Cavagna Et Al IFASD-0093

7/27/2019 Cavagna Et Al IFASD-0093

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EFFICIENT APPLICATION OF CFD AEROELASTIC

METHODS USING COMMERCIAL SOFTWARE

Luca Cavagna, Giuseppe Quaranta,

Gian Luca Ghiringhelli and Paolo Mantegazza

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano,via La Masa 34, 20156 Milano, Italy

email: {cavagna,quaranta,ghiringhelli,mantegazza}@aero.polimi.it

Key words: Computational Fluid Dynamics, Transonic Aeroelasticity, Flutter Analysis.

Abstract. Aeroelastic analyses in transonic regime require the adoption of accurateaerodynamics physical models, such as Euler or Navier-Stokes equations. To move theapplication of these type of analyses from a pure academical environment to an industrialone, it is necessary to show that the technology is mature enough to be implementedwithout using specialized pieces of software. This paper presents a numerical procedure

defined to solve Fluid-Structure Interactions (FSI) for aeroelastic problems using par-titioned procedures based on the adoption of black-box commercial software for thesolution of each field. A special attention is given to the efficiency of the procedure, keep-ing in mind the high number of analyses that have to be run during the development ofa new aircraft.

1 INTRODUCTION

Aeroelastic phenomena in transonic speed range may become extremely complex, becauseunder these conditions shock waves appear and move in the flow field as consequence of air-

craft unsteady flexible motions. Usually, the appearance of shock waves may cause a dropof the flutter velocity, the well known transonic dip effect, which is under-predictedby classical potential methods used for unsteady aerodynamic loads description. Im-provements in the flutter boundary evaluation can be obtained by using more complexdescriptions of the aerodynamic domain, capable of predicting shock waves in the flow,such as those based on the Euler or Navier-Stokes equations. Starting from the pioneeringworks of Lee-Raush and Batina 1;2, given the fast increase in the computer performances ofthe last few years, the application of unsteady CFD solutions for aeroelastic problems hasgrown into a large and successful research field, with applications to complete aircraft con-figurations 3. However, up to now the complexity of the procedures and the high amount

of specialized computational resources required for the application of these methodologiesprecludes them from being extensively used in industrial aeroelastic analysis 4;5.

The purpose of this paper is to show that times are mature for trying to define pro-cedures to solve transonic aeroelastic problems effectively in an industrial environment.To do so, we tested the possibility to create specific procedures for aeroelastic analysesusing off-the-shelf software products, such as the commercially available ComputationalFluid Dynamics (CFD) software FLUENT. Figure 1 shows a block diagram with all theelements needed for conducting an aeroelastic assessment with CFD. Three main elementsof this block diagram have a key role in generating an efficient and robust solution pro-cedure: the first is the grid interpolation between the structural and the aerodynamicdiscretization; the second is the grid deformation which must be used in order to adapt itto the motion of the aircraft under investigation, and third is the definition of a smart

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CFD Grid

Definition

InterfaceMovement

Grid

CFDSolver

Transfer Matrix

Identification

Structural

Modes

Coupling

Algorithm

Direct Integration

Figure 1: Block diagram for CFD aeroelastic analyses

numerical test procedure to be used for the identification of the unsteady aerodynamicbehavior in terms of a state space Reduced Order Model (ROM). For the first point aninterfacing procedure based on a mesh-free Moving Least Square method is proposed.It ensures the conservation of the momentum and energy transfer between the fluid andstructure and is suitable for the treatment of geometrically complex configurations, whichmay include not only wings, but fuselage, nacelles and so forth. The methodology givesthe user an high level of freedom to achieve the required fidelity and smoothness of the in-terpolated movements, and it is highly portable since it ensures a complete independencefrom the details of the numerical solvers adopted.

Grid motion is another task which has a great impact on the time required by CFD-CSD aeroelastic simulations. To keep a good quality of the grid through the wholesimulation is very daunting job. We propose here to use a simple but robust algorithmbased on the modeling of the volume grid as an arbitrary elastic solid with a prescribedstiffness distribution used to to preserve the grid quality even for large movements. Whenreasonably small linear movements are required, the grid deformation can be computedonce for the whole time transient and than simply scaled using the input variation law atthe different time steps. In this way great computational savings are achieved.

