AIAA 2003–1347Parallel Computation of Wing Flutterwith a Coupled Navier-Stokes/CSDMethod

M. Sadeghi, S. Yang, F. LiuDepartment of Mechanical and Aerospace EngineeringUniversity of California, Irvine, CA 92697-3975

H. M. TsaiTemasek LaboratoriesNational University of SingaporeKent Ridge Crescent, Singapore 119260

41th AIAA Aerospace SciencesMeeting and Exhibit

January 6–9, 2003/Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

Parallel Computation of Wing Flutter with aCoupled Navier-Stokes/CSD Method

M. Sadeghi∗, S. Yang†, F. Liu‡

Department of Mechanical and Aerospace EngineeringUniversity of California, Irvine, CA 92697-3975

H. M. Tsai§

Temasek LaboratoriesNational University of Singapore

Kent Ridge Crescent, Singapore 119260

A code is developed for the computation of three-dimensional aeroelastic problemssuch as wing flutter. The unsteady Navier-Stokes flow solver is based on a finite-volumeapproach with centered flux discretization and artificial diffusion. For the structuraldisplacements a modal approach is applied. The temporal discretization is implicit forboth the flow equations and the structural equations. An explicit dual-time methodis used to integrate the coupled governing equations. A multigrid method is applied toadvance the flow solution, and the computation is performed in parallel with a multiblockapproach. A supercritical 2-D wing and the AGARD 445.6 wing serve as test cases forflutter investigations. Results for inviscid flow are compared with results obtained bysolving the Navier-Stokes equations with the Baldwin-Lomax and k-ω turbulence models,respectively. Inclusion of viscous effects is critical for the 2-D wing. LCO of the 2-Dwing is predicted, but with larger amplitude compared to experimental measurements.Predicted flutter boundary for the AGARD wing agrees well with experimental data insubsonic and transonic range but deviates significantly from experimental data in thesupersonic range. Inclusion of viscous effects only slightly improves the result for thiscase.

I. Introduction

Numerical investigations of aeroelastic phenomenasuch as wing flutter require considerations of bothaerodynamics and structural dynamics. The struc-tural deformation of interest is usually small enough toassume linear structural equations which can be solvedby linear methods. On the other hand, the flow sys-tem is complex owing to complex boundary conditions,compressibility, existence of shock waves, and effectsof viscosity and turbulence. Thus, a large variety ofmethods have been applied to solve for the unsteadyflow around moving or deforming structures.

Early approaches to solve for the unsteady flow overoscillating airfoils applied linear potential theory toboth the mean flow and the unsteady perturbations.Assuming the flow is linear, it can be viewed as the su-perposition of a time independent mean solution and aharmonic unsteady perturbation. The unsteady per-turbation potential is solved by a small disturbance

Copyright c© 2003 by the authors. Published by the AmericanInstitute of Aeronautics and Astronautics, Inc. with permission.∗Graduate Researcher, AIAA Student Member.†Post-Doctoral Researcher.‡Associate Professor, AIAA Member.§Principal Research Scientist, AIAA Member.

theory for each frequency and mode of interest, regard-ing the flow equations as uncoupled from the structuralequations.

However, the most severe limitations on the stabil-ity of wings are usually caused by shock motion undertransonic flow conditions. Linear theory has beenapplied for the unsteady perturbations, even wherethe steady mean flow is nonlinear. The superposi-tion holds under the assumption that the shock motioncaused by the wing oscillation is so small that nonlin-ear higher order terms can be neglected. Such meth-ods are not capable of capturing nonlinear phenomenasuch as limit-cycle oscillations.

Ballhaus and Goorjian1 present results for airfoilflutter obtained with three different methods: the lin-ear harmonic approach, the indicial method, and acoupled time-domain approach using their LTRAN2code and the structural equation for a pitching air-foil. The authors find that the indicial response givescomparable results to the coupled approach only if theshock waves are sufficiently weak. LTRAN2 solves thenonlinear transonic small disturbance (TSD) equationin 2-D with an implicit finite-difference scheme in con-servative form.

Since flutter is a phenomenon that arises from the

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interaction between aerodynamics and structure dy-namics, the flow equations and structural equationsshould be solved as a coupled system of equations.

By solving the transonic small disturbance equationcoupled to a spring-mass model for typical wing sec-tions, Isogai2 finds that the phase delay of the shockwave motion plays the dominant role in the transonicdip phenomenon of swept-back wings.

With their XTRAN3S code, Borland and Rizzetta3

solve the three-dimensional TSD equation by an im-plicit finite-difference scheme using ADI, coupled withthe modal structural equations. The authors predictthe transonic dip for a thin rectangular wing with aparabolic-arc profile.

Edwards4,5 developed a novel interactive boundary-layer method coupled with the CAP-TSD (Computa-tional Aeroelasticity Program – Transonic Small Dis-turbance) code developed at NASA Langley.6

These methods are limited to potential flow withsmall disturbances. In order to fully account for non-linear flow behavior, the Euler or Navier-Stokes equa-tions have to be applied. Shock motion plays a majorrole in transonic wing flutter. Viscosity and turbulencebecome important where shock-boundary-layer inter-ference has an effect on the shock motion, or wherethe flutter behavior is dominated by separation (stallflutter).

