ψ

Cc*M

2

0 90 180 270 360−0.02

0

0.02r/R=0.775

ψ0 90 180 270 360

−0.02

0

0.02r/R=0.865

ψ

0 90 180 270 360−0.02

0

0.02r/R=0.920

ψ

Cc*M

2

0 90 180 270 360−0.02

0

0.02r/R=0.965

ψ0 90 180 270 360

−0.02

0

0.02r/R=0.990

ψ

expTATS (15)

Fig. 21 Section chord force of flight 8534 ( four blade case (red-TA; green-TS)).

18 of 19

0 90 180 270 360−0.02

0

0.02

0.04r/R=0.225

Cm

*M2

0 90 180 270 360−5

0

5x 10

−3r/R=0.400

0 90 180 270 360−0.01

0

0.01r/R=0.550

ψ

0 90 180 270 360−0.01

0

0.01r/R=0.675

ψ

Cm

*M2

0 90 180 270 360−0.02

−0.01

0

0.01r/R=0.775

ψ0 90 180 270 360

−0.02

0

0.02r/R=0.865

ψ

0 90 180 270 360−0.02

0

0.02r/R=0.920

ψ

Cm

*M2

0 90 180 270 360−0.04

−0.02

0

0.02r/R=0.965

ψ0 90 180 270 360

−0.02

0

0.02r/R=0.990

ψ

expTATS(15)

Fig. 22 Section pitching moment of flight 8534 (four blade case (red-TA; green-TS)).

19 of 19