Flutter assessment can be conducted analyzing a linearized model of the equationsabout a certain condition found by a complete nonlinear aircraft flight trim. A linear state-space model for the unsteady aeroelastic forces can be build starting form the knowledge

of the frequency domain transfer matrices using the classical Pade approach presented inRef. 6, or the one shown in Ref. 7. Consequently, a certain number of numerical experimentsare needed to obtain this frequency data. It will be shown how the best trade-off betweenthe necessity to reduce the computational time and the accuracy is obtained by computingthe response of the aerodynamic system to a trimmed step input signal for each elasticmode to be analyzed. Then, by means of the Fast Fourier Transform (FFT) the completetransfer matrix in the desired range of reduced frequency is obtained with a very smallnumber of transient computations being required. A similar identification approach isadopted by Raveh 8 which uses a wite noise modal displacements input as excitation.

To assess the quality of the proposed method, a comparison with the classical exper-

imental results for the AGARD 445.6 will be shown 9. Furthermore, as an example of

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Figure 2: Piaggio P-180 turboprop aircraft

application to a real life industrial case, the analysis of the main wing of the Piaggio

P-180 aircraft (Figure 2) is presented. P-180 is a turboprop aircraft which must be clearedfor flutter significantly beyond cruise condition, where fully transonic condition are met.

2 DEFINITION OF THE LINEARIZED AEROELASTIC PROBLEM

The primary assessment necessary for the aeroelastic certification of aircraft is related tothe analysis of the local linearized solutions to find instabilities, and specifically flutterpoints. Differently from what can be done in the flight regimes where a linear represen-tation of the aerodynamic field can be adopted (i.e. potential flows), in case of strongnonlinearities in the field such has shock waves, it is necessary to assess the stability of

each movement associated with each equilibrium point of the aeroelastic system10

. Con-sequently, each flight configuration may potentially assume a different stability behavior.However, if there are not abrupt changes in the fluid flow, it is reasonable to considerthe linearization around a certain flight point sufficiently stable for being representativeof the behavior also for nearby configurations (i.e. small differences in the mass andstiffness distribution, and consequently small variation in the aircraft attitude). As aconsequence, to speed up the analysis, expecially when a large number of configurationsneeds to be tested, it is much easier to try to extract a linearized model of the unsteadyaerodynamic forces from the CFD solutions instead of running for each flight condition anew nonlinear coupled numerical flutter test. The result of the linearization is a ROMfor aerodynamic unsteady forces. As structural model a linear modal representation ofthe structure is used, as it is usually done in classic aeroelastic analysis. In any case it isnecessary to have a backup procedure to run the coupled nonlinear analysis in order toverify and validate key instability points obtained by using linearized models.

2.1 Creation of the time domain reduced order model

Aerodynamic forces can be modeled as a state space dynamic system which receives asinput the structural displacements, velocities, and gusts, and gives the associated gener-alized aerodynamic forces as output. Using a sufficiently large modal basis to representthe structural displacements, it is possible to identify a small set of boundary movement

inputs for the aerodynamic field. In case of structural changes which cause a variation

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of the modal frequencies, if the modal basis is well chosen, and possibly hybridized withappropriate static branch modes, it is possible to see the new modes as a combinationof primitive modes 11. As a consequence, the identified model can be easily adopted forparametric analysis of the aircraft stability for different operative conditions.