A method for computing unsteady flows and aeroe-lastic response for wings using the 3-D Euler equationsis presented by Guruswamy.7 An implicit finite dif-ference scheme is used to solve the Euler equationssimultaneously to the modal structural equations. Ineach time step, the aerodynamic grid is regeneratedby an analytical method. Results obtained for a semi-infinite wing compare better to experimental resultsthan results obtained by solving the TSD equation.The code ENSAERO is extended by implementing asolver for the thin-layer Navier-Stokes (TLNS) equa-tions.8 The viscous terms are treated by an explicitformulation.

Sisto et al.9,10 apply a coupled method to studystall flutter in a linear cascade. The authors use avortex and boundary layer method for incompressibleflow coupled with a spring model for the blade motion.A similar torsional spring and linear spring model forrigid profiles is used by Bakhle et al.,11 by Hwangand Fang,12 and by Alonso and Jameson.13 Bakhleet al.11 investigate potential flow through a cascade.A case with linear flow behavior was chosen in orderto validate the coupled method by comparison withuncoupled results. Hwang and Fang solve the Eulerand Navier-Stokes equations through a cascade on un-structured grids including a transonic test case and acase of stall flutter. Alonso and Jameson13 apply animplicit finite-volume method with Jameson’s artificialdiffusion to solve the two-dimensional Euler equations,coupled to the modal equations for Isogai’s pitching

airfoil.

Investigations of mistuning effects on cascade flutterare performed by Sadeghi and Liu14 solving the quasi-three-dimensional Euler equations on multiple oscillat-ing blades. The influence of fluid-structure couplingon mistuning effects is studied in Ref. 15. A nonlineartype of cascade flutter is observed in transonic flow,using the quasi-three-dimensional Navier-Stokes equa-tions coupled with a structural model in Ref. 16.

Three-dimensional Euler and thin-layer Navier-Stokes computations are performed by Lee-Rauschand Batina.17–19 Results by time-marching coupledcomputations, using the modal structural equations,are compared to results by a linear stability analysis(V-g method).

A closely coupled approach is developed byBendiksen and Hwang,20 using an Eulerian-Lagrangian method to solve the fluid and structuraldynamics as single system. A finite-element methodis applied for both fluid and structure, marched intime simultaneously by a Runge-Kutta scheme. Theefficiency of this rather time consuming approach isimproved by parallel computation.

With time-marching grid-based computations, theflow grid has to be adjusted in each time step in orderto match the boundaries of the deformed structure.Tsai et al.21 present a method for the deformation ofa multiblock grid. A spring-analogy method is used todetermine displacements of block corners. Transfiniteinterpolation is then applied to interpolate boundarydisplacements in the interior domain. The original gridangles are approximately preserved near wall surfacesby blending the deformed angles with the original an-gles in near-wall regions. For a pitching airfoil, theunsteady results by the grid deformation method areshown to be identical to results by rigid grid motion.

Liu et al.22 apply a coupled approach as well as theindicial response method on a two-dimensional airfoiland on a wing. The three-dimensional Euler equationsare solved on deforming multiblock grids by parallelcomputation. The flutter boundary of Isogai’s airfoil isobtained by time-marching calculations and compareswell to Alonso’s13 results. For the AGARD 445.6 wing,the results by the indicial response method agree withthose obtained by coupled computations.

This work presents a numerical method for parallelcomputation of aeroelasticity (PARCAE). The codesolves the unsteady three-dimensional RANS equa-tions on structured multi-block grids with a finite-volume method coupled to the modal structural equa-tions. The basic numerical scheme and grid deforma-tion method are the same as those in Refs. 21 and 22but they are implemented in a different computer codeand extended to solve the Reynolds-averaged Navier-Stokes equations with either the Baldwin-Lomax tur-bulence model or the k-ω two-equation turbulencemodel. Pseudo-time subiterations on flow and struc-

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ture ensure a strongly coupled solution in the time-domain. Flow computations on multiple grid blocksare performed in parallel.

II. Flow Solver

Fluid motion is governed by the fundamental con-servation laws for mass, momentum and energy. Fora flow without internal heat or mass sources, and ne-glecting effects of body forces, the governing equationscan be written in integral form as

∂

∂t

∫∫∫

V

WdV +

∫∫

S

[F ] · ndS = 0 (1)

where V is an arbitrary control volume with closedboundary surface S, and n is the unit normal vectorin outward direction. The vector of state variables Win Eq. (1) is defined as follows:

W =

ρρuρE

(2)

where ρ is the density, u = {u, v, w}T is the velocityvector, and E is the total energy of the flow. The fluxtensor [F ] in Eq. (1) consists of a convective (inviscid)part [Fc] and a diffusive (viscous and thermal) part[Fd], with

[F ] = [Fc]− [Fd] (3)

The convective fluxes are given by

[Fc] =

ρuT

ρ [uu] + p [I]

(ρEu + pu)T

(4)

Since the control volume and its surface in Eq. (1)may generally move in the fixed coordinate system,the fluxes through the surface are expressed in termsof the contravariant velocity u = {u, v, w}T :

u = u− ug (5)

where ug = {ug, vg, wg}T is the grid velocity vector.The fluxes arising from viscous shear stresses and ther-mal diffusion are

[Fd] =

0[τ ]

([τ ] · u− q)T

(6)

where the shear stress tensor and the heat flux vectorare defined as

τij = (µ+ µt)

[(∂ui∂xj

+∂uj∂xi

)− 2

3(∇ · u) δij

](7)

q = − (k + kt)∇T

with the laminar viscosity µ, the turbulent eddy vis-cosity µt, the laminar thermal conductivity k, the

turbulent eddy thermal conductivity kt, and the tem-perature T .

The basic numerical algorithm for solving theNavier-Stokes equations with the k-ω turbulencemodel follows that presented by Liu & Zheng23 andLiu & Ji.24 A cell-centered finite-volume method withartificial dissipation as proposed by Jameson et al.25

is used. In semi-discrete form the governing equationscan be written for each cell as

d

dt(W∆V ) = R (W) (8)

where the residual R (W) is given by the discretizedconvective and viscous fluxes and artificial dissipation.

The time derivative is discretized by an implicitbackward-difference scheme of second-order accuracyto obtain

Dt (W∆V )n+1

= R(Wn+1

)(9)

with

Dt (W∆V )n+1

=

1

2∆t

[3 (W∆V )

n+1− 4 (W∆V )n+ (W∆V )

n−1] (10)

where n + 1 denotes the current time level, and thetwo previous time levels are denoted by superscripts nand n− 1.

We reformulate the problem at each time step as thefollowing steady-state problem in a pseudo-time t∗

d

dt∗Wn+1 =

1

∆V n+1R∗(Wn+1) (11)

where

R∗(Wn+1) = R(Wn+1)−Dt (W∆V )n+1

(12)

For each real-time level, the solution to Eqs. (9) isfound by iterating Eqs. (11) to steady state in pseudo-time.

A hybrid multistage Runge-Kutta scheme is usedto integrate the semi-discrete Equation (11). For mstages, the integration is carried out as follows:

W(0)i,j,k = Wn

i,j,k

W(1)i,j,k = W

(0)i,j,k + α1

∆t∗i,j,k∆V n+1

i,j,k

R∗(W

(0)i,j,k

)

· · ·

W(m)i,j,k = W

(0)i,j,k + αm

∆t∗i,j,k∆V n+1

i,j,k

R∗(W

(m−1)i,j,k

)

Wn+1i,j,k = W

(m)i,j,k

(13)

A 5-stage scheme with the following coefficients isused:

α1 =1

4, α2 =

1

6, α3 =

3

8, α4 =

1

2, α5 = 1

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The artificial dissipation is updated at stages 1, 3and 5. Local pseudo-time stepping is used in orderto advance the flow solution at the local maximumspeed. Residual smoothing is applied at stages 1, 3 and5 in order to increase the stability limit. A multigridmethod is adopted to accelerate the convergence of thesolution.

III. Parallel Multiblock Method

The application of multiblock grids allows for com-plex geometries and provides domain decompositionfor parallel computation. Two ghost-cell layers aroundeach block serve for the implementation of boundaryconditions as well as for the exchange of informationat block-to-block interfaces. In case of an interface,the ghost-cell geometry coincides with the geometryof the corresponding cells in the neighboring block.The values of the flow variables in these cells are ex-changed after each multigrid step. If both blocks aretreated by the same processor, the flow variables aresimply copied from one array to the ghost-elements ofthe other array. On the other hand, if the blocks arecalculated on different processors in parallel computa-tions, MPI is used to exchange the information.

IV. Structural Solver

The general form of the structural equations for amechanical system with a finite number of degrees offreedom is given by

[M ]q + [C]q + [K]q = F (14)

where [M ] is the mass matrix, [C] the damping matrix,[K] the stiffness matrix, q the vector of displacements,and F the forcing vector.

The linear structural equations can be solved usinga modal approach, composing the solution with theeigenvectors of the free vibration problem. With thefirst N modes, the approximate description of the dis-placement vector is given by

q =

N∑

i=1

ηiφi (15)

where φi is the i-th eigenvector of the generalizedeigenvalue problem, and ηi is the corresponding gen-eralized coordinate.

The eigenvectors are orthogonal with respect toboth the mass and stiffness matrices. If we assumeclassical damping (e.g. Rayleigh damping) the eigen-vectors are also orthogonal with respect to the damp-ing matrix [C]. Thus, pre-multiplying Eq. (14) by [φ]T

(normalized such that the eigenvectors are orthonor-mal with respect to the mass matrix) yields a set ofuncoupled equations in generalized coordinates of theform

ηi + 2ζiωiηi + ω2i ηi = Qi, i = 1, ..., N (16)

where

Qi = φTi F, ω2i = φTi [K]φi, φTi [M ]φi = 1

and ζi are the modal damping parameters.For each mode i, the second-order differential equa-

tion (16) is transformed into two first-order equations

x1i = ηi

x1i = x2i

x2i = Qi − 2ζiωix2i − ω2i x1i

(17)

which can be written in matrix form as

Xi = [Ai]Xi + Qi, i = 1, 2 (18)

where

[Ai] =

[0 1−ω2

i −2ωiζi

]