It is necessary to define a simple excitation method which requires a reasonable com-putational cost but permits a good identification of the principal dynamics of aerodynamic

forces, remembering that the aerodynamic system is usually over-damped. Among pos-sible input signal the classical are: sinusoidal, impulse and step. The characterizationthrough sinusoidal input seems the most natural but it is extremely expensive in termsof computational costs, because each modal form needs to be tested for a set of imposedfrequency. The other two cases, at least ideally, require just one test for each input tocharacterize completely the system in the whole range of frequencies of interest. Differenttests made have shown a great sensibility to the sampling time of the numerical discreteinput realizations, expecially for the impulse case, usually requiring to increment a lotthe time sampling points near discontinuities. However, it is often necessary to charac-terize the dynamics of generalized forces only in an assigned range of reduced frequency

[0, kmax] and not in the whole range, so it is not really necessary to adopt discontinuousinput signals. As a matter of fact, it seems reasonable to apply as input a trimmed stepas the following one, smoothing the discontinuities

q() =

q2

(1 cos0) 0 < max,

q max(1)

where = tV/La is the non-dimensional time, 0 = /max and max = 2/kmax. Inthis way only the frequencies in the range of interest are effectively excited. Results havea very good accuracy and do not incur in numerical integration problems caused by high

frequency oscillations induced in the aerodynamic field. Furthermore, the adoption of stepsignals allows to compute the asymptotic value for aerodynamic forces due to a changein the boundary condition, which represent an essential data for the correct evaluation ofthe static gains of the transfer matrix. Of course a linearity test is always necessary todecide the correct amplitude scaling of the input signal which ensures a linear behaviorof aerodynamic forces.

During the simulation the aerodynamic generalized forces vector w associated withmodal forms is computed. Using the FFT transformation is easy to get each column ofthe aerodynamic transfer matrix after each step simulation as

Ham(jk, M)i =

F(w(, M)i)

F(q(, M)i) . (2)

Better results in the numerical transformation, which reduce the classical Gibbs oscilla-tions effect near discontinuities, can be obtained expressing any generic signal s as a sumof the asymptotic value s and the deficiency function Ds, which is so defined as

Ds(t) = s(t) s. (3)

The deficiency function for the input signal can be easily computed as

Dq() =

q

2

(1 + cos 0) 0 < max,

0 max(4)

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and the Fourier transform is equal to

F(q(, M)i) =qjk

+ F(Dq). (5)

As a consequence Eq. (2) becomes

Ham(jk, M)i =w + jkF(Dw(, M)i)qi + jkF(Dq(, M)i)

. (6)

The transfer matrix Ham obtained in this way can be used for frequency domainanalysis or can be transformed in a state space time domain system using any of thetechniques currently adopted (see Ref. 6;7;12). The result is a state space system

xa = Axa + Bq,

fa = Cxa + D0q + D1q + D2q,(7)

which can be connected in feedback with the structural model and so used for all types ofdynamic analyses and stability investigations. A special care must be taken to correctlyrepresent the quasi-steady coefficients which are contained in the matrices D0, D1 andD2. They represent the system behavior at very low frequency, which corresponds tothe behavior of the excited time responses near the tail of the simulations, while thesystem is reaching the stationary values. It must be stressed that the knowledge of theasymptotic values of the step response allows a correct static residualization of generalizedforces, which is an essential ingredient for a correct comprehensive modeling of the wholedynamics of a deformable aircraft.

2.2 Direct integration of the coupled problem

The direct time integration can be easily implemented using a partitioned loosely coupledalgorithm. Both systems, the structural and the aerodynamic are integrated using animplicit algorithm, leaving the time step size decision just to accuracy and sampling issuesand not to numerical stability issues. For the aerodynamics, the implicit backward Eulerintegrator is imposed by FLUENT as the only implicit solver usable for ALE solutions.The structure is instead represented by a modal description, so the contained frequencyspectrum is perfectly known and no numerical higher frequencies are present in the model.The classical partitioned scheme 13 requires for each time step: 1) a prediction of thestructural displacement, which gives the new position for the structural interface; 2)

the solution of the aerodynamic field, which gives the new loads; 3) a correction of thestructural time integration using the new computed loads. In this work we adopted oneof the methods proposed by Giles 14 which is based on the adoption of a predictor and acorrector derived form Crank-Nicholson algorithm:

Predictor q =

I +

1

2hA

1

I 1

2hA

q(n) + hp(n)

, (8)

Corrector q(n+1) =

I +

1

2hA

1

I 1

2hA

q(n) +

1

2h

p(n+1) + p(n)

, (9)

where A is the state space matrix used to represent the modal structural model, q is thevector of the structural modal states (i.e. modal position and velocity), and p is the nodal

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aerodynamic loads vector. Even though partitioned loosely coupled methods may createa net energy loss/increment during the simulation, because

B

(p(n) x(n+1)s p(n+1) x

(n+1)f ) dA =

B

(p(n) p(n+1)) x(n+1)s dA = 0 (10)

Giles14

shown that choosing a sufficiently small time step ensures the overall stability ofthe system.

3 INTERFACING DATA BETWEEN FLUID AND STRUCTURE

The adoption of a partitioned approach for the solution of Fluid-Structure Interactionproblems requires the definition of an interface scheme to exchange displacements andvelocities from the structural grid to the aerodynamic wet surfaces of the CFD grid and totransfer back aerodynamic forces on the structural nodes. The two models are discretizedin a very different and often not compatible way; this is especially true in an industrial

environment, where they usually come from different departments.Structures are represented by complex definition volumes, often very discontinuous.Their numerical representations are based on the adoption of schematic models, whichhave a long tradition in the aerospace industry, made by elements with very differenttopologies, such has beams, plates and solid elements, which usually are not coincidentwith the real geometrical representation of the aircraft (see Figure 3). It is the authorsopinion that these simplified models will be used for some time to come in aerospaceindustry for dynamic analysis, so it is essential to be able to cope with them.

On the other side, aerodynamic grid requires an exact representation of the wet sur-faces, so it is necessary to make these two representations of the same aircraft compatiblein order to transfer information between them.

A correct and efficient interface scheme for partitioned analysis must possess all thoseproperties:

possibility to interface both non-matching surfaces or non-matching topologies;

capability to deal with situations where a control point fall outside the range of thesource mesh (extrapolation);

exact treatment of rigid translations and rotations;

capability to deal correctly with situations having a wide variation of the node

density of the source mesh;

independence from the numerical formulation of the Computational Fluid Dynamics(CFD) and Computational Structural Dynamics (CSD) solvers;

conservation of the exchanged quantities (in particular momentum and energy);

possibility to control the smoothness of the resulting surface.

The last two points are essential when stability analysis have to be carried out. Thecomparison of spurious energy created or canceled by the interface scheme may alterthe stability boundary of the system. Furthermore, when highly accurate aerodynamics

models are used, such has Euler or Navier-Stokes, a non correct smoothness of the wet

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Figure 3: Results of the interface procedure for the first two Piaggio P180 wing out-of-plane modes

surface may cause numerical convergence problems or unphysical local instabilities of theflux.

In order to guarantee the conservation between the two models, the correct strategywould be to enforce the coupling conditions only in a weak sense, through the use of simplevariation principles such as that of Virtual Work. Let yf and ys be two admissiblevirtual displacements for each field. Admissible means that the trace of these two fieldson , which can be either a newly defined virtual interface surface or simply the surfaceof the fluid field f which is always present, must be equal

Tr (yf)| = Tr (ys)|. (11)

After computing the nodal loads (Ff)i for the aerodynamic boundary grid points usingthe correct approximation space, the loads on the structural nodes (Fs)j, can be obtainedby simply multiplying the formers by the transpose of the interpolation matrix H thatconnects the two grid displacements

(yf)i =

jsj=1

hij (ys)j, (12)

(Fs)j =

ifi=1

hij(Ff)i. (13)

This is a well known result, reported in almost all works about the implementation of aninterface algorithm 15;16, that ensure the balance of the energy exchanged between the fluidand the structure. However, this point does not completely solve the conservation issue,because there is no conservation of the velocity transmitted from the structure to fluidboundaries, so no guarantee about the conservation of the momentum transferred to fluid.The problem of conservation is now shifted on the definition of the correct interpolation

matrix H.