Xi =

{x1i

x2i

}, Qi =

{0Qi

}

Decoupled by the transformation Zi = [Pi]−1Xi, with

[Pi] =

[− ζi+

√ζ2i−1

ωi

−ζi+√ζ2i−1

ωi1 1

]

the equations of motion take the form

Zi = ([Pi]−1[Ai][Pi])Zi + [Pi]

−1Qi (19)

or, in component notation,

dz(1,2)i

dτ=

ωi

(−ζi ±

√ζ2i − 1

)z(1,2)i +

√ζ2i − 1∓ ζi

2√ζ2i − 1

Qi

(20)

with Zi = {z1i, z2i}T .The time derivative operator is now discretized with

the same second-order scheme that is used to discretizethe Navier-Stokes equations in Eq. (10), and the resultis the following set of two finite-difference equations foreach mode

R∗s,i(Zn+1i ) =

3zn+1(1,2)i − 4zn(1,2)i + zn−1

(1,2)i

2∆τ

− ωi(−ζi ±

√ζ2i − 1

)zn+1

(1,2)i

+

√ζ2i − 1∓ ζi

2√ζ2i − 1

Qn+1i

= 0

(21)

which can be integrated to steady state in pseudo-time t∗

dZn+1i

dt∗+ R∗s,i(Z

n+1i ) = 0 (22)

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V. Fluid-Structure Coupling

The aeroelastic Eqs. (22) are coupled with the flowEqs. (11) since the aerodynamic residual R∗(W) de-pends implicitly on the blade motion Z through theflow boundary conditions, and the structural residualR∗s(Z) depends on the flow variables W through aero-dynamic forcing.

Equations (11) and (22) form a coupled system inpseudo-time which can be solved by the same explicitRunge-Kutta scheme (Eq. 13). In principle, Eqs. (11)and (22) can be marched in pseudo-time simultane-ously, i.e. when Eq. (11) is marched one pseudo-timestep, Eq. (22) is also marched by one pseudo-timestep. In practice, it is found that this procedure maylead to divergence because the flow equations usuallyconverge slower in pseudo-time than the structuralequations. Intermediate flow solutions may lead toinaccurate aerodynamic forcing, which in turn wouldcause a large deformation in the structures, resultingin potential divergence. In view of the above, severalpseudo-time iterations are performed on the flow equa-tions before the structural equations are also marchedby several pseudo-time steps, followed by an update ofthe grid coordinates, grid velocities and the forcing. Inthis way, both iterations, for the flow model and thestructural model, are converging in a coupled mannerwithin each real-time step.

The structural mode shapes are provided for a struc-tural grid that does not necessarily coincide with thewalls of the flow grid. Therefore, a spline interpolationmethod is applied in order to determine the structuralforcing from the aerodynamic forcing:

Fs = [G]TFa (23)

where Fs denotes the forcing to be applied on thestructural grid, Fa are the forces obtained from theflow solution on the flow grid. The matrix [G] is the thespline matrix that is used to obtain the deformationof the flow grid ∆xa from the structural displacements∆xs:

∆xa = [G]∆xs (24)

VI. Grid Deformation

The solution of the Navier-Stokes flow around amoving and deforming structure requires an efficientalgorithm for grid deformation. If the structural mo-tion is prescribed and therefore known a priori, thegrid has to be deformed only once per time step. How-ever, if the structural motion itself is a part of theaeroelastic solution, the iterative coupled computationinvolves several grid updates per time step.

The grid deformation is performed in three steps,adopting a method by Tsai et al.21 :

1. Structural displacements are imposed on theboundaries between structure and fluid. Thestructural displacements are obtained by thestructural solver and then interpolated onto theflow grid as described above.

2. The corner points of all grid blocks are displacedusing a spring-analogy method.

3. When the displacements of all corner points havebeen determined, they are interpolated along thesurface edges. Hermite polynomials are usedin order to be able to specify the displacementderivatives and thereby control the angle of theedges. This is done only where an edge touchesa wall surface in order to preserve the near-wallgrid angles. Subsequently, the same technique isapplied in two and three dimensions in order toobtain the surface displacements and the displace-ments of the interior grid points. This is doneby 2-D and 3-D transfinite interpolation, usingHermite polynomials in order to specify the gridangles close to walls. These grid angles are speci-fied as a blend between the angles of the originalgrid and the angles of neighboring edges or sur-faces, respectively. In this way we avoid largeangular discontinuities which may lead to over-lapping grid lines. This is particularly importantfor Navier-Stokes grids.

In parallel computations, the displacement of theunstructured spring network is most efficiently calcu-lated on the master node, which carries informationabout all blocks. Since the correct angle of the sur-face edges may require information from neighboringblocks, the 1-D interpolation along the edges is alsoperformed on the master node. The edge displace-ments are then distributed to the other processors, andthe following 2-D and 3-D interpolations are performedin parallel. Finally, all deformations are superimposedon the original grid.

For the flux calculation on a moving grid we needthe grid velocity ug of each grid point. These veloci-ties are not known exactly, therefore they have to beobtained in discrete form. Applying the same differ-ence operator that is used for the time derivative ofthe flow variables, we obtain the grid velocities usingthe grids from the current time level and two previoustime levels as

un+1g =

1

2∆t

(3xn+1 − 4xn + xn−1

)(25)

where x is the vector of grid coordinates.