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To build a conservative interpolation matrix which enforces the compatibility, Eq. (11),a weak/variational formulation can be used. The idea is to express the problem as aweighted least-square problem

Minimize

(Tr (yf)| Tr (ys)|)2 dA. (14)

In addition to this, additional properties can be sought, like smoothness of the resultinginterpolated field, computational efficiency and some control on the interpolation error. Asolution which possesses all these qualities can be obtained using the MLS technique. Theorigin of this approximation is connected to the surface or data reconstruction field (seeLancaster and Salkauskas 17, and Schaback 18). The problem can be mathematically statedas follows. Given a compact space Rn, the object of the analysis is the reconstructionof a function f Cd() from its values f(x1), f(x2), . . . , f (xN) on scattered distinctcenters X = {x1, x2, . . . , xN}. Of course, it is not necessary to derive an analyticalexpression for f; it is sufficient to have an efficient method to compute the value of fon a different set of centers Y = {y1, y2, . . . , yN}. The method should ideally have theseproperties: a) computational efficiency; b) correct smoothness of the resulting surface; c)quality of reproduction.

The first step is to build a local approximation of f as a sum of monomial basisfunctions pi(x) Pd

f =mi=1

pi(x)ai(x) pT(x) a(x), (15)

where m is the number of basis functions, and ai(x) are their coefficients. Pd Cd()

is a finite dimensional space of basis functions; usually it is spanned by polynomials, butother forms can be adopted. In this case the adopted basis functions are either linear or

quadratic polynomials

pT(x) = (1, x , y , z )T C1(R3), (16)

pT(x) = (1, x , y, z, x2, x y, y2, yz, z 2, zx)T C2(R3). (17)

The coefficients ai(x) are obtained by performing a weighted least square fit for theapproximation

Minimize J(x) =

(x x)

f f(x)2

d(x), (18)

under the linear constraint

f(x) =

mi=1

pi(x)ai(x). (19)

This equation is completely equivalent to Eq. (14) which expresses the interface problem.The great advantage of the problem expressed in this form is that it can be localizedby choosing compact support weight functions such as smooth nonnegative Radial BasisFunctions (RBF). Usually the weight RBF are written as (r/), where delta is a scalingfactor that allows to change the function support at different space centers. The parameter allows the user to adapt the support radius to the problem, being sure that, on theone hand, enough points are covered, and, on the other hand, far away points have noinfluence. More details on the implementation of this interface scheme together with few

application results may be found in Ref. 19.

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4 EFFICIENT GRID DEFORMATION

To correctly represent the structural deformation of the aircraft, the CFD computationalgrid must be modified at each time step in order to be compatible with the structuraldeformation. Using an Arbitrary Lagrangian Eulerian formulation of the fluid equation20,it is possible to keep into account any possible movement of grid nodes. Consequently,

by simply deforming the fluid grid, it is possible to solve the problem easily withoutgenerating a new grid at each time step. Of course, the deformed grid must follow thestructural movements, but, at the same time, must keep a good quality in order to avoidany numerical problem during the simulation. There is a large literature about methodsto obtain compatible deformed grid. Batina 21 introduced the elastic analogy representingeach side of the grid as a spring with a nonlinear stiffness proportional to the side length.To avoid the occurrence of invalid elements with negative volumes during the simulation,Degand and Farhat 22 introduced torsional springs at each vertex, making the algorithmeven most expensive in terms of computational costs. If not treated in the right way,the deformation problem may become one of the most expensive task of this types of

simulations.In this work a different way is followed. The first goal sought is the overall compu-tational efficiency, so we tried to avoid any nonlinear model for grid deformation. Alsoin this case an elastic analogy is exploited, but the grid is represented as a linear elasticcontinuum with a local Young modulus proportional to the minimal dimension of eachelement following a law of this type

Eel =1

minj,k el

xj xk. (20)

A Poisson coefficient [0;0.35] is chosen in order to avoid numerical bad conditioningof the problem. This distribution of stiffness allows to relieve the effects of structural de-formations from inner small elements near the aircraft surface leaving the burden on outerlarger elements, which can be deformed without large distortions. This method works wellfor any element shape and also for Navier-Stokes hybrid meshes made by tetrahedronsand hexahedrons. An example of local grid deformation for very large structural defor-mation is shown in Figure 4. In this case the AGARD 445.6 wing is moved following thepattern of the fifth mode. In the shown cases the grid is still valid, in the sense that noelement volume is negative and both the overall and local quality is kept almost constant.In fact, the structural analogy gives to the user a large freedom in choosing the materialconstitutive properties to rule the grid deformation behavior during the simulation. By

simply choosing the structural properties it is possible to work on the quality of the gridduring the deformation phase.