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VII. Results

A. Nonlinear Flutter of a Supercritical Airfoil

A test case for the investigation of nonlinear flutterphenomena is presented by Schewe et al.26 Attachedto pitching and heaving springs, a two-dimensionalwing with the supercritical NLR 7301 profile is stud-ied experimentally and is shown to exhibit a sharptransonic dip at a Mach number of about 0.77. Limit-cycle flutter is observed in the region of the transonicdip. Numerical investigations on the transonic testcase MP77 (nominal M∞ = 0.768, α = 1.28◦ andRe = 1.7·106) have been performed in the time-domainby a number of authors.27–30 The common result inthose investigations is the prediction of limit-cycle flut-ter at amplitudes one order of magnitude larger thanthose observed in the experiments.

By taking the wind-tunnel walls into account in theircomputation, Castro et al.28,29 are able to show someeffects of the wall porosity on steady and unsteadyflow. These effects do not, however, fully explain thelarge difference between the predicted and measuredlimit-cycle amplitudes.

To account for wall interference, Schewe et al.26 sug-gest to correct the Mach number to 0.759 and the angleof attack to 0.74◦ for test case MP77. Figure 1 showsthe steady-state distribution of the pressure coefficientcomputed by the present method running in 2-D modeon a 257× 33 grid for the Euler computations and ona 257 × 97 grid for the Navier-Stokes computations.with the corrected Mach number and angle of attacksuggested by Schewe et al.. The experimental steady-state results (test case MP2084) were obtained with afixed wing.26 Even with the corrections, the numericalresults do not agree well with the experimental datain Fig. 1 except for the rear loading and around thetrailing edge. The pressure over the front half-chordof the pressure side is too large, and the shock on thepressure side is too weak. With inviscid flow the po-sition and strength of the shock on the suction side isvery different from the results of the viscous compu-tation, which agree better with the experiment. Thetwo-equation k-ω turbulence model yields slightly bet-ter results than the Baldwin-Lomax model. The lift isoverpredicted in all cases.

In an attempt to obtain better steady-state resultsfor this test case before proceeding to the aeroelasticcomputation, the Mach number and angle of attack areadjusted to yield the best possible match with the ex-perimental data. For the Euler computation the bestresults were obtained with M∞ = 0.735, α = −0.5◦,and for the Navier-Stokes computations M∞ = 0.75,α = 0◦. The results are shown in Fig. 2. The fluid-structure coupled computations are performed withthese adjusted values.

Figures 3 and 4 show the time histories of the heaveand pitch displacements, obtained by Euler calcula-

0 0.2 0.4 0.6 0.8 1x/c

-1.5

-1

-0.5

0

0.5

1

Cp

Exp., Pressure SideExp., Suction SideEulerBaldwin-Lomaxk-ω

Fig. 1 Steady-state pressure coefficient calculatedfor the corrected Mach number M∞ = 0.759 andangle of attack α = 0.74◦, compared to experimentMP2084.

0 0.2 0.4 0.6 0.8 1x/c

-1.5

-1

-0.5

0

0.5

1

Cp

Exp., Pressure SideExp., Suction Side

Euler, M = 0.735, α = -0.5o

Baldwin-Lomax, M = 0.750, α = 0o

k-ω, Μ = 0.750, α = 0o

Fig. 2 Steady-state pressure coefficient for ad-justed Mach numbers and angles of attack.

tions and Navier-Stokes calculations with the Baldwin-Lomax and the k-ω turbulence models, respectively.Note that the angular displacement is defined nose-uppositive, and the translational displacement is defineddownward positive and is non-dimensionalized by thehalf-chord length b. By comparing Figs. 3 and 4 wecan see that the pitch and plunge motions (as definedabove) are almost in phase, which is in agreement withthe experimental results.

Limit-cycle oscillations are observed with inviscidas well as viscous flow. However, the LCO amplitudesare overpredicted in all cases. Similar to the steadystate, there are large discrepancies between inviscidflow and viscous flow in the unsteady behavior. TheEuler computations predict very large displacements.Considerable improvement is achieved by using theNavier-Stokes equations with the algebraic turbulencemodel. The flutter amplitudes are further decreasedwhen the k-ω model is used. However, even the low-est amplitudes shown in Figs. 3 and 4 are still aboutone order of magnitude larger than those found inthe experiments. Furthermore, the flutter frequency is

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American Institute of Aeronautics and Astronautics Paper 2003–1347

0 10 20 30 40Structural Time t/Tα

-0.4

-0.2

0

0.2

0.4

Plun

ge h

/bEulerB-Lk-ω

Fig. 3 Plunging motion for various flow models.

0 10 20 30 40Structural Time t/Tα

-20o

-15o

-10o

-5o

0o

5o

10o

15o

20o

Pitc

h ∆

α

EulerB-Lk-ω

Fig. 4 Pitching motion for various flow models.

underpredicted. Table 1 gives a summary of the limit-cycle amplitudes and reduced frequencies (defined asκf = ωfc/u∞).

∆α/◦ h/b κf

Euler 15.6 0.323 0.199B–L 4.8 0.088 0.206k−ω 3.1 0.055 0.205Exp.26 0.2 0.005 0.242

Table 1 LCO amplitudes and frequencies.