Furthermore, the linearity of the problem allows more room for computational savings.Using a direct method for the solution of related linear algebra problems, the matrixfactorization can be done once for all. The new deformed grid at each time step may befound by a simple and fast application of the backward and forward substitution steps.In the tests presented here, the structural model is always represented as a set of modalforms. Consequently, also the CFD grid deformation can be represented, exploiting thelinearity of the problem, as a superposition of grid deformations computed for each modalform. In this way the grid deformation is almost a trivial task which requires very small

computational resources and time.

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Figure 4: CFD unstructured grid section deformation along the fifth mode of AFGARD 445.6 wing

5 VALIDATION ON AGARD 445.6 WING

A well-known three-dimensional standard aeroelastic configuration is considered to vali-date the whole procedure: the AGARD 445.6 weakened wing which was tested in wind-tunnel at NASA Langley. The wing semispan model is made of laminated mahogany, withNACA 65A004 airfoil, a quarter-chord sweep angle of 45 deg, and an aspect ratio of 1.65and a taper ratio of 0.66. To reduce the stiffness, the wing was weakened by holes drilledthrough it and filled with foam. In this section we refer to the weakened model number3 since all flow-conditions (subsonic, transonic and supersonic) were tested. Further dataconcerning the models, test setups and conditions are reported in Ref9.

The structural model is represented, by the first four normal undamped modes. The

first two are primarily involved in the flutter mechanism, identified respectively as firstbending and first-torsional modes by a finite-element analysis (Figure 5). Since no struc-tural damping is available for the wing model, a estimate value of modal damping coeffi-cient equal to 0.01 is applied.

The aerodynamic system is based on the Euler equations representing a good compro-mise between accuracy in description of the transonic effects and computational burden.Moreover, no information is available on the experimental transition location. A C-Htype mesh of 124.160 cells is built, extending 7 root chord lengths from the wing to theupstream and downstream boundaries and 3 semi-span lengths from the tip to the sideboundary.

Stability behavior is determined at null incidence for three different Mach numberconditions, 0.678, 0.96 and 1.141, in order to assess the quality of the defined procedurein all flow-conditions, subsonic, transonic and weakly supersonic with a detached curvedshock bump. The airfoil section is symmetric, so no static aeroelastic trimming procedurefor wing deflections is required. The computational of each column of the transfer matrixrequires about 2 hours with a single Pentium IV 2.8 MHz.

Flutter results for the subsonic and transonic cases agree with experimental datawithin 1% (Figure 6). In the supersonic case, similarly to Refs. 1;23, 20% higher fluttervelocity than those determined in experiment are found. Although Ref.2 showed someimprovements by a Navier-Stokes model, the degree of improvements was small comparedto the still large difference with experimental values. For this case our tests have shown

that a better grid space resolution near the leading edge with inviscid model leads to

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Figure 5: AGARD 445.6 structuredd grid deformed along first two structural normal modes

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Mach number

VF

Experiment

CFDEuler

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Mach number

f

/

Experiment

CFDEuler

Figure 6: Flutter speed index and frequency ratio for the AGARD 445.6 wing

improvements in correlation with experiments. The flow behind the detached shock bump

needs to be correctly evaluated because has an high influence on pressure loads on thewhole wing behind. For this reason we suppose Navier-Stokes improvements are due tothe mandatory better grid resolution in order to correctly model the viscous layer. Futureworks will presents results of this research.

5.1 Analysis by means of direct integration

Unlike linearized flutter analysis where motion is prescribed using a trimmed step inputlaw, direct integration is used to study the time evolution of the aeroelastic system by adirect CSD-CFD coupled simulation. The wing motion is excited with a user-definable

initial modal velocity perturbing a steady trimmed aerodynamic solution. Input am-plitude is determined by maximum local incidence variation based on maximum modaldisplacement. As Figures 7 - 8 show, the wing starts oscillating resulting in damped,neutral or diverging vibrations. The diverging oscillations evolve very fast toward a limitcycle. This test case shows how the grid deformation routines, and the entire procedure,are robust enough to solve the problem even with large deformations, such as those metin this simulation.

6 APPLICATION TO THE PIAGGIO P-180 AVANTI WING

Piaggio P180 (Figure 2) is a twin turbo-prop pushed aircraft whose flight envelope, ac-cording to current regulations, must be clear from flutter instability up to Mach 0.92,

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0 4 8 12 16 20 240.015

0.01

0.005

0

0.005

0.01

0.015

Structural adimensional time

Mo

dalamplitude

Mode 1

Mode 2

Figure 7: Modal amplitudes history for the AGARD 445.6wing, M = 0.678, VF = 0.34

0 4 8 12 16 20 24 28 32 36 40

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15


Mo

dalamplitude

Mode 1

Mode 2

Mode 3

Mode 4

Figure 8: Modal amplitudes history for the AGARD 445.6wing using 4 modes, M = 0.678, VF = 0.50

far beyond Mach cruise approximately set to Mach 0.7. Cruise conditions are signifi-

cantly marked by transonic effects manifesting as a shock on the aft portion of the liftingsurfaces. P180s layout is not conventional since three lifting surfaces are used in orderto reduce induced drag. Moreover its wing has a full-immersed nacelle giving rise tophenomena which are rather complex to be analytically modeled: engine-inlet and out-let, pushing props influence on local flow, body-interferences with the surrounding wing.In this work,we present the first steps taken to model the P180 as an aeroelastic system,starting from the isolated wing model. In this way it is possible to lay out the foundationsfor a full-plane procedure.

6.1 Model description

A finite-element model of the wing-box is used to describe P180s structural dynamicsystem. First two undamped normal modes, respectively the first bending and first tor-sional, are used for wing flutter analysis after considering different simplified Doublet-Lattice Models (DLM). The aerodynamic model is based on Euler equations. Given thegeometric complexity, an unstructured grid consisting of 274.740 cells is built to modelthe flow-field around the wing. Rear-pushing propeller aerodynamic over-velocity effectsare neglected since they are small and considered not fundamental for the analysis offlutter mechanism. Correct aerodynamic propeller-modeling, besides adding work termsin the flutter equation considering thrust as a follower force, will be the object of futureworks. In order to add respectively inlet and outlet effects, two iterative procedures are

used while solving reference steady aerodynamic solution: the former changes local staticpressure in order to get a fixed mass flow for the engine; the latter changes local totaltemperature to get a fixed thrust. Good inlet modeling turned out to improve steadyaerodynamics analysis convergence.

6.2 Aerodynamic validation

Various steady cases are investigated in order to validate the correct pressure loads ofthe CFD model, neglecting aeroelastic static deflections. Figure 9 shows good agreementwith transonic wind-tunnel tests. Despite using an inviscid model, shock locations arewell defined and so is the overall pressure load, especially if we consider that the reference

measured profile section is particularly close to the nacelle. An exception is represented

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by the discrepancy between the last but three sample point and CFD pressure coefficient:flap gap has been closed in grid building phase in order to avoid unuseful increase in cellnumber and because local solution nearby control surfaces are not considered importantfor a overall stability. Lift curve slope agrees with experimental wind-tunnel tests. A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.5

1.25

1

0.75

0.5

0.25

0

0.25

0.5

0.75

1

x / c

Cp

CFDEulerExperiment

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1.5

1.25

1

0.75

0.5

0.25

0

0.25

0.5

0.75

1

x / c

Cp

CFDEulerExperiment

Figure 9: Comparison of pressure distributions, y/b=0.2, M = 0.7, = 2.259 and 4.383 respectively

first dynamic example is shown in Figure 11 representing aerodynamic generalized forcecoefficient due to plunge motion. The experimental steady rigid value is confirmed by theresult extrapolated from numerical tests.