While the above comparisons between inviscid andviscous results suggest that this test case exhibitsstrong viscous effects and it is imperative that aNavier-Stokes code be used, the results also indicatethat the current computations do not capture all sig-nificant flow features. Schewe et al. suggest thatthere might be a small separation region beneath theshock, which would have significant influence on theshock motion and might cause the flutter amplitudesto be limited to remarkably small values. However,the pressure distributions obtained by the experimentsare not sufficient to prove the existence of such a smallseparation region and it is not revealed in our numer-ical simulations. Since there is no reliable information

available about the transition point, the current cal-culations assume fully turbulent flow. In addition, itis possible (as noted in Ref. 26) that the thin rear ofthe airfoil is subject to some deformation, which mightintroduce unsteady effects on the shock or the trailingedge separation, not accounted for in the computa-tions.

B. Flutter of the AGARD 445.6 Wing

The AGARD 445.6 wing has a quarter-chord sweepangle of 45 degrees and its cross-section is given bythe NACA 65A004 airfoil. The flutter characteris-tics of this wing were investigated experimentally overa wide range of Mach numbers.31 The results werepresented as an AGARD standard aeroelastic config-uration32 and have since been widely used to test andvalidate flutter calculations.

Lee-Rausch and Batina17 find that at subsonic Machnumbers the flutter boundary of this wing can be pre-dicted by coupled time-marching computations usingthe Euler equations and modal structural equations.However, in the supersonic regime, the flutter bound-ary is overpredicted by the inviscid computations. Us-ing the thin-layer Navier-Stokes equations, the authorsare able to achieve some improvement. However, theviscous results still overpredict the flutter boundary.

Liu et al.22 apply a time-marching method for solv-ing the Euler equations coupled to the modal equationsfor the first four modes. Their results confirm thelarge overprediction in case of inviscid flow, while goodagreement with the experiments is achieved in the sub-sonic domain.

In an attempt to demonstrate differences betweenthe inviscid and viscous flow, this work presents re-sults by using the Euler and Navier-Stokes computa-tions. The Euler computations were performed on a129 × 33 × 33 grid. The Navier-Stokes computationswere performed on a 129×49×33 grid. Figure 5 showsthe flutter speed index as a function of the freestreamMach number. Experimental results are plotted forcomparison with those obtained by the coupled aeroe-lastic calculation. The first five structural modes aretaken into account. Results are shown for inviscid flowand viscous flow using the Baldwin-Lomax and the k-ωturbulence models. The flutter speed index is definedas the value

Vf =u∞

bωα√µ

∣∣∣∣neutral

(26)

at which an initial oscillation is neither amplified nordamped by aerodynamic forcing. If the speed index isabove the flutter speed index, the motion is unstable,if the speed index V is smaller than Vf , the motionis stable. Here, u∞ is the freestream velocity, b is thehalf-chord length at the wing root, ωα is the eigenfre-quency of the first torsional mode, and µ is the massratio of the wing.

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0.2 0.4 0.6 0.8 1 1.2 1.4Freestream Mach number

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Flut

ter

Spee

d In

dex

ExperimentEulerN-S, Baldwin-LomaxN-S, k-ω

Fig. 5 Flutter speed of the AGARD 445.6 wing.

In the subsonic and transonic range the results ofthe computation match the experimental data reason-ably well in Fig. 5. Since this is true even for theinviscid results, it appears that in the subsonic to tran-sonic range the aeroelastic behavior is not significantlyinfluenced by viscous effects for this wing. However,for supersonic flow the calculations clearly overpredictthe flutter speed. The overprediction is most apparentwith inviscid flow. Part of the reason may be that theeffect of viscosity on the shock motion is not accountedfor by the inviscid flow calculation. The viscous calcu-lations with both turbulence models show only smallimprovement over the inviscid results. It seems likelythat with the given computational grid, neither tur-bulence model is able to predict all significant flowfeatures.

Another possible cause for the discrepancy at super-sonic speeds might be given by the nonlinear flutterbehavior. If the Mach number is low, the oscillationamplitude increases monotonically in an unstable sit-uation, as shown in Fig. 6. In the supersonic range,however, the amplitudes may initially increase butthen decay to yield a stable solution after some time,as shown in Fig. 7. The wind-tunnel tests might havebeen terminated at an early stage, and the behaviormisinterpreted as unstable.

Figure 8 shows the flutter frequency ratio, i.e.ωf/ωα, for various Mach numbers. Despite the goodagreement on the flutter speed index in the subsonicrange (Fig. 5), the frequency is overpredicted in all thecomputations. Considering some scatter in the experi-mental data, the computational results are reasonablein the transonic Mach number range, while the pre-dicted frequencies in the supersonic region are againtoo large. It is interesting to note that the flutter speedindex is accurately predicted despite the overpredic-tion of the frequency in the subsonic regime. Sincestability is directly related to the phase difference be-tween structural motion and forcing, one might inferthat in the supersonic regime the phase is more sensi-tive to the frequency than in subsonic flow.

0 5 10 15 20Structural Time

-0.01

-0.005

0

0.005

0.01

Gen

eral

ized

Dis

plac

emen

t

1st bending mode

1st torsional mode

Fig. 6 Time history of the wing displacements ofthe first two modes for M∞ = 0.338 and a speedindex of 0.475 in inviscid flow.