Figure 10: Pressure distributions on the wing, M =0.7, = 0

0 0.1 0.2 0.3 0.4 0.51

0.75

0.5

0.25

0

0.25

0.5

0.75

Reduced frequency k

Qhh[

m2]

Real CFDImaginary CFD

Real DLM

Imaginary DLM

Figure 11: Generalized plunge aerodynamic forcesfor Piaggio P180 wing compared to DLM, M = 0.7

6.3 Aeroelastic flutter analysis

Figure 12 shows the computed coefficient of the aerodynamic transfer matrix at Mach0.7, comparing them with the linear results of DLM. Discrepancy among coefficientsgrows as Mach number increases. The CFD results are compared with two linear models:a simple model of the lifting surface with equivalent panels in the nacelle zone; andone using slender and interference bodies implemented in MSC-NASTRAN aeroelasticsolver in order to account for wing-nacelle interactions and their effect on the pressureload. Despite a transonic flight condition is studied, DLM results agree with CFD flutter

procedure within 2%; no transonic dip effect is shown. Table 1 reports flutter results(not Mach-aligned) at sea level for different flight conditions. Two different effects are

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M 0.7 0.8 0.9

VFpanels 203.6 203.9 205.7VFCFD 207.8 209.2 210.4

VFbody 210.2 212.4 210.5

Table 1: Flutter velocity (m/s) at different Mach numbers, z = 0m

0 0.1 0.2 0.3 0.4 0.5

2

1.75

1.5

1.25

1

0.75

0.5

0.25

0

0.25

0.5

Reduced frequency k

Q1

1

[m

2]

Real CFD

Imaginary CFD

Real DLM

Imaginary DLM

0 0.1 0.2 0.3 0.4 0.5

1

0.75

0.5

0.25

0

0.25

0.5

0.75

1

Reduced frequency k

Q12[

m2]

Real CFD

Imaginary CFD

Real DLM

Imaginary DLM

0 0.1 0.2 0.3 0.4 0.51

0.75

0.5

0.25

0

0.25

0.5

Reduced frequency k

Q21[

m2]

Real CFD

Imaginary CFD

Real DLM

Imaginary DLM

0 0.1 0.2 0.3 0.4 0.51

0.75

0.5

0.25

0

0.25

0.5

0.75

1

Reduced frequency k

Q22[

m2]

Real CFD

Imaginary CFD

Real DLM

Imaginary DLM

Figure 12: Generalized aerodynamic forces for Piaggio P180 wing compared to Doublet Lattice Method, M = 0.7

combined to give results which are so similar with such different aerodynamic models.On one side is the effect of the shock wave which cause an anti-stabilizing effect. On theother side, as can be seen from Figure 10, nacelle interference plays an important role inload distribution: besides a sharp drop in pressure load, aerodynamic pressure resultantis placed far backward from the elastic axis creating a stabilizing terms that cannot be

caught with the simplified DLM geometry and that results in a delayed flutter condition.The results obtained by means of the linearized analysis are confirmed by direct anal-

ysis of the coupled systems. Figure 13 shows the unstable movement of the two modalcomponents at a velocity slightly above the flutter condition for Mach 0 .7. The initialexcitation is automatically generated since the system is not starting from a trimmedcondition for the elastic deformations.

7 CONCLUSIONS

In this paper a procedure to apply CFD solutions for aeroelastic stability evaluation has

been presented. It has been displayed how, by a careful choice of the basic algorithmicelements necessary for this type of analyses, it is possible to obtain a methodology rou-

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0 2 4 6 8 10101

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2


Modalamplitu

de

Mode 1

Mode 2

Figure 13: Post flutter direct simulation of P-180 at M = 0.7: modal amplitudes history

tinely applicable for analyses in an industrial environment. To further demonstrate thepotential of the proposed technique, applications to complete aircraft configurations arecurrently under way and will be presented soon.

8 ACKNOWLEDGMENTS

The authors wish to acknowledge Piaggio Aero Industries for providing P-180 data, SaraBozzini, Raffaele Ponzini and Paolo Ramieri, members of Cilea Consortium, for the tech-

nical support with Avogadro computing cluster.

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