0 5 10 15 20Structural Time

-0.01

-0.005

0

0.005

0.01

Gen

eral

ized

Dis

plac

emen

t

1st bending mode

1st torsional mode

Fig. 7 Time history of the wing displacements ofthe first two modes for M∞ = 1.141 and a speedindex of 0.670 in inviscid flow.

0.2 0.4 0.6 0.8 1 1.2 1.4Freestream Mach number

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Flut

ter

Freq

uenc

y R

atio

ExperimentEulerN-S, Baldwin-LomaxN-S, k-ω

Fig. 8 Flutter frequency (normalized with the fre-quency of the first torsional mode) of the AGARD445.6 wing.

The largest discrepancies between the experimentalresults, the Euler results and the Navier-Stokes resultsappear at the largest Mach number, M∞ = 1.141. Forcomparison between the inviscid and viscous results,the steady pressure distributions at α = 0◦ are shown

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cp: -2.3E-01 -7.6E-02 2.1E-01 3.7E-01 5.2E-01 6.8E-01 8.4E-01 9.9E-01

a) Euler b) N-S, Baldwin-Lomax c) N-S, k-ω

Fig. 9 Steady-state pressure coefficient for M∞ =1.141, α = 0◦ with a) inviscid flow, and viscous flowusing b) the Baldwin-Lomax model and c) the k-ωmodel.

7.4E-03 3.0E-02 5.2E-02 7.4E-02 9.7E-02 1.2E-01 1.4E-01

-

-

-

-+

+

+

- -

+

+ +

+

+

- -

-

a) V = 0.670 b) V=Vf=0.685 c) V=0.700

Magnitude (shade) and Phase (+/-) of Cp(t) , Inviscid∆

Fig. 10 Unsteady pressure coefficient for inviscidflow at various speed indices.

in Figs. 9a-c. Contours of the pressure coefficient areshown over the wing surface as well as in a cross-section at 95% span, for a) inviscid flow, b) viscousflow using the Baldwin-Lomax model, and c) viscousflow using the k-ω model. While there seems to belittle difference between the two viscous solutions, theoblique shock near the trailing edge is stronger in theinviscid flow solution.

For the same Mach number of M∞ = 1.141,Figs. 10a-c show the magnitude and phase of the un-steady ∆Cp over the wing for inviscid flow at threedifferent speed indices. Here ∆Cp is the difference be-tween the pressure coefficients on the lower and upperwing surfaces:

∆Cp(t) = Cp,l(t)− Cp,u(t) =pl(t)− pu(t)

12γp∞M

2∞(27)

The unsteady pressure has a destabilizing effect wherethe phase difference between ∆Cp and the displace-ment is positive. This phase difference is zero alongthe black contour lines in Figs. 10a-c, therefore, thelines separate stabilizing regions (-) from destabilizingregions (+). The magnitude of the oscillating pressuredifference is shown as underlying shades of gray.

Three different results are shown: a) a stable solu-tion at a speed index slightly below the flutter limit,b) the neutral solution at V = Vf , and c) a slightly

7.4E-03 3.0E-02 5.2E-02 7.4E-02 9.7E-02 1.2E-01 1.4E-01

-

-

-

-+

+

+

--

+

+

++

+

- -

-

a) V = 0.625 b) V=Vf=0.630 c) V=0.640

Magnitude (shade) and Phase (+/-) of Cp(t) , Viscous, k-∆ ω

Fig. 11 Unsteady pressure coefficient for viscousflow (k-ω) at various speed indices.

unstable solution. In all cases, the pressure magni-tude is dominant in the region of the oblique shocknear the wing tip. In that region the phase is positive(destabilizing). In Fig. 10a there is also an unstableregion close to the wing root, which does not, how-ever, contribute much to the bending forces, becausethe magnitude in that region is very small. Over alarge part of the wing, a small region about the shockexcluded, the phase angle is negative, so that overallthe wing is stable at the speed index of V = 0.67. Asthe speed index is increased to 0.685, the unstable re-gion about the shock extends all the way to the wingroot as is shown in Fig. 10b. In addition, there is agrowing region of instability about the leading edgenear the wing tip. At this speed index, the unstableand stable regions seem to eliminate each other to yieldoverall neutral stability for the wing. The two unsta-ble regions at the leading edge and around the shockgrow together to cover most of the wing area whenthe speed index is further increased (Fig. 10c). At thespeed index of V = 0.7 the wing exhibits flutter.

For comparison with the inviscid results, Figs. 11a-cshow the unsteady pressure difference obtained by theviscous computation using the k-ω turbulence model.There is no fundamental difference between the invis-cid and the viscous result. The instability seems toarise from the shock motion (Fig. 11a) and at theleading edge near the tip (Fig. 11b). The destabi-lizing regions extend over a major part of the wingas the speed index is increased (Fig. 11c). Thoughqualitatively similar to the inviscid case, the viscouscomputation predicts flutter at a lower speed index,closer to but still higher than the experimental valuefor this supersonic Mach number.

In order to visualize the shock motion in relationto the wing deformation, Fig. 12 shows the time his-tories of the pressure coefficient around the shock aswell as the time histories of the generalized displace-ments of the first three modes. For both inviscidand viscous flow, the pressure contours and displace-ments are obtained with M∞ = 1.141 for the caseof neutral stability, i.e. at Vf = 0.685 (inviscid) andVf = 0.63 (viscous), respectively. The pressure coef-ficient is shown over a fraction of the chord at 95%

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-0.01 0 0.01

1. Mode Displacement2. Mode3. Mode

-0.01 0 0.01

(x/c)rel at 95% span

Str

uct

ural

Tim

e

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

CP: -0.24 -0.21 -0.19 -0.16 -0.13 -0.10 -0.08 -0.05 -0.02 0.01

70% 84% 66% 86%

Euler N-S, k- Euler N-S, k- ωω

Fig. 12 Shock motion at 95% span and modalstructural deformations for M∞ = 1.141 at neutralstability.

span, i.e. close to the wing tip, where the shock ismost visible.

It is apparent in Fig. 12, that the inviscid flow yieldsa more confined and stronger shock than the viscousflow. This has already been observed with the steadystate in Fig. 9. In viscous flow the shock appearsto move sinusoidally, whereas in the inviscid case thetime-history of the shock position exhibits various fre-quencies. The modal displacements are shown on theright hand side of Fig. 12. The deflections of the fourthand fifth modes are not shown here, because of theirinsignificant contribution to the total deformation.

The Euler computation yields a slightly higher fre-quency than the Navier-Stokes computation for thefirst-mode deflections. This was shown more clearly inFig. 8. A difference between the inviscid and viscouscases can be seen in the third-mode deflection (secondbending mode). While the third mode is almost sup-pressed in the viscous case, it is more clearly apparentin the inviscid result, where it exhibits a frequencywhich is about twice that of the dominant first mode.This harmonic frequency is reflected in the shock mo-tion only in the inviscid case.

VIII. Conclusions

A numerical method for the computation of three-dimensional aeroelastic problems is presented and isapplied to test cases of airfoil and wing flutter. The un-steady Euler or Navier-Stokes equations are solved bya multigrid finite-volume method on structured multi-block grids. The Baldwin-Lomax model and a k-ωturbulence model are implemented. Using a modalapproach, the structural equations are solved simulta-neously with the flow equations. Coupled convergenceis achieved by pseudo-time stepping with several up-dates of the forcing and deformation in each time step.The flow in multiple grid blocks is calculated in paral-lel.

The flutter behavior of a pitching and plunging su-percritical airfoil is investigated and compared to ex-perimental results in Ref. 26. For a transonic test case,the calculated steady-state distribution of the pressurecoefficient is in poor agreement with the experiments.Comparisons between inviscid and viscous computa-tions in the current work confirm that viscous effectsare significant. However, even with viscous computa-tions the Mach number and angle of attack have to beadjusted in order to improve the agreement with theexperimental data.

Aeroelastic computations are performed using theMach number and angle of attack that yields the bestmatch to the experimental steady-state results. Limit-cycle flutter is observed with both inviscid and viscousflow, qualitatively confirming the experiments. How-ever, the oscillation amplitudes are largely overpre-dicted in all cases. The flutter frequency is underpre-dicted. As in the steady-state calculations, consider-able improvement is achieved by taking viscosity intoaccount. The best result is achieved by using the k-ωmodel.

The discrepancies between computation and exper-iments may be due to a number of physical factorsthat are not accounted for in the computations. Po-tentially, these include shock induced boundary-layerseparation, deformation of the airfoil near the trailingedge, and wind-tunnel interference. Similar observa-tions are made in Refs. 26–30. Further investigationsare needed to resolve those issues, given that generallyspeaking flows over many other airfoils have been wellpredicted by modern Navier-Stokes computations, atleast for steady-state solutions.

The AGARD 445.6 wing is used as a three-dimensional test case. Aeroelastic computations areperformed over a large range of Mach numbers in orderto obtain the flutter boundary. The five most domi-nant structural modes are taken into account. In thesubsonic and transonic regimes, the flutter speed in-dex for inviscid as well as viscous flow agrees well withthe experimental data in Ref. 32, although the flutterfrequency is slightly overpredicted. However, at super-sonic Mach numbers the flutter speed index as well asthe flutter frequency are overpredicted by all compu-tations. The Navier-Stokes computations, assumingfully turbulent flow, achieve little improvement overthe Euler results. The most visible effect of viscos-ity is to weaken the oblique shock on the wing. Eventhough the dynamics of this shock are shown to besomewhat different in the inviscid and viscous cases,the effect of viscosity on flutter stability seems to besmall for this test case.

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17Lee-Rausch, E. and Batina, J. T., “Calculation of AgardWing 445.6 Flutter Using Navier-Stokes Aerodynamics,” AIAAPaper 93–3476, 1993.

18Lee-Rausch, E. and Batina, J. T., “Wing Flutter Bound-ary Prediction Using Unsteady Euler Aerodynamic Method,”Journal of Aircraft , Vol. 32, 1995, pp. 416–422.

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28Castro, B. M., Ekaterinaris, J. A., and Platzer, M. F.,“Transonic Flutter Computations for the NLR 7301 Airfoil In-side a Wind Tunnel,” AIAA Paper 2000–0984, 2000.